[ { "path": ".dockerignore", "content": "# Don't include the .git in the image. It's big!\n.git\n" }, { "path": ".github/actions/install-dependencies/action.yml", "content": "name: Install dependencies\n\nruns:\n using: composite\n steps:\n - name: Install apt dependencies and upgrade pip\n shell: bash -el {0}\n run: |\n apt-get update && apt-get install -y libxrender1 libgl1-mesa-dev mesa-utils gzip" }, { "path": ".github/dependabot.yml", "content": "# To get started with Dependabot version updates, you'll need to specify which\n# package ecosystems to update and where the package manifests are located.\n# Please see the documentation for all configuration options:\n# https://docs.github.com/code-security/dependabot/dependabot-version-updates/configuration-options-for-the-dependabot.yml-file\n\nversion: 2\nupdates:\n - package-ecosystem: \"github-actions\" # See documentation for possible values\n directory: \"/\" # Location of package manifests\n schedule:\n interval: \"weekly\"\n - package-ecosystem: \"pip\" # See documentation for possible values\n directory: \"python/\"\n schedule:\n interval: \"weekly\"\n" }, { "path": ".github/workflows/book_stable.yml", "content": "name: Test stable build of book\n\non:\n workflow_dispatch:\n workflow_call:\n pull_request:\n branches: [\"release\"]\n push:\n branches: [\"release\"]\nenv:\n HDF5_MPI: \"ON\"\n HDF5_DIR: \"/usr/local/\"\n H5PY_SETUP_REQUIRES: 0\n DEB_PYTHON_INSTALL_LAYOUT: deb_system\n LIBGL_ALWAYS_SOFTWARE: 1\n\njobs:\n build-book:\n runs-on: ubuntu-latest\n container: ghcr.io/fenics/dolfinx/lab:stable\n\n env:\n PYVISTA_OFF_SCREEN: false\n PYVISTA_JUPYTER_BACKEND: html\n\n steps:\n - uses: actions/checkout@v6\n\n - name: Install common packages\n uses: ./.github/actions/install-dependencies\n\n - name: Install book deps\n run: |\n python3 -m pip install --break-system-packages -U pip setuptools pkgconfig poetry-core\n python3 -m pip install --no-build-isolation --no-binary=h5py .[netgen]\n\n - name: Build the book\n run: jupyter-book build .\n\n - uses: actions/upload-artifact@v7\n if: always()\n with:\n name: webpage\n path: ./_build/html\n retention-days: 2\n if-no-files-found: error\n" }, { "path": ".github/workflows/deploy.yml", "content": "name: Publish book\n\non:\n push:\n branches:\n - \"release\"\n workflow_dispatch:\n\n # Weekly build on Mondays at 8 am\n schedule:\n - cron: \"0 8 * * 1\"\n\npermissions:\n contents: read\n pages: write\n id-token: write\n\nconcurrency:\n group: \"pages\"\n cancel-in-progress: true\n\njobs:\n run-tests:\n uses: ./.github/workflows/test_stable.yml\n\n build-book:\n uses: ./.github/workflows/book_stable.yml\n\n deploy:\n runs-on: ubuntu-22.04\n needs: [build-book, run-tests]\n environment:\n name: github-pages\n url: ${{ steps.deployment.outputs.page_url }}\n\n steps:\n - name: Checkout\n uses: actions/checkout@v6\n\n - name: Setup Pages\n uses: actions/configure-pages@v6\n\n - name: Download docs artifact\n uses: actions/download-artifact@v8\n with:\n name: webpage\n path: \"./public\"\n\n - name: Upload page artifact\n uses: actions/upload-pages-artifact@v5\n with:\n path: \"./public\"\n\n - name: Deploy coverage report to GH Pages\n id: deployment\n uses: actions/deploy-pages@v5\n" }, { "path": ".github/workflows/publish_docker.yml", "content": "# Recipe based on: https://docs.docker.com/build/ci/github-actions/multi-platform/#distribute-build-across-multiple-runners\nname: Build and publish platform dependent docker image\non:\n push:\n branches:\n - \"release\"\n tags:\n - \"v*\"\n pull_request:\n branches:\n - \"release\"\n workflow_dispatch:\n\nenv:\n REGISTRY: ghcr.io\n IMAGE_NAME: ${{ github.repository }}\n\njobs:\n build:\n strategy:\n matrix:\n os: [\"ubuntu-24.04\", \"ubuntu-24.04-arm\"]\n runs-on: ${{ matrix.os }}\n\n permissions:\n contents: read\n packages: write\n\n steps:\n - name: Checkout repository\n uses: actions/checkout@v6\n\n - name: Log in to the Container registry\n uses: docker/login-action@v4\n with:\n registry: ${{ env.REGISTRY }}\n username: ${{ github.actor }}\n password: ${{ secrets.GITHUB_TOKEN }}\n\n - name: Set up Docker Buildx\n uses: docker/setup-buildx-action@v4\n\n - name: Extract metadata (tags, labels) for Docker\n id: meta\n uses: docker/metadata-action@v6\n with:\n images: ${{ env.REGISTRY }}/${{ env.IMAGE_NAME }}\n\n - name: Set architecture tag (amd64)\n if: ${{ matrix.os == 'ubuntu-24.04' }}\n run: echo \"ARCH_TAG=amd64\" >> $GITHUB_ENV\n\n - name: Set architecture tag (arm)\n if: ${{ contains(matrix.os, 'arm') }}\n run: echo \"ARCH_TAG=arm64\" >> $GITHUB_ENV\n\n - name: Build and push by digest\n id: build\n uses: docker/build-push-action@v7\n with:\n file: docker/Dockerfile\n platforms: ${{ env.ARCH_TAG }}\n labels: ${{ steps.meta.outputs.labels }}\n outputs: type=image,\"name=${{ env.REGISTRY }}/${{ env.IMAGE_NAME}}\",push-by-digest=true,name-canonical=true,push=true\n\n - name: Export digest\n run: |\n mkdir -p ${{ runner.temp }}/digests\n digest=\"${{ steps.build.outputs.digest }}\"\n touch \"${{ runner.temp }}/digests/${digest#sha256:}\" \n\n - name: Upload digest\n if: github.event_name == 'push'\n uses: actions/upload-artifact@v7\n with:\n name: digests-${{ env.ARCH_TAG }}\n path: ${{ runner.temp }}/digests/*\n if-no-files-found: error\n retention-days: 1\n \n merge-and-publish:\n if: github.event_name == 'push'\n runs-on: ubuntu-latest\n needs:\n - build\n steps:\n - name: Download digests\n uses: actions/download-artifact@v8\n with:\n path: ${{ runner.temp }}/digests\n pattern: digests-*\n merge-multiple: true\n\n - name: Log in to the Container registry\n uses: docker/login-action@v4\n with:\n registry: ${{ env.REGISTRY }}\n username: ${{ github.actor }}\n password: ${{ secrets.GITHUB_TOKEN }}\n\n - name: Extract metadata (tags, labels) for Docker\n id: meta\n uses: docker/metadata-action@v6\n with:\n images: ${{ env.REGISTRY }}/${{ env.IMAGE_NAME }}\n \n\n - name: Create manifest list and push\n working-directory: ${{ runner.temp }}/digests\n run: |\n docker buildx imagetools create $(jq -cr '.tags | map(\"-t \" + .) | join(\" \")' <<< \"$DOCKER_METADATA_OUTPUT_JSON\") \\\n $(printf '${{ env.REGISTRY }}/${{ env.IMAGE_NAME}}@sha256:%s ' *) \n\n - name: Inspect image\n run: |\n docker buildx imagetools inspect ${{ env.REGISTRY }}/${{ env.IMAGE_NAME}}:${{ steps.meta.outputs.version }} \n\n" }, { "path": ".github/workflows/test_nightly.yml", "content": "name: Test against DOLFINx nightly build\n\n# Controls when the action will run.\non:\n pull_request:\n branches:\n - main\n\n # Allows you to run this workflow manually from the Actions tab\n workflow_dispatch:\n workflow_call:\n schedule:\n - cron: \"0 9 * * *\"\n\n# A workflow run is made up of one or more jobs that can run sequentially or in parallel\njobs:\n test-nightly:\n # The type of runner that the job will run on\n runs-on: ubuntu-latest\n container: ghcr.io/fenics/dolfinx/lab:nightly\n\n env:\n HDF5_MPI: \"ON\"\n H5PY_SETUP_REQUIRES: 0\n HDF5_DIR: \"/usr/local/\"\n PYVISTA_OFF_SCREEN: true\n PYVISTA_JUPYTER_BACKEND: html\n LIBGL_ALWAYS_SOFTWARE: 1\n\n steps:\n - uses: actions/checkout@v6\n\n - name: Special handling of some installation\n uses: ./.github/actions/install-dependencies\n\n - name: Install requirements\n run: |\n python3 -m pip install --break-system-packages -U pip setuptools pkgconfig poetry-core\n python3 -m pip install --no-build-isolation --break-system-packages --no-cache-dir --no-binary=h5py .[netgen] --upgrade\n\n - name: Test building the book\n run: PYVISTA_OFF_SCREEN=false jupyter-book build .\n\n - name: Test complex notebooks in parallel\n working-directory: chapter1\n run: |\n export PKG_CONFIG_PATH=/usr/local/dolfinx-complex/lib/pkgconfig:$PKG_CONFIG_PATH\n export PETSC_ARCH=linux-gnu-complex128-32\n export PYTHONPATH=/usr/local/dolfinx-complex/lib/python3.12/dist-packages:$PYTHONPATH\n export LD_LIBRARY_PATH=/usr/local/dolfinx-complex/lib:$LD_LIBRARY_PATH\n python3 complex_mode.py\n mpirun -n 2 python3 complex_mode.py\n\n - name: Test chapter 1\n working-directory: chapter1\n run: |\n mpirun -n 2 python3 fundamentals_code.py\n mpirun -n 2 python3 nitsche.py\n mpirun -n 2 python3 membrane_code.py\n\n - name: Test chapter 2\n working-directory: chapter2\n run: |\n mpirun -n 2 python3 diffusion_code.py\n mpirun -n 2 python3 heat_code.py\n mpirun -n 2 python3 linearelasticity_code.py\n mpirun -n 2 python3 hyperelasticity.py\n mpirun -n 2 python3 nonlinpoisson_code.py\n mpirun -n 2 python3 ns_code1.py\n mpirun -n 2 python3 ns_code2.py\n - name: Test chapter 3\n working-directory: chapter3\n run: |\n mpirun -n 2 python3 neumann_dirichlet_code.py\n mpirun -n 2 python3 multiple_dirichlet.py\n mpirun -n 2 python3 subdomains.py\n mpirun -n 2 python3 robin_neumann_dirichlet.py\n mpirun -n 2 python3 component_bc.py\n mpirun -n 2 python3 em.py\n - name: Test chapter 4\n working-directory: chapter4\n run: |\n mpirun -n 2 python3 solvers.py\n mpirun -n 2 python3 convergence.py\n mpirun -n 2 python3 compiler_parameters.py\n mpirun -n 2 python3 newton-solver.py\n\n - uses: actions/upload-artifact@v7\n if: always()\n with:\n name: webpage\n path: ./_build/html\n retention-days: 2\n if-no-files-found: error\n" }, { "path": ".github/workflows/test_stable.yml", "content": "name: Test stable release\n\non:\n workflow_dispatch:\n workflow_call:\n pull_request:\n branches: [\"release\"]\n\njobs:\n test:\n runs-on: ubuntu-latest\n container: ghcr.io/fenics/dolfinx/lab:stable\n env:\n HDF5_MPI: \"ON\"\n HDF5_DIR: \"/usr/local/\"\n H5PY_SETUP_REQUIRES: \"0\"\n DEB_PYTHON_INSTALL_LAYOUT: deb_system\n PYVISTA_OFF_SCREEN: true\n PYVISTA_JUPYTER_BACKEND: html\n LIBGL_ALWAYS_SOFTWARE: 1\n\n # Steps represent a sequence of tasks that will be executed as part of the job\n steps:\n - uses: actions/checkout@v6\n with:\n ref: release\n\n - uses: ./.github/actions/install-dependencies\n\n - name: Install additional deps\n run: |\n python3 -m pip install -U pip setuptools pkgconfig poetry-core\n python3 -m pip install --no-binary=h5py --no-build-isolation .[netgen]\n\n - name: Test complex notebooks in parallel\n working-directory: chapter1\n run: |\n export PKG_CONFIG_PATH=/usr/local/dolfinx-complex/lib/pkgconfig:$PKG_CONFIG_PATH\n export PETSC_ARCH=linux-gnu-complex128-32\n export PYTHONPATH=/usr/local/dolfinx-complex/lib/python3.12/dist-packages:$PYTHONPATH\n export LD_LIBRARY_PATH=/usr/local/dolfinx-complex/lib:$LD_LIBRARY_PATH\n python3 complex_mode.py\n mpirun -n 2 python3 complex_mode.py\n\n - name: Test chapter 1\n working-directory: chapter1\n run: |\n mpirun -n 2 python3 fundamentals_code.py\n mpirun -n 2 python3 nitsche.py\n mpirun -n 2 python3 membrane_code.py\n\n - name: Test chapter 2\n working-directory: chapter2\n run: |\n mpirun -n 2 python3 diffusion_code.py\n mpirun -n 2 python3 heat_code.py\n mpirun -n 2 python3 linearelasticity_code.py\n mpirun -n 2 python3 hyperelasticity.py\n mpirun -n 2 python3 nonlinpoisson_code.py\n mpirun -n 2 python3 ns_code1.py\n mpirun -n 2 python3 ns_code2.py\n\n - name: Test chapter 3\n working-directory: chapter3\n run: |\n mpirun -n 2 python3 neumann_dirichlet_code.py\n mpirun -n 2 python3 multiple_dirichlet.py\n mpirun -n 2 python3 subdomains.py\n mpirun -n 2 python3 robin_neumann_dirichlet.py\n mpirun -n 2 python3 component_bc.py\n mpirun -n 2 python3 em.py\n\n - name: Test chapter 4\n working-directory: chapter4\n run: |\n mpirun -n 2 python3 solvers.py\n mpirun -n 2 python3 convergence.py\n mpirun -n 2 python3 compiler_parameters.py\n mpirun -n 2 python3 newton-solver.py\n\n - name: Upload Navier-Stokes DFG 2D 3 plots\n uses: actions/upload-artifact@v7\n with:\n name: DFG2D-3\n path: chapter2/figures\n retention-days: 2\n if-no-files-found: error\n" }, { "path": ".gitignore", "content": "_build\n*.pvd\n*.h5\n*.xdmf\n*.vtu\n*/.ipynb_checkpoints/*\n.ipynb_checkpoints/*\n**/.cache\n*.png\n*.pvtu\n*.msh\n*.bp" }, { "path": ".pre-commit-config.yaml", "content": "repos:\n- repo: https://github.com/kynan/nbstripout\n rev: 0.7.1\n hooks:\n - id: nbstripout" }, { "path": ".vscode/c_cpp_properties.json", "content": "{\n \"configurations\": [\n {\n \"name\": \"Linux\",\n \"includePath\": [\n \"/usr/include/python3.10/\",\n \"${workspaceFolder}/\",\n \"/usr/include/eigen3/\",\n \"/usr/local/petsc/include/\",\n \"/home/shared/dolfinx_src/ffcX/ffcx/codegeneration/\",\n \"/usr/include/x86_64-linux-gnu/mpich/\",\n \"/usr/local/dolfinx-real/include/\"\n ],\n \"defines\": [],\n \"compilerPath\": \"/usr/bin/gcc\",\n \"cStandard\": \"c11\",\n \"cppStandard\": \"c++20\",\n \"intelliSenseMode\": \"clang-x64\",\n \"configurationProvider\": \"vector-of-bool.cmake-tools\"\n }\n ],\n \"version\": 4\n}" }, { "path": ".vscode/settings.json", "content": "{\n \"clang_format_style set\": \"file\",\n \"editor.formatOnSave\": true,\n \"cornflakes.linter.executablePath\": \"/usr/local/bin/flake8\",\n \"cSpell.ignoreWords\": [\n \"dolfinx\",\n \"meshio\",\n \"petsc\",\n \"py\",\n \"pygmsh\",\n \"gmsh\"\n ],\n // whitelist numpy to remove lint errors\n \"python.linting.pylintArgs\": [\n \"--ignored-modules=petsc4py.PETSc\",\n \"--ignored-classes=petsc4py.PETSc\",\n \"--extension-pkg-whitelist=petsc4py.PETSc\"\n ],\n \"python.linting.flake8Enabled\": true,\n \"python.linting.pylintEnabled\": false,\n \"python.formatting.autopep8Path\": \"/usr/local/bin/autopep8\",\n \"python.formatting.provider\": \"autopep8\",\n \"python.formatting.autopep8Args\": [\n \"--ignore=W503\",\n \"--max-line-length=120\"\n ],\n \"python.linting.enabled\": true,\n \"python.pythonPath\": \"/usr/bin/python3\",\n \"spellright.language\": [\n \"en_GB\"\n ],\n \"spellright.documentTypes\": [\n \"markdown\",\n \"latex\",\n \"plaintext\"\n ],\n}" }, { "path": "Changelog.md", "content": "# Changelog\n\n## v0.10.0\n\n- Full refactoring of {py:class}`dolfinx.fem.petsc.NonlinearProblem`, which now uses the PETSc SNES backend. See [the non-linear poisson demo](./chapter2/nonlinpoisson_code.ipynb) for details.\n- {py:class}`dolfinx.fem.petsc.LinearProblem` now requires an additional argument, `petsc_options_prefix`. This should be a unique string identifier for each `LinearProblem` that is created.\n- Change how one reads in GMSH data with `gmshio`. See [the membrane code](./chapter1/membrane_code.ipynb) for more details.\n- {py:meth}`dolfinx.fem.FiniteElement.interpolation_points` -> {py:attr}`dolfinx.fem.FiniteElement.interpolation_points`.\n- {py:mod}`dolfinx.io.gmshio` has been renamed to {py:mod}`dolfinx.io.gmsh`\n- Input to {py:func}`dolfinx.fem.petsc.create_vector` has changed. One should now call {py:func}`dolfinx.fem.extract_function_spaces` on the input form first.\n\n## v0.9.0\n\n- `scale` in {py:func}`apply_lifting` has been renamed to `alpha`\n- Use `dolfinx.fem.Function.x.petsc_vec` as opposed to `dolfinx.fem.Function.vector`\n\n## v0.8.0\n\n- Replace all `ufl.FiniteElement` and `ufl.VectorElement` with the appropriate {py:func}`basix.ufl.element`\n- Replace {py:class}`dolfinx.fem.FunctionSpace` with {py:func}`dolfinx.fem.functionspace`\n\n## v0.7.2\n\n- Change pyvista backend to `html`, using Pyvista main branch\n- Using DOLFINx v0.7.2 https://github.com/FEniCS/dolfinx/releases/tag/v0.7.2 as base\n\n## v0.7.1\n\n- No API changes, release due to various bug-fixes from the 0.7.0 release, see:\n https://github.com/FEniCS/dolfinx/releases/tag/v0.7.1 for more information\n\n## v0.7.0\n\n- Renamed `dolfinx.graph.create_adjacencylist` to {py:func}`dolfinx.graph.adjacencylist`\n- Renamed `dolfinx.plot.create_vtk_mesh` to {py:func}`dolfinx.plot.vtk_mesh`\n- Initialization of {py:class}`dolfinx.geometry.BoundingBoxTree` has been changed to {py:func}`dolfinx.geometry.bb_tree`\n- `create_mesh` with Meshio has been modified. Note that you now need to pass dtype `np.int32` to the cell_data.\n- Update dolfinx petsc API. Now one needs to explicitly import {py:mod}`dolfinx.fem.petsc` and {py:mod}`dolfinx.fem.nls`, as PETSc is no longer a strict requirement.\n Replace `petsc4py.PETSc.ScalarType` with `dolfinx.default_scalar_type` in demos where we do not use {py:mod}`petsc4py` explicitly.\n \n## v0.6.0\n\n- Remove `ipygany` and `pythreejs` as plotting backends. Using `panel`.\n- Add gif-output to [chapter2/diffusion_code] and [chapter2/hyperelasticity].\n- Replace `dolfinx.fem.Function.geometric_dimension` with `len(dolfinx.fem.Function)`\n- Improve [chapter2/ns_code2] to have better splitting scheme and density.\n- Improve mesh quality in [chapter3/em].\n- `jit_params` and `form_compiler_params` renamed to `*_options`.\n\n## v0.5.0\n\n- Using new GMSH interface in DOLFINx (`dolfinx.io.gmshio`) in all demos using GMSH\n- Added a section on custom Newton-solvers, see [chapter4/newton-solver].\n- Various minor DOLFINx API updates. `dolfinx.mesh.compute_boundary_facets` -> {py:func}`dolfinx.mesh.exterior_facet_indices` with slightly different functionality.\n Use `dolfinx.mesh.MeshTagsMetaClass.find` instead of `mt.indices[mt.values==value]`.\n- Various numpy updates, use `np.full_like`.\n- Change all notebooks to use [jupytext](https://jupytext.readthedocs.io/en/latest/install.html) to automatically sync `.ipynb` with `.py` files.\n- Add example of how to use `DOLFINx` in complex mode, see [chapter1/complex_mode].\n\n## 0.4.1\n\n- No changes\n\n## 0.4.0 (05.02.2021)\n\n- All `pyvista` plotting has been rewritten to use `ipygany` and `pythreejs` as well as using a cleaner interface.\n- `dolfinx.plot.create_vtk_topology` has been renamed to `dolfinx.plot.create_vtk_mesh` and can now be directly used as input\n to {py:class}`pyvista.UnstructuredGrid`.\n- `dolfinx.fem.Function.compute_point_values` has been deprecated. Interpolation into a CG-1 is now the way of getting vertex values.\n- Instead of initializing class with {py:class}`Form`, use {py:func}`form`.\n- Instead of initializing class with {py:class}`DirichletBC` use {py:func}`dirichletbc`.\n- Updates on error computations in [Error control: Computing convergence rates](chapter4/convergence).\n- Added tutorial on interpolation of {py:class}`ufl.core.expr.Expr` in [Deflection of a membrane](chapter1/membrane_code).\n- Added tutorial on how to apply constant-valued Dirichlet conditions in [Deflection of a membrane](chapter1/membrane_code).\n- Various API changes relating to the import structure of DOLFINx\n\n## 0.3.0 (09.09.2021)\n\n- Major improvements in [Form compiler parameters](chapter4/compiler_parameters), using pandas and seaborn for visualization of speed-ups gained using form compiler parameters.\n- API change: `dolfinx.cpp.la.scatter_forward(u.x)` -> `u.x.scatter_forward`\n- Various plotting updates due to new version of pyvista.\n- Updating of the [Hyperelasticity demo](chapter2/hyperelasticity), now using DOLFINx wrappers to create the non-linear problem\n- Internal updates due to bumping of jupyter-book versions\n- Various typos and capitalizations fixed by @mscroggs in [PR 35](https://github.com/jorgensd/dolfinx-tutorial/pull/35).\n\n## 0.1.0 (11.05.2021)\n\n- First tagged release of DOLFINx Tutorial, compatible with [DOLFINx 0.1.0](https://github.com/FEniCS/dolfinx/releases/tag/0.1.0).\n" }, { "path": "Dockerfile", "content": "FROM ghcr.io/jorgensd/dolfinx-tutorial:v0.10.0\n\n# create user with a home directory\nARG NB_USER=jovyan\nARG NB_UID=1000\n# 24.04 adds `ubuntu` as uid 1000;\n# remove it if it already exists before creating our user\nRUN id -nu ${NB_UID} && userdel --force $(id -nu ${NB_UID}) || true; \\\n useradd -m ${NB_USER} -u ${NB_UID}\nENV HOME=/home/${NB_USER}\n\n# Copy home directory for usage in binder\nWORKDIR ${HOME}\nCOPY --chown=${NB_UID} . ${HOME}\n\nUSER ${NB_USER}\nENTRYPOINT []\n" }, { "path": "README.md", "content": "# The DOLFINx tutorial\n\n[![Test, build and publish](https://github.com/jorgensd/dolfinx-tutorial/actions/workflows/deploy.yml/badge.svg)](https://github.com/jorgensd/dolfinx-tutorial/actions/workflows/deploy.yml)\n[![Test release branch against DOLFINx nightly build](https://github.com/jorgensd/dolfinx-tutorial/actions/workflows/test_nightly.yml/badge.svg)](https://github.com/jorgensd/dolfinx-tutorial/actions/workflows/test_nightly.yml)\n\nAuthor: Jørgen S. Dokken\n\nThis is the source code for the dolfinx-tutorial [webpage](https://jorgensd.github.io/dolfinx-tutorial/).\nIf you have any comments, corrections or questions, please submit an issue in the issue tracker.\n\n## Contributing\n\nIf you want to contribute to this tutorial, please make a fork of the repository, make your changes, and test that the CI passes.\n\nAlternatively, if you want to add a separate chapter, a Jupyter notebook can be added to a pull request, without integrating it into the tutorial. If so, the notebook will be reviewed and modified to be included in the tutorial.\n\nAny code added to the tutorial should work in parallel. If any changes are made to `ipynb` files, please ensure that these changes are reflected in the corresponding `py` files by using [`jupytext`](https://jupytext.readthedocs.io/en/latest/faq.html#can-i-use-jupytext-with-jupyterhub-binder-nteract-colab-saturn-or-azure):\n\n## Building the book and running code\n\nThe book is built using [jupyterbook](https://jupyterbook.org/). The following environment variables should be set if you want to build the book\n\n```bash\nPYVISTA_OFF_SCREEN=false\nPYVISTA_JUPYTER_BACKEND=\"html\"\nJUPYTER_EXTENSION_ENABLED=true\nLIBGL_ALWAYS_SOFTWARE=1\n```\n\nIf you run the tutorial using `jupyter-lab`, for instance through `conda`, one should set the following environment variables\n\n```bash\nPYVISTA_OFF_SCREEN=false\nPYVISTA_JUPYTER_BACKEND=\"trame\"\nJUPYTER_EXTENSION_ENABLED=true\nLIBGL_ALWAYS_SOFTWARE=1\n```\n\nIf you use docker to run your code, you should set the following variables:\n\n```bash\ndocker run -ti -e DISPLAY=$DISPLAY -e LIBGL_ALWAYS_SOFTWARE=1 -e PYVISTA_OFF_SCREEN=false -e PYVISTA_JUPYTER_BACKEND=\"trame\" -e JUPYTER_EXTENSION_ENABLED=true --network=host -v $(pwd):/root/shared -w /root/shared ....\n```\n\nTo run python scripts, either choose `PYVISTA_OFF_SCREEN=True` to get screenshots, or render interactive plots with `PYVISTA_OFF_SCREEN=False`\n\n```bash\npython3 -m jupytext --sync */*.ipynb --set-formats ipynb,py:light\n```\n\nor\n\n```bash\npython3 -m jupytext --sync */*.py --set-formats ipynb,py:light\n```\n\nAny code added to the tutorial should work in parallel.\n\nTo strip notebook output, one can use pre-commit.\n\n```bash\npre-commit run --all-files\n```\n\n## Dependencies\n\nIt is advised to use a pre-installed version of DOLFINx, for instance through conda or docker. Remaining dependencies can be installed with\n\n```bash\npython3 -m pip install --no-binary=h5py --no-build-isolation -e .\n```\n\n# Docker images\n\nDocker images for this tutorial can be found in the [packages tab](https://github.com/jorgensd/dolfinx-tutorial/pkgs/container/dolfinx-tutorial)\n\nAdditional requirements on top of the `dolfinx/lab:nightly` images can be found at [Dockerfile](docker/Dockerfile) and [pyproject.toml](./pyproject.toml)\n\n##\n\nAn image building DOLFINx, Basix, UFL and FFCx from source can be built using:\n\n```bash\ndocker build -f ./docker/Dockerfile -t local_lab_env .\n```\n\nfrom the root of this repository, and run\n\n```bash\n docker run --rm -ti -v $(pwd):/root/shared -w /root/shared --init -p 8888:8888 local_lab_env\n```\n\nfrom the main directory.\n" }, { "path": "_config.yml", "content": "# Book settings\n# Learn more at https://jupyterbook.org/customize/config.html\n\ntitle: FEniCSx tutorial\nauthor: Jørgen S. Dokken\nlogo: fenics_logo.png\n\n# Force re-execution of notebooks on each build.\n# See https://jupyterbook.org/content/execute.html\nexecute:\n execute_notebooks: cache\n\n # Set timeout for any example to 20 minutes\n timeout: 1800\n# Define the name of the latex output file for PDF builds\n# latex:\n# latex_documents:\n# targetname: book.tex\n\n# Information about where the book exists on the web\nrepository:\n url: https://github.com/jorgensd/dolfinx-tutorial # Online location of your book\n path_to_book: . # Optional path to your book, relative to the repository root\n branch: release # Which branch of the repository should be used when creating links (optional)\n\n # Add a bibtex file so that we can create citations\nbibtex_bibfiles:\n - references.bib\n\nlaunch_buttons:\n notebook_interface: \"jupyterlab\"\n binderhub_url: \"https://mybinder.org\"\n\nsphinx:\n config:\n html_last_updated_fmt: \"%b %d, %Y\"\n suppress_warnings: [\"mystnb.unknown_mime_type\"]\n\n # To avoid warning about default changing due to\n # https://github.com/pydata/pydata-sphinx-theme/issues/1492\n # html_theme_options:\n # navigation_with_keys: false\n codeautolink_concat_default: True\n intersphinx_mapping:\n basix: [\"https://docs.fenicsproject.org/basix/main/python/\", null]\n ffcx: [\"https://docs.fenicsproject.org/ffcx/main/\", null]\n ufl: [\"https://docs.fenicsproject.org/ufl/main/\", null]\n dolfinx: [\"https://docs.fenicsproject.org/dolfinx/main/python\", null]\n petsc4py: [\"https://petsc.org/release/petsc4py\", null]\n mpi4py: [\"https://mpi4py.readthedocs.io/en/stable\", null]\n numpy: [\"https://numpy.org/doc/stable/\", null]\n pyvista: [\"https://docs.pyvista.org/\", null]\n packaging: [\"https://packaging.pypa.io/en/stable/\", null]\n matplotlib: [\"https://matplotlib.org/stable/\", null]\n\n extra_extensions:\n - \"sphinx.ext.autodoc\"\n - \"sphinx.ext.intersphinx\"\n - \"sphinx_codeautolink\"\n\nparse:\n myst_enable_extensions:\n - \"amsmath\"\n - \"colon_fence\"\n - \"deflist\"\n - \"dollarmath\"\n - \"html_admonition\"\n - \"html_image\"\n - \"linkify\"\n - \"replacements\"\n - \"smartquotes\"\n - \"substitution\"\n# Add GitHub buttons to your book\n# See https://jupyterbook.org/customize/config.html#add-a-link-to-your-repository\nhtml:\n use_issues_button: true\n use_repository_button: true\n use_edit_page_button: true\n\n extra_footer: |\n
\n This webpage is an adaptation of The FEniCS tutorial and\n is distributed under the terms of the Creative Commons Attribution 4.0 International License \n which permits use, duplication, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source,\n provide a link to the Creative Commons license and indicate if changes were made.\n
\n\nexclude_patterns: [README.md, chapter2/advdiffreac.md]\nonly_build_toc_files: true\n" }, { "path": "_toc.yml", "content": "format: jb-book\nroot: index\nparts:\n - caption: Introduction\n chapters:\n - file: fem\n - file: Changelog\n - caption: Fundamentals\n chapters:\n - file: chapter1/fundamentals\n sections:\n - file: chapter1/fundamentals_code\n - file: chapter1/complex_mode\n - file: chapter1/nitsche\n - file: chapter1/membrane\n sections:\n - file: chapter1/membrane_code\n - file: chapter1/membrane_paraview\n\n - caption: A Gallery of finite element solvers\n chapters:\n - file: chapter2/intro\n - file: chapter2/heat_equation\n sections:\n - file: chapter2/diffusion_code\n - file: chapter2/heat_code\n - file: chapter2/singular_poisson\n - file: chapter2/nonlinpoisson\n sections:\n - file: chapter2/nonlinpoisson_code\n - file: chapter2/linearelasticity\n sections:\n - file: chapter2/linearelasticity_code\n - file: chapter2/elasticity_scaling\n - file: chapter2/navierstokes\n sections:\n - file: chapter2/ns_code1\n - file: chapter2/ns_code2\n - file: chapter2/hyperelasticity\n - file: chapter2/helmholtz\n sections:\n - file: chapter2/helmholtz_code\n - file: chapter2/amr\n\n - caption: Subdomains and boundary conditions\n chapters:\n - file: chapter3/neumann_dirichlet_code\n - file: chapter3/multiple_dirichlet\n - file: chapter3/subdomains\n - file: chapter3/robin_neumann_dirichlet\n - file: chapter3/component_bc\n - file: chapter3/em\n - caption: Improving your FEniCSx code\n chapters:\n - file: chapter4/mixed_poisson\n - file: chapter4/solvers\n - file: chapter4/compiler_parameters\n - file: chapter4/convergence\n - file: chapter4/newton-solver\n" }, { "path": "chapter1/complex_mode.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"markdown\",\n \"id\": \"0\",\n \"metadata\": {},\n \"source\": [\n \"# The Poisson problem with complex numbers\\n\",\n \"\\n\",\n \"Author: Jørgen S. Dokken\\n\",\n \"\\n\",\n \"Many PDEs, such as the [Helmholtz equation](https://docs.fenicsproject.org/dolfinx/main/python/demos/demo_helmholtz.html)\\n\",\n \"require complex-valued fields.\\n\",\n \"\\n\",\n \"For simplicity, let us consider a Poisson equation of the form:\\n\",\n \"\\n\",\n \"$$\\n\",\n \"\\\\begin{align}\\n\",\n \"-\\\\Delta u &= f &&\\\\text{in } \\\\Omega,\\\\\\\\\\n\",\n \"f &= -1 - 2j &&\\\\text{in } \\\\Omega,\\\\\\\\\\n\",\n \"u &= u_{exact} &&\\\\text{on } \\\\partial\\\\Omega,\\\\\\\\\\n\",\n \"u_{exact}(x, y) &= \\\\frac{1}{2}x^2 + 1j\\\\cdot y^2,\\n\",\n \"\\\\end{align}\\n\",\n \"$$\\n\",\n \"\\n\",\n \"As in [Solving the Poisson equation](./fundamentals) we want to express our partial differential equation\\n\",\n \"as a weak formulation.\\n\",\n \"\\n\",\n \"We start by defining our discrete function space $V_h$, such that $u_h\\\\in V_h$ and\\n\",\n \"$u_h = \\\\sum_{i=1}^N c_i \\\\phi_i(x, y)$ where $\\\\phi_i$ are **real valued** global basis\\n\",\n \"functions of our space $V_h$, and $c_i \\\\in \\\\mathcal{C}$ are the **complex valued** degrees of freedom.\\n\",\n \"\\n\",\n \"Next, we choose a test function $v\\\\in \\\\hat V_h$ where $\\\\hat V_h\\\\subset V_h$ such that $v\\\\vert_{\\\\partial\\\\Omega}=0$, as done in the [first tutorial](./fundamentals).\\n\",\n \"We now need to define our inner product space.\\n\",\n \"We choose the $L^2$ inner product spaces, which is a _[sesquilinear](https://en.wikipedia.org/wiki/Sesquilinear_form) 2-form_,\\n\",\n \"meaning that $\\\\langle u, v\\\\rangle$ is a map from $V_h\\\\times V_h\\\\mapsto K$, and\\n\",\n \"$\\\\langle u, v \\\\rangle = \\\\int_\\\\Omega u \\\\cdot \\\\bar v ~\\\\mathrm{d} x$. As it is sesquilinear, we have the following properties:\\n\",\n \"\\n\",\n \"$$\\n\",\n \"\\\\begin{align}\\n\",\n \"\\\\langle u , v \\\\rangle &= \\\\overline{\\\\langle v, u \\\\rangle},\\\\\\\\\\n\",\n \"\\\\langle u , u \\\\rangle &\\\\geq 0.\\n\",\n \"\\\\end{align}\\n\",\n \"$$\\n\",\n \"\\n\",\n \"We can now use this inner product space to do integration by parts\\n\",\n \"\\n\",\n \"$$\\n\",\n \"\\\\int_\\\\Omega \\\\nabla u_h \\\\cdot \\\\nabla \\\\overline{v}~\\\\mathrm{dx} =\\n\",\n \"\\\\int_{\\\\Omega} f \\\\cdot \\\\overline{v} ~\\\\mathrm{d} s \\\\qquad \\\\forall v \\\\in \\\\hat{V}_h.\\n\",\n \"$$\\n\",\n \"\\n\",\n \"## Installation of FEniCSx with complex number support\\n\",\n \"\\n\",\n \"FEniCSx supports both real and complex numbers, so we can create a {py:class}`function space `\\n\",\n \"with either real valued or complex valued coefficients.\\n\",\n \"```{admonition} Function or Coefficient\\n\",\n \"In FEniCSx, the term *function* and *coefficient* are used interchangeably.\\n\",\n \"A function is a linear combination of basis functions with coefficients, and the coefficients can be real or complex numbers.\\n\",\n \"In {py:mod}`ufl`, the term {py:class}`Coefficient `, while in {py:mod}`dolfinx` we use {py:class}`Function`\\n\",\n \"to represent the same concept (through inheritance). This is because most people think of finding the **unknown** function that solves a PDE,\\n\",\n \"while the coefficients are the set of values that define the function.\\n\",\n \"```\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"1\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"from mpi4py import MPI\\n\",\n \"import dolfinx\\n\",\n \"import numpy as np\\n\",\n \"\\n\",\n \"mesh = dolfinx.mesh.create_unit_square(MPI.COMM_WORLD, 10, 10)\\n\",\n \"V = dolfinx.fem.functionspace(mesh, (\\\"Lagrange\\\", 1))\\n\",\n \"u_r = dolfinx.fem.Function(V, dtype=np.float64)\\n\",\n \"u_r.interpolate(lambda x: x[0])\\n\",\n \"u_c = dolfinx.fem.Function(V, dtype=np.complex128)\\n\",\n \"u_c.interpolate(lambda x: 0.5 * x[0] ** 2 + 1j * x[1] ** 2)\\n\",\n \"print(u_r.x.array.dtype)\\n\",\n \"print(u_c.x.array.dtype)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"2\",\n \"metadata\": {},\n \"source\": [\n \"However, as we would like to solve linear algebra problems of the form $Ax=b$, we need to be able to use matrices and vectors that support real and complex numbers.\\n\",\n \"As {[PETSc](https://petsc.org/release/)} is the most popular interfaces to linear algebra packages, we need to be able to work with their matrix and vector structures.\\n\",\n \"\\n\",\n \"Unfortunately, PETSc only supports one floating type in their matrices, thus we need to install two versions of PETSc,\\n\",\n \"one that supports `float64` and one that supports `complex128`.\\n\",\n \"In the [Docker images]https://github.com/orgs/FEniCS/packages/container/package/dolfinx%2Fdolfinx) for DOLFINx, both versions are installed,\\n\",\n \"and one can switch between them by calling `source dolfinx-real-mode` or `source dolfinx-complex-mode`.\\n\",\n \"For the [dolfinx/lab](https://github.com/FEniCS/dolfinx/pkgs/container/dolfinx%2Flab) images,\\n\",\n \"one can change the Python kernel to be either the real or complex mode, by going to\\n\",\n \"`Kernel->Change Kernel...` and choosing `Python3 (ipykernel)` (for real mode) or `Python3 (DOLFINx complex)` (for complex mode).\\n\",\n \"\\n\",\n \"We check that we are using the correct installation of PETSc by inspecting the scalar type.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"3\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"from petsc4py import PETSc\\n\",\n \"from dolfinx.fem.petsc import assemble_vector\\n\",\n \"\\n\",\n \"print(PETSc.ScalarType)\\n\",\n \"assert np.dtype(PETSc.ScalarType).kind == \\\"c\\\"\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"4\",\n \"metadata\": {},\n \"source\": [\n \"## Variational problem\\n\",\n \"We are now ready to define our variational problem\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"5\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"import ufl\\n\",\n \"\\n\",\n \"u = ufl.TrialFunction(V)\\n\",\n \"v = ufl.TestFunction(V)\\n\",\n \"f = dolfinx.fem.Constant(mesh, PETSc.ScalarType(-1 - 2j))\\n\",\n \"a = ufl.inner(ufl.grad(u), ufl.grad(v)) * ufl.dx\\n\",\n \"L = ufl.inner(f, v) * ufl.dx\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"6\",\n \"metadata\": {},\n \"source\": [\n \"Note that we have used the `PETSc.ScalarType` to wrap the constant source on the right hand side.\\n\",\n \"This is because we want the integration kernels to assemble into the correct floating type.\\n\",\n \"\\n\",\n \"Secondly, note that we are using {py:func}`ufl.inner` to describe multiplication of $f$ and $v$,\\n\",\n \"even if they are scalar values.\\n\",\n \"This is because {py:func}`ufl.inner` takes the conjugate of the second argument,\\n\",\n \"as decribed by the $L^2$ inner product.\\n\",\n \"One could alternatively write this out explicitly\\n\",\n \"\\n\",\n \"### Inner-products and derivatives\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"7\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"L2 = f * ufl.conj(v) * ufl.dx\\n\",\n \"print(L)\\n\",\n \"print(L2)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"8\",\n \"metadata\": {},\n \"source\": [\n \"Similarly, if we want to use the function {py:func}`ufl.derivative` to take derivatives of functionals,\\n\",\n \"we need to take some special care.\\n\",\n \"As {py:func}`ufl.derivative` inserts a {py:func}`ufl.TestFunction` to represent the variation,\\n\",\n \"we need to take the conjugate of this to be able to use it to assemble vectors.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"9\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"J = u_c**2 * ufl.dx\\n\",\n \"F = ufl.derivative(J, u_c, ufl.conj(v))\\n\",\n \"residual = assemble_vector(dolfinx.fem.form(F))\\n\",\n \"print(residual.array)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"10\",\n \"metadata\": {},\n \"source\": [\n \"We define our Dirichlet condition and setup and solve the variational problem.\\n\",\n \"## Solve variational problem\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"11\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"mesh.topology.create_connectivity(mesh.topology.dim - 1, mesh.topology.dim)\\n\",\n \"boundary_facets = dolfinx.mesh.exterior_facet_indices(mesh.topology)\\n\",\n \"boundary_dofs = dolfinx.fem.locate_dofs_topological(\\n\",\n \" V, mesh.topology.dim - 1, boundary_facets\\n\",\n \")\\n\",\n \"bc = dolfinx.fem.dirichletbc(u_c, boundary_dofs)\\n\",\n \"problem = dolfinx.fem.petsc.LinearProblem(\\n\",\n \" a, L, bcs=[bc], petsc_options_prefix=\\\"complex_poisson\\\"\\n\",\n \")\\n\",\n \"uh = problem.solve()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"12\",\n \"metadata\": {},\n \"source\": [\n \"We compute the $L^2$ error and the max error.\\n\",\n \"\\n\",\n \"## Error computation\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"13\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"x = ufl.SpatialCoordinate(mesh)\\n\",\n \"u_ex = 0.5 * x[0] ** 2 + 1j * x[1] ** 2\\n\",\n \"L2_error = dolfinx.fem.form(\\n\",\n \" ufl.dot(uh - u_ex, uh - u_ex) * ufl.dx(metadata={\\\"quadrature_degree\\\": 5})\\n\",\n \")\\n\",\n \"local_error = dolfinx.fem.assemble_scalar(L2_error)\\n\",\n \"global_error = np.sqrt(mesh.comm.allreduce(local_error, op=MPI.SUM))\\n\",\n \"max_error = mesh.comm.allreduce(np.max(np.abs(u_c.x.array - uh.x.array)))\\n\",\n \"print(global_error, max_error)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"14\",\n \"metadata\": {},\n \"source\": [\n \"## Plotting\\n\",\n \"\\n\",\n \"Finally, we plot the real and imaginary solutions.\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"15\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"import pyvista\\n\",\n \"\\n\",\n \"mesh.topology.create_connectivity(mesh.topology.dim, mesh.topology.dim)\\n\",\n \"p_mesh = pyvista.UnstructuredGrid(*dolfinx.plot.vtk_mesh(mesh, mesh.topology.dim))\\n\",\n \"pyvista_cells, cell_types, geometry = dolfinx.plot.vtk_mesh(V)\\n\",\n \"grid = pyvista.UnstructuredGrid(pyvista_cells, cell_types, geometry)\\n\",\n \"grid.point_data[\\\"u_real\\\"] = uh.x.array.real\\n\",\n \"grid.point_data[\\\"u_imag\\\"] = uh.x.array.imag\\n\",\n \"_ = grid.set_active_scalars(\\\"u_real\\\")\\n\",\n \"\\n\",\n \"p_real = pyvista.Plotter()\\n\",\n \"p_real.add_text(\\\"uh real\\\", position=\\\"upper_edge\\\", font_size=14, color=\\\"black\\\")\\n\",\n \"p_real.add_mesh(grid, show_edges=True)\\n\",\n \"p_real.view_xy()\\n\",\n \"if not pyvista.OFF_SCREEN:\\n\",\n \" p_real.show()\\n\",\n \"\\n\",\n \"grid.set_active_scalars(\\\"u_imag\\\")\\n\",\n \"p_imag = pyvista.Plotter()\\n\",\n \"p_imag.add_text(\\\"uh imag\\\", position=\\\"upper_edge\\\", font_size=14, color=\\\"black\\\")\\n\",\n \"p_imag.add_mesh(grid, show_edges=True)\\n\",\n \"p_imag.view_xy()\\n\",\n \"if not pyvista.OFF_SCREEN:\\n\",\n \" p_imag.show()\"\n ]\n }\n ],\n \"metadata\": {\n \"jupytext\": {\n \"formats\": \"ipynb,py:light\"\n },\n \"kernelspec\": {\n \"display_name\": \"Python 3 (DOLFINx complex)\",\n \"language\": \"python\",\n \"name\": \"python3-complex\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 3\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython3\",\n \"version\": \"3.10.12\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 5\n}\n" }, { "path": "chapter1/complex_mode.py", "content": "# ---\n# jupyter:\n# jupytext:\n# formats: ipynb,py:light\n# text_representation:\n# extension: .py\n# format_name: light\n# format_version: '1.5'\n# jupytext_version: 1.18.1\n# kernelspec:\n# display_name: Python 3 (DOLFINx complex)\n# language: python\n# name: python3-complex\n# ---\n\n# # The Poisson problem with complex numbers\n#\n# Author: Jørgen S. Dokken\n#\n# Many PDEs, such as the [Helmholtz equation](https://docs.fenicsproject.org/dolfinx/main/python/demos/demo_helmholtz.html)\n# require complex-valued fields.\n#\n# For simplicity, let us consider a Poisson equation of the form:\n#\n# $$\n# \\begin{align}\n# -\\Delta u &= f &&\\text{in } \\Omega,\\\\\n# f &= -1 - 2j &&\\text{in } \\Omega,\\\\\n# u &= u_{exact} &&\\text{on } \\partial\\Omega,\\\\\n# u_{exact}(x, y) &= \\frac{1}{2}x^2 + 1j\\cdot y^2,\n# \\end{align}\n# $$\n#\n# As in [Solving the Poisson equation](./fundamentals) we want to express our partial differential equation\n# as a weak formulation.\n#\n# We start by defining our discrete function space $V_h$, such that $u_h\\in V_h$ and\n# $u_h = \\sum_{i=1}^N c_i \\phi_i(x, y)$ where $\\phi_i$ are **real valued** global basis\n# functions of our space $V_h$, and $c_i \\in \\mathcal{C}$ are the **complex valued** degrees of freedom.\n#\n# Next, we choose a test function $v\\in \\hat V_h$ where $\\hat V_h\\subset V_h$ such that $v\\vert_{\\partial\\Omega}=0$, as done in the [first tutorial](./fundamentals).\n# We now need to define our inner product space.\n# We choose the $L^2$ inner product spaces, which is a _[sesquilinear](https://en.wikipedia.org/wiki/Sesquilinear_form) 2-form_,\n# meaning that $\\langle u, v\\rangle$ is a map from $V_h\\times V_h\\mapsto K$, and\n# $\\langle u, v \\rangle = \\int_\\Omega u \\cdot \\bar v ~\\mathrm{d} x$. As it is sesquilinear, we have the following properties:\n#\n# $$\n# \\begin{align}\n# \\langle u , v \\rangle &= \\overline{\\langle v, u \\rangle},\\\\\n# \\langle u , u \\rangle &\\geq 0.\n# \\end{align}\n# $$\n#\n# We can now use this inner product space to do integration by parts\n#\n# $$\n# \\int_\\Omega \\nabla u_h \\cdot \\nabla \\overline{v}~\\mathrm{dx} =\n# \\int_{\\Omega} f \\cdot \\overline{v} ~\\mathrm{d} s \\qquad \\forall v \\in \\hat{V}_h.\n# $$\n#\n# ## Installation of FEniCSx with complex number support\n#\n# FEniCSx supports both real and complex numbers, so we can create a {py:class}`function space `\n# with either real valued or complex valued coefficients.\n# ```{admonition} Function or Coefficient\n# In FEniCSx, the term *function* and *coefficient* are used interchangeably.\n# A function is a linear combination of basis functions with coefficients, and the coefficients can be real or complex numbers.\n# In {py:mod}`ufl`, the term {py:class}`Coefficient `, while in {py:mod}`dolfinx` we use {py:class}`Function`\n# to represent the same concept (through inheritance). This is because most people think of finding the **unknown** function that solves a PDE,\n# while the coefficients are the set of values that define the function.\n# ```\n\n# +\nfrom mpi4py import MPI\nimport dolfinx\nimport numpy as np\n\nmesh = dolfinx.mesh.create_unit_square(MPI.COMM_WORLD, 10, 10)\nV = dolfinx.fem.functionspace(mesh, (\"Lagrange\", 1))\nu_r = dolfinx.fem.Function(V, dtype=np.float64)\nu_r.interpolate(lambda x: x[0])\nu_c = dolfinx.fem.Function(V, dtype=np.complex128)\nu_c.interpolate(lambda x: 0.5 * x[0] ** 2 + 1j * x[1] ** 2)\nprint(u_r.x.array.dtype)\nprint(u_c.x.array.dtype)\n# -\n\n# However, as we would like to solve linear algebra problems of the form $Ax=b$, we need to be able to use matrices and vectors that support real and complex numbers.\n# As {[PETSc](https://petsc.org/release/)} is the most popular interfaces to linear algebra packages, we need to be able to work with their matrix and vector structures.\n#\n# Unfortunately, PETSc only supports one floating type in their matrices, thus we need to install two versions of PETSc,\n# one that supports `float64` and one that supports `complex128`.\n# In the [Docker images]https://github.com/orgs/FEniCS/packages/container/package/dolfinx%2Fdolfinx) for DOLFINx, both versions are installed,\n# and one can switch between them by calling `source dolfinx-real-mode` or `source dolfinx-complex-mode`.\n# For the [dolfinx/lab](https://github.com/FEniCS/dolfinx/pkgs/container/dolfinx%2Flab) images,\n# one can change the Python kernel to be either the real or complex mode, by going to\n# `Kernel->Change Kernel...` and choosing `Python3 (ipykernel)` (for real mode) or `Python3 (DOLFINx complex)` (for complex mode).\n#\n# We check that we are using the correct installation of PETSc by inspecting the scalar type.\n\n# +\nfrom petsc4py import PETSc\nfrom dolfinx.fem.petsc import assemble_vector\n\nprint(PETSc.ScalarType)\nassert np.dtype(PETSc.ScalarType).kind == \"c\"\n# -\n\n# ## Variational problem\n# We are now ready to define our variational problem\n\n# +\nimport ufl\n\nu = ufl.TrialFunction(V)\nv = ufl.TestFunction(V)\nf = dolfinx.fem.Constant(mesh, PETSc.ScalarType(-1 - 2j))\na = ufl.inner(ufl.grad(u), ufl.grad(v)) * ufl.dx\nL = ufl.inner(f, v) * ufl.dx\n# -\n\n# Note that we have used the `PETSc.ScalarType` to wrap the constant source on the right hand side.\n# This is because we want the integration kernels to assemble into the correct floating type.\n#\n# Secondly, note that we are using {py:func}`ufl.inner` to describe multiplication of $f$ and $v$,\n# even if they are scalar values.\n# This is because {py:func}`ufl.inner` takes the conjugate of the second argument,\n# as decribed by the $L^2$ inner product.\n# One could alternatively write this out explicitly\n#\n# ### Inner-products and derivatives\n\nL2 = f * ufl.conj(v) * ufl.dx\nprint(L)\nprint(L2)\n\n# Similarly, if we want to use the function {py:func}`ufl.derivative` to take derivatives of functionals,\n# we need to take some special care.\n# As {py:func}`ufl.derivative` inserts a {py:func}`ufl.TestFunction` to represent the variation,\n# we need to take the conjugate of this to be able to use it to assemble vectors.\n\nJ = u_c**2 * ufl.dx\nF = ufl.derivative(J, u_c, ufl.conj(v))\nresidual = assemble_vector(dolfinx.fem.form(F))\nprint(residual.array)\n\n# We define our Dirichlet condition and setup and solve the variational problem.\n# ## Solve variational problem\n\nmesh.topology.create_connectivity(mesh.topology.dim - 1, mesh.topology.dim)\nboundary_facets = dolfinx.mesh.exterior_facet_indices(mesh.topology)\nboundary_dofs = dolfinx.fem.locate_dofs_topological(\n V, mesh.topology.dim - 1, boundary_facets\n)\nbc = dolfinx.fem.dirichletbc(u_c, boundary_dofs)\nproblem = dolfinx.fem.petsc.LinearProblem(\n a, L, bcs=[bc], petsc_options_prefix=\"complex_poisson\"\n)\nuh = problem.solve()\n\n# We compute the $L^2$ error and the max error.\n#\n# ## Error computation\n#\n\nx = ufl.SpatialCoordinate(mesh)\nu_ex = 0.5 * x[0] ** 2 + 1j * x[1] ** 2\nL2_error = dolfinx.fem.form(\n ufl.dot(uh - u_ex, uh - u_ex) * ufl.dx(metadata={\"quadrature_degree\": 5})\n)\nlocal_error = dolfinx.fem.assemble_scalar(L2_error)\nglobal_error = np.sqrt(mesh.comm.allreduce(local_error, op=MPI.SUM))\nmax_error = mesh.comm.allreduce(np.max(np.abs(u_c.x.array - uh.x.array)))\nprint(global_error, max_error)\n\n# ## Plotting\n#\n# Finally, we plot the real and imaginary solutions.\n#\n\n# +\nimport pyvista\n\nmesh.topology.create_connectivity(mesh.topology.dim, mesh.topology.dim)\np_mesh = pyvista.UnstructuredGrid(*dolfinx.plot.vtk_mesh(mesh, mesh.topology.dim))\npyvista_cells, cell_types, geometry = dolfinx.plot.vtk_mesh(V)\ngrid = pyvista.UnstructuredGrid(pyvista_cells, cell_types, geometry)\ngrid.point_data[\"u_real\"] = uh.x.array.real\ngrid.point_data[\"u_imag\"] = uh.x.array.imag\n_ = grid.set_active_scalars(\"u_real\")\n\np_real = pyvista.Plotter()\np_real.add_text(\"uh real\", position=\"upper_edge\", font_size=14, color=\"black\")\np_real.add_mesh(grid, show_edges=True)\np_real.view_xy()\nif not pyvista.OFF_SCREEN:\n p_real.show()\n\ngrid.set_active_scalars(\"u_imag\")\np_imag = pyvista.Plotter()\np_imag.add_text(\"uh imag\", position=\"upper_edge\", font_size=14, color=\"black\")\np_imag.add_mesh(grid, show_edges=True)\np_imag.view_xy()\nif not pyvista.OFF_SCREEN:\n p_imag.show()\n" }, { "path": "chapter1/fundamentals.md", "content": "# Solving the Poisson equation\n\nAuthors: Hans Petter Langtangen, Anders Logg\n\nAdapted to FEniCSx by Jørgen S. Dokken\n\nThe goal of this tutorial is to solve one of the most basic PDEs, the Poisson equation, with a few lines of code in FEniCSx.\nWe start by introducing some fundamental FEniCSx objects, such as {py:class}`Function`,\n{py:func}`functionspace`, {py:func}`TrialFunction` and {py:func}`TestFunction`,\nand learn how to write a basic PDE solver.\nThis will include:\n\n- How to formulate a mathematical variational problem\n- How to apply boundary conditions\n- How to solve the discrete linear system\n- How to visualize the solution\n\nThe Poisson equation is the following boundary-value problem\n\\begin{align}\n-\\nabla^2 u(\\mathbf{x}) &= f(\\mathbf{x})&&\\mathbf{x} \\in \\Omega\\\\\nu(\\mathbf{x}) &= u_D(\\mathbf{x})&& \\mathbf{x} \\in \\partial\\Omega\n\\end{align}\n\nHere, $u=u(\\mathbf{x})$ is the unknown function, $f=f(\\mathbf{x})$ a prescribed function, $\\nabla^2$ the Laplace operator\n(often written as $\\Delta$), $\\Omega$ the spatial domain, and $\\partial\\Omega$ the boundary of $\\Omega$.\nThe Poisson problem, including both the PDE, $-\\nabla^2 u = f$, and the boundary condition, $u=u_D$ on $\\partial\\Omega$, is an example of a _boundary-value problem_, which must be precisely stated before we can start solving it numerically with FEniCSx.\n\nIn the two-dimensional space with coordinates $x$ and $y$, we can expand the Poisson equation as\n\n$$-\\frac{\\partial^2 u}{\\partial x^2} - \\frac{\\partial^2 u}{\\partial y^2} = f(x,y)$$\n\nThe unknown $u$ is now a function of two variables, $u=u(x,y)$, defined over the two-dimensional domain $\\Omega$.\n\nThe Poisson equation arises in numerous physical contexts, including\nheat conduction, electrostatics, diffusion of substances, twisting of\nelastic rods, inviscid fluid flow, and water waves. Moreover, the\nequation appears in numerical splitting strategies for more complicated\nsystems of PDEs, in particular the Navier--Stokes equations.\n\nSolving a boundary value problem in FEniCSx consists of the following steps:\n\n1. Identify the computational domain $\\Omega$, the PDE, and its corresponding boundary conditions and source terms $f$.\n2. Reformulate the PDE as a finite element variational problem.\n3. Write a Python program defining the computational domain, the boundary conditions, the variational problem, and the source terms, using FEniCSx.\n4. Run the Python program to solve the boundary-value problem. Optionally, you can extend the program to derive quantities such as fluxes and averages,\n and visualize the results.\n\nAs we have already covered step 1, we shall now cover steps 2-4.\n\n## Finite element variational formulation\n\nFEniCSx is based on the finite element method, which is a general and\nefficient mathematical technique for the numerical solution of\nPDEs. The starting point for finite element methods is a PDE\nexpressed in _variational form_. For readers not familiar with variational problems, we suggest reading a proper treatment on the finite element method, as this tutorial is meant as a brief introduction to the subject. See the original tutorial {cite}`fd-FenicsTutorial` (Chapter 1.6.2).\n\nThe basic recipe for turning a PDE into a variational problem is:\n\n- Multiply the PDE by a function $v$\n- Integrate the resulting equation over the domain $\\Omega$\n- Perform integration by parts of those terms with second order derivatives\n\nThe function $v$ which multiplies the PDE is called a _test function_. The unknown function $u$ that is to be approximated is referred to as a _trial function_.\nThe terms trial and test functions are used in FEniCSx too. The test and trial functions belong to certain _function spaces_ that specify the properties of the functions.\n\nIn the present case, we multiply the Poisson equation by a test function $v$ and integrate over $\\Omega$:\n\n$$\\int_\\Omega (-\\nabla^2 u) v~\\mathrm{d} x = \\int_\\Omega f v~\\mathrm{d} x.$$\n\nHere $\\mathrm{d} x$ denotes the differential element for integration over the domain $\\Omega$. We will later let $\\mathrm{d} s$ denote the differential element for integration over $\\partial\\Omega$, the boundary of $\\Omega$.\n\nA rule of thumb when deriving variational formulations is that one tries to keep the order of derivatives of $u$ and $v$ as small as possible.\nHere, we have a second-order differential of $u$, which can be transformed to a first derivative by employing the technique of\n[integration by parts](https://en.wikipedia.org/wiki/Integration_by_parts).\nThe formula reads\n\n$$-\\int_\\Omega (\\nabla^2 u)v~\\mathrm{d}x\n= \\int_\\Omega\\nabla u\\cdot\\nabla v~\\mathrm{d}x- \n\\int_{\\partial\\Omega}\\frac{\\partial u}{\\partial n}v~\\mathrm{d}s,$$\n\nwhere $\\dfrac{\\partial u}{\\partial n}=\\nabla u \\cdot \\vec{n}$ is the derivative of $u$ in the outward normal direction $\\vec{n}$ on the boundary.\n\nAnother feature of variational formulations is that the test function $v$ is required to vanish on the parts of the boundary where the solution $u$ is known. See for instance {cite}`fd-Langtangen_Mardal_FEM_2019`.\n\nIn the present problem, this means that $v$ is $0$ on the whole boundary $\\partial\\Omega$. Thus, the second term in the integration by parts formula vanishes, and we have that\n\n$$\\int_\\Omega \\nabla u \\cdot \\nabla v~\\mathrm{d} x = \\int_\\Omega f v~\\mathrm{d} x.$$\n\nIf we require that this equation holds for all test functions $v$ in some suitable space $\\hat{V}$, the so-called _test space_, we obtain a well-defined mathematical problem that uniquely determines the solution $u$ which lies in some function space $V$. Note that $V$ does not have to be the same space as\n$\\hat{V}$. We call the space $V$ the _trial space_. We refer to the equation above as the _weak form_/_variational form_ of the original boundary-value problem. We now properly state our variational problem:\nFind $u\\in V$ such that\n\n$$\\int_\\Omega \\nabla u \\cdot \\nabla v~\\mathrm{d} x = \\int_\\Omega f v~\\mathrm{d} x\\qquad \\forall v \\in \\hat{V}.$$\n\nFor the present problem, the trial and test spaces $V$ and $\\hat{V}$ are defined as\n\\begin{equation}\n\\begin{alignedat}{2}\nV &= \\{v \\in H^1(\\Omega) \\mid v = u_D && \\quad \\text{on } \\partial \\Omega \\}, \\\\\n\\hat{V} &= \\{v \\in H^1(\\Omega) \\mid v = 0 && \\quad \\text{on } \\partial \\Omega \\}.\n\\end{alignedat}\n\\end{equation}\nIn short, $H^1(\\Omega)$ is the Sobolev space containing functions $v$ such that $v^2$ and $\\vert \\nabla v \\vert ^2$ have finite integrals over $\\Omega$. The solution of the underlying\nPDE must lie in a function space where the derivatives are\nalso continuous, but the Sobolev space $H^1(\\Omega)$ allows functions with discontinuous derivatives.\nThis weaker continuity requirement in our weak formulation (caused by the integration by parts) is of great importance when it comes to constructing the finite element function space. In particular, it allows the use of piecewise polynomial function spaces. This means that the function spaces are constructed\nby stitching together polynomial functions on simple domains\nsuch as intervals, triangles, quadrilaterals, tetrahedra and\nhexahedra.\n\nThe variational problem is a _continuous problem_: it defines the solution $u$ in the infinite-dimensional function space $V$.\nThe finite element method for the Poisson equation finds an approximate solution of the variational problem by replacing the infinite-dimensional function spaces $V$ and $\\hat{V}$ by _discrete_ (finite dimensional) trial and test spaces $V_h\\subset V$ and $\\hat{V}_h \\subset \\hat{V}$. The discrete\nvariational problem reads: Find $u_h\\in V_h$ such that\n\\begin{align}\n\\int_\\Omega \\nabla u_h \\cdot \\nabla v~\\mathrm{d} x &= \\int_\\Omega fv~\\mathrm{d} x && \\forall v \\in \\hat{V}_h.\n\\end{align}\nThis variational problem, together with suitable definitions of $V_h$ and $\\hat{V}_h$ uniquely define our approximate numerical solution of the Poisson equation.\nNote that the boundary condition is encoded as part of the test and trial spaces. This might seem complicated at first glance,\nbut means that the finite element variational problem and the continuous variational problem look the same.\n\n## Abstract finite element variational formulation\n\nWe will introduce the following notation for variational problems:\nFind $u\\in V$ such that\n\\begin{align}\na(u,v)&=L(v)&& \\forall v \\in \\hat{V}.\n\\end{align}\nFor the Poisson equation, we have:\n\\begin{align}\na(u,v) &= \\int_{\\Omega} \\nabla u \\cdot \\nabla v~\\mathrm{d} x,\\\\\nL(v) &= \\int_{\\Omega} fv~\\mathrm{d} x.\n\\end{align}\nIn the literature $a(u,v)$ is known as the _bilinear form_ and $L(v)$ as a _linear form_.\nFor every linear problem, we will identify all terms with the unknown $u$ and collect them in $a(u,v)$, and collect all terms with only known functions in $L(v)$.\n\nTo solve a linear PDE in FEniCSx, such as the Poisson equation, a user thus needs to perform two steps:\n\n1. Choose the finite element spaces $V$ and $\\hat{V}$ by specifying the domain (the mesh) and the type of function space (polynomial degree and type).\n2. Express the PDE as a (discrete) variational problem: Find $u\\in V$ such that $a(u,v)=L(v)$ for all $v \\in \\hat{V}$.\n\n## References\n\n```{bibliography}\n :filter: cited\n :labelprefix:\n :keyprefix: fd-\n```\n" }, { "path": "chapter1/fundamentals_code.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"markdown\",\n \"id\": \"0\",\n \"metadata\": {},\n \"source\": [\n \"# Implementation\\n\",\n \"\\n\",\n \"Author: Jørgen Schartum Dokken\\n\",\n \"\\n\",\n \"This implementation is an adaptation of the work in {cite}`fundamentals-FenicsTutorial` to DOLFINx.\\n\",\n \"\\n\",\n \"In this section, you will learn:\\n\",\n \"- How to use the built-in meshes in DOLFINx\\n\",\n \"- How to create a spatially varying Dirichlet boundary conditions on the whole domain boundary\\n\",\n \"- How to define a weak formulation of your PDE\\n\",\n \"- How to solve the resulting system of linear equations\\n\",\n \"- How to visualize the solution using a variety of tools\\n\",\n \"- How to compute the $L^2(\\\\Omega)$ error and the error at mesh vertices\\n\",\n \"\\n\",\n \"## Interactive tutorials\\n\",\n \"```{admonition} Run the tutorial as Jupyter notebook in browser\\n\",\n \"As this book has been published as a Jupyter Book, each code can be run in your browser as a Jupyter notebook.\\n\",\n \"To start such a notebook click the rocket symbol in the top right corner of the relevant tutorial.\\n\",\n \"```\\n\",\n \"\\n\",\n \"The Poisson problem has so far featured a general domain $\\\\Omega$ and general functions $u_D$ for\\n\",\n \"the boundary conditions and $f$ for the right hand side.\\n\",\n \"Therefore, we need to make specific choices of $\\\\Omega, u_D$ and $f$.\\n\",\n \"A wise choice is to construct a problem with a known analytical solution,\\n\",\n \"so that we can check that the computed solution is correct.\\n\",\n \"The primary candidates are lower-order polynomials.\\n\",\n \"The continuous Galerkin finite element spaces of degree $r$ will exactly reproduce polynomials of degree $r$.\\n\",\n \"\\n\",\n \" We use this fact to construct a quadratic function in $2D$. In particular we choose\\n\",\n \"\\n\",\n \"$$\\n\",\n \"\\\\begin{align}\\n\",\n \" u_e(x,y)=1+x^2+2y^2\\n\",\n \" \\\\end{align}\\n\",\n \"$$\\n\",\n \"\\n\",\n \"Inserting $u_e$ in the original boundary problem, we find that\\n\",\n \"\\n\",\n \"$$\\n\",\n \"\\\\begin{align}\\n\",\n \" f(x,y)= -6,\\\\qquad u_D(x,y)=u_e(x,y)=1+x^2+2y^2,\\n\",\n \"\\\\end{align}\\n\",\n \"$$\\n\",\n \"\\n\",\n \"regardless of the shape of the domain as long as we prescribe\\n\",\n \"$u_e$ on the boundary.\\n\",\n \"\\n\",\n \"For simplicity, we choose the domain to be a unit square $\\\\Omega=[0,1]\\\\times [0,1]$\\n\",\n \"\\n\",\n \"This simple but very powerful method for constructing test problems is called _the method of manufactured solutions_.\\n\",\n \"First pick a simple expression for the exact solution, plug into\\n\",\n \"the equation to obtain the right-hand side (source term $f$).\\n\",\n \"Then solve the equation with this right hand side, and using the exact solution as boundary condition.\\n\",\n \"Finally, we create a program that tries to reproduce the exact solution.\\n\",\n \"\\n\",\n \"Note that in many cases, it can be hard to determine if the program works if it produces an error of size\\n\",\n \"$10^{-5}$ on a $20 \\\\times 20$ grid.\\n\",\n \"However, since we are using Sobolev spaces, we usually know about the numerical errors _asymptotic properties_.\\n\",\n \"For instance that it is proportional to $h^2$ if $h$ is the size of a cell in the mesh.\\n\",\n \"We can then compare the error on meshes with different $h$-values to see if the asymptotic behavior is correct.\\n\",\n \"This technique will be explained in detail in the chapter [Improving your fenics code](./../chapter4/convergence).\\n\",\n \"\\n\",\n \"However, in cases where we have a solution we know that should have no approximation error,\\n\",\n \"we know that the solution should be produced to machine precision by the program.\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"1\",\n \"metadata\": {},\n \"source\": [\n \"A major difference between a traditional FEniCS code and a FEniCSx code,\\n\",\n \"is that one is not advised to use the wildcard import.\\n\",\n \"We will see this throughout this first example.\\n\",\n \"\\n\",\n \"## Generating simple meshes\\n\",\n \"The next step is to define the discrete domain, _the mesh_.\\n\",\n \"We do this by importing one of the built-in mesh generators.\\n\",\n \"We will build a {py:func}`unit square mesh`, i.e. a mesh spanning $[0,1]\\\\times[0,1]$.\\n\",\n \"It can consist of either triangles or quadrilaterals.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"2\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"from mpi4py import MPI\\n\",\n \"from dolfinx import mesh\\n\",\n \"import numpy\\n\",\n \"\\n\",\n \"domain = mesh.create_unit_square(MPI.COMM_WORLD, 8, 8, mesh.CellType.quadrilateral)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"3\",\n \"metadata\": {},\n \"source\": [\n \"Note that in addition to give how many elements we would like to have in each direction,\\n\",\n \"we also have to supply the _MPI-communicator_.\\n\",\n \"This is to specify how we would like the program to behave in parallel.\\n\",\n \"If we supply {py:data}`MPI.COMM_WORLD` we create a single mesh,\\n\",\n \"whose data is distributed over the number of processors we would like to use.\\n\",\n \"We can for instance run the program in parallel on two processors by using `mpirun`, as:\\n\",\n \"``` bash\\n\",\n \" mpirun -n 2 python3 t1.py\\n\",\n \"```\\n\",\n \"However, if we would like to create a separate mesh on each processor,\\n\",\n \"we can use {py:data}`MPI.COMM_SELF`.\\n\",\n \"This is for instance useful if we run a small problem, and would like to run it with multiple parameters.\\n\",\n \"\\n\",\n \"## Defining the finite element function space\\n\",\n \" Once the mesh has been created, we can create the finite element function space $V$.\\n\",\n \"The finite element function space does not need to be the same as the one used to describe the mesh.\\n\",\n \"DOLFINx supports a wide range of arbitrary order finite element function spaces, see:\\n\",\n \"[Supported elements in DOLFINx](https://defelement.org/lists/implementations/basix.ufl.html)\\n\",\n \"for an extensive list.\\n\",\n \"To create a function space, we need to specify what mesh the space is defined on,\\n\",\n \"what element famil the space is based on, and the degree of the element.\\n\",\n \"These can for instance be defned through a tuple `(\\\"family\\\", degree)`, as shown below\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"4\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"from dolfinx import fem\\n\",\n \"\\n\",\n \"V = fem.functionspace(domain, (\\\"Lagrange\\\", 1))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"960048ad\",\n \"metadata\": {},\n \"source\": [\n \"Further details about specification/customization of this tuple, see {py:class}`dolfinx.fem.ElementMetaData`.\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"5\",\n \"metadata\": {},\n \"source\": [\n \"## Dirichlet boundary conditions\\n\",\n \"Next, we create a function that will hold the Dirichlet boundary data, and use interpolation to\\n\",\n \"fill it with the appropriate data.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"6\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"uD = fem.Function(V)\\n\",\n \"uD.interpolate(lambda x: 1 + x[0] ** 2 + 2 * x[1] ** 2)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"7\",\n \"metadata\": {},\n \"source\": [\n \"We now have the boundary data (and in this case the solution of the finite element problem)\\n\",\n \"represented in the discrete function space.\\n\",\n \"Next we would like to apply the boundary values to all degrees of freedom that are on the\\n\",\n \"boundary of the discrete domain.\\n\",\n \"We start by identifying the facets (line-segments) representing the outer boundary,\\n\",\n \"using {py:func}`dolfinx.mesh.exterior_facet_indices`.\\n\",\n \"We start by creating the facet to cell connectivity required to determine boundary facets by\\n\",\n \"calling {py:meth}`dolfinx.mesh.Topology.create_connectivity`.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"10\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"tdim = domain.topology.dim\\n\",\n \"fdim = tdim - 1\\n\",\n \"domain.topology.create_connectivity(fdim, tdim)\\n\",\n \"boundary_facets = mesh.exterior_facet_indices(domain.topology)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"11\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"For the current problem, as we are using the first order Lagrange function space,\\n\",\n \"the degrees of freedom are located at the vertices of each cell, thus each facet contains two degrees of freedom.\\n\",\n \"\\n\",\n \"To find the local indices of these degrees of freedom, we use {py:func}`dolfinx.fem.locate_dofs_topological`\\n\",\n \"which takes in the function space, the dimension of entities in the mesh we would like to identify and the local entities.\\n\",\n \"```{admonition} Local ordering of degrees of freedom and mesh vertices\\n\",\n \"Many people expect there to be a 1-1 correspondence between the mesh coordinates and the coordinates of the degrees of freedom.\\n\",\n \"However, this is only true in the case of `Lagrange` 1 elements on a first order mesh.\\n\",\n \"Therefore, in DOLFINx we use separate local numbering for the mesh coordinates and the dof coordinates.\\n\",\n \"To obtain the local dof coordinates we can use\\n\",\n \"{py:meth}`V.tabulate_dof_coordinates()`,\\n\",\n \"while the ordering of the local vertices can be obtained by {py:attr}`mesh.geometry.x`.\\n\",\n \"```\\n\",\n \"With this data at hand, we can create the Dirichlet boundary condition\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"12\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"boundary_dofs = fem.locate_dofs_topological(V, fdim, boundary_facets)\\n\",\n \"bc = fem.dirichletbc(uD, boundary_dofs)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"13\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"## Defining the trial and test function\\n\",\n \"\\n\",\n \"In mathematics, we distinguish between trial and test spaces $V$ and $\\\\hat{V}$.\\n\",\n \"The only difference in the present problem is the boundary conditions.\\n\",\n \"In FEniCSx, we do not specify boundary conditions as part of the function space,\\n\",\n \"so it is sufficient to use a common space for the trial and test function.\\n\",\n \"\\n\",\n \"We use the {py:mod}`Unified Form Language` (UFL) to specify the variational formulations.\\n\",\n \"See {cite}`fundamentals-ufl2014` for more details.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"14\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"import ufl\\n\",\n \"\\n\",\n \"u = ufl.TrialFunction(V)\\n\",\n \"v = ufl.TestFunction(V)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"15\",\n \"metadata\": {},\n \"source\": [\n \"## Defining the source term\\n\",\n \"As the source term is constant over the domain, we use {py:class}`dolfinx.fem.Constant`\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"16\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"from dolfinx import default_scalar_type\\n\",\n \"\\n\",\n \"f = fem.Constant(domain, default_scalar_type(-6))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"17\",\n \"metadata\": {},\n \"source\": [\n \"```{admonition} Compilation speed-up\\n\",\n \"Instead of wrapping $-6$ in a {py:class}`dolfinx.fem.Constant`, we could simply define $f$ as `f=-6`.\\n\",\n \"However, if we would like to change this parameter later in the simulation,\\n\",\n \"we would have to redefine our variational formulation.\\n\",\n \"The {py:attr}`dolfinx.fem.Constant.value` allows us to update the value in $f$ by using `f.value=5`.\\n\",\n \"Additionally, by indicating that $f$ is a constant, we speed up compilation of the variational\\n\",\n \"formulations required for the created linear system.\\n\",\n \"```\\n\",\n \"\\n\",\n \"## Defining the variational problem\\n\",\n \"As we now have defined all variables used to describe our variational problem, we can create the weak formulation\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"18\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"a = ufl.dot(ufl.grad(u), ufl.grad(v)) * ufl.dx\\n\",\n \"L = f * v * ufl.dx\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"19\",\n \"metadata\": {},\n \"source\": [\n \"Note that there is a very close correspondence between the Python syntax and the mathematical syntax\\n\",\n \"$\\\\int_{\\\\Omega} \\\\nabla u \\\\cdot \\\\nabla v ~\\\\mathrm{d} x$ and $\\\\int_{\\\\Omega}fv~\\\\mathrm{d} x$.\\n\",\n \"The integration over the domain $\\\\Omega$ is defined by using {py:func}`ufl.dx`, an integration\\n\",\n \"{py:class}`measure` over all cells of the mesh.\\n\",\n \"\\n\",\n \"This is the key strength of FEniCSx:\\n\",\n \"the formulas in the variational formulation translate directly to very similar Python code,\\n\",\n \"a feature that makes it easy to specify and solve complicated PDE problems.\\n\",\n \"\\n\",\n \"## Expressing inner products\\n\",\n \"The inner product $\\\\int_\\\\Omega \\\\nabla u \\\\cdot \\\\nabla v ~\\\\mathrm{d} x$ can be expressed in various ways in UFL.\\n\",\n \"We have used the notation `ufl.dot(ufl.grad(u), ufl.grad(v))*ufl.dx`.\\n\",\n \"The {py:func}`dot` product in UFL computes the sum (contraction) over the last index\\n\",\n \"of the first factor and first index of the second factor.\\n\",\n \"In this case, both factors are tensors of rank one (vectors) and so the sum is just over\\n\",\n \"the single index of both $\\\\nabla u$ and $\\\\nabla v$.\\n\",\n \"To compute an inner product of matrices (with two indices),\\n\",\n \"one must use the function {py:func}`ufl.inner` instead of {py:func}`ufl.dot`.\\n\",\n \"For real-valued vectors, {py:func}`ufl.dot` and {py:func}`ufl.inner` are equivalent.\\n\",\n \"\\n\",\n \"```{admonition} Complex numbers\\n\",\n \"In DOLFINx, one can solve complex number problems by using an installation of PETSc using complex numbers.\\n\",\n \"For variational formulations with complex numbers, one cannot use {py:func}`ufl.dot` to compute inner products.\\n\",\n \"One has to use {py:func}`ufl.inner`, with the test-function as the second input argument for {py:func}`ufl.inner`.\\n\",\n \"See [Running DOLFINx in complex mode](./complex_mode) for more information.\\n\",\n \"```\\n\",\n \"\\n\",\n \"\\n\",\n \"## Forming and solving the linear system\\n\",\n \"\\n\",\n \"Having defined the finite element variational problem and boundary condition,\\n\",\n \"we can create our {py:class}`LinearProblem` to solve the variational problem:\\n\",\n \"Find $u_h\\\\in V$ such that $a(u_h, v)==L(v) \\\\quad \\\\forall v \\\\in \\\\hat{V}$.\\n\",\n \"We will use {py:mod}`PETSc` as our linear algebra backend, using a direct solver (LU-factorization).\\n\",\n \"See the [PETSc-documentation](https://petsc.org/main/docs/manual/ksp/?highlight=ksp#ksp-linear-system-solvers) of the method for more information.\\n\",\n \"PETSc is not a required dependency of DOLFINx, and therefore we explicitly import the DOLFINx wrapper for interfacing with PETSc.\\n\",\n \"To ensure that the options passed to the {py:class}`LinearProblem`\\n\",\n \"is only used for the given KSP solver, we pass a **unique** option prefix as well.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"20\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"from dolfinx.fem.petsc import LinearProblem\\n\",\n \"\\n\",\n \"problem = LinearProblem(\\n\",\n \" a,\\n\",\n \" L,\\n\",\n \" bcs=[bc],\\n\",\n \" petsc_options={\\\"ksp_type\\\": \\\"preonly\\\", \\\"pc_type\\\": \\\"lu\\\"},\\n\",\n \" petsc_options_prefix=\\\"Poisson\\\",\\n\",\n \")\\n\",\n \"uh = problem.solve()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"c82e1ec7\",\n \"metadata\": {},\n \"source\": [\n \"Using {py:meth}`problem.solve()` we solve the linear system of equations and\\n\",\n \"return a {py:class}`Function` containing the solution.\\n\",\n \"\\n\",\n \"(error-norm)=\\n\",\n \"## Computing the error\\n\",\n \"Finally, we want to compute the error to check the accuracy of the solution.\\n\",\n \"We do this by comparing the finite element solution `u` with the exact solution.\\n\",\n \"First we interpolate the exact solution into a function space that contains it\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"21\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"V2 = fem.functionspace(domain, (\\\"Lagrange\\\", 2))\\n\",\n \"uex = fem.Function(V2, name=\\\"u_exact\\\")\\n\",\n \"uex.interpolate(lambda x: 1 + x[0] ** 2 + 2 * x[1] ** 2)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"22\",\n \"metadata\": {},\n \"source\": [\n \"We compute the error in two different ways.\\n\",\n \"First, we compute the $L^2$-norm of the error, defined by $E=\\\\sqrt{\\\\int_\\\\Omega (u_D-u_h)^2\\\\mathrm{d} x}$.\\n\",\n \"We use UFL to express the $L^2$-error, and use {py:func}`dolfinx.fem.assemble_scalar` to compute the scalar value.\\n\",\n \"In DOLFINx, {py:func}`assemble_scalar`\\n\",\n \"only assembles over the cells on the local process.\\n\",\n \"This means that if we use 2 processes to solve our problem,\\n\",\n \"we need to accumulate the local contributions to get the global error (on one or all processes).\\n\",\n \"We can do this with the {py:meth}`Comm.allreduce` function.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"25\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"L2_error = fem.form(ufl.inner(uh - uex, uh - uex) * ufl.dx)\\n\",\n \"error_local = fem.assemble_scalar(L2_error)\\n\",\n \"error_L2 = numpy.sqrt(domain.comm.allreduce(error_local, op=MPI.SUM))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"26\",\n \"metadata\": {},\n \"source\": [\n \"Secondly, we compute the maximum error at any degree of freedom.\\n\",\n \"As the finite element function $u$ can be expressed as a linear combination of basis functions $\\\\phi_j$,\\n\",\n \"spanning the space $V$: $ u = \\\\sum_{j=1}^N U_j\\\\phi_j.$\\n\",\n \"By writing {py:meth}`problem.solve()`\\n\",\n \"we compute all the coefficients $U_1,\\\\dots, U_N$.\\n\",\n \"These values are known as the _degrees of freedom_ (dofs).\\n\",\n \"We can access the degrees of freedom by accessing the underlying vector in `uh`.\\n\",\n \"However, as a second order function space has more dofs than a linear function space,\\n\",\n \"we cannot compare these arrays directly.\\n\",\n \"As we already have interpolated the exact solution into the first order space when creating the boundary condition,\\n\",\n \"we can compare the maximum values at any degree of freedom of the approximation space.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"27\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"error_max = numpy.max(numpy.abs(uD.x.array - uh.x.array))\\n\",\n \"if domain.comm.rank == 0: # Only print the error on one process\\n\",\n \" print(f\\\"Error_L2 : {error_L2:.2e}\\\")\\n\",\n \" print(f\\\"Error_max : {error_max:.2e}\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"28\",\n \"metadata\": {},\n \"source\": [\n \"## Plotting the mesh using pyvista\\n\",\n \"We will visualizing the mesh using [pyvista](https://docs.pyvista.org/), an interface to the VTK toolkit.\\n\",\n \"We start by converting the mesh to a format that can be used with {py:mod}`pyvista`.\\n\",\n \"To do this we use the function {py:func}`dolfinx.plot.vtk_mesh`.\\n\",\n \"It creates the data required to create a {py:class}`pyvista.UnstructuredGrid`.\\n\",\n \"You can print the current backend and change it with {py:func}`pyvista.set_jupyter_backend`.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"29\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"import pyvista\\n\",\n \"\\n\",\n \"print(pyvista.global_theme.jupyter_backend)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"30\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"from dolfinx import plot\\n\",\n \"\\n\",\n \"domain.topology.create_connectivity(tdim, tdim)\\n\",\n \"topology, cell_types, geometry = plot.vtk_mesh(domain, tdim)\\n\",\n \"grid = pyvista.UnstructuredGrid(topology, cell_types, geometry)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"31\",\n \"metadata\": {},\n \"source\": [\n \"There are several backends that can be used with pyvista, and they have different benefits and drawbacks.\\n\",\n \"See the [pyvista documentation](https://docs.pyvista.org/user-guide/jupyter/index.html#state-of-3d-interactive-jupyterlab-plotting)\\n\",\n \"for more information and installation details.\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"32\",\n \"metadata\": {},\n \"source\": [\n \"We can now use the {py:class}`pyvista.Plotter` to visualize the mesh. We visualize it by showing it in 2D and warped in 3D.\\n\",\n \"In the jupyter notebook environment, we use the default setting of `pyvista.OFF_SCREEN=False`,\\n\",\n \"which will render plots directly in the notebook.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"33\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"plotter = pyvista.Plotter()\\n\",\n \"plotter.add_mesh(grid, show_edges=True)\\n\",\n \"plotter.view_xy()\\n\",\n \"if not pyvista.OFF_SCREEN:\\n\",\n \" plotter.show()\\n\",\n \"else:\\n\",\n \" figure = plotter.screenshot(\\\"fundamentals_mesh.png\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"34\",\n \"metadata\": {},\n \"source\": [\n \"## Plotting a function using pyvista\\n\",\n \"We want to plot the solution `uh`.\\n\",\n \"As the function space used to defined the mesh is decoupled from the representation of the mesh,\\n\",\n \"we create a mesh based on the dof coordinates for the function space `V`.\\n\",\n \"We use {py:func}`dolfinx.plot.vtk_mesh` with the function space as input to create a mesh with\\n\",\n \"mesh geometry based on the dof coordinates.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"35\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"u_topology, u_cell_types, u_geometry = plot.vtk_mesh(V)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"36\",\n \"metadata\": {},\n \"source\": [\n \"Next, we create the {py:class}`pyvista.UnstructuredGrid` and add the dof-values to the mesh.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"37\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"u_grid = pyvista.UnstructuredGrid(u_topology, u_cell_types, u_geometry)\\n\",\n \"u_grid.point_data[\\\"u\\\"] = uh.x.array.real\\n\",\n \"u_grid.set_active_scalars(\\\"u\\\")\\n\",\n \"u_plotter = pyvista.Plotter()\\n\",\n \"u_plotter.add_mesh(u_grid, show_edges=True)\\n\",\n \"u_plotter.view_xy()\\n\",\n \"if not pyvista.OFF_SCREEN:\\n\",\n \" u_plotter.show()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"38\",\n \"metadata\": {},\n \"source\": [\n \"We can also warp the mesh by scalar to make use of the 3D plotting.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"39\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"warped = u_grid.warp_by_scalar()\\n\",\n \"plotter2 = pyvista.Plotter()\\n\",\n \"plotter2.add_mesh(warped, show_edges=True, show_scalar_bar=True)\\n\",\n \"if not pyvista.OFF_SCREEN:\\n\",\n \" plotter2.show()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"40\",\n \"metadata\": {},\n \"source\": [\n \"## External post-processing\\n\",\n \"For post-processing outside the python code, it is suggested to save the solution to file using either\\n\",\n \"{py:class}`dolfinx.io.VTXWriter` or {py:class}`dolfinx.io.XDMFFile` and using [Paraview](https://www.paraview.org/).\\n\",\n \"This is especially suggested for 3D visualization.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"41\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"from dolfinx import io\\n\",\n \"from pathlib import Path\\n\",\n \"\\n\",\n \"results_folder = Path(\\\"results\\\")\\n\",\n \"results_folder.mkdir(exist_ok=True, parents=True)\\n\",\n \"filename = results_folder / \\\"fundamentals\\\"\\n\",\n \"with io.VTXWriter(domain.comm, filename.with_suffix(\\\".bp\\\"), [uh]) as vtx:\\n\",\n \" vtx.write(0.0)\\n\",\n \"with io.XDMFFile(domain.comm, filename.with_suffix(\\\".xdmf\\\"), \\\"w\\\") as xdmf:\\n\",\n \" xdmf.write_mesh(domain)\\n\",\n \" xdmf.write_function(uh)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"42\",\n \"metadata\": {},\n \"source\": [\n \"```{bibliography}\\n\",\n \" :filter: cited\\n\",\n \" :labelprefix:\\n\",\n \" :keyprefix: fundamentals-\\n\",\n \"```\"\n ]\n }\n ],\n \"metadata\": {\n \"jupytext\": {\n \"formats\": \"ipynb,py:light\"\n },\n \"kernelspec\": {\n \"display_name\": \"Python 3 (ipykernel)\",\n \"language\": \"python\",\n \"name\": \"python3\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 3\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython3\",\n \"version\": \"3.10.12\"\n },\n \"vscode\": {\n \"interpreter\": {\n \"hash\": \"31f2aee4e71d21fbe5cf8b01ff0e069b9275f58929596ceb00d14d90e3e16cd6\"\n }\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 5\n}\n" }, { "path": "chapter1/fundamentals_code.py", "content": "# ---\n# jupyter:\n# jupytext:\n# formats: ipynb,py:light\n# text_representation:\n# extension: .py\n# format_name: light\n# format_version: '1.5'\n# jupytext_version: 1.19.1\n# kernelspec:\n# display_name: Python 3 (ipykernel)\n# language: python\n# name: python3\n# ---\n\n# # Implementation\n#\n# Author: Jørgen Schartum Dokken\n#\n# This implementation is an adaptation of the work in {cite}`fundamentals-FenicsTutorial` to DOLFINx.\n#\n# In this section, you will learn:\n# - How to use the built-in meshes in DOLFINx\n# - How to create a spatially varying Dirichlet boundary conditions on the whole domain boundary\n# - How to define a weak formulation of your PDE\n# - How to solve the resulting system of linear equations\n# - How to visualize the solution using a variety of tools\n# - How to compute the $L^2(\\Omega)$ error and the error at mesh vertices\n#\n# ## Interactive tutorials\n# ```{admonition} Run the tutorial as Jupyter notebook in browser\n# As this book has been published as a Jupyter Book, each code can be run in your browser as a Jupyter notebook.\n# To start such a notebook click the rocket symbol in the top right corner of the relevant tutorial.\n# ```\n#\n# The Poisson problem has so far featured a general domain $\\Omega$ and general functions $u_D$ for\n# the boundary conditions and $f$ for the right hand side.\n# Therefore, we need to make specific choices of $\\Omega, u_D$ and $f$.\n# A wise choice is to construct a problem with a known analytical solution,\n# so that we can check that the computed solution is correct.\n# The primary candidates are lower-order polynomials.\n# The continuous Galerkin finite element spaces of degree $r$ will exactly reproduce polynomials of degree $r$.\n# \n# We use this fact to construct a quadratic function in $2D$. In particular we choose\n#\n# $$\n# \\begin{align}\n# u_e(x,y)=1+x^2+2y^2\n# \\end{align}\n# $$\n#\n# Inserting $u_e$ in the original boundary problem, we find that\n#\n# $$\n# \\begin{align}\n# f(x,y)= -6,\\qquad u_D(x,y)=u_e(x,y)=1+x^2+2y^2,\n# \\end{align}\n# $$\n#\n# regardless of the shape of the domain as long as we prescribe\n# $u_e$ on the boundary.\n#\n# For simplicity, we choose the domain to be a unit square $\\Omega=[0,1]\\times [0,1]$\n#\n# This simple but very powerful method for constructing test problems is called _the method of manufactured solutions_.\n# First pick a simple expression for the exact solution, plug into\n# the equation to obtain the right-hand side (source term $f$).\n# Then solve the equation with this right hand side, and using the exact solution as boundary condition.\n# Finally, we create a program that tries to reproduce the exact solution.\n#\n# Note that in many cases, it can be hard to determine if the program works if it produces an error of size\n# $10^{-5}$ on a $20 \\times 20$ grid.\n# However, since we are using Sobolev spaces, we usually know about the numerical errors _asymptotic properties_.\n# For instance that it is proportional to $h^2$ if $h$ is the size of a cell in the mesh.\n# We can then compare the error on meshes with different $h$-values to see if the asymptotic behavior is correct.\n# This technique will be explained in detail in the chapter [Improving your fenics code](./../chapter4/convergence).\n#\n# However, in cases where we have a solution we know that should have no approximation error,\n# we know that the solution should be produced to machine precision by the program.\n\n# A major difference between a traditional FEniCS code and a FEniCSx code,\n# is that one is not advised to use the wildcard import.\n# We will see this throughout this first example.\n#\n# ## Generating simple meshes\n# The next step is to define the discrete domain, _the mesh_.\n# We do this by importing one of the built-in mesh generators.\n# We will build a {py:func}`unit square mesh`, i.e. a mesh spanning $[0,1]\\times[0,1]$.\n# It can consist of either triangles or quadrilaterals.\n\n# +\nfrom mpi4py import MPI\nfrom dolfinx import mesh\nimport numpy\n\ndomain = mesh.create_unit_square(MPI.COMM_WORLD, 8, 8, mesh.CellType.quadrilateral)\n# -\n\n# Note that in addition to give how many elements we would like to have in each direction,\n# we also have to supply the _MPI-communicator_.\n# This is to specify how we would like the program to behave in parallel.\n# If we supply {py:data}`MPI.COMM_WORLD` we create a single mesh,\n# whose data is distributed over the number of processors we would like to use.\n# We can for instance run the program in parallel on two processors by using `mpirun`, as:\n# ``` bash\n# mpirun -n 2 python3 t1.py\n# ```\n# However, if we would like to create a separate mesh on each processor,\n# we can use {py:data}`MPI.COMM_SELF`.\n# This is for instance useful if we run a small problem, and would like to run it with multiple parameters.\n#\n# ## Defining the finite element function space\n# Once the mesh has been created, we can create the finite element function space $V$.\n# The finite element function space does not need to be the same as the one used to describe the mesh.\n# DOLFINx supports a wide range of arbitrary order finite element function spaces, see:\n# [Supported elements in DOLFINx](https://defelement.org/lists/implementations/basix.ufl.html)\n# for an extensive list.\n# To create a function space, we need to specify what mesh the space is defined on,\n# what element famil the space is based on, and the degree of the element.\n# These can for instance be defned through a tuple `(\"family\", degree)`, as shown below\n\n# +\nfrom dolfinx import fem\n\nV = fem.functionspace(domain, (\"Lagrange\", 1))\n# -\n\n# Further details about specification/customization of this tuple, see {py:class}`dolfinx.fem.ElementMetaData`.\n\n# ## Dirichlet boundary conditions\n# Next, we create a function that will hold the Dirichlet boundary data, and use interpolation to\n# fill it with the appropriate data.\n\nuD = fem.Function(V)\nuD.interpolate(lambda x: 1 + x[0] ** 2 + 2 * x[1] ** 2)\n\n# We now have the boundary data (and in this case the solution of the finite element problem)\n# represented in the discrete function space.\n# Next we would like to apply the boundary values to all degrees of freedom that are on the\n# boundary of the discrete domain.\n# We start by identifying the facets (line-segments) representing the outer boundary,\n# using {py:func}`dolfinx.mesh.exterior_facet_indices`.\n# We start by creating the facet to cell connectivity required to determine boundary facets by\n# calling {py:meth}`dolfinx.mesh.Topology.create_connectivity`.\n\ntdim = domain.topology.dim\nfdim = tdim - 1\ndomain.topology.create_connectivity(fdim, tdim)\nboundary_facets = mesh.exterior_facet_indices(domain.topology)\n\n# For the current problem, as we are using the first order Lagrange function space,\n# the degrees of freedom are located at the vertices of each cell, thus each facet contains two degrees of freedom.\n#\n# To find the local indices of these degrees of freedom, we use {py:func}`dolfinx.fem.locate_dofs_topological`\n# which takes in the function space, the dimension of entities in the mesh we would like to identify and the local entities.\n# ```{admonition} Local ordering of degrees of freedom and mesh vertices\n# Many people expect there to be a 1-1 correspondence between the mesh coordinates and the coordinates of the degrees of freedom.\n# However, this is only true in the case of `Lagrange` 1 elements on a first order mesh.\n# Therefore, in DOLFINx we use separate local numbering for the mesh coordinates and the dof coordinates.\n# To obtain the local dof coordinates we can use\n# {py:meth}`V.tabulate_dof_coordinates()`,\n# while the ordering of the local vertices can be obtained by {py:attr}`mesh.geometry.x`.\n# ```\n# With this data at hand, we can create the Dirichlet boundary condition\n\n\nboundary_dofs = fem.locate_dofs_topological(V, fdim, boundary_facets)\nbc = fem.dirichletbc(uD, boundary_dofs)\n\n# ## Defining the trial and test function\n#\n# In mathematics, we distinguish between trial and test spaces $V$ and $\\hat{V}$.\n# The only difference in the present problem is the boundary conditions.\n# In FEniCSx, we do not specify boundary conditions as part of the function space,\n# so it is sufficient to use a common space for the trial and test function.\n#\n# We use the {py:mod}`Unified Form Language` (UFL) to specify the variational formulations.\n# See {cite}`fundamentals-ufl2014` for more details.\n\n\n# +\nimport ufl\n\nu = ufl.TrialFunction(V)\nv = ufl.TestFunction(V)\n# -\n\n# ## Defining the source term\n# As the source term is constant over the domain, we use {py:class}`dolfinx.fem.Constant`\n\n# +\nfrom dolfinx import default_scalar_type\n\nf = fem.Constant(domain, default_scalar_type(-6))\n# -\n\n# ```{admonition} Compilation speed-up\n# Instead of wrapping $-6$ in a {py:class}`dolfinx.fem.Constant`, we could simply define $f$ as `f=-6`.\n# However, if we would like to change this parameter later in the simulation,\n# we would have to redefine our variational formulation.\n# The {py:attr}`dolfinx.fem.Constant.value` allows us to update the value in $f$ by using `f.value=5`.\n# Additionally, by indicating that $f$ is a constant, we speed up compilation of the variational\n# formulations required for the created linear system.\n# ```\n#\n# ## Defining the variational problem\n# As we now have defined all variables used to describe our variational problem, we can create the weak formulation\n\na = ufl.dot(ufl.grad(u), ufl.grad(v)) * ufl.dx\nL = f * v * ufl.dx\n\n# Note that there is a very close correspondence between the Python syntax and the mathematical syntax\n# $\\int_{\\Omega} \\nabla u \\cdot \\nabla v ~\\mathrm{d} x$ and $\\int_{\\Omega}fv~\\mathrm{d} x$.\n# The integration over the domain $\\Omega$ is defined by using {py:func}`ufl.dx`, an integration\n# {py:class}`measure` over all cells of the mesh.\n#\n# This is the key strength of FEniCSx:\n# the formulas in the variational formulation translate directly to very similar Python code,\n# a feature that makes it easy to specify and solve complicated PDE problems.\n#\n# ## Expressing inner products\n# The inner product $\\int_\\Omega \\nabla u \\cdot \\nabla v ~\\mathrm{d} x$ can be expressed in various ways in UFL.\n# We have used the notation `ufl.dot(ufl.grad(u), ufl.grad(v))*ufl.dx`.\n# The {py:func}`dot` product in UFL computes the sum (contraction) over the last index\n# of the first factor and first index of the second factor.\n# In this case, both factors are tensors of rank one (vectors) and so the sum is just over\n# the single index of both $\\nabla u$ and $\\nabla v$.\n# To compute an inner product of matrices (with two indices),\n# one must use the function {py:func}`ufl.inner` instead of {py:func}`ufl.dot`.\n# For real-valued vectors, {py:func}`ufl.dot` and {py:func}`ufl.inner` are equivalent.\n#\n# ```{admonition} Complex numbers\n# In DOLFINx, one can solve complex number problems by using an installation of PETSc using complex numbers.\n# For variational formulations with complex numbers, one cannot use {py:func}`ufl.dot` to compute inner products.\n# One has to use {py:func}`ufl.inner`, with the test-function as the second input argument for {py:func}`ufl.inner`.\n# See [Running DOLFINx in complex mode](./complex_mode) for more information.\n# ```\n#\n#\n# ## Forming and solving the linear system\n#\n# Having defined the finite element variational problem and boundary condition,\n# we can create our {py:class}`LinearProblem` to solve the variational problem:\n# Find $u_h\\in V$ such that $a(u_h, v)==L(v) \\quad \\forall v \\in \\hat{V}$.\n# We will use {py:mod}`PETSc` as our linear algebra backend, using a direct solver (LU-factorization).\n# See the [PETSc-documentation](https://petsc.org/main/docs/manual/ksp/?highlight=ksp#ksp-linear-system-solvers) of the method for more information.\n# PETSc is not a required dependency of DOLFINx, and therefore we explicitly import the DOLFINx wrapper for interfacing with PETSc.\n# To ensure that the options passed to the {py:class}`LinearProblem`\n# is only used for the given KSP solver, we pass a **unique** option prefix as well.\n\n# +\nfrom dolfinx.fem.petsc import LinearProblem\n\nproblem = LinearProblem(\n a,\n L,\n bcs=[bc],\n petsc_options={\"ksp_type\": \"preonly\", \"pc_type\": \"lu\"},\n petsc_options_prefix=\"Poisson\",\n)\nuh = problem.solve()\n# -\n\n# Using {py:meth}`problem.solve()` we solve the linear system of equations and\n# return a {py:class}`Function` containing the solution.\n#\n# (error-norm)=\n# ## Computing the error\n# Finally, we want to compute the error to check the accuracy of the solution.\n# We do this by comparing the finite element solution `u` with the exact solution.\n# First we interpolate the exact solution into a function space that contains it\n\nV2 = fem.functionspace(domain, (\"Lagrange\", 2))\nuex = fem.Function(V2, name=\"u_exact\")\nuex.interpolate(lambda x: 1 + x[0] ** 2 + 2 * x[1] ** 2)\n\n# We compute the error in two different ways.\n# First, we compute the $L^2$-norm of the error, defined by $E=\\sqrt{\\int_\\Omega (u_D-u_h)^2\\mathrm{d} x}$.\n# We use UFL to express the $L^2$-error, and use {py:func}`dolfinx.fem.assemble_scalar` to compute the scalar value.\n# In DOLFINx, {py:func}`assemble_scalar`\n# only assembles over the cells on the local process.\n# This means that if we use 2 processes to solve our problem,\n# we need to accumulate the local contributions to get the global error (on one or all processes).\n# We can do this with the {py:meth}`Comm.allreduce` function.\n\nL2_error = fem.form(ufl.inner(uh - uex, uh - uex) * ufl.dx)\nerror_local = fem.assemble_scalar(L2_error)\nerror_L2 = numpy.sqrt(domain.comm.allreduce(error_local, op=MPI.SUM))\n\n# Secondly, we compute the maximum error at any degree of freedom.\n# As the finite element function $u$ can be expressed as a linear combination of basis functions $\\phi_j$,\n# spanning the space $V$: $ u = \\sum_{j=1}^N U_j\\phi_j.$\n# By writing {py:meth}`problem.solve()`\n# we compute all the coefficients $U_1,\\dots, U_N$.\n# These values are known as the _degrees of freedom_ (dofs).\n# We can access the degrees of freedom by accessing the underlying vector in `uh`.\n# However, as a second order function space has more dofs than a linear function space,\n# we cannot compare these arrays directly.\n# As we already have interpolated the exact solution into the first order space when creating the boundary condition,\n# we can compare the maximum values at any degree of freedom of the approximation space.\n\nerror_max = numpy.max(numpy.abs(uD.x.array - uh.x.array))\nif domain.comm.rank == 0: # Only print the error on one process\n print(f\"Error_L2 : {error_L2:.2e}\")\n print(f\"Error_max : {error_max:.2e}\")\n\n# ## Plotting the mesh using pyvista\n# We will visualizing the mesh using [pyvista](https://docs.pyvista.org/), an interface to the VTK toolkit.\n# We start by converting the mesh to a format that can be used with {py:mod}`pyvista`.\n# To do this we use the function {py:func}`dolfinx.plot.vtk_mesh`.\n# It creates the data required to create a {py:class}`pyvista.UnstructuredGrid`.\n# You can print the current backend and change it with {py:func}`pyvista.set_jupyter_backend`.\n\n# +\nimport pyvista\n\nprint(pyvista.global_theme.jupyter_backend)\n\n# +\nfrom dolfinx import plot\n\ndomain.topology.create_connectivity(tdim, tdim)\ntopology, cell_types, geometry = plot.vtk_mesh(domain, tdim)\ngrid = pyvista.UnstructuredGrid(topology, cell_types, geometry)\n# -\n\n# There are several backends that can be used with pyvista, and they have different benefits and drawbacks.\n# See the [pyvista documentation](https://docs.pyvista.org/user-guide/jupyter/index.html#state-of-3d-interactive-jupyterlab-plotting)\n# for more information and installation details.\n\n# We can now use the {py:class}`pyvista.Plotter` to visualize the mesh. We visualize it by showing it in 2D and warped in 3D.\n# In the jupyter notebook environment, we use the default setting of `pyvista.OFF_SCREEN=False`,\n# which will render plots directly in the notebook.\n\nplotter = pyvista.Plotter()\nplotter.add_mesh(grid, show_edges=True)\nplotter.view_xy()\nif not pyvista.OFF_SCREEN:\n plotter.show()\nelse:\n figure = plotter.screenshot(\"fundamentals_mesh.png\")\n\n# ## Plotting a function using pyvista\n# We want to plot the solution `uh`.\n# As the function space used to defined the mesh is decoupled from the representation of the mesh,\n# we create a mesh based on the dof coordinates for the function space `V`.\n# We use {py:func}`dolfinx.plot.vtk_mesh` with the function space as input to create a mesh with\n# mesh geometry based on the dof coordinates.\n\nu_topology, u_cell_types, u_geometry = plot.vtk_mesh(V)\n\n# Next, we create the {py:class}`pyvista.UnstructuredGrid` and add the dof-values to the mesh.\n\nu_grid = pyvista.UnstructuredGrid(u_topology, u_cell_types, u_geometry)\nu_grid.point_data[\"u\"] = uh.x.array.real\nu_grid.set_active_scalars(\"u\")\nu_plotter = pyvista.Plotter()\nu_plotter.add_mesh(u_grid, show_edges=True)\nu_plotter.view_xy()\nif not pyvista.OFF_SCREEN:\n u_plotter.show()\n\n# We can also warp the mesh by scalar to make use of the 3D plotting.\n\nwarped = u_grid.warp_by_scalar()\nplotter2 = pyvista.Plotter()\nplotter2.add_mesh(warped, show_edges=True, show_scalar_bar=True)\nif not pyvista.OFF_SCREEN:\n plotter2.show()\n\n# ## External post-processing\n# For post-processing outside the python code, it is suggested to save the solution to file using either\n# {py:class}`dolfinx.io.VTXWriter` or {py:class}`dolfinx.io.XDMFFile` and using [Paraview](https://www.paraview.org/).\n# This is especially suggested for 3D visualization.\n\n# +\nfrom dolfinx import io\nfrom pathlib import Path\n\nresults_folder = Path(\"results\")\nresults_folder.mkdir(exist_ok=True, parents=True)\nfilename = results_folder / \"fundamentals\"\nwith io.VTXWriter(domain.comm, filename.with_suffix(\".bp\"), [uh]) as vtx:\n vtx.write(0.0)\nwith io.XDMFFile(domain.comm, filename.with_suffix(\".xdmf\"), \"w\") as xdmf:\n xdmf.write_mesh(domain)\n xdmf.write_function(uh)\n# -\n\n# ```{bibliography}\n# :filter: cited\n# :labelprefix:\n# :keyprefix: fundamentals-\n# ```\n" }, { "path": "chapter1/membrane.md", "content": "# Deflection of a membrane\nAuthors: Hans Petter Langtangen and Anders Logg.\n\nModified for DOLFINx by Jørgen S. Dokken\n\nIn the first FEniCSx program, we solved a simple problem which we could easily use to verify the implementation.\nIn this section, we will turn our attentition to a physically more relevant problem with solutions of a somewhat more exciting shape.\n\nWe would like to compute the deflection $D(x,y)$ of a two-dimensional, circular membrane of radius $R$, subject to a load $p$ over the membrane. The appropriate PDE model is \n\\begin{align}\n -T \\nabla^2D&=p \\quad\\text{in }\\quad \\Omega=\\{(x,y)\\vert x^2+y^2\\leq R^2 \\}.\n\\end{align}\nHere, $T$ is the tension in the membrane (constant), and $p$ is the external pressure load. The boundary of the membrane has no deflection. This implies that $D=0$ is the boundary condition. We model a localized load as a Gaussian function:\n\\begin{align}\n p(x,y)&=\\frac{A}{2\\pi\\sigma}e^{-\\frac{1}{2}\\left(\\frac{x-x_0}{\\sigma}\\right)^2-\\frac{1}{2}\\left(\\frac{y-y_0}{\\sigma}\\right)^2}.\n\\end{align}\nThe parameter $A$ is the amplitude of the pressure, $(x_0, y_0)$ the location of the maximum point of the load, and $\\sigma$ the \"width\" of $p$. We will take the center $(x_0,y_0)$ to be $(0,R_0)$ for some $0`\\n\",\n \"- Use {py:class}`ufl.SpatialCoordinate` to create a spatially varying function\\n\",\n \"- Interpolate a {py:class}`ufl-Expression` into an appropriate function space\\n\",\n \"- Evaluate a {py:class}`dolfinx.fem.Function` at any point $x$\\n\",\n \"- Use Paraview to visualize the solution of a PDE\\n\",\n \"\\n\",\n \"## Creating the mesh\\n\",\n \"\\n\",\n \"To create the computational geometry, we use the Python-API of [GMSH](https://gmsh.info/).\\n\",\n \"We start by importing the gmsh-module and initializing it.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"1\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"import gmsh\\n\",\n \"\\n\",\n \"gmsh.initialize()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"2\",\n \"metadata\": {},\n \"source\": [\n \"The next step is to create the membrane and start the computations by the GMSH CAD kernel,\\n\",\n \"to generate the relevant underlying data structures.\\n\",\n \"The first arguments of `addDisk` are the x, y and z coordinate of the center of the circle,\\n\",\n \"while the two last arguments are the x-radius and y-radius.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"3\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"membrane = gmsh.model.occ.addDisk(0, 0, 0, 1, 1)\\n\",\n \"gmsh.model.occ.synchronize()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"4\",\n \"metadata\": {},\n \"source\": [\n \"After that, we make the membrane a physical surface, such that it is recognized by `gmsh` when generating the mesh.\\n\",\n \"As a surface is a two-dimensional entity, we add `2` as the first argument,\\n\",\n \"the entity tag of the membrane as the second argument, and the physical tag as the last argument.\\n\",\n \"In a later demo, we will get into when this tag matters.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"5\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"gdim = 2\\n\",\n \"gmsh.model.addPhysicalGroup(gdim, [membrane], 1)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"6\",\n \"metadata\": {},\n \"source\": [\n \"Finally, we generate the two-dimensional mesh.\\n\",\n \"We set a uniform mesh size by modifying the GMSH options.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"7\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"gmsh.option.setNumber(\\\"Mesh.CharacteristicLengthMin\\\", 0.05)\\n\",\n \"gmsh.option.setNumber(\\\"Mesh.CharacteristicLengthMax\\\", 0.05)\\n\",\n \"gmsh.model.mesh.generate(gdim)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"8\",\n \"metadata\": {},\n \"source\": [\n \"# Interfacing with GMSH in DOLFINx\\n\",\n \"We will import the GMSH-mesh directly from GMSH into DOLFINx via the {py:mod}`dolfinx.io.gmsh` interface.\\n\",\n \"The {py:mod}`dolfinx.io.gmsh` module contains two functions\\n\",\n \"1. {py:func}`model_to_mesh` which takes in a `gmsh.model`\\n\",\n \" and returns a {py:class}`dolfinx.io.gmsh.MeshData` object.\\n\",\n \"2. {py:func}`read_from_msh` which takes in a path to a `.msh`-file\\n\",\n \" and returns a {py:class}`dolfinx.io.gmsh.MeshData` object.\\n\",\n \"\\n\",\n \"The {py:class}`MeshData` object will contain a {py:class}`dolfinx.mesh.Mesh`,\\n\",\n \"under the attribute {py:attr}`mesh`.\\n\",\n \"This mesh will contain all GMSH Physical Groups of the highest topological dimension.\\n\",\n \"```{note}\\n\",\n \"If you do not use `gmsh.model.addPhysicalGroup` when creating the mesh with GMSH, it can not be read into DOLFINx.\\n\",\n \"```\\n\",\n \"The {py:class}`MeshData` object can also contain tags for\\n\",\n \"all other `PhysicalGroups` that has been added to the mesh, that being\\n\",\n \"{py:attr}`cell_tags`, {py:attr}`facet_tags`,\\n\",\n \"{py:attr}`ridge_tags` and\\n\",\n \"{py:attr}`peak_tags`.\\n\",\n \"To read either `gmsh.model` or a `.msh`-file, one has to distribute the mesh to all processes used by DOLFINx.\\n\",\n \"As GMSH does not support mesh creation with MPI, we currently have a `gmsh.model.mesh` on each process.\\n\",\n \"To distribute the mesh, we have to specify which process the mesh was created on,\\n\",\n \"and which communicator rank should distribute the mesh.\\n\",\n \"The {py:func}`model_to_mesh` will then load the mesh on the specified rank,\\n\",\n \"and distribute it to the communicator using a mesh partitioner.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"9\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"from dolfinx.io import gmsh as gmshio\\n\",\n \"from dolfinx.fem.petsc import LinearProblem\\n\",\n \"from mpi4py import MPI\\n\",\n \"\\n\",\n \"gmsh_model_rank = 0\\n\",\n \"mesh_comm = MPI.COMM_WORLD\\n\",\n \"mesh_data = gmshio.model_to_mesh(gmsh.model, mesh_comm, gmsh_model_rank, gdim=gdim)\\n\",\n \"assert mesh_data.cell_tags is not None\\n\",\n \"cell_markers = mesh_data.cell_tags\\n\",\n \"domain = mesh_data.mesh\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"10\",\n \"metadata\": {},\n \"source\": [\n \"We define the function space as in the previous tutorial\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"11\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"from dolfinx import fem\\n\",\n \"\\n\",\n \"V = fem.functionspace(domain, (\\\"Lagrange\\\", 1))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"12\",\n \"metadata\": {},\n \"source\": [\n \"## Defining a spatially varying load\\n\",\n \"The right hand side pressure function is represented using {py:class}`ufl.SpatialCoordinate` and two constants,\\n\",\n \"one for $\\\\beta$ and one for $R_0$.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"13\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"import ufl\\n\",\n \"from dolfinx import default_scalar_type\\n\",\n \"\\n\",\n \"x = ufl.SpatialCoordinate(domain)\\n\",\n \"beta = fem.Constant(domain, default_scalar_type(12))\\n\",\n \"R0 = fem.Constant(domain, default_scalar_type(0.3))\\n\",\n \"p = 4 * ufl.exp(-(beta**2) * (x[0] ** 2 + (x[1] - R0) ** 2))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"9f059bd4\",\n \"metadata\": {},\n \"source\": [\n \"## Create a Dirichlet boundary condition using geometrical conditions\\n\",\n \"The next step is to create the homogeneous boundary condition.\\n\",\n \"As opposed to the [first tutorial](./fundamentals_code.ipynb) we will use\\n\",\n \"{py:func}`locate_dofs_geometrical` to locate the degrees of freedom on the boundary.\\n\",\n \"As we know that our domain is a circle with radius 1, we know that any degree of freedom should be\\n\",\n \"located at a coordinate $(x,y)$ such that $\\\\sqrt{x^2+y^2}=1$.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"16\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"import numpy as np\\n\",\n \"\\n\",\n \"\\n\",\n \"def on_boundary(x):\\n\",\n \" return np.isclose(np.sqrt(x[0] ** 2 + x[1] ** 2), 1)\\n\",\n \"\\n\",\n \"\\n\",\n \"boundary_dofs = fem.locate_dofs_geometrical(V, on_boundary)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"17\",\n \"metadata\": {},\n \"source\": [\n \"As our Dirichlet condition is homogeneous (`u=0` on the whole boundary), we can initialize the\\n\",\n \"{py:class}`dolfinx.fem.DirichletBC` with a constant value, the degrees of freedom and the function\\n\",\n \"space to apply the boundary condition on. We use the constructor {py:func}`dolfinx.fem.dirichletbc`.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"18\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"bc = fem.dirichletbc(default_scalar_type(0), boundary_dofs, V)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"19\",\n \"metadata\": {},\n \"source\": [\n \"## Defining the variational problem\\n\",\n \"The variational problem is the same as in our first Poisson problem, where `f` is replaced by `p`.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"20\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"u = ufl.TrialFunction(V)\\n\",\n \"v = ufl.TestFunction(V)\\n\",\n \"a = ufl.dot(ufl.grad(u), ufl.grad(v)) * ufl.dx\\n\",\n \"L = p * v * ufl.dx\\n\",\n \"problem = LinearProblem(\\n\",\n \" a,\\n\",\n \" L,\\n\",\n \" bcs=[bc],\\n\",\n \" petsc_options={\\\"ksp_type\\\": \\\"preonly\\\", \\\"pc_type\\\": \\\"lu\\\"},\\n\",\n \" petsc_options_prefix=\\\"membrane_\\\",\\n\",\n \")\\n\",\n \"uh = problem.solve()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"21\",\n \"metadata\": {},\n \"source\": [\n \"## Interpolation of a UFL-expression\\n\",\n \"As we previously defined the load `p` as a spatially varying function,\\n\",\n \"we would like to interpolate this function into an appropriate function space for visualization.\\n\",\n \"To do this we use the class {py:class}`Expression`.\\n\",\n \"The expression takes in any UFL-expression, and a set of points on the reference element.\\n\",\n \"We will use the {py:attr}`interpolation points`\\n\",\n \"of the space we want to interpolate in to.\\n\",\n \"We choose a high order function space to represent the function `p`, as it is rapidly varying in space.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"22\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"Q = fem.functionspace(domain, (\\\"Lagrange\\\", 5))\\n\",\n \"expr = fem.Expression(p, Q.element.interpolation_points)\\n\",\n \"pressure = fem.Function(Q)\\n\",\n \"pressure.interpolate(expr)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"23\",\n \"metadata\": {},\n \"source\": [\n \"## Plotting the solution over a line\\n\",\n \"We first plot the deflection $u_h$ over the domain $\\\\Omega$.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"24\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"from dolfinx.plot import vtk_mesh\\n\",\n \"import pyvista\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"55b28bcd\",\n \"metadata\": {},\n \"source\": [\n \"Extract topology from mesh and create {py:class}`pyvista.UnstructuredGrid`\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"26\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"topology, cell_types, x = vtk_mesh(V)\\n\",\n \"grid = pyvista.UnstructuredGrid(topology, cell_types, x)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"b2b183e2\",\n \"metadata\": {},\n \"source\": [\n \"Set deflection values and add it to plotter\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"28\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"grid.point_data[\\\"u\\\"] = uh.x.array\\n\",\n \"warped = grid.warp_by_scalar(\\\"u\\\", factor=25)\\n\",\n \"\\n\",\n \"plotter = pyvista.Plotter()\\n\",\n \"plotter.add_mesh(warped, show_edges=True, show_scalar_bar=True, scalars=\\\"u\\\")\\n\",\n \"if not pyvista.OFF_SCREEN:\\n\",\n \" plotter.show()\\n\",\n \"else:\\n\",\n \" plotter.screenshot(\\\"deflection.png\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"29\",\n \"metadata\": {},\n \"source\": [\n \"We next plot the load on the domain\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"30\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"load_plotter = pyvista.Plotter()\\n\",\n \"p_grid = pyvista.UnstructuredGrid(*vtk_mesh(Q))\\n\",\n \"p_grid.point_data[\\\"p\\\"] = pressure.x.array.real\\n\",\n \"warped_p = p_grid.warp_by_scalar(\\\"p\\\", factor=0.5)\\n\",\n \"warped_p.set_active_scalars(\\\"p\\\")\\n\",\n \"load_plotter.add_mesh(warped_p, show_scalar_bar=True)\\n\",\n \"load_plotter.view_xy()\\n\",\n \"if not pyvista.OFF_SCREEN:\\n\",\n \" load_plotter.show()\\n\",\n \"else:\\n\",\n \" load_plotter.screenshot(\\\"load.png\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"31\",\n \"metadata\": {},\n \"source\": [\n \"## Making curve plots throughout the domain\\n\",\n \"Another way to compare the deflection and the load is to make a plot along the line $x=0$.\\n\",\n \"This is just a matter of defining a set of points along the $y$-axis and evaluating the\\n\",\n \"finite element functions $u$ and $p$ at these points.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"32\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"tol = 0.001 # Avoid hitting the outside of the domain\\n\",\n \"y = np.linspace(-1 + tol, 1 - tol, 101)\\n\",\n \"points = np.zeros((3, 101))\\n\",\n \"points[1] = y\\n\",\n \"u_values = []\\n\",\n \"p_values = []\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"33\",\n \"metadata\": {},\n \"source\": [\n \"As a finite element function is the linear combination of all degrees of freedom,\\n\",\n \"$u_h(x)=\\\\sum_{i=1}^N c_i \\\\phi_i(x)$ where $c_i$ are the coefficients of $u_h$ and $\\\\phi_i$\\n\",\n \"is the $i$-th basis function, we can compute the exact solution at any point in $\\\\Omega$.\\n\",\n \"However, as a mesh consists of a large set of degrees of freedom (i.e. $N$ is large),\\n\",\n \"we want to reduce the number of evaluations of the basis function $\\\\phi_i(x)$.\\n\",\n \"We do this by identifying which cell of the mesh $x$ is in.\\n\",\n \"This is efficiently done by creating a {py:class}`bounding box tree`.\\n\",\n \"However, as the bounding box of a cell spans more of $\\\\mathbb{R}^n$ than the actual cell,\\n\",\n \"we check that the actual cell collides with the input point using\\n\",\n \"{py:func}`dolfinx.geometry.compute_colliding_cells`,\\n\",\n \"which measures the exact distance between the point and the cell\\n\",\n \"(approximated as a convex hull for higher order geometries).\\n\",\n \"This function also returns an adjacency-list, as the point might align with a facet,\\n\",\n \"edge or vertex that is shared between multiple cells in the mesh.\\n\",\n \"\\n\",\n \"Finally, we would like the code below to run in parallel,\\n\",\n \"when the mesh is distributed over multiple processors.\\n\",\n \"In that case, it is not guaranteed that every point in `points` is on each processor.\\n\",\n \"Therefore we create a subset `points_on_proc` only containing the points found on the current processor.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"36\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"cells = []\\n\",\n \"points_on_proc = []\\n\",\n \"# Find cells whose bounding-box collide with the the points\\n\",\n \"cell_candidates = geometry.compute_collisions_points(bb_tree, points.T)\\n\",\n \"# Choose one of the cells that contains the point\\n\",\n \"colliding_cells = geometry.compute_colliding_cells(domain, cell_candidates, points.T)\\n\",\n \"for i, point in enumerate(points.T):\\n\",\n \" if len(colliding_cells.links(i)) > 0:\\n\",\n \" points_on_proc.append(point)\\n\",\n \" cells.append(colliding_cells.links(i)[0])\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"37\",\n \"metadata\": {},\n \"source\": [\n \"We now have a list of points on the processor, on in which cell each point belongs.\\n\",\n \"We can then call {py:meth}`uh.eval` and\\n\",\n \"{py:meth}`pressure.eval` to obtain the set of values for all the points.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"38\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"points_on_proc = np.array(points_on_proc, dtype=np.float64)\\n\",\n \"u_values = uh.eval(points_on_proc, cells)\\n\",\n \"p_values = pressure.eval(points_on_proc, cells)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"39\",\n \"metadata\": {},\n \"source\": [\n \"As we now have an array of coordinates and two arrays of function values,\\n\",\n \"we can use {py:mod}`matplotlib` to plot them\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"40\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"import matplotlib.pyplot as plt\\n\",\n \"\\n\",\n \"fig = plt.figure()\\n\",\n \"plt.plot(\\n\",\n \" points_on_proc[:, 1],\\n\",\n \" 50 * u_values,\\n\",\n \" \\\"k\\\",\\n\",\n \" linewidth=2,\\n\",\n \" label=\\\"Deflection ($\\\\\\\\times 50$)\\\",\\n\",\n \")\\n\",\n \"plt.plot(points_on_proc[:, 1], p_values, \\\"b--\\\", linewidth=2, label=\\\"Load\\\")\\n\",\n \"plt.grid(True)\\n\",\n \"plt.xlabel(\\\"y\\\")\\n\",\n \"plt.legend()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"41\",\n \"metadata\": {},\n \"source\": [\n \"If executed in parallel as a Python file, we save a plot per processor\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"42\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"plt.savefig(f\\\"membrane_rank{MPI.COMM_WORLD.rank:d}.png\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"43\",\n \"metadata\": {},\n \"source\": [\n \"## Saving functions to file\\n\",\n \"As mentioned in the previous section, we can also use Paraview to visualize the solution.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"44\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"import dolfinx.io\\n\",\n \"from pathlib import Path\\n\",\n \"\\n\",\n \"pressure.name = \\\"Load\\\"\\n\",\n \"uh.name = \\\"Deflection\\\"\\n\",\n \"results_folder = Path(\\\"results\\\")\\n\",\n \"results_folder.mkdir(exist_ok=True, parents=True)\\n\",\n \"with dolfinx.io.VTXWriter(\\n\",\n \" MPI.COMM_WORLD, results_folder / \\\"membrane_pressure.bp\\\", [pressure], engine=\\\"BP4\\\"\\n\",\n \") as vtx:\\n\",\n \" vtx.write(0.0)\\n\",\n \"with dolfinx.io.VTXWriter(\\n\",\n \" MPI.COMM_WORLD, results_folder / \\\"membrane_deflection.bp\\\", [uh], engine=\\\"BP4\\\"\\n\",\n \") as vtx:\\n\",\n \" vtx.write(0.0)\"\n ]\n }\n ],\n \"metadata\": {\n \"jupytext\": {\n \"formats\": \"ipynb,py:light\"\n },\n \"kernelspec\": {\n \"display_name\": \"Python 3 (ipykernel)\",\n \"language\": \"python\",\n \"name\": \"python3\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 5\n}\n" }, { "path": "chapter1/membrane_code.py", "content": "# ---\n# jupyter:\n# jupytext:\n# formats: ipynb,py:light\n# text_representation:\n# extension: .py\n# format_name: light\n# format_version: '1.5'\n# jupytext_version: 1.18.1\n# kernelspec:\n# display_name: Python 3 (ipykernel)\n# language: python\n# name: python3\n# ---\n\n# # Implementation\n# Author: Jørgen S. Dokken\n#\n# In this section, we will solve the deflection of the membrane problem.\n# After finishing this section, you should be able to:\n# - Create a simple mesh using the GMSH Python API and load it into DOLFINx\n# - Create constant boundary conditions using a {py:func}`geometrical identifier`\n# - Use {py:class}`ufl.SpatialCoordinate` to create a spatially varying function\n# - Interpolate a {py:class}`ufl-Expression` into an appropriate function space\n# - Evaluate a {py:class}`dolfinx.fem.Function` at any point $x$\n# - Use Paraview to visualize the solution of a PDE\n#\n# ## Creating the mesh\n#\n# To create the computational geometry, we use the Python-API of [GMSH](https://gmsh.info/).\n# We start by importing the gmsh-module and initializing it.\n\n# +\nimport gmsh\n\ngmsh.initialize()\n# -\n\n# The next step is to create the membrane and start the computations by the GMSH CAD kernel,\n# to generate the relevant underlying data structures.\n# The first arguments of `addDisk` are the x, y and z coordinate of the center of the circle,\n# while the two last arguments are the x-radius and y-radius.\n\nmembrane = gmsh.model.occ.addDisk(0, 0, 0, 1, 1)\ngmsh.model.occ.synchronize()\n\n# After that, we make the membrane a physical surface, such that it is recognized by `gmsh` when generating the mesh.\n# As a surface is a two-dimensional entity, we add `2` as the first argument,\n# the entity tag of the membrane as the second argument, and the physical tag as the last argument.\n# In a later demo, we will get into when this tag matters.\n\ngdim = 2\ngmsh.model.addPhysicalGroup(gdim, [membrane], 1)\n\n# Finally, we generate the two-dimensional mesh.\n# We set a uniform mesh size by modifying the GMSH options.\n\ngmsh.option.setNumber(\"Mesh.CharacteristicLengthMin\", 0.05)\ngmsh.option.setNumber(\"Mesh.CharacteristicLengthMax\", 0.05)\ngmsh.model.mesh.generate(gdim)\n\n# # Interfacing with GMSH in DOLFINx\n# We will import the GMSH-mesh directly from GMSH into DOLFINx via the {py:mod}`dolfinx.io.gmsh` interface.\n# The {py:mod}`dolfinx.io.gmsh` module contains two functions\n# 1. {py:func}`model_to_mesh` which takes in a `gmsh.model`\n# and returns a {py:class}`dolfinx.io.gmsh.MeshData` object.\n# 2. {py:func}`read_from_msh` which takes in a path to a `.msh`-file\n# and returns a {py:class}`dolfinx.io.gmsh.MeshData` object.\n#\n# The {py:class}`MeshData` object will contain a {py:class}`dolfinx.mesh.Mesh`,\n# under the attribute {py:attr}`mesh`.\n# This mesh will contain all GMSH Physical Groups of the highest topological dimension.\n# ```{note}\n# If you do not use `gmsh.model.addPhysicalGroup` when creating the mesh with GMSH, it can not be read into DOLFINx.\n# ```\n# The {py:class}`MeshData` object can also contain tags for\n# all other `PhysicalGroups` that has been added to the mesh, that being\n# {py:attr}`cell_tags`, {py:attr}`facet_tags`,\n# {py:attr}`ridge_tags` and\n# {py:attr}`peak_tags`.\n# To read either `gmsh.model` or a `.msh`-file, one has to distribute the mesh to all processes used by DOLFINx.\n# As GMSH does not support mesh creation with MPI, we currently have a `gmsh.model.mesh` on each process.\n# To distribute the mesh, we have to specify which process the mesh was created on,\n# and which communicator rank should distribute the mesh.\n# The {py:func}`model_to_mesh` will then load the mesh on the specified rank,\n# and distribute it to the communicator using a mesh partitioner.\n\n# +\nfrom dolfinx.io import gmsh as gmshio\nfrom dolfinx.fem.petsc import LinearProblem\nfrom mpi4py import MPI\n\ngmsh_model_rank = 0\nmesh_comm = MPI.COMM_WORLD\nmesh_data = gmshio.model_to_mesh(gmsh.model, mesh_comm, gmsh_model_rank, gdim=gdim)\nassert mesh_data.cell_tags is not None\ncell_markers = mesh_data.cell_tags\ndomain = mesh_data.mesh\n# -\n\n# We define the function space as in the previous tutorial\n\n# +\nfrom dolfinx import fem\n\nV = fem.functionspace(domain, (\"Lagrange\", 1))\n# -\n\n# ## Defining a spatially varying load\n# The right hand side pressure function is represented using {py:class}`ufl.SpatialCoordinate` and two constants,\n# one for $\\beta$ and one for $R_0$.\n\n# +\nimport ufl\nfrom dolfinx import default_scalar_type\n\nx = ufl.SpatialCoordinate(domain)\nbeta = fem.Constant(domain, default_scalar_type(12))\nR0 = fem.Constant(domain, default_scalar_type(0.3))\np = 4 * ufl.exp(-(beta**2) * (x[0] ** 2 + (x[1] - R0) ** 2))\n# -\n\n# ## Create a Dirichlet boundary condition using geometrical conditions\n# The next step is to create the homogeneous boundary condition.\n# As opposed to the [first tutorial](./fundamentals_code.ipynb) we will use\n# {py:func}`locate_dofs_geometrical` to locate the degrees of freedom on the boundary.\n# As we know that our domain is a circle with radius 1, we know that any degree of freedom should be\n# located at a coordinate $(x,y)$ such that $\\sqrt{x^2+y^2}=1$.\n\n# +\nimport numpy as np\n\n\ndef on_boundary(x):\n return np.isclose(np.sqrt(x[0] ** 2 + x[1] ** 2), 1)\n\n\nboundary_dofs = fem.locate_dofs_geometrical(V, on_boundary)\n# -\n\n# As our Dirichlet condition is homogeneous (`u=0` on the whole boundary), we can initialize the\n# {py:class}`dolfinx.fem.DirichletBC` with a constant value, the degrees of freedom and the function\n# space to apply the boundary condition on. We use the constructor {py:func}`dolfinx.fem.dirichletbc`.\n\nbc = fem.dirichletbc(default_scalar_type(0), boundary_dofs, V)\n\n# ## Defining the variational problem\n# The variational problem is the same as in our first Poisson problem, where `f` is replaced by `p`.\n\nu = ufl.TrialFunction(V)\nv = ufl.TestFunction(V)\na = ufl.dot(ufl.grad(u), ufl.grad(v)) * ufl.dx\nL = p * v * ufl.dx\nproblem = LinearProblem(\n a,\n L,\n bcs=[bc],\n petsc_options={\"ksp_type\": \"preonly\", \"pc_type\": \"lu\"},\n petsc_options_prefix=\"membrane_\",\n)\nuh = problem.solve()\n\n# ## Interpolation of a UFL-expression\n# As we previously defined the load `p` as a spatially varying function,\n# we would like to interpolate this function into an appropriate function space for visualization.\n# To do this we use the class {py:class}`Expression`.\n# The expression takes in any UFL-expression, and a set of points on the reference element.\n# We will use the {py:attr}`interpolation points`\n# of the space we want to interpolate in to.\n# We choose a high order function space to represent the function `p`, as it is rapidly varying in space.\n\nQ = fem.functionspace(domain, (\"Lagrange\", 5))\nexpr = fem.Expression(p, Q.element.interpolation_points)\npressure = fem.Function(Q)\npressure.interpolate(expr)\n\n# ## Plotting the solution over a line\n# We first plot the deflection $u_h$ over the domain $\\Omega$.\n\nfrom dolfinx.plot import vtk_mesh\nimport pyvista\n\n# Extract topology from mesh and create {py:class}`pyvista.UnstructuredGrid`\n\ntopology, cell_types, x = vtk_mesh(V)\ngrid = pyvista.UnstructuredGrid(topology, cell_types, x)\n\n# Set deflection values and add it to plotter\n\n# +\ngrid.point_data[\"u\"] = uh.x.array\nwarped = grid.warp_by_scalar(\"u\", factor=25)\n\nplotter = pyvista.Plotter()\nplotter.add_mesh(warped, show_edges=True, show_scalar_bar=True, scalars=\"u\")\nif not pyvista.OFF_SCREEN:\n plotter.show()\nelse:\n plotter.screenshot(\"deflection.png\")\n# -\n\n# We next plot the load on the domain\n\nload_plotter = pyvista.Plotter()\np_grid = pyvista.UnstructuredGrid(*vtk_mesh(Q))\np_grid.point_data[\"p\"] = pressure.x.array.real\nwarped_p = p_grid.warp_by_scalar(\"p\", factor=0.5)\nwarped_p.set_active_scalars(\"p\")\nload_plotter.add_mesh(warped_p, show_scalar_bar=True)\nload_plotter.view_xy()\nif not pyvista.OFF_SCREEN:\n load_plotter.show()\nelse:\n load_plotter.screenshot(\"load.png\")\n\n# ## Making curve plots throughout the domain\n# Another way to compare the deflection and the load is to make a plot along the line $x=0$.\n# This is just a matter of defining a set of points along the $y$-axis and evaluating the\n# finite element functions $u$ and $p$ at these points.\n\ntol = 0.001 # Avoid hitting the outside of the domain\ny = np.linspace(-1 + tol, 1 - tol, 101)\npoints = np.zeros((3, 101))\npoints[1] = y\nu_values = []\np_values = []\n\n# As a finite element function is the linear combination of all degrees of freedom,\n# $u_h(x)=\\sum_{i=1}^N c_i \\phi_i(x)$ where $c_i$ are the coefficients of $u_h$ and $\\phi_i$\n# is the $i$-th basis function, we can compute the exact solution at any point in $\\Omega$.\n# However, as a mesh consists of a large set of degrees of freedom (i.e. $N$ is large),\n# we want to reduce the number of evaluations of the basis function $\\phi_i(x)$.\n# We do this by identifying which cell of the mesh $x$ is in.\n# This is efficiently done by creating a {py:class}`bounding box tree`.\n# However, as the bounding box of a cell spans more of $\\mathbb{R}^n$ than the actual cell,\n# we check that the actual cell collides with the input point using\n# {py:func}`dolfinx.geometry.compute_colliding_cells`,\n# which measures the exact distance between the point and the cell\n# (approximated as a convex hull for higher order geometries).\n# This function also returns an adjacency-list, as the point might align with a facet,\n# edge or vertex that is shared between multiple cells in the mesh.\n#\n# Finally, we would like the code below to run in parallel,\n# when the mesh is distributed over multiple processors.\n# In that case, it is not guaranteed that every point in `points` is on each processor.\n# Therefore we create a subset `points_on_proc` only containing the points found on the current processor.\n\ncells = []\npoints_on_proc = []\n# Find cells whose bounding-box collide with the the points\ncell_candidates = geometry.compute_collisions_points(bb_tree, points.T)\n# Choose one of the cells that contains the point\ncolliding_cells = geometry.compute_colliding_cells(domain, cell_candidates, points.T)\nfor i, point in enumerate(points.T):\n if len(colliding_cells.links(i)) > 0:\n points_on_proc.append(point)\n cells.append(colliding_cells.links(i)[0])\n\n# We now have a list of points on the processor, on in which cell each point belongs.\n# We can then call {py:meth}`uh.eval` and\n# {py:meth}`pressure.eval` to obtain the set of values for all the points.\n\npoints_on_proc = np.array(points_on_proc, dtype=np.float64)\nu_values = uh.eval(points_on_proc, cells)\np_values = pressure.eval(points_on_proc, cells)\n\n# As we now have an array of coordinates and two arrays of function values,\n# we can use {py:mod}`matplotlib` to plot them\n\n# +\nimport matplotlib.pyplot as plt\n\nfig = plt.figure()\nplt.plot(\n points_on_proc[:, 1],\n 50 * u_values,\n \"k\",\n linewidth=2,\n label=\"Deflection ($\\\\times 50$)\",\n)\nplt.plot(points_on_proc[:, 1], p_values, \"b--\", linewidth=2, label=\"Load\")\nplt.grid(True)\nplt.xlabel(\"y\")\nplt.legend()\n# -\n\n# If executed in parallel as a Python file, we save a plot per processor\n\nplt.savefig(f\"membrane_rank{MPI.COMM_WORLD.rank:d}.png\")\n\n# ## Saving functions to file\n# As mentioned in the previous section, we can also use Paraview to visualize the solution.\n\n# +\nimport dolfinx.io\nfrom pathlib import Path\n\npressure.name = \"Load\"\nuh.name = \"Deflection\"\nresults_folder = Path(\"results\")\nresults_folder.mkdir(exist_ok=True, parents=True)\nwith dolfinx.io.VTXWriter(\n MPI.COMM_WORLD, results_folder / \"membrane_pressure.bp\", [pressure], engine=\"BP4\"\n) as vtx:\n vtx.write(0.0)\nwith dolfinx.io.VTXWriter(\n MPI.COMM_WORLD, results_folder / \"membrane_deflection.bp\", [uh], engine=\"BP4\"\n) as vtx:\n vtx.write(0.0)\n" }, { "path": "chapter1/membrane_paraview.md", "content": "# Using Paraview for visualization\n\nWe start by opening [Paraview](https://www.paraview.org/). We start by opening the file by pressing `File->Open`. Next you should choose the `Xdmf3ReaderT` to open the data with.\n\nThe next step is to visualize each of the functions. To do this, we choose `Filters->Alphabetic->ExtractBlock`. The next step is to select the first Unstructured Grid and press `Apply` as shown below:\n\n![Select Block](select_block.png)\n\nWe can select the other block to visualize the deflection. There are also options to visualize the deflection in three dimensions using `Filters->Alphabetical->Warp By Scalar`, and change the layout to 3D by pressing the `2D`-button.\n\n![Change interactive layout](2Dto3D.png)\n\nFinally, press the `Set view direction button`.\n\n![Set view direction](axis.png)\n\nWith these instructions, you can obtain the following figures\n\n![Results for membrane](result_membrane.png)" }, { "path": "chapter1/nitsche.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"markdown\",\n \"id\": \"0\",\n \"metadata\": {},\n \"source\": [\n \"# Weak imposition of Dirichlet conditions for the Poisson problem\\n\",\n \"Author: Jørgen S. Dokken\\n\",\n \"\\n\",\n \"In this section, we will go through how to solve the Poisson problem from the\\n\",\n \"[Fundamentals](./fundamentals_code.ipynb) tutorial using Nitsche's method {cite}`nitsche-Nitsche1971`.\\n\",\n \"The idea of weak imposition is that we add additional terms to the variational formulation to\\n\",\n \"impose the boundary condition, instead of modifying the matrix system using strong imposition (lifting).\\n\",\n \"\\n\",\n \"We start by importing the required modules and creating the mesh and function space for our solution\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"1\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"from dolfinx import fem, mesh, plot, default_scalar_type\\n\",\n \"from dolfinx.fem.petsc import LinearProblem\\n\",\n \"import numpy\\n\",\n \"from mpi4py import MPI\\n\",\n \"from ufl import (\\n\",\n \" Circumradius,\\n\",\n \" FacetNormal,\\n\",\n \" SpatialCoordinate,\\n\",\n \" TrialFunction,\\n\",\n \" TestFunction,\\n\",\n \" div,\\n\",\n \" dx,\\n\",\n \" ds,\\n\",\n \" grad,\\n\",\n \" inner,\\n\",\n \")\\n\",\n \"\\n\",\n \"N = 8\\n\",\n \"domain = mesh.create_unit_square(MPI.COMM_WORLD, N, N)\\n\",\n \"V = fem.functionspace(domain, (\\\"Lagrange\\\", 1))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"2\",\n \"metadata\": {},\n \"source\": [\n \"Next, we create a function containing the exact solution (which will also be used in the Dirichlet boundary condition)\\n\",\n \"and the corresponding source function for the right hand side. Note that we use {py:class}`ufl.SpatialCoordinate`\\n\",\n \"to define the exact solution, which in turn is interpolated into {py:class}`uD`\\n\",\n \"by wrapping the {py:class}`ufl-expression` as a {py:class}`dolfinx.fem.Expression`.\\n\",\n \"For the source function, we use the symbolic differentiation capabilities of UFL to\\n\",\n \"compute the negative Laplacian of the exact solution, which we can use directly in the variational formulation.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"3\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"uD = fem.Function(V)\\n\",\n \"x = SpatialCoordinate(domain)\\n\",\n \"u_ex = 1 + x[0] ** 2 + 2 * x[1] ** 2\\n\",\n \"uD.interpolate(fem.Expression(u_ex, V.element.interpolation_points))\\n\",\n \"f = -div(grad(u_ex))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"4\",\n \"metadata\": {},\n \"source\": [\n \"As opposed to the first tutorial, we now have to have another look at the variational form.\\n\",\n \"We start by integrating the problem by parts, to obtain\\n\",\n \"\\n\",\n \"$$\\n\",\n \"\\\\begin{align}\\n\",\n \" \\\\int_{\\\\Omega} \\\\nabla u \\\\cdot \\\\nabla v~\\\\mathrm{d}x\\n\",\n \" - \\\\int_{\\\\partial\\\\Omega}\\\\nabla u \\\\cdot n v~\\\\mathrm{d}s = \\\\int_{\\\\Omega} f v~\\\\mathrm{d}x.\\n\",\n \"\\\\end{align}\\n\",\n \"$$\\n\",\n \"\\n\",\n \"As we are not using strong enforcement, we do not set the trace of the test function to $0$ on the outer boundary.\\n\",\n \"Instead, we add the following two terms to the variational formulation\\n\",\n \"\\n\",\n \"$$\\n\",\n \"\\\\begin{align}\\n\",\n \" -\\\\int_{\\\\partial\\\\Omega} \\\\nabla v \\\\cdot n (u-u_D)~\\\\mathrm{d}s\\n\",\n \" + \\\\frac{\\\\alpha}{h} \\\\int_{\\\\partial\\\\Omega} (u-u_D)v~\\\\mathrm{d}s.\\n\",\n \"\\\\end{align}\\n\",\n \"$$\\n\",\n \"\\n\",\n \"where the first term enforces symmetry to the bilinear form, while the latter term enforces coercivity.\\n\",\n \"$u_D$ is the known Dirichlet condition, and $h$ is the diameter of the circumscribed sphere of the mesh element.\\n\",\n \"We create bilinear and linear form, $a$ and $L$\\n\",\n \"\\n\",\n \"$$\\n\",\n \"\\\\begin{align}\\n\",\n \" a(u, v) &= \\\\int_{\\\\Omega} \\\\nabla u \\\\cdot \\\\nabla v~\\\\mathrm{d}x + \\\\int_{\\\\partial\\\\Omega}-(n \\\\cdot\\\\nabla u) v - (n \\\\cdot \\\\nabla v) u + \\\\frac{\\\\alpha}{h} uv~\\\\mathrm{d}s,\\\\\\\\\\n\",\n \" L(v) &= \\\\int_{\\\\Omega} fv~\\\\mathrm{d}x + \\\\int_{\\\\partial\\\\Omega} -(n \\\\cdot \\\\nabla v) u_D + \\\\frac{\\\\alpha}{h} u_Dv~\\\\mathrm{d}s\\n\",\n \"\\\\end{align}\\n\",\n \"$$\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"5\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"u = TrialFunction(V)\\n\",\n \"v = TestFunction(V)\\n\",\n \"n = FacetNormal(domain)\\n\",\n \"h = 2 * Circumradius(domain)\\n\",\n \"alpha = fem.Constant(domain, default_scalar_type(10))\\n\",\n \"a = inner(grad(u), grad(v)) * dx - inner(n, grad(u)) * v * ds\\n\",\n \"a += -inner(n, grad(v)) * u * ds + alpha / h * inner(u, v) * ds\\n\",\n \"L = inner(f, v) * dx\\n\",\n \"L += -inner(n, grad(v)) * uD * ds + alpha / h * inner(uD, v) * ds\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"6\",\n \"metadata\": {},\n \"source\": [\n \"As we now have the variational form, we can solve the arising\\n\",\n \"{py:class}`linear problem`\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"7\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"problem = LinearProblem(a, L, petsc_options_prefix=\\\"nitsche_poisson\\\")\\n\",\n \"uh = problem.solve()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"8\",\n \"metadata\": {},\n \"source\": [\n \"We compute the error of the computation by comparing it to the analytical solution\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"9\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"error_form = fem.form(inner(uh - uD, uh - uD) * dx)\\n\",\n \"error_local = fem.assemble_scalar(error_form)\\n\",\n \"errorL2 = numpy.sqrt(domain.comm.allreduce(error_local, op=MPI.SUM))\\n\",\n \"if domain.comm.rank == 0:\\n\",\n \" print(rf\\\"$L^2$-error: {errorL2:.2e}\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"10\",\n \"metadata\": {},\n \"source\": [\n \"We observe that the $L^2$-error is of the same magnitude as in the first tutorial.\\n\",\n \"As in the previous tutorial, we also compute the maximal error for all the degrees of freedom.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"11\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"error_max = domain.comm.allreduce(\\n\",\n \" numpy.max(numpy.abs(uD.x.array - uh.x.array)), op=MPI.MAX\\n\",\n \")\\n\",\n \"if domain.comm.rank == 0:\\n\",\n \" print(f\\\"Error_max : {error_max:.2e}\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"12\",\n \"metadata\": {},\n \"source\": [\n \"We observe that as we weakly impose the boundary condition,\\n\",\n \"we no longer fullfill the equation to machine precision at the mesh vertices.\\n\",\n \"We also plot the solution using {py:mod}`pyvista`\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"13\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"import pyvista\\n\",\n \"\\n\",\n \"grid = pyvista.UnstructuredGrid(*plot.vtk_mesh(V))\\n\",\n \"grid.point_data[\\\"u\\\"] = uh.x.array.real\\n\",\n \"grid.set_active_scalars(\\\"u\\\")\\n\",\n \"plotter = pyvista.Plotter()\\n\",\n \"plotter.add_mesh(grid, show_edges=True, show_scalar_bar=True)\\n\",\n \"plotter.view_xy()\\n\",\n \"if not pyvista.OFF_SCREEN:\\n\",\n \" plotter.show()\\n\",\n \"else:\\n\",\n \" figure = plotter.screenshot(\\\"nitsche.png\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"14\",\n \"metadata\": {},\n \"source\": [\n \"```{bibliography}\\n\",\n \" :filter: cited\\n\",\n \" :labelprefix:\\n\",\n \" :keyprefix: nitsche-\\n\",\n \"```\"\n ]\n }\n ],\n \"metadata\": {\n \"jupytext\": {\n \"formats\": \"ipynb,py:light\"\n },\n \"kernelspec\": {\n \"display_name\": \"Python 3 (ipykernel)\",\n \"language\": \"python\",\n \"name\": \"python3\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 3\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython3\",\n \"version\": \"3.10.12\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 5\n}\n" }, { "path": "chapter1/nitsche.py", "content": "# ---\n# jupyter:\n# jupytext:\n# formats: ipynb,py:light\n# text_representation:\n# extension: .py\n# format_name: light\n# format_version: '1.5'\n# jupytext_version: 1.18.1\n# kernelspec:\n# display_name: Python 3 (ipykernel)\n# language: python\n# name: python3\n# ---\n\n# # Weak imposition of Dirichlet conditions for the Poisson problem\n# Author: Jørgen S. Dokken\n#\n# In this section, we will go through how to solve the Poisson problem from the\n# [Fundamentals](./fundamentals_code.ipynb) tutorial using Nitsche's method {cite}`nitsche-Nitsche1971`.\n# The idea of weak imposition is that we add additional terms to the variational formulation to\n# impose the boundary condition, instead of modifying the matrix system using strong imposition (lifting).\n#\n# We start by importing the required modules and creating the mesh and function space for our solution\n\n# +\nfrom dolfinx import fem, mesh, plot, default_scalar_type\nfrom dolfinx.fem.petsc import LinearProblem\nimport numpy\nfrom mpi4py import MPI\nfrom ufl import (\n Circumradius,\n FacetNormal,\n SpatialCoordinate,\n TrialFunction,\n TestFunction,\n div,\n dx,\n ds,\n grad,\n inner,\n)\n\nN = 8\ndomain = mesh.create_unit_square(MPI.COMM_WORLD, N, N)\nV = fem.functionspace(domain, (\"Lagrange\", 1))\n# -\n\n# Next, we create a function containing the exact solution (which will also be used in the Dirichlet boundary condition)\n# and the corresponding source function for the right hand side. Note that we use {py:class}`ufl.SpatialCoordinate`\n# to define the exact solution, which in turn is interpolated into {py:class}`uD`\n# by wrapping the {py:class}`ufl-expression` as a {py:class}`dolfinx.fem.Expression`.\n# For the source function, we use the symbolic differentiation capabilities of UFL to\n# compute the negative Laplacian of the exact solution, which we can use directly in the variational formulation.\n\nuD = fem.Function(V)\nx = SpatialCoordinate(domain)\nu_ex = 1 + x[0] ** 2 + 2 * x[1] ** 2\nuD.interpolate(fem.Expression(u_ex, V.element.interpolation_points))\nf = -div(grad(u_ex))\n\n# As opposed to the first tutorial, we now have to have another look at the variational form.\n# We start by integrating the problem by parts, to obtain\n#\n# $$\n# \\begin{align}\n# \\int_{\\Omega} \\nabla u \\cdot \\nabla v~\\mathrm{d}x\n# - \\int_{\\partial\\Omega}\\nabla u \\cdot n v~\\mathrm{d}s = \\int_{\\Omega} f v~\\mathrm{d}x.\n# \\end{align}\n# $$\n#\n# As we are not using strong enforcement, we do not set the trace of the test function to $0$ on the outer boundary.\n# Instead, we add the following two terms to the variational formulation\n#\n# $$\n# \\begin{align}\n# -\\int_{\\partial\\Omega} \\nabla v \\cdot n (u-u_D)~\\mathrm{d}s\n# + \\frac{\\alpha}{h} \\int_{\\partial\\Omega} (u-u_D)v~\\mathrm{d}s.\n# \\end{align}\n# $$\n#\n# where the first term enforces symmetry to the bilinear form, while the latter term enforces coercivity.\n# $u_D$ is the known Dirichlet condition, and $h$ is the diameter of the circumscribed sphere of the mesh element.\n# We create bilinear and linear form, $a$ and $L$\n#\n# $$\n# \\begin{align}\n# a(u, v) &= \\int_{\\Omega} \\nabla u \\cdot \\nabla v~\\mathrm{d}x + \\int_{\\partial\\Omega}-(n \\cdot\\nabla u) v - (n \\cdot \\nabla v) u + \\frac{\\alpha}{h} uv~\\mathrm{d}s,\\\\\n# L(v) &= \\int_{\\Omega} fv~\\mathrm{d}x + \\int_{\\partial\\Omega} -(n \\cdot \\nabla v) u_D + \\frac{\\alpha}{h} u_Dv~\\mathrm{d}s\n# \\end{align}\n# $$\n\nu = TrialFunction(V)\nv = TestFunction(V)\nn = FacetNormal(domain)\nh = 2 * Circumradius(domain)\nalpha = fem.Constant(domain, default_scalar_type(10))\na = inner(grad(u), grad(v)) * dx - inner(n, grad(u)) * v * ds\na += -inner(n, grad(v)) * u * ds + alpha / h * inner(u, v) * ds\nL = inner(f, v) * dx\nL += -inner(n, grad(v)) * uD * ds + alpha / h * inner(uD, v) * ds\n\n# As we now have the variational form, we can solve the arising\n# {py:class}`linear problem`\n\nproblem = LinearProblem(a, L, petsc_options_prefix=\"nitsche_poisson\")\nuh = problem.solve()\n\n# We compute the error of the computation by comparing it to the analytical solution\n\nerror_form = fem.form(inner(uh - uD, uh - uD) * dx)\nerror_local = fem.assemble_scalar(error_form)\nerrorL2 = numpy.sqrt(domain.comm.allreduce(error_local, op=MPI.SUM))\nif domain.comm.rank == 0:\n print(rf\"$L^2$-error: {errorL2:.2e}\")\n\n# We observe that the $L^2$-error is of the same magnitude as in the first tutorial.\n# As in the previous tutorial, we also compute the maximal error for all the degrees of freedom.\n\nerror_max = domain.comm.allreduce(\n numpy.max(numpy.abs(uD.x.array - uh.x.array)), op=MPI.MAX\n)\nif domain.comm.rank == 0:\n print(f\"Error_max : {error_max:.2e}\")\n\n# We observe that as we weakly impose the boundary condition,\n# we no longer fullfill the equation to machine precision at the mesh vertices.\n# We also plot the solution using {py:mod}`pyvista`\n\n# +\nimport pyvista\n\ngrid = pyvista.UnstructuredGrid(*plot.vtk_mesh(V))\ngrid.point_data[\"u\"] = uh.x.array.real\ngrid.set_active_scalars(\"u\")\nplotter = pyvista.Plotter()\nplotter.add_mesh(grid, show_edges=True, show_scalar_bar=True)\nplotter.view_xy()\nif not pyvista.OFF_SCREEN:\n plotter.show()\nelse:\n figure = plotter.screenshot(\"nitsche.png\")\n# -\n\n# ```{bibliography}\n# :filter: cited\n# :labelprefix:\n# :keyprefix: nitsche-\n# ```\n" }, { "path": "chapter2/advdiffreac.md", "content": "# TO BE IMPLEMETED: A system of advection-diffusion-reaction equations\nAuthors: Hans Petter Langtangen and Anders Logg \n\nMost of the problems we have encountered so far have a common feature: they all invlove models expressed by a single scalar or vector PDE.\nIn many situations the model is instead expressed as a system of PDEs, describing different quantities possibly govered by (very) different physics. \nAs we saw for the Navier-Stokes equations, one way to solve one equation a system of PDEs in FEniCSx is to use a splitting method where we solve one equation at a time and feed the solution from one equation into the next. However, one of the strengths with FEniCSx is the ease by which one can instead define variational problems that couple several PDEs into one compound system. In this system, we will look at how to use FEniCSx to write solvers for such a system of coupled PDEs. The goal is to demonstrate how easy it is to implement fully implicit, also known as monolithic, solvers in FEniCSx.\n\n## The PDE problem\nOur model problem is the following system of advection-diffusion-reaction equations:\n```{math}\n:label: adv-diff-reac\n\\frac{\\partial u_1}{\\partial t} + w \\cdot \\nabla u_1 - \\nabla \\cdot (\\epsilon \\nabla u_1) &= f_1 - Ku_1 u_2,\\\\\n\\frac{\\partial u_2}{\\partial t} + w \\cdot \\nabla u_2 - \\nabla \\cdot (\\epsilon \\nabla u_2) &= f_2 - Ku_1 u_2,\\\\\n\\frac{\\partial u_3}{\\partial t} + w \\cdot \\nabla u_3 - \\nabla \\cdot (\\epsilon \\nabla u_3) &= f_3 - Ku_1 u_2 - K u_3,\\\\\n```\nThis system models the chemical reaction between two species $A$ and $B$ in some domain $\\Omega$:\n```{math}\n A + B \\rightarrow C.\n```\nWe assume that the reaction is *first-order*, meaning that the reaction rate is proportional to concentrations $[A]$ and $[B]$ of the two species $A$ and $B$:\n```{math}\n\\frac{\\mathrm{d}}{\\mathrm{d}t}[C]= K[A][B].\n```\nWe also assume that the formed species $C$ spontaneously decaas with a rate proportional to the concentration [C]. In the [PDE system](adv-diff-reac), we use the variatbles $u_1, u_2$ and $u_3$ to denote the concentrations of the three species:\n```{math}\n u_1=[A],\\qquad u_2=[B], \\qquad u_3=[C].\n```\nWe see that the chemical reactions are accounted for in the right-hand sides of the PDE system.\n\nThe chemical reactions takes part at each point in the domain $\\Omega$. \nIn addition, we assume that the species $A, B$ and $C$ diffuse throughout the domain with diffusivity $\\epsilon$ (the terms $-\\nabla \\cdot (\\epsilon \\nabla u_i)$) and are advected with velocity $w$ (the terms $w\\cdot \\nabla u_i$).\n\n```{admonition} Implementation note\nFor this demo to work, we need to have checkpointing implemented\n```\n" }, { "path": "chapter2/amr.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"markdown\",\n \"id\": \"0\",\n \"metadata\": {},\n \"source\": [\n \"# Adaptive mesh refinement with NetGen and DOLFINx\\n\",\n \"\\n\",\n \"Author: Jørgen S. Dokken\\n\",\n \"\\n\",\n \"```{admonition} NetGen and linux/arm64\\n\",\n \"NetGen is not available on PyPi on linux/arm64, so to run this tutorial on such machine, please use the\\n\",\n \"docker image [ghcr.io/jorgensd/dolfinx-tutorial:release](https://github.com/jorgensd/dolfinx-tutorial/pkgs/container/dolfinx-tutorial/489387776?tag=release).\\n\",\n \"You can also install NetGen from source. See the [Dockerfile](https://github.com/jorgensd/dolfinx-tutorial/blob/main/docker/Dockerfile) for instructions.\\n\",\n \"```\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"1\",\n \"metadata\": {},\n \"source\": [\n \"In this tutorial, we will consider an adaptive mesh refinement method, applied to\\n\",\n \"the Laplace eigenvalue problem.\\n\",\n \"This demo is an adaptation of [Firedrake - Adaptive Mesh Refinement](https://www.firedrakeproject.org/firedrake/demos/netgen_mesh.py.html).\\n\",\n \"In this tutorial we will use the mesh generator [NetGen](https://ngsolve.org/) from NGSolve.\\n\",\n \"First, we import the packages needed for this demo:\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"2\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"from mpi4py import MPI\\n\",\n \"from petsc4py import PETSc\\n\",\n \"from slepc4py import SLEPc\\n\",\n \"from packaging.version import Version\\n\",\n \"import dolfinx.fem.petsc\\n\",\n \"import numpy as np\\n\",\n \"import ufl\\n\",\n \"import pyvista\\n\",\n \"import ngsPETSc.utils.fenicsx as ngfx\\n\",\n \"\\n\",\n \"from netgen.geom2d import SplineGeometry\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"3\",\n \"metadata\": {},\n \"source\": [\n \"## Generating a higher-order mesh with NetGen\\n\",\n \"Next, we generate a PacMan-like geometry using NetGen.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"4\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"geo = SplineGeometry()\\n\",\n \"pnts = [(0, 0), (1, 0), (1, 1), (0, 1), (-1, 1), (-1, 0), (-1, -1), (0, -1)]\\n\",\n \"p1, p2, p3, p4, p5, p6, p7, p8 = [geo.AppendPoint(*pnt) for pnt in pnts]\\n\",\n \"curves = [\\n\",\n \" [[\\\"line\\\", p1, p2], \\\"line\\\"],\\n\",\n \" [[\\\"spline3\\\", p2, p3, p4], \\\"curve\\\"],\\n\",\n \" [[\\\"spline3\\\", p4, p5, p6], \\\"curve\\\"],\\n\",\n \" [[\\\"spline3\\\", p6, p7, p8], \\\"curve\\\"],\\n\",\n \" [[\\\"line\\\", p8, p1], \\\"line\\\"],\\n\",\n \"]\\n\",\n \"for c, bc in curves:\\n\",\n \" geo.Append(c, bc=bc)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"5\",\n \"metadata\": {},\n \"source\": [\n \"## Loading a mesh into DOLFINx\\n\",\n \"The ngsPETSc package provides a communication layer between NetGen and DOLFINx.\\n\",\n \"We initialize this layer by passing in a NetGen-model, as well as an MPI communicator,\\n\",\n \"which will be used to distribute the mesh.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"6\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"geoModel = ngfx.GeometricModel(geo, MPI.COMM_WORLD)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"7\",\n \"metadata\": {},\n \"source\": [\n \"Next, we generate the mesh with the function :py:func:`ngsPETSc.utils.fenicsx.GeometricModel.model_to_mesh`.\\n\",\n \"Which takes in the target geometric dimension of the mesh (2 for triangular meshes, 3 for tetrahedral), the\\n\",\n \"maximum mesh size (`hmax`) and a few optional parameters.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"8\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"mesh, (ct, ft), region_map = geoModel.model_to_mesh(gdim=2, hmax=0.5)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"9\",\n \"metadata\": {},\n \"source\": [\n \"We use pyvista to visualize the mesh.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"10\",\n \"metadata\": {\n \"tags\": [\n \"hide-input\"\n ]\n },\n \"outputs\": [],\n \"source\": [\n \"grid = pyvista.UnstructuredGrid(*dolfinx.plot.vtk_mesh(mesh))\\n\",\n \"grid.cell_data[\\\"ct\\\"] = ct.values\\n\",\n \"\\n\",\n \"plotter = pyvista.Plotter()\\n\",\n \"plotter.add_mesh(\\n\",\n \" grid, show_edges=True, scalars=\\\"ct\\\", cmap=\\\"blues\\\", show_scalar_bar=False\\n\",\n \")\\n\",\n \"plotter.view_xy()\\n\",\n \"if not pyvista.OFF_SCREEN:\\n\",\n \" plotter.show()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"11\",\n \"metadata\": {},\n \"source\": [\n \"We have read in any cell and facet markers that have been defined in the NetGen model,\\n\",\n \"as well as a map from their names to their integer ids in `ct`, `ft` and `region_map` respectively.\\n\",\n \"We can curve the grids with the command `curveField`.\\n\",\n \"In this example, we use third order Lagrange elements to represent the geometry.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"12\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"order = 3\\n\",\n \"curved_mesh = geoModel.curveField(order)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"13\",\n \"metadata\": {},\n \"source\": [\n \"Again, we visualize the curved mesh with pyvista.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"14\",\n \"metadata\": {\n \"tags\": [\n \"hide-input\"\n ]\n },\n \"outputs\": [],\n \"source\": [\n \"curved_grid = pyvista.UnstructuredGrid(*dolfinx.plot.vtk_mesh(curved_mesh))\\n\",\n \"curved_grid.cell_data[\\\"ct\\\"] = ct.values\\n\",\n \"plotter = pyvista.Plotter()\\n\",\n \"plotter.add_mesh(\\n\",\n \" curved_grid, show_edges=False, scalars=\\\"ct\\\", cmap=\\\"blues\\\", show_scalar_bar=False\\n\",\n \")\\n\",\n \"plotter.add_mesh(grid, style=\\\"wireframe\\\", color=\\\"black\\\")\\n\",\n \"plotter.view_xy()\\n\",\n \"if not pyvista.OFF_SCREEN:\\n\",\n \" plotter.show()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"15\",\n \"metadata\": {},\n \"source\": [\n \"## Solving the eigenvalue problem\\n\",\n \"In this section we will solve the eigenvalue problem:\\n\",\n \"\\n\",\n \"Find $u_h\\\\in H_0^1(\\\\Omega)$ and $\\\\lambda\\\\in\\\\mathbb{R}$ such that\\n\",\n \"\\n\",\n \"$$\\n\",\n \"\\\\begin{align}\\n\",\n \"\\\\int_\\\\Omega \\\\nabla u \\\\cdot \\\\nabla v~\\\\mathrm{d} x &= \\\\lambda \\\\int_\\\\Omega u v~\\\\mathrm{d} x \\\\qquad\\n\",\n \"\\\\forall v \\\\in H_0^1(\\\\Omega).\\n\",\n \"\\\\end{align}\\n\",\n \"$$\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"16\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"Next, we define a convenience function to solve the eigenvalue problem using [SLEPc](https://slepc.upv.es/)\\n\",\n \"given a discretized domain, its facet markers and the region map.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"17\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"outputs\": [],\n \"source\": [\n \"def solve(\\n\",\n \" mesh: dolfinx.mesh.Mesh,\\n\",\n \" facet_tags: dolfinx.mesh.MeshTags,\\n\",\n \" region_map: dict[tuple[int, str], tuple[int, ...]],\\n\",\n \") -> tuple[float, dolfinx.fem.Function, dolfinx.fem.Function]:\\n\",\n \" # We define the lhs and rhs bilinear forms\\n\",\n \" V = dolfinx.fem.functionspace(mesh, (\\\"Lagrange\\\", 3))\\n\",\n \" u = ufl.TrialFunction(V)\\n\",\n \" v = ufl.TestFunction(V)\\n\",\n \" a = ufl.inner(ufl.grad(u), ufl.grad(v)) * ufl.dx\\n\",\n \" m = ufl.inner(u, v) * ufl.dx\\n\",\n \"\\n\",\n \" # We identify the boundary facets and their corresponding dofs\\n\",\n \" straight_facets = facet_tags.indices[\\n\",\n \" np.isin(facet_tags.values, region_map[(1, \\\"line\\\")])\\n\",\n \" ]\\n\",\n \" curved_facets = facet_tags.indices[\\n\",\n \" np.isin(facet_tags.values, region_map[(1, \\\"curve\\\")])\\n\",\n \" ]\\n\",\n \" boundary_facets = np.concatenate([straight_facets, curved_facets])\\n\",\n \" mesh.topology.create_connectivity(mesh.topology.dim - 1, mesh.topology.dim)\\n\",\n \" boundary_dofs = dolfinx.fem.locate_dofs_topological(\\n\",\n \" V, mesh.topology.dim - 1, boundary_facets\\n\",\n \" )\\n\",\n \"\\n\",\n \" # We create a zero boundary condition for these dofs to be in the suitable space, and\\n\",\n \" # set up the discrete matrices `A` and `M`\\n\",\n \" bc = dolfinx.fem.dirichletbc(0.0, boundary_dofs, V)\\n\",\n \" A = dolfinx.fem.petsc.assemble_matrix(dolfinx.fem.form(a), bcs=[bc])\\n\",\n \" A.assemble()\\n\",\n \" if Version(dolfinx.__version__) < Version(\\\"0.10.0\\\"):\\n\",\n \" diag_kwargs = {\\\"diagonal\\\": 0.0}\\n\",\n \" else:\\n\",\n \" diag_kwargs = {\\\"diag\\\": 0.0}\\n\",\n \"\\n\",\n \" M = dolfinx.fem.petsc.assemble_matrix(dolfinx.fem.form(m), bcs=[bc], **diag_kwargs)\\n\",\n \" M.assemble()\\n\",\n \"\\n\",\n \" # Next, we define the SLEPc Eigenvalue Problem Solver (EPS), and set up to use a shift\\n\",\n \" # and invert (SINVERT) spectral transformation where the preconditioner factorisation\\n\",\n \" # is computed using [MUMPS](https://mumps-solver.org/index.php).\\n\",\n \"\\n\",\n \" E = SLEPc.EPS().create(mesh.comm)\\n\",\n \" E.setType(SLEPc.EPS.Type.ARNOLDI)\\n\",\n \" E.setProblemType(SLEPc.EPS.ProblemType.GHEP)\\n\",\n \" E.setDimensions(1, SLEPc.DECIDE)\\n\",\n \" E.setOperators(A, M)\\n\",\n \" ST = E.getST()\\n\",\n \" ST.setType(SLEPc.ST.Type.SINVERT)\\n\",\n \" PC = ST.getKSP().getPC()\\n\",\n \" PC.setType(\\\"lu\\\")\\n\",\n \" PC.setFactorSolverType(\\\"mumps\\\")\\n\",\n \" E.setST(ST)\\n\",\n \" E.solve()\\n\",\n \" assert E.getConvergedReason() >= 0, \\\"Eigenvalue solver did not converge\\\"\\n\",\n \"\\n\",\n \" # We get the real and imaginary parts of the first eigenvector along with the eigenvalue.\\n\",\n \" uh_r = dolfinx.fem.Function(V)\\n\",\n \" uh_i = dolfinx.fem.Function(V)\\n\",\n \" lam = E.getEigenpair(0, uh_r.x.petsc_vec, uh_i.x.petsc_vec)\\n\",\n \" E.destroy()\\n\",\n \" uh_r.x.scatter_forward()\\n\",\n \" uh_i.x.scatter_forward()\\n\",\n \" return (lam, uh_r, uh_i)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"18\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"## Error-indicator\\n\",\n \"In this example, we will use an error-indicator $\\\\eta$ to decide what cells should be refined.\\n\",\n \"Specifically, the estimator $\\\\eta$ is defined as:\\n\",\n \"\\n\",\n \"\\\\begin{align*}\\n\",\n \" \\\\eta^2 = \\\\sum_{K\\\\in \\\\mathcal{T}_h(\\\\Omega)}\\\\left(h^2\\\\int_K \\\\vert \\\\lambda u_h + \\\\Delta u_h\\\\vert^2~\\\\mathrm{d}x\\\\right)\\n\",\n \"+ \\\\sum_{E\\\\in\\\\mathcal{F}_i}\\\\frac{h}{2} \\\\vert [\\\\nabla \\\\cdot \\\\mathbf{n}_E ]\\\\vert^2~\\\\mathrm{d}s\\n\",\n \"\\\\end{align*}\\n\",\n \"\\n\",\n \"where $\\\\mathcal{T}_h$ is the collection of cells in the mesh, $\\\\mathcal{F}_i$ the collection of interior facets\\n\",\n \"(those connected to two cells).\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"19\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"def mark_cells(uh_r: dolfinx.fem.Function, lam: float):\\n\",\n \" mesh = uh_r.function_space.mesh\\n\",\n \" W = dolfinx.fem.functionspace(mesh, (\\\"DG\\\", 0))\\n\",\n \" w = ufl.TestFunction(W)\\n\",\n \" eta_squared = dolfinx.fem.Function(W)\\n\",\n \" f = dolfinx.fem.Constant(mesh, 1.0)\\n\",\n \" h = dolfinx.fem.Function(W)\\n\",\n \" h.x.array[:] = mesh.h(mesh.topology.dim, np.arange(len(h.x.array), dtype=np.int32))\\n\",\n \" n = ufl.FacetNormal(mesh)\\n\",\n \"\\n\",\n \" G = ( # compute cellwise error estimator\\n\",\n \" ufl.inner(h**2 * (f + ufl.div(ufl.grad(uh_r))) ** 2, w) * ufl.dx\\n\",\n \" + ufl.inner(h(\\\"+\\\") / 2 * ufl.jump(ufl.grad(uh_r), n) ** 2, w(\\\"+\\\")) * ufl.dS\\n\",\n \" + ufl.inner(h(\\\"-\\\") / 2 * ufl.jump(ufl.grad(uh_r), n) ** 2, w(\\\"-\\\")) * ufl.dS\\n\",\n \" )\\n\",\n \" dolfinx.fem.petsc.assemble_vector(eta_squared.x.petsc_vec, dolfinx.fem.form(G))\\n\",\n \" eta = dolfinx.fem.Function(W)\\n\",\n \" eta.x.array[:] = np.sqrt(eta_squared.x.array[:])\\n\",\n \"\\n\",\n \" eta_max = eta.x.petsc_vec.max()[1]\\n\",\n \"\\n\",\n \" theta = 0.5\\n\",\n \" should_refine = ufl.conditional(ufl.gt(eta, theta * eta_max), 1, 0)\\n\",\n \" markers = dolfinx.fem.Function(W)\\n\",\n \" ip = W.element.interpolation_points\\n\",\n \" if Version(dolfinx.__version__) < Version(\\\"0.10.0\\\"):\\n\",\n \" ip = ip()\\n\",\n \" markers.interpolate(dolfinx.fem.Expression(should_refine, ip))\\n\",\n \" return np.flatnonzero(np.isclose(markers.x.array.astype(np.int32), 1))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"20\",\n \"metadata\": {},\n \"source\": [\n \"## Running the adaptive refinement algorithm\\n\",\n \"Next, we will run the adaptive mesh refinement algorithm.\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"21\",\n \"metadata\": {},\n \"source\": [\n \"We will track the progress of the adaptive mesh refinement as a GIF.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"22\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"plotter = pyvista.Plotter()\\n\",\n \"plotter.open_gif(\\\"amr.gif\\\", fps=1)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"23\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"We make a convenience function to attach the relevant data to the plotter at a given\\n\",\n \"refinement step.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"24\",\n \"metadata\": {\n \"tags\": [\n \"hide-input\"\n ]\n },\n \"outputs\": [],\n \"source\": [\n \"def write_frame(plotter: pyvista.Plotter, uh_r: dolfinx.fem.Function):\\n\",\n \" # Scale uh_r to be consistent between refinement steps, as it can be multiplied by -1\\n\",\n \" uh_r_min = curved_mesh.comm.allreduce(uh_r.x.array.min(), op=MPI.MIN)\\n\",\n \" uh_r_max = curved_mesh.comm.allreduce(uh_r.x.array.max(), op=MPI.MAX)\\n\",\n \" uh_sign = np.sign(uh_r_min)\\n\",\n \" if np.isclose(uh_sign, 0):\\n\",\n \" uh_sign = np.sign(uh_r_max)\\n\",\n \" assert not np.isclose(uh_sign, 0), \\\"uh_r has zero values, cannot determine sign.\\\"\\n\",\n \" uh_r.x.array[:] *= uh_sign\\n\",\n \"\\n\",\n \" # Update plot with refined mesh\\n\",\n \" grid = pyvista.UnstructuredGrid(*dolfinx.plot.vtk_mesh(mesh))\\n\",\n \" curved_grid = pyvista.UnstructuredGrid(*dolfinx.plot.vtk_mesh(uh_r.function_space))\\n\",\n \" curved_grid.point_data[\\\"u\\\"] = uh_r.x.array\\n\",\n \" curved_grid = curved_grid.tessellate()\\n\",\n \" curved_actor = plotter.add_mesh(\\n\",\n \" curved_grid,\\n\",\n \" show_edges=False,\\n\",\n \" )\\n\",\n \"\\n\",\n \" actor = plotter.add_mesh(grid, style=\\\"wireframe\\\", color=\\\"black\\\")\\n\",\n \" plotter.view_xy()\\n\",\n \" plotter.write_frame()\\n\",\n \" plotter.remove_actor(actor)\\n\",\n \" plotter.remove_actor(curved_actor)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"25\",\n \"metadata\": {},\n \"source\": [\n \"We set some parameters for checking convergence of the algorithm, and provide the exact eigenvalue\\n\",\n \"for comparison.\\n\",\n \"```{admonition} Using ngsPETSc for mesh refinement\\n\",\n \"In `ngsPETSc`, we provide the function `GeometricModel.refineMarkedElements` which we\\n\",\n \"pass the entities we would like to refine, and the topological dimensions of those entities.\\n\",\n \"The function returns a refined mesh, with corresponding cell and facet markers extracted from\\n\",\n \"the NetGen model.\\n\",\n \"```\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"26\",\n \"metadata\": {\n \"tags\": [\n \"scroll-output\"\n ]\n },\n \"outputs\": [],\n \"source\": [\n \"max_iterations = 15\\n\",\n \"exact = 3.375610652693620492628**2\\n\",\n \"termination_criteria = 1e-5\\n\",\n \"for i in range(max_iterations):\\n\",\n \" lam, uh_r, _ = solve(curved_mesh, ft, region_map)\\n\",\n \"\\n\",\n \" relative_error = (lam - exact) / abs(exact)\\n\",\n \" PETSc.Sys.Print(\\n\",\n \" f\\\"Iteration {i + 1}/{max_iterations}, {lam=:.5e}, {exact=:.5e}, {relative_error=:.2e}\\\"\\n\",\n \" )\\n\",\n \"\\n\",\n \" cells_to_mark = mark_cells(uh_r, lam)\\n\",\n \" mesh, (_, ft) = geoModel.refineMarkedElements(mesh.topology.dim, cells_to_mark)\\n\",\n \" curved_mesh = geoModel.curveField(order)\\n\",\n \" write_frame(plotter, uh_r)\\n\",\n \"\\n\",\n \" if relative_error < termination_criteria:\\n\",\n \" PETSc.Sys.Print(f\\\"Converged in {i + 1} iterations.\\\")\\n\",\n \" break\\n\",\n \"plotter.close()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"27\",\n \"metadata\": {},\n \"source\": [\n \"\\\"gif\\\"\"\n ]\n }\n ],\n \"metadata\": {\n \"jupytext\": {\n \"cell_metadata_filter\": \"tags,-all\",\n \"formats\": \"ipynb,py:light\",\n \"main_language\": \"python\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 5\n}\n" }, { "path": "chapter2/amr.py", "content": "# ---\n# jupyter:\n# jupytext:\n# cell_metadata_filter: tags,-all\n# formats: ipynb,py:light\n# text_representation:\n# extension: .py\n# format_name: light\n# format_version: '1.5'\n# jupytext_version: 1.19.1\n# ---\n\n# # Adaptive mesh refinement with NetGen and DOLFINx\n#\n# Author: Jørgen S. Dokken\n#\n# ```{admonition} NetGen and linux/arm64\n# NetGen is not available on PyPi on linux/arm64, so to run this tutorial on such machine, please use the\n# docker image [ghcr.io/jorgensd/dolfinx-tutorial:release](https://github.com/jorgensd/dolfinx-tutorial/pkgs/container/dolfinx-tutorial/489387776?tag=release).\n# You can also install NetGen from source. See the [Dockerfile](https://github.com/jorgensd/dolfinx-tutorial/blob/main/docker/Dockerfile) for instructions.\n# ```\n\n# In this tutorial, we will consider an adaptive mesh refinement method, applied to\n# the Laplace eigenvalue problem.\n# This demo is an adaptation of [Firedrake - Adaptive Mesh Refinement](https://www.firedrakeproject.org/firedrake/demos/netgen_mesh.py.html).\n# In this tutorial we will use the mesh generator [NetGen](https://ngsolve.org/) from NGSolve.\n# First, we import the packages needed for this demo:\n\n# +\nfrom mpi4py import MPI\nfrom petsc4py import PETSc\nfrom slepc4py import SLEPc\nfrom packaging.version import Version\nimport dolfinx.fem.petsc\nimport numpy as np\nimport ufl\nimport pyvista\nimport ngsPETSc.utils.fenicsx as ngfx\n\nfrom netgen.geom2d import SplineGeometry\n# -\n\n# ## Generating a higher-order mesh with NetGen\n# Next, we generate a PacMan-like geometry using NetGen.\n\ngeo = SplineGeometry()\npnts = [(0, 0), (1, 0), (1, 1), (0, 1), (-1, 1), (-1, 0), (-1, -1), (0, -1)]\np1, p2, p3, p4, p5, p6, p7, p8 = [geo.AppendPoint(*pnt) for pnt in pnts]\ncurves = [\n [[\"line\", p1, p2], \"line\"],\n [[\"spline3\", p2, p3, p4], \"curve\"],\n [[\"spline3\", p4, p5, p6], \"curve\"],\n [[\"spline3\", p6, p7, p8], \"curve\"],\n [[\"line\", p8, p1], \"line\"],\n]\nfor c, bc in curves:\n geo.Append(c, bc=bc)\n\n# ## Loading a mesh into DOLFINx\n# The ngsPETSc package provides a communication layer between NetGen and DOLFINx.\n# We initialize this layer by passing in a NetGen-model, as well as an MPI communicator,\n# which will be used to distribute the mesh.\n\ngeoModel = ngfx.GeometricModel(geo, MPI.COMM_WORLD)\n\n# Next, we generate the mesh with the function :py:func:`ngsPETSc.utils.fenicsx.GeometricModel.model_to_mesh`.\n# Which takes in the target geometric dimension of the mesh (2 for triangular meshes, 3 for tetrahedral), the\n# maximum mesh size (`hmax`) and a few optional parameters.\n\nmesh, (ct, ft), region_map = geoModel.model_to_mesh(gdim=2, hmax=0.5)\n\n# We use pyvista to visualize the mesh.\n\n# + tags=[\"hide-input\"]\ngrid = pyvista.UnstructuredGrid(*dolfinx.plot.vtk_mesh(mesh))\ngrid.cell_data[\"ct\"] = ct.values\n\nplotter = pyvista.Plotter()\nplotter.add_mesh(\n grid, show_edges=True, scalars=\"ct\", cmap=\"blues\", show_scalar_bar=False\n)\nplotter.view_xy()\nif not pyvista.OFF_SCREEN:\n plotter.show()\n# -\n\n# We have read in any cell and facet markers that have been defined in the NetGen model,\n# as well as a map from their names to their integer ids in `ct`, `ft` and `region_map` respectively.\n# We can curve the grids with the command `curveField`.\n# In this example, we use third order Lagrange elements to represent the geometry.\n\norder = 3\ncurved_mesh = geoModel.curveField(order)\n\n# Again, we visualize the curved mesh with pyvista.\n\n# + tags=[\"hide-input\"]\ncurved_grid = pyvista.UnstructuredGrid(*dolfinx.plot.vtk_mesh(curved_mesh))\ncurved_grid.cell_data[\"ct\"] = ct.values\nplotter = pyvista.Plotter()\nplotter.add_mesh(\n curved_grid, show_edges=False, scalars=\"ct\", cmap=\"blues\", show_scalar_bar=False\n)\nplotter.add_mesh(grid, style=\"wireframe\", color=\"black\")\nplotter.view_xy()\nif not pyvista.OFF_SCREEN:\n plotter.show()\n# -\n\n# ## Solving the eigenvalue problem\n# In this section we will solve the eigenvalue problem:\n#\n# Find $u_h\\in H_0^1(\\Omega)$ and $\\lambda\\in\\mathbb{R}$ such that\n#\n# $$\n# \\begin{align}\n# \\int_\\Omega \\nabla u \\cdot \\nabla v~\\mathrm{d} x &= \\lambda \\int_\\Omega u v~\\mathrm{d} x \\qquad\n# \\forall v \\in H_0^1(\\Omega).\n# \\end{align}\n# $$\n\n# Next, we define a convenience function to solve the eigenvalue problem using [SLEPc](https://slepc.upv.es/)\n# given a discretized domain, its facet markers and the region map.\n\n\ndef solve(\n mesh: dolfinx.mesh.Mesh,\n facet_tags: dolfinx.mesh.MeshTags,\n region_map: dict[tuple[int, str], tuple[int, ...]],\n) -> tuple[float, dolfinx.fem.Function, dolfinx.fem.Function]:\n # We define the lhs and rhs bilinear forms\n V = dolfinx.fem.functionspace(mesh, (\"Lagrange\", 3))\n u = ufl.TrialFunction(V)\n v = ufl.TestFunction(V)\n a = ufl.inner(ufl.grad(u), ufl.grad(v)) * ufl.dx\n m = ufl.inner(u, v) * ufl.dx\n\n # We identify the boundary facets and their corresponding dofs\n straight_facets = facet_tags.indices[\n np.isin(facet_tags.values, region_map[(1, \"line\")])\n ]\n curved_facets = facet_tags.indices[\n np.isin(facet_tags.values, region_map[(1, \"curve\")])\n ]\n boundary_facets = np.concatenate([straight_facets, curved_facets])\n mesh.topology.create_connectivity(mesh.topology.dim - 1, mesh.topology.dim)\n boundary_dofs = dolfinx.fem.locate_dofs_topological(\n V, mesh.topology.dim - 1, boundary_facets\n )\n\n # We create a zero boundary condition for these dofs to be in the suitable space, and\n # set up the discrete matrices `A` and `M`\n bc = dolfinx.fem.dirichletbc(0.0, boundary_dofs, V)\n A = dolfinx.fem.petsc.assemble_matrix(dolfinx.fem.form(a), bcs=[bc])\n A.assemble()\n if Version(dolfinx.__version__) < Version(\"0.10.0\"):\n diag_kwargs = {\"diagonal\": 0.0}\n else:\n diag_kwargs = {\"diag\": 0.0}\n\n M = dolfinx.fem.petsc.assemble_matrix(dolfinx.fem.form(m), bcs=[bc], **diag_kwargs)\n M.assemble()\n\n # Next, we define the SLEPc Eigenvalue Problem Solver (EPS), and set up to use a shift\n # and invert (SINVERT) spectral transformation where the preconditioner factorisation\n # is computed using [MUMPS](https://mumps-solver.org/index.php).\n\n E = SLEPc.EPS().create(mesh.comm)\n E.setType(SLEPc.EPS.Type.ARNOLDI)\n E.setProblemType(SLEPc.EPS.ProblemType.GHEP)\n E.setDimensions(1, SLEPc.DECIDE)\n E.setOperators(A, M)\n ST = E.getST()\n ST.setType(SLEPc.ST.Type.SINVERT)\n PC = ST.getKSP().getPC()\n PC.setType(\"lu\")\n PC.setFactorSolverType(\"mumps\")\n E.setST(ST)\n E.solve()\n assert E.getConvergedReason() >= 0, \"Eigenvalue solver did not converge\"\n\n # We get the real and imaginary parts of the first eigenvector along with the eigenvalue.\n uh_r = dolfinx.fem.Function(V)\n uh_i = dolfinx.fem.Function(V)\n lam = E.getEigenpair(0, uh_r.x.petsc_vec, uh_i.x.petsc_vec)\n E.destroy()\n uh_r.x.scatter_forward()\n uh_i.x.scatter_forward()\n return (lam, uh_r, uh_i)\n\n\n# ## Error-indicator\n# In this example, we will use an error-indicator $\\eta$ to decide what cells should be refined.\n# Specifically, the estimator $\\eta$ is defined as:\n#\n# \\begin{align*}\n# \\eta^2 = \\sum_{K\\in \\mathcal{T}_h(\\Omega)}\\left(h^2\\int_K \\vert \\lambda u_h + \\Delta u_h\\vert^2~\\mathrm{d}x\\right)\n# + \\sum_{E\\in\\mathcal{F}_i}\\frac{h}{2} \\vert [\\nabla \\cdot \\mathbf{n}_E ]\\vert^2~\\mathrm{d}s\n# \\end{align*}\n#\n# where $\\mathcal{T}_h$ is the collection of cells in the mesh, $\\mathcal{F}_i$ the collection of interior facets\n# (those connected to two cells).\n\n\ndef mark_cells(uh_r: dolfinx.fem.Function, lam: float):\n mesh = uh_r.function_space.mesh\n W = dolfinx.fem.functionspace(mesh, (\"DG\", 0))\n w = ufl.TestFunction(W)\n eta_squared = dolfinx.fem.Function(W)\n f = dolfinx.fem.Constant(mesh, 1.0)\n h = dolfinx.fem.Function(W)\n h.x.array[:] = mesh.h(mesh.topology.dim, np.arange(len(h.x.array), dtype=np.int32))\n n = ufl.FacetNormal(mesh)\n\n G = ( # compute cellwise error estimator\n ufl.inner(h**2 * (f + ufl.div(ufl.grad(uh_r))) ** 2, w) * ufl.dx\n + ufl.inner(h(\"+\") / 2 * ufl.jump(ufl.grad(uh_r), n) ** 2, w(\"+\")) * ufl.dS\n + ufl.inner(h(\"-\") / 2 * ufl.jump(ufl.grad(uh_r), n) ** 2, w(\"-\")) * ufl.dS\n )\n dolfinx.fem.petsc.assemble_vector(eta_squared.x.petsc_vec, dolfinx.fem.form(G))\n eta = dolfinx.fem.Function(W)\n eta.x.array[:] = np.sqrt(eta_squared.x.array[:])\n\n eta_max = eta.x.petsc_vec.max()[1]\n\n theta = 0.5\n should_refine = ufl.conditional(ufl.gt(eta, theta * eta_max), 1, 0)\n markers = dolfinx.fem.Function(W)\n ip = W.element.interpolation_points\n if Version(dolfinx.__version__) < Version(\"0.10.0\"):\n ip = ip()\n markers.interpolate(dolfinx.fem.Expression(should_refine, ip))\n return np.flatnonzero(np.isclose(markers.x.array.astype(np.int32), 1))\n\n\n# ## Running the adaptive refinement algorithm\n# Next, we will run the adaptive mesh refinement algorithm.\n\n# We will track the progress of the adaptive mesh refinement as a GIF.\n\nplotter = pyvista.Plotter()\nplotter.open_gif(\"amr.gif\", fps=1)\n\n# We make a convenience function to attach the relevant data to the plotter at a given\n# refinement step.\n\n\n# + tags=[\"hide-input\"]\ndef write_frame(plotter: pyvista.Plotter, uh_r: dolfinx.fem.Function):\n # Scale uh_r to be consistent between refinement steps, as it can be multiplied by -1\n uh_r_min = curved_mesh.comm.allreduce(uh_r.x.array.min(), op=MPI.MIN)\n uh_r_max = curved_mesh.comm.allreduce(uh_r.x.array.max(), op=MPI.MAX)\n uh_sign = np.sign(uh_r_min)\n if np.isclose(uh_sign, 0):\n uh_sign = np.sign(uh_r_max)\n assert not np.isclose(uh_sign, 0), \"uh_r has zero values, cannot determine sign.\"\n uh_r.x.array[:] *= uh_sign\n\n # Update plot with refined mesh\n grid = pyvista.UnstructuredGrid(*dolfinx.plot.vtk_mesh(mesh))\n curved_grid = pyvista.UnstructuredGrid(*dolfinx.plot.vtk_mesh(uh_r.function_space))\n curved_grid.point_data[\"u\"] = uh_r.x.array\n curved_grid = curved_grid.tessellate()\n curved_actor = plotter.add_mesh(\n curved_grid,\n show_edges=False,\n )\n\n actor = plotter.add_mesh(grid, style=\"wireframe\", color=\"black\")\n plotter.view_xy()\n plotter.write_frame()\n plotter.remove_actor(actor)\n plotter.remove_actor(curved_actor)\n\n\n# -\n\n# We set some parameters for checking convergence of the algorithm, and provide the exact eigenvalue\n# for comparison.\n# ```{admonition} Using ngsPETSc for mesh refinement\n# In `ngsPETSc`, we provide the function `GeometricModel.refineMarkedElements` which we\n# pass the entities we would like to refine, and the topological dimensions of those entities.\n# The function returns a refined mesh, with corresponding cell and facet markers extracted from\n# the NetGen model.\n# ```\n\n# + tags=[\"scroll-output\"]\nmax_iterations = 15\nexact = 3.375610652693620492628**2\ntermination_criteria = 1e-5\nfor i in range(max_iterations):\n lam, uh_r, _ = solve(curved_mesh, ft, region_map)\n\n relative_error = (lam - exact) / abs(exact)\n PETSc.Sys.Print(\n f\"Iteration {i + 1}/{max_iterations}, {lam=:.5e}, {exact=:.5e}, {relative_error=:.2e}\"\n )\n\n cells_to_mark = mark_cells(uh_r, lam)\n mesh, (_, ft) = geoModel.refineMarkedElements(mesh.topology.dim, cells_to_mark)\n curved_mesh = geoModel.curveField(order)\n write_frame(plotter, uh_r)\n\n if relative_error < termination_criteria:\n PETSc.Sys.Print(f\"Converged in {i + 1} iterations.\")\n break\nplotter.close()\n# -\n\n# \"gif\"\n" }, { "path": "chapter2/bdforces_lv4", "content": "# timestep time bdc horiz vert\n0 0.0000000000E+00 2 0.0000000000E+00 0.0000000000E+00\n1 3.1250000000E-04 2 1.3726547634E-01 -2.3688715430E-04\n2 9.3750000000E-04 2 1.4039968314E-01 -2.3605983416E-04\n3 1.5625000000E-03 2 1.4235406913E-01 -2.3460352800E-04\n4 2.1875000000E-03 2 1.4402969709E-01 -2.3336938713E-04\n5 2.8125000000E-03 2 1.4545447740E-01 -2.3229564448E-04\n6 3.4375000000E-03 2 1.4674958894E-01 -2.3140917781E-04\n7 4.0625000000E-03 2 1.4793014505E-01 -2.3065928772E-04\n8 4.6875000000E-03 2 1.4903278329E-01 -2.3003337267E-04\n9 5.3125000000E-03 2 1.5006676841E-01 -2.2950123817E-04\n10 5.9375000000E-03 2 1.5104753385E-01 -2.2905035339E-04\n11 6.5625000000E-03 2 1.5198113488E-01 -2.2866272925E-04\n12 7.1875000000E-03 2 1.5287551488E-01 -2.2832946794E-04\n13 7.8125000000E-03 2 1.5373481415E-01 -2.2803967427E-04\n14 8.4375000000E-03 2 1.5456370627E-01 -2.2778743940E-04\n15 9.0625000000E-03 2 1.5536512932E-01 -2.2756604346E-04\n16 9.6875000000E-03 2 1.5614210551E-01 -2.2737160850E-04\n17 1.0312500000E-02 2 1.5689678714E-01 -2.2719989899E-04\n18 1.0937500000E-02 2 1.5763126806E-01 -2.2704834106E-04\n19 1.1562500000E-02 2 1.5834716907E-01 -2.2691419297E-04\n20 1.2187500000E-02 2 1.5904601739E-01 -2.2679570183E-04\n21 1.2812500000E-02 2 1.5972906332E-01 -2.2669103229E-04\n22 1.3437500000E-02 2 1.6039746526E-01 -2.2659894817E-04\n23 1.4062500000E-02 2 1.6105220830E-01 -2.2651817004E-04\n24 1.4687500000E-02 2 1.6169419817E-01 -2.2644778862E-04\n25 1.5312500000E-02 2 1.6232422557E-01 -2.2638687157E-04\n26 1.5937500000E-02 2 1.6294301590E-01 -2.2633471829E-04\n27 1.6562500000E-02 2 1.6355121432E-01 -2.2629061796E-04\n28 1.7187500000E-02 2 1.6414941324E-01 -2.2625400753E-04\n29 1.7812500000E-02 2 1.6473814670E-01 -2.2622432031E-04\n30 1.8437500000E-02 2 1.6531790634E-01 -2.2620108588E-04\n31 1.9062500000E-02 2 1.6588913991E-01 -2.2618383554E-04\n32 1.9687500000E-02 2 1.6645226099E-01 -2.2617216494E-04\n33 2.0312500000E-02 2 1.6700764924E-01 -2.2616567391E-04\n34 2.0937500000E-02 2 1.6755565662E-01 -2.2616400691E-04\n35 2.1562500000E-02 2 1.6809660830E-01 -2.2616681393E-04\n36 2.2187500000E-02 2 1.6863080685E-01 -2.2617377840E-04\n37 2.2812500000E-02 2 1.6915853325E-01 -2.2618458963E-04\n38 2.3437500000E-02 2 1.6968004991E-01 -2.2619896194E-04\n39 2.4062500000E-02 2 1.7019560161E-01 -2.2621661684E-04\n40 2.4687500000E-02 2 1.7070541769E-01 -2.2623729490E-04\n41 2.5312500000E-02 2 1.7120971295E-01 -2.2626074541E-04\n42 2.5937500000E-02 2 1.7170868924E-01 -2.2628673222E-04\n43 2.6562500000E-02 2 1.7220253626E-01 -2.2631502838E-04\n44 2.7187500000E-02 2 1.7269143279E-01 -2.2634541917E-04\n45 2.7812500000E-02 2 1.7317554736E-01 -2.2637769763E-04\n46 2.8437500000E-02 2 1.7365503923E-01 -2.2641167014E-04\n47 2.9062500000E-02 2 1.7413005892E-01 -2.2644714809E-04\n48 2.9687500000E-02 2 1.7460074904E-01 -2.2648395541E-04\n49 3.0312500000E-02 2 1.7506724469E-01 -2.2652192213E-04\n50 3.0937500000E-02 2 1.7552967416E-01 -2.2656088834E-04\n51 3.1562500000E-02 2 1.7598815929E-01 -2.2660070022E-04\n52 3.2187500000E-02 2 1.7644281596E-01 -2.2664121261E-04\n53 3.2812500000E-02 2 1.7689375450E-01 -2.2668228795E-04\n54 3.3437500000E-02 2 1.7734108004E-01 -2.2672379515E-04\n55 3.4062500000E-02 2 1.7778489285E-01 -2.2676560970E-04\n56 3.4687500000E-02 2 1.7822528866E-01 -2.2680761406E-04\n57 3.5312500000E-02 2 1.7866235892E-01 -2.2684969704E-04\n58 3.5937500000E-02 2 1.7909619110E-01 -2.2689175341E-04\n59 3.6562500000E-02 2 1.7952686888E-01 -2.2693368274E-04\n60 3.7187500000E-02 2 1.7995447244E-01 -2.2697539213E-04\n61 3.7812500000E-02 2 1.8037907860E-01 -2.2701679157E-04\n62 3.8437500000E-02 2 1.8080076106E-01 -2.2705779864E-04\n63 3.9062500000E-02 2 1.8121959054E-01 -2.2709833337E-04\n64 3.9687500000E-02 2 1.8163563497E-01 -2.2713832234E-04\n65 4.0312500000E-02 2 1.8204895965E-01 -2.2717769603E-04\n66 4.0937500000E-02 2 1.8245962736E-01 -2.2721638818E-04\n67 4.1562500000E-02 2 1.8286769852E-01 -2.2725433832E-04\n68 4.2187500000E-02 2 1.8327323129E-01 -2.2729148856E-04\n69 4.2812500000E-02 2 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2 -1.8093864431E-01 1.8283034329E-02\n1569 7.9803125000E+00 2 -1.8120305631E-01 1.8240544971E-02\n1570 7.9809375000E+00 2 -1.8146728786E-01 1.8197868407E-02\n1571 7.9815625000E+00 2 -1.8173133926E-01 1.8155004827E-02\n1572 7.9821875000E+00 2 -1.8199521078E-01 1.8111954421E-02\n1573 7.9828125000E+00 2 -1.8225890272E-01 1.8068717375E-02\n1574 7.9834375000E+00 2 -1.8252241536E-01 1.8025293877E-02\n1575 7.9840625000E+00 2 -1.8278574899E-01 1.7981684111E-02\n1576 7.9846875000E+00 2 -1.8304890389E-01 1.7937888261E-02\n1577 7.9853125000E+00 2 -1.8331188037E-01 1.7893906510E-02\n1578 7.9859375000E+00 2 -1.8357467869E-01 1.7849739039E-02\n1579 7.9865625000E+00 2 -1.8383729916E-01 1.7805386029E-02\n1580 7.9871875000E+00 2 -1.8409974208E-01 1.7760847659E-02\n1581 7.9878125000E+00 2 -1.8436200772E-01 1.7716124105E-02\n1582 7.9884375000E+00 2 -1.8462409638E-01 1.7671215545E-02\n1583 7.9890625000E+00 2 -1.8488600836E-01 1.7626122154E-02\n1584 7.9896875000E+00 2 -1.8514774395E-01 1.7580844105E-02\n1585 7.9903125000E+00 2 -1.8540930344E-01 1.7535381572E-02\n1586 7.9909375000E+00 2 -1.8567068713E-01 1.7489734726E-02\n1587 7.9915625000E+00 2 -1.8593189532E-01 1.7443903738E-02\n1588 7.9921875000E+00 2 -1.8619292830E-01 1.7397888775E-02\n1589 7.9928125000E+00 2 -1.8645378637E-01 1.7351690007E-02\n1590 7.9934375000E+00 2 -1.8671446982E-01 1.7305307601E-02\n1591 7.9940625000E+00 2 -1.8697497897E-01 1.7258741720E-02\n1592 7.9946875000E+00 2 -1.8723531410E-01 1.7211992531E-02\n1593 7.9953125000E+00 2 -1.8749547552E-01 1.7165060196E-02\n1594 7.9959375000E+00 2 -1.8775546352E-01 1.7117944876E-02\n1595 7.9965625000E+00 2 -1.8801527842E-01 1.7070646734E-02\n1596 7.9971875000E+00 2 -1.8827492051E-01 1.7023165928E-02\n1597 7.9978125000E+00 2 -1.8853439009E-01 1.6975502616E-02\n1598 7.9984375000E+00 2 -1.8879368748E-01 1.6927656957E-02\n1599 7.9990625000E+00 2 -1.8905281297E-01 1.6879629106E-02\n1600 7.9996875000E+00 2 -1.8931176686E-01 1.6831419217E-02\n" }, { "path": "chapter2/diffusion_code.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Diffusion of a Gaussian function\\n\",\n \"\\n\",\n \"Author: Jørgen S. Dokken\\n\",\n \"\\n\",\n \"Let us now solve a more interesting problem, namely the diffusion of a Gaussian hill.\\n\",\n \"We take the initial value to be\\n\",\n \"\\n\",\n \"$$\\n\",\n \"\\\\begin{align}\\n\",\n \" u_0(x,y)&= e^{-ax^2-ay^2}\\n\",\n \"\\\\end{align}\\n\",\n \"$$\\n\",\n \"\\n\",\n \"for $a=5$ on the domain $[-2,2]\\\\times[-2,2]$.\\n\",\n \"For this problem we will use homogeneous Dirichlet boundary conditions ($u_D=0$).\\n\",\n \"\\n\",\n \"The first difference from the previous problem is that we are not using a unit square.\\n\",\n \"We create the rectangular domain with {py:func}`dolfinx.mesh.create_rectangle`.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"1\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"import matplotlib as mpl\\n\",\n \"import pyvista\\n\",\n \"import ufl\\n\",\n \"import numpy as np\\n\",\n \"\\n\",\n \"from petsc4py import PETSc\\n\",\n \"from mpi4py import MPI\\n\",\n \"\\n\",\n \"from dolfinx import fem, mesh, io, plot\\n\",\n \"from dolfinx.fem.petsc import (\\n\",\n \" assemble_vector,\\n\",\n \" assemble_matrix,\\n\",\n \" create_vector,\\n\",\n \" apply_lifting,\\n\",\n \" set_bc,\\n\",\n \")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"2\",\n \"metadata\": {},\n \"source\": [\n \"We define the time discretization parameters\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"3\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"t = 0.0 # Start time\\n\",\n \"T = 1.0 # Final time\\n\",\n \"num_steps = 50\\n\",\n \"dt = T / num_steps # time step size\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"4\",\n \"metadata\": {},\n \"source\": [\n \"Next, we define the computational domain\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"outputs\": [],\n \"source\": [\n \"nx, ny = 50, 50\\n\",\n \"domain = mesh.create_rectangle(\\n\",\n \" MPI.COMM_WORLD,\\n\",\n \" [np.array([-2, -2]), np.array([2, 2])],\\n\",\n \" [nx, ny],\\n\",\n \" mesh.CellType.triangle,\\n\",\n \")\\n\",\n \"V = fem.functionspace(domain, (\\\"Lagrange\\\", 1))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"6\",\n \"metadata\": {},\n \"source\": [\n \"-\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"Note that we have used a much higher resolution than before to better resolve features of the solution.\\n\",\n \"We also easily update the intial and boundary conditions.\\n\",\n \"Instead of using a class to define the initial condition, we simply use a function\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"def initial_condition(x, a=5):\\n\",\n \" return np.exp(-a * (x[0] ** 2 + x[1] ** 2))\\n\",\n \"\\n\",\n \"\\n\",\n \"u_n = fem.Function(V)\\n\",\n \"u_n.name = \\\"u_n\\\"\\n\",\n \"u_n.interpolate(initial_condition)\\n\",\n \"\\n\",\n \"# Create boundary condition\\n\",\n \"fdim = domain.topology.dim - 1\\n\",\n \"boundary_facets = mesh.locate_entities_boundary(\\n\",\n \" domain, fdim, lambda x: np.full(x.shape[1], True, dtype=bool)\\n\",\n \")\\n\",\n \"bc = fem.dirichletbc(\\n\",\n \" PETSc.ScalarType(0), fem.locate_dofs_topological(V, fdim, boundary_facets), V\\n\",\n \")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Time-dependent output\\n\",\n \"To visualize the solution in an external program such as Paraview,\\n\",\n \"we create a an {py:class}`XDMFFile` which we can store multiple solutions in.\\n\",\n \"The main advantage with an XDMFFile is that we only need to store the mesh once and that we can\\n\",\n \"append multiple solutions to the same grid, reducing the storage space.\\n\",\n \"The first argument to the XDMFFile is the {py:class}`communicator`\\n\",\n \"which should be used to write data to file in parallel.\\n\",\n \"As we would like one output, independent of the number of processors,\\n\",\n \"we use the {py:data}`COMM_WORLD`.\\n\",\n \"The second argument is the file name of the output file,\\n\",\n \"while the third argument is the state of the file,\\n\",\n \"this could be read (`\\\"r\\\"`), write (`\\\"w\\\"`) or append (`\\\"a\\\"`).\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"10\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"xdmf = io.XDMFFile(domain.comm, \\\"diffusion.xdmf\\\", \\\"w\\\")\\n\",\n \"xdmf.write_mesh(domain)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"11\",\n \"metadata\": {},\n \"source\": [\n \"Define solution variable, and interpolate initial solution for visualization in Paraview\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"uh = fem.Function(V)\\n\",\n \"uh.name = \\\"uh\\\"\\n\",\n \"uh.interpolate(initial_condition)\\n\",\n \"xdmf.write_function(uh, t)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Variational problem and solver\\n\",\n \"As in the previous example, we prepare objects for time dependent problems,\\n\",\n \"such that we do not have to recreate data-structures.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"u, v = ufl.TrialFunction(V), ufl.TestFunction(V)\\n\",\n \"f = fem.Constant(domain, PETSc.ScalarType(0))\\n\",\n \"a = u * v * ufl.dx + dt * ufl.dot(ufl.grad(u), ufl.grad(v)) * ufl.dx\\n\",\n \"L = (u_n + dt * f) * v * ufl.dx\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Preparing linear algebra structures for time dependent problems\\n\",\n \"We note that even if `u_n` is time dependent, we will reuse the same function for\\n\",\n \"`f` and `u_n` at every time step.\\n\",\n \"We therefore call {py:func}`dolfinx.fem.form` to generate assembly kernels for\\n\",\n \"the matrix and vector. This function creates a {py:class}`dolfinx.fem.Form`.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"bilinear_form = fem.form(a)\\n\",\n \"linear_form = fem.form(L)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We observe that the left hand side of the system, the matrix {py:class}`A`\\n\",\n \"does not change from one time step to another, thus we only need to assemble it once.\\n\",\n \"We call {py:meth}`A.assemble()` to finalize the assembly process,\\n\",\n \"which means communicating local contributions from each process to other processes that shares the\\n\",\n \"same degrees of freedom.\\n\",\n \"However, the right hand side, which is dependent on the previous time step `u_n`,\\n\",\n \"we have to assemble it at every time step.\\n\",\n \"Therefore, we only create a {py:class}`vector` `b` based on `L`,\\n\",\n \"which we will reuse at every time step.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"A = assemble_matrix(bilinear_form, bcs=[bc])\\n\",\n \"A.assemble()\\n\",\n \"b = create_vector(fem.extract_function_spaces(linear_form))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Using petsc4py to create a linear solver\\n\",\n \"As we have already assembled `a` into the matrix `A`, we can no longer\\n\",\n \"use the {py:class}`dolfinx.fem.petsc.LinearProblem` class to solve the problem.\\n\",\n \"Therefore, we create a {py:class}`krylov subspace solver` using PETSc,\\n\",\n \"assign the matrix `A` to the solver, and choose the solution strategy.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"solver = PETSc.KSP().create(domain.comm)\\n\",\n \"solver.setOperators(A)\\n\",\n \"solver.setType(PETSc.KSP.Type.PREONLY)\\n\",\n \"solver.getPC().setType(PETSc.PC.Type.LU)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Visualization of time dependent problem using pyvista\\n\",\n \"We use the DOLFINx plotting functionality, which is based on {py:mod}`pyvista`\\n\",\n \"to plot the solution at every $15$th time step.\\n\",\n \"We would also like to visualize a colorbar reflecting the minimal and maximum\\n\",\n \"value of $u$ at each time step.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"grid = pyvista.UnstructuredGrid(*plot.vtk_mesh(V))\\n\",\n \"\\n\",\n \"plotter = pyvista.Plotter()\\n\",\n \"plotter.open_gif(\\\"u_time.gif\\\", fps=10)\\n\",\n \"\\n\",\n \"grid.point_data[\\\"uh\\\"] = uh.x.array\\n\",\n \"warped = grid.warp_by_scalar(\\\"uh\\\", factor=1)\\n\",\n \"\\n\",\n \"viridis = mpl.colormaps.get_cmap(\\\"viridis\\\").resampled(25)\\n\",\n \"sargs = dict(\\n\",\n \" title_font_size=25,\\n\",\n \" label_font_size=20,\\n\",\n \" fmt=\\\"%.2e\\\",\\n\",\n \" color=\\\"black\\\",\\n\",\n \" position_x=0.1,\\n\",\n \" position_y=0.8,\\n\",\n \" width=0.8,\\n\",\n \" height=0.1,\\n\",\n \")\\n\",\n \"\\n\",\n \"renderer = plotter.add_mesh(\\n\",\n \" warped,\\n\",\n \" show_edges=True,\\n\",\n \" lighting=False,\\n\",\n \" cmap=viridis,\\n\",\n \" scalar_bar_args=sargs,\\n\",\n \" clim=[0, max(uh.x.array)],\\n\",\n \")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {\n \"tags\": []\n },\n \"source\": [\n \"(time-dep-assembly)=\\n\",\n \"## Updating the solution and right hand side per time step\\n\",\n \"To be able to solve the variation problem at each time step,\\n\",\n \"we have to assemble the right hand side and apply the boundary condition before calling\\n\",\n \"{py:meth}`solver.solve(b, uh.x.petsc_vec)`.\\n\",\n \"We start by resetting the values in `b` as we are reusing the vector at every time step.\\n\",\n \"The next step is to assemble the vector calling\\n\",\n \"{py:func}`dolfinx.fem.petsc.assemble_vector(b, L)`,\\n\",\n \"which means that we are assembling the linear form `L(v)` into the vector `b`.\\n\",\n \"Note that we do not supply the boundary conditions for assembly, as opposed to the left hand side.\\n\",\n \"This is because we want to use {py:func}`lifting` to apply the boundary condition,\\n\",\n \"which preserves symmetry of the matrix $A$ in the bilinear form $a(u,v)=a(v,u)$ without Dirichlet boundary conditions.\\n\",\n \"Once we have performed the lifting, we accumulate values from degrees of freedom that are shared between processes\\n\",\n \"using {py:meth}`b.ghostUpdate()`.\\n\",\n \"Finally, we apply the boundary condition to the fixed degrees of freedom with\\n\",\n \"{py:func}`dolfinx.fem.petsc.set_bc`.\\n\",\n \"Next, we can {py:meth}`solve` the linear system and\\n\",\n \"{py:meth}`update` degrees of freedom shared between processors.\\n\",\n \"Finally, before moving to the next time step, we update the solution at the previous time step\\n\",\n \"to the solution at this time step.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"for i in range(num_steps):\\n\",\n \" t += dt\\n\",\n \"\\n\",\n \" # Update the right hand side reusing the initial vector\\n\",\n \" with b.localForm() as loc_b:\\n\",\n \" loc_b.set(0)\\n\",\n \" assemble_vector(b, linear_form)\\n\",\n \"\\n\",\n \" # Apply Dirichlet boundary condition to the vector\\n\",\n \" apply_lifting(b, [bilinear_form], [[bc]])\\n\",\n \" b.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)\\n\",\n \" set_bc(b, [bc])\\n\",\n \"\\n\",\n \" # Solve linear problem\\n\",\n \" solver.solve(b, uh.x.petsc_vec)\\n\",\n \" uh.x.scatter_forward()\\n\",\n \"\\n\",\n \" # Update solution at previous time step (u_n)\\n\",\n \" u_n.x.array[:] = uh.x.array\\n\",\n \"\\n\",\n \" # Write solution to file\\n\",\n \" xdmf.write_function(uh, t)\\n\",\n \" # Update plot\\n\",\n \" new_warped = grid.warp_by_scalar(\\\"uh\\\", factor=1)\\n\",\n \" warped.points[:, :] = new_warped.points\\n\",\n \" warped.point_data[\\\"uh\\\"][:] = uh.x.array\\n\",\n \" plotter.write_frame()\\n\",\n \"plotter.close()\\n\",\n \"xdmf.close()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"25\",\n \"metadata\": {},\n \"source\": [\n \"We {py:meth}`destroy` the PETSc objects to avoid memory leaks.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"26\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"A.destroy()\\n\",\n \"b.destroy()\\n\",\n \"solver.destroy()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"\\\"gif\\\"\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Animation with Paraview\\n\",\n \"We can also use Paraview to create an animation. We open the file in paraview with `File->Open`,\\n\",\n \"and then press `Apply` in the properties panel.\\n\",\n \"\\n\",\n \"Then, we add a time-annotation to the figure, pressing: `Sources->Alphabetical->Annotate Time`\\n\",\n \"and `Apply` in the properties panel.\\n\",\n \"It Is also a good idea to select an output resolution, by pressing `View->Preview->1280 x 720 (HD)`.\\n\",\n \"\\n\",\n \"Then finally, click `File->Save Animation`, and save the animation to the desired format,\\n\",\n \"such as `avi`, `ogv` or a sequence of `png`s. Make sure to set the frame rate to something sensible,\\n\",\n \"in the range of $5-10$ frames per second.\"\n ]\n }\n ],\n \"metadata\": {\n \"jupytext\": {\n \"formats\": \"ipynb,py:light\"\n },\n \"kernelspec\": {\n \"display_name\": \"Python 3 (ipykernel)\",\n \"language\": \"python\",\n \"name\": \"python3\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 3\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython3\",\n \"version\": \"3.10.12\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 4\n}\n" }, { "path": "chapter2/diffusion_code.py", "content": "# ---\n# jupyter:\n# jupytext:\n# formats: ipynb,py:light\n# text_representation:\n# extension: .py\n# format_name: light\n# format_version: '1.5'\n# jupytext_version: 1.18.1\n# kernelspec:\n# display_name: Python 3 (ipykernel)\n# language: python\n# name: python3\n# ---\n\n# # Diffusion of a Gaussian function\n#\n# Author: Jørgen S. Dokken\n#\n# Let us now solve a more interesting problem, namely the diffusion of a Gaussian hill.\n# We take the initial value to be\n#\n# $$\n# \\begin{align}\n# u_0(x,y)&= e^{-ax^2-ay^2}\n# \\end{align}\n# $$\n#\n# for $a=5$ on the domain $[-2,2]\\times[-2,2]$.\n# For this problem we will use homogeneous Dirichlet boundary conditions ($u_D=0$).\n#\n# The first difference from the previous problem is that we are not using a unit square.\n# We create the rectangular domain with {py:func}`dolfinx.mesh.create_rectangle`.\n\n# +\nimport matplotlib as mpl\nimport pyvista\nimport ufl\nimport numpy as np\n\nfrom petsc4py import PETSc\nfrom mpi4py import MPI\n\nfrom dolfinx import fem, mesh, io, plot\nfrom dolfinx.fem.petsc import (\n assemble_vector,\n assemble_matrix,\n create_vector,\n apply_lifting,\n set_bc,\n)\n# -\n\n# We define the time discretization parameters\n\nt = 0.0 # Start time\nT = 1.0 # Final time\nnum_steps = 50\ndt = T / num_steps # time step size\n\n# Next, we define the computational domain\n\nnx, ny = 50, 50\ndomain = mesh.create_rectangle(\n MPI.COMM_WORLD,\n [np.array([-2, -2]), np.array([2, 2])],\n [nx, ny],\n mesh.CellType.triangle,\n)\nV = fem.functionspace(domain, (\"Lagrange\", 1))\n\n\n# -\n\n# Note that we have used a much higher resolution than before to better resolve features of the solution.\n# We also easily update the intial and boundary conditions.\n# Instead of using a class to define the initial condition, we simply use a function\n\n\n# +\ndef initial_condition(x, a=5):\n return np.exp(-a * (x[0] ** 2 + x[1] ** 2))\n\n\nu_n = fem.Function(V)\nu_n.name = \"u_n\"\nu_n.interpolate(initial_condition)\n\n# Create boundary condition\nfdim = domain.topology.dim - 1\nboundary_facets = mesh.locate_entities_boundary(\n domain, fdim, lambda x: np.full(x.shape[1], True, dtype=bool)\n)\nbc = fem.dirichletbc(\n PETSc.ScalarType(0), fem.locate_dofs_topological(V, fdim, boundary_facets), V\n)\n# -\n\n# ## Time-dependent output\n# To visualize the solution in an external program such as Paraview,\n# we create a an {py:class}`XDMFFile` which we can store multiple solutions in.\n# The main advantage with an XDMFFile is that we only need to store the mesh once and that we can\n# append multiple solutions to the same grid, reducing the storage space.\n# The first argument to the XDMFFile is the {py:class}`communicator`\n# which should be used to write data to file in parallel.\n# As we would like one output, independent of the number of processors,\n# we use the {py:data}`COMM_WORLD`.\n# The second argument is the file name of the output file,\n# while the third argument is the state of the file,\n# this could be read (`\"r\"`), write (`\"w\"`) or append (`\"a\"`).\n\nxdmf = io.XDMFFile(domain.comm, \"diffusion.xdmf\", \"w\")\nxdmf.write_mesh(domain)\n\n# Define solution variable, and interpolate initial solution for visualization in Paraview\n\nuh = fem.Function(V)\nuh.name = \"uh\"\nuh.interpolate(initial_condition)\nxdmf.write_function(uh, t)\n\n# ## Variational problem and solver\n# As in the previous example, we prepare objects for time dependent problems,\n# such that we do not have to recreate data-structures.\n\nu, v = ufl.TrialFunction(V), ufl.TestFunction(V)\nf = fem.Constant(domain, PETSc.ScalarType(0))\na = u * v * ufl.dx + dt * ufl.dot(ufl.grad(u), ufl.grad(v)) * ufl.dx\nL = (u_n + dt * f) * v * ufl.dx\n\n# ## Preparing linear algebra structures for time dependent problems\n# We note that even if `u_n` is time dependent, we will reuse the same function for\n# `f` and `u_n` at every time step.\n# We therefore call {py:func}`dolfinx.fem.form` to generate assembly kernels for\n# the matrix and vector. This function creates a {py:class}`dolfinx.fem.Form`.\n\nbilinear_form = fem.form(a)\nlinear_form = fem.form(L)\n\n# We observe that the left hand side of the system, the matrix {py:class}`A`\n# does not change from one time step to another, thus we only need to assemble it once.\n# We call {py:meth}`A.assemble()` to finalize the assembly process,\n# which means communicating local contributions from each process to other processes that shares the\n# same degrees of freedom.\n# However, the right hand side, which is dependent on the previous time step `u_n`,\n# we have to assemble it at every time step.\n# Therefore, we only create a {py:class}`vector` `b` based on `L`,\n# which we will reuse at every time step.\n\nA = assemble_matrix(bilinear_form, bcs=[bc])\nA.assemble()\nb = create_vector(fem.extract_function_spaces(linear_form))\n\n# ## Using petsc4py to create a linear solver\n# As we have already assembled `a` into the matrix `A`, we can no longer\n# use the {py:class}`dolfinx.fem.petsc.LinearProblem` class to solve the problem.\n# Therefore, we create a {py:class}`krylov subspace solver` using PETSc,\n# assign the matrix `A` to the solver, and choose the solution strategy.\n\nsolver = PETSc.KSP().create(domain.comm)\nsolver.setOperators(A)\nsolver.setType(PETSc.KSP.Type.PREONLY)\nsolver.getPC().setType(PETSc.PC.Type.LU)\n\n# ## Visualization of time dependent problem using pyvista\n# We use the DOLFINx plotting functionality, which is based on {py:mod}`pyvista`\n# to plot the solution at every $15$th time step.\n# We would also like to visualize a colorbar reflecting the minimal and maximum\n# value of $u$ at each time step.\n\n# +\ngrid = pyvista.UnstructuredGrid(*plot.vtk_mesh(V))\n\nplotter = pyvista.Plotter()\nplotter.open_gif(\"u_time.gif\", fps=10)\n\ngrid.point_data[\"uh\"] = uh.x.array\nwarped = grid.warp_by_scalar(\"uh\", factor=1)\n\nviridis = mpl.colormaps.get_cmap(\"viridis\").resampled(25)\nsargs = dict(\n title_font_size=25,\n label_font_size=20,\n fmt=\"%.2e\",\n color=\"black\",\n position_x=0.1,\n position_y=0.8,\n width=0.8,\n height=0.1,\n)\n\nrenderer = plotter.add_mesh(\n warped,\n show_edges=True,\n lighting=False,\n cmap=viridis,\n scalar_bar_args=sargs,\n clim=[0, max(uh.x.array)],\n)\n# -\n\n# (time-dep-assembly)=\n# ## Updating the solution and right hand side per time step\n# To be able to solve the variation problem at each time step,\n# we have to assemble the right hand side and apply the boundary condition before calling\n# {py:meth}`solver.solve(b, uh.x.petsc_vec)`.\n# We start by resetting the values in `b` as we are reusing the vector at every time step.\n# The next step is to assemble the vector calling\n# {py:func}`dolfinx.fem.petsc.assemble_vector(b, L)`,\n# which means that we are assembling the linear form `L(v)` into the vector `b`.\n# Note that we do not supply the boundary conditions for assembly, as opposed to the left hand side.\n# This is because we want to use {py:func}`lifting` to apply the boundary condition,\n# which preserves symmetry of the matrix $A$ in the bilinear form $a(u,v)=a(v,u)$ without Dirichlet boundary conditions.\n# Once we have performed the lifting, we accumulate values from degrees of freedom that are shared between processes\n# using {py:meth}`b.ghostUpdate()`.\n# Finally, we apply the boundary condition to the fixed degrees of freedom with\n# {py:func}`dolfinx.fem.petsc.set_bc`.\n# Next, we can {py:meth}`solve` the linear system and\n# {py:meth}`update` degrees of freedom shared between processors.\n# Finally, before moving to the next time step, we update the solution at the previous time step\n# to the solution at this time step.\n\nfor i in range(num_steps):\n t += dt\n\n # Update the right hand side reusing the initial vector\n with b.localForm() as loc_b:\n loc_b.set(0)\n assemble_vector(b, linear_form)\n\n # Apply Dirichlet boundary condition to the vector\n apply_lifting(b, [bilinear_form], [[bc]])\n b.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)\n set_bc(b, [bc])\n\n # Solve linear problem\n solver.solve(b, uh.x.petsc_vec)\n uh.x.scatter_forward()\n\n # Update solution at previous time step (u_n)\n u_n.x.array[:] = uh.x.array\n\n # Write solution to file\n xdmf.write_function(uh, t)\n # Update plot\n new_warped = grid.warp_by_scalar(\"uh\", factor=1)\n warped.points[:, :] = new_warped.points\n warped.point_data[\"uh\"][:] = uh.x.array\n plotter.write_frame()\nplotter.close()\nxdmf.close()\n\n# We {py:meth}`destroy` the PETSc objects to avoid memory leaks.\n\nA.destroy()\nb.destroy()\nsolver.destroy()\n\n# \"gif\"\n\n# ## Animation with Paraview\n# We can also use Paraview to create an animation. We open the file in paraview with `File->Open`,\n# and then press `Apply` in the properties panel.\n#\n# Then, we add a time-annotation to the figure, pressing: `Sources->Alphabetical->Annotate Time`\n# and `Apply` in the properties panel.\n# It Is also a good idea to select an output resolution, by pressing `View->Preview->1280 x 720 (HD)`.\n#\n# Then finally, click `File->Save Animation`, and save the animation to the desired format,\n# such as `avi`, `ogv` or a sequence of `png`s. Make sure to set the frame rate to something sensible,\n# in the range of $5-10$ frames per second.\n" }, { "path": "chapter2/elasticity_scaling.md", "content": "# Scaling\nAuthors: Anders Logg and Hans Petter Langtangen\n\nIt is often advantageous to scale a problem as it reduces the need for setting physical parameters, and one obtains dimensionless numbers that reflect the competition of parameters and physical effects. We develop the code for the original model with dimensions, and run the scaled problem by tweaking parameters appropriately. Scaling reduces the number of active parameters from $6$ to $2$ for the present application.\n\nIn Navier's equation for $u$, arising from insertion of $\\sigma(u)$ in [](elasticity-PDE),\n\n```{math}\n -(\\lambda + \\mu)\\nabla (\\nabla \\cdot u) - \\mu \\nabla^2 u = f,\n```\nwe insert coordinates made dimensionless by $L$, and $\\bar{u}=\\frac{u}{U}$, which results in the dimensionless governing equations\n```{math}\n - \\beta \\bar{\\nabla}(\\bar{\\nabla}\\cdot \\bar{u})-\\bar{\\nabla}^2\\bar{u} = \\bar{f}, \\qquad \\bar{f} = (0,0,\\gamma)\n```\nwhere $\\beta = 1+\\frac{\\lambda}{\\mu}$ is a dimensionless elasticity parameter and where\n```{math}\n \\gamma=\\frac{\\rho g L^2}{\\mu U}\n```\nis a dimensionless variable reflecting the ratio of the load $\\rho g$ and the shear stress term $\\mu \\nabla^2u \\sim \\mu \\frac{U}{L^2}$ in the PDE.\n\nOne option for the scaling is to choose $U$ such that $\\gamma$ is of unit size ($U=\\frac{\\rho g L^2}{\\mu}$). However, in elasticity, this leads to displacements of the size of the geometry. This can be achieved by choosing $U$ equal to the maximum deflection of a clamped beam, for which there actually exists a formula: $U=\\frac{3}{2} \\rho g L^2\\frac{\\delta^2}{E}$ where $\\delta=\\frac{L}{W}$ is a parameter reflecting how slender the beam is, and $E$ is the modulus of elasticity. Thus the dimensionless parameter $\\delta$ is very important in the problem (as expected $\\delta\\gg 1$ is what gives beam theory!). Taking $E$ to be of the same order as $\\mu$, which in this case and for many materials, we realize that $\\gamma \\sim \\delta^{-2}$ is an appropriate choice. Experimenting with the code to find a displacement that \"looks right\" in the plots of the deformed geometry, points to $\\gamma=0.4\\delta^{-2}$ as our final choice of $\\gamma$.\n\nThe simulation code implements the problem with dimensions and physical parameters $\\lambda, \\mu, \\rho, g, L$ and $W$. However, we can easily reuse this code for a scaled problem: Just set $\\mu=\\rho=L=1$, $W$ as $W/L(\\delta^{-1})$, $g=\\gamma$ and $\\lambda=\\beta$.\n" }, { "path": "chapter2/heat_code.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# A known analytical solution\\n\",\n \"Author: Jørgen S. Dokken\\n\",\n \"\\n\",\n \"Just as for the [Poisson problem](./../chapter1/fundamentals_code), we construct a test problem\\n\",\n \"which makes it easy to determine if the calculations are correct.\\n\",\n \"\\n\",\n \"Since we know that our first-order time-stepping scheme is exact for linear functions,\\n\",\n \"we create a problem which has linear variation in time.\\n\",\n \"We combine this with a quadratic variation in space.\\n\",\n \"Therefore, we choose the analytical solution to be\\n\",\n \"\\n\",\n \"$$\\n\",\n \"\\\\begin{align}\\n\",\n \"u = 1 + x^2+\\\\alpha y^2 + \\\\beta t\\n\",\n \"\\\\end{align}\\n\",\n \"$$\\n\",\n \"\\n\",\n \"which yields a function whose computed values at the degrees of freedom will be exact,\\n\",\n \"regardless of the mesh size and $\\\\Delta t$ as long as the mesh is uniformly partitioned.\\n\",\n \"## Method of manufactured solutions\\n\",\n \"By inserting this into our original PDE, we find that the right hand side $f=\\\\beta-2-2\\\\alpha$.\\n\",\n \"The boundary value $u_d(x,y,t)=1+x^2+\\\\alpha y^2 + \\\\beta t$ and the initial value $u_0(x,y)=1+x^2+\\\\alpha y^2$.\\n\",\n \"\\n\",\n \"We start by defining the temporal discretization parameters, along with the parameters for $\\\\alpha$ and $\\\\beta$.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"from petsc4py import PETSc\\n\",\n \"from mpi4py import MPI\\n\",\n \"import ufl\\n\",\n \"from dolfinx import mesh, fem\\n\",\n \"from dolfinx.fem.petsc import (\\n\",\n \" assemble_matrix,\\n\",\n \" assemble_vector,\\n\",\n \" apply_lifting,\\n\",\n \" create_vector,\\n\",\n \" set_bc,\\n\",\n \")\\n\",\n \"import numpy\\n\",\n \"import numpy.typing as npt\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"2\",\n \"metadata\": {},\n \"source\": [\n \"We define the problem specific parameters\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"3\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"t = 0.0 # Start time\\n\",\n \"T = 2.0 # End time\\n\",\n \"num_steps = 20 # Number of time steps\\n\",\n \"dt = (T - t) / num_steps # Time step size\\n\",\n \"alpha = 3.0\\n\",\n \"beta = 1.2\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"As for the [previous example](./diffusion_code), we define the mesh and appropriate function spaces\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"nx, ny = 5, 5\\n\",\n \"domain = mesh.create_unit_square(MPI.COMM_WORLD, nx, ny, mesh.CellType.triangle)\\n\",\n \"V = fem.functionspace(domain, (\\\"Lagrange\\\", 1))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"## Defining the exact solution\\n\",\n \"As in the [membrane problem](../chapter1/membrane_code), we create a Python-class to\\n\",\n \"represent the exact solution\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"class ExactSolution:\\n\",\n \" def __init__(self, alpha: float, beta: float, t: float):\\n\",\n \" self.alpha = alpha\\n\",\n \" self.beta = beta\\n\",\n \" self.t = t\\n\",\n \"\\n\",\n \" def __call__(self, x: npt.NDArray[numpy.floating]) -> npt.NDArray[numpy.floating]:\\n\",\n \" return 1 + x[0] ** 2 + self.alpha * x[1] ** 2 + self.beta * self.t\\n\",\n \"\\n\",\n \"\\n\",\n \"u_exact = ExactSolution(alpha, beta, t)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Defining the boundary condition\\n\",\n \"As in the previous chapters, we define a Dirichlet boundary condition over the whole boundary\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"u_D = fem.Function(V)\\n\",\n \"u_D.interpolate(u_exact)\\n\",\n \"tdim = domain.topology.dim\\n\",\n \"fdim = tdim - 1\\n\",\n \"domain.topology.create_connectivity(fdim, tdim)\\n\",\n \"boundary_facets = mesh.exterior_facet_indices(domain.topology)\\n\",\n \"bc = fem.dirichletbc(u_D, fem.locate_dofs_topological(V, fdim, boundary_facets))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Defining the variational formualation\\n\",\n \"As we have set $t=0$ in `u_exact`, we can reuse this variable to obtain $u_n$ for the first time step.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"u_n = fem.Function(V)\\n\",\n \"u_n.interpolate(u_exact)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"As $f$ is a constant independent of $t$, we can define it as a {py:class}`constant`.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"f = fem.Constant(domain, beta - 2 - 2 * alpha)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We can now create our variational formulation, with the bilinear form `a` and linear form `L`.\\n\",\n \"Note that we write the variational form on residual form, and use {py:func}`ufl.lhs` and {py:func}`ufl.rhs`\\n\",\n \"to extract the bilinear and linear forms.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"u, v = ufl.TrialFunction(V), ufl.TestFunction(V)\\n\",\n \"F = (\\n\",\n \" u * v * ufl.dx\\n\",\n \" + dt * ufl.dot(ufl.grad(u), ufl.grad(v)) * ufl.dx\\n\",\n \" - (u_n + dt * f) * v * ufl.dx\\n\",\n \")\\n\",\n \"a = fem.form(ufl.lhs(F))\\n\",\n \"L = fem.form(ufl.rhs(F))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Create the matrix and vector for the linear problem\\n\",\n \"To ensure that we are solving the variational problem efficiently,\\n\",\n \"we will create several structures which can reuse data, such as matrix sparsity patterns.\\n\",\n \"See {ref}`time-dep-assembly` for more details.\\n\",\n \"Especially note as the bilinear form `a` is independent of time, we only need to assemble the matrix once.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"A = assemble_matrix(a, bcs=[bc])\\n\",\n \"A.assemble()\\n\",\n \"b = create_vector(fem.extract_function_spaces(L))\\n\",\n \"uh = fem.Function(V)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Define a linear variational solver\\n\",\n \"We will use {py:mod}`petsc4py.PETSc` to solve the resulting linear algebra problem.\\n\",\n \"We can choose either a direct or iterative solver.\\n\",\n \"For the given problem we choose a direct solver, as we can then reuse the\\n\",\n \"{py:attr}`LU` factorization at every time step.\\n\",\n \"Once can choose between different factorization backends by calling\\n\",\n \"{py:meth}`pc.setFactorSolverType`\\n\",\n \"to for instance {py:attr}`mumps`,\\n\",\n \"{py:attr}`petsc` or\\n\",\n \"{py:attr}`superlu_dist`.\\n\",\n \"See {py:class}`petsc4py.PETSc.Mat.SolverType` for a full list of available direct solvers.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"solver = PETSc.KSP().create(domain.comm)\\n\",\n \"solver.setOperators(A)\\n\",\n \"solver.setType(PETSc.KSP.Type.PREONLY)\\n\",\n \"pc = solver.getPC()\\n\",\n \"pc.setType(PETSc.PC.Type.LU)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Solving the time-dependent problem\\n\",\n \"With these structures in place, we create our time-stepping loop.\\n\",\n \"In this loop, we first update the Dirichlet boundary condition by interpolating the updated\\n\",\n \"expression `u_exact` into `V`. The next step is to re-assemble the vector `b`, with the update `u_n`.\\n\",\n \"Then, we need to apply the boundary condition to this vector. We do this by using the lifting operation,\\n\",\n \"which applies the boundary condition such that symmetry of the matrix is preserved.\\n\",\n \"Then we solve the problem using PETSc and update `u_n` with the data from `uh`.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"for n in range(num_steps):\\n\",\n \" # Update Diriclet boundary condition\\n\",\n \" u_exact.t += dt\\n\",\n \" u_D.interpolate(u_exact)\\n\",\n \"\\n\",\n \" # Update the right hand side reusing the initial vector\\n\",\n \" with b.localForm() as loc_b:\\n\",\n \" loc_b.set(0)\\n\",\n \" assemble_vector(b, L)\\n\",\n \"\\n\",\n \" # Apply Dirichlet boundary condition to the vector\\n\",\n \" apply_lifting(b, [a], [[bc]])\\n\",\n \" b.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)\\n\",\n \" set_bc(b, [bc])\\n\",\n \"\\n\",\n \" # Solve linear problem\\n\",\n \" solver.solve(b, uh.x.petsc_vec)\\n\",\n \" uh.x.scatter_forward()\\n\",\n \"\\n\",\n \" # Update solution at previous time step (u_n)\\n\",\n \" u_n.x.array[:] = uh.x.array\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"22\",\n \"metadata\": {},\n \"source\": [\n \"We free the PETSc object to avoid memory leaks.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"23\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"A.destroy()\\n\",\n \"b.destroy()\\n\",\n \"solver.destroy()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Verifying the numerical solution\\n\",\n \"As in the {ref}`error-norm`, we compute the L2-error and the error at the mesh vertices for the last time step.\\n\",\n \"to verify our implementation.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"tags\": []\n },\n \"outputs\": [],\n \"source\": [\n \"# Compute L2 error and error at nodes\\n\",\n \"V_ex = fem.functionspace(domain, (\\\"Lagrange\\\", 2))\\n\",\n \"u_ex = fem.Function(V_ex)\\n\",\n \"u_ex.interpolate(u_exact)\\n\",\n \"error_L2 = numpy.sqrt(\\n\",\n \" domain.comm.allreduce(\\n\",\n \" fem.assemble_scalar(fem.form((uh - u_ex) ** 2 * ufl.dx)), op=MPI.SUM\\n\",\n \" )\\n\",\n \")\\n\",\n \"if domain.comm.rank == 0:\\n\",\n \" print(f\\\"L2-error: {error_L2:.2e}\\\")\\n\",\n \"\\n\",\n \"# Compute values at mesh vertices\\n\",\n \"error_max = domain.comm.allreduce(\\n\",\n \" numpy.max(numpy.abs(uh.x.array - u_D.x.array)), op=MPI.MAX\\n\",\n \")\\n\",\n \"if domain.comm.rank == 0:\\n\",\n \" print(f\\\"Error_max: {error_max:.2e}\\\")\"\n ]\n }\n ],\n \"metadata\": {\n \"jupytext\": {\n \"formats\": \"ipynb,py:light\"\n },\n \"kernelspec\": {\n \"display_name\": \"Python 3 (ipykernel)\",\n \"language\": \"python\",\n \"name\": \"python3\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 3\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython3\",\n \"version\": \"3.10.12\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 4\n}\n" }, { "path": "chapter2/heat_code.py", "content": "# ---\n# jupyter:\n# jupytext:\n# formats: ipynb,py:light\n# text_representation:\n# extension: .py\n# format_name: light\n# format_version: '1.5'\n# jupytext_version: 1.18.1\n# kernelspec:\n# display_name: Python 3 (ipykernel)\n# language: python\n# name: python3\n# ---\n\n# # A known analytical solution\n# Author: Jørgen S. Dokken\n#\n# Just as for the [Poisson problem](./../chapter1/fundamentals_code), we construct a test problem\n# which makes it easy to determine if the calculations are correct.\n#\n# Since we know that our first-order time-stepping scheme is exact for linear functions,\n# we create a problem which has linear variation in time.\n# We combine this with a quadratic variation in space.\n# Therefore, we choose the analytical solution to be\n#\n# $$\n# \\begin{align}\n# u = 1 + x^2+\\alpha y^2 + \\beta t\n# \\end{align}\n# $$\n#\n# which yields a function whose computed values at the degrees of freedom will be exact,\n# regardless of the mesh size and $\\Delta t$ as long as the mesh is uniformly partitioned.\n# ## Method of manufactured solutions\n# By inserting this into our original PDE, we find that the right hand side $f=\\beta-2-2\\alpha$.\n# The boundary value $u_d(x,y,t)=1+x^2+\\alpha y^2 + \\beta t$ and the initial value $u_0(x,y)=1+x^2+\\alpha y^2$.\n#\n# We start by defining the temporal discretization parameters, along with the parameters for $\\alpha$ and $\\beta$.\n\nfrom petsc4py import PETSc\nfrom mpi4py import MPI\nimport ufl\nfrom dolfinx import mesh, fem\nfrom dolfinx.fem.petsc import (\n assemble_matrix,\n assemble_vector,\n apply_lifting,\n create_vector,\n set_bc,\n)\nimport numpy\nimport numpy.typing as npt\n\n# We define the problem specific parameters\n\nt = 0.0 # Start time\nT = 2.0 # End time\nnum_steps = 20 # Number of time steps\ndt = (T - t) / num_steps # Time step size\nalpha = 3.0\nbeta = 1.2\n\n# As for the [previous example](./diffusion_code), we define the mesh and appropriate function spaces\n\nnx, ny = 5, 5\ndomain = mesh.create_unit_square(MPI.COMM_WORLD, nx, ny, mesh.CellType.triangle)\nV = fem.functionspace(domain, (\"Lagrange\", 1))\n\n# ## Defining the exact solution\n# As in the [membrane problem](../chapter1/membrane_code), we create a Python-class to\n# represent the exact solution\n\n\n# +\nclass ExactSolution:\n def __init__(self, alpha: float, beta: float, t: float):\n self.alpha = alpha\n self.beta = beta\n self.t = t\n\n def __call__(self, x: npt.NDArray[numpy.floating]) -> npt.NDArray[numpy.floating]:\n return 1 + x[0] ** 2 + self.alpha * x[1] ** 2 + self.beta * self.t\n\n\nu_exact = ExactSolution(alpha, beta, t)\n# -\n\n# ## Defining the boundary condition\n# As in the previous chapters, we define a Dirichlet boundary condition over the whole boundary\n\nu_D = fem.Function(V)\nu_D.interpolate(u_exact)\ntdim = domain.topology.dim\nfdim = tdim - 1\ndomain.topology.create_connectivity(fdim, tdim)\nboundary_facets = mesh.exterior_facet_indices(domain.topology)\nbc = fem.dirichletbc(u_D, fem.locate_dofs_topological(V, fdim, boundary_facets))\n\n# ## Defining the variational formualation\n# As we have set $t=0$ in `u_exact`, we can reuse this variable to obtain $u_n$ for the first time step.\n\nu_n = fem.Function(V)\nu_n.interpolate(u_exact)\n\n# As $f$ is a constant independent of $t$, we can define it as a {py:class}`constant`.\n\nf = fem.Constant(domain, beta - 2 - 2 * alpha)\n\n# We can now create our variational formulation, with the bilinear form `a` and linear form `L`.\n# Note that we write the variational form on residual form, and use {py:func}`ufl.lhs` and {py:func}`ufl.rhs`\n# to extract the bilinear and linear forms.\n\nu, v = ufl.TrialFunction(V), ufl.TestFunction(V)\nF = (\n u * v * ufl.dx\n + dt * ufl.dot(ufl.grad(u), ufl.grad(v)) * ufl.dx\n - (u_n + dt * f) * v * ufl.dx\n)\na = fem.form(ufl.lhs(F))\nL = fem.form(ufl.rhs(F))\n\n# ## Create the matrix and vector for the linear problem\n# To ensure that we are solving the variational problem efficiently,\n# we will create several structures which can reuse data, such as matrix sparsity patterns.\n# See {ref}`time-dep-assembly` for more details.\n# Especially note as the bilinear form `a` is independent of time, we only need to assemble the matrix once.\n\nA = assemble_matrix(a, bcs=[bc])\nA.assemble()\nb = create_vector(fem.extract_function_spaces(L))\nuh = fem.Function(V)\n\n# ## Define a linear variational solver\n# We will use {py:mod}`petsc4py.PETSc` to solve the resulting linear algebra problem.\n# We can choose either a direct or iterative solver.\n# For the given problem we choose a direct solver, as we can then reuse the\n# {py:attr}`LU` factorization at every time step.\n# Once can choose between different factorization backends by calling\n# {py:meth}`pc.setFactorSolverType`\n# to for instance {py:attr}`mumps`,\n# {py:attr}`petsc` or\n# {py:attr}`superlu_dist`.\n# See {py:class}`petsc4py.PETSc.Mat.SolverType` for a full list of available direct solvers.\n\nsolver = PETSc.KSP().create(domain.comm)\nsolver.setOperators(A)\nsolver.setType(PETSc.KSP.Type.PREONLY)\npc = solver.getPC()\npc.setType(PETSc.PC.Type.LU)\n\n# ## Solving the time-dependent problem\n# With these structures in place, we create our time-stepping loop.\n# In this loop, we first update the Dirichlet boundary condition by interpolating the updated\n# expression `u_exact` into `V`. The next step is to re-assemble the vector `b`, with the update `u_n`.\n# Then, we need to apply the boundary condition to this vector. We do this by using the lifting operation,\n# which applies the boundary condition such that symmetry of the matrix is preserved.\n# Then we solve the problem using PETSc and update `u_n` with the data from `uh`.\n\nfor n in range(num_steps):\n # Update Diriclet boundary condition\n u_exact.t += dt\n u_D.interpolate(u_exact)\n\n # Update the right hand side reusing the initial vector\n with b.localForm() as loc_b:\n loc_b.set(0)\n assemble_vector(b, L)\n\n # Apply Dirichlet boundary condition to the vector\n apply_lifting(b, [a], [[bc]])\n b.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)\n set_bc(b, [bc])\n\n # Solve linear problem\n solver.solve(b, uh.x.petsc_vec)\n uh.x.scatter_forward()\n\n # Update solution at previous time step (u_n)\n u_n.x.array[:] = uh.x.array\n\n# We free the PETSc object to avoid memory leaks.\n\nA.destroy()\nb.destroy()\nsolver.destroy()\n\n# ## Verifying the numerical solution\n# As in the {ref}`error-norm`, we compute the L2-error and the error at the mesh vertices for the last time step.\n# to verify our implementation.\n\n# +\n# Compute L2 error and error at nodes\nV_ex = fem.functionspace(domain, (\"Lagrange\", 2))\nu_ex = fem.Function(V_ex)\nu_ex.interpolate(u_exact)\nerror_L2 = numpy.sqrt(\n domain.comm.allreduce(\n fem.assemble_scalar(fem.form((uh - u_ex) ** 2 * ufl.dx)), op=MPI.SUM\n )\n)\nif domain.comm.rank == 0:\n print(f\"L2-error: {error_L2:.2e}\")\n\n# Compute values at mesh vertices\nerror_max = domain.comm.allreduce(\n numpy.max(numpy.abs(uh.x.array - u_D.x.array)), op=MPI.MAX\n)\nif domain.comm.rank == 0:\n print(f\"Error_max: {error_max:.2e}\")\n" }, { "path": "chapter2/heat_equation.md", "content": "# The heat equation\nAuthors: Anders Logg and Hans Petter Langtangen\n\nMinor modifications by: Jørgen S. Dokken\n\nAs a first extension of the Poisson problem from the previous chapter, we consider the time-dependent heat equation, or the time-dependent diffusion equation. This is the natural extension of the Poisson equation describing the stationary distribution of heat in a body to a time-dependent problem. We will see that by discretizing time into small time intervals and applying standard time-stepping methods, we can solve the heat equation by solving a sequence of variational problems, much like the one we encountered for the Poisson equation.\n\n## The PDE problem\nThe model problem for the time-dependent PDE reads\n\\begin{align}\n \\frac{\\partial u}{\\partial t}&=\\nabla^2 u + f && \\text{in } \\Omega \\times (0, T],\\\\\n u &= u_D && \\text{on } \\partial\\Omega \\times (0,T],\\\\\n u &= u_0 && \\text{at } t=0.\n\\end{align}\n\nHere $u$ varies with space and time, e.g. $u=u(x,y,t)$ if the spatial domain $\\Omega$ is two-dimensional. The source function $f$ and the boundary values $u_D$ may also vary with space and time. The initial condition $u_0$ is a function of space only.\n\n## The variational formulation\nA straightforward approach to solving time-dependent PDEs by the finite element method is to first discretize the time derivative by a finite difference approximation, which yields a sequence of stationary problems, and then turn each stationary problem into a variational formulation. \nWe will let the superscript $n$ denote a quantity at time $t_n$, where $n$ is an integer counting time levels. For example, $u^n$ means $u$ at time level $n$. The first step of a finite difference discretization in time consists of sampling the PDE at some time level, for instance $t_{n+1}$\n\\begin{align}\n \\left(\\frac{\\partial u }{\\partial t}\\right)^{n+1}= \\nabla^2 u^{n+1}+ f^{n+1}.\n\\end{align}\nThe time-derivative can be approximated by a difference quotient. For simplicity and stability reasons, we choose a simple backward difference:\n\\begin{align}\n \\left(\\frac{\\partial u }{\\partial t}\\right)^{n+1}\\approx \\frac{u^{n+1}-u^n}{\\Delta t},\n\\end{align}\nwhere $\\Delta t$ is the time discretization parameter. Inserting the latter expression into our equation at time step $n+1$ yields\n\\begin{align}\n \\frac{u^{n+1}-u^n}{\\Delta t}= \\nabla^2 u^{n+1}+ f^{n+1}.\n\\end{align}\nThis is our time-discrete version of the heat equation. It is called a *backward Euler* or a *implicit Euler* discretization.\n\nWe reorder the equation such that the left-hand side contains the terms with only the unknown $u^{n+1}$ and right-hand side contains only computed terms. The resulting equation is a sequence of stationary problems for $u^{n+1}$, assuming $u^{n}$ is known from the previous time step:\n\\begin{align}\n u^0&=u_0 &&\\\\\n u^{n+1}-\\Delta t \\nabla^2 u^{n+1}&= u^{n} + \\Delta t f^{n+1}, && n = 0,1,2,\\dots\n\\end{align}\nGiven $u_0$, we can solve for $u^0, u^1, u^2$ and so on.\n\nWe then in turn use the finite element method. This means that we have to turn the equation into its weak formulation. We multiply by the test-function of $v\\in \\hat{V}$ and integrate second-order derivatives by parts. We now introduce the symbol $u$ for $u^{n+1}$ and we write the resulting weak formulation as\n\n\\begin{align}\n a(u,v)&=L_{n+1}(v),\n\\end{align}\nwhere \n\\begin{align}\n a(u,v)&=\\int_{\\Omega}(uv + \\Delta t \\nabla u \\cdot \\nabla v )~\\mathrm{d} x\\\\\n L_{n+1}(v)&=\\int_{\\Omega} (u^n+\\Delta t f^{n+1}) \\cdot v~\\mathrm{d} x.\n\\end{align}\n\n\n## Projection or interpolation of the initial condition\nIn addition to the variational problem to be solved in each time step, we also need to approximate the initial condition. This equation can also be turned into a variational problem\n\\begin{align}\n a_0(u,v)&=L_0(V),\n\\end{align}\nwith \n\\begin{align}\n a_0(u,v)&=\\int_{\\Omega}uv~\\mathrm{d} x,\\\\\n L_0(v)&=\\int_{\\Omega}u_0v~\\mathrm{d} x.\n\\end{align}\nWhen solving this variational problem $u^0$ becomes the $L^2$-projection of the given initial value $u_0$ into the finite element space. \n\nThe alternative is to construct $u^0$ by just interpolating the initial value $u_0$. We covered how to use interpolation in DOLFINx in the {doc}`membrane chapter <../chapter1/membrane_code>`.\n\nWe can use DOLFINx to either project or interpolate the initial condition. The most common choice is to use a projection, which computes an approximation to $u_0$. However, in some applications where we want to verify the code by reproducing exact solutions, one must use interpolation. In this chapter, we will use such a problem.\n" }, { "path": "chapter2/helmholtz.md", "content": "# The Helmholtz equation\nAuthor: Antonio Baiano Svizzero \n \nThe study of computational acoustics is fundamental in fields such as noise, vibration, and harshness (NVH), noise control, and acoustic design. In this chapter, we focus on the theoretical foundations of the Helmholtz equation - valid for noise problems with harmonic time dependency - and its implementation in FEniCSx to compute the sound pressure for any acoustic system.\n\n## The PDE problem\nThe acoustic Helmholtz equation in its general form reads\n\n$$\n\\begin{align}\n\\nabla^2 p + k^2 p = -j \\omega \\rho_0 q \\qquad\\text{in } \\Omega,\n\\end{align}\n$$\n\nwhere $k$ is the acoustic wavenumber, $\\omega$ is the angular frequency, $j$ the imaginary unit and $q$ is the volume velocity ($m^3/s$) of a generic source field.\nIn case of a monopole source, we can write $q=Q \\delta(x_s,y_s,z_s)$, where $\\delta(x_s,y_s,z_s)$ is the 3D Dirac Delta centered at the monopole location. \n\nThis equation is coupled with the following boundary conditions: \n\n- Dirichlet BC: \n\n $$\n \\begin{align}\n p = \\bar{p} \\qquad \\text{on } \\partial\\Omega_p,\n \\end{align}\n $$\n\n- Neumann BC: \n\n $$\n \\begin{align}\n \\frac{\\partial p}{\\partial n} = - j \\omega \\rho_0 \\bar{v}_n\\qquad \\text{on } \\partial\\Omega_v,\n \\end{align}\n $$\n\n- Robin BC: \n\n $$\n \\begin{align}\n \\frac{\\partial p}{\\partial n} = - \\frac{j \\omega \\rho_0 }{\\bar{Z}} p \\qquad \\text{on } \\partial\\Omega_Z,\n \\end{align}\n $$\n\nwhere we prescribe, respectively, an acoustic pressure $\\bar{p}$ on the boundary $\\partial\\Omega_p$,\na sound particle velocity $\\bar{v}_n$ on the boundary $\\partial\\Omega_v$ and\nan acoustic impedance $\\bar{Z}$ on the boundary $\\partial\\Omega_Z$ where $n$ is the outward normal.\nIn general, any BC can also be frequency dependant, as it happens in real-world applications.\n\n## The variational formulation\nNow we have to turn the equation in its weak formulation.\nThe first step is to multiplicate the equation by a *test function* $v\\in \\hat V$,\nwhere $\\hat V$ is the *test function space*, after which we integrate over the whole domain, $\\Omega$:\n\n$$\n\\begin{align}\n\\int_{\\Omega}\\left(\\nabla^2 p + k^2 p \\right) \\bar v ~\\mathrm{d}x = -\\int_{\\Omega} j \\omega \\rho_0 q \\bar v ~\\mathrm{d}x.\n\\end{align}\n$$\n\nHere, the unknown function $p$ is referred to as *trial function* and the $\\bar{\\cdot}$ is the complex conjugate operator.\n\nIn order to keep the order of derivatives as low as possible, we use integration by parts on the Laplacian term: \n\n$$\n\\begin{align}\n\\int_{\\Omega}(\\nabla^2 p) \\bar v ~\\mathrm{d}x =\n-\\int_{\\Omega} \\nabla p \\cdot \\nabla \\bar v ~\\mathrm{d}x\n+ \\int_{\\partial \\Omega} \\frac{\\partial p}{\\partial n} \\bar v ~\\mathrm{d}s.\n\\end{align}\n$$\n\nSubstituting in the original version and rearranging we get: \n\n$$\n\\begin{align}\n\\int_{\\Omega} \\nabla p \\cdot \\nabla \\bar v ~\\mathrm{d}x\n- k^2 \\int_{\\Omega} p \\bar v ~\\mathrm{d} x = \\int_{\\Omega} j \\omega \\rho_0 q \\bar v ~\\mathrm{d}x\n+ \\int_{\\partial \\Omega} \\frac{\\partial p}{\\partial n} \\bar v ~\\mathrm{d}s.\n\\end{align}\n$$\n\nSince we are dealing with complex values, the inner product in the first equation is *sesquilinear*,\nmeaning it is linear in one argument and conjugate-linear in the other,\nas explained in [The Poisson problem with complex numbers](../chapter1/complex_mode).\n\nThe last term can be written using the Neumann and Robin BCs, that is: \n\n$$\n\\begin{align}\n\\int_{\\partial \\Omega} \\frac{\\partial p}{\\partial n} \\bar v ~\\mathrm{d}s =\n-\\int_{\\partial \\Omega_v} j \\omega \\rho_0 \\bar{v}_n \\bar v ~\\mathrm{d}s\n- \\int_{\\partial \\Omega_Z} \\frac{j \\omega \\rho_0}{\\bar{Z}} p \\bar v ~\\mathrm{d}s.\n\\end{align}\n$$\n\nSubstituting, rearranging and taking out of integrals the terms with $j$ and $\\omega$ we get the variational formulation of the Helmholtz.\nFind $u \\in V$ such that: \n\n$$\n\\begin{align}\n\\int_{\\Omega} \\nabla p \\cdot \\nabla \\bar v ~\\mathrm{d}x\n+ \\frac{j \\omega }{\\bar{Z}} \\int_{\\partial \\Omega_Z} \\rho_0 p \\bar v ~\\mathrm{d}s \n- k^2 \\int_{\\Omega} p \\bar v ~\\mathrm{d}x\n= j \\omega \\int_{\\Omega} \\rho_0 q \\bar v ~\\mathrm{d}x\n-j \\omega\\int_{\\partial \\Omega_v} \\rho_0 \\bar{v}_n \\bar v ~\\mathrm{d}s \\qquad \\forall v \\in \\hat{V}.\n\\end{align}\n$$\n\nWe define the sesquilinear form $a(p,v)$ is\n\n$$\n\\begin{align}\na(p,v) = \\int_{\\Omega} \\nabla p \\cdot \\nabla \\bar v ~\\mathrm{d}x\n+ \\frac{j \\omega }{\\bar{Z}} \\int_{\\partial \\Omega_Z} \\rho_0 p \\bar v ~\\mathrm{d}s\n- k^2 \\int_{\\Omega} p \\bar v ~\\mathrm{d}x \n\\end{align}\n$$\n\nand the linear form $L(v)$ reads\n\n$$\n\\begin{align}\nL(v) = j \\omega \\int_{\\Omega}\\rho_0 q \\bar v ~\\mathrm{d}x - j \\omega \\int_{\\partial \\Omega_v} \\rho_0 \\bar{v}_n \\bar v ~\\mathrm{d}s.\n\\end{align}\n$$\n" }, { "path": "chapter2/helmholtz_code.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Implementation\\n\",\n \"Author: Antonio Baiano Svizzero and Jørgen S. Dokken\\n\",\n \"\\n\",\n \"In this tutorial, you will learn how to:\\n\",\n \"- Define acoustic velocity and impedance boundary conditions\\n\",\n \"- Compute acoustic sound pressure for multiple frequencies\\n\",\n \"- Compute the Sound Pressure Level (SPL) at a given microphone position\\n\",\n \"\\n\",\n \"## Test problem\\n\",\n \"As an example, we will model a plane wave propagating in a tube.\\n\",\n \"While it is a basic test case, the code can be adapted to way more complex problems where\\n\",\n \"velocity and impedance boundary conditions are needed.\\n\",\n \"We will apply a velocity boundary condition $v_n = 0.001$ to one end of the tube\\n\",\n \"(for the sake of simplicity, in this basic example, we are ignoring the point source, which can be applied with scifem)\\n\",\n \"and an impedance $Z$ computed with the Delaney-Bazley model,\\n\",\n \"supposing that a layer of thickness $d = 0.02$ and flow resistivity $\\\\sigma = 1e4$ is\\n\",\n \"placed at the second end of the tube.\\n\",\n \"The choice of such impedance (the one of a plane wave propagating in free field) will give, as a result,\\n\",\n \"a solution with no reflections.\\n\",\n \"\\n\",\n \"First, we create the mesh with gmsh, also setting the physical group for velocity and impedance boundary\\n\",\n \"conditions and the respective tags.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"import gmsh\\n\",\n \"\\n\",\n \"gmsh.initialize()\\n\",\n \"\\n\",\n \"# meshsize settings\\n\",\n \"meshsize = 0.02\\n\",\n \"gmsh.option.setNumber(\\\"Mesh.MeshSizeMax\\\", meshsize)\\n\",\n \"gmsh.option.setNumber(\\\"Mesh.MeshSizeMax\\\", meshsize)\\n\",\n \"\\n\",\n \"\\n\",\n \"# create geometry\\n\",\n \"L = 1\\n\",\n \"W = 0.1\\n\",\n \"\\n\",\n \"gmsh.model.occ.addBox(0, 0, 0, L, W, W)\\n\",\n \"gmsh.model.occ.synchronize()\\n\",\n \"\\n\",\n \"# setup physical groups\\n\",\n \"v_bc_tag = 2\\n\",\n \"Z_bc_tag = 3\\n\",\n \"gmsh.model.addPhysicalGroup(3, [1], 1, \\\"air_volume\\\")\\n\",\n \"gmsh.model.addPhysicalGroup(2, [1], v_bc_tag, \\\"velocity_BC\\\")\\n\",\n \"gmsh.model.addPhysicalGroup(2, [2], Z_bc_tag, \\\"impedance\\\")\\n\",\n \"\\n\",\n \"# mesh generation\\n\",\n \"gmsh.model.mesh.generate(3)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"Then we import the gmsh mesh with the {py:mod}`dolfinx.io.gmsh` module.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"from mpi4py import MPI\\n\",\n \"from dolfinx import (\\n\",\n \" fem,\\n\",\n \" default_scalar_type,\\n\",\n \" geometry,\\n\",\n \" __version__ as dolfinx_version,\\n\",\n \")\\n\",\n \"from dolfinx.io import gmsh as gmshio\\n\",\n \"from dolfinx.fem.petsc import LinearProblem\\n\",\n \"import ufl\\n\",\n \"import numpy as np\\n\",\n \"import numpy.typing as npt\\n\",\n \"from packaging.version import Version\\n\",\n \"\\n\",\n \"mesh_data = gmshio.model_to_mesh(gmsh.model, MPI.COMM_WORLD, 0, gdim=3)\\n\",\n \"if Version(dolfinx_version) > Version(\\\"0.9.0\\\"):\\n\",\n \" domain = mesh_data.mesh\\n\",\n \" assert mesh_data.facet_tags is not None\\n\",\n \" facet_tags = mesh_data.facet_tags\\n\",\n \"else:\\n\",\n \" domain, _, facet_tags = mesh_data\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"4\",\n \"metadata\": {},\n \"source\": [\n \"We define the function space for our unknown $p$ and define the range of frequencies we want to solve the Helmholtz equation for.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"5\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"V = fem.functionspace(domain, (\\\"Lagrange\\\", 1))\\n\",\n \"\\n\",\n \"# Discrete frequency range\\n\",\n \"freq = np.arange(10, 1000, 5) # Hz\\n\",\n \"\\n\",\n \"# Air parameters\\n\",\n \"rho0 = 1.225 # kg/m^3\\n\",\n \"c = 340 # m/s\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"## Boundary conditions\\n\",\n \"\\n\",\n \"The Delaney-Bazley model is used to compute the characteristic impedance and wavenumber of the porous layer,\\n\",\n \"treated as an equivalent fluid with complex valued properties\\n\",\n \"\\n\",\n \"\\\\begin{align}\\n\",\n \"Z_c(\\\\omega) &= \\\\rho_0 c_0 \\\\left[1 + 0.0571 X^{-0.754} - j 0.087 X^{-0.732}\\\\right],\\\\\\\\\\n\",\n \"k_c(\\\\omega) &= \\\\frac{\\\\omega}{c_0} \\\\left[1 + 0.0978 X^{-0.700} - j 0.189 X^{-0.595}\\\\right],\\\\\\\\\\n\",\n \"\\\\end{align}\\n\",\n \"\\n\",\n \"where $X = \\\\frac{\\\\rho_0 f}{\\\\sigma}$.\\n\",\n \"\\n\",\n \"With these, we can compute the surface impedance, that in the case of a rigid passive absorber placed on a rigid wall is given by the formula\\n\",\n \"\\n\",\n \"$$\\n\",\n \"Z_s = -j Z_c cot(k_c d).\\n\",\n \"$$\\n\",\n \"\\n\",\n \"Let's create a function to compute it.\\n\",\n \"\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"# Impedance calculation\\n\",\n \"def delany_bazley_layer(f, rho0, c, sigma):\\n\",\n \" X = rho0 * f / sigma\\n\",\n \" Zc = rho0 * c * (1 + 0.0571 * X**-0.754 - 1j * 0.087 * X**-0.732)\\n\",\n \" kc = 2 * np.pi * f / c * (1 + 0.0978 * (X**-0.700) - 1j * 0.189 * (X**-0.595))\\n\",\n \" Z_s = -1j * Zc * (1 / np.tan(kc * d))\\n\",\n \" return Z_s\\n\",\n \"\\n\",\n \"\\n\",\n \"sigma = 1.5e4\\n\",\n \"d = 0.01\\n\",\n \"Z_s = delany_bazley_layer(freq, rho0, c, sigma)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"Since we are going to compute a sound pressure spectrum, all the variables that depend on frequency\\n\",\n \"($\\\\omega$, $k$ and $Z$) need to be updated in the frequency loop.\\n\",\n \"To make this possible, we will initialize them as dolfinx constants.\\n\",\n \"Then, we define the value for the normal velocity on the first end of the tube\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"omega = fem.Constant(domain, default_scalar_type(0))\\n\",\n \"k = fem.Constant(domain, default_scalar_type(0))\\n\",\n \"Z = fem.Constant(domain, default_scalar_type(0))\\n\",\n \"v_n = 1e-5\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We also need to specify the integration measure $ds$, by using `ufl`, and its built in integration measures\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"ds = ufl.Measure(\\\"ds\\\", domain=domain, subdomain_data=facet_tags)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Variational Formulation\\n\",\n \"We can now write the variational formulation.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"p = ufl.TrialFunction(V)\\n\",\n \"v = ufl.TestFunction(V)\\n\",\n \"\\n\",\n \"a = (\\n\",\n \" ufl.inner(ufl.grad(p), ufl.grad(v)) * ufl.dx\\n\",\n \" + 1j * rho0 * omega / Z * ufl.inner(p, v) * ds(Z_bc_tag)\\n\",\n \" - k**2 * ufl.inner(p, v) * ufl.dx\\n\",\n \")\\n\",\n \"L = -1j * omega * rho0 * ufl.inner(v_n, v) * ds(v_bc_tag)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"The class ```LinearProblem``` is used to setup the PETSc backend and assemble the system vector and matrices.\\n\",\n \"The solution will be stored in a `dolfinx.fem.Function`, ```p_a```.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"lines_to_end_of_cell_marker\": 2\n },\n \"outputs\": [],\n \"source\": [\n \"p_a = fem.Function(V)\\n\",\n \"p_a.name = \\\"pressure\\\"\\n\",\n \"\\n\",\n \"problem = LinearProblem(\\n\",\n \" a,\\n\",\n \" L,\\n\",\n \" u=p_a,\\n\",\n \" petsc_options={\\n\",\n \" \\\"ksp_type\\\": \\\"preonly\\\",\\n\",\n \" \\\"pc_type\\\": \\\"lu\\\",\\n\",\n \" \\\"pc_factor_mat_solver_type\\\": \\\"mumps\\\",\\n\",\n \" },\\n\",\n \" petsc_options_prefix=\\\"helmholtz\\\",\\n\",\n \")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"## Computing the pressure at a given location\\n\",\n \"Before starting our frequency loop, we can build a function that, given a microphone position,\\n\",\n \"computes the sound pressure at its location.\\n\",\n \"We will use the a similar method as in [Deflection of a membrane](../chapter1/membrane_code).\\n\",\n \"However, as the domain doesn't deform in time, we cache the collision detection\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"class MicrophonePressure:\\n\",\n \" def __init__(self, domain, microphone_position):\\n\",\n \" \\\"\\\"\\\"Initialize microphone(s).\\n\",\n \"\\n\",\n \" Args:\\n\",\n \" domain: The domain to insert microphones on\\n\",\n \" microphone_position: Position of the microphone(s).\\n\",\n \" Assumed to be ordered as ``(mic0_x, mic1_x, ..., mic0_y, mic1_y, ..., mic0_z, mic1_z, ...)``\\n\",\n \"\\n\",\n \" \\\"\\\"\\\"\\n\",\n \" self._domain = domain\\n\",\n \" self._position = np.asarray(\\n\",\n \" microphone_position, dtype=domain.geometry.x.dtype\\n\",\n \" ).reshape(3, -1)\\n\",\n \" self._local_cells, self._local_position = self.compute_local_microphones()\\n\",\n \"\\n\",\n \" def compute_local_microphones(\\n\",\n \" self,\\n\",\n \" ) -> tuple[npt.NDArray[np.int32], npt.NDArray[np.floating]]:\\n\",\n \" \\\"\\\"\\\"\\n\",\n \" Compute the local microphone positions for a distributed mesh\\n\",\n \"\\n\",\n \" Returns:\\n\",\n \" Two lists (local_cells, local_points) containing the local cell indices and the local points\\n\",\n \" \\\"\\\"\\\"\\n\",\n \" points = self._position.T\\n\",\n \" bb_tree = geometry.bb_tree(self._domain, self._domain.topology.dim)\\n\",\n \"\\n\",\n \" cells = []\\n\",\n \" points_on_proc = []\\n\",\n \"\\n\",\n \" cell_candidates = geometry.compute_collisions_points(bb_tree, points)\\n\",\n \" colliding_cells = geometry.compute_colliding_cells(\\n\",\n \" domain, cell_candidates, points\\n\",\n \" )\\n\",\n \"\\n\",\n \" for i, point in enumerate(points):\\n\",\n \" if len(colliding_cells.links(i)) > 0:\\n\",\n \" points_on_proc.append(point)\\n\",\n \" cells.append(colliding_cells.links(i)[0])\\n\",\n \"\\n\",\n \" return np.asarray(cells, dtype=np.int32), np.asarray(\\n\",\n \" points_on_proc, dtype=domain.geometry.x.dtype\\n\",\n \" )\\n\",\n \"\\n\",\n \" def listen(\\n\",\n \" self, recompute_collisions: bool = False\\n\",\n \" ) -> npt.NDArray[np.complexfloating]:\\n\",\n \" if recompute_collisions:\\n\",\n \" self._local_cells, self._local_position = self.compute_local_microphones()\\n\",\n \" if len(self._local_cells) > 0:\\n\",\n \" return p_a.eval(self._local_position, self._local_cells)\\n\",\n \" else:\\n\",\n \" return np.zeros(0, dtype=default_scalar_type)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"The pressure spectrum is initialized as a numpy array and the microphone location is assigned\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"p_mic = np.zeros((len(freq), 1), dtype=complex)\\n\",\n \"\\n\",\n \"mic = np.array([0.5, 0.05, 0.05])\\n\",\n \"microphone = MicrophonePressure(domain, mic)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Frequency loop\\n\",\n \"\\n\",\n \"Finally, we can write the frequency loop, where we update the values of the frequency-dependent variables and solve the system for each frequency\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"for nf in range(0, len(freq)):\\n\",\n \" k.value = 2 * np.pi * freq[nf] / c\\n\",\n \" omega.value = 2 * np.pi * freq[nf]\\n\",\n \" Z.value = Z_s[nf]\\n\",\n \"\\n\",\n \" problem.solve()\\n\",\n \" p_a.x.scatter_forward()\\n\",\n \"\\n\",\n \" p_f = microphone.listen()\\n\",\n \" p_f = domain.comm.gather(p_f, root=0)\\n\",\n \"\\n\",\n \" if domain.comm.rank == 0:\\n\",\n \" assert p_f is not None\\n\",\n \" p_mic[nf] = np.hstack(p_f)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## SPL spectrum\\n\",\n \"After the computation, the pressure spectrum at the prescribed location is available.\\n\",\n \"Such a spectrum is usually shown using the decibel (dB) scale to obtain the SPL, with the RMS pressure as input,\\n\",\n \"defined as $p_{rms} = \\\\frac{p}{\\\\sqrt{2}}$.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"if domain.comm.rank == 0:\\n\",\n \" import matplotlib.pyplot as plt\\n\",\n \"\\n\",\n \" fig = plt.figure(figsize=(25, 8))\\n\",\n \" plt.plot(freq, 20 * np.log10(np.abs(p_mic) / np.sqrt(2) / 2e-5), linewidth=2)\\n\",\n \" plt.grid(True)\\n\",\n \" plt.xlabel(\\\"Frequency [Hz]\\\")\\n\",\n \" plt.ylabel(\\\"SPL [dB]\\\")\\n\",\n \" plt.xlim([freq[0], freq[-1]])\\n\",\n \" plt.ylim([0, 90])\\n\",\n \" plt.legend()\\n\",\n \" plt.show()\"\n ]\n }\n ],\n \"metadata\": {\n \"jupytext\": {\n \"formats\": \"ipynb,py:light\"\n },\n \"kernelspec\": {\n \"display_name\": \"Python 3 (DOLFINx complex)\",\n \"language\": \"python\",\n \"name\": \"python3-complex\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 3\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython3\",\n \"version\": \"3.12.7\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 2\n}\n" }, { "path": "chapter2/helmholtz_code.py", "content": "# ---\n# jupyter:\n# jupytext:\n# formats: ipynb,py:light\n# text_representation:\n# extension: .py\n# format_name: light\n# format_version: '1.5'\n# jupytext_version: 1.18.1\n# kernelspec:\n# display_name: Python 3 (DOLFINx complex)\n# language: python\n# name: python3-complex\n# ---\n\n# # Implementation\n# Author: Antonio Baiano Svizzero and Jørgen S. Dokken\n#\n# In this tutorial, you will learn how to:\n# - Define acoustic velocity and impedance boundary conditions\n# - Compute acoustic sound pressure for multiple frequencies\n# - Compute the Sound Pressure Level (SPL) at a given microphone position\n#\n# ## Test problem\n# As an example, we will model a plane wave propagating in a tube.\n# While it is a basic test case, the code can be adapted to way more complex problems where\n# velocity and impedance boundary conditions are needed.\n# We will apply a velocity boundary condition $v_n = 0.001$ to one end of the tube\n# (for the sake of simplicity, in this basic example, we are ignoring the point source, which can be applied with scifem)\n# and an impedance $Z$ computed with the Delaney-Bazley model,\n# supposing that a layer of thickness $d = 0.02$ and flow resistivity $\\sigma = 1e4$ is\n# placed at the second end of the tube.\n# The choice of such impedance (the one of a plane wave propagating in free field) will give, as a result,\n# a solution with no reflections.\n#\n# First, we create the mesh with gmsh, also setting the physical group for velocity and impedance boundary\n# conditions and the respective tags.\n\n# +\nimport gmsh\n\ngmsh.initialize()\n\n# meshsize settings\nmeshsize = 0.02\ngmsh.option.setNumber(\"Mesh.MeshSizeMax\", meshsize)\ngmsh.option.setNumber(\"Mesh.MeshSizeMax\", meshsize)\n\n\n# create geometry\nL = 1\nW = 0.1\n\ngmsh.model.occ.addBox(0, 0, 0, L, W, W)\ngmsh.model.occ.synchronize()\n\n# setup physical groups\nv_bc_tag = 2\nZ_bc_tag = 3\ngmsh.model.addPhysicalGroup(3, [1], 1, \"air_volume\")\ngmsh.model.addPhysicalGroup(2, [1], v_bc_tag, \"velocity_BC\")\ngmsh.model.addPhysicalGroup(2, [2], Z_bc_tag, \"impedance\")\n\n# mesh generation\ngmsh.model.mesh.generate(3)\n# -\n\n# Then we import the gmsh mesh with the {py:mod}`dolfinx.io.gmsh` module.\n\n# +\nfrom mpi4py import MPI\nfrom dolfinx import (\n fem,\n default_scalar_type,\n geometry,\n __version__ as dolfinx_version,\n)\nfrom dolfinx.io import gmsh as gmshio\nfrom dolfinx.fem.petsc import LinearProblem\nimport ufl\nimport numpy as np\nimport numpy.typing as npt\nfrom packaging.version import Version\n\nmesh_data = gmshio.model_to_mesh(gmsh.model, MPI.COMM_WORLD, 0, gdim=3)\nif Version(dolfinx_version) > Version(\"0.9.0\"):\n domain = mesh_data.mesh\n assert mesh_data.facet_tags is not None\n facet_tags = mesh_data.facet_tags\nelse:\n domain, _, facet_tags = mesh_data\n# -\n\n# We define the function space for our unknown $p$ and define the range of frequencies we want to solve the Helmholtz equation for.\n\n# +\nV = fem.functionspace(domain, (\"Lagrange\", 1))\n\n# Discrete frequency range\nfreq = np.arange(10, 1000, 5) # Hz\n\n# Air parameters\nrho0 = 1.225 # kg/m^3\nc = 340 # m/s\n# -\n\n# ## Boundary conditions\n#\n# The Delaney-Bazley model is used to compute the characteristic impedance and wavenumber of the porous layer,\n# treated as an equivalent fluid with complex valued properties\n#\n# \\begin{align}\n# Z_c(\\omega) &= \\rho_0 c_0 \\left[1 + 0.0571 X^{-0.754} - j 0.087 X^{-0.732}\\right],\\\\\n# k_c(\\omega) &= \\frac{\\omega}{c_0} \\left[1 + 0.0978 X^{-0.700} - j 0.189 X^{-0.595}\\right],\\\\\n# \\end{align}\n#\n# where $X = \\frac{\\rho_0 f}{\\sigma}$.\n#\n# With these, we can compute the surface impedance, that in the case of a rigid passive absorber placed on a rigid wall is given by the formula\n#\n# $$\n# Z_s = -j Z_c cot(k_c d).\n# $$\n#\n# Let's create a function to compute it.\n#\n#\n\n\n# +\n# Impedance calculation\ndef delany_bazley_layer(f, rho0, c, sigma):\n X = rho0 * f / sigma\n Zc = rho0 * c * (1 + 0.0571 * X**-0.754 - 1j * 0.087 * X**-0.732)\n kc = 2 * np.pi * f / c * (1 + 0.0978 * (X**-0.700) - 1j * 0.189 * (X**-0.595))\n Z_s = -1j * Zc * (1 / np.tan(kc * d))\n return Z_s\n\n\nsigma = 1.5e4\nd = 0.01\nZ_s = delany_bazley_layer(freq, rho0, c, sigma)\n# -\n\n# Since we are going to compute a sound pressure spectrum, all the variables that depend on frequency\n# ($\\omega$, $k$ and $Z$) need to be updated in the frequency loop.\n# To make this possible, we will initialize them as dolfinx constants.\n# Then, we define the value for the normal velocity on the first end of the tube\n\nomega = fem.Constant(domain, default_scalar_type(0))\nk = fem.Constant(domain, default_scalar_type(0))\nZ = fem.Constant(domain, default_scalar_type(0))\nv_n = 1e-5\n\n# We also need to specify the integration measure $ds$, by using `ufl`, and its built in integration measures\n\nds = ufl.Measure(\"ds\", domain=domain, subdomain_data=facet_tags)\n\n# ## Variational Formulation\n# We can now write the variational formulation.\n\n# +\np = ufl.TrialFunction(V)\nv = ufl.TestFunction(V)\n\na = (\n ufl.inner(ufl.grad(p), ufl.grad(v)) * ufl.dx\n + 1j * rho0 * omega / Z * ufl.inner(p, v) * ds(Z_bc_tag)\n - k**2 * ufl.inner(p, v) * ufl.dx\n)\nL = -1j * omega * rho0 * ufl.inner(v_n, v) * ds(v_bc_tag)\n# -\n\n# The class ```LinearProblem``` is used to setup the PETSc backend and assemble the system vector and matrices.\n# The solution will be stored in a `dolfinx.fem.Function`, ```p_a```.\n\n# +\np_a = fem.Function(V)\np_a.name = \"pressure\"\n\nproblem = LinearProblem(\n a,\n L,\n u=p_a,\n petsc_options={\n \"ksp_type\": \"preonly\",\n \"pc_type\": \"lu\",\n \"pc_factor_mat_solver_type\": \"mumps\",\n },\n petsc_options_prefix=\"helmholtz\",\n)\n\n\n# -\n\n# ## Computing the pressure at a given location\n# Before starting our frequency loop, we can build a function that, given a microphone position,\n# computes the sound pressure at its location.\n# We will use the a similar method as in [Deflection of a membrane](../chapter1/membrane_code).\n# However, as the domain doesn't deform in time, we cache the collision detection\n\n\nclass MicrophonePressure:\n def __init__(self, domain, microphone_position):\n \"\"\"Initialize microphone(s).\n\n Args:\n domain: The domain to insert microphones on\n microphone_position: Position of the microphone(s).\n Assumed to be ordered as ``(mic0_x, mic1_x, ..., mic0_y, mic1_y, ..., mic0_z, mic1_z, ...)``\n\n \"\"\"\n self._domain = domain\n self._position = np.asarray(\n microphone_position, dtype=domain.geometry.x.dtype\n ).reshape(3, -1)\n self._local_cells, self._local_position = self.compute_local_microphones()\n\n def compute_local_microphones(\n self,\n ) -> tuple[npt.NDArray[np.int32], npt.NDArray[np.floating]]:\n \"\"\"\n Compute the local microphone positions for a distributed mesh\n\n Returns:\n Two lists (local_cells, local_points) containing the local cell indices and the local points\n \"\"\"\n points = self._position.T\n bb_tree = geometry.bb_tree(self._domain, self._domain.topology.dim)\n\n cells = []\n points_on_proc = []\n\n cell_candidates = geometry.compute_collisions_points(bb_tree, points)\n colliding_cells = geometry.compute_colliding_cells(\n domain, cell_candidates, points\n )\n\n for i, point in enumerate(points):\n if len(colliding_cells.links(i)) > 0:\n points_on_proc.append(point)\n cells.append(colliding_cells.links(i)[0])\n\n return np.asarray(cells, dtype=np.int32), np.asarray(\n points_on_proc, dtype=domain.geometry.x.dtype\n )\n\n def listen(\n self, recompute_collisions: bool = False\n ) -> npt.NDArray[np.complexfloating]:\n if recompute_collisions:\n self._local_cells, self._local_position = self.compute_local_microphones()\n if len(self._local_cells) > 0:\n return p_a.eval(self._local_position, self._local_cells)\n else:\n return np.zeros(0, dtype=default_scalar_type)\n\n\n# The pressure spectrum is initialized as a numpy array and the microphone location is assigned\n\n# +\np_mic = np.zeros((len(freq), 1), dtype=complex)\n\nmic = np.array([0.5, 0.05, 0.05])\nmicrophone = MicrophonePressure(domain, mic)\n# -\n\n# ## Frequency loop\n#\n# Finally, we can write the frequency loop, where we update the values of the frequency-dependent variables and solve the system for each frequency\n\nfor nf in range(0, len(freq)):\n k.value = 2 * np.pi * freq[nf] / c\n omega.value = 2 * np.pi * freq[nf]\n Z.value = Z_s[nf]\n\n problem.solve()\n p_a.x.scatter_forward()\n\n p_f = microphone.listen()\n p_f = domain.comm.gather(p_f, root=0)\n\n if domain.comm.rank == 0:\n assert p_f is not None\n p_mic[nf] = np.hstack(p_f)\n\n# ## SPL spectrum\n# After the computation, the pressure spectrum at the prescribed location is available.\n# Such a spectrum is usually shown using the decibel (dB) scale to obtain the SPL, with the RMS pressure as input,\n# defined as $p_{rms} = \\frac{p}{\\sqrt{2}}$.\n\nif domain.comm.rank == 0:\n import matplotlib.pyplot as plt\n\n fig = plt.figure(figsize=(25, 8))\n plt.plot(freq, 20 * np.log10(np.abs(p_mic) / np.sqrt(2) / 2e-5), linewidth=2)\n plt.grid(True)\n plt.xlabel(\"Frequency [Hz]\")\n plt.ylabel(\"SPL [dB]\")\n plt.xlim([freq[0], freq[-1]])\n plt.ylim([0, 90])\n plt.legend()\n plt.show()\n" }, { "path": "chapter2/hyperelasticity.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"markdown\",\n \"id\": \"0\",\n \"metadata\": {},\n \"source\": [\n \"# Hyperelasticity\\n\",\n \"Author: Jørgen S. Dokken and Garth N. Wells\\n\",\n \"\\n\",\n \"This section shows how to solve the hyperelasticity problem for deformation of a beam.\\n\",\n \"\\n\",\n \"We will also show how to create a constant boundary condition for a vector function space.\\n\",\n \"\\n\",\n \"We start by importing DOLFINx and some additional dependencies.\\n\",\n \"Then, we create a slender cantilever consisting of hexahedral elements and create the function space `V` for our unknown.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"1\",\n \"metadata\": {\n \"lines_to_end_of_cell_marker\": 2,\n \"tags\": []\n },\n \"outputs\": [],\n \"source\": [\n \"from dolfinx import log, default_scalar_type\\n\",\n \"from dolfinx.fem.petsc import NonlinearProblem\\n\",\n \"import pyvista\\n\",\n \"import numpy as np\\n\",\n \"import ufl\\n\",\n \"\\n\",\n \"from mpi4py import MPI\\n\",\n \"from dolfinx import fem, mesh, plot\\n\",\n \"\\n\",\n \"L = 20.0\\n\",\n \"domain = mesh.create_box(\\n\",\n \" MPI.COMM_WORLD, [[0.0, 0.0, 0.0], [L, 1, 1]], [20, 5, 5], mesh.CellType.hexahedron\\n\",\n \")\\n\",\n \"V = fem.functionspace(domain, (\\\"Lagrange\\\", 2, (domain.geometry.dim,)))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"2\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"We create two python functions for determining the facets to apply boundary conditions to\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"3\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"def left(x):\\n\",\n \" return np.isclose(x[0], 0)\\n\",\n \"\\n\",\n \"\\n\",\n \"def right(x):\\n\",\n \" return np.isclose(x[0], L)\\n\",\n \"\\n\",\n \"\\n\",\n \"fdim = domain.topology.dim - 1\\n\",\n \"left_facets = mesh.locate_entities_boundary(domain, fdim, left)\\n\",\n \"right_facets = mesh.locate_entities_boundary(domain, fdim, right)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"4\",\n \"metadata\": {},\n \"source\": [\n \"Next, we create a marker based on these two functions\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"5\",\n \"metadata\": {},\n \"source\": [\n \"Concatenate and sort the arrays based on facet indices. Left facets marked with 1, right facets with two\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"6\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"marked_facets = np.hstack([left_facets, right_facets])\\n\",\n \"marked_values = np.hstack([np.full_like(left_facets, 1), np.full_like(right_facets, 2)])\\n\",\n \"sorted_facets = np.argsort(marked_facets)\\n\",\n \"facet_tag = mesh.meshtags(\\n\",\n \" domain, fdim, marked_facets[sorted_facets], marked_values[sorted_facets]\\n\",\n \")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"7\",\n \"metadata\": {},\n \"source\": [\n \"We then create a function for supplying the boundary condition on the left side, which is fixed.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"8\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"u_bc = np.array((0,) * domain.geometry.dim, dtype=default_scalar_type)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"9\",\n \"metadata\": {},\n \"source\": [\n \"To apply the boundary condition, we identity the dofs located on the facets marked by the `MeshTag`.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"10\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"left_dofs = fem.locate_dofs_topological(V, facet_tag.dim, facet_tag.find(1))\\n\",\n \"bcs = [fem.dirichletbc(u_bc, left_dofs, V)]\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"11\",\n \"metadata\": {},\n \"source\": [\n \"Next, we define the body force on the reference configuration (`B`), and nominal (first Piola-Kirchhoff) traction (`T`).\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"12\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"B = fem.Constant(domain, default_scalar_type((0, 0, 0)))\\n\",\n \"T = fem.Constant(domain, default_scalar_type((0, 0, 0)))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"13\",\n \"metadata\": {},\n \"source\": [\n \"Define the test and solution functions on the space $V$\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"14\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"v = ufl.TestFunction(V)\\n\",\n \"u = fem.Function(V)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"15\",\n \"metadata\": {},\n \"source\": [\n \"Define kinematic quantities used in the problem\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"16\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"# Spatial dimension\\n\",\n \"d = len(u)\\n\",\n \"\\n\",\n \"# Identity tensor\\n\",\n \"I = ufl.variable(ufl.Identity(d))\\n\",\n \"\\n\",\n \"# Deformation gradient\\n\",\n \"F = ufl.variable(I + ufl.grad(u))\\n\",\n \"\\n\",\n \"# Right Cauchy-Green tensor\\n\",\n \"C = ufl.variable(F.T * F)\\n\",\n \"\\n\",\n \"# Invariants of deformation tensors\\n\",\n \"Ic = ufl.variable(ufl.tr(C))\\n\",\n \"J = ufl.variable(ufl.det(F))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"17\",\n \"metadata\": {},\n \"source\": [\n \"Define the elasticity model via a stored strain energy density function $\\\\psi$,\\n\",\n \"and create the expression for the first Piola-Kirchhoff stress:\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"18\",\n \"metadata\": {},\n \"source\": [\n \"Elasticity parameters\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"19\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"E = default_scalar_type(1.0e4)\\n\",\n \"nu = default_scalar_type(0.3)\\n\",\n \"mu = fem.Constant(domain, E / (2 * (1 + nu)))\\n\",\n \"lmbda = fem.Constant(domain, E * nu / ((1 + nu) * (1 - 2 * nu)))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"20\",\n \"metadata\": {},\n \"source\": [\n \"Stored strain energy density (compressible neo-Hookean model)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"21\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"psi = (mu / 2) * (Ic - 3) - mu * ufl.ln(J) + (lmbda / 2) * (ufl.ln(J)) ** 2\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"22\",\n \"metadata\": {},\n \"source\": [\n \"Hyper-elasticity\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"23\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"P = ufl.diff(psi, F)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"24\",\n \"metadata\": {},\n \"source\": [\n \"```{admonition} Comparison to linear elasticity\\n\",\n \"To illustrate the difference between linear and hyperelasticity,\\n\",\n \"the following lines can be uncommented to solve the linear elasticity problem.\\n\",\n \"```\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"25\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"# P = 2.0 * mu * ufl.sym(ufl.grad(u)) + lmbda * ufl.tr(ufl.sym(ufl.grad(u))) * I\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"26\",\n \"metadata\": {},\n \"source\": [\n \"Define the variational form with traction integral over all facets with value 2.\\n\",\n \"We set the quadrature degree for the integrals to 4.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"27\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"metadata = {\\\"quadrature_degree\\\": 4}\\n\",\n \"ds = ufl.Measure(\\\"ds\\\", domain=domain, subdomain_data=facet_tag, metadata=metadata)\\n\",\n \"dx = ufl.Measure(\\\"dx\\\", domain=domain, metadata=metadata)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"28\",\n \"metadata\": {},\n \"source\": [\n \"Define the residual of the equation (we want to find u such that residual(u) = 0)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"29\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"residual = (\\n\",\n \" ufl.inner(ufl.grad(v), P) * dx - ufl.inner(v, B) * dx - ufl.inner(v, T) * ds(2)\\n\",\n \")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"30\",\n \"metadata\": {},\n \"source\": [\n \"As the varitional form is non-linear and written on residual form,\\n\",\n \"we use the non-linear problem class from DOLFINx to set up required structures to use a Newton solver.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"31\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"petsc_options = {\\n\",\n \" \\\"snes_type\\\": \\\"newtonls\\\",\\n\",\n \" \\\"snes_linesearch_type\\\": \\\"none\\\",\\n\",\n \" \\\"snes_monitor\\\": None,\\n\",\n \" \\\"snes_atol\\\": 1e-8,\\n\",\n \" \\\"snes_rtol\\\": 1e-8,\\n\",\n \" \\\"snes_stol\\\": 1e-8,\\n\",\n \" \\\"ksp_type\\\": \\\"preonly\\\",\\n\",\n \" \\\"pc_type\\\": \\\"lu\\\",\\n\",\n \" \\\"pc_factor_mat_solver_type\\\": \\\"mumps\\\",\\n\",\n \"}\\n\",\n \"problem = NonlinearProblem(\\n\",\n \" residual,\\n\",\n \" u,\\n\",\n \" bcs=bcs,\\n\",\n \" petsc_options=petsc_options,\\n\",\n \" petsc_options_prefix=\\\"hyperelasticity\\\",\\n\",\n \")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"32\",\n \"metadata\": {},\n \"source\": [\n \"We create a function to plot the solution at each time step.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"33\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"plotter = pyvista.Plotter()\\n\",\n \"plotter.open_gif(\\\"deformation.gif\\\", fps=3)\\n\",\n \"\\n\",\n \"topology, cells, geometry = plot.vtk_mesh(u.function_space)\\n\",\n \"function_grid = pyvista.UnstructuredGrid(topology, cells, geometry)\\n\",\n \"\\n\",\n \"values = np.zeros((geometry.shape[0], 3))\\n\",\n \"values[:, : len(u)] = u.x.array.reshape(geometry.shape[0], len(u))\\n\",\n \"function_grid[\\\"u\\\"] = values\\n\",\n \"function_grid.set_active_vectors(\\\"u\\\")\\n\",\n \"\\n\",\n \"# Warp mesh by deformation\\n\",\n \"warped = function_grid.warp_by_vector(\\\"u\\\", factor=1)\\n\",\n \"warped.set_active_vectors(\\\"u\\\")\\n\",\n \"\\n\",\n \"# Add mesh to plotter and visualize\\n\",\n \"actor = plotter.add_mesh(warped, show_edges=True, lighting=False, clim=[0, 10])\\n\",\n \"\\n\",\n \"# Compute magnitude of displacement to visualize in GIF\\n\",\n \"Vs = fem.functionspace(domain, (\\\"Lagrange\\\", 2))\\n\",\n \"magnitude = fem.Function(Vs)\\n\",\n \"us = fem.Expression(\\n\",\n \" ufl.sqrt(sum([u[i] ** 2 for i in range(len(u))])), Vs.element.interpolation_points\\n\",\n \")\\n\",\n \"magnitude.interpolate(us)\\n\",\n \"warped[\\\"mag\\\"] = magnitude.x.array\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"34\",\n \"metadata\": {},\n \"source\": [\n \"Finally, we solve the problem over several time steps, updating the z-component of the traction\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"35\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"log.set_log_level(log.LogLevel.INFO)\\n\",\n \"tval0 = -1.5\\n\",\n \"for n in range(1, 10):\\n\",\n \" T.value[2] = n * tval0\\n\",\n \" problem.solve()\\n\",\n \" converged = problem.solver.getConvergedReason()\\n\",\n \" num_its = problem.solver.getIterationNumber()\\n\",\n \" assert converged > 0, f\\\"Solver did not converge with reason {converged}.\\\"\\n\",\n \"\\n\",\n \" print(f\\\"Time step {n}, Number of iterations {num_its}, Load {T.value}\\\")\\n\",\n \" function_grid[\\\"u\\\"][:, : len(u)] = u.x.array.reshape(geometry.shape[0], len(u))\\n\",\n \" magnitude.interpolate(us)\\n\",\n \" warped.set_active_scalars(\\\"mag\\\")\\n\",\n \" warped_n = function_grid.warp_by_vector(factor=1)\\n\",\n \" warped.points[:, :] = warped_n.points\\n\",\n \" warped.point_data[\\\"mag\\\"][:] = magnitude.x.array\\n\",\n \" plotter.update_scalar_bar_range([0, 10])\\n\",\n \" plotter.write_frame()\\n\",\n \"plotter.close()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"36\",\n \"metadata\": {},\n \"source\": [\n \"\\\"gif\\\"\"\n ]\n }\n ],\n \"metadata\": {\n \"jupytext\": {\n \"formats\": \"ipynb,py:light\"\n },\n \"kernelspec\": {\n \"display_name\": \"Python 3 (ipykernel)\",\n \"language\": \"python\",\n \"name\": \"python3\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 3\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython3\",\n \"version\": \"3.10.12\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 5\n}\n" }, { "path": "chapter2/hyperelasticity.py", "content": "# ---\n# jupyter:\n# jupytext:\n# formats: ipynb,py:light\n# text_representation:\n# extension: .py\n# format_name: light\n# format_version: '1.5'\n# jupytext_version: 1.18.1\n# kernelspec:\n# display_name: Python 3 (ipykernel)\n# language: python\n# name: python3\n# ---\n\n# # Hyperelasticity\n# Author: Jørgen S. Dokken and Garth N. Wells\n#\n# This section shows how to solve the hyperelasticity problem for deformation of a beam.\n#\n# We will also show how to create a constant boundary condition for a vector function space.\n#\n# We start by importing DOLFINx and some additional dependencies.\n# Then, we create a slender cantilever consisting of hexahedral elements and create the function space `V` for our unknown.\n\n# +\nfrom dolfinx import log, default_scalar_type\nfrom dolfinx.fem.petsc import NonlinearProblem\nimport pyvista\nimport numpy as np\nimport ufl\n\nfrom mpi4py import MPI\nfrom dolfinx import fem, mesh, plot\n\nL = 20.0\ndomain = mesh.create_box(\n MPI.COMM_WORLD, [[0.0, 0.0, 0.0], [L, 1, 1]], [20, 5, 5], mesh.CellType.hexahedron\n)\nV = fem.functionspace(domain, (\"Lagrange\", 2, (domain.geometry.dim,)))\n\n\n# -\n\n# We create two python functions for determining the facets to apply boundary conditions to\n\n\n# +\ndef left(x):\n return np.isclose(x[0], 0)\n\n\ndef right(x):\n return np.isclose(x[0], L)\n\n\nfdim = domain.topology.dim - 1\nleft_facets = mesh.locate_entities_boundary(domain, fdim, left)\nright_facets = mesh.locate_entities_boundary(domain, fdim, right)\n# -\n\n# Next, we create a marker based on these two functions\n\n# Concatenate and sort the arrays based on facet indices. Left facets marked with 1, right facets with two\n\nmarked_facets = np.hstack([left_facets, right_facets])\nmarked_values = np.hstack([np.full_like(left_facets, 1), np.full_like(right_facets, 2)])\nsorted_facets = np.argsort(marked_facets)\nfacet_tag = mesh.meshtags(\n domain, fdim, marked_facets[sorted_facets], marked_values[sorted_facets]\n)\n\n# We then create a function for supplying the boundary condition on the left side, which is fixed.\n\nu_bc = np.array((0,) * domain.geometry.dim, dtype=default_scalar_type)\n\n# To apply the boundary condition, we identity the dofs located on the facets marked by the `MeshTag`.\n\nleft_dofs = fem.locate_dofs_topological(V, facet_tag.dim, facet_tag.find(1))\nbcs = [fem.dirichletbc(u_bc, left_dofs, V)]\n\n# Next, we define the body force on the reference configuration (`B`), and nominal (first Piola-Kirchhoff) traction (`T`).\n\nB = fem.Constant(domain, default_scalar_type((0, 0, 0)))\nT = fem.Constant(domain, default_scalar_type((0, 0, 0)))\n\n# Define the test and solution functions on the space $V$\n\nv = ufl.TestFunction(V)\nu = fem.Function(V)\n\n# Define kinematic quantities used in the problem\n\n# +\n# Spatial dimension\nd = len(u)\n\n# Identity tensor\nI = ufl.variable(ufl.Identity(d))\n\n# Deformation gradient\nF = ufl.variable(I + ufl.grad(u))\n\n# Right Cauchy-Green tensor\nC = ufl.variable(F.T * F)\n\n# Invariants of deformation tensors\nIc = ufl.variable(ufl.tr(C))\nJ = ufl.variable(ufl.det(F))\n# -\n\n# Define the elasticity model via a stored strain energy density function $\\psi$,\n# and create the expression for the first Piola-Kirchhoff stress:\n\n# Elasticity parameters\n\nE = default_scalar_type(1.0e4)\nnu = default_scalar_type(0.3)\nmu = fem.Constant(domain, E / (2 * (1 + nu)))\nlmbda = fem.Constant(domain, E * nu / ((1 + nu) * (1 - 2 * nu)))\n\n# Stored strain energy density (compressible neo-Hookean model)\n\npsi = (mu / 2) * (Ic - 3) - mu * ufl.ln(J) + (lmbda / 2) * (ufl.ln(J)) ** 2\n\n# Hyper-elasticity\n\nP = ufl.diff(psi, F)\n\n# ```{admonition} Comparison to linear elasticity\n# To illustrate the difference between linear and hyperelasticity,\n# the following lines can be uncommented to solve the linear elasticity problem.\n# ```\n\n# +\n# P = 2.0 * mu * ufl.sym(ufl.grad(u)) + lmbda * ufl.tr(ufl.sym(ufl.grad(u))) * I\n# -\n\n# Define the variational form with traction integral over all facets with value 2.\n# We set the quadrature degree for the integrals to 4.\n\nmetadata = {\"quadrature_degree\": 4}\nds = ufl.Measure(\"ds\", domain=domain, subdomain_data=facet_tag, metadata=metadata)\ndx = ufl.Measure(\"dx\", domain=domain, metadata=metadata)\n\n# Define the residual of the equation (we want to find u such that residual(u) = 0)\n\nresidual = (\n ufl.inner(ufl.grad(v), P) * dx - ufl.inner(v, B) * dx - ufl.inner(v, T) * ds(2)\n)\n\n# As the varitional form is non-linear and written on residual form,\n# we use the non-linear problem class from DOLFINx to set up required structures to use a Newton solver.\n\npetsc_options = {\n \"snes_type\": \"newtonls\",\n \"snes_linesearch_type\": \"none\",\n \"snes_monitor\": None,\n \"snes_atol\": 1e-8,\n \"snes_rtol\": 1e-8,\n \"snes_stol\": 1e-8,\n \"ksp_type\": \"preonly\",\n \"pc_type\": \"lu\",\n \"pc_factor_mat_solver_type\": \"mumps\",\n}\nproblem = NonlinearProblem(\n residual,\n u,\n bcs=bcs,\n petsc_options=petsc_options,\n petsc_options_prefix=\"hyperelasticity\",\n)\n\n# We create a function to plot the solution at each time step.\n\n# +\nplotter = pyvista.Plotter()\nplotter.open_gif(\"deformation.gif\", fps=3)\n\ntopology, cells, geometry = plot.vtk_mesh(u.function_space)\nfunction_grid = pyvista.UnstructuredGrid(topology, cells, geometry)\n\nvalues = np.zeros((geometry.shape[0], 3))\nvalues[:, : len(u)] = u.x.array.reshape(geometry.shape[0], len(u))\nfunction_grid[\"u\"] = values\nfunction_grid.set_active_vectors(\"u\")\n\n# Warp mesh by deformation\nwarped = function_grid.warp_by_vector(\"u\", factor=1)\nwarped.set_active_vectors(\"u\")\n\n# Add mesh to plotter and visualize\nactor = plotter.add_mesh(warped, show_edges=True, lighting=False, clim=[0, 10])\n\n# Compute magnitude of displacement to visualize in GIF\nVs = fem.functionspace(domain, (\"Lagrange\", 2))\nmagnitude = fem.Function(Vs)\nus = fem.Expression(\n ufl.sqrt(sum([u[i] ** 2 for i in range(len(u))])), Vs.element.interpolation_points\n)\nmagnitude.interpolate(us)\nwarped[\"mag\"] = magnitude.x.array\n# -\n\n# Finally, we solve the problem over several time steps, updating the z-component of the traction\n\nlog.set_log_level(log.LogLevel.INFO)\ntval0 = -1.5\nfor n in range(1, 10):\n T.value[2] = n * tval0\n problem.solve()\n converged = problem.solver.getConvergedReason()\n num_its = problem.solver.getIterationNumber()\n assert converged > 0, f\"Solver did not converge with reason {converged}.\"\n\n print(f\"Time step {n}, Number of iterations {num_its}, Load {T.value}\")\n function_grid[\"u\"][:, : len(u)] = u.x.array.reshape(geometry.shape[0], len(u))\n magnitude.interpolate(us)\n warped.set_active_scalars(\"mag\")\n warped_n = function_grid.warp_by_vector(factor=1)\n warped.points[:, :] = warped_n.points\n warped.point_data[\"mag\"][:] = magnitude.x.array\n plotter.update_scalar_bar_range([0, 10])\n plotter.write_frame()\nplotter.close()\n\n# \"gif\"\n" }, { "path": "chapter2/intro.md", "content": "# A Gallery of finite element solvers\n\nThe goal of this chapter is to demonstrate how a range of important PDEs from science and engineering can be quickly solved with a few lines of DOLFINx code. \nWe will start with the heat equation, then continue with the nonlinear Poisson equation, the equations for linear elasticity, hyperelasticity, the Navier-Stokes equations and the Helmholtz equations.\nThese problems illustrate how to solve time-dependent problems, nonlinear problems, vector-valued problems and systems of PDEs.\nFor each problem, we derive the variational formulation and express the problem in Python in a way that closely resembles the mathematics.\n" }, { "path": "chapter2/linearelasticity.md", "content": "# The equations of linear elasticity\n\nAuthors: Anders Logg and Hans Petter Langtangen\n\nAnalysis of structures is one of the major activities of modern engineering, which likely makes the PDE modelling the deformation of elastic bodies the most popular PDE in the world. It takes just one page of code to solve the equations of 2D or 3D elasticity in DOLFINx, and shown in this section.\n\n## The PDE problem\nThe equations governing small elastic deformations of a body $\\Omega$ can be written as\n```{math}\n:label: elasticity-PDE\n -\\nabla \\cdot \\sigma (u) &= f && \\text{in } \\Omega\\\\\n \\sigma(u)&= \\lambda \\mathrm{tr}(\\epsilon(u))I + 2 \\mu \\epsilon(u)\\\\\n \\epsilon(u) &= \\frac{1}{2}\\left(\\nabla u + (\\nabla u )^T\\right)\n```\nwhere $\\sigma$ is the stress tensor, $f$ is the body force per unit volume, $\\lambda$ and $\\mu$ are Lamé's elasticity parameters for the material in $\\Omega$, $I$ is the identity tensor, $\\mathrm{tr}$ is the trace operator on a tensor, $\\epsilon$ is the symmetric strain tensor (symmetric gradient), and $u$ is the displacement vector field. Above we have assumed isotropic elastic conditions.\nBy inserting $\\epsilon(u)$ into $\\sigma$ we obtain \n\\begin{align}\n \\sigma(u)&=\\lambda(\\nabla \\cdot u)I + \\mu(\\nabla u + (\\nabla u)^T)\n\\end{align}\nNote that we could have written the PDE above as a single vector PDE for $u$, which is the governing PDE for the unknown $u$ (Navier's) equation. However, it is convenient to keep the current representation of the PDE for the derivation of the variational formulation.\n\n## The variational formulation\nThe variational formulation of the PDE consists of forming the inner product of the PDE [](elasticity-PDE) with a *vector* test function $v\\in\\hat{V}$, where $\\hat{V}$ is a vector-valued test function space, and integrating over the domain $\\Omega$:\n```{math}\n -\\int_{\\Omega}(\\nabla \\cdot \\sigma)\\cdot v ~\\mathrm{d} x = \\int_{\\Omega} f\\cdot v ~\\mathrm{d} x.\n```\nSince $\\nabla \\cdot \\sigma$ contains second-order derivatives of our unknown $u$, we integrate this term by parts\n```{math}\n -\\int_{\\Omega}(\\nabla \\cdot \\sigma)\\cdot v ~\\mathrm{d} x =\\int_{\\Omega}\\sigma : \\nabla v ~\\mathrm{d}x - \\int_{\\partial\\Omega} (\\sigma \\cdot n)\\cdot v~\\mathrm{d}s,\n```\nwhere the colon operator is the inner product between tensors (summed pairwise product of all elements), and $n$ is the outward unit normal at the boundary. The quantity $\\sigma \\cdot n$ is known as the *traction* or stress vector at the boundary, and often prescribed as a boundary condition. We here assume that it is prescribed on a part $\\partial \\Omega_T$ of the boundary as $\\sigma \\cdot n=T$. On the remaining part of the boundary, we assume that the value of the displacement is given as Dirichlet condition (and hence the boundary integral on those boundaries are $0$). We thus obtain\n```{math}\n \\int_{\\Omega} \\sigma : \\nabla v ~\\mathrm{d} x = \\int_{\\Omega} f\\cdot v ~\\mathrm{d} x + \\int_{\\partial\\Omega_T}T\\cdot v~\\mathrm{d} s.\n```\nIf we now insert for $\\sigma$ its representation with the unknown $u$, we can obtain our variational formulation:\nFind $u\\in V$ such that \n```{math}\n a(u,v) = L(v)\\qquad \\forall v \\in \\hat{V},\n```\nwhere\n```{math}\n:label: elasticity\n a(u,v)&=\\int_{\\Omega}\\sigma(u):\\nabla v ~\\mathrm{d}x\\\\\n \\sigma(u)&=\\lambda(\\nabla \\cdot u)I+\\mu (\\nabla u + (\\nabla u)^T),\\\\\n L(v)&=\\int_{\\Omega}f\\cdot v~\\mathrm{d} x + \\int_{\\partial\\Omega_T}T\\cdot v~\\mathrm{d}s.\n```\nOne can show that the inner product of a symmetric tensor $A$ and an anti-symmetric tensor $B$ vanishes. If we express $\\nabla v$ as a sum of its symmetric and anti-symmetric parts, only the symmetric part will survive in the product $\\sigma : \\nabla v$ since $\\sigma$ is a symmetric tensor. Thus replacing $\\nabla v$ by the symmetric gradient $\\epsilon(v)$ gives rise to a slightly different variational form\n```{math}\n:label: elasticity-alternative\n a(u,v)= \\int_{\\Omega}\\sigma(u):\\epsilon(v)~\\mathrm{d} x,\n```\nwhere $\\epsilon(v)$ is the symmetric part of $\\nabla v$:\n```{math}\n \\epsilon(v)=\\frac{1}{2}\\left(\\nabla v + (\\nabla v)^T\\right)\n```\nThe formulation [](elasticity-alternative) is what naturally arises from minimization of elastic potential energy and is a more popular formulation than [](elasticity).\n" }, { "path": "chapter2/linearelasticity_code.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Implementation\\n\",\n \"Author: Jørgen S. Dokken\\n\",\n \"\\n\",\n \"In this tutorial, you will learn how to:\\n\",\n \"- Use a vector function space\\n\",\n \"- Create a constant boundary condition on a vector space\\n\",\n \"- Visualize cell-wise constant functions\\n\",\n \"- Compute Von Mises stresses\\n\",\n \"\\n\",\n \"## Test problem\\n\",\n \"As a test example, we will model a clamped beam deformed under its own weight in 3D.\\n\",\n \"This can be modeled, by setting the right-hand side body force per unit volume to $f=(0,0,-\\\\rho g)$,\\n\",\n \"where $\\\\rho$ the density of the beam and $g$ the acceleration of gravity.\\n\",\n \"The beam is box-shaped with length $L$ and has a square cross section of width $W$.\\n\",\n \"We set $u=u_D=(0,0,0)$ at the clamped end, x=0. The rest of the boundary is traction free, that is, we set $T=0$.\\n\",\n \"We start by defining the physical variables used in the program.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"1\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"import pyvista\\n\",\n \"from dolfinx import mesh, fem, plot, io, default_scalar_type\\n\",\n \"from dolfinx.fem.petsc import LinearProblem\\n\",\n \"from mpi4py import MPI\\n\",\n \"import ufl\\n\",\n \"import numpy as np\\n\",\n \"\\n\",\n \"L = 1.0\\n\",\n \"W = 0.2\\n\",\n \"mu = 1.0\\n\",\n \"rho = 1.0\\n\",\n \"delta = W / L\\n\",\n \"gamma = 0.4 * delta**2\\n\",\n \"beta = 1.25\\n\",\n \"lambda_ = beta\\n\",\n \"g = gamma\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We then create the mesh, which will consist of hexahedral elements, along with the function space.\\n\",\n \"As we want a vector element with three components, we add `(3, )` or `(domain.geometry.dim, )` to the element tuple to make it a triplet.\\n\",\n \"However, we also could have used `basix.ufl`s functionality,\\n\",\n \"creating a vector element `el = basix.ufl.element(\\\"Lagrange\\\", domain.basix_cell(), 1, shape=(domain.geometry.dim,))`,\\n\",\n \"and initializing the function space as `V = dolfinx.fem.functionspace(domain, el)`.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"lines_to_next_cell\": 2,\n \"tags\": []\n },\n \"outputs\": [],\n \"source\": [\n \"domain = mesh.create_box(\\n\",\n \" MPI.COMM_WORLD,\\n\",\n \" [np.array([0, 0, 0]), np.array([L, W, W])],\\n\",\n \" [20, 6, 6],\\n\",\n \" cell_type=mesh.CellType.hexahedron,\\n\",\n \")\\n\",\n \"V = fem.functionspace(domain, (\\\"Lagrange\\\", 1, (domain.geometry.dim,)))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"## Boundary conditions\\n\",\n \"As we would like to clamp the boundary at $x=0$, we do this by using a marker function,\\n\",\n \"which locates the facets where $x$ is close to zero by machine precision.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"def clamped_boundary(x):\\n\",\n \" return np.isclose(x[0], 0)\\n\",\n \"\\n\",\n \"\\n\",\n \"fdim = domain.topology.dim - 1\\n\",\n \"boundary_facets = mesh.locate_entities_boundary(domain, fdim, clamped_boundary)\\n\",\n \"\\n\",\n \"u_D = np.array([0, 0, 0], dtype=default_scalar_type)\\n\",\n \"bc = fem.dirichletbc(u_D, fem.locate_dofs_topological(V, fdim, boundary_facets), V)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"As we want the traction $T$ over the remaining boundary to be $0$, we create a `dolfinx.fem.Constant`\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"T = fem.Constant(domain, default_scalar_type((0, 0, 0)))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We also want to specify the integration measure $\\\\mathrm{d}s$, which should be the integral over the boundary of our domain.\\n\",\n \"We do this by using `ufl`, and its built in integration measures\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"outputs\": [],\n \"source\": [\n \"ds = ufl.Measure(\\\"ds\\\", domain=domain)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"## Variational formulation\\n\",\n \"We are now ready to create our variational formulation in close to mathematical syntax, as for the previous problems.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"def epsilon(u):\\n\",\n \" return ufl.sym(\\n\",\n \" ufl.grad(u)\\n\",\n \" ) # Equivalent to 0.5*(ufl.nabla_grad(u) + ufl.nabla_grad(u).T)\\n\",\n \"\\n\",\n \"\\n\",\n \"def sigma(u):\\n\",\n \" return lambda_ * ufl.nabla_div(u) * ufl.Identity(len(u)) + 2 * mu * epsilon(u)\\n\",\n \"\\n\",\n \"\\n\",\n \"u = ufl.TrialFunction(V)\\n\",\n \"v = ufl.TestFunction(V)\\n\",\n \"f = fem.Constant(domain, default_scalar_type((0, 0, -rho * g)))\\n\",\n \"a = ufl.inner(sigma(u), epsilon(v)) * ufl.dx\\n\",\n \"L = ufl.dot(f, v) * ufl.dx + ufl.dot(T, v) * ds\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"```{note}\\n\",\n \"Note that we used `nabla_grad` and optionally `nabla_div` for the variational formulation, as opposed to our previous usage of\\n\",\n \"`div` and `grad`. This is because for scalar functions $\\\\nabla u$ has a clear meaning\\n\",\n \"$\\\\nabla u = \\\\left(\\\\frac{\\\\partial u}{\\\\partial x}, \\\\frac{\\\\partial u}{\\\\partial y}, \\\\frac{\\\\partial u}{\\\\partial z} \\\\right)$.\\n\",\n \"\\n\",\n \"However, if $u$ is vector valued, the meaning is less clear.\\n\",\n \"Some sources define $\\\\nabla u$ as a matrix with the elements $\\\\frac{\\\\partial u_j}{\\\\partial x_i}$, while other sources prefer\\n\",\n \"$\\\\frac{\\\\partial u_i}{\\\\partial x_j}$.\\n\",\n \"In DOLFINx `grad(u)` is defined as the matrix with elements $\\\\frac{\\\\partial u_i}{\\\\partial x_j}$.\\n\",\n \"However, as it is common in continuum mechanics to use the other definition, `ufl` supplies us with `nabla_grad` for this purpose.\\n\",\n \"```\\n\",\n \"\\n\",\n \"## Solve the linear variational problem\\n\",\n \"As in the previous demos, we assemble the matrix and right hand side vector and use PETSc to solve our variational problem\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"problem = LinearProblem(\\n\",\n \" a,\\n\",\n \" L,\\n\",\n \" bcs=[bc],\\n\",\n \" petsc_options={\\\"ksp_type\\\": \\\"preonly\\\", \\\"pc_type\\\": \\\"lu\\\"},\\n\",\n \" petsc_options_prefix=\\\"linear_elasticity\\\",\\n\",\n \")\\n\",\n \"uh = problem.solve()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Visualization\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"As in the previous demos, we can either use Pyvista or Paraview for visualization.\\n\",\n \"We start by using Pyvista.\\n\",\n \"In previous tutorials, we have considered scalar values, while the following section considers vectors.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"# Create plotter and pyvista grid\\n\",\n \"p = pyvista.Plotter()\\n\",\n \"topology, cell_types, geometry = plot.vtk_mesh(V)\\n\",\n \"grid = pyvista.UnstructuredGrid(topology, cell_types, geometry)\\n\",\n \"\\n\",\n \"# Attach vector values to grid and warp grid by vector\\n\",\n \"grid[\\\"u\\\"] = uh.x.array.reshape((geometry.shape[0], 3))\\n\",\n \"actor_0 = p.add_mesh(grid, style=\\\"wireframe\\\", color=\\\"k\\\")\\n\",\n \"warped = grid.warp_by_vector(\\\"u\\\", factor=1.5)\\n\",\n \"actor_1 = p.add_mesh(warped, show_edges=True)\\n\",\n \"p.show_axes()\\n\",\n \"if not pyvista.OFF_SCREEN:\\n\",\n \" p.show()\\n\",\n \"else:\\n\",\n \" figure_as_array = p.screenshot(\\\"deflection.png\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We could also use Paraview for visualizing this.\\n\",\n \"As explained in previous sections, we save the solution with `XDMFFile`.\\n\",\n \"After opening the file `deformation.xdmf` in Paraview and pressing `Apply`,\\n\",\n \"one can press the `Warp by vector button` ![Warp by vector](warp_by_vector.png)\\n\",\n \"or go through the top menu (`Filters->Alphabetical->Warp by Vector`) and press `Apply`.\\n\",\n \"We can also change the color of the deformed beam by changing the value in the\\n\",\n \"color menu ![color](color.png) from `Solid Color` to `Deformation`.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"with io.XDMFFile(domain.comm, \\\"deformation.xdmf\\\", \\\"w\\\") as xdmf:\\n\",\n \" xdmf.write_mesh(domain)\\n\",\n \" uh.name = \\\"Deformation\\\"\\n\",\n \" xdmf.write_function(uh)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Stress computation\\n\",\n \"As soon as the displacement is computed, we can compute various stress measures.\\n\",\n \"We will compute the von Mises stress defined as $\\\\sigma_m=\\\\sqrt{\\\\frac{3}{2}s:s}$ where\\n\",\n \"$s$ is the deviatoric stress tensor $s(u)=\\\\sigma(u)-\\\\frac{1}{3}\\\\mathrm{tr}(\\\\sigma(u))I$.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"s = sigma(uh) - 1.0 / 3 * ufl.tr(sigma(uh)) * ufl.Identity(len(uh))\\n\",\n \"von_Mises = ufl.sqrt(3.0 / 2 * ufl.inner(s, s))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"The `von_Mises` variable is now an expression that must be projected into an appropriate\\n\",\n \"function space so that we can visualize it.\\n\",\n \"As `uh` is a linear combination of first order piecewise continuous functions,\\n\",\n \"the von Mises stresses will be a cell-wise constant function.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"V_von_mises = fem.functionspace(domain, (\\\"DG\\\", 0))\\n\",\n \"stress_expr = fem.Expression(von_Mises, V_von_mises.element.interpolation_points)\\n\",\n \"stresses = fem.Function(V_von_mises)\\n\",\n \"stresses.interpolate(stress_expr)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"In the previous sections, we have only visualized first order Lagrangian functions.\\n\",\n \"However, the Von Mises stresses are piecewise constant on each cell.\\n\",\n \"Therefore, we modify our plotting routine slightly.\\n\",\n \"The first thing we notice is that we now set values for each cell,\\n\",\n \"which has a one to one correspondence with the degrees of freedom in the function space.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"tags\": []\n },\n \"outputs\": [],\n \"source\": [\n \"warped.cell_data[\\\"VonMises\\\"] = stresses.x.petsc_vec.array\\n\",\n \"warped.set_active_scalars(\\\"VonMises\\\")\\n\",\n \"p = pyvista.Plotter()\\n\",\n \"p.add_mesh(warped)\\n\",\n \"p.show_axes()\\n\",\n \"if not pyvista.OFF_SCREEN:\\n\",\n \" p.show()\\n\",\n \"else:\\n\",\n \" stress_figure = p.screenshot(\\\"stresses.png\\\")\"\n ]\n }\n ],\n \"metadata\": {\n \"jupytext\": {\n \"formats\": \"ipynb,py:light\"\n },\n \"kernelspec\": {\n \"display_name\": \"Python 3 (ipykernel)\",\n \"language\": \"python\",\n \"name\": \"python3\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 3\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython3\",\n \"version\": \"3.10.12\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 4\n}\n" }, { "path": "chapter2/linearelasticity_code.py", "content": "# ---\n# jupyter:\n# jupytext:\n# formats: ipynb,py:light\n# text_representation:\n# extension: .py\n# format_name: light\n# format_version: '1.5'\n# jupytext_version: 1.18.1\n# kernelspec:\n# display_name: Python 3 (ipykernel)\n# language: python\n# name: python3\n# ---\n\n# # Implementation\n# Author: Jørgen S. Dokken\n#\n# In this tutorial, you will learn how to:\n# - Use a vector function space\n# - Create a constant boundary condition on a vector space\n# - Visualize cell-wise constant functions\n# - Compute Von Mises stresses\n#\n# ## Test problem\n# As a test example, we will model a clamped beam deformed under its own weight in 3D.\n# This can be modeled, by setting the right-hand side body force per unit volume to $f=(0,0,-\\rho g)$,\n# where $\\rho$ the density of the beam and $g$ the acceleration of gravity.\n# The beam is box-shaped with length $L$ and has a square cross section of width $W$.\n# We set $u=u_D=(0,0,0)$ at the clamped end, x=0. The rest of the boundary is traction free, that is, we set $T=0$.\n# We start by defining the physical variables used in the program.\n\n# +\nimport pyvista\nfrom dolfinx import mesh, fem, plot, io, default_scalar_type\nfrom dolfinx.fem.petsc import LinearProblem\nfrom mpi4py import MPI\nimport ufl\nimport numpy as np\n\nL = 1.0\nW = 0.2\nmu = 1.0\nrho = 1.0\ndelta = W / L\ngamma = 0.4 * delta**2\nbeta = 1.25\nlambda_ = beta\ng = gamma\n# -\n\n# We then create the mesh, which will consist of hexahedral elements, along with the function space.\n# As we want a vector element with three components, we add `(3, )` or `(domain.geometry.dim, )` to the element tuple to make it a triplet.\n# However, we also could have used `basix.ufl`s functionality,\n# creating a vector element `el = basix.ufl.element(\"Lagrange\", domain.basix_cell(), 1, shape=(domain.geometry.dim,))`,\n# and initializing the function space as `V = dolfinx.fem.functionspace(domain, el)`.\n\ndomain = mesh.create_box(\n MPI.COMM_WORLD,\n [np.array([0, 0, 0]), np.array([L, W, W])],\n [20, 6, 6],\n cell_type=mesh.CellType.hexahedron,\n)\nV = fem.functionspace(domain, (\"Lagrange\", 1, (domain.geometry.dim,)))\n\n\n# ## Boundary conditions\n# As we would like to clamp the boundary at $x=0$, we do this by using a marker function,\n# which locates the facets where $x$ is close to zero by machine precision.\n\n\n# +\ndef clamped_boundary(x):\n return np.isclose(x[0], 0)\n\n\nfdim = domain.topology.dim - 1\nboundary_facets = mesh.locate_entities_boundary(domain, fdim, clamped_boundary)\n\nu_D = np.array([0, 0, 0], dtype=default_scalar_type)\nbc = fem.dirichletbc(u_D, fem.locate_dofs_topological(V, fdim, boundary_facets), V)\n# -\n\n# As we want the traction $T$ over the remaining boundary to be $0$, we create a `dolfinx.fem.Constant`\n\nT = fem.Constant(domain, default_scalar_type((0, 0, 0)))\n\n# We also want to specify the integration measure $\\mathrm{d}s$, which should be the integral over the boundary of our domain.\n# We do this by using `ufl`, and its built in integration measures\n\nds = ufl.Measure(\"ds\", domain=domain)\n\n\n# ## Variational formulation\n# We are now ready to create our variational formulation in close to mathematical syntax, as for the previous problems.\n\n\n# +\ndef epsilon(u):\n return ufl.sym(\n ufl.grad(u)\n ) # Equivalent to 0.5*(ufl.nabla_grad(u) + ufl.nabla_grad(u).T)\n\n\ndef sigma(u):\n return lambda_ * ufl.nabla_div(u) * ufl.Identity(len(u)) + 2 * mu * epsilon(u)\n\n\nu = ufl.TrialFunction(V)\nv = ufl.TestFunction(V)\nf = fem.Constant(domain, default_scalar_type((0, 0, -rho * g)))\na = ufl.inner(sigma(u), epsilon(v)) * ufl.dx\nL = ufl.dot(f, v) * ufl.dx + ufl.dot(T, v) * ds\n# -\n\n# ```{note}\n# Note that we used `nabla_grad` and optionally `nabla_div` for the variational formulation, as opposed to our previous usage of\n# `div` and `grad`. This is because for scalar functions $\\nabla u$ has a clear meaning\n# $\\nabla u = \\left(\\frac{\\partial u}{\\partial x}, \\frac{\\partial u}{\\partial y}, \\frac{\\partial u}{\\partial z} \\right)$.\n#\n# However, if $u$ is vector valued, the meaning is less clear.\n# Some sources define $\\nabla u$ as a matrix with the elements $\\frac{\\partial u_j}{\\partial x_i}$, while other sources prefer\n# $\\frac{\\partial u_i}{\\partial x_j}$.\n# In DOLFINx `grad(u)` is defined as the matrix with elements $\\frac{\\partial u_i}{\\partial x_j}$.\n# However, as it is common in continuum mechanics to use the other definition, `ufl` supplies us with `nabla_grad` for this purpose.\n# ```\n#\n# ## Solve the linear variational problem\n# As in the previous demos, we assemble the matrix and right hand side vector and use PETSc to solve our variational problem\n\nproblem = LinearProblem(\n a,\n L,\n bcs=[bc],\n petsc_options={\"ksp_type\": \"preonly\", \"pc_type\": \"lu\"},\n petsc_options_prefix=\"linear_elasticity\",\n)\nuh = problem.solve()\n\n# ## Visualization\n\n# As in the previous demos, we can either use Pyvista or Paraview for visualization.\n# We start by using Pyvista.\n# In previous tutorials, we have considered scalar values, while the following section considers vectors.\n\n# +\n# Create plotter and pyvista grid\np = pyvista.Plotter()\ntopology, cell_types, geometry = plot.vtk_mesh(V)\ngrid = pyvista.UnstructuredGrid(topology, cell_types, geometry)\n\n# Attach vector values to grid and warp grid by vector\ngrid[\"u\"] = uh.x.array.reshape((geometry.shape[0], 3))\nactor_0 = p.add_mesh(grid, style=\"wireframe\", color=\"k\")\nwarped = grid.warp_by_vector(\"u\", factor=1.5)\nactor_1 = p.add_mesh(warped, show_edges=True)\np.show_axes()\nif not pyvista.OFF_SCREEN:\n p.show()\nelse:\n figure_as_array = p.screenshot(\"deflection.png\")\n# -\n\n# We could also use Paraview for visualizing this.\n# As explained in previous sections, we save the solution with `XDMFFile`.\n# After opening the file `deformation.xdmf` in Paraview and pressing `Apply`,\n# one can press the `Warp by vector button` ![Warp by vector](warp_by_vector.png)\n# or go through the top menu (`Filters->Alphabetical->Warp by Vector`) and press `Apply`.\n# We can also change the color of the deformed beam by changing the value in the\n# color menu ![color](color.png) from `Solid Color` to `Deformation`.\n\nwith io.XDMFFile(domain.comm, \"deformation.xdmf\", \"w\") as xdmf:\n xdmf.write_mesh(domain)\n uh.name = \"Deformation\"\n xdmf.write_function(uh)\n\n# ## Stress computation\n# As soon as the displacement is computed, we can compute various stress measures.\n# We will compute the von Mises stress defined as $\\sigma_m=\\sqrt{\\frac{3}{2}s:s}$ where\n# $s$ is the deviatoric stress tensor $s(u)=\\sigma(u)-\\frac{1}{3}\\mathrm{tr}(\\sigma(u))I$.\n\ns = sigma(uh) - 1.0 / 3 * ufl.tr(sigma(uh)) * ufl.Identity(len(uh))\nvon_Mises = ufl.sqrt(3.0 / 2 * ufl.inner(s, s))\n\n# The `von_Mises` variable is now an expression that must be projected into an appropriate\n# function space so that we can visualize it.\n# As `uh` is a linear combination of first order piecewise continuous functions,\n# the von Mises stresses will be a cell-wise constant function.\n\nV_von_mises = fem.functionspace(domain, (\"DG\", 0))\nstress_expr = fem.Expression(von_Mises, V_von_mises.element.interpolation_points)\nstresses = fem.Function(V_von_mises)\nstresses.interpolate(stress_expr)\n\n# In the previous sections, we have only visualized first order Lagrangian functions.\n# However, the Von Mises stresses are piecewise constant on each cell.\n# Therefore, we modify our plotting routine slightly.\n# The first thing we notice is that we now set values for each cell,\n# which has a one to one correspondence with the degrees of freedom in the function space.\n\nwarped.cell_data[\"VonMises\"] = stresses.x.petsc_vec.array\nwarped.set_active_scalars(\"VonMises\")\np = pyvista.Plotter()\np.add_mesh(warped)\np.show_axes()\nif not pyvista.OFF_SCREEN:\n p.show()\nelse:\n stress_figure = p.screenshot(\"stresses.png\")\n" }, { "path": "chapter2/navierstokes.md", "content": "# The Navier-Stokes equations\nAuthors: Anders Logg and Hans Petter Langtangen\n\nMinor modifications: Jørgen S. Dokken\n\nIn this section, we will solve the incompressible Navier-Stokes equations. This problem combines many of the challenges from our previously studied problems: time-dependencies, non-linearity, and vector-valued variables.\n\n## The PDE problem\n\nThe incompressible Navier-Stokes equations form a system of equations for the velocity $u$ and pressure $p$ in an incompressible fluid\n```{math}\n:label: navier-stokes\n\\rho \\left( \\frac{\\partial u }{\\partial t} + u \\cdot \\nabla u \\right) &= \\nabla \\cdot \\sigma (u, p) + f,\\\\\n\\nabla \\cdot u &= 0 \n```\nThe right-hand side of $f$ is a given force per unit volume and just as for the equations of linear elasticity, $\\sigma(u,p)$ denotes the stress tensor, which for a Newtonian fluid is given by\n```{math}\n:label: navier-stokes-stress\n\\sigma(u, p)=2\\mu \\epsilon (u) - pI,\n```\nwhere $\\epsilon(u)$ is the strain-rate tensor\n```{math}\n \\epsilon(u)=\\frac{1}{2}\\left(\\nabla u + (\\nabla u)^T\\right).\n```\nThe parameter $\\mu$ is the dynamic viscosity. Note that the momentum equation [](navier-stokes) is very similar to the elasticity equation [](elasticity-PDE). The difference is in the two additional terms $\\rho\\left(\\frac{\\partial u}{\\partial t}+ u\\cdot \\nabla u\\right)$ and the different expression for the stress tensor. The two extra terms express the acceleration balanced by the force $F=\\nabla \\cdot \\sigma + f$ per unit volume in Newton's second law of motion.\n\n## Variational formulation\nThe Navier-Stokes equations are different from the time-dependent heat equation in that we need to solve a system of equations and the system of a special type. If we apply the same technique as for the heat equation; that is, replacing the time derivative with a simple difference quotient, we obtain a non-linear system of equations. This in itself is not a problem as we saw for the [non-linear Poisson equation](./nonlinpoisson.md), but the system has a so-called *saddle point structure* and requires special techniques (special preconditioners and iterative methods) to be solved efficiently.\n\nInstead we will apply a simpler and often very efficient approach, known as a *splitting method*. The idea is to consider the two equations in [](navier-stokes) separately. There exist many splitting strategies for the incompressible Navier-Stokes equations. One of the oldest is the method proposed by Chorin {cite}`chorin1968numerical` and Temam {cite}`Temam1969`, often referred to as *Chorin's method*. We will use a modified version of Chorin's method, the so-called incremental pressure correction scheme (IPCS) due to {cite}`goda1979multistep` which gives improved accuracy compared to the original scheme at little extra cost.\n\nThe IPCS scheme involves three steps. First, we compute a *tentative velocity $u$* by advancing the momentum equation by a midpoint finite difference scheme in time, but using $p^n$ from the previous interval. We will also linearize the nonlinear convective term by using the known velocity $u^n$ from the previous time step: $u^n\\cdot \\nabla u^n$. Note that there exists several other methods to linearize this term, such as the Adams-Bashforth method, see {cite}`Guermond1999` and {cite}`QuarteroniSaccoSaleri2010`. The variational problem for the first step is: For the $n+1$th step, find $u^*$ such that\n```{math}\n:label: ipcs-one\n &\\left\\langle \\rho \\frac{u^*-u^n}{\\Delta t}, v\\right\\rangle\n + \\left\\langle \\rho u^n\\cdot \\nabla u^n, v \\right\\rangle\n +\\left\\langle \\sigma(u^{n+\\frac{1}{2}}, p^n), \\epsilon(v)\\right\\rangle\\\\\n &+ \\left\\langle p^n n, v \\right\\rangle_{\\partial\\Omega}\n -\\left\\langle \\mu \\nabla u^{n+\\frac{1}{2}}\\cdot n, v \\right \\rangle_{\\partial\\Omega}=\n \\left\\langle f^{n+1}, v \\right\\rangle.\n```\nThis notation, suitable for problems with many terms in the variational formulations, requires some explanation. \nFirst, we use the short-hand notation\n```{math}\n\\langle v, w \\rangle = \\int_{\\Omega} vw~\\mathrm{d}x, \\qquad\n\\langle v, w \\rangle_{\\partial\\Omega}=\\int_{\\partial\\Omega}vw~\\mathrm{d}s.\n```\nThis allows us to express the variational problem in a more compact way. Second, we use the notation $u^{n+\\frac{1}{2}}$. This notation refers to the value of $u$ at the midpoint of the interval, usually approximated by an arithmetic mean:\n```{math}\n u^{n+\\frac{1}{2}}\\approx \\frac{u^{n}+ u^{n+1}}{2}.\n```\nThird, we notice that the variational problem [](ipcs-one) arises from the integration by parts of the term \n$\\langle -\\nabla \\cdot \\sigma, v\\rangle$. Just as for the [linear elasticity problem](./linearelasticity.md), we obtain\n```{math}\n \\langle -\\nabla \\cdot \\sigma, v\\rangle =\n \\langle \\sigma, \\epsilon(v) \\rangle \n - \\langle T, v\\rangle_{\\partial \\Omega},\n```\nwhere $T=\\sigma \\cdot n$ is the boundary traction. If we solve a problem with a free boundary, we can take $T=0$ on the boundary. However, if we compute the flow through a channel or a pipe and want to model flow that continues into an \"imaginary channel\" at the outflow, we need to treat this term with some care. \nThe assumption we then can make is that the derivative of the velocity in the direction of the channel is zero at the outflow, corresponding to that the flow is \"fully developed\" or doesn't change significantly downstream at the outflow.\nDoing so, the remaining boundary term at the outflow becomes \n$pn - \\mu \\nabla u \\cdot n$, which is the term appearing in the variational problem [](ipcs-one). Note that this argument and the implementation depend exactly on the definition of $\\nabla u$, as either the matrix with components $\\frac{\\partial u_i}{\\partial x_j}$ or $\\frac{\\partial u_j}{\\partial x_i}$.\nWe here choose the latter, $\\frac{\\partial u_j}{\\partial x_i}$,\nwhich means that we must use the UFL-operator `nabla_grad`. If we use the operator `grad` and the definition $\\frac{\\partial u_i}{\\partial x_j}$, we must instead keep the terms $pn-\\mu(\\nabla u)^T \\cdot n$.\n\n```{admonition} The usage of \"nabla_grad\" and \"grad\"\nAs mentioned in the note in [Linear elasticity implementation](./linearelasticity_code) the usage of `nabla_grad` and `grad` has to be interpreted with care. For the Navier-Stokes equations it is important to consider the term $u\\cdot \\nabla u$ which should be interpreted as the vector $w$ with elements\n$w_i=\\sum_{j}\\left(u_j\\frac{\\partial}{\\partial x_j}\\right)u_i = \\sum_j u_j\\frac{\\partial u_i}{\\partial x_j}$. \nThis term can be implemented in FEniCSx as either \n`grad(u)*u`, since this expression becomes $\\sum_j\\frac{\\partial u_i}{\\partial x_j}u_j$, or as `dot(u, nabla_grad(u))` since this \nexpression becomes $\\sum_i u_i\\frac{\\partial u_j}{\\partial x_i}$. We will use the notation `dot(u, nabla_grad(u))` below since it corresponds more closely to the standard notation $u\\cdot \\nabla u$.\n```\n\nWe now move on to the second step in our splitting scheme for the incompressible Navier-Stokes equations. In the first step, we computed the *tentative velocity* $u^*$ based on the pressure from the previous time step. \nWe may now use the computed tentative velocity to compute the new pressure $p^{n+1}$:\n```{math}\n:label: ipcs-two\n \\langle \\nabla p^{n+1}, \\nabla q \\rangle = \\langle \\nabla p^n, \\nabla q\\rangle - \\frac{\\rho}{\\Delta t}\\langle \\nabla \\cdot u^*, q\\rangle.\n```\nNote here that $q$ is a scalar-valued test function from the pressure space, whereas the test function $v$ in [](ipcs-one) is a vector-valued test function from the velocity space.\n\nOne way to think about this step is to subtract the Navier-Stokes momentum equation [](navier-stokes) expressed in terms of the tentative velocity $u^*$ and the pressure $p^n$ from the momentum equation expressed in terms of the velocity $u^{n+1}$ and pressure $p^{n+1}$. This results in the equation\n```{math}\n\\frac{\\rho (u^{n+1}-u^*)}{\\Delta t}+\\nabla p^{n+1}- \\nabla p^n = 0.\n```\nTaking the divergence and requiring that $\\nabla \\cdot u^{n+1}=0$ by the Navier-Stokes continuity equation, we obtain the equation\n```{math}\n:label: ipcs-tmp\n - \\frac{\\rho \\nabla\\cdot u^*}{\\Delta t}+ \\nabla^2p^{n+1}-\\nabla^2p^n=0,\n```\nwhich is the Poisson problem for the pressure $p^{n+1}$ resulting in the variational formulation [](ipcs-two).\n\nFinally, we compute the corrected velocity $u^{n+1}$ from the equation [](ipcs-tmp). Multiplying this equation by a test function $v$, we obtain\n```{math}\n \\rho \\langle (u^{n+1} - u^*), v\\rangle= -\\Delta t\\langle \\nabla(p^{n+1}-p^n), v\\rangle\n```\n\nIn summary, we may thus solve the incompressible Navier-Stokes equations efficiently by solving a sequence of three linear variational problems in each step.\n\n## References\n```{bibliography}\n:filter: docname in docnames\n```\n" }, { "path": "chapter2/nonlinpoisson.md", "content": "# A nonlinear Poisson equation\nAuthors: Anders Logg and Hans Petter Langtangen\n\nWe shall now address how to solve nonlinear PDEs. We will see that nonlinear problems introduce some subtle differences in how we define the variational form.\n\n## The PDE problem\nAs a model for the solution of nonlinear PDEs, we take the following nonlinear Poisson equation\n\\begin{align}\n - \\nabla \\cdot (q(u) \\nabla u)&=f && \\text{in } \\Omega,\\\\\n u&=u_D && \\text{on } \\partial \\Omega.\n\\end{align}\nThe coefficients $q(u)$ make the problem nonlinear (unless $q(u)$ is constant in $u$).\n\n## Variational formulation\nAs usual, we multiply the PDE by a test function $v\\in \\hat{V}$, integrate over the domain, and integrate second-order derivatives by parts. The boundary integrals arising from integration by parts vanish wherever we employ Dirichlet conditions. The resulting variational formulation of our model problem becomes:\n\nFind $u\\in V$ such that\n\\begin{align}\n F(u; v)&=0 && \\forall v \\in \\hat{V},\n\\end{align}\nwhere\n\\begin{align}\n F(u; v)&=\\int_{\\Omega}(q(u)\\nabla u \\cdot \\nabla v - fv)\\mathrm{d}x,\n\\end{align}\nand \n\\begin{align}\n V&=\\left\\{v\\in H^1(\\Omega)\\vert v=u_D \\text{ on } \\partial \\Omega \\right\\}\\\\\n \\hat{V}&=\\left\\{v\\in H^1(\\Omega)\\vert v=0 \\text{ on } \\partial \\Omega \\right\\}\n\\end{align}\n\nThe discrete problem arises as usual by restricting $V$ and $\\hat{V}$ to a pair of discrete spaces. The discrete nonlinear problem can therefore be written as:\n\nFind $u_h \\in V_h$ such that\n\\begin{align}\nF(u_h; v) &=0 \\quad \\forall v \\in \\hat{V}_h,\n\\end{align}\nwith $u_h=\\sum_{j=1}^N U_j\\phi_j$. Since $F$ is nonlinear in $u$, the variational statement gives rise to a system of nonlinear algebraic equations in the unknowns $U_1,\\dots,U_N$.\n" }, { "path": "chapter2/nonlinpoisson_code.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Implementation\\n\",\n \"\\n\",\n \"Author: Jørgen S. Dokken\\n\",\n \"\\n\",\n \"## Test problem\\n\",\n \"To solve a test problem, we need to choose the right hand side $f$, the coefficient $q(u)$, and the boundary $u_D$.\\n\",\n \"Previously, we have worked with manufactured solutions that can be reproduced without approximation errors.\\n\",\n \"This is more difficult in nonlinear problems, and the algebra is more tedious.\\n\",\n \"However, we will utilize the UFL differentiation capabilities to obtain a manufactured solution.\\n\",\n \"\\n\",\n \"For this problem, we will choose $q(u) = 1 + u^2$ and define a two dimensional manufactured solution\\n\",\n \"that is linear in $x$ and $y$:\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"import ufl\\n\",\n \"import numpy\\n\",\n \"\\n\",\n \"from mpi4py import MPI\\n\",\n \"\\n\",\n \"from dolfinx import mesh, fem\\n\",\n \"from dolfinx.fem.petsc import NonlinearProblem\\n\",\n \"\\n\",\n \"\\n\",\n \"def q(u):\\n\",\n \" return 1 + u**2\\n\",\n \"\\n\",\n \"\\n\",\n \"domain = mesh.create_unit_square(MPI.COMM_WORLD, 10, 10)\\n\",\n \"x = ufl.SpatialCoordinate(domain)\\n\",\n \"u_ufl = 1 + x[0] + 2 * x[1]\\n\",\n \"f = -ufl.div(q(u_ufl) * ufl.grad(u_ufl))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"Note that since `x` is a 2D vector, the first component (index 0) represents $x$,\\n\",\n \"while the second component (index 1) represents $y$.\\n\",\n \"The resulting function `f` can be directly used in variational formulations in DOLFINx.\\n\",\n \"\\n\",\n \"As we now have defined our source term and an exact solution,\\n\",\n \"we can create the appropriate function space and boundary conditions.\\n\",\n \"Note that as we have already defined the exact solution,\\n\",\n \"we only have to convert it to a Python function that can be evaluated in the interpolation function.\\n\",\n \"We do this by employing the Python `eval` and `lambda`-functions.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"3\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"V = fem.functionspace(domain, (\\\"Lagrange\\\", 1))\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"def u_exact(x):\\n\",\n \" return eval(str(u_ufl))\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"u_D = fem.Function(V)\\n\",\n \"u_D.interpolate(u_exact)\\n\",\n \"fdim = domain.topology.dim - 1\\n\",\n \"boundary_facets = mesh.locate_entities_boundary(\\n\",\n \" domain, fdim, lambda x: numpy.full(x.shape[1], True, dtype=bool)\\n\",\n \")\\n\",\n \"bc = fem.dirichletbc(u_D, fem.locate_dofs_topological(V, fdim, boundary_facets))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We are now ready to define the variational formulation.\\n\",\n \"Note that as the problem is nonlinear, we have to replace the `TrialFunction` with a `Function`,\\n\",\n \"which serves as the unknown of our problem.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"uh = fem.Function(V)\\n\",\n \"v = ufl.TestFunction(V)\\n\",\n \"F = q(uh) * ufl.dot(ufl.grad(uh), ufl.grad(v)) * ufl.dx - f * v * ufl.dx\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Newton's method\\n\",\n \"The next step is to define the non-linear problem.\\n\",\n \"As it is non-linear we will use [Newtons method](https://en.wikipedia.org/wiki/Newton%27s_method).\\n\",\n \"For details about how to implement a Newton solver, see [Custom Newton solvers](../chapter4/newton-solver.ipynb).\\n\",\n \"Newton's method requires methods for evaluating the residual `F` (including application of boundary conditions),\\n\",\n \"as well as a method for computing the Jacobian matrix.\\n\",\n \"DOLFINx provides the function `NonlinearProblem` that implements these methods.\\n\",\n \"In addition to the boundary conditions, you can supply the variational form for the Jacobian\\n\",\n \"(computed if not supplied), and form and JIT parameters,\\n\",\n \"see the [JIT parameters section](../chapter4/compiler_parameters.ipynb).\\n\",\n \"The DOLFINx `NonlinearProblem` is an interface to the [PETSc SNES solver](https://petsc.org/release/manual/snes/),\\n\",\n \"which provides a large variety of options.\\n\",\n \"In this example, we will turn of line-search, to run the problem with a standard Newton method.\\n\",\n \"We can also provide PETSc options for the underlying linear solver (KSP) and preconditioner (PC).\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"petsc_options = {\\n\",\n \" \\\"snes_type\\\": \\\"newtonls\\\",\\n\",\n \" \\\"snes_linesearch_type\\\": \\\"none\\\",\\n\",\n \" \\\"snes_atol\\\": 1e-6,\\n\",\n \" \\\"snes_rtol\\\": 1e-6,\\n\",\n \" \\\"snes_monitor\\\": None,\\n\",\n \" \\\"ksp_error_if_not_converged\\\": True,\\n\",\n \" \\\"ksp_type\\\": \\\"gmres\\\",\\n\",\n \" \\\"ksp_rtol\\\": 1e-8,\\n\",\n \" \\\"ksp_monitor\\\": None,\\n\",\n \" \\\"pc_type\\\": \\\"hypre\\\",\\n\",\n \" \\\"pc_hypre_type\\\": \\\"boomeramg\\\",\\n\",\n \" \\\"pc_hypre_boomeramg_max_iter\\\": 1,\\n\",\n \" \\\"pc_hypre_boomeramg_cycle_type\\\": \\\"v\\\",\\n\",\n \"}\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"10\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"outputs\": [],\n \"source\": [\n \"problem = NonlinearProblem(\\n\",\n \" F,\\n\",\n \" uh,\\n\",\n \" bcs=[bc],\\n\",\n \" petsc_options=petsc_options,\\n\",\n \" petsc_options_prefix=\\\"nonlinpoisson\\\",\\n\",\n \")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"11\",\n \"metadata\": {},\n \"source\": [\n \"We are now ready to solve the non-linear problem.\\n\",\n \"We assert that the solver has converged and print the number of iterations.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"12\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"problem.solve()\\n\",\n \"converged = problem.solver.getConvergedReason()\\n\",\n \"num_iter = problem.solver.getIterationNumber()\\n\",\n \"assert converged > 0, f\\\"Solver did not converge, got {converged}.\\\"\\n\",\n \"print(\\n\",\n \" f\\\"Solver converged after {num_iter} iterations with converged reason {converged}.\\\"\\n\",\n \")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"```{admonition} Convergence checks\\n\",\n \"We can remove the assertion above, and let PETSc do the error handling by adding\\n\",\n \"`snes_error_if_not_converged: True` to the `petsc_options` dictionary.\\n\",\n \"This will raise an exception if the solver does not converge.\\n\",\n \"We can also set the `snes_atol` and `snes_rtol` or `snes_stol` to control the convergence criteria\\n\",\n \"or create custom convergence checks, see [SNES: Convergence checks](https://petsc.org/main/manual/snes/#convergence-tests)\\n\",\n \"for more details.\\n\",\n \"```\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We observe that the solver converges after $8$ iterations.\\n\",\n \"If we think of the problem in terms of finite differences on a uniform mesh,\\n\",\n \"$\\\\mathcal{P}_1$ elements mimic standard second-order finite differences,\\n\",\n \"which compute the derivative of a linear or quadratic funtion exactly.\\n\",\n \"Here $\\\\nabla u$ is a constant vector, which is multiplied by $1+u^2$,\\n\",\n \"giving a second order polynomial in $x$ and $y$, which the finite difference operator would compute exactly.\\n\",\n \"We can therefore, even with $\\\\mathcal{P}_1$ elements, expect the manufactured solution to be\\n\",\n \"reproduced by the numerical method.\\n\",\n \"However, if we had chosen a nonlinearity, such as $1+u^4$, this would not be the case,\\n\",\n \"and we would need to verify convergence rates.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"tags\": []\n },\n \"outputs\": [],\n \"source\": [\n \"# Compute L2 error and error at nodes\\n\",\n \"V_ex = fem.functionspace(domain, (\\\"Lagrange\\\", 2))\\n\",\n \"u_ex = fem.Function(V_ex)\\n\",\n \"u_ex.interpolate(u_exact)\\n\",\n \"error_local = fem.assemble_scalar(fem.form((uh - u_ex) ** 2 * ufl.dx))\\n\",\n \"error_L2 = numpy.sqrt(domain.comm.allreduce(error_local, op=MPI.SUM))\\n\",\n \"if domain.comm.rank == 0:\\n\",\n \" print(f\\\"L2-error: {error_L2:.2e}\\\")\\n\",\n \"\\n\",\n \"# Compute values at mesh vertices\\n\",\n \"error_max = domain.comm.allreduce(\\n\",\n \" numpy.max(numpy.abs(uh.x.array - u_D.x.array)), op=MPI.MAX\\n\",\n \")\\n\",\n \"if domain.comm.rank == 0:\\n\",\n \" print(f\\\"Error_max: {error_max:.2e}\\\")\"\n ]\n }\n ],\n \"metadata\": {\n \"jupytext\": {\n \"formats\": \"ipynb,py:light\"\n },\n \"kernelspec\": {\n \"display_name\": \"Python 3 (ipykernel)\",\n \"language\": \"python\",\n \"name\": \"python3\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 3\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython3\",\n \"version\": \"3.10.12\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 4\n}\n" }, { "path": "chapter2/nonlinpoisson_code.py", "content": "# ---\n# jupyter:\n# jupytext:\n# formats: ipynb,py:light\n# text_representation:\n# extension: .py\n# format_name: light\n# format_version: '1.5'\n# jupytext_version: 1.18.1\n# kernelspec:\n# display_name: Python 3 (ipykernel)\n# language: python\n# name: python3\n# ---\n\n# # Implementation\n#\n# Author: Jørgen S. Dokken\n#\n# ## Test problem\n# To solve a test problem, we need to choose the right hand side $f$, the coefficient $q(u)$, and the boundary $u_D$.\n# Previously, we have worked with manufactured solutions that can be reproduced without approximation errors.\n# This is more difficult in nonlinear problems, and the algebra is more tedious.\n# However, we will utilize the UFL differentiation capabilities to obtain a manufactured solution.\n#\n# For this problem, we will choose $q(u) = 1 + u^2$ and define a two dimensional manufactured solution\n# that is linear in $x$ and $y$:\n\n# +\nimport ufl\nimport numpy\n\nfrom mpi4py import MPI\n\nfrom dolfinx import mesh, fem\nfrom dolfinx.fem.petsc import NonlinearProblem\n\n\ndef q(u):\n return 1 + u**2\n\n\ndomain = mesh.create_unit_square(MPI.COMM_WORLD, 10, 10)\nx = ufl.SpatialCoordinate(domain)\nu_ufl = 1 + x[0] + 2 * x[1]\nf = -ufl.div(q(u_ufl) * ufl.grad(u_ufl))\n# -\n\n# Note that since `x` is a 2D vector, the first component (index 0) represents $x$,\n# while the second component (index 1) represents $y$.\n# The resulting function `f` can be directly used in variational formulations in DOLFINx.\n#\n# As we now have defined our source term and an exact solution,\n# we can create the appropriate function space and boundary conditions.\n# Note that as we have already defined the exact solution,\n# we only have to convert it to a Python function that can be evaluated in the interpolation function.\n# We do this by employing the Python `eval` and `lambda`-functions.\n\nV = fem.functionspace(domain, (\"Lagrange\", 1))\n\n\ndef u_exact(x):\n return eval(str(u_ufl))\n\n\nu_D = fem.Function(V)\nu_D.interpolate(u_exact)\nfdim = domain.topology.dim - 1\nboundary_facets = mesh.locate_entities_boundary(\n domain, fdim, lambda x: numpy.full(x.shape[1], True, dtype=bool)\n)\nbc = fem.dirichletbc(u_D, fem.locate_dofs_topological(V, fdim, boundary_facets))\n\n# We are now ready to define the variational formulation.\n# Note that as the problem is nonlinear, we have to replace the `TrialFunction` with a `Function`,\n# which serves as the unknown of our problem.\n\nuh = fem.Function(V)\nv = ufl.TestFunction(V)\nF = q(uh) * ufl.dot(ufl.grad(uh), ufl.grad(v)) * ufl.dx - f * v * ufl.dx\n\n# ## Newton's method\n# The next step is to define the non-linear problem.\n# As it is non-linear we will use [Newtons method](https://en.wikipedia.org/wiki/Newton%27s_method).\n# For details about how to implement a Newton solver, see [Custom Newton solvers](../chapter4/newton-solver.ipynb).\n# Newton's method requires methods for evaluating the residual `F` (including application of boundary conditions),\n# as well as a method for computing the Jacobian matrix.\n# DOLFINx provides the function `NonlinearProblem` that implements these methods.\n# In addition to the boundary conditions, you can supply the variational form for the Jacobian\n# (computed if not supplied), and form and JIT parameters,\n# see the [JIT parameters section](../chapter4/compiler_parameters.ipynb).\n# The DOLFINx `NonlinearProblem` is an interface to the [PETSc SNES solver](https://petsc.org/release/manual/snes/),\n# which provides a large variety of options.\n# In this example, we will turn of line-search, to run the problem with a standard Newton method.\n# We can also provide PETSc options for the underlying linear solver (KSP) and preconditioner (PC).\n\npetsc_options = {\n \"snes_type\": \"newtonls\",\n \"snes_linesearch_type\": \"none\",\n \"snes_atol\": 1e-6,\n \"snes_rtol\": 1e-6,\n \"snes_monitor\": None,\n \"ksp_error_if_not_converged\": True,\n \"ksp_type\": \"gmres\",\n \"ksp_rtol\": 1e-8,\n \"ksp_monitor\": None,\n \"pc_type\": \"hypre\",\n \"pc_hypre_type\": \"boomeramg\",\n \"pc_hypre_boomeramg_max_iter\": 1,\n \"pc_hypre_boomeramg_cycle_type\": \"v\",\n}\n\nproblem = NonlinearProblem(\n F,\n uh,\n bcs=[bc],\n petsc_options=petsc_options,\n petsc_options_prefix=\"nonlinpoisson\",\n)\n\n\n# We are now ready to solve the non-linear problem.\n# We assert that the solver has converged and print the number of iterations.\n\nproblem.solve()\nconverged = problem.solver.getConvergedReason()\nnum_iter = problem.solver.getIterationNumber()\nassert converged > 0, f\"Solver did not converge, got {converged}.\"\nprint(\n f\"Solver converged after {num_iter} iterations with converged reason {converged}.\"\n)\n\n# ```{admonition} Convergence checks\n# We can remove the assertion above, and let PETSc do the error handling by adding\n# `snes_error_if_not_converged: True` to the `petsc_options` dictionary.\n# This will raise an exception if the solver does not converge.\n# We can also set the `snes_atol` and `snes_rtol` or `snes_stol` to control the convergence criteria\n# or create custom convergence checks, see [SNES: Convergence checks](https://petsc.org/main/manual/snes/#convergence-tests)\n# for more details.\n# ```\n\n\n# We observe that the solver converges after $8$ iterations.\n# If we think of the problem in terms of finite differences on a uniform mesh,\n# $\\mathcal{P}_1$ elements mimic standard second-order finite differences,\n# which compute the derivative of a linear or quadratic funtion exactly.\n# Here $\\nabla u$ is a constant vector, which is multiplied by $1+u^2$,\n# giving a second order polynomial in $x$ and $y$, which the finite difference operator would compute exactly.\n# We can therefore, even with $\\mathcal{P}_1$ elements, expect the manufactured solution to be\n# reproduced by the numerical method.\n# However, if we had chosen a nonlinearity, such as $1+u^4$, this would not be the case,\n# and we would need to verify convergence rates.\n\n# +\n# Compute L2 error and error at nodes\nV_ex = fem.functionspace(domain, (\"Lagrange\", 2))\nu_ex = fem.Function(V_ex)\nu_ex.interpolate(u_exact)\nerror_local = fem.assemble_scalar(fem.form((uh - u_ex) ** 2 * ufl.dx))\nerror_L2 = numpy.sqrt(domain.comm.allreduce(error_local, op=MPI.SUM))\nif domain.comm.rank == 0:\n print(f\"L2-error: {error_L2:.2e}\")\n\n# Compute values at mesh vertices\nerror_max = domain.comm.allreduce(\n numpy.max(numpy.abs(uh.x.array - u_D.x.array)), op=MPI.MAX\n)\nif domain.comm.rank == 0:\n print(f\"Error_max: {error_max:.2e}\")\n" }, { "path": "chapter2/ns_code1.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Test problem 1: Channel flow (Poiseuille flow)\\n\",\n \"\\n\",\n \"Authors: Anders Logg and Hans Petter Langtangen\\n\",\n \"\\n\",\n \"Adapted to DOLFINx by: Jørgen S. Dokken\\n\",\n \"\\n\",\n \"In this section, you will learn how to:\\n\",\n \"- Solve the Navier-Stokes problem using a splitting scheme\\n\",\n \"- Visualize functions from higher order Lagrangian spaces\\n\",\n \"\\n\",\n \"In this section, we will compute the flow between two infinite plates, so-called channel or Poiseuille flow.\\n\",\n \"As we shall see, this problem has an analytical solution.\\n\",\n \"Let $H$ be the distance between the plates and $L$ the length of the channel. There are no body forces.\\n\",\n \"\\n\",\n \"We may scale the problem first to get rid of seemingly independent physical parameters.\\n\",\n \"The physics of this problem are governed by viscous effects only, in the direction perpendicular to the flow,\\n\",\n \"so a time scale should be based on diffusion across the channel: $t_v=H^2/\\\\nu$.\\n\",\n \"We let $U$, some characteristic inflow velocity, be the velocity scale and $H$ the spatial scale.\\n\",\n \"The pressure scale is taken as the characteristic shear stress, $\\\\mu U/H$, as this is a primary example of shear flow.\\n\",\n \"Inserting $\\\\bar{x}=x/H, \\\\bar{y}=y/H, \\\\bar{z}=z/H, \\\\bar{u}=u/U, \\\\bar{p}=Hp/{\\\\mu U}$, and $\\\\bar{t}=H^2/\\\\nu$\\n\",\n \"in the equations results in the scaled Navier-Stokes equations (dropping the bars after scaling)\\n\",\n \"```{math}\\n\",\n \":label: ns-scaled\\n\",\n \"\\\\frac{\\\\partial u}{\\\\partial t}+ \\\\mathrm{Re} u \\\\cdot \\\\nabla u &= -\\\\nabla p + \\\\nabla^2 u,\\\\\\\\\\n\",\n \"\\\\nabla \\\\cdot u &=0.\\n\",\n \"```\\n\",\n \"A detailed derivation for scaling of the Navier-Stokes equation for a large variety of physical situations\\n\",\n \"can be found in {cite}`Langtangen2016scaling` (Chapter 4.2) by Hans Petter Langtangen and Geir K. Pedersen.\\n\",\n \"\\n\",\n \"Here, $\\\\mathrm{Re}=\\\\rho UH/\\\\mu$ is the Reynolds number.\\n\",\n \"Because of the time and pressure scales, which are different from convection dominated fluid flow,\\n\",\n \"the Reynolds number is associated with the convective term and not the viscosity term.\\n\",\n \"\\n\",\n \"The exact solution is derived by assuming $u=(u_x(x,y,z),0,0)$ with the $x$-axis pointing along the channel.\\n\",\n \"Since $\\\\nabla \\\\cdot u = 0$, $u$ cannot be dependent on $x$.\\n\",\n \"\\n\",\n \"The physics of channel flow is also two-dimensional so we can omit the $z$-coordinate\\n\",\n \"(more precisely: $\\\\partial/\\\\partial z = 0$).\\n\",\n \"Inserting $u=(u_x, 0, 0)$ in the (scaled) governing equations gives $u_x''(y)=\\\\frac{\\\\partial p}{\\\\partial x}$.\\n\",\n \"Differentiating this equation with respect to $x$ shows that\\n\",\n \"$\\\\frac{\\\\partial^2p}{\\\\partial x^2}=0$ so $\\\\partial p/\\\\partial x$ is a constant here called $-\\\\beta$.\\n\",\n \"This is the driving force of the flow and can be specified as a known parameter in the problem.\\n\",\n \"Integrating $u_x''(x,y)=-\\\\beta$ over the width of the channel, $[0,1]$,\\n\",\n \"and requiring $u=(0,0,0)$ at the channel walls, results in $u_x=\\\\frac{1}{2}\\\\beta y(1-y)$.\\n\",\n \"The characteristic inlet velocity $U$ can be taken as the maximum inflow at $y=0.5$, implying $\\\\beta=8$.\\n\",\n \"The length of the channel, $L/H$ in the scaled model, has no impact on the result,\\n\",\n \"so for simplicity we just compute on the unit square.\\n\",\n \"Mathematically, the pressure must be prescribed at a point, but since $p$ does not depend on $y$,\\n\",\n \"we can set $p$ to a known value, e.g. zero, along the outlet boundary $x=1$.\\n\",\n \"The result is $p(x)=8(1-x)$ and $u_x=4y(1-y)$.\\n\",\n \"\\n\",\n \"The boundary conditions can be set as $p=8$ at $x=0$, $p=0$ at $x=1$ and $u=(0,0,0)$ on the walls $y=0,1$.\\n\",\n \"This defines the pressure drop and should result in unit maximum velocity at the inlet and outlet and\\n\",\n \" a parabolic velocity profile without no further specifications.\\n\",\n \"Note that it is only meaningful to solve the Navier-Stokes equations in 2D or 3D geometries,\\n\",\n \"although the underlying mathematical problem collapses to two $1D$ problems, one for $u_x(y)$ and one for $p(x)$.\\n\",\n \"\\n\",\n \"The scaled model is not so easy to simulate using a standard Navier-Stokes solver with dimensions.\\n\",\n \"However, one can argue that the convection term is zero, so the Re coefficient in front of this term in\\n\",\n \"the scaled PDEs is not important and can be set to unity.\\n\",\n \"In that case, setting $\\\\rho=\\\\mu=1$ in the original Navier-Stokes equations resembles the scaled model.\\n\",\n \"\\n\",\n \"For a specific engineering problem one wants to simulate a specific fluid and set corresponding parameters.\\n\",\n \"A general solver is therefore most naturally implemented with dimensions and using the original physical parameters.\\n\",\n \"However, scaling may greatly simplify numerical simulations.\\n\",\n \"First of all, it shows that all fluids behave in the same way;\\n\",\n \"it does not matter whether we have oil, gas, or water flowing between two plates.\\n\",\n \"Secondly, it does not matter how fast the flow is, up to some critical value of the Reynolds number where the\\n\",\n \"flow becomes unstable and transitions to a complicated turbulent flow of totally different nature.\\n\",\n \"This means that one simulation is enough to cover all types of channel flow!\\n\",\n \"In other applications, scaling shows that it might be necessary to just set the fraction of some parameters\\n\",\n \"(dimensionless numbers) rather than the parameters themselves.\\n\",\n \"This simplifies exploring the input parameter space which is often the purpose of simulation.\\n\",\n \"Frequently, the scaled problem is run by setting some of the input parameters with dimension\\n\",\n \"to fixed values (often unity).\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Implementation\\n\",\n \"\\n\",\n \"Author: Jørgen S. Dokken\\n\",\n \"\\n\",\n \"As in the previous example, we load the DOLFINx module, along with the `mpi4py` module,\\n\",\n \"and create the unit square mesh and define the run-time and temporal discretization\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"tags\": []\n },\n \"outputs\": [],\n \"source\": [\n \"from mpi4py import MPI\\n\",\n \"from petsc4py import PETSc\\n\",\n \"import numpy as np\\n\",\n \"import pyvista\\n\",\n \"\\n\",\n \"from dolfinx.fem import (\\n\",\n \" Constant,\\n\",\n \" Function,\\n\",\n \" extract_function_spaces,\\n\",\n \" functionspace,\\n\",\n \" assemble_scalar,\\n\",\n \" dirichletbc,\\n\",\n \" form,\\n\",\n \" locate_dofs_geometrical,\\n\",\n \")\\n\",\n \"from dolfinx.fem.petsc import (\\n\",\n \" assemble_matrix,\\n\",\n \" assemble_vector,\\n\",\n \" apply_lifting,\\n\",\n \" create_vector,\\n\",\n \" set_bc,\\n\",\n \")\\n\",\n \"from dolfinx.io import VTXWriter\\n\",\n \"from dolfinx.mesh import create_unit_square\\n\",\n \"from dolfinx.plot import vtk_mesh\\n\",\n \"from basix.ufl import element\\n\",\n \"from ufl import (\\n\",\n \" FacetNormal,\\n\",\n \" Identity,\\n\",\n \" TestFunction,\\n\",\n \" TrialFunction,\\n\",\n \" div,\\n\",\n \" dot,\\n\",\n \" ds,\\n\",\n \" dx,\\n\",\n \" inner,\\n\",\n \" lhs,\\n\",\n \" nabla_grad,\\n\",\n \" rhs,\\n\",\n \" sym,\\n\",\n \")\\n\",\n \"\\n\",\n \"mesh = create_unit_square(MPI.COMM_WORLD, 10, 10)\\n\",\n \"t = 0.0\\n\",\n \"T = 10.0\\n\",\n \"num_steps = 500\\n\",\n \"dt = T / num_steps\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"As opposed to the previous demos, we will create our two function spaces using the `ufl` element definitions as input\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"tags\": []\n },\n \"outputs\": [],\n \"source\": [\n \"v_cg2 = element(\\\"Lagrange\\\", mesh.basix_cell(), 2, shape=(mesh.geometry.dim,))\\n\",\n \"s_cg1 = element(\\\"Lagrange\\\", mesh.basix_cell(), 1)\\n\",\n \"V = functionspace(mesh, v_cg2)\\n\",\n \"Q = functionspace(mesh, s_cg1)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"The first space `V` is a vector valued function space for the velocity,\\n\",\n \"while `Q` is a scalar valued function space for pressure.\\n\",\n \"We use piecewise quadratic elements for the velocity and piecewise linear elements for the pressure.\\n\",\n \"One can easily create vector-valued function spaces with other dimensions by replacing\\n\",\n \"`shape=(mesh.geometry.dim, )` with something else, like\\n\",\n \"```\\n\",\n \"v_cg basix.ufl.element(\\\"Lagrange\\\", mesh.basix_cell(), 2, shape=(10,))\\n\",\n \"```\\n\",\n \"or\\n\",\n \"```\\n\",\n \"tensor_element = basix.ufl.element(\\\"Lagrange\\\", mesh.basix_cell(), 2, shape=(3, 3))\\n\",\n \"```\\n\",\n \"or\\n\",\n \"```\\n\",\n \"tensor_element = basix.ufl.element(\\\"Lagrange\\\", mesh.basix_cell(), 2, shape=(3, 2, 4))\\n\",\n \"```\\n\",\n \"\\n\",\n \"\\n\",\n \"```{admonition} Stable finite element spaces for the Navier-Stokes equation\\n\",\n \"It is well-known that certain finite element spaces are not *stable* for the Navier-Stokes equations,\\n\",\n \"or even for the simpler Stokes equation.\\n\",\n \"The prime example of an unstable pair of finite element spaces is to use first order degree continuous\\n\",\n \"piecewise polynomials for both the velocity and the pressure.\\n\",\n \"Using an unstable pair of spaces typically results in a solution with *spurious* (unwanted, non-physical)\\n\",\n \"oscillations in the pressure solution.\\n\",\n \"The simple remedy is to use continuous piecewise quadratic elements for the velocity and continuous\\n\",\n \"piecewise linear elements for the pressure.\\n\",\n \"Together, these elements form the so-called *Taylor-Hood* element.\\n\",\n \"Spurious oscillations may occur also for splitting methods if an unstable element pair is used.\\n\",\n \"\\n\",\n \"Since we have two different function spaces, we need to create two sets of trial and test functions:\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"outputs\": [],\n \"source\": [\n \"u = TrialFunction(V)\\n\",\n \"v = TestFunction(V)\\n\",\n \"p = TrialFunction(Q)\\n\",\n \"q = TestFunction(Q)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"As we have seen in [Linear elasticity problem](./linearelasticity_code) we can use Python-functions\\n\",\n \"to create the different Dirichlet conditions.\\n\",\n \"For this problem, we have three Dirichlet condition:\\n\",\n \"First, we will set $u=0$ at the walls of the channel, that is at $y=0$ and $y=1$.\\n\",\n \"In this case, we will use `dolfinx.fem.locate_dofs_geometrical`\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"def walls(x):\\n\",\n \" return np.logical_or(np.isclose(x[1], 0), np.isclose(x[1], 1))\\n\",\n \"\\n\",\n \"\\n\",\n \"wall_dofs = locate_dofs_geometrical(V, walls)\\n\",\n \"u_noslip = np.array((0,) * mesh.geometry.dim, dtype=PETSc.ScalarType)\\n\",\n \"bc_noslip = dirichletbc(u_noslip, wall_dofs, V)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"Second, we will set $p=8$ at the inflow ($x=0$)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"def inflow(x):\\n\",\n \" return np.isclose(x[0], 0)\\n\",\n \"\\n\",\n \"\\n\",\n \"inflow_dofs = locate_dofs_geometrical(Q, inflow)\\n\",\n \"bc_inflow = dirichletbc(PETSc.ScalarType(8), inflow_dofs, Q)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"11\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"And finally, $p=0$ at the outflow ($x=1$).\\n\",\n \"This will result in a pressure gradient that will accelerate the flow from the initial state with zero velocity.\\n\",\n \"At the end, we collect the boundary conditions for the velocity and pressure in Python lists so we\\n\",\n \"can easily access them in the following computation.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"12\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"def outflow(x):\\n\",\n \" return np.isclose(x[0], 1)\\n\",\n \"\\n\",\n \"\\n\",\n \"outflow_dofs = locate_dofs_geometrical(Q, outflow)\\n\",\n \"bc_outflow = dirichletbc(PETSc.ScalarType(0), outflow_dofs, Q)\\n\",\n \"bcu = [bc_noslip]\\n\",\n \"bcp = [bc_inflow, bc_outflow]\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"13\",\n \"metadata\": {},\n \"source\": [\n \"We now move on to the definition of the three variational forms, one for each step in the IPCS scheme.\\n\",\n \"Let us look at the definition of the first variational problem and the relevant parameters.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"14\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"outputs\": [],\n \"source\": [\n \"u_n = Function(V)\\n\",\n \"u_n.name = \\\"u_n\\\"\\n\",\n \"U = 0.5 * (u_n + u)\\n\",\n \"n = FacetNormal(mesh)\\n\",\n \"f = Constant(mesh, PETSc.ScalarType((0, 0)))\\n\",\n \"k = Constant(mesh, PETSc.ScalarType(dt))\\n\",\n \"mu = Constant(mesh, PETSc.ScalarType(1))\\n\",\n \"rho = Constant(mesh, PETSc.ScalarType(1))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"15\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"```{admonition} Usage of \\\"dolfinx.fem.Constant\\\"\\n\",\n \"Note that we have wrapped several parameters as constants.\\n\",\n \"This is to reduce the compilation-time of the variational formulations.\\n\",\n \"By wrapping them as a constant, we can change the variable\\n\",\n \"```\\n\",\n \"The next step is to set up the variational form of the first step.\\n\",\n \"As the variational problem contains a mix of known and unknown quantities,\\n\",\n \"we will use the following naming convention: `u` (mathematically $u^{n+1}$) is known as a trial function\\n\",\n \"in the variational form. `u_` is the most recently computed approximation\\n\",\n \"($u^{n+1}$ available as a `Function` object), `u_n` is $u^n$, and the same convention\\n\",\n \"goes for `p,p_` ($p^{n+1}$) and `p_n` (p^n).\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"16\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"def epsilon(u):\\n\",\n \" \\\"\\\"\\\"Strain-rate tensor.\\\"\\\"\\\"\\n\",\n \" return sym(nabla_grad(u))\\n\",\n \"\\n\",\n \"\\n\",\n \"def sigma(u, p):\\n\",\n \" \\\"\\\"\\\"Stress tensor.\\\"\\\"\\\"\\n\",\n \" return 2 * mu * epsilon(u) - p * Identity(len(u))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"17\",\n \"metadata\": {},\n \"source\": [\n \"Define the variational problem for the first step\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"p_n = Function(Q)\\n\",\n \"p_n.name = \\\"p_n\\\"\\n\",\n \"F1 = rho * dot((u - u_n) / k, v) * dx\\n\",\n \"F1 += rho * dot(dot(u_n, nabla_grad(u_n)), v) * dx\\n\",\n \"F1 += inner(sigma(U, p_n), epsilon(v)) * dx\\n\",\n \"F1 += dot(p_n * n, v) * ds - dot(mu * nabla_grad(U) * n, v) * ds\\n\",\n \"F1 -= dot(f, v) * dx\\n\",\n \"a1 = form(lhs(F1))\\n\",\n \"L1 = form(rhs(F1))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"Note that we have used the `ufl`-functions `lhs` and `rhs` to sort out the bilinear form\\n\",\n \"$a(u,v)$ and linear form $L(v)$.\\n\",\n \"This is particulary convenient in longer and more complicated variational forms.\\n\",\n \"With our particular discretization $a(u,v)$ `a1` is not time dependent,\\n\",\n \"and only has to be assembled once, while the right hand side is dependent on the solution\\n\",\n \"from the previous time step (`u_n`).\\n\",\n \"Thus, we do as for the [](./heat_code), and create the matrix outside the time-loop.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"A1 = assemble_matrix(a1, bcs=bcu)\\n\",\n \"A1.assemble()\\n\",\n \"b1 = create_vector(extract_function_spaces(L1))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We now set up similar variational formulations and structures for the second and third step\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"# Define variational problem for step 2\\n\",\n \"u_ = Function(V)\\n\",\n \"a2 = form(dot(nabla_grad(p), nabla_grad(q)) * dx)\\n\",\n \"L2 = form(dot(nabla_grad(p_n), nabla_grad(q)) * dx - (rho / k) * div(u_) * q * dx)\\n\",\n \"A2 = assemble_matrix(a2, bcs=bcp)\\n\",\n \"A2.assemble()\\n\",\n \"b2 = create_vector(extract_function_spaces(L2))\\n\",\n \"\\n\",\n \"# Define variational problem for step 3\\n\",\n \"p_ = Function(Q)\\n\",\n \"a3 = form(rho * dot(u, v) * dx)\\n\",\n \"L3 = form(rho * dot(u_, v) * dx - k * dot(nabla_grad(p_ - p_n), v) * dx)\\n\",\n \"A3 = assemble_matrix(a3)\\n\",\n \"A3.assemble()\\n\",\n \"b3 = create_vector(extract_function_spaces(L3))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"As we have create all the linear structures for the problem, we can now create a solver for each of them using PETSc.\\n\",\n \"We can therefore customize the solution strategy for each step.\\n\",\n \"For the tentative velocity step and pressure correction step,\\n\",\n \"we will use the Stabilized version of BiConjugate Gradient to solve the linear system,\\n\",\n \"and using algebraic multigrid for preconditioning.\\n\",\n \"For the last step, the velocity update, we use a conjugate gradient method with successive over relaxation,\\n\",\n \"Gauss Seidel (SOR) preconditioning.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"# Solver for step 1\\n\",\n \"solver1 = PETSc.KSP().create(mesh.comm)\\n\",\n \"solver1.setOperators(A1)\\n\",\n \"solver1.setType(PETSc.KSP.Type.BCGS)\\n\",\n \"pc1 = solver1.getPC()\\n\",\n \"pc1.setType(PETSc.PC.Type.HYPRE)\\n\",\n \"pc1.setHYPREType(\\\"boomeramg\\\")\\n\",\n \"\\n\",\n \"# Solver for step 2\\n\",\n \"solver2 = PETSc.KSP().create(mesh.comm)\\n\",\n \"solver2.setOperators(A2)\\n\",\n \"solver2.setType(PETSc.KSP.Type.BCGS)\\n\",\n \"pc2 = solver2.getPC()\\n\",\n \"pc2.setType(PETSc.PC.Type.HYPRE)\\n\",\n \"pc2.setHYPREType(\\\"boomeramg\\\")\\n\",\n \"\\n\",\n \"# Solver for step 3\\n\",\n \"solver3 = PETSc.KSP().create(mesh.comm)\\n\",\n \"solver3.setOperators(A3)\\n\",\n \"solver3.setType(PETSc.KSP.Type.CG)\\n\",\n \"pc3 = solver3.getPC()\\n\",\n \"pc3.setType(PETSc.PC.Type.SOR)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We prepare output files for the velocity and pressure data, and write the mesh and initial conditions to file\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"26\",\n \"metadata\": {\n \"lines_to_end_of_cell_marker\": 2\n },\n \"outputs\": [],\n \"source\": [\n \"from pathlib import Path\\n\",\n \"\\n\",\n \"folder = Path(\\\"results\\\")\\n\",\n \"folder.mkdir(exist_ok=True, parents=True)\\n\",\n \"vtx_u = VTXWriter(mesh.comm, folder / \\\"poiseuille_u.bp\\\", u_n, engine=\\\"BP4\\\")\\n\",\n \"vtx_p = VTXWriter(mesh.comm, folder / \\\"poiseuille_p.bp\\\", p_n, engine=\\\"BP4\\\")\\n\",\n \"vtx_u.write(t)\\n\",\n \"vtx_p.write(t)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"We also interpolate the analytical solution into our function-space and create a variational formulation for the $L^2$-error.\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"def u_exact(x):\\n\",\n \" values = np.zeros((2, x.shape[1]), dtype=PETSc.ScalarType)\\n\",\n \" values[0] = 4 * x[1] * (1.0 - x[1])\\n\",\n \" return values\\n\",\n \"\\n\",\n \"\\n\",\n \"u_ex = Function(V)\\n\",\n \"u_ex.interpolate(u_exact)\\n\",\n \"\\n\",\n \"L2_error = form(dot(u_ - u_ex, u_ - u_ex) * dx)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"The next step is to create the loop over time. Note that we for all three steps only have to assemble the right hand side and apply the boundary condition using lifting.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"tags\": []\n },\n \"outputs\": [],\n \"source\": [\n \"for i in range(num_steps):\\n\",\n \" # Update current time step\\n\",\n \" t += dt\\n\",\n \"\\n\",\n \" # Step 1: Tentative veolcity step\\n\",\n \" with b1.localForm() as loc_1:\\n\",\n \" loc_1.set(0)\\n\",\n \" assemble_vector(b1, L1)\\n\",\n \" apply_lifting(b1, [a1], [bcu])\\n\",\n \" b1.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)\\n\",\n \" set_bc(b1, bcu)\\n\",\n \" solver1.solve(b1, u_.x.petsc_vec)\\n\",\n \" u_.x.scatter_forward()\\n\",\n \"\\n\",\n \" # Step 2: Pressure corrrection step\\n\",\n \" with b2.localForm() as loc_2:\\n\",\n \" loc_2.set(0)\\n\",\n \" assemble_vector(b2, L2)\\n\",\n \" apply_lifting(b2, [a2], [bcp])\\n\",\n \" b2.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)\\n\",\n \" set_bc(b2, bcp)\\n\",\n \" solver2.solve(b2, p_.x.petsc_vec)\\n\",\n \" p_.x.scatter_forward()\\n\",\n \"\\n\",\n \" # Step 3: Velocity correction step\\n\",\n \" with b3.localForm() as loc_3:\\n\",\n \" loc_3.set(0)\\n\",\n \" assemble_vector(b3, L3)\\n\",\n \" b3.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)\\n\",\n \" solver3.solve(b3, u_.x.petsc_vec)\\n\",\n \" u_.x.scatter_forward()\\n\",\n \" # Update variable with solution form this time step\\n\",\n \" u_n.x.array[:] = u_.x.array[:]\\n\",\n \" p_n.x.array[:] = p_.x.array[:]\\n\",\n \"\\n\",\n \" # Write solutions to file\\n\",\n \" vtx_u.write(t)\\n\",\n \" vtx_p.write(t)\\n\",\n \"\\n\",\n \" # Compute error at current time-step\\n\",\n \" error_L2 = np.sqrt(mesh.comm.allreduce(assemble_scalar(L2_error), op=MPI.SUM))\\n\",\n \" error_max = mesh.comm.allreduce(\\n\",\n \" np.max(u_.x.petsc_vec.array - u_ex.x.petsc_vec.array), op=MPI.MAX\\n\",\n \" )\\n\",\n \" # Print error only every 20th step and at the last step\\n\",\n \" if (i % 20 == 0) or (i == num_steps - 1):\\n\",\n \" print(f\\\"Time {t:.2f}, L2-error {error_L2:.2e}, Max error {error_max:.2e}\\\")\\n\",\n \"# Close xmdf file\\n\",\n \"vtx_u.close()\\n\",\n \"vtx_p.close()\\n\",\n \"b1.destroy()\\n\",\n \"b2.destroy()\\n\",\n \"b3.destroy()\\n\",\n \"solver1.destroy()\\n\",\n \"solver2.destroy()\\n\",\n \"solver3.destroy()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"31\",\n \"metadata\": {},\n \"source\": [\n \"## Verification\\n\",\n \"As for the previous problems we compute the error at each degree of freedom and the $L^2(\\\\Omega)$-error.\\n\",\n \"We start with the initial condition $u=(0,0)$.\\n\",\n \"We have not specified the initial condition explicitly, and FEniCSx will initialize all\\n\",\n \"`Function`s including `u_n` and `u_` to zero.\\n\",\n \"Since the exact solution is quadratic, we expect to reach machine precision within finite time.\\n\",\n \"For our implementation, we observe that the error quickly approaches zero, and is of order $10^{-6}$ at $T=10$\\n\",\n \"\\n\",\n \"## Visualization of vectors\\n\",\n \"We have already looked at how to plot higher order functions and vector functions.\\n\",\n \"In this section we will look at how to visualize vector functions with glyphs, instead of warping the mesh.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"32\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"topology, cell_types, geometry = vtk_mesh(V)\\n\",\n \"values = np.zeros((geometry.shape[0], 3), dtype=np.float64)\\n\",\n \"values[:, : len(u_n)] = u_n.x.array.real.reshape((geometry.shape[0], len(u_n)))\\n\",\n \"\\n\",\n \"# Create a point cloud of glyphs\\n\",\n \"function_grid = pyvista.UnstructuredGrid(topology, cell_types, geometry)\\n\",\n \"function_grid[\\\"u\\\"] = values\\n\",\n \"glyphs = function_grid.glyph(orient=\\\"u\\\", factor=0.2)\\n\",\n \"\\n\",\n \"# Create a pyvista-grid for the mesh\\n\",\n \"tdim = mesh.topology.dim\\n\",\n \"mesh.topology.create_connectivity(tdim, tdim)\\n\",\n \"grid = pyvista.UnstructuredGrid(*vtk_mesh(mesh, tdim))\\n\",\n \"\\n\",\n \"# Create plotter\\n\",\n \"plotter = pyvista.Plotter()\\n\",\n \"plotter.add_mesh(grid, style=\\\"wireframe\\\", color=\\\"k\\\")\\n\",\n \"plotter.add_mesh(glyphs)\\n\",\n \"plotter.view_xy()\\n\",\n \"\\n\",\n \"if not pyvista.OFF_SCREEN:\\n\",\n \" plotter.show()\\n\",\n \"else:\\n\",\n \" fig_as_array = plotter.screenshot(\\\"glyphs.png\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## References\\n\",\n \"```{bibliography}\\n\",\n \":filter: docname in docnames\\n\",\n \"```\"\n ]\n }\n ],\n \"metadata\": {\n \"jupytext\": {\n \"formats\": \"ipynb,py:light\"\n },\n \"kernelspec\": {\n \"display_name\": \"Python 3 (ipykernel)\",\n \"language\": \"python\",\n \"name\": \"python3\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 3\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython3\",\n \"version\": \"3.10.12\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 4\n}\n" }, { "path": "chapter2/ns_code1.py", "content": "# ---\n# jupyter:\n# jupytext:\n# formats: ipynb,py:light\n# text_representation:\n# extension: .py\n# format_name: light\n# format_version: '1.5'\n# jupytext_version: 1.18.1\n# kernelspec:\n# display_name: Python 3 (ipykernel)\n# language: python\n# name: python3\n# ---\n\n# # Test problem 1: Channel flow (Poiseuille flow)\n#\n# Authors: Anders Logg and Hans Petter Langtangen\n#\n# Adapted to DOLFINx by: Jørgen S. Dokken\n#\n# In this section, you will learn how to:\n# - Solve the Navier-Stokes problem using a splitting scheme\n# - Visualize functions from higher order Lagrangian spaces\n#\n# In this section, we will compute the flow between two infinite plates, so-called channel or Poiseuille flow.\n# As we shall see, this problem has an analytical solution.\n# Let $H$ be the distance between the plates and $L$ the length of the channel. There are no body forces.\n#\n# We may scale the problem first to get rid of seemingly independent physical parameters.\n# The physics of this problem are governed by viscous effects only, in the direction perpendicular to the flow,\n# so a time scale should be based on diffusion across the channel: $t_v=H^2/\\nu$.\n# We let $U$, some characteristic inflow velocity, be the velocity scale and $H$ the spatial scale.\n# The pressure scale is taken as the characteristic shear stress, $\\mu U/H$, as this is a primary example of shear flow.\n# Inserting $\\bar{x}=x/H, \\bar{y}=y/H, \\bar{z}=z/H, \\bar{u}=u/U, \\bar{p}=Hp/{\\mu U}$, and $\\bar{t}=H^2/\\nu$\n# in the equations results in the scaled Navier-Stokes equations (dropping the bars after scaling)\n# ```{math}\n# :label: ns-scaled\n# \\frac{\\partial u}{\\partial t}+ \\mathrm{Re} u \\cdot \\nabla u &= -\\nabla p + \\nabla^2 u,\\\\\n# \\nabla \\cdot u &=0.\n# ```\n# A detailed derivation for scaling of the Navier-Stokes equation for a large variety of physical situations\n# can be found in {cite}`Langtangen2016scaling` (Chapter 4.2) by Hans Petter Langtangen and Geir K. Pedersen.\n#\n# Here, $\\mathrm{Re}=\\rho UH/\\mu$ is the Reynolds number.\n# Because of the time and pressure scales, which are different from convection dominated fluid flow,\n# the Reynolds number is associated with the convective term and not the viscosity term.\n#\n# The exact solution is derived by assuming $u=(u_x(x,y,z),0,0)$ with the $x$-axis pointing along the channel.\n# Since $\\nabla \\cdot u = 0$, $u$ cannot be dependent on $x$.\n#\n# The physics of channel flow is also two-dimensional so we can omit the $z$-coordinate\n# (more precisely: $\\partial/\\partial z = 0$).\n# Inserting $u=(u_x, 0, 0)$ in the (scaled) governing equations gives $u_x''(y)=\\frac{\\partial p}{\\partial x}$.\n# Differentiating this equation with respect to $x$ shows that\n# $\\frac{\\partial^2p}{\\partial x^2}=0$ so $\\partial p/\\partial x$ is a constant here called $-\\beta$.\n# This is the driving force of the flow and can be specified as a known parameter in the problem.\n# Integrating $u_x''(x,y)=-\\beta$ over the width of the channel, $[0,1]$,\n# and requiring $u=(0,0,0)$ at the channel walls, results in $u_x=\\frac{1}{2}\\beta y(1-y)$.\n# The characteristic inlet velocity $U$ can be taken as the maximum inflow at $y=0.5$, implying $\\beta=8$.\n# The length of the channel, $L/H$ in the scaled model, has no impact on the result,\n# so for simplicity we just compute on the unit square.\n# Mathematically, the pressure must be prescribed at a point, but since $p$ does not depend on $y$,\n# we can set $p$ to a known value, e.g. zero, along the outlet boundary $x=1$.\n# The result is $p(x)=8(1-x)$ and $u_x=4y(1-y)$.\n#\n# The boundary conditions can be set as $p=8$ at $x=0$, $p=0$ at $x=1$ and $u=(0,0,0)$ on the walls $y=0,1$.\n# This defines the pressure drop and should result in unit maximum velocity at the inlet and outlet and\n# a parabolic velocity profile without no further specifications.\n# Note that it is only meaningful to solve the Navier-Stokes equations in 2D or 3D geometries,\n# although the underlying mathematical problem collapses to two $1D$ problems, one for $u_x(y)$ and one for $p(x)$.\n#\n# The scaled model is not so easy to simulate using a standard Navier-Stokes solver with dimensions.\n# However, one can argue that the convection term is zero, so the Re coefficient in front of this term in\n# the scaled PDEs is not important and can be set to unity.\n# In that case, setting $\\rho=\\mu=1$ in the original Navier-Stokes equations resembles the scaled model.\n#\n# For a specific engineering problem one wants to simulate a specific fluid and set corresponding parameters.\n# A general solver is therefore most naturally implemented with dimensions and using the original physical parameters.\n# However, scaling may greatly simplify numerical simulations.\n# First of all, it shows that all fluids behave in the same way;\n# it does not matter whether we have oil, gas, or water flowing between two plates.\n# Secondly, it does not matter how fast the flow is, up to some critical value of the Reynolds number where the\n# flow becomes unstable and transitions to a complicated turbulent flow of totally different nature.\n# This means that one simulation is enough to cover all types of channel flow!\n# In other applications, scaling shows that it might be necessary to just set the fraction of some parameters\n# (dimensionless numbers) rather than the parameters themselves.\n# This simplifies exploring the input parameter space which is often the purpose of simulation.\n# Frequently, the scaled problem is run by setting some of the input parameters with dimension\n# to fixed values (often unity).\n\n# ## Implementation\n#\n# Author: Jørgen S. Dokken\n#\n# As in the previous example, we load the DOLFINx module, along with the `mpi4py` module,\n# and create the unit square mesh and define the run-time and temporal discretization\n\n# +\nfrom mpi4py import MPI\nfrom petsc4py import PETSc\nimport numpy as np\nimport pyvista\n\nfrom dolfinx.fem import (\n Constant,\n Function,\n extract_function_spaces,\n functionspace,\n assemble_scalar,\n dirichletbc,\n form,\n locate_dofs_geometrical,\n)\nfrom dolfinx.fem.petsc import (\n assemble_matrix,\n assemble_vector,\n apply_lifting,\n create_vector,\n set_bc,\n)\nfrom dolfinx.io import VTXWriter\nfrom dolfinx.mesh import create_unit_square\nfrom dolfinx.plot import vtk_mesh\nfrom basix.ufl import element\nfrom ufl import (\n FacetNormal,\n Identity,\n TestFunction,\n TrialFunction,\n div,\n dot,\n ds,\n dx,\n inner,\n lhs,\n nabla_grad,\n rhs,\n sym,\n)\n\nmesh = create_unit_square(MPI.COMM_WORLD, 10, 10)\nt = 0.0\nT = 10.0\nnum_steps = 500\ndt = T / num_steps\n# -\n\n# As opposed to the previous demos, we will create our two function spaces using the `ufl` element definitions as input\n\nv_cg2 = element(\"Lagrange\", mesh.basix_cell(), 2, shape=(mesh.geometry.dim,))\ns_cg1 = element(\"Lagrange\", mesh.basix_cell(), 1)\nV = functionspace(mesh, v_cg2)\nQ = functionspace(mesh, s_cg1)\n\n# The first space `V` is a vector valued function space for the velocity,\n# while `Q` is a scalar valued function space for pressure.\n# We use piecewise quadratic elements for the velocity and piecewise linear elements for the pressure.\n# One can easily create vector-valued function spaces with other dimensions by replacing\n# `shape=(mesh.geometry.dim, )` with something else, like\n# ```\n# v_cg basix.ufl.element(\"Lagrange\", mesh.basix_cell(), 2, shape=(10,))\n# ```\n# or\n# ```\n# tensor_element = basix.ufl.element(\"Lagrange\", mesh.basix_cell(), 2, shape=(3, 3))\n# ```\n# or\n# ```\n# tensor_element = basix.ufl.element(\"Lagrange\", mesh.basix_cell(), 2, shape=(3, 2, 4))\n# ```\n#\n#\n# ```{admonition} Stable finite element spaces for the Navier-Stokes equation\n# It is well-known that certain finite element spaces are not *stable* for the Navier-Stokes equations,\n# or even for the simpler Stokes equation.\n# The prime example of an unstable pair of finite element spaces is to use first order degree continuous\n# piecewise polynomials for both the velocity and the pressure.\n# Using an unstable pair of spaces typically results in a solution with *spurious* (unwanted, non-physical)\n# oscillations in the pressure solution.\n# The simple remedy is to use continuous piecewise quadratic elements for the velocity and continuous\n# piecewise linear elements for the pressure.\n# Together, these elements form the so-called *Taylor-Hood* element.\n# Spurious oscillations may occur also for splitting methods if an unstable element pair is used.\n#\n# Since we have two different function spaces, we need to create two sets of trial and test functions:\n\nu = TrialFunction(V)\nv = TestFunction(V)\np = TrialFunction(Q)\nq = TestFunction(Q)\n\n\n# As we have seen in [Linear elasticity problem](./linearelasticity_code) we can use Python-functions\n# to create the different Dirichlet conditions.\n# For this problem, we have three Dirichlet condition:\n# First, we will set $u=0$ at the walls of the channel, that is at $y=0$ and $y=1$.\n# In this case, we will use `dolfinx.fem.locate_dofs_geometrical`\n\n\n# +\ndef walls(x):\n return np.logical_or(np.isclose(x[1], 0), np.isclose(x[1], 1))\n\n\nwall_dofs = locate_dofs_geometrical(V, walls)\nu_noslip = np.array((0,) * mesh.geometry.dim, dtype=PETSc.ScalarType)\nbc_noslip = dirichletbc(u_noslip, wall_dofs, V)\n# -\n\n# Second, we will set $p=8$ at the inflow ($x=0$)\n\n\n# +\ndef inflow(x):\n return np.isclose(x[0], 0)\n\n\ninflow_dofs = locate_dofs_geometrical(Q, inflow)\nbc_inflow = dirichletbc(PETSc.ScalarType(8), inflow_dofs, Q)\n# -\n\n# And finally, $p=0$ at the outflow ($x=1$).\n# This will result in a pressure gradient that will accelerate the flow from the initial state with zero velocity.\n# At the end, we collect the boundary conditions for the velocity and pressure in Python lists so we\n# can easily access them in the following computation.\n\n\n# +\ndef outflow(x):\n return np.isclose(x[0], 1)\n\n\noutflow_dofs = locate_dofs_geometrical(Q, outflow)\nbc_outflow = dirichletbc(PETSc.ScalarType(0), outflow_dofs, Q)\nbcu = [bc_noslip]\nbcp = [bc_inflow, bc_outflow]\n# -\n\n# We now move on to the definition of the three variational forms, one for each step in the IPCS scheme.\n# Let us look at the definition of the first variational problem and the relevant parameters.\n\nu_n = Function(V)\nu_n.name = \"u_n\"\nU = 0.5 * (u_n + u)\nn = FacetNormal(mesh)\nf = Constant(mesh, PETSc.ScalarType((0, 0)))\nk = Constant(mesh, PETSc.ScalarType(dt))\nmu = Constant(mesh, PETSc.ScalarType(1))\nrho = Constant(mesh, PETSc.ScalarType(1))\n\n\n# ```{admonition} Usage of \"dolfinx.fem.Constant\"\n# Note that we have wrapped several parameters as constants.\n# This is to reduce the compilation-time of the variational formulations.\n# By wrapping them as a constant, we can change the variable\n# ```\n# The next step is to set up the variational form of the first step.\n# As the variational problem contains a mix of known and unknown quantities,\n# we will use the following naming convention: `u` (mathematically $u^{n+1}$) is known as a trial function\n# in the variational form. `u_` is the most recently computed approximation\n# ($u^{n+1}$ available as a `Function` object), `u_n` is $u^n$, and the same convention\n# goes for `p,p_` ($p^{n+1}$) and `p_n` (p^n).\n\n\n# +\ndef epsilon(u):\n \"\"\"Strain-rate tensor.\"\"\"\n return sym(nabla_grad(u))\n\n\ndef sigma(u, p):\n \"\"\"Stress tensor.\"\"\"\n return 2 * mu * epsilon(u) - p * Identity(len(u))\n\n\n# -\n\n# Define the variational problem for the first step\n\np_n = Function(Q)\np_n.name = \"p_n\"\nF1 = rho * dot((u - u_n) / k, v) * dx\nF1 += rho * dot(dot(u_n, nabla_grad(u_n)), v) * dx\nF1 += inner(sigma(U, p_n), epsilon(v)) * dx\nF1 += dot(p_n * n, v) * ds - dot(mu * nabla_grad(U) * n, v) * ds\nF1 -= dot(f, v) * dx\na1 = form(lhs(F1))\nL1 = form(rhs(F1))\n\n# Note that we have used the `ufl`-functions `lhs` and `rhs` to sort out the bilinear form\n# $a(u,v)$ and linear form $L(v)$.\n# This is particulary convenient in longer and more complicated variational forms.\n# With our particular discretization $a(u,v)$ `a1` is not time dependent,\n# and only has to be assembled once, while the right hand side is dependent on the solution\n# from the previous time step (`u_n`).\n# Thus, we do as for the [](./heat_code), and create the matrix outside the time-loop.\n\nA1 = assemble_matrix(a1, bcs=bcu)\nA1.assemble()\nb1 = create_vector(extract_function_spaces(L1))\n\n# We now set up similar variational formulations and structures for the second and third step\n\n# +\n# Define variational problem for step 2\nu_ = Function(V)\na2 = form(dot(nabla_grad(p), nabla_grad(q)) * dx)\nL2 = form(dot(nabla_grad(p_n), nabla_grad(q)) * dx - (rho / k) * div(u_) * q * dx)\nA2 = assemble_matrix(a2, bcs=bcp)\nA2.assemble()\nb2 = create_vector(extract_function_spaces(L2))\n\n# Define variational problem for step 3\np_ = Function(Q)\na3 = form(rho * dot(u, v) * dx)\nL3 = form(rho * dot(u_, v) * dx - k * dot(nabla_grad(p_ - p_n), v) * dx)\nA3 = assemble_matrix(a3)\nA3.assemble()\nb3 = create_vector(extract_function_spaces(L3))\n# -\n\n# As we have create all the linear structures for the problem, we can now create a solver for each of them using PETSc.\n# We can therefore customize the solution strategy for each step.\n# For the tentative velocity step and pressure correction step,\n# we will use the Stabilized version of BiConjugate Gradient to solve the linear system,\n# and using algebraic multigrid for preconditioning.\n# For the last step, the velocity update, we use a conjugate gradient method with successive over relaxation,\n# Gauss Seidel (SOR) preconditioning.\n\n# +\n# Solver for step 1\nsolver1 = PETSc.KSP().create(mesh.comm)\nsolver1.setOperators(A1)\nsolver1.setType(PETSc.KSP.Type.BCGS)\npc1 = solver1.getPC()\npc1.setType(PETSc.PC.Type.HYPRE)\npc1.setHYPREType(\"boomeramg\")\n\n# Solver for step 2\nsolver2 = PETSc.KSP().create(mesh.comm)\nsolver2.setOperators(A2)\nsolver2.setType(PETSc.KSP.Type.BCGS)\npc2 = solver2.getPC()\npc2.setType(PETSc.PC.Type.HYPRE)\npc2.setHYPREType(\"boomeramg\")\n\n# Solver for step 3\nsolver3 = PETSc.KSP().create(mesh.comm)\nsolver3.setOperators(A3)\nsolver3.setType(PETSc.KSP.Type.CG)\npc3 = solver3.getPC()\npc3.setType(PETSc.PC.Type.SOR)\n# -\n\n# We prepare output files for the velocity and pressure data, and write the mesh and initial conditions to file\n\n# +\nfrom pathlib import Path\n\nfolder = Path(\"results\")\nfolder.mkdir(exist_ok=True, parents=True)\nvtx_u = VTXWriter(mesh.comm, folder / \"poiseuille_u.bp\", u_n, engine=\"BP4\")\nvtx_p = VTXWriter(mesh.comm, folder / \"poiseuille_p.bp\", p_n, engine=\"BP4\")\nvtx_u.write(t)\nvtx_p.write(t)\n\n\n# -\n\n# We also interpolate the analytical solution into our function-space and create a variational formulation for the $L^2$-error.\n#\n\n\n# +\ndef u_exact(x):\n values = np.zeros((2, x.shape[1]), dtype=PETSc.ScalarType)\n values[0] = 4 * x[1] * (1.0 - x[1])\n return values\n\n\nu_ex = Function(V)\nu_ex.interpolate(u_exact)\n\nL2_error = form(dot(u_ - u_ex, u_ - u_ex) * dx)\n# -\n\n# The next step is to create the loop over time. Note that we for all three steps only have to assemble the right hand side and apply the boundary condition using lifting.\n\nfor i in range(num_steps):\n # Update current time step\n t += dt\n\n # Step 1: Tentative veolcity step\n with b1.localForm() as loc_1:\n loc_1.set(0)\n assemble_vector(b1, L1)\n apply_lifting(b1, [a1], [bcu])\n b1.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)\n set_bc(b1, bcu)\n solver1.solve(b1, u_.x.petsc_vec)\n u_.x.scatter_forward()\n\n # Step 2: Pressure corrrection step\n with b2.localForm() as loc_2:\n loc_2.set(0)\n assemble_vector(b2, L2)\n apply_lifting(b2, [a2], [bcp])\n b2.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)\n set_bc(b2, bcp)\n solver2.solve(b2, p_.x.petsc_vec)\n p_.x.scatter_forward()\n\n # Step 3: Velocity correction step\n with b3.localForm() as loc_3:\n loc_3.set(0)\n assemble_vector(b3, L3)\n b3.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)\n solver3.solve(b3, u_.x.petsc_vec)\n u_.x.scatter_forward()\n # Update variable with solution form this time step\n u_n.x.array[:] = u_.x.array[:]\n p_n.x.array[:] = p_.x.array[:]\n\n # Write solutions to file\n vtx_u.write(t)\n vtx_p.write(t)\n\n # Compute error at current time-step\n error_L2 = np.sqrt(mesh.comm.allreduce(assemble_scalar(L2_error), op=MPI.SUM))\n error_max = mesh.comm.allreduce(\n np.max(u_.x.petsc_vec.array - u_ex.x.petsc_vec.array), op=MPI.MAX\n )\n # Print error only every 20th step and at the last step\n if (i % 20 == 0) or (i == num_steps - 1):\n print(f\"Time {t:.2f}, L2-error {error_L2:.2e}, Max error {error_max:.2e}\")\n# Close xmdf file\nvtx_u.close()\nvtx_p.close()\nb1.destroy()\nb2.destroy()\nb3.destroy()\nsolver1.destroy()\nsolver2.destroy()\nsolver3.destroy()\n\n# ## Verification\n# As for the previous problems we compute the error at each degree of freedom and the $L^2(\\Omega)$-error.\n# We start with the initial condition $u=(0,0)$.\n# We have not specified the initial condition explicitly, and FEniCSx will initialize all\n# `Function`s including `u_n` and `u_` to zero.\n# Since the exact solution is quadratic, we expect to reach machine precision within finite time.\n# For our implementation, we observe that the error quickly approaches zero, and is of order $10^{-6}$ at $T=10$\n#\n# ## Visualization of vectors\n# We have already looked at how to plot higher order functions and vector functions.\n# In this section we will look at how to visualize vector functions with glyphs, instead of warping the mesh.\n\n# +\ntopology, cell_types, geometry = vtk_mesh(V)\nvalues = np.zeros((geometry.shape[0], 3), dtype=np.float64)\nvalues[:, : len(u_n)] = u_n.x.array.real.reshape((geometry.shape[0], len(u_n)))\n\n# Create a point cloud of glyphs\nfunction_grid = pyvista.UnstructuredGrid(topology, cell_types, geometry)\nfunction_grid[\"u\"] = values\nglyphs = function_grid.glyph(orient=\"u\", factor=0.2)\n\n# Create a pyvista-grid for the mesh\ntdim = mesh.topology.dim\nmesh.topology.create_connectivity(tdim, tdim)\ngrid = pyvista.UnstructuredGrid(*vtk_mesh(mesh, tdim))\n\n# Create plotter\nplotter = pyvista.Plotter()\nplotter.add_mesh(grid, style=\"wireframe\", color=\"k\")\nplotter.add_mesh(glyphs)\nplotter.view_xy()\n\nif not pyvista.OFF_SCREEN:\n plotter.show()\nelse:\n fig_as_array = plotter.screenshot(\"glyphs.png\")\n# -\n\n# ## References\n# ```{bibliography}\n# :filter: docname in docnames\n# ```\n" }, { "path": "chapter2/ns_code2.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"markdown\",\n \"id\": \"0\",\n \"metadata\": {},\n \"source\": [\n \"# Test problem 2: Flow past a cylinder (DFG 2D-3 benchmark)\\n\",\n \"\\n\",\n \"Author: Jørgen S. Dokken\\n\",\n \"\\n\",\n \"In this section, we will turn our attention to a slightly more challenging problem: flow past a cylinder.\\n\",\n \"The geometry and parameters are taken from the\\n\",\n \"[DFG 2D-3 benchmark](https://wwwold.mathematik.tu-dortmund.de/~featflow/en/benchmarks/cfdbenchmarking/flow/dfg_benchmark3_re100.html) in FeatFlow.\\n\",\n \"\\n\",\n \"To be able to solve this problem efficiently and ensure numerical stability,\\n\",\n \"we will substitute our first order backward difference scheme with a Crank-Nicholson discretization in time,\\n\",\n \"and a semi-implicit Adams-Bashforth approximation of the non-linear term.\\n\",\n \"\\n\",\n \"```{admonition} Computationally demanding demo\\n\",\n \"This demo is computationally demanding, with a run-time up to 15 minutes,\\n\",\n \"as it is using parameters from the DFG 2D-3 benchmark, which consists of 12800 time steps.\\n\",\n \"It is adviced to download this demo and not run it in a browser.\\n\",\n \"This runtime of the demo can be decreased by using 2 or 3 mpi processes.\\n\",\n \"```\\n\",\n \"\\n\",\n \"The computational geometry we would like to use is\\n\",\n \"![Fluid channel with a circular obstacle](turek.png)\\n\",\n \"\\n\",\n \"The kinematic velocity is given by $\\\\nu=0.001=\\\\frac{\\\\mu}{\\\\rho}$ and the inflow velocity profile is specified as\\n\",\n \"\\n\",\n \"\\\\begin{align*}\\n\",\n \" u(x,y,t) &= \\\\left( \\\\frac{4Uy(0.41-y)}{0.41^2}, 0 \\\\right)\\\\\\\\\\n\",\n \" U &= U(t) = 1.5\\\\sin(\\\\pi t/8)\\n\",\n \"\\\\end{align*}\\n\",\n \"\\n\",\n \"which has a maximum magnitude of $1.5$ at $y=0.41/2$.\\n\",\n \"We do not use any scaling for this problem since all exact parameters are known.\\n\",\n \"\\n\",\n \"## Mesh generation\\n\",\n \"\\n\",\n \"As in the [Deflection of a membrane](./../chapter1/membrane_code.ipynb) we use GMSH to generate the mesh.\\n\",\n \"We fist create the rectangle and obstacle.\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"1\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"import gmsh\\n\",\n \"import os\\n\",\n \"import numpy as np\\n\",\n \"import matplotlib.pyplot as plt\\n\",\n \"import tqdm.autonotebook\\n\",\n \"\\n\",\n \"from mpi4py import MPI\\n\",\n \"from petsc4py import PETSc\\n\",\n \"\\n\",\n \"from basix.ufl import element\\n\",\n \"\\n\",\n \"from dolfinx.fem import (\\n\",\n \" Constant,\\n\",\n \" Function,\\n\",\n \" functionspace,\\n\",\n \" assemble_scalar,\\n\",\n \" dirichletbc,\\n\",\n \" extract_function_spaces,\\n\",\n \" form,\\n\",\n \" locate_dofs_topological,\\n\",\n \" set_bc,\\n\",\n \")\\n\",\n \"from dolfinx.fem.petsc import (\\n\",\n \" apply_lifting,\\n\",\n \" assemble_matrix,\\n\",\n \" assemble_vector,\\n\",\n \" create_vector,\\n\",\n \" create_matrix,\\n\",\n \" set_bc,\\n\",\n \")\\n\",\n \"from dolfinx.geometry import bb_tree, compute_collisions_points, compute_colliding_cells\\n\",\n \"from dolfinx.io import VTXWriter, gmsh as gmshio\\n\",\n \"from ufl import (\\n\",\n \" FacetNormal,\\n\",\n \" Measure,\\n\",\n \" TestFunction,\\n\",\n \" TrialFunction,\\n\",\n \" as_vector,\\n\",\n \" div,\\n\",\n \" dot,\\n\",\n \" dx,\\n\",\n \" inner,\\n\",\n \" lhs,\\n\",\n \" grad,\\n\",\n \" nabla_grad,\\n\",\n \" rhs,\\n\",\n \")\\n\",\n \"\\n\",\n \"gmsh.initialize()\\n\",\n \"\\n\",\n \"L = 2.2\\n\",\n \"H = 0.41\\n\",\n \"c_x = c_y = 0.2\\n\",\n \"r = 0.05\\n\",\n \"gdim = 2\\n\",\n \"mesh_comm = MPI.COMM_WORLD\\n\",\n \"model_rank = 0\\n\",\n \"if mesh_comm.rank == model_rank:\\n\",\n \" rectangle = gmsh.model.occ.addRectangle(0, 0, 0, L, H, tag=1)\\n\",\n \" obstacle = gmsh.model.occ.addDisk(c_x, c_y, 0, r, r)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"2\",\n \"metadata\": {},\n \"source\": [\n \"The next step is to subtract the obstacle from the channel, such that we do not mesh the interior of the circle.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"3\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"if mesh_comm.rank == model_rank:\\n\",\n \" fluid = gmsh.model.occ.cut([(gdim, rectangle)], [(gdim, obstacle)])\\n\",\n \" gmsh.model.occ.synchronize()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"4\",\n \"metadata\": {},\n \"source\": [\n \"To get GMSH to mesh the fluid, we add a physical volume marker\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"5\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"fluid_marker = 1\\n\",\n \"if mesh_comm.rank == model_rank:\\n\",\n \" volumes = gmsh.model.getEntities(dim=gdim)\\n\",\n \" assert len(volumes) == 1\\n\",\n \" gmsh.model.addPhysicalGroup(volumes[0][0], [volumes[0][1]], fluid_marker)\\n\",\n \" gmsh.model.setPhysicalName(volumes[0][0], fluid_marker, \\\"Fluid\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"6\",\n \"metadata\": {},\n \"source\": [\n \"To tag the different surfaces of the mesh, we tag the inflow (left hand side) with marker 2,\\n\",\n \"the outflow (right hand side) with marker 3 and the fluid walls with 4 and obstacle with 5.\\n\",\n \"We will do this by computing the center of mass for each geometrical entity.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"7\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"inlet_marker, outlet_marker, wall_marker, obstacle_marker = 2, 3, 4, 5\\n\",\n \"inflow, outflow, walls, obstacle = [], [], [], []\\n\",\n \"if mesh_comm.rank == model_rank:\\n\",\n \" boundaries = gmsh.model.getBoundary(volumes, oriented=False)\\n\",\n \" for boundary in boundaries:\\n\",\n \" center_of_mass = gmsh.model.occ.getCenterOfMass(boundary[0], boundary[1])\\n\",\n \" if np.allclose(center_of_mass, [0, H / 2, 0]):\\n\",\n \" inflow.append(boundary[1])\\n\",\n \" elif np.allclose(center_of_mass, [L, H / 2, 0]):\\n\",\n \" outflow.append(boundary[1])\\n\",\n \" elif np.allclose(center_of_mass, [L / 2, H, 0]) or np.allclose(\\n\",\n \" center_of_mass, [L / 2, 0, 0]\\n\",\n \" ):\\n\",\n \" walls.append(boundary[1])\\n\",\n \" else:\\n\",\n \" obstacle.append(boundary[1])\\n\",\n \" gmsh.model.addPhysicalGroup(1, walls, wall_marker)\\n\",\n \" gmsh.model.setPhysicalName(1, wall_marker, \\\"Walls\\\")\\n\",\n \" gmsh.model.addPhysicalGroup(1, inflow, inlet_marker)\\n\",\n \" gmsh.model.setPhysicalName(1, inlet_marker, \\\"Inlet\\\")\\n\",\n \" gmsh.model.addPhysicalGroup(1, outflow, outlet_marker)\\n\",\n \" gmsh.model.setPhysicalName(1, outlet_marker, \\\"Outlet\\\")\\n\",\n \" gmsh.model.addPhysicalGroup(1, obstacle, obstacle_marker)\\n\",\n \" gmsh.model.setPhysicalName(1, obstacle_marker, \\\"Obstacle\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"8\",\n \"metadata\": {},\n \"source\": [\n \"In our previous meshes, we have used uniform mesh sizes.\\n\",\n \"In this example, we will have variable mesh sizes to resolve the flow solution in the area of interest;\\n\",\n \"close to the circular obstacle. To do this, we use GMSH Fields.\\n\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"9\",\n \"metadata\": {},\n \"source\": [\n \"Create distance field from obstacle.\\n\",\n \"Add threshold of mesh sizes based on the distance field\\n\",\n \"```\\n\",\n \"LcMax - /--------\\n\",\n \" /\\n\",\n \"LcMin -o---------/\\n\",\n \" | | |\\n\",\n \" Point DistMin DistMax\\n\",\n \"```\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"10\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"res_min = r / 3\\n\",\n \"if mesh_comm.rank == model_rank:\\n\",\n \" distance_field = gmsh.model.mesh.field.add(\\\"Distance\\\")\\n\",\n \" gmsh.model.mesh.field.setNumbers(distance_field, \\\"EdgesList\\\", obstacle)\\n\",\n \" threshold_field = gmsh.model.mesh.field.add(\\\"Threshold\\\")\\n\",\n \" gmsh.model.mesh.field.setNumber(threshold_field, \\\"IField\\\", distance_field)\\n\",\n \" gmsh.model.mesh.field.setNumber(threshold_field, \\\"LcMin\\\", res_min)\\n\",\n \" gmsh.model.mesh.field.setNumber(threshold_field, \\\"LcMax\\\", 0.25 * H)\\n\",\n \" gmsh.model.mesh.field.setNumber(threshold_field, \\\"DistMin\\\", r)\\n\",\n \" gmsh.model.mesh.field.setNumber(threshold_field, \\\"DistMax\\\", 2 * H)\\n\",\n \" min_field = gmsh.model.mesh.field.add(\\\"Min\\\")\\n\",\n \" gmsh.model.mesh.field.setNumbers(min_field, \\\"FieldsList\\\", [threshold_field])\\n\",\n \" gmsh.model.mesh.field.setAsBackgroundMesh(min_field)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"11\",\n \"metadata\": {},\n \"source\": [\n \"## Generating the mesh\\n\",\n \"\\n\",\n \"We are now ready to generate the mesh.\\n\",\n \"However, we have to decide if our mesh should consist of triangles or quadrilaterals.\\n\",\n \"In this demo, to match the DFG 2D-3 benchmark, we use second order quadrilateral elements.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"12\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"if mesh_comm.rank == model_rank:\\n\",\n \" gmsh.option.setNumber(\\\"Mesh.Algorithm\\\", 8)\\n\",\n \" gmsh.option.setNumber(\\\"Mesh.RecombinationAlgorithm\\\", 2)\\n\",\n \" gmsh.option.setNumber(\\\"Mesh.RecombineAll\\\", 1)\\n\",\n \" gmsh.option.setNumber(\\\"Mesh.SubdivisionAlgorithm\\\", 1)\\n\",\n \" gmsh.model.mesh.generate(gdim)\\n\",\n \" gmsh.model.mesh.setOrder(2)\\n\",\n \" gmsh.model.mesh.optimize(\\\"Netgen\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"13\",\n \"metadata\": {},\n \"source\": [\n \"## Loading mesh and boundary markers\\n\",\n \"\\n\",\n \"As we have generated the mesh, we now need to load the mesh and corresponding facet markers into DOLFINx.\\n\",\n \"To load the mesh, we follow the same structure as in [Deflection of a membrane](./../chapter1/membrane_code.ipynb),\\n\",\n \"with the difference being that we will load in facet markers as well.\\n\",\n \"To learn more about the specifics of the function below,\\n\",\n \"see [A GMSH tutorial for DOLFINx](https://jsdokken.com/src/tutorial_gmsh.html).\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"14\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"mesh_data = gmshio.model_to_mesh(gmsh.model, mesh_comm, model_rank, gdim=gdim)\\n\",\n \"mesh = mesh_data.mesh\\n\",\n \"assert mesh_data.facet_tags is not None\\n\",\n \"ft = mesh_data.facet_tags\\n\",\n \"ft.name = \\\"Facet markers\\\"\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"15\",\n \"metadata\": {},\n \"source\": [\n \"## Physical and discretization parameters\\n\",\n \"\\n\",\n \"Following the DGF-2 benchmark, we define our problem specific parameters\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"16\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"t = 0.0\\n\",\n \"T = 8.0 # Final time\\n\",\n \"dt = 1 / 1600 # Time step size\\n\",\n \"num_steps = int(T / dt)\\n\",\n \"k = Constant(mesh, PETSc.ScalarType(dt))\\n\",\n \"mu = Constant(mesh, PETSc.ScalarType(0.001)) # Dynamic viscosity\\n\",\n \"rho = Constant(mesh, PETSc.ScalarType(1)) # Density\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"17\",\n \"metadata\": {},\n \"source\": [\n \"```{admonition} Reduce runtime of problem\\n\",\n \"This problem takes about 15 minutes to run in serial, due to the large amount of time steps.\\n\",\n \"If you convert the notebook to a python file and use `mpirun`, you can reduce the runtime of the problem.\\n\",\n \"```\\n\",\n \"\\n\",\n \"## Boundary conditions\\n\",\n \"\\n\",\n \"As we have created the mesh and relevant mesh tags, we can now specify the\\n\",\n \"function spaces `V` and `Q` along with the boundary conditions.\\n\",\n \"As the `ft` contains markers for facets, we use this class to find the facets for the inlet and walls.\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"18\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"v_cg2 = element(\\\"Lagrange\\\", mesh.basix_cell(), 2, shape=(mesh.geometry.dim,))\\n\",\n \"s_cg1 = element(\\\"Lagrange\\\", mesh.basix_cell(), 1)\\n\",\n \"V = functionspace(mesh, v_cg2)\\n\",\n \"Q = functionspace(mesh, s_cg1)\\n\",\n \"\\n\",\n \"fdim = mesh.topology.dim - 1\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"19\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"Define boundary conditions\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"20\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"class InletVelocity:\\n\",\n \" def __init__(self, t):\\n\",\n \" self.t = t\\n\",\n \"\\n\",\n \" def __call__(self, x):\\n\",\n \" values = np.zeros((gdim, x.shape[1]), dtype=PETSc.ScalarType)\\n\",\n \" values[0] = (\\n\",\n \" 4 * 1.5 * np.sin(self.t * np.pi / 8) * x[1] * (0.41 - x[1]) / (0.41**2)\\n\",\n \" )\\n\",\n \" return values\\n\",\n \"\\n\",\n \"\\n\",\n \"# Inlet\\n\",\n \"u_inlet = Function(V)\\n\",\n \"inlet_velocity = InletVelocity(t)\\n\",\n \"u_inlet.interpolate(inlet_velocity)\\n\",\n \"bcu_inflow = dirichletbc(\\n\",\n \" u_inlet, locate_dofs_topological(V, fdim, ft.find(inlet_marker))\\n\",\n \")\\n\",\n \"# Walls\\n\",\n \"u_nonslip = np.array((0,) * mesh.geometry.dim, dtype=PETSc.ScalarType)\\n\",\n \"bcu_walls = dirichletbc(\\n\",\n \" u_nonslip, locate_dofs_topological(V, fdim, ft.find(wall_marker)), V\\n\",\n \")\\n\",\n \"# Obstacle\\n\",\n \"bcu_obstacle = dirichletbc(\\n\",\n \" u_nonslip, locate_dofs_topological(V, fdim, ft.find(obstacle_marker)), V\\n\",\n \")\\n\",\n \"bcu = [bcu_inflow, bcu_obstacle, bcu_walls]\\n\",\n \"# Outlet\\n\",\n \"bcp_outlet = dirichletbc(\\n\",\n \" PETSc.ScalarType(0), locate_dofs_topological(Q, fdim, ft.find(outlet_marker)), Q\\n\",\n \")\\n\",\n \"bcp = [bcp_outlet]\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"21\",\n \"metadata\": {},\n \"source\": [\n \"## Variational form\\n\",\n \"\\n\",\n \"As opposed to [Pouseille flow](./ns_code1.ipynb), we will use a Crank-Nicolson discretization,\\n\",\n \"and an semi-implicit Adams-Bashforth approximation. The first step can be written as\\n\",\n \"\\n\",\n \"$$\\n\",\n \"\\\\rho\\\\left(\\\\frac{u^*- u^n}{\\\\delta t} + \\\\left(\\\\frac{3}{2}u^{n}\\n\",\n \"- \\\\frac{1}{2} u^{n-1}\\\\right)\\\\cdot \\\\frac{1}{2}\\\\nabla (u^*+u^n) \\\\right)\\n\",\n \"- \\\\frac{1}{2}\\\\mu \\\\Delta( u^*+ u^n )+ \\\\nabla p^{n-1/2} = f^{n+\\\\frac{1}{2}} \\\\qquad \\\\text{ in } \\\\Omega\\n\",\n \"$$\\n\",\n \"\\n\",\n \"$$\\n\",\n \"u^{*}=g(\\\\cdot, t^{n+1}) \\\\qquad \\\\text{ on } \\\\partial \\\\Omega_{D}\\n\",\n \"$$\\n\",\n \"\\n\",\n \"$$\\n\",\n \"\\\\frac{1}{2}\\\\nu \\\\nabla (u^*+u^n) \\\\cdot n = p^{n-\\\\frac{1}{2}} \\\\qquad \\\\text{ on } \\\\partial \\\\Omega_{N}\\n\",\n \"$$\\n\",\n \"\\n\",\n \"where we have used the two previous time steps in the temporal derivative for the velocity,\\n\",\n \"and compute the pressure staggered in time, at the time between the previous and current solution.\\n\",\n \"The second step becomes\\n\",\n \"\\n\",\n \"$$\\n\",\n \"\\\\nabla^2 \\\\phi = \\\\frac{\\\\rho}{\\\\delta t} \\\\nabla \\\\cdot u^* \\\\qquad\\\\text{in } \\\\Omega,\\n\",\n \"$$\\n\",\n \"\\n\",\n \"$$\\n\",\n \"\\\\nabla \\\\phi \\\\cdot n = 0 \\\\qquad \\\\text{on } \\\\partial \\\\Omega_D,\\n\",\n \"$$\\n\",\n \"\\n\",\n \"$$\\n\",\n \"\\\\phi = 0 \\\\qquad\\\\text{on } \\\\partial\\\\Omega_N\\n\",\n \"$$\\n\",\n \"\\n\",\n \"where $p^{n+\\\\frac{1}{2}}=p^{n-\\\\frac{1}{2}} + \\\\phi$.\\n\",\n \"Finally, the third step is\\n\",\n \"\\n\",\n \"$$\\n\",\n \"\\\\rho (u^{n+1}-u^{*}) = -\\\\delta t \\\\nabla\\\\phi.\\n\",\n \"$$\\n\",\n \"\\n\",\n \"We start by defining all the variables used in the variational formulations.\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"22\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"u = TrialFunction(V)\\n\",\n \"v = TestFunction(V)\\n\",\n \"u_ = Function(V, name=\\\"u\\\")\\n\",\n \"u_s = Function(V, name=\\\"u_tentative\\\")\\n\",\n \"u_n = Function(V)\\n\",\n \"u_n1 = Function(V)\\n\",\n \"p = TrialFunction(Q)\\n\",\n \"q = TestFunction(Q)\\n\",\n \"p_ = Function(Q, name=\\\"p\\\")\\n\",\n \"phi = Function(Q, name=\\\"phi\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"23\",\n \"metadata\": {},\n \"source\": [\n \"Next, we define the variational formulation for the first step,\\n\",\n \"where we have integrated the diffusion term, as well as the pressure term by parts.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"24\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"f = Constant(mesh, PETSc.ScalarType((0, 0)))\\n\",\n \"F1 = rho / k * dot(u - u_n, v) * dx\\n\",\n \"F1 += inner(dot(1.5 * u_n - 0.5 * u_n1, 0.5 * nabla_grad(u + u_n)), v) * dx\\n\",\n \"F1 += 0.5 * mu * inner(grad(u + u_n), grad(v)) * dx - dot(p_, div(v)) * dx\\n\",\n \"F1 += dot(f, v) * dx\\n\",\n \"a1 = form(lhs(F1))\\n\",\n \"L1 = form(rhs(F1))\\n\",\n \"A1 = create_matrix(a1)\\n\",\n \"b1 = create_vector(extract_function_spaces(L1))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"25\",\n \"metadata\": {},\n \"source\": [\n \"Next we define the second step\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"26\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"a2 = form(dot(grad(p), grad(q)) * dx)\\n\",\n \"L2 = form(-rho / k * dot(div(u_s), q) * dx)\\n\",\n \"A2 = assemble_matrix(a2, bcs=bcp)\\n\",\n \"A2.assemble()\\n\",\n \"b2 = create_vector(extract_function_spaces(L2))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"27\",\n \"metadata\": {},\n \"source\": [\n \"We finally create the last step\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"28\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"a3 = form(rho * dot(u, v) * dx)\\n\",\n \"L3 = form(rho * dot(u_s, v) * dx - k * dot(nabla_grad(phi), v) * dx)\\n\",\n \"A3 = assemble_matrix(a3)\\n\",\n \"A3.assemble()\\n\",\n \"b3 = create_vector(extract_function_spaces(L3))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"29\",\n \"metadata\": {},\n \"source\": [\n \"As in the previous tutorials, we use PETSc as a linear algebra backend.\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"30\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"# Solver for step 1\\n\",\n \"solver1 = PETSc.KSP().create(mesh.comm)\\n\",\n \"solver1.setOperators(A1)\\n\",\n \"solver1.setType(PETSc.KSP.Type.BCGS)\\n\",\n \"pc1 = solver1.getPC()\\n\",\n \"pc1.setType(PETSc.PC.Type.JACOBI)\\n\",\n \"\\n\",\n \"# Solver for step 2\\n\",\n \"solver2 = PETSc.KSP().create(mesh.comm)\\n\",\n \"solver2.setOperators(A2)\\n\",\n \"solver2.setType(PETSc.KSP.Type.MINRES)\\n\",\n \"pc2 = solver2.getPC()\\n\",\n \"pc2.setType(PETSc.PC.Type.HYPRE)\\n\",\n \"pc2.setHYPREType(\\\"boomeramg\\\")\\n\",\n \"\\n\",\n \"# Solver for step 3\\n\",\n \"solver3 = PETSc.KSP().create(mesh.comm)\\n\",\n \"solver3.setOperators(A3)\\n\",\n \"solver3.setType(PETSc.KSP.Type.CG)\\n\",\n \"pc3 = solver3.getPC()\\n\",\n \"pc3.setType(PETSc.PC.Type.SOR)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"31\",\n \"metadata\": {},\n \"source\": [\n \"## Verification of the implementation compute known physical quantities\\n\",\n \"\\n\",\n \"As a further verification of our implementation, we compute the drag and lift\\n\",\n \"coefficients over the obstacle, defined as\\n\",\n \"\\n\",\n \"\\\\begin{align*}\\n\",\n \" C_{\\\\text{D}}(u,p,t,\\\\partial\\\\Omega_S) &=\\n\",\n \"\\\\frac{2}{\\\\rho L U_{mean}^2}\\\\int_{\\\\partial\\\\Omega_S}\\\\rho \\\\nu n \\\\cdot \\\\nabla u_{t_S}(t)n_y -p(t)n_x~\\\\mathrm{d} s,\\\\\\\\\\n\",\n \" C_{\\\\text{L}}(u,p,t,\\\\partial\\\\Omega_S) &= -\\\\frac{2}{\\\\rho L U_{mean}^2}\\\\int_{\\\\partial\\\\Omega_S}\\\\rho \\\\nu n \\\\cdot \\\\nabla u_{t_S}(t)n_x + p(t)n_y~\\\\mathrm{d} s,\\n\",\n \"\\\\end{align*}\\n\",\n \"\\n\",\n \"where $u_{t_S}$ is the tangential velocity component at the interface of the obstacle $\\\\partial\\\\Omega_S$,\\n\",\n \"defined as $u_{t_S}=u\\\\cdot (n_y,-n_x)$, $U_{mean}=1$ the average inflow velocity, and $L$ the length of the channel.\\n\",\n \"We use `UFL` to create the relevant integrals, and assemble them at each time step.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"32\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"n = -FacetNormal(mesh) # Normal pointing out of obstacle\\n\",\n \"dObs = Measure(\\\"ds\\\", domain=mesh, subdomain_data=ft, subdomain_id=obstacle_marker)\\n\",\n \"u_t = inner(as_vector((n[1], -n[0])), u_)\\n\",\n \"drag = form(2 / 0.1 * (mu / rho * inner(grad(u_t), n) * n[1] - p_ * n[0]) * dObs)\\n\",\n \"lift = form(-2 / 0.1 * (mu / rho * inner(grad(u_t), n) * n[0] + p_ * n[1]) * dObs)\\n\",\n \"if mesh.comm.rank == 0:\\n\",\n \" C_D = np.zeros(num_steps, dtype=PETSc.ScalarType)\\n\",\n \" C_L = np.zeros(num_steps, dtype=PETSc.ScalarType)\\n\",\n \" t_u = np.zeros(num_steps, dtype=np.float64)\\n\",\n \" t_p = np.zeros(num_steps, dtype=np.float64)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"33\",\n \"metadata\": {},\n \"source\": [\n \"We will also evaluate the pressure at two points, one in front of the obstacle, $(0.15, 0.2)$,\\n\",\n \"and one behind the obstacle, $(0.25, 0.2)$.\\n\",\n \"To do this, we have to find which cell contains each of the points,\\n\",\n \"so that we can create a linear combination of the local basis functions and coefficients.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"34\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"tree = bb_tree(mesh, mesh.geometry.dim)\\n\",\n \"points = np.array([[0.15, 0.2, 0], [0.25, 0.2, 0]])\\n\",\n \"cell_candidates = compute_collisions_points(tree, points)\\n\",\n \"colliding_cells = compute_colliding_cells(mesh, cell_candidates, points)\\n\",\n \"front_cells = colliding_cells.links(0)\\n\",\n \"back_cells = colliding_cells.links(1)\\n\",\n \"if mesh.comm.rank == 0:\\n\",\n \" p_diff = np.zeros(num_steps, dtype=PETSc.ScalarType)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"35\",\n \"metadata\": {},\n \"source\": [\n \"## Solving the time-dependent problem\\n\",\n \"\\n\",\n \"```{admonition} Stability of the Navier-Stokes equation\\n\",\n \"Note that the current splitting scheme has to fullfil the a\\n\",\n \"[Courant–Friedrichs–Lewy condition](https://en.wikipedia.org/wiki/Courant%E2%80%93Friedrichs%E2%80%93Lewy_condition).\\n\",\n \"This limits the spatial discretization with respect to the inlet velocity and temporal discretization.\\n\",\n \"Other temporal discretization schemes such as the second order backward difference discretization or Crank-Nicholson\\n\",\n \"discretization with Adams-Bashforth linearization are better behaved than our simple backward difference scheme.\\n\",\n \"```\\n\",\n \"\\n\",\n \"As in the previous example, we create output files for the velocity and pressure and solve the time-dependent problem.\\n\",\n \"As we are solving a time dependent problem with many time steps, we use the `tqdm`-package to visualize the progress.\\n\",\n \"This package can be installed with `pip`.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"36\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"from pathlib import Path\\n\",\n \"\\n\",\n \"folder = Path(\\\"results\\\")\\n\",\n \"folder.mkdir(exist_ok=True, parents=True)\\n\",\n \"vtx_u = VTXWriter(mesh.comm, folder / \\\"dfg2D-3-u.bp\\\", [u_], engine=\\\"BP4\\\")\\n\",\n \"vtx_p = VTXWriter(mesh.comm, folder / \\\"dfg2D-3-p.bp\\\", [p_], engine=\\\"BP4\\\")\\n\",\n \"vtx_u.write(t)\\n\",\n \"vtx_p.write(t)\\n\",\n \"progress = tqdm.autonotebook.tqdm(desc=\\\"Solving PDE\\\", total=num_steps)\\n\",\n \"for i in range(num_steps):\\n\",\n \" progress.update(1)\\n\",\n \" # Update current time step\\n\",\n \" t += dt\\n\",\n \" # Update inlet velocity\\n\",\n \" inlet_velocity.t = t\\n\",\n \" u_inlet.interpolate(inlet_velocity)\\n\",\n \"\\n\",\n \" # Step 1: Tentative velocity step\\n\",\n \" A1.zeroEntries()\\n\",\n \" assemble_matrix(A1, a1, bcs=bcu)\\n\",\n \" A1.assemble()\\n\",\n \" with b1.localForm() as loc:\\n\",\n \" loc.set(0)\\n\",\n \" assemble_vector(b1, L1)\\n\",\n \" apply_lifting(b1, [a1], [bcu])\\n\",\n \" b1.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)\\n\",\n \" set_bc(b1, bcu)\\n\",\n \" solver1.solve(b1, u_s.x.petsc_vec)\\n\",\n \" u_s.x.scatter_forward()\\n\",\n \"\\n\",\n \" # Step 2: Pressure corrrection step\\n\",\n \" with b2.localForm() as loc:\\n\",\n \" loc.set(0)\\n\",\n \" assemble_vector(b2, L2)\\n\",\n \" apply_lifting(b2, [a2], [bcp])\\n\",\n \" b2.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)\\n\",\n \" set_bc(b2, bcp)\\n\",\n \" solver2.solve(b2, phi.x.petsc_vec)\\n\",\n \" phi.x.scatter_forward()\\n\",\n \"\\n\",\n \" p_.x.petsc_vec.axpy(1, phi.x.petsc_vec)\\n\",\n \" p_.x.scatter_forward()\\n\",\n \"\\n\",\n \" # Step 3: Velocity correction step\\n\",\n \" with b3.localForm() as loc:\\n\",\n \" loc.set(0)\\n\",\n \" assemble_vector(b3, L3)\\n\",\n \" b3.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)\\n\",\n \" solver3.solve(b3, u_.x.petsc_vec)\\n\",\n \" u_.x.scatter_forward()\\n\",\n \"\\n\",\n \" # Write solutions to file\\n\",\n \" vtx_u.write(t)\\n\",\n \" vtx_p.write(t)\\n\",\n \"\\n\",\n \" # Update variable with solution form this time step\\n\",\n \" with (\\n\",\n \" u_.x.petsc_vec.localForm() as loc_,\\n\",\n \" u_n.x.petsc_vec.localForm() as loc_n,\\n\",\n \" u_n1.x.petsc_vec.localForm() as loc_n1,\\n\",\n \" ):\\n\",\n \" loc_n.copy(loc_n1)\\n\",\n \" loc_.copy(loc_n)\\n\",\n \"\\n\",\n \" # Compute physical quantities\\n\",\n \" # For this to work in paralell, we gather contributions from all processors\\n\",\n \" # to processor zero and sum the contributions.\\n\",\n \" drag_coeff = mesh.comm.gather(assemble_scalar(drag), root=0)\\n\",\n \" lift_coeff = mesh.comm.gather(assemble_scalar(lift), root=0)\\n\",\n \" p_front = None\\n\",\n \" if len(front_cells) > 0:\\n\",\n \" p_front = p_.eval(points[0], front_cells[:1])\\n\",\n \" p_front = mesh.comm.gather(p_front, root=0)\\n\",\n \" p_back = None\\n\",\n \" if len(back_cells) > 0:\\n\",\n \" p_back = p_.eval(points[1], back_cells[:1])\\n\",\n \" p_back = mesh.comm.gather(p_back, root=0)\\n\",\n \" if mesh.comm.rank == 0:\\n\",\n \" t_u[i] = t\\n\",\n \" t_p[i] = t - dt / 2\\n\",\n \" C_D[i] = sum(drag_coeff)\\n\",\n \" C_L[i] = sum(lift_coeff)\\n\",\n \" # Choose first pressure that is found from the different processors\\n\",\n \" for pressure in p_front:\\n\",\n \" if pressure is not None:\\n\",\n \" p_diff[i] = pressure[0]\\n\",\n \" break\\n\",\n \" for pressure in p_back:\\n\",\n \" if pressure is not None:\\n\",\n \" p_diff[i] -= pressure[0]\\n\",\n \" break\\n\",\n \"progress.close()\\n\",\n \"vtx_u.close()\\n\",\n \"vtx_p.close()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"37\",\n \"metadata\": {},\n \"source\": [\n \"Destroy PETSc objects to free memory\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"38\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"A1.destroy()\\n\",\n \"A2.destroy()\\n\",\n \"A3.destroy()\\n\",\n \"b1.destroy()\\n\",\n \"b2.destroy()\\n\",\n \"b3.destroy()\\n\",\n \"solver1.destroy()\\n\",\n \"solver2.destroy()\\n\",\n \"solver3.destroy()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"39\",\n \"metadata\": {},\n \"source\": [\n \"## Verification using data from FEATFLOW\\n\",\n \"\\n\",\n \"As FEATFLOW has provided data for different discretization levels,\\n\",\n \"we compare our numerical data with the data provided using `matplotlib`.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"40\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"if mesh.comm.rank == 0:\\n\",\n \" if not os.path.exists(\\\"figures\\\"):\\n\",\n \" os.mkdir(\\\"figures\\\")\\n\",\n \" num_velocity_dofs = V.dofmap.index_map_bs * V.dofmap.index_map.size_global\\n\",\n \" num_pressure_dofs = Q.dofmap.index_map_bs * V.dofmap.index_map.size_global\\n\",\n \"\\n\",\n \" turek = np.loadtxt(\\\"bdforces_lv4\\\")\\n\",\n \" turek_p = np.loadtxt(\\\"pointvalues_lv4\\\")\\n\",\n \" fig = plt.figure(figsize=(25, 8))\\n\",\n \" l1 = plt.plot(\\n\",\n \" t_u,\\n\",\n \" C_D,\\n\",\n \" label=r\\\"FEniCSx ({0:d} dofs)\\\".format(num_velocity_dofs + num_pressure_dofs),\\n\",\n \" linewidth=2,\\n\",\n \" )\\n\",\n \" l2 = plt.plot(\\n\",\n \" turek[1:, 1],\\n\",\n \" turek[1:, 3],\\n\",\n \" marker=\\\"x\\\",\\n\",\n \" markevery=50,\\n\",\n \" linestyle=\\\"\\\",\\n\",\n \" markersize=4,\\n\",\n \" label=\\\"FEATFLOW (42016 dofs)\\\",\\n\",\n \" )\\n\",\n \" plt.title(\\\"Drag coefficient\\\")\\n\",\n \" plt.grid()\\n\",\n \" plt.legend()\\n\",\n \" plt.savefig(\\\"figures/drag_comparison.png\\\")\\n\",\n \"\\n\",\n \" fig = plt.figure(figsize=(25, 8))\\n\",\n \" l1 = plt.plot(\\n\",\n \" t_u,\\n\",\n \" C_L,\\n\",\n \" label=r\\\"FEniCSx ({0:d} dofs)\\\".format(num_velocity_dofs + num_pressure_dofs),\\n\",\n \" linewidth=2,\\n\",\n \" )\\n\",\n \" l2 = plt.plot(\\n\",\n \" turek[1:, 1],\\n\",\n \" turek[1:, 4],\\n\",\n \" marker=\\\"x\\\",\\n\",\n \" markevery=50,\\n\",\n \" linestyle=\\\"\\\",\\n\",\n \" markersize=4,\\n\",\n \" label=\\\"FEATFLOW (42016 dofs)\\\",\\n\",\n \" )\\n\",\n \" plt.title(\\\"Lift coefficient\\\")\\n\",\n \" plt.grid()\\n\",\n \" plt.legend()\\n\",\n \" plt.savefig(\\\"figures/lift_comparison.png\\\")\\n\",\n \"\\n\",\n \" fig = plt.figure(figsize=(25, 8))\\n\",\n \" l1 = plt.plot(\\n\",\n \" t_p,\\n\",\n \" p_diff,\\n\",\n \" label=r\\\"FEniCSx ({0:d} dofs)\\\".format(num_velocity_dofs + num_pressure_dofs),\\n\",\n \" linewidth=2,\\n\",\n \" )\\n\",\n \" l2 = plt.plot(\\n\",\n \" turek[1:, 1],\\n\",\n \" turek_p[1:, 6] - turek_p[1:, -1],\\n\",\n \" marker=\\\"x\\\",\\n\",\n \" markevery=50,\\n\",\n \" linestyle=\\\"\\\",\\n\",\n \" markersize=4,\\n\",\n \" label=\\\"FEATFLOW (42016 dofs)\\\",\\n\",\n \" )\\n\",\n \" plt.title(\\\"Pressure difference\\\")\\n\",\n \" plt.grid()\\n\",\n \" plt.legend()\\n\",\n \" plt.savefig(\\\"figures/pressure_comparison.png\\\")\"\n ]\n }\n ],\n \"metadata\": {\n \"jupytext\": {\n \"formats\": \"ipynb,py:light\"\n },\n \"kernelspec\": {\n \"display_name\": \"Python 3 (ipykernel)\",\n \"language\": \"python\",\n \"name\": \"python3\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 5\n}\n" }, { "path": "chapter2/ns_code2.py", "content": "# ---\n# jupyter:\n# jupytext:\n# formats: ipynb,py:light\n# text_representation:\n# extension: .py\n# format_name: light\n# format_version: '1.5'\n# jupytext_version: 1.19.1\n# kernelspec:\n# display_name: Python 3 (ipykernel)\n# language: python\n# name: python3\n# ---\n\n# # Test problem 2: Flow past a cylinder (DFG 2D-3 benchmark)\n#\n# Author: Jørgen S. Dokken\n#\n# In this section, we will turn our attention to a slightly more challenging problem: flow past a cylinder.\n# The geometry and parameters are taken from the\n# [DFG 2D-3 benchmark](https://wwwold.mathematik.tu-dortmund.de/~featflow/en/benchmarks/cfdbenchmarking/flow/dfg_benchmark3_re100.html) in FeatFlow.\n#\n# To be able to solve this problem efficiently and ensure numerical stability,\n# we will substitute our first order backward difference scheme with a Crank-Nicholson discretization in time,\n# and a semi-implicit Adams-Bashforth approximation of the non-linear term.\n#\n# ```{admonition} Computationally demanding demo\n# This demo is computationally demanding, with a run-time up to 15 minutes,\n# as it is using parameters from the DFG 2D-3 benchmark, which consists of 12800 time steps.\n# It is adviced to download this demo and not run it in a browser.\n# This runtime of the demo can be decreased by using 2 or 3 mpi processes.\n# ```\n#\n# The computational geometry we would like to use is\n# ![Fluid channel with a circular obstacle](turek.png)\n#\n# The kinematic velocity is given by $\\nu=0.001=\\frac{\\mu}{\\rho}$ and the inflow velocity profile is specified as\n#\n# \\begin{align*}\n# u(x,y,t) &= \\left( \\frac{4Uy(0.41-y)}{0.41^2}, 0 \\right)\\\\\n# U &= U(t) = 1.5\\sin(\\pi t/8)\n# \\end{align*}\n#\n# which has a maximum magnitude of $1.5$ at $y=0.41/2$.\n# We do not use any scaling for this problem since all exact parameters are known.\n#\n# ## Mesh generation\n#\n# As in the [Deflection of a membrane](./../chapter1/membrane_code.ipynb) we use GMSH to generate the mesh.\n# We fist create the rectangle and obstacle.\n#\n\n# +\nimport gmsh\nimport os\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport tqdm.autonotebook\n\nfrom mpi4py import MPI\nfrom petsc4py import PETSc\n\nfrom basix.ufl import element\n\nfrom dolfinx.fem import (\n Constant,\n Function,\n functionspace,\n assemble_scalar,\n dirichletbc,\n extract_function_spaces,\n form,\n locate_dofs_topological,\n set_bc,\n)\nfrom dolfinx.fem.petsc import (\n apply_lifting,\n assemble_matrix,\n assemble_vector,\n create_vector,\n create_matrix,\n set_bc,\n)\nfrom dolfinx.geometry import bb_tree, compute_collisions_points, compute_colliding_cells\nfrom dolfinx.io import VTXWriter, gmsh as gmshio\nfrom ufl import (\n FacetNormal,\n Measure,\n TestFunction,\n TrialFunction,\n as_vector,\n div,\n dot,\n dx,\n inner,\n lhs,\n grad,\n nabla_grad,\n rhs,\n)\n\ngmsh.initialize()\n\nL = 2.2\nH = 0.41\nc_x = c_y = 0.2\nr = 0.05\ngdim = 2\nmesh_comm = MPI.COMM_WORLD\nmodel_rank = 0\nif mesh_comm.rank == model_rank:\n rectangle = gmsh.model.occ.addRectangle(0, 0, 0, L, H, tag=1)\n obstacle = gmsh.model.occ.addDisk(c_x, c_y, 0, r, r)\n# -\n\n# The next step is to subtract the obstacle from the channel, such that we do not mesh the interior of the circle.\n\nif mesh_comm.rank == model_rank:\n fluid = gmsh.model.occ.cut([(gdim, rectangle)], [(gdim, obstacle)])\n gmsh.model.occ.synchronize()\n\n# To get GMSH to mesh the fluid, we add a physical volume marker\n\nfluid_marker = 1\nif mesh_comm.rank == model_rank:\n volumes = gmsh.model.getEntities(dim=gdim)\n assert len(volumes) == 1\n gmsh.model.addPhysicalGroup(volumes[0][0], [volumes[0][1]], fluid_marker)\n gmsh.model.setPhysicalName(volumes[0][0], fluid_marker, \"Fluid\")\n\n# To tag the different surfaces of the mesh, we tag the inflow (left hand side) with marker 2,\n# the outflow (right hand side) with marker 3 and the fluid walls with 4 and obstacle with 5.\n# We will do this by computing the center of mass for each geometrical entity.\n\ninlet_marker, outlet_marker, wall_marker, obstacle_marker = 2, 3, 4, 5\ninflow, outflow, walls, obstacle = [], [], [], []\nif mesh_comm.rank == model_rank:\n boundaries = gmsh.model.getBoundary(volumes, oriented=False)\n for boundary in boundaries:\n center_of_mass = gmsh.model.occ.getCenterOfMass(boundary[0], boundary[1])\n if np.allclose(center_of_mass, [0, H / 2, 0]):\n inflow.append(boundary[1])\n elif np.allclose(center_of_mass, [L, H / 2, 0]):\n outflow.append(boundary[1])\n elif np.allclose(center_of_mass, [L / 2, H, 0]) or np.allclose(\n center_of_mass, [L / 2, 0, 0]\n ):\n walls.append(boundary[1])\n else:\n obstacle.append(boundary[1])\n gmsh.model.addPhysicalGroup(1, walls, wall_marker)\n gmsh.model.setPhysicalName(1, wall_marker, \"Walls\")\n gmsh.model.addPhysicalGroup(1, inflow, inlet_marker)\n gmsh.model.setPhysicalName(1, inlet_marker, \"Inlet\")\n gmsh.model.addPhysicalGroup(1, outflow, outlet_marker)\n gmsh.model.setPhysicalName(1, outlet_marker, \"Outlet\")\n gmsh.model.addPhysicalGroup(1, obstacle, obstacle_marker)\n gmsh.model.setPhysicalName(1, obstacle_marker, \"Obstacle\")\n\n# In our previous meshes, we have used uniform mesh sizes.\n# In this example, we will have variable mesh sizes to resolve the flow solution in the area of interest;\n# close to the circular obstacle. To do this, we use GMSH Fields.\n#\n\n# Create distance field from obstacle.\n# Add threshold of mesh sizes based on the distance field\n# ```\n# LcMax - /--------\n# /\n# LcMin -o---------/\n# | | |\n# Point DistMin DistMax\n# ```\n\nres_min = r / 3\nif mesh_comm.rank == model_rank:\n distance_field = gmsh.model.mesh.field.add(\"Distance\")\n gmsh.model.mesh.field.setNumbers(distance_field, \"EdgesList\", obstacle)\n threshold_field = gmsh.model.mesh.field.add(\"Threshold\")\n gmsh.model.mesh.field.setNumber(threshold_field, \"IField\", distance_field)\n gmsh.model.mesh.field.setNumber(threshold_field, \"LcMin\", res_min)\n gmsh.model.mesh.field.setNumber(threshold_field, \"LcMax\", 0.25 * H)\n gmsh.model.mesh.field.setNumber(threshold_field, \"DistMin\", r)\n gmsh.model.mesh.field.setNumber(threshold_field, \"DistMax\", 2 * H)\n min_field = gmsh.model.mesh.field.add(\"Min\")\n gmsh.model.mesh.field.setNumbers(min_field, \"FieldsList\", [threshold_field])\n gmsh.model.mesh.field.setAsBackgroundMesh(min_field)\n\n# ## Generating the mesh\n#\n# We are now ready to generate the mesh.\n# However, we have to decide if our mesh should consist of triangles or quadrilaterals.\n# In this demo, to match the DFG 2D-3 benchmark, we use second order quadrilateral elements.\n\nif mesh_comm.rank == model_rank:\n gmsh.option.setNumber(\"Mesh.Algorithm\", 8)\n gmsh.option.setNumber(\"Mesh.RecombinationAlgorithm\", 2)\n gmsh.option.setNumber(\"Mesh.RecombineAll\", 1)\n gmsh.option.setNumber(\"Mesh.SubdivisionAlgorithm\", 1)\n gmsh.model.mesh.generate(gdim)\n gmsh.model.mesh.setOrder(2)\n gmsh.model.mesh.optimize(\"Netgen\")\n\n# ## Loading mesh and boundary markers\n#\n# As we have generated the mesh, we now need to load the mesh and corresponding facet markers into DOLFINx.\n# To load the mesh, we follow the same structure as in [Deflection of a membrane](./../chapter1/membrane_code.ipynb),\n# with the difference being that we will load in facet markers as well.\n# To learn more about the specifics of the function below,\n# see [A GMSH tutorial for DOLFINx](https://jsdokken.com/src/tutorial_gmsh.html).\n\nmesh_data = gmshio.model_to_mesh(gmsh.model, mesh_comm, model_rank, gdim=gdim)\nmesh = mesh_data.mesh\nassert mesh_data.facet_tags is not None\nft = mesh_data.facet_tags\nft.name = \"Facet markers\"\n\n# ## Physical and discretization parameters\n#\n# Following the DGF-2 benchmark, we define our problem specific parameters\n\nt = 0.0\nT = 8.0 # Final time\ndt = 1 / 1600 # Time step size\nnum_steps = int(T / dt)\nk = Constant(mesh, PETSc.ScalarType(dt))\nmu = Constant(mesh, PETSc.ScalarType(0.001)) # Dynamic viscosity\nrho = Constant(mesh, PETSc.ScalarType(1)) # Density\n\n# ```{admonition} Reduce runtime of problem\n# This problem takes about 15 minutes to run in serial, due to the large amount of time steps.\n# If you convert the notebook to a python file and use `mpirun`, you can reduce the runtime of the problem.\n# ```\n#\n# ## Boundary conditions\n#\n# As we have created the mesh and relevant mesh tags, we can now specify the\n# function spaces `V` and `Q` along with the boundary conditions.\n# As the `ft` contains markers for facets, we use this class to find the facets for the inlet and walls.\n#\n\n# +\nv_cg2 = element(\"Lagrange\", mesh.basix_cell(), 2, shape=(mesh.geometry.dim,))\ns_cg1 = element(\"Lagrange\", mesh.basix_cell(), 1)\nV = functionspace(mesh, v_cg2)\nQ = functionspace(mesh, s_cg1)\n\nfdim = mesh.topology.dim - 1\n# -\n\n# Define boundary conditions\n\n\n# +\nclass InletVelocity:\n def __init__(self, t):\n self.t = t\n\n def __call__(self, x):\n values = np.zeros((gdim, x.shape[1]), dtype=PETSc.ScalarType)\n values[0] = (\n 4 * 1.5 * np.sin(self.t * np.pi / 8) * x[1] * (0.41 - x[1]) / (0.41**2)\n )\n return values\n\n\n# Inlet\nu_inlet = Function(V)\ninlet_velocity = InletVelocity(t)\nu_inlet.interpolate(inlet_velocity)\nbcu_inflow = dirichletbc(\n u_inlet, locate_dofs_topological(V, fdim, ft.find(inlet_marker))\n)\n# Walls\nu_nonslip = np.array((0,) * mesh.geometry.dim, dtype=PETSc.ScalarType)\nbcu_walls = dirichletbc(\n u_nonslip, locate_dofs_topological(V, fdim, ft.find(wall_marker)), V\n)\n# Obstacle\nbcu_obstacle = dirichletbc(\n u_nonslip, locate_dofs_topological(V, fdim, ft.find(obstacle_marker)), V\n)\nbcu = [bcu_inflow, bcu_obstacle, bcu_walls]\n# Outlet\nbcp_outlet = dirichletbc(\n PETSc.ScalarType(0), locate_dofs_topological(Q, fdim, ft.find(outlet_marker)), Q\n)\nbcp = [bcp_outlet]\n# -\n\n# ## Variational form\n#\n# As opposed to [Pouseille flow](./ns_code1.ipynb), we will use a Crank-Nicolson discretization,\n# and an semi-implicit Adams-Bashforth approximation. The first step can be written as\n#\n# $$\n# \\rho\\left(\\frac{u^*- u^n}{\\delta t} + \\left(\\frac{3}{2}u^{n}\n# - \\frac{1}{2} u^{n-1}\\right)\\cdot \\frac{1}{2}\\nabla (u^*+u^n) \\right)\n# - \\frac{1}{2}\\mu \\Delta( u^*+ u^n )+ \\nabla p^{n-1/2} = f^{n+\\frac{1}{2}} \\qquad \\text{ in } \\Omega\n# $$\n#\n# $$\n# u^{*}=g(\\cdot, t^{n+1}) \\qquad \\text{ on } \\partial \\Omega_{D}\n# $$\n#\n# $$\n# \\frac{1}{2}\\nu \\nabla (u^*+u^n) \\cdot n = p^{n-\\frac{1}{2}} \\qquad \\text{ on } \\partial \\Omega_{N}\n# $$\n#\n# where we have used the two previous time steps in the temporal derivative for the velocity,\n# and compute the pressure staggered in time, at the time between the previous and current solution.\n# The second step becomes\n#\n# $$\n# \\nabla^2 \\phi = \\frac{\\rho}{\\delta t} \\nabla \\cdot u^* \\qquad\\text{in } \\Omega,\n# $$\n#\n# $$\n# \\nabla \\phi \\cdot n = 0 \\qquad \\text{on } \\partial \\Omega_D,\n# $$\n#\n# $$\n# \\phi = 0 \\qquad\\text{on } \\partial\\Omega_N\n# $$\n#\n# where $p^{n+\\frac{1}{2}}=p^{n-\\frac{1}{2}} + \\phi$.\n# Finally, the third step is\n#\n# $$\n# \\rho (u^{n+1}-u^{*}) = -\\delta t \\nabla\\phi.\n# $$\n#\n# We start by defining all the variables used in the variational formulations.\n#\n\nu = TrialFunction(V)\nv = TestFunction(V)\nu_ = Function(V, name=\"u\")\nu_s = Function(V, name=\"u_tentative\")\nu_n = Function(V)\nu_n1 = Function(V)\np = TrialFunction(Q)\nq = TestFunction(Q)\np_ = Function(Q, name=\"p\")\nphi = Function(Q, name=\"phi\")\n\n# Next, we define the variational formulation for the first step,\n# where we have integrated the diffusion term, as well as the pressure term by parts.\n\nf = Constant(mesh, PETSc.ScalarType((0, 0)))\nF1 = rho / k * dot(u - u_n, v) * dx\nF1 += inner(dot(1.5 * u_n - 0.5 * u_n1, 0.5 * nabla_grad(u + u_n)), v) * dx\nF1 += 0.5 * mu * inner(grad(u + u_n), grad(v)) * dx - dot(p_, div(v)) * dx\nF1 += dot(f, v) * dx\na1 = form(lhs(F1))\nL1 = form(rhs(F1))\nA1 = create_matrix(a1)\nb1 = create_vector(extract_function_spaces(L1))\n\n# Next we define the second step\n\na2 = form(dot(grad(p), grad(q)) * dx)\nL2 = form(-rho / k * dot(div(u_s), q) * dx)\nA2 = assemble_matrix(a2, bcs=bcp)\nA2.assemble()\nb2 = create_vector(extract_function_spaces(L2))\n\n# We finally create the last step\n\na3 = form(rho * dot(u, v) * dx)\nL3 = form(rho * dot(u_s, v) * dx - k * dot(nabla_grad(phi), v) * dx)\nA3 = assemble_matrix(a3)\nA3.assemble()\nb3 = create_vector(extract_function_spaces(L3))\n\n# As in the previous tutorials, we use PETSc as a linear algebra backend.\n#\n\n# +\n# Solver for step 1\nsolver1 = PETSc.KSP().create(mesh.comm)\nsolver1.setOperators(A1)\nsolver1.setType(PETSc.KSP.Type.BCGS)\npc1 = solver1.getPC()\npc1.setType(PETSc.PC.Type.JACOBI)\n\n# Solver for step 2\nsolver2 = PETSc.KSP().create(mesh.comm)\nsolver2.setOperators(A2)\nsolver2.setType(PETSc.KSP.Type.MINRES)\npc2 = solver2.getPC()\npc2.setType(PETSc.PC.Type.HYPRE)\npc2.setHYPREType(\"boomeramg\")\n\n# Solver for step 3\nsolver3 = PETSc.KSP().create(mesh.comm)\nsolver3.setOperators(A3)\nsolver3.setType(PETSc.KSP.Type.CG)\npc3 = solver3.getPC()\npc3.setType(PETSc.PC.Type.SOR)\n# -\n\n# ## Verification of the implementation compute known physical quantities\n#\n# As a further verification of our implementation, we compute the drag and lift\n# coefficients over the obstacle, defined as\n#\n# \\begin{align*}\n# C_{\\text{D}}(u,p,t,\\partial\\Omega_S) &=\n# \\frac{2}{\\rho L U_{mean}^2}\\int_{\\partial\\Omega_S}\\rho \\nu n \\cdot \\nabla u_{t_S}(t)n_y -p(t)n_x~\\mathrm{d} s,\\\\\n# C_{\\text{L}}(u,p,t,\\partial\\Omega_S) &= -\\frac{2}{\\rho L U_{mean}^2}\\int_{\\partial\\Omega_S}\\rho \\nu n \\cdot \\nabla u_{t_S}(t)n_x + p(t)n_y~\\mathrm{d} s,\n# \\end{align*}\n#\n# where $u_{t_S}$ is the tangential velocity component at the interface of the obstacle $\\partial\\Omega_S$,\n# defined as $u_{t_S}=u\\cdot (n_y,-n_x)$, $U_{mean}=1$ the average inflow velocity, and $L$ the length of the channel.\n# We use `UFL` to create the relevant integrals, and assemble them at each time step.\n\nn = -FacetNormal(mesh) # Normal pointing out of obstacle\ndObs = Measure(\"ds\", domain=mesh, subdomain_data=ft, subdomain_id=obstacle_marker)\nu_t = inner(as_vector((n[1], -n[0])), u_)\ndrag = form(2 / 0.1 * (mu / rho * inner(grad(u_t), n) * n[1] - p_ * n[0]) * dObs)\nlift = form(-2 / 0.1 * (mu / rho * inner(grad(u_t), n) * n[0] + p_ * n[1]) * dObs)\nif mesh.comm.rank == 0:\n C_D = np.zeros(num_steps, dtype=PETSc.ScalarType)\n C_L = np.zeros(num_steps, dtype=PETSc.ScalarType)\n t_u = np.zeros(num_steps, dtype=np.float64)\n t_p = np.zeros(num_steps, dtype=np.float64)\n\n# We will also evaluate the pressure at two points, one in front of the obstacle, $(0.15, 0.2)$,\n# and one behind the obstacle, $(0.25, 0.2)$.\n# To do this, we have to find which cell contains each of the points,\n# so that we can create a linear combination of the local basis functions and coefficients.\n\ntree = bb_tree(mesh, mesh.geometry.dim)\npoints = np.array([[0.15, 0.2, 0], [0.25, 0.2, 0]])\ncell_candidates = compute_collisions_points(tree, points)\ncolliding_cells = compute_colliding_cells(mesh, cell_candidates, points)\nfront_cells = colliding_cells.links(0)\nback_cells = colliding_cells.links(1)\nif mesh.comm.rank == 0:\n p_diff = np.zeros(num_steps, dtype=PETSc.ScalarType)\n\n# ## Solving the time-dependent problem\n#\n# ```{admonition} Stability of the Navier-Stokes equation\n# Note that the current splitting scheme has to fullfil the a\n# [Courant–Friedrichs–Lewy condition](https://en.wikipedia.org/wiki/Courant%E2%80%93Friedrichs%E2%80%93Lewy_condition).\n# This limits the spatial discretization with respect to the inlet velocity and temporal discretization.\n# Other temporal discretization schemes such as the second order backward difference discretization or Crank-Nicholson\n# discretization with Adams-Bashforth linearization are better behaved than our simple backward difference scheme.\n# ```\n#\n# As in the previous example, we create output files for the velocity and pressure and solve the time-dependent problem.\n# As we are solving a time dependent problem with many time steps, we use the `tqdm`-package to visualize the progress.\n# This package can be installed with `pip`.\n\n# +\nfrom pathlib import Path\n\nfolder = Path(\"results\")\nfolder.mkdir(exist_ok=True, parents=True)\nvtx_u = VTXWriter(mesh.comm, folder / \"dfg2D-3-u.bp\", [u_], engine=\"BP4\")\nvtx_p = VTXWriter(mesh.comm, folder / \"dfg2D-3-p.bp\", [p_], engine=\"BP4\")\nvtx_u.write(t)\nvtx_p.write(t)\nprogress = tqdm.autonotebook.tqdm(desc=\"Solving PDE\", total=num_steps)\nfor i in range(num_steps):\n progress.update(1)\n # Update current time step\n t += dt\n # Update inlet velocity\n inlet_velocity.t = t\n u_inlet.interpolate(inlet_velocity)\n\n # Step 1: Tentative velocity step\n A1.zeroEntries()\n assemble_matrix(A1, a1, bcs=bcu)\n A1.assemble()\n with b1.localForm() as loc:\n loc.set(0)\n assemble_vector(b1, L1)\n apply_lifting(b1, [a1], [bcu])\n b1.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)\n set_bc(b1, bcu)\n solver1.solve(b1, u_s.x.petsc_vec)\n u_s.x.scatter_forward()\n\n # Step 2: Pressure corrrection step\n with b2.localForm() as loc:\n loc.set(0)\n assemble_vector(b2, L2)\n apply_lifting(b2, [a2], [bcp])\n b2.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)\n set_bc(b2, bcp)\n solver2.solve(b2, phi.x.petsc_vec)\n phi.x.scatter_forward()\n\n p_.x.petsc_vec.axpy(1, phi.x.petsc_vec)\n p_.x.scatter_forward()\n\n # Step 3: Velocity correction step\n with b3.localForm() as loc:\n loc.set(0)\n assemble_vector(b3, L3)\n b3.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)\n solver3.solve(b3, u_.x.petsc_vec)\n u_.x.scatter_forward()\n\n # Write solutions to file\n vtx_u.write(t)\n vtx_p.write(t)\n\n # Update variable with solution form this time step\n with (\n u_.x.petsc_vec.localForm() as loc_,\n u_n.x.petsc_vec.localForm() as loc_n,\n u_n1.x.petsc_vec.localForm() as loc_n1,\n ):\n loc_n.copy(loc_n1)\n loc_.copy(loc_n)\n\n # Compute physical quantities\n # For this to work in paralell, we gather contributions from all processors\n # to processor zero and sum the contributions.\n drag_coeff = mesh.comm.gather(assemble_scalar(drag), root=0)\n lift_coeff = mesh.comm.gather(assemble_scalar(lift), root=0)\n p_front = None\n if len(front_cells) > 0:\n p_front = p_.eval(points[0], front_cells[:1])\n p_front = mesh.comm.gather(p_front, root=0)\n p_back = None\n if len(back_cells) > 0:\n p_back = p_.eval(points[1], back_cells[:1])\n p_back = mesh.comm.gather(p_back, root=0)\n if mesh.comm.rank == 0:\n t_u[i] = t\n t_p[i] = t - dt / 2\n C_D[i] = sum(drag_coeff)\n C_L[i] = sum(lift_coeff)\n # Choose first pressure that is found from the different processors\n for pressure in p_front:\n if pressure is not None:\n p_diff[i] = pressure[0]\n break\n for pressure in p_back:\n if pressure is not None:\n p_diff[i] -= pressure[0]\n break\nprogress.close()\nvtx_u.close()\nvtx_p.close()\n# -\n\n# Destroy PETSc objects to free memory\n\nA1.destroy()\nA2.destroy()\nA3.destroy()\nb1.destroy()\nb2.destroy()\nb3.destroy()\nsolver1.destroy()\nsolver2.destroy()\nsolver3.destroy()\n\n# ## Verification using data from FEATFLOW\n#\n# As FEATFLOW has provided data for different discretization levels,\n# we compare our numerical data with the data provided using `matplotlib`.\n\nif mesh.comm.rank == 0:\n if not os.path.exists(\"figures\"):\n os.mkdir(\"figures\")\n num_velocity_dofs = V.dofmap.index_map_bs * V.dofmap.index_map.size_global\n num_pressure_dofs = Q.dofmap.index_map_bs * V.dofmap.index_map.size_global\n\n turek = np.loadtxt(\"bdforces_lv4\")\n turek_p = np.loadtxt(\"pointvalues_lv4\")\n fig = plt.figure(figsize=(25, 8))\n l1 = plt.plot(\n t_u,\n C_D,\n label=r\"FEniCSx ({0:d} dofs)\".format(num_velocity_dofs + num_pressure_dofs),\n linewidth=2,\n )\n l2 = plt.plot(\n turek[1:, 1],\n turek[1:, 3],\n marker=\"x\",\n markevery=50,\n linestyle=\"\",\n markersize=4,\n label=\"FEATFLOW (42016 dofs)\",\n )\n plt.title(\"Drag coefficient\")\n plt.grid()\n plt.legend()\n plt.savefig(\"figures/drag_comparison.png\")\n\n fig = plt.figure(figsize=(25, 8))\n l1 = plt.plot(\n t_u,\n C_L,\n label=r\"FEniCSx ({0:d} dofs)\".format(num_velocity_dofs + num_pressure_dofs),\n linewidth=2,\n )\n l2 = plt.plot(\n turek[1:, 1],\n turek[1:, 4],\n marker=\"x\",\n markevery=50,\n linestyle=\"\",\n markersize=4,\n label=\"FEATFLOW (42016 dofs)\",\n )\n plt.title(\"Lift coefficient\")\n plt.grid()\n plt.legend()\n plt.savefig(\"figures/lift_comparison.png\")\n\n fig = plt.figure(figsize=(25, 8))\n l1 = plt.plot(\n t_p,\n p_diff,\n label=r\"FEniCSx ({0:d} dofs)\".format(num_velocity_dofs + num_pressure_dofs),\n linewidth=2,\n )\n l2 = plt.plot(\n turek[1:, 1],\n turek_p[1:, 6] - turek_p[1:, -1],\n marker=\"x\",\n markevery=50,\n linestyle=\"\",\n markersize=4,\n label=\"FEATFLOW (42016 dofs)\",\n )\n plt.title(\"Pressure difference\")\n plt.grid()\n plt.legend()\n plt.savefig(\"figures/pressure_comparison.png\")\n" }, { "path": "chapter2/pointvalues_lv4", "content": "# timestep time x y type deriv value x y type deriv value ...\n0 0.0000000000E+00 1.50000E-01 1.99999E-01 3 0 0.0000000000E+00 2.50000E-01 1.99999E-01 3 0 0.0000000000E+00\n1 3.1250000000E-04 1.50000E-01 1.99999E-01 3 0 4.1073820907E-01 2.50000E-01 1.99999E-01 3 0 3.2371300233E-01\n2 9.3750000000E-04 1.50000E-01 1.99999E-01 3 0 4.1179414118E-01 2.50000E-01 1.99999E-01 3 0 3.2370602894E-01\n3 1.5625000000E-03 1.50000E-01 1.99999E-01 3 0 4.1256633721E-01 2.50000E-01 1.99999E-01 3 0 3.2380174869E-01\n4 2.1875000000E-03 1.50000E-01 1.99999E-01 3 0 4.1325564289E-01 2.50000E-01 1.99999E-01 3 0 3.2390469754E-01\n5 2.8125000000E-03 1.50000E-01 1.99999E-01 3 0 4.1387410188E-01 2.50000E-01 1.99999E-01 3 0 3.2401879251E-01\n6 3.4375000000E-03 1.50000E-01 1.99999E-01 3 0 4.1444905341E-01 2.50000E-01 1.99999E-01 3 0 3.2413237153E-01\n7 4.0625000000E-03 1.50000E-01 1.99999E-01 3 0 4.1498642050E-01 2.50000E-01 1.99999E-01 3 0 3.2424674641E-01\n8 4.6875000000E-03 1.50000E-01 1.99999E-01 3 0 4.1549545373E-01 2.50000E-01 1.99999E-01 3 0 3.2435899352E-01\n9 5.3125000000E-03 1.50000E-01 1.99999E-01 3 0 4.1597950784E-01 2.50000E-01 1.99999E-01 3 0 3.2446972396E-01\n10 5.9375000000E-03 1.50000E-01 1.99999E-01 3 0 4.1644301016E-01 2.50000E-01 1.99999E-01 3 0 3.2457803507E-01\n11 6.5625000000E-03 1.50000E-01 1.99999E-01 3 0 4.1688811167E-01 2.50000E-01 1.99999E-01 3 0 3.2468427243E-01\n12 7.1875000000E-03 1.50000E-01 1.99999E-01 3 0 4.1731733907E-01 2.50000E-01 1.99999E-01 3 0 3.2478815868E-01\n13 7.8125000000E-03 1.50000E-01 1.99999E-01 3 0 4.1773218946E-01 2.50000E-01 1.99999E-01 3 0 3.2488991818E-01\n14 8.4375000000E-03 1.50000E-01 1.99999E-01 3 0 4.1813426798E-01 2.50000E-01 1.99999E-01 3 0 3.2498949629E-01\n15 9.0625000000E-03 1.50000E-01 1.99999E-01 3 0 4.1852466992E-01 2.50000E-01 1.99999E-01 3 0 3.2508705293E-01\n16 9.6875000000E-03 1.50000E-01 1.99999E-01 3 0 4.1890449466E-01 2.50000E-01 1.99999E-01 3 0 3.2518261694E-01\n17 1.0312500000E-02 1.50000E-01 1.99999E-01 3 0 4.1927457025E-01 2.50000E-01 1.99999E-01 3 0 3.2527630967E-01\n18 1.0937500000E-02 1.50000E-01 1.99999E-01 3 0 4.1963569293E-01 2.50000E-01 1.99999E-01 3 0 3.2536818964E-01\n19 1.1562500000E-02 1.50000E-01 1.99999E-01 3 0 4.1998850466E-01 2.50000E-01 1.99999E-01 3 0 3.2545835334E-01\n20 1.2187500000E-02 1.50000E-01 1.99999E-01 3 0 4.2033360699E-01 2.50000E-01 1.99999E-01 3 0 3.2554686689E-01\n21 1.2812500000E-02 1.50000E-01 1.99999E-01 3 0 4.2067150782E-01 2.50000E-01 1.99999E-01 3 0 3.2563380940E-01\n22 1.3437500000E-02 1.50000E-01 1.99999E-01 3 0 4.2100267705E-01 2.50000E-01 1.99999E-01 3 0 3.2571924601E-01\n23 1.4062500000E-02 1.50000E-01 1.99999E-01 3 0 4.2132752358E-01 2.50000E-01 1.99999E-01 3 0 3.2580324297E-01\n24 1.4687500000E-02 1.50000E-01 1.99999E-01 3 0 4.2164642400E-01 2.50000E-01 1.99999E-01 3 0 3.2588586086E-01\n25 1.5312500000E-02 1.50000E-01 1.99999E-01 3 0 4.2195971294E-01 2.50000E-01 1.99999E-01 3 0 3.2596715640E-01\n26 1.5937500000E-02 1.50000E-01 1.99999E-01 3 0 4.2226769819E-01 2.50000E-01 1.99999E-01 3 0 3.2604718425E-01\n27 1.6562500000E-02 1.50000E-01 1.99999E-01 3 0 4.2257065762E-01 2.50000E-01 1.99999E-01 3 0 3.2612599377E-01\n28 1.7187500000E-02 1.50000E-01 1.99999E-01 3 0 4.2286884728E-01 2.50000E-01 1.99999E-01 3 0 3.2620363401E-01\n29 1.7812500000E-02 1.50000E-01 1.99999E-01 3 0 4.2316250060E-01 2.50000E-01 1.99999E-01 3 0 3.2628014818E-01\n30 1.8437500000E-02 1.50000E-01 1.99999E-01 3 0 4.2345183340E-01 2.50000E-01 1.99999E-01 3 0 3.2635557996E-01\n31 1.9062500000E-02 1.50000E-01 1.99999E-01 3 0 4.2373704427E-01 2.50000E-01 1.99999E-01 3 0 3.2642996773E-01\n32 1.9687500000E-02 1.50000E-01 1.99999E-01 3 0 4.2401831737E-01 2.50000E-01 1.99999E-01 3 0 3.2650335049E-01\n33 2.0312500000E-02 1.50000E-01 1.99999E-01 3 0 4.2429582317E-01 2.50000E-01 1.99999E-01 3 0 3.2657576245E-01\n34 2.0937500000E-02 1.50000E-01 1.99999E-01 3 0 4.2456972041E-01 2.50000E-01 1.99999E-01 3 0 3.2664723842E-01\n35 2.1562500000E-02 1.50000E-01 1.99999E-01 3 0 4.2484015675E-01 2.50000E-01 1.99999E-01 3 0 3.2671780908E-01\n36 2.2187500000E-02 1.50000E-01 1.99999E-01 3 0 4.2510727035E-01 2.50000E-01 1.99999E-01 3 0 3.2678750572E-01\n37 2.2812500000E-02 1.50000E-01 1.99999E-01 3 0 4.2537119016E-01 2.50000E-01 1.99999E-01 3 0 3.2685635603E-01\n38 2.3437500000E-02 1.50000E-01 1.99999E-01 3 0 4.2563203729E-01 2.50000E-01 1.99999E-01 3 0 3.2692438814E-01\n39 2.4062500000E-02 1.50000E-01 1.99999E-01 3 0 4.2588992516E-01 2.50000E-01 1.99999E-01 3 0 3.2699162712E-01\n40 2.4687500000E-02 1.50000E-01 1.99999E-01 3 0 4.2614496060E-01 2.50000E-01 1.99999E-01 3 0 3.2705809830E-01\n41 2.5312500000E-02 1.50000E-01 1.99999E-01 3 0 4.2639724417E-01 2.50000E-01 1.99999E-01 3 0 3.2712382462E-01\n42 2.5937500000E-02 1.50000E-01 1.99999E-01 3 0 4.2664687081E-01 2.50000E-01 1.99999E-01 3 0 3.2718882904E-01\n43 2.6562500000E-02 1.50000E-01 1.99999E-01 3 0 4.2689392998E-01 2.50000E-01 1.99999E-01 3 0 3.2725313243E-01\n44 2.7187500000E-02 1.50000E-01 1.99999E-01 3 0 4.2713850649E-01 2.50000E-01 1.99999E-01 3 0 3.2731675567E-01\n45 2.7812500000E-02 1.50000E-01 1.99999E-01 3 0 4.2738068042E-01 2.50000E-01 1.99999E-01 3 0 3.2737971783E-01\n46 2.8437500000E-02 1.50000E-01 1.99999E-01 3 0 4.2762052808E-01 2.50000E-01 1.99999E-01 3 0 3.2744203813E-01\n47 2.9062500000E-02 1.50000E-01 1.99999E-01 3 0 4.2785812142E-01 2.50000E-01 1.99999E-01 3 0 3.2750373397E-01\n48 2.9687500000E-02 1.50000E-01 1.99999E-01 3 0 4.2809352919E-01 2.50000E-01 1.99999E-01 3 0 3.2756482286E-01\n49 3.0312500000E-02 1.50000E-01 1.99999E-01 3 0 4.2832681659E-01 2.50000E-01 1.99999E-01 3 0 3.2762532099E-01\n50 3.0937500000E-02 1.50000E-01 1.99999E-01 3 0 4.2855804589E-01 2.50000E-01 1.99999E-01 3 0 3.2768524443E-01\n51 3.1562500000E-02 1.50000E-01 1.99999E-01 3 0 4.2878727616E-01 2.50000E-01 1.99999E-01 3 0 3.2774460803E-01\n52 3.2187500000E-02 1.50000E-01 1.99999E-01 3 0 4.2901456391E-01 2.50000E-01 1.99999E-01 3 0 3.2780342645E-01\n53 3.2812500000E-02 1.50000E-01 1.99999E-01 3 0 4.2923996327E-01 2.50000E-01 1.99999E-01 3 0 3.2786171375E-01\n54 3.3437500000E-02 1.50000E-01 1.99999E-01 3 0 4.2946352567E-01 2.50000E-01 1.99999E-01 3 0 3.2791948334E-01\n55 3.4062500000E-02 1.50000E-01 1.99999E-01 3 0 4.2968530047E-01 2.50000E-01 1.99999E-01 3 0 3.2797674809E-01\n56 3.4687500000E-02 1.50000E-01 1.99999E-01 3 0 4.2990533486E-01 2.50000E-01 1.99999E-01 3 0 3.2803352053E-01\n57 3.5312500000E-02 1.50000E-01 1.99999E-01 3 0 4.3012367411E-01 2.50000E-01 1.99999E-01 3 0 3.2808981262E-01\n58 3.5937500000E-02 1.50000E-01 1.99999E-01 3 0 4.3034036165E-01 2.50000E-01 1.99999E-01 3 0 3.2814563606E-01\n59 3.6562500000E-02 1.50000E-01 1.99999E-01 3 0 4.3055543888E-01 2.50000E-01 1.99999E-01 3 0 3.2820100175E-01\n60 3.7187500000E-02 1.50000E-01 1.99999E-01 3 0 4.3076894604E-01 2.50000E-01 1.99999E-01 3 0 3.2825592079E-01\n61 3.7812500000E-02 1.50000E-01 1.99999E-01 3 0 4.3098092115E-01 2.50000E-01 1.99999E-01 3 0 3.2831040317E-01\n62 3.8437500000E-02 1.50000E-01 1.99999E-01 3 0 4.3119140148E-01 2.50000E-01 1.99999E-01 3 0 3.2836445932E-01\n63 3.9062500000E-02 1.50000E-01 1.99999E-01 3 0 4.3140042212E-01 2.50000E-01 1.99999E-01 3 0 3.2841809855E-01\n64 3.9687500000E-02 1.50000E-01 1.99999E-01 3 0 4.3160801745E-01 2.50000E-01 1.99999E-01 3 0 3.2847133042E-01\n65 4.0312500000E-02 1.50000E-01 1.99999E-01 3 0 4.3181422041E-01 2.50000E-01 1.99999E-01 3 0 3.2852416405E-01\n66 4.0937500000E-02 1.50000E-01 1.99999E-01 3 0 4.3201906243E-01 2.50000E-01 1.99999E-01 3 0 3.2857660793E-01\n67 4.1562500000E-02 1.50000E-01 1.99999E-01 3 0 4.3222257425E-01 2.50000E-01 1.99999E-01 3 0 3.2862867070E-01\n68 4.2187500000E-02 1.50000E-01 1.99999E-01 3 0 4.3242478530E-01 2.50000E-01 1.99999E-01 3 0 3.2868036053E-01\n69 4.2812500000E-02 1.50000E-01 1.99999E-01 3 0 4.3262572398E-01 2.50000E-01 1.99999E-01 3 0 3.2873168529E-01\n70 4.3437500000E-02 1.50000E-01 1.99999E-01 3 0 4.3282541776E-01 2.50000E-01 1.99999E-01 3 0 3.2878265268E-01\n71 4.4062500000E-02 1.50000E-01 1.99999E-01 3 0 4.3302389315E-01 2.50000E-01 1.99999E-01 3 0 3.2883327012E-01\n72 4.4687500000E-02 1.50000E-01 1.99999E-01 3 0 4.3322117579E-01 2.50000E-01 1.99999E-01 3 0 3.2888354483E-01\n73 4.5312500000E-02 1.50000E-01 1.99999E-01 3 0 4.3341729046E-01 2.50000E-01 1.99999E-01 3 0 3.2893348377E-01\n74 4.5937500000E-02 1.50000E-01 1.99999E-01 3 0 4.3361226116E-01 2.50000E-01 1.99999E-01 3 0 3.2898309374E-01\n75 4.6562500000E-02 1.50000E-01 1.99999E-01 3 0 4.3380611097E-01 2.50000E-01 1.99999E-01 3 0 3.2903238119E-01\n76 4.7187500000E-02 1.50000E-01 1.99999E-01 3 0 4.3399886254E-01 2.50000E-01 1.99999E-01 3 0 3.2908135269E-01\n77 4.7812500000E-02 1.50000E-01 1.99999E-01 3 0 4.3419053742E-01 2.50000E-01 1.99999E-01 3 0 3.2913001419E-01\n78 4.8437500000E-02 1.50000E-01 1.99999E-01 3 0 4.3438115689E-01 2.50000E-01 1.99999E-01 3 0 3.2917837190E-01\n79 4.9062500000E-02 1.50000E-01 1.99999E-01 3 0 4.3457074121E-01 2.50000E-01 1.99999E-01 3 0 3.2922643148E-01\n80 4.9687500000E-02 1.50000E-01 1.99999E-01 3 0 4.3475931027E-01 2.50000E-01 1.99999E-01 3 0 3.2927419868E-01\n81 5.0312500000E-02 1.50000E-01 1.99999E-01 3 0 4.3494688324E-01 2.50000E-01 1.99999E-01 3 0 3.2932167892E-01\n82 5.0937500000E-02 1.50000E-01 1.99999E-01 3 0 4.3513347889E-01 2.50000E-01 1.99999E-01 3 0 3.2936887767E-01\n83 5.1562500000E-02 1.50000E-01 1.99999E-01 3 0 4.3531911514E-01 2.50000E-01 1.99999E-01 3 0 3.2941579995E-01\n84 5.2187500000E-02 1.50000E-01 1.99999E-01 3 0 4.3550380965E-01 2.50000E-01 1.99999E-01 3 0 3.2946245088E-01\n85 5.2812500000E-02 1.50000E-01 1.99999E-01 3 0 4.3568757955E-01 2.50000E-01 1.99999E-01 3 0 3.2950883544E-01\n86 5.3437500000E-02 1.50000E-01 1.99999E-01 3 0 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5.7690508595E-01 2.50000E-01 1.99999E-01 3 0 3.3484226187E-01\n1070 6.6843750000E-01 1.50000E-01 1.99999E-01 3 0 5.7712397380E-01 2.50000E-01 1.99999E-01 3 0 3.3479519603E-01\n1071 6.6906250000E-01 1.50000E-01 1.99999E-01 3 0 5.7734310284E-01 2.50000E-01 1.99999E-01 3 0 3.3474797476E-01\n1072 6.6968750000E-01 1.50000E-01 1.99999E-01 3 0 5.7756247268E-01 2.50000E-01 1.99999E-01 3 0 3.3470059798E-01\n1073 6.7031250000E-01 1.50000E-01 1.99999E-01 3 0 5.7778208290E-01 2.50000E-01 1.99999E-01 3 0 3.3465306564E-01\n1074 6.7093750000E-01 1.50000E-01 1.99999E-01 3 0 5.7800193309E-01 2.50000E-01 1.99999E-01 3 0 3.3460537764E-01\n1075 6.7156250000E-01 1.50000E-01 1.99999E-01 3 0 5.7822202284E-01 2.50000E-01 1.99999E-01 3 0 3.3455753393E-01\n1076 6.7218750000E-01 1.50000E-01 1.99999E-01 3 0 5.7844235172E-01 2.50000E-01 1.99999E-01 3 0 3.3450953442E-01\n1077 6.7281250000E-01 1.50000E-01 1.99999E-01 3 0 5.7866291931E-01 2.50000E-01 1.99999E-01 3 0 3.3446137905E-01\n1078 6.7343750000E-01 1.50000E-01 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7.3406250000E-01 1.50000E-01 1.99999E-01 3 0 6.0135358314E-01 2.50000E-01 1.99999E-01 3 0 3.2897663514E-01\n1176 7.3468750000E-01 1.50000E-01 1.99999E-01 3 0 6.0159512877E-01 2.50000E-01 1.99999E-01 3 0 3.2891278202E-01\n1177 7.3531250000E-01 1.50000E-01 1.99999E-01 3 0 6.0183685574E-01 2.50000E-01 1.99999E-01 3 0 3.2884876874E-01\n1178 7.3593750000E-01 1.50000E-01 1.99999E-01 3 0 6.0207876204E-01 2.50000E-01 1.99999E-01 3 0 3.2878459401E-01\n1179 7.3656250000E-01 1.50000E-01 1.99999E-01 3 0 6.0232084994E-01 2.50000E-01 1.99999E-01 3 0 3.2872026057E-01\n1180 7.3718750000E-01 1.50000E-01 1.99999E-01 3 0 6.0256311650E-01 2.50000E-01 1.99999E-01 3 0 3.2865576627E-01\n1181 7.3781250000E-01 1.50000E-01 1.99999E-01 3 0 6.0280556117E-01 2.50000E-01 1.99999E-01 3 0 3.2859111118E-01\n1182 7.3843750000E-01 1.50000E-01 1.99999E-01 3 0 6.0304818274E-01 2.50000E-01 1.99999E-01 3 0 3.2852629442E-01\n1183 7.3906250000E-01 1.50000E-01 1.99999E-01 3 0 6.0329098362E-01 2.50000E-01 1.99999E-01 3 0 3.2846131979E-01\n1184 7.3968750000E-01 1.50000E-01 1.99999E-01 3 0 6.0353395980E-01 2.50000E-01 1.99999E-01 3 0 3.2839618334E-01\n1185 7.4031250000E-01 1.50000E-01 1.99999E-01 3 0 6.0377711181E-01 2.50000E-01 1.99999E-01 3 0 3.2833088640E-01\n1186 7.4093750000E-01 1.50000E-01 1.99999E-01 3 0 6.0402044048E-01 2.50000E-01 1.99999E-01 3 0 3.2826543079E-01\n1187 7.4156250000E-01 1.50000E-01 1.99999E-01 3 0 6.0426394036E-01 2.50000E-01 1.99999E-01 3 0 3.2819981084E-01\n1188 7.4218750000E-01 1.50000E-01 1.99999E-01 3 0 6.0450761479E-01 2.50000E-01 1.99999E-01 3 0 3.2813403125E-01\n1189 7.4281250000E-01 1.50000E-01 1.99999E-01 3 0 6.0475146585E-01 2.50000E-01 1.99999E-01 3 0 3.2806809445E-01\n1190 7.4343750000E-01 1.50000E-01 1.99999E-01 3 0 6.0499548573E-01 2.50000E-01 1.99999E-01 3 0 3.2800199351E-01\n1191 7.4406250000E-01 1.50000E-01 1.99999E-01 3 0 6.0523967855E-01 2.50000E-01 1.99999E-01 3 0 3.2793573304E-01\n1192 7.4468750000E-01 1.50000E-01 1.99999E-01 3 0 6.0548404365E-01 2.50000E-01 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6.7682847386E-01 2.50000E-01 1.99999E-01 3 0 3.0387525426E-01\n1466 9.1593750000E-01 1.50000E-01 1.99999E-01 3 0 6.7710284665E-01 2.50000E-01 1.99999E-01 3 0 3.0376724308E-01\n1467 9.1656250000E-01 1.50000E-01 1.99999E-01 3 0 6.7737730722E-01 2.50000E-01 1.99999E-01 3 0 3.0365910413E-01\n1468 9.1718750000E-01 1.50000E-01 1.99999E-01 3 0 6.7765184986E-01 2.50000E-01 1.99999E-01 3 0 3.0355083214E-01\n1469 9.1781250000E-01 1.50000E-01 1.99999E-01 3 0 6.7792647757E-01 2.50000E-01 1.99999E-01 3 0 3.0344242999E-01\n1470 9.1843750000E-01 1.50000E-01 1.99999E-01 3 0 6.7820119351E-01 2.50000E-01 1.99999E-01 3 0 3.0333390115E-01\n1471 9.1906250000E-01 1.50000E-01 1.99999E-01 3 0 6.7847598952E-01 2.50000E-01 1.99999E-01 3 0 3.0322523786E-01\n1472 9.1968750000E-01 1.50000E-01 1.99999E-01 3 0 6.7875087427E-01 2.50000E-01 1.99999E-01 3 0 3.0311644868E-01\n1473 9.2031250000E-01 1.50000E-01 1.99999E-01 3 0 6.7902584309E-01 2.50000E-01 1.99999E-01 3 0 3.0300752933E-01\n1474 9.2093750000E-01 1.50000E-01 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9.2656250000E-01 1.50000E-01 1.99999E-01 3 0 6.8178022511E-01 2.50000E-01 1.99999E-01 3 0 3.0191128662E-01\n1484 9.2718750000E-01 1.50000E-01 1.99999E-01 3 0 6.8205613314E-01 2.50000E-01 1.99999E-01 3 0 3.0180096217E-01\n1485 9.2781250000E-01 1.50000E-01 1.99999E-01 3 0 6.8233212746E-01 2.50000E-01 1.99999E-01 3 0 3.0169051202E-01\n1486 9.2843750000E-01 1.50000E-01 1.99999E-01 3 0 6.8260820751E-01 2.50000E-01 1.99999E-01 3 0 3.0157993583E-01\n1487 9.2906250000E-01 1.50000E-01 1.99999E-01 3 0 6.8288437333E-01 2.50000E-01 1.99999E-01 3 0 3.0146923382E-01\n1488 9.2968750000E-01 1.50000E-01 1.99999E-01 3 0 6.8316062530E-01 2.50000E-01 1.99999E-01 3 0 3.0135840656E-01\n1489 9.3031250000E-01 1.50000E-01 1.99999E-01 3 0 6.8343696242E-01 2.50000E-01 1.99999E-01 3 0 3.0124745328E-01\n1490 9.3093750000E-01 1.50000E-01 1.99999E-01 3 0 6.8371338913E-01 2.50000E-01 1.99999E-01 3 0 3.0113637844E-01\n1491 9.3156250000E-01 1.50000E-01 1.99999E-01 3 0 6.8398989492E-01 2.50000E-01 1.99999E-01 3 0 3.0102517218E-01\n1492 9.3218750000E-01 1.50000E-01 1.99999E-01 3 0 6.8426649341E-01 2.50000E-01 1.99999E-01 3 0 3.0091384762E-01\n1493 9.3281250000E-01 1.50000E-01 1.99999E-01 3 0 6.8454317587E-01 2.50000E-01 1.99999E-01 3 0 3.0080239671E-01\n1494 9.3343750000E-01 1.50000E-01 1.99999E-01 3 0 6.8481994456E-01 2.50000E-01 1.99999E-01 3 0 3.0069082176E-01\n1495 9.3406250000E-01 1.50000E-01 1.99999E-01 3 0 6.8509679989E-01 2.50000E-01 1.99999E-01 3 0 3.0057912336E-01\n1496 9.3468750000E-01 1.50000E-01 1.99999E-01 3 0 6.8537374171E-01 2.50000E-01 1.99999E-01 3 0 3.0046730156E-01\n1497 9.3531250000E-01 1.50000E-01 1.99999E-01 3 0 6.8565076840E-01 2.50000E-01 1.99999E-01 3 0 3.0035535497E-01\n1498 9.3593750000E-01 1.50000E-01 1.99999E-01 3 0 6.8592788567E-01 2.50000E-01 1.99999E-01 3 0 3.0024328927E-01\n1499 9.3656250000E-01 1.50000E-01 1.99999E-01 3 0 6.8620508326E-01 2.50000E-01 1.99999E-01 3 0 3.0013109484E-01\n1500 9.3718750000E-01 1.50000E-01 1.99999E-01 3 0 6.8648237318E-01 2.50000E-01 1.99999E-01 3 0 3.0001878336E-01\n1501 9.3781250000E-01 1.50000E-01 1.99999E-01 3 0 6.8675974772E-01 2.50000E-01 1.99999E-01 3 0 2.9990634757E-01\n1502 9.3843750000E-01 1.50000E-01 1.99999E-01 3 0 6.8703721106E-01 2.50000E-01 1.99999E-01 3 0 2.9979379176E-01\n1503 9.3906250000E-01 1.50000E-01 1.99999E-01 3 0 6.8731475416E-01 2.50000E-01 1.99999E-01 3 0 2.9968110738E-01\n1504 9.3968750000E-01 1.50000E-01 1.99999E-01 3 0 6.8759239537E-01 2.50000E-01 1.99999E-01 3 0 2.9956831226E-01\n1505 9.4031250000E-01 1.50000E-01 1.99999E-01 3 0 6.8787011383E-01 2.50000E-01 1.99999E-01 3 0 2.9945538653E-01\n1506 9.4093750000E-01 1.50000E-01 1.99999E-01 3 0 6.8814792464E-01 2.50000E-01 1.99999E-01 3 0 2.9934234489E-01\n1507 9.4156250000E-01 1.50000E-01 1.99999E-01 3 0 6.8842581828E-01 2.50000E-01 1.99999E-01 3 0 2.9922917842E-01\n1508 9.4218750000E-01 1.50000E-01 1.99999E-01 3 0 6.8870380427E-01 2.50000E-01 1.99999E-01 3 0 2.9911589637E-01\n1509 9.4281250000E-01 1.50000E-01 1.99999E-01 3 0 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7.9609375000E+00 1.50000E-01 1.99999E-01 3 0 -4.3090464835E-01 2.50000E-01 1.99999E-01 3 0 -3.2742303581E-01\n1539 7.9615625000E+00 1.50000E-01 1.99999E-01 3 0 -4.3103483451E-01 2.50000E-01 1.99999E-01 3 0 -3.2742083025E-01\n1540 7.9621875000E+00 1.50000E-01 1.99999E-01 3 0 -4.3116486590E-01 2.50000E-01 1.99999E-01 3 0 -3.2741861315E-01\n1541 7.9628125000E+00 1.50000E-01 1.99999E-01 3 0 -4.3129474279E-01 2.50000E-01 1.99999E-01 3 0 -3.2741638453E-01\n1542 7.9634375000E+00 1.50000E-01 1.99999E-01 3 0 -4.3142446524E-01 2.50000E-01 1.99999E-01 3 0 -3.2741414418E-01\n1543 7.9640625000E+00 1.50000E-01 1.99999E-01 3 0 -4.3155403327E-01 2.50000E-01 1.99999E-01 3 0 -3.2741189185E-01\n1544 7.9646875000E+00 1.50000E-01 1.99999E-01 3 0 -4.3168344719E-01 2.50000E-01 1.99999E-01 3 0 -3.2740962761E-01\n1545 7.9653125000E+00 1.50000E-01 1.99999E-01 3 0 -4.3181270703E-01 2.50000E-01 1.99999E-01 3 0 -3.2740735120E-01\n1546 7.9659375000E+00 1.50000E-01 1.99999E-01 3 0 -4.3194181292E-01 2.50000E-01 1.99999E-01 3 0 -3.2740506250E-01\n1547 7.9665625000E+00 1.50000E-01 1.99999E-01 3 0 -4.3207076500E-01 2.50000E-01 1.99999E-01 3 0 -3.2740276139E-01\n1548 7.9671875000E+00 1.50000E-01 1.99999E-01 3 0 -4.3219956337E-01 2.50000E-01 1.99999E-01 3 0 -3.2740044769E-01\n1549 7.9678125000E+00 1.50000E-01 1.99999E-01 3 0 -4.3232820816E-01 2.50000E-01 1.99999E-01 3 0 -3.2739812129E-01\n1550 7.9684375000E+00 1.50000E-01 1.99999E-01 3 0 -4.3245669963E-01 2.50000E-01 1.99999E-01 3 0 -3.2739578217E-01\n1551 7.9690625000E+00 1.50000E-01 1.99999E-01 3 0 -4.3258503768E-01 2.50000E-01 1.99999E-01 3 0 -3.2739342997E-01\n1552 7.9696875000E+00 1.50000E-01 1.99999E-01 3 0 -4.3271322260E-01 2.50000E-01 1.99999E-01 3 0 -3.2739106471E-01\n1553 7.9703125000E+00 1.50000E-01 1.99999E-01 3 0 -4.3284125451E-01 2.50000E-01 1.99999E-01 3 0 -3.2738868625E-01\n1554 7.9709375000E+00 1.50000E-01 1.99999E-01 3 0 -4.3296913350E-01 2.50000E-01 1.99999E-01 3 0 -3.2738629442E-01\n1555 7.9715625000E+00 1.50000E-01 1.99999E-01 3 0 -4.3309685964E-01 2.50000E-01 1.99999E-01 3 0 -3.2738388903E-01\n1556 7.9721875000E+00 1.50000E-01 1.99999E-01 3 0 -4.3322443322E-01 2.50000E-01 1.99999E-01 3 0 -3.2738147009E-01\n1557 7.9728125000E+00 1.50000E-01 1.99999E-01 3 0 -4.3335185423E-01 2.50000E-01 1.99999E-01 3 0 -3.2737903733E-01\n1558 7.9734375000E+00 1.50000E-01 1.99999E-01 3 0 -4.3347912288E-01 2.50000E-01 1.99999E-01 3 0 -3.2737659070E-01\n1559 7.9740625000E+00 1.50000E-01 1.99999E-01 3 0 -4.3360623926E-01 2.50000E-01 1.99999E-01 3 0 -3.2737413001E-01\n1560 7.9746875000E+00 1.50000E-01 1.99999E-01 3 0 -4.3373320348E-01 2.50000E-01 1.99999E-01 3 0 -3.2737165512E-01\n1561 7.9753125000E+00 1.50000E-01 1.99999E-01 3 0 -4.3386001578E-01 2.50000E-01 1.99999E-01 3 0 -3.2736916600E-01\n1562 7.9759375000E+00 1.50000E-01 1.99999E-01 3 0 -4.3398667617E-01 2.50000E-01 1.99999E-01 3 0 -3.2736666238E-01\n1563 7.9765625000E+00 1.50000E-01 1.99999E-01 3 0 -4.3411318485E-01 2.50000E-01 1.99999E-01 3 0 -3.2736414421E-01\n1564 7.9771875000E+00 1.50000E-01 1.99999E-01 3 0 -4.3423954190E-01 2.50000E-01 1.99999E-01 3 0 -3.2736161131E-01\n1565 7.9778125000E+00 1.50000E-01 1.99999E-01 3 0 -4.3436574757E-01 2.50000E-01 1.99999E-01 3 0 -3.2735906363E-01\n1566 7.9784375000E+00 1.50000E-01 1.99999E-01 3 0 -4.3449180189E-01 2.50000E-01 1.99999E-01 3 0 -3.2735650096E-01\n1567 7.9790625000E+00 1.50000E-01 1.99999E-01 3 0 -4.3461770499E-01 2.50000E-01 1.99999E-01 3 0 -3.2735392316E-01\n1568 7.9796875000E+00 1.50000E-01 1.99999E-01 3 0 -4.3474345708E-01 2.50000E-01 1.99999E-01 3 0 -3.2735133016E-01\n1569 7.9803125000E+00 1.50000E-01 1.99999E-01 3 0 -4.3486905820E-01 2.50000E-01 1.99999E-01 3 0 -3.2734872175E-01\n1570 7.9809375000E+00 1.50000E-01 1.99999E-01 3 0 -4.3499450859E-01 2.50000E-01 1.99999E-01 3 0 -3.2734609788E-01\n1571 7.9815625000E+00 1.50000E-01 1.99999E-01 3 0 -4.3511980830E-01 2.50000E-01 1.99999E-01 3 0 -3.2734345836E-01\n1572 7.9821875000E+00 1.50000E-01 1.99999E-01 3 0 -4.3524495751E-01 2.50000E-01 1.99999E-01 3 0 -3.2734080308E-01\n1573 7.9828125000E+00 1.50000E-01 1.99999E-01 3 0 -4.3536995639E-01 2.50000E-01 1.99999E-01 3 0 -3.2733813194E-01\n1574 7.9834375000E+00 1.50000E-01 1.99999E-01 3 0 -4.3549480495E-01 2.50000E-01 1.99999E-01 3 0 -3.2733544470E-01\n1575 7.9840625000E+00 1.50000E-01 1.99999E-01 3 0 -4.3561950349E-01 2.50000E-01 1.99999E-01 3 0 -3.2733274136E-01\n1576 7.9846875000E+00 1.50000E-01 1.99999E-01 3 0 -4.3574405203E-01 2.50000E-01 1.99999E-01 3 0 -3.2733002170E-01\n1577 7.9853125000E+00 1.50000E-01 1.99999E-01 3 0 -4.3586845078E-01 2.50000E-01 1.99999E-01 3 0 -3.2732728565E-01\n1578 7.9859375000E+00 1.50000E-01 1.99999E-01 3 0 -4.3599269984E-01 2.50000E-01 1.99999E-01 3 0 -3.2732453304E-01\n1579 7.9865625000E+00 1.50000E-01 1.99999E-01 3 0 -4.3611679930E-01 2.50000E-01 1.99999E-01 3 0 -3.2732176369E-01\n1580 7.9871875000E+00 1.50000E-01 1.99999E-01 3 0 -4.3624074948E-01 2.50000E-01 1.99999E-01 3 0 -3.2731897765E-01\n1581 7.9878125000E+00 1.50000E-01 1.99999E-01 3 0 -4.3636455026E-01 2.50000E-01 1.99999E-01 3 0 -3.2731617453E-01\n1582 7.9884375000E+00 1.50000E-01 1.99999E-01 3 0 -4.3648820202E-01 2.50000E-01 1.99999E-01 3 0 -3.2731335443E-01\n1583 7.9890625000E+00 1.50000E-01 1.99999E-01 3 0 -4.3661170473E-01 2.50000E-01 1.99999E-01 3 0 -3.2731051706E-01\n1584 7.9896875000E+00 1.50000E-01 1.99999E-01 3 0 -4.3673505866E-01 2.50000E-01 1.99999E-01 3 0 -3.2730766242E-01\n1585 7.9903125000E+00 1.50000E-01 1.99999E-01 3 0 -4.3685826387E-01 2.50000E-01 1.99999E-01 3 0 -3.2730479029E-01\n1586 7.9909375000E+00 1.50000E-01 1.99999E-01 3 0 -4.3698132050E-01 2.50000E-01 1.99999E-01 3 0 -3.2730190055E-01\n1587 7.9915625000E+00 1.50000E-01 1.99999E-01 3 0 -4.3710422877E-01 2.50000E-01 1.99999E-01 3 0 -3.2729899315E-01\n1588 7.9921875000E+00 1.50000E-01 1.99999E-01 3 0 -4.3722698869E-01 2.50000E-01 1.99999E-01 3 0 -3.2729606781E-01\n1589 7.9928125000E+00 1.50000E-01 1.99999E-01 3 0 -4.3734960050E-01 2.50000E-01 1.99999E-01 3 0 -3.2729312450E-01\n1590 7.9934375000E+00 1.50000E-01 1.99999E-01 3 0 -4.3747206438E-01 2.50000E-01 1.99999E-01 3 0 -3.2729016314E-01\n1591 7.9940625000E+00 1.50000E-01 1.99999E-01 3 0 -4.3759438037E-01 2.50000E-01 1.99999E-01 3 0 -3.2728718350E-01\n1592 7.9946875000E+00 1.50000E-01 1.99999E-01 3 0 -4.3771654871E-01 2.50000E-01 1.99999E-01 3 0 -3.2728418553E-01\n1593 7.9953125000E+00 1.50000E-01 1.99999E-01 3 0 -4.3783856942E-01 2.50000E-01 1.99999E-01 3 0 -3.2728116900E-01\n1594 7.9959375000E+00 1.50000E-01 1.99999E-01 3 0 -4.3796044278E-01 2.50000E-01 1.99999E-01 3 0 -3.2727813389E-01\n1595 7.9965625000E+00 1.50000E-01 1.99999E-01 3 0 -4.3808216887E-01 2.50000E-01 1.99999E-01 3 0 -3.2727508003E-01\n1596 7.9971875000E+00 1.50000E-01 1.99999E-01 3 0 -4.3820374783E-01 2.50000E-01 1.99999E-01 3 0 -3.2727200727E-01\n1597 7.9978125000E+00 1.50000E-01 1.99999E-01 3 0 -4.3832517981E-01 2.50000E-01 1.99999E-01 3 0 -3.2726891550E-01\n1598 7.9984375000E+00 1.50000E-01 1.99999E-01 3 0 -4.3844646495E-01 2.50000E-01 1.99999E-01 3 0 -3.2726580457E-01\n1599 7.9990625000E+00 1.50000E-01 1.99999E-01 3 0 -4.3856760349E-01 2.50000E-01 1.99999E-01 3 0 -3.2726267445E-01\n1600 7.9996875000E+00 1.50000E-01 1.99999E-01 3 0 -4.3868859542E-01 2.50000E-01 1.99999E-01 3 0 -3.2725952487E-01\n" }, { "path": "chapter2/singular_poisson.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"markdown\",\n \"id\": \"0\",\n \"metadata\": {},\n \"source\": [\n \"# Singular Poisson problem\\n\",\n \"Author: Jørgen S. Dokken\\n\",\n \"\\n\",\n \"In this example, we will solve the singular Poisson problem by attaching information about the\\n\",\n \"nullspace of the discretized problem to the matrix system.\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"1\",\n \"metadata\": {},\n \"source\": [\n \"The problem is defined as\\n\",\n \"\\n\",\n \"\\\\begin{align}\\n\",\n \" -\\\\Delta u &= f &&\\\\text{in } \\\\Omega,\\\\\\\\\\n\",\n \" -\\\\nabla u \\\\cdot \\\\mathbf{n} &= \\\\mathbf{g} &&\\\\text{on } \\\\partial\\\\Omega.\\n\",\n \"\\\\end{align}\\n\",\n \"\\n\",\n \"This problem has a nullspace, i.e. if we take a solution of the problem above, say $\\\\tilde u$ and\\n\",\n \"add a constant $c$ to it, $u_c=\\\\tilde u + c$, we still have a solution to the problem.\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"2\",\n \"metadata\": {},\n \"source\": [\n \"We will use a manufactured solution on a unit square to investigate this problem, namely\\n\",\n \"\\n\",\n \"\\\\begin{align}\\n\",\n \" u(x, y) &= \\\\sin(2\\\\pi x)\\\\\\\\\\n\",\n \" f(x, y) &= -4\\\\pi^2\\\\sin(2\\\\pi x)\\\\\\\\\\n\",\n \" g(x, y) &=\\n\",\n \" \\\\begin{cases}\\n\",\n \" -2\\\\pi & \\\\text{if } x=0,\\\\\\\\\\n\",\n \" 2\\\\pi & \\\\text{if } x=1,\\\\\\\\\\n\",\n \" 0 & \\\\text{otherwise.}\\n\",\n \" \\\\end{cases}\\n\",\n \"\\\\end{align}\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"3\",\n \"metadata\": {},\n \"source\": [\n \"As we have discretized the Poisson problem in other tutorials, we create a simple wrapper function to set up the variational problem,\\n\",\n \"given a manufactured solution\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"4\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"import dolfinx.fem.petsc\\n\",\n \"from mpi4py import MPI\\n\",\n \"import numpy as np\\n\",\n \"import typing\\n\",\n \"import ufl\\n\",\n \"\\n\",\n \"\\n\",\n \"def u_ex(mod, x):\\n\",\n \" return mod.sin(2 * mod.pi * x[0])\\n\",\n \"\\n\",\n \"\\n\",\n \"def setup_problem(\\n\",\n \" N: int,\\n\",\n \") -> typing.Tuple[dolfinx.fem.FunctionSpace, dolfinx.fem.Form, dolfinx.fem.Form]:\\n\",\n \" \\\"\\\"\\\"Set up bilinear and linear form of the singular Poisson problem\\n\",\n \"\\n\",\n \" Args:\\n\",\n \" N (int): Number of elements in each direction of the mesh.\\n\",\n \"\\n\",\n \" Returns:\\n\",\n \" The function space, the bilinear form and the linear form of the problem.\\n\",\n \"\\n\",\n \" \\\"\\\"\\\"\\n\",\n \"\\n\",\n \" domain = dolfinx.mesh.create_unit_square(\\n\",\n \" MPI.COMM_WORLD, N, N, cell_type=dolfinx.mesh.CellType.quadrilateral\\n\",\n \" )\\n\",\n \" V = dolfinx.fem.functionspace(domain, (\\\"Lagrange\\\", 1))\\n\",\n \" u = ufl.TrialFunction(V)\\n\",\n \" v = ufl.TestFunction(V)\\n\",\n \"\\n\",\n \" x = ufl.SpatialCoordinate(domain)\\n\",\n \" u_exact = u_ex(ufl, x)\\n\",\n \" f = -ufl.div(ufl.grad(u_exact))\\n\",\n \" n = ufl.FacetNormal(domain)\\n\",\n \" g = -ufl.dot(ufl.grad(u_exact), n)\\n\",\n \"\\n\",\n \" F = ufl.dot(ufl.grad(u), ufl.grad(v)) * ufl.dx\\n\",\n \" F += ufl.inner(g, v) * ufl.ds\\n\",\n \" F -= f * v * ufl.dx\\n\",\n \" return V, *dolfinx.fem.form(ufl.system(F))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"5\",\n \"metadata\": {},\n \"source\": [\n \"With the above convenience function set up, we can now address the nullspace.\\n\",\n \"We will use PETSc for this, by attaching additional information to the assembled matrices.\\n\",\n \"PETSc has a convenience functon for creating constant nullspaces, which we will use here.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"6\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"from petsc4py import PETSc\\n\",\n \"\\n\",\n \"nullspace = PETSc.NullSpace().create(constant=True, comm=MPI.COMM_WORLD)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"7\",\n \"metadata\": {},\n \"source\": [\n \"## Direct solver\\n\",\n \"We start by considering the singular problem using a direct solver (MUMPS).\\n\",\n \"Mumps has some additional options to support singular matrices, which we will use.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"8\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"petsc_options = {\\n\",\n \" \\\"ksp_error_if_not_converged\\\": True,\\n\",\n \" \\\"ksp_type\\\": \\\"preonly\\\",\\n\",\n \" \\\"pc_type\\\": \\\"lu\\\",\\n\",\n \" \\\"pc_factor_mat_solver_type\\\": \\\"mumps\\\",\\n\",\n \" \\\"ksp_monitor\\\": None,\\n\",\n \"}\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"9\",\n \"metadata\": {},\n \"source\": [\n \"Next, we set up the the KSP solver\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"10\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"ksp = PETSc.KSP().create(MPI.COMM_WORLD)\\n\",\n \"ksp.setOptionsPrefix(\\\"singular_direct\\\")\\n\",\n \"opts = PETSc.Options()\\n\",\n \"opts.prefixPush(ksp.getOptionsPrefix())\\n\",\n \"for key, value in petsc_options.items():\\n\",\n \" opts[key] = value\\n\",\n \"ksp.setFromOptions()\\n\",\n \"for key, value in petsc_options.items():\\n\",\n \" del opts[key]\\n\",\n \"opts.prefixPop()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"11\",\n \"metadata\": {},\n \"source\": [\n \"and we assemble the bilinear and linear forms, and create the matrix `A` and right hand side vector `b`.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"12\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"V, a, L = setup_problem(40)\\n\",\n \"A = dolfinx.fem.petsc.assemble_matrix(a)\\n\",\n \"A.assemble()\\n\",\n \"b = dolfinx.fem.petsc.assemble_vector(L)\\n\",\n \"b.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)\\n\",\n \"ksp.setOperators(A)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"13\",\n \"metadata\": {},\n \"source\": [\n \"Next, We first check that this indeed is the nullspace of `A`, then attach the nullspace to the matrix `A`.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"14\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"assert nullspace.test(A)\\n\",\n \"A.setNullSpace(nullspace)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"15\",\n \"metadata\": {},\n \"source\": [\n \"Then, we can solve the linear system of equations\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"16\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"uh = dolfinx.fem.Function(V)\\n\",\n \"ksp.solve(b, uh.x.petsc_vec)\\n\",\n \"uh.x.scatter_forward()\\n\",\n \"\\n\",\n \"ksp.destroy()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"17\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"We can now check the $L^2$-error against the analytical solution\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"18\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"def compute_L2_error(uh: dolfinx.fem.Function) -> float:\\n\",\n \" mesh = uh.function_space.mesh\\n\",\n \" u_exact = u_ex(ufl, ufl.SpatialCoordinate(mesh))\\n\",\n \" error_L2 = dolfinx.fem.form(ufl.inner(uh - u_exact, uh - u_exact) * ufl.dx)\\n\",\n \" error_local = dolfinx.fem.assemble_scalar(error_L2)\\n\",\n \" return np.sqrt(mesh.comm.allreduce(error_local, op=MPI.SUM))\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"19\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"print(f\\\"Direct solver L2 error {compute_L2_error(uh):.5e}\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"20\",\n \"metadata\": {},\n \"source\": [\n \"We also check that the mean value of the solution is equal to the mean value of the manufactured solution.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"21\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"u_exact = u_ex(ufl, ufl.SpatialCoordinate(V.mesh))\\n\",\n \"ex_mean = V.mesh.comm.allreduce(\\n\",\n \" dolfinx.fem.assemble_scalar(dolfinx.fem.form(u_exact * ufl.dx)), op=MPI.SUM\\n\",\n \")\\n\",\n \"approx_mean = V.mesh.comm.allreduce(\\n\",\n \" dolfinx.fem.assemble_scalar(dolfinx.fem.form(uh * ufl.dx)), op=MPI.SUM\\n\",\n \")\\n\",\n \"print(f\\\"Mean value of manufactured solution: {ex_mean}\\\")\\n\",\n \"print(f\\\"Mean value of computed solution (direct solver): {approx_mean}\\\")\\n\",\n \"assert np.isclose(ex_mean, approx_mean), \\\"Mean values do not match!\\\"\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"22\",\n \"metadata\": {},\n \"source\": [\n \"## Iterative solver\\n\",\n \"We can also solve the problem above using an iterative solver,\\n\",\n \"for instance GMRES with AMG preconditioning.\\n\",\n \"We therefore select a new set of PETSc options, and create a new KSP solver.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"23\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"ksp_iterative = PETSc.KSP().create(MPI.COMM_WORLD)\\n\",\n \"ksp_iterative.setOptionsPrefix(\\\"singular_iterative\\\")\\n\",\n \"petsc_options_iterative = {\\n\",\n \" \\\"ksp_error_if_not_converged\\\": True,\\n\",\n \" \\\"ksp_monitor\\\": None,\\n\",\n \" \\\"ksp_type\\\": \\\"gmres\\\",\\n\",\n \" \\\"pc_type\\\": \\\"hypre\\\",\\n\",\n \" \\\"pc_hypre_type\\\": \\\"boomeramg\\\",\\n\",\n \" \\\"pc_hypre_boomeramg_max_iter\\\": 1,\\n\",\n \" \\\"pc_hypre_boomeramg_cycle_type\\\": \\\"v\\\",\\n\",\n \" \\\"ksp_rtol\\\": 1.0e-13,\\n\",\n \"}\\n\",\n \"opts.prefixPush(ksp_iterative.getOptionsPrefix())\\n\",\n \"for key, value in petsc_options_iterative.items():\\n\",\n \" opts[key] = value\\n\",\n \"ksp_iterative.setFromOptions()\\n\",\n \"for key, value in petsc_options_iterative.items():\\n\",\n \" del opts[key]\\n\",\n \"opts.prefixPop()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"24\",\n \"metadata\": {},\n \"source\": [\n \"Instead of setting the nullspace, we attach it as a near nullspace, for the multigrid preconditioner.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"25\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"A_iterative = dolfinx.fem.petsc.assemble_matrix(a)\\n\",\n \"A_iterative.assemble()\\n\",\n \"A_iterative.setNearNullSpace(nullspace)\\n\",\n \"ksp_iterative.setOperators(A_iterative)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"26\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"uh_iterative = dolfinx.fem.Function(V)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"27\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"ksp_iterative.solve(b, uh_iterative.x.petsc_vec)\\n\",\n \"uh_iterative.x.scatter_forward()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"28\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"For the iterative solver, we subtract the mean value of the approximated solution,\\n\",\n \"and add the mean value of manufactured solution before computing the error.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"29\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"approx_mean = V.mesh.comm.allreduce(\\n\",\n \" dolfinx.fem.assemble_scalar(dolfinx.fem.form(uh_iterative * ufl.dx)), op=MPI.SUM\\n\",\n \")\\n\",\n \"print(\\\"Mean value of computed solution (iterative solver):\\\", approx_mean)\\n\",\n \"uh_iterative.x.array[:] += ex_mean - approx_mean\\n\",\n \"approx_mean = V.mesh.comm.allreduce(\\n\",\n \" dolfinx.fem.assemble_scalar(dolfinx.fem.form(uh_iterative * ufl.dx)), op=MPI.SUM\\n\",\n \")\\n\",\n \"print(\\n\",\n \" \\\"Mean value of computed solution (iterative solver) post normalization:\\\",\\n\",\n \" approx_mean,\\n\",\n \")\\n\",\n \"print(f\\\"Iterative solver L2 error {compute_L2_error(uh_iterative):.5e}\\\")\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"30\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"np.testing.assert_allclose(uh.x.array, uh_iterative.x.array, rtol=1e-10, atol=1e-12)\"\n ]\n }\n ],\n \"metadata\": {\n \"jupytext\": {\n \"cell_metadata_filter\": \"-all\",\n \"formats\": \"ipynb,py:light\",\n \"main_language\": \"python\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 5\n}\n" }, { "path": "chapter2/singular_poisson.py", "content": "# ---\n# jupyter:\n# jupytext:\n# cell_metadata_filter: -all\n# formats: ipynb,py:light\n# text_representation:\n# extension: .py\n# format_name: light\n# format_version: '1.5'\n# jupytext_version: 1.18.1\n# ---\n\n# # Singular Poisson problem\n# Author: Jørgen S. Dokken\n#\n# In this example, we will solve the singular Poisson problem by attaching information about the\n# nullspace of the discretized problem to the matrix system.\n\n# The problem is defined as\n#\n# \\begin{align}\n# -\\Delta u &= f &&\\text{in } \\Omega,\\\\\n# -\\nabla u \\cdot \\mathbf{n} &= \\mathbf{g} &&\\text{on } \\partial\\Omega.\n# \\end{align}\n#\n# This problem has a nullspace, i.e. if we take a solution of the problem above, say $\\tilde u$ and\n# add a constant $c$ to it, $u_c=\\tilde u + c$, we still have a solution to the problem.\n\n# We will use a manufactured solution on a unit square to investigate this problem, namely\n#\n# \\begin{align}\n# u(x, y) &= \\sin(2\\pi x)\\\\\n# f(x, y) &= -4\\pi^2\\sin(2\\pi x)\\\\\n# g(x, y) &=\n# \\begin{cases}\n# -2\\pi & \\text{if } x=0,\\\\\n# 2\\pi & \\text{if } x=1,\\\\\n# 0 & \\text{otherwise.}\n# \\end{cases}\n# \\end{align}\n\n# As we have discretized the Poisson problem in other tutorials, we create a simple wrapper function to set up the variational problem,\n# given a manufactured solution\n\n# +\nimport dolfinx.fem.petsc\nfrom mpi4py import MPI\nimport numpy as np\nimport typing\nimport ufl\n\n\ndef u_ex(mod, x):\n return mod.sin(2 * mod.pi * x[0])\n\n\ndef setup_problem(\n N: int,\n) -> typing.Tuple[dolfinx.fem.FunctionSpace, dolfinx.fem.Form, dolfinx.fem.Form]:\n \"\"\"Set up bilinear and linear form of the singular Poisson problem\n\n Args:\n N (int): Number of elements in each direction of the mesh.\n\n Returns:\n The function space, the bilinear form and the linear form of the problem.\n\n \"\"\"\n\n domain = dolfinx.mesh.create_unit_square(\n MPI.COMM_WORLD, N, N, cell_type=dolfinx.mesh.CellType.quadrilateral\n )\n V = dolfinx.fem.functionspace(domain, (\"Lagrange\", 1))\n u = ufl.TrialFunction(V)\n v = ufl.TestFunction(V)\n\n x = ufl.SpatialCoordinate(domain)\n u_exact = u_ex(ufl, x)\n f = -ufl.div(ufl.grad(u_exact))\n n = ufl.FacetNormal(domain)\n g = -ufl.dot(ufl.grad(u_exact), n)\n\n F = ufl.dot(ufl.grad(u), ufl.grad(v)) * ufl.dx\n F += ufl.inner(g, v) * ufl.ds\n F -= f * v * ufl.dx\n return V, *dolfinx.fem.form(ufl.system(F))\n\n\n# -\n\n# With the above convenience function set up, we can now address the nullspace.\n# We will use PETSc for this, by attaching additional information to the assembled matrices.\n# PETSc has a convenience functon for creating constant nullspaces, which we will use here.\n\n# +\nfrom petsc4py import PETSc\n\nnullspace = PETSc.NullSpace().create(constant=True, comm=MPI.COMM_WORLD)\n# -\n\n# ## Direct solver\n# We start by considering the singular problem using a direct solver (MUMPS).\n# Mumps has some additional options to support singular matrices, which we will use.\n\npetsc_options = {\n \"ksp_error_if_not_converged\": True,\n \"ksp_type\": \"preonly\",\n \"pc_type\": \"lu\",\n \"pc_factor_mat_solver_type\": \"mumps\",\n \"ksp_monitor\": None,\n}\n\n# Next, we set up the the KSP solver\n\nksp = PETSc.KSP().create(MPI.COMM_WORLD)\nksp.setOptionsPrefix(\"singular_direct\")\nopts = PETSc.Options()\nopts.prefixPush(ksp.getOptionsPrefix())\nfor key, value in petsc_options.items():\n opts[key] = value\nksp.setFromOptions()\nfor key, value in petsc_options.items():\n del opts[key]\nopts.prefixPop()\n\n# and we assemble the bilinear and linear forms, and create the matrix `A` and right hand side vector `b`.\n\nV, a, L = setup_problem(40)\nA = dolfinx.fem.petsc.assemble_matrix(a)\nA.assemble()\nb = dolfinx.fem.petsc.assemble_vector(L)\nb.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)\nksp.setOperators(A)\n\n# Next, We first check that this indeed is the nullspace of `A`, then attach the nullspace to the matrix `A`.\n\nassert nullspace.test(A)\nA.setNullSpace(nullspace)\n\n# Then, we can solve the linear system of equations\n\n# +\nuh = dolfinx.fem.Function(V)\nksp.solve(b, uh.x.petsc_vec)\nuh.x.scatter_forward()\n\nksp.destroy()\n# -\n\n# We can now check the $L^2$-error against the analytical solution\n\n\ndef compute_L2_error(uh: dolfinx.fem.Function) -> float:\n mesh = uh.function_space.mesh\n u_exact = u_ex(ufl, ufl.SpatialCoordinate(mesh))\n error_L2 = dolfinx.fem.form(ufl.inner(uh - u_exact, uh - u_exact) * ufl.dx)\n error_local = dolfinx.fem.assemble_scalar(error_L2)\n return np.sqrt(mesh.comm.allreduce(error_local, op=MPI.SUM))\n\n\nprint(f\"Direct solver L2 error {compute_L2_error(uh):.5e}\")\n\n# We also check that the mean value of the solution is equal to the mean value of the manufactured solution.\n\nu_exact = u_ex(ufl, ufl.SpatialCoordinate(V.mesh))\nex_mean = V.mesh.comm.allreduce(\n dolfinx.fem.assemble_scalar(dolfinx.fem.form(u_exact * ufl.dx)), op=MPI.SUM\n)\napprox_mean = V.mesh.comm.allreduce(\n dolfinx.fem.assemble_scalar(dolfinx.fem.form(uh * ufl.dx)), op=MPI.SUM\n)\nprint(f\"Mean value of manufactured solution: {ex_mean}\")\nprint(f\"Mean value of computed solution (direct solver): {approx_mean}\")\nassert np.isclose(ex_mean, approx_mean), \"Mean values do not match!\"\n\n# ## Iterative solver\n# We can also solve the problem above using an iterative solver,\n# for instance GMRES with AMG preconditioning.\n# We therefore select a new set of PETSc options, and create a new KSP solver.\n\nksp_iterative = PETSc.KSP().create(MPI.COMM_WORLD)\nksp_iterative.setOptionsPrefix(\"singular_iterative\")\npetsc_options_iterative = {\n \"ksp_error_if_not_converged\": True,\n \"ksp_monitor\": None,\n \"ksp_type\": \"gmres\",\n \"pc_type\": \"hypre\",\n \"pc_hypre_type\": \"boomeramg\",\n \"pc_hypre_boomeramg_max_iter\": 1,\n \"pc_hypre_boomeramg_cycle_type\": \"v\",\n \"ksp_rtol\": 1.0e-13,\n}\nopts.prefixPush(ksp_iterative.getOptionsPrefix())\nfor key, value in petsc_options_iterative.items():\n opts[key] = value\nksp_iterative.setFromOptions()\nfor key, value in petsc_options_iterative.items():\n del opts[key]\nopts.prefixPop()\n\n# Instead of setting the nullspace, we attach it as a near nullspace, for the multigrid preconditioner.\n\nA_iterative = dolfinx.fem.petsc.assemble_matrix(a)\nA_iterative.assemble()\nA_iterative.setNearNullSpace(nullspace)\nksp_iterative.setOperators(A_iterative)\n\nuh_iterative = dolfinx.fem.Function(V)\n\nksp_iterative.solve(b, uh_iterative.x.petsc_vec)\nuh_iterative.x.scatter_forward()\n\n# For the iterative solver, we subtract the mean value of the approximated solution,\n# and add the mean value of manufactured solution before computing the error.\n\n\napprox_mean = V.mesh.comm.allreduce(\n dolfinx.fem.assemble_scalar(dolfinx.fem.form(uh_iterative * ufl.dx)), op=MPI.SUM\n)\nprint(\"Mean value of computed solution (iterative solver):\", approx_mean)\nuh_iterative.x.array[:] += ex_mean - approx_mean\napprox_mean = V.mesh.comm.allreduce(\n dolfinx.fem.assemble_scalar(dolfinx.fem.form(uh_iterative * ufl.dx)), op=MPI.SUM\n)\nprint(\n \"Mean value of computed solution (iterative solver) post normalization:\",\n approx_mean,\n)\nprint(f\"Iterative solver L2 error {compute_L2_error(uh_iterative):.5e}\")\n\nnp.testing.assert_allclose(uh.x.array, uh_iterative.x.array, rtol=1e-10, atol=1e-12)\n" }, { "path": "chapter3/component_bc.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Component-wise Dirichlet BC\\n\",\n \"Author: Jørgen S. Dokken\\n\",\n \"\\n\",\n \"In this section, we will learn how to prescribe Dirichlet boundary conditions on a component of your unknown $u_h$.\\n\",\n \"We will illustrate the problem using a vector element. However, the method generalizes to any mixed element.\\n\",\n \"\\n\",\n \"We will use a slightly modified version of [the linear elasticity demo](./../chapter2/linearelasticity_code), namely\\n\",\n \"\\n\",\n \"$$\\n\",\n \"-\\\\nabla \\\\cdot \\\\sigma (u) = f\\\\quad \\\\text{in } \\\\Omega,\\n\",\n \"$$\\n\",\n \"\\n\",\n \"$$\\n\",\n \"\\\\sigma \\\\cdot n = 0 \\\\quad \\\\text{on } \\\\partial \\\\Omega_N,\\n\",\n \"$$\\n\",\n \"\\n\",\n \"$$\\n\",\n \"u= 0\\\\quad \\\\text{at } \\\\partial\\\\Omega_{D},\\n\",\n \"$$\\n\",\n \"\\n\",\n \"$$\\n\",\n \"u_x=0 \\\\quad \\\\text{at } \\\\partial\\\\Omega_{Dx},\\n\",\n \"$$\\n\",\n \"\\n\",\n \"$$\\n\",\n \"\\\\sigma(u)= \\\\lambda \\\\mathrm{tr}(\\\\epsilon(u))I + 2 \\\\mu \\\\epsilon(u), \\\\qquad \\\\epsilon(u) = \\\\frac{1}{2}\\\\left(\\\\nabla u + (\\\\nabla u )^T\\\\right).\\n\",\n \"$$\\n\",\n \"We will consider a two dimensional box spanning $[0,L]\\\\times[0,H]$, where\\n\",\n \"$\\\\partial\\\\Omega_N$ is the left and right side of the beam, $\\\\partial\\\\Omega_D$ the bottom of the beam, while $\\\\partial\\\\Omega_{Dx}$ is the right side of the beam.\\n\",\n \"We will prescribe a displacement $u_x=0$ on the right side of the beam, while the beam is being deformed under its own weight. The sides of the box are traction free.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"tags\": []\n },\n \"outputs\": [],\n \"source\": [\n \"import pyvista\\n\",\n \"import numpy as np\\n\",\n \"from mpi4py import MPI\\n\",\n \"from ufl import (\\n\",\n \" Identity,\\n\",\n \" Measure,\\n\",\n \" TestFunction,\\n\",\n \" TrialFunction,\\n\",\n \" dot,\\n\",\n \" dx,\\n\",\n \" inner,\\n\",\n \" grad,\\n\",\n \" nabla_div,\\n\",\n \" sym,\\n\",\n \")\\n\",\n \"from dolfinx import default_scalar_type\\n\",\n \"from dolfinx.mesh import CellType, create_rectangle, locate_entities_boundary\\n\",\n \"from dolfinx.fem.petsc import LinearProblem\\n\",\n \"from dolfinx.fem import (\\n\",\n \" Constant,\\n\",\n \" dirichletbc,\\n\",\n \" functionspace,\\n\",\n \" locate_dofs_geometrical,\\n\",\n \" locate_dofs_topological,\\n\",\n \")\\n\",\n \"from dolfinx.plot import vtk_mesh\\n\",\n \"\\n\",\n \"L = 1\\n\",\n \"H = 1.3\\n\",\n \"lambda_ = 1.25\\n\",\n \"mu = 1\\n\",\n \"rho = 1\\n\",\n \"g = 1\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"As in the previous demos, we define our mesh and function space.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"lines_to_next_cell\": 2,\n \"tags\": []\n },\n \"outputs\": [],\n \"source\": [\n \"mesh = create_rectangle(\\n\",\n \" MPI.COMM_WORLD, np.array([[0, 0], [L, H]]), [30, 30], cell_type=CellType.triangle\\n\",\n \")\\n\",\n \"V = functionspace(mesh, (\\\"Lagrange\\\", 1, (mesh.geometry.dim,)))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"## Boundary conditions\\n\",\n \"As we would like to clamp the boundary at $x=0$, we do this by using a marker function, we use `dolfinx.fem.locate_dofs_geometrical` to identify the relevant degrees of freedom.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"tags\": []\n },\n \"outputs\": [],\n \"source\": [\n \"def clamped_boundary(x):\\n\",\n \" return np.isclose(x[1], 0)\\n\",\n \"\\n\",\n \"\\n\",\n \"u_zero = np.array((0,) * mesh.geometry.dim, dtype=default_scalar_type)\\n\",\n \"bc = dirichletbc(u_zero, locate_dofs_geometrical(V, clamped_boundary), V)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"Next we would like to constrain the $x$-component of our solution at $x=L$ to $0$. We start by creating the sub space only containing the $x$\\n\",\n \"-component.\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"Next, we locate the degrees of freedom on the top boundary.\\n\",\n \"However, as the boundary condition is in a sub space of our solution,\\n\",\n \"we have to carefully decide whoe to locate the degrees of freedom.\\n\",\n \"In the example below, we will use a constant value as the prescribed value.\\n\",\n \"This means that we can use `dolfinx.fem.locate_dofs_topological`\\n\",\n \"on the (un-collapsed) sub space to locate the degrees of freedom on the boundary.\\n\",\n \"If you want to use a spatially dependent function, see\\n\",\n \"[FEniCS Workshop: Dirichlet conditions in mixed spaces](https://jsdokken.com/FEniCS-workshop/src/deep_dive/mixed_problems.html#dirichlet-conditions-in-mixed-spaces)\\n\",\n \"for a detailed discussion.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"tags\": []\n },\n \"outputs\": [],\n \"source\": [\n \"def right(x):\\n\",\n \" return np.logical_and(np.isclose(x[0], L), x[1] < H)\\n\",\n \"\\n\",\n \"\\n\",\n \"boundary_facets = locate_entities_boundary(mesh, mesh.topology.dim - 1, right)\\n\",\n \"boundary_dofs_x = locate_dofs_topological(\\n\",\n \" V.sub(0), mesh.topology.dim - 1, boundary_facets\\n\",\n \")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We can now create our Dirichlet condition\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"tags\": []\n },\n \"outputs\": [],\n \"source\": [\n \"bcx = dirichletbc(default_scalar_type(0), boundary_dofs_x, V.sub(0))\\n\",\n \"bcs = [bc, bcx]\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"As we want the traction $T$ over the remaining boundary to be $0$, we create a `dolfinx.fem.Constant`\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"tags\": []\n },\n \"outputs\": [],\n \"source\": [\n \"T = Constant(mesh, default_scalar_type((0, 0)))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We also want to specify the integration measure $\\\\mathrm{d}s$, which should be the integral over the boundary of our domain. We do this by using `ufl`, and its built in integration measures\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"lines_to_next_cell\": 2,\n \"tags\": []\n },\n \"outputs\": [],\n \"source\": [\n \"ds = Measure(\\\"ds\\\", domain=mesh)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"## Variational formulation\\n\",\n \"We are now ready to create our variational formulation in close to mathematical syntax, as for the previous problems.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"tags\": []\n },\n \"outputs\": [],\n \"source\": [\n \"def epsilon(u):\\n\",\n \" return sym(grad(u))\\n\",\n \"\\n\",\n \"\\n\",\n \"def sigma(u):\\n\",\n \" return lambda_ * nabla_div(u) * Identity(len(u)) + 2 * mu * epsilon(u)\\n\",\n \"\\n\",\n \"\\n\",\n \"u = TrialFunction(V)\\n\",\n \"v = TestFunction(V)\\n\",\n \"f = Constant(mesh, default_scalar_type((0, -rho * g)))\\n\",\n \"a = inner(sigma(u), epsilon(v)) * dx\\n\",\n \"L = dot(f, v) * dx + dot(T, v) * ds\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Solve the linear variational problem\\n\",\n \"As in the previous demos, we assemble the matrix and right hand side vector and use PETSc to solve our variational problem\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"tags\": []\n },\n \"outputs\": [],\n \"source\": [\n \"problem = LinearProblem(\\n\",\n \" a,\\n\",\n \" L,\\n\",\n \" bcs=bcs,\\n\",\n \" petsc_options={\\\"ksp_type\\\": \\\"preonly\\\", \\\"pc_type\\\": \\\"lu\\\"},\\n\",\n \" petsc_options_prefix=\\\"component_bc_\\\",\\n\",\n \")\\n\",\n \"uh = problem.solve()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Visualization\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"tags\": []\n },\n \"outputs\": [],\n \"source\": [\n \"# Create plotter and pyvista grid\\n\",\n \"p = pyvista.Plotter()\\n\",\n \"topology, cell_types, x = vtk_mesh(V)\\n\",\n \"grid = pyvista.UnstructuredGrid(topology, cell_types, x)\\n\",\n \"\\n\",\n \"# Attach vector values to grid and warp grid by vector\\n\",\n \"\\n\",\n \"vals = np.zeros((x.shape[0], 3))\\n\",\n \"vals[:, : len(uh)] = uh.x.array.reshape((x.shape[0], len(uh)))\\n\",\n \"grid[\\\"u\\\"] = vals\\n\",\n \"actor_0 = p.add_mesh(grid, style=\\\"wireframe\\\", color=\\\"k\\\")\\n\",\n \"warped = grid.warp_by_vector(\\\"u\\\", factor=1.5)\\n\",\n \"actor_1 = p.add_mesh(warped, opacity=0.8)\\n\",\n \"p.view_xy()\\n\",\n \"if not pyvista.OFF_SCREEN:\\n\",\n \" p.show()\\n\",\n \"else:\\n\",\n \" fig_array = p.screenshot(\\\"component.png\\\")\"\n ]\n }\n ],\n \"metadata\": {\n \"jupytext\": {\n \"formats\": \"ipynb,py:light\"\n },\n \"kernelspec\": {\n \"display_name\": \"Python 3 (ipykernel)\",\n \"language\": \"python\",\n \"name\": \"python3\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 3\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython3\",\n \"version\": \"3.10.12\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 4\n}\n" }, { "path": "chapter3/component_bc.py", "content": "# ---\n# jupyter:\n# jupytext:\n# formats: ipynb,py:light\n# text_representation:\n# extension: .py\n# format_name: light\n# format_version: '1.5'\n# jupytext_version: 1.18.1\n# kernelspec:\n# display_name: Python 3 (ipykernel)\n# language: python\n# name: python3\n# ---\n\n# # Component-wise Dirichlet BC\n# Author: Jørgen S. Dokken\n#\n# In this section, we will learn how to prescribe Dirichlet boundary conditions on a component of your unknown $u_h$.\n# We will illustrate the problem using a vector element. However, the method generalizes to any mixed element.\n#\n# We will use a slightly modified version of [the linear elasticity demo](./../chapter2/linearelasticity_code), namely\n#\n# $$\n# -\\nabla \\cdot \\sigma (u) = f\\quad \\text{in } \\Omega,\n# $$\n#\n# $$\n# \\sigma \\cdot n = 0 \\quad \\text{on } \\partial \\Omega_N,\n# $$\n#\n# $$\n# u= 0\\quad \\text{at } \\partial\\Omega_{D},\n# $$\n#\n# $$\n# u_x=0 \\quad \\text{at } \\partial\\Omega_{Dx},\n# $$\n#\n# $$\n# \\sigma(u)= \\lambda \\mathrm{tr}(\\epsilon(u))I + 2 \\mu \\epsilon(u), \\qquad \\epsilon(u) = \\frac{1}{2}\\left(\\nabla u + (\\nabla u )^T\\right).\n# $$\n# We will consider a two dimensional box spanning $[0,L]\\times[0,H]$, where\n# $\\partial\\Omega_N$ is the left and right side of the beam, $\\partial\\Omega_D$ the bottom of the beam, while $\\partial\\Omega_{Dx}$ is the right side of the beam.\n# We will prescribe a displacement $u_x=0$ on the right side of the beam, while the beam is being deformed under its own weight. The sides of the box are traction free.\n\n# +\nimport pyvista\nimport numpy as np\nfrom mpi4py import MPI\nfrom ufl import (\n Identity,\n Measure,\n TestFunction,\n TrialFunction,\n dot,\n dx,\n inner,\n grad,\n nabla_div,\n sym,\n)\nfrom dolfinx import default_scalar_type\nfrom dolfinx.mesh import CellType, create_rectangle, locate_entities_boundary\nfrom dolfinx.fem.petsc import LinearProblem\nfrom dolfinx.fem import (\n Constant,\n dirichletbc,\n functionspace,\n locate_dofs_geometrical,\n locate_dofs_topological,\n)\nfrom dolfinx.plot import vtk_mesh\n\nL = 1\nH = 1.3\nlambda_ = 1.25\nmu = 1\nrho = 1\ng = 1\n# -\n\n# As in the previous demos, we define our mesh and function space.\n\nmesh = create_rectangle(\n MPI.COMM_WORLD, np.array([[0, 0], [L, H]]), [30, 30], cell_type=CellType.triangle\n)\nV = functionspace(mesh, (\"Lagrange\", 1, (mesh.geometry.dim,)))\n\n\n# ## Boundary conditions\n# As we would like to clamp the boundary at $x=0$, we do this by using a marker function, we use `dolfinx.fem.locate_dofs_geometrical` to identify the relevant degrees of freedom.\n\n\n# +\ndef clamped_boundary(x):\n return np.isclose(x[1], 0)\n\n\nu_zero = np.array((0,) * mesh.geometry.dim, dtype=default_scalar_type)\nbc = dirichletbc(u_zero, locate_dofs_geometrical(V, clamped_boundary), V)\n# -\n\n# Next we would like to constrain the $x$-component of our solution at $x=L$ to $0$. We start by creating the sub space only containing the $x$\n# -component.\n\n# Next, we locate the degrees of freedom on the top boundary.\n# However, as the boundary condition is in a sub space of our solution,\n# we have to carefully decide whoe to locate the degrees of freedom.\n# In the example below, we will use a constant value as the prescribed value.\n# This means that we can use `dolfinx.fem.locate_dofs_topological`\n# on the (un-collapsed) sub space to locate the degrees of freedom on the boundary.\n# If you want to use a spatially dependent function, see\n# [FEniCS Workshop: Dirichlet conditions in mixed spaces](https://jsdokken.com/FEniCS-workshop/src/deep_dive/mixed_problems.html#dirichlet-conditions-in-mixed-spaces)\n# for a detailed discussion.\n\n\n# +\ndef right(x):\n return np.logical_and(np.isclose(x[0], L), x[1] < H)\n\n\nboundary_facets = locate_entities_boundary(mesh, mesh.topology.dim - 1, right)\nboundary_dofs_x = locate_dofs_topological(\n V.sub(0), mesh.topology.dim - 1, boundary_facets\n)\n# -\n\n# We can now create our Dirichlet condition\n\nbcx = dirichletbc(default_scalar_type(0), boundary_dofs_x, V.sub(0))\nbcs = [bc, bcx]\n\n# As we want the traction $T$ over the remaining boundary to be $0$, we create a `dolfinx.fem.Constant`\n\nT = Constant(mesh, default_scalar_type((0, 0)))\n\n# We also want to specify the integration measure $\\mathrm{d}s$, which should be the integral over the boundary of our domain. We do this by using `ufl`, and its built in integration measures\n\nds = Measure(\"ds\", domain=mesh)\n\n\n# ## Variational formulation\n# We are now ready to create our variational formulation in close to mathematical syntax, as for the previous problems.\n\n\n# +\ndef epsilon(u):\n return sym(grad(u))\n\n\ndef sigma(u):\n return lambda_ * nabla_div(u) * Identity(len(u)) + 2 * mu * epsilon(u)\n\n\nu = TrialFunction(V)\nv = TestFunction(V)\nf = Constant(mesh, default_scalar_type((0, -rho * g)))\na = inner(sigma(u), epsilon(v)) * dx\nL = dot(f, v) * dx + dot(T, v) * ds\n# -\n\n# ## Solve the linear variational problem\n# As in the previous demos, we assemble the matrix and right hand side vector and use PETSc to solve our variational problem\n\nproblem = LinearProblem(\n a,\n L,\n bcs=bcs,\n petsc_options={\"ksp_type\": \"preonly\", \"pc_type\": \"lu\"},\n petsc_options_prefix=\"component_bc_\",\n)\nuh = problem.solve()\n\n# ## Visualization\n\n# +\n# Create plotter and pyvista grid\np = pyvista.Plotter()\ntopology, cell_types, x = vtk_mesh(V)\ngrid = pyvista.UnstructuredGrid(topology, cell_types, x)\n\n# Attach vector values to grid and warp grid by vector\n\nvals = np.zeros((x.shape[0], 3))\nvals[:, : len(uh)] = uh.x.array.reshape((x.shape[0], len(uh)))\ngrid[\"u\"] = vals\nactor_0 = p.add_mesh(grid, style=\"wireframe\", color=\"k\")\nwarped = grid.warp_by_vector(\"u\", factor=1.5)\nactor_1 = p.add_mesh(warped, opacity=0.8)\np.view_xy()\nif not pyvista.OFF_SCREEN:\n p.show()\nelse:\n fig_array = p.screenshot(\"component.png\")\n" }, { "path": "chapter3/em.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Electromagnetics example\\n\",\n \"\\n\",\n \"Theoretical introduction by: Hans Petter Langtangen and Anders Logg\\n\",\n \"\\n\",\n \"Implementation by: Jørgen S. Dokken\\n\",\n \"\\n\",\n \"In this example, we will consider an iron cylinder with copper wires wound around the cylinder, as shown below\\n\",\n \"\\n\",\n \"![Cross section of wires](wire.png)\\n\",\n \"\\n\",\n \"Through the copper wires a static current of $J=1A$ is flowing.\\n\",\n \"We would like to compute the magnetic field $B$ in the iron cylinder, the copper wires, and the surrounding vaccum.\\n\",\n \"\\n\",\n \"We start by simplifying the problem to a 2D problem.\\n\",\n \"We can do this by assuming that the cylinder extends far along the z-axis and\\n\",\n \"as a consequence the field is virtually independent of the z-coordinate.\\n\",\n \"Next, we consider Maxwell's equation to derive a Poisson equation for the magnetic field (or rather its potential)\\n\",\n \"\\n\",\n \"$$\\n\",\n \"\\\\nabla \\\\cdot D = \\\\rho,\\n\",\n \"$$\\n\",\n \"$$\\n\",\n \"\\\\nabla \\\\cdot B = 0,\\n\",\n \"$$\\n\",\n \"$$\\n\",\n \"\\\\nabla \\\\times E = -\\\\frac{\\\\partial B}{\\\\partial t},\\n\",\n \"$$\\n\",\n \"$$\\n\",\n \"\\\\nabla \\\\times H = \\\\frac{\\\\partial D}{\\\\partial t}+ J.\\n\",\n \"$$\\n\",\n \"\\n\",\n \"Here, $D$ is the displacement field, $B$ is the magnetic field, $E$ is the electric field, and $H$ is the magnetizing field.\\n\",\n \"In addition to Maxwell's equation, we need a constitutive relation between $B$ and $H$,\\n\",\n \"\\n\",\n \"$$\\n\",\n \"B =\\\\mu H,\\n\",\n \"$$\\n\",\n \"\\n\",\n \"which holds for an isotropic linear magnetic medium.\\n\",\n \"Here, $\\\\mu$ is the magnetic permability of the material.\\n\",\n \"Now, since $B$ is solenodial (divergence free) according to Maxwell's equations, we known that $B$ must be the curl of some vector field $A$. This field is called the magnetic vector potential. Since the problem is static and thus $\\\\frac{\\\\partial D}{\\\\partial t}=0$, it follows that\\n\",\n \"\\n\",\n \"$$\\n\",\n \"J = \\\\nabla \\\\times H = \\\\nabla \\\\times(\\\\mu^{-1} B)=\\\\nabla \\\\times (\\\\mu^{-1}\\\\nabla \\\\times A ) = -\\\\nabla \\\\cdot (\\\\mu^{-1}\\\\nabla A).\\n\",\n \"$$\\n\",\n \"\\n\",\n \"In the last step, we have expanded the second derivatives and used the gauge freedom of $A$ to simplify the equations to a simple vector-valued Poisson equation for the magnetic vector potential; if $B=\\\\nabla \\\\times A$, then $B=\\\\nabla \\\\times (A+\\\\nabla \\\\phi)$ for any scalar field $\\\\phi$ (the gauge function).\\n\",\n \"For the current problem, we thus need to solve the following 2D Poisson problem for the $z$-component $A_z$ of the magnetic vector potential\\n\",\n \"\\n\",\n \"$$\\n\",\n \" - \\\\nabla \\\\cdot (\\\\mu^{-1} \\\\nabla A_z) = J_z \\\\qquad \\\\text{in } \\\\mathbb{R}^2,\\\\\\\\\\n\",\n \"$$\\n\",\n \"$$\\n\",\n \"\\\\lim_{\\\\vert(x,y)\\\\vert\\\\to \\\\infty}A_z = 0.\\n\",\n \"$$\\n\",\n \"\\n\",\n \"Since we cannot solve the problem on an infinite domain, we will truncate the domain using a large disk, and set $A_z=0$ on the boundary. The current $J_z$ is set to $+1$A in the interior set of the circles (copper-wire cross sections) and to $-1$ A in the exterior set of circles in the cross section figure.\\n\",\n \"Once the magnetic field vector potential has been computed, we can compute the magnetic field $B=B(x,y)$ by\\n\",\n \"\\n\",\n \"$$\\n\",\n \" B(x,y)=\\\\left(\\\\frac{\\\\partial A_z}{\\\\partial y}, - \\\\frac{\\\\partial A_z}{\\\\partial x} \\\\right).\\n\",\n \"$$\\n\",\n \"\\n\",\n \"The weak formulation is easily obtained by multiplication of a test function $v$, followed by integration by parts, where all boundary integrals vanish due to the Dirichlet condition, we obtain $a(A_z,v)=L(v)$ with\\n\",\n \"\\n\",\n \"$$\\n\",\n \"a(A_z, v)=\\\\int_\\\\Omega \\\\mu^{-1}\\\\nabla A_z \\\\cdot \\\\nabla v ~\\\\mathrm{d}x,\\n\",\n \"$$\\n\",\n \"$$\\n\",\n \"L(v)=\\\\int_\\\\Omega J_z v~\\\\mathrm{d} x.\\n\",\n \"$$\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Meshing a complex structure with subdomains\\n\",\n \"\\n\",\n \"We create the domain visualized in the cross section figure above using gmsh. Note that we are using the `gmsh.model.occ.fragment` commands to ensure that the boundaries of the wires are resolved in the mesh.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"tags\": []\n },\n \"outputs\": [],\n \"source\": [\n \"from dolfinx import default_scalar_type\\n\",\n \"from dolfinx.fem import (\\n\",\n \" dirichletbc,\\n\",\n \" Expression,\\n\",\n \" Function,\\n\",\n \" functionspace,\\n\",\n \" locate_dofs_topological,\\n\",\n \")\\n\",\n \"from dolfinx.fem.petsc import LinearProblem\\n\",\n \"from dolfinx.io import XDMFFile\\n\",\n \"from dolfinx.io import gmsh as gmshio\\n\",\n \"from dolfinx.mesh import compute_midpoints, locate_entities_boundary\\n\",\n \"from dolfinx.plot import vtk_mesh\\n\",\n \"\\n\",\n \"from ufl import TestFunction, TrialFunction, as_vector, dot, dx, grad\\n\",\n \"from mpi4py import MPI\\n\",\n \"\\n\",\n \"import gmsh\\n\",\n \"import numpy as np\\n\",\n \"import pyvista\\n\",\n \"\\n\",\n \"rank = MPI.COMM_WORLD.rank\\n\",\n \"\\n\",\n \"gmsh.initialize()\\n\",\n \"r = 0.1 # Radius of copper wires\\n\",\n \"R = 5 # Radius of domain\\n\",\n \"a = 1 # Radius of inner iron cylinder\\n\",\n \"b = 1.2 # Radius of outer iron cylinder\\n\",\n \"N = 8 # Number of windings\\n\",\n \"c_1 = 0.8 # Radius of inner copper wires\\n\",\n \"c_2 = 1.4 # Radius of outer copper wires\\n\",\n \"gdim = 2 # Geometric dimension of the mesh\\n\",\n \"model_rank = 0\\n\",\n \"mesh_comm = MPI.COMM_WORLD\\n\",\n \"if mesh_comm.rank == model_rank:\\n\",\n \" # Define geometry for iron cylinder\\n\",\n \" outer_iron = gmsh.model.occ.addCircle(0, 0, 0, b)\\n\",\n \" inner_iron = gmsh.model.occ.addCircle(0, 0, 0, a)\\n\",\n \" gmsh.model.occ.addCurveLoop([outer_iron], 5)\\n\",\n \" gmsh.model.occ.addCurveLoop([inner_iron], 6)\\n\",\n \" iron = gmsh.model.occ.addPlaneSurface([5, 6])\\n\",\n \" gmsh.model.occ.synchronize()\\n\",\n \"\\n\",\n \" # Define geometry for background\\n\",\n \" background = gmsh.model.occ.addDisk(0, 0, 0, R, R)\\n\",\n \" gmsh.model.occ.synchronize()\\n\",\n \"\\n\",\n \" # Define the copper-wires inside iron cylinder\\n\",\n \" angles_N = [i * 2 * np.pi / N for i in range(N)]\\n\",\n \" wires_N = [\\n\",\n \" (2, gmsh.model.occ.addDisk(c_1 * np.cos(v), c_1 * np.sin(v), 0, r, r))\\n\",\n \" for v in angles_N\\n\",\n \" ]\\n\",\n \"\\n\",\n \" # Define the copper-wires outside the iron cylinder\\n\",\n \" angles_S = [(i + 0.5) * 2 * np.pi / N for i in range(N)]\\n\",\n \" wires_S = [\\n\",\n \" (2, gmsh.model.occ.addDisk(c_2 * np.cos(v), c_2 * np.sin(v), 0, r, r))\\n\",\n \" for v in angles_S\\n\",\n \" ]\\n\",\n \" gmsh.model.occ.synchronize()\\n\",\n \" # Resolve all boundaries of the different wires in the background domain\\n\",\n \" all_surfaces = [(2, iron)]\\n\",\n \" all_surfaces.extend(wires_S)\\n\",\n \" all_surfaces.extend(wires_N)\\n\",\n \" whole_domain = gmsh.model.occ.fragment([(2, background)], all_surfaces)\\n\",\n \" gmsh.model.occ.synchronize()\\n\",\n \" # Create physical markers for the different wires.\\n\",\n \" # We use the following markers:\\n\",\n \" # - Vacuum: 0\\n\",\n \" # - Iron cylinder: 1\\n\",\n \" # - Inner copper wires: $[2,3,\\\\dots,N+1]$\\n\",\n \" # - Outer copper wires: $[N+2,\\\\dots, 2\\\\cdot N+1]\\n\",\n \" inner_tag = 2\\n\",\n \" outer_tag = 2 + N\\n\",\n \" background_surfaces = []\\n\",\n \" other_surfaces = []\\n\",\n \" for domain in whole_domain[0]:\\n\",\n \" com = gmsh.model.occ.getCenterOfMass(domain[0], domain[1])\\n\",\n \" mass = gmsh.model.occ.getMass(domain[0], domain[1])\\n\",\n \" # Identify iron circle by its mass\\n\",\n \" if np.isclose(mass, np.pi * (b**2 - a**2)):\\n\",\n \" gmsh.model.addPhysicalGroup(domain[0], [domain[1]], tag=1)\\n\",\n \" other_surfaces.append(domain)\\n\",\n \" # Identify the background circle by its center of mass\\n\",\n \" elif np.allclose(com, [0, 0, 0]):\\n\",\n \" background_surfaces.append(domain[1])\\n\",\n \"\\n\",\n \" # Identify the inner circles by their center of mass\\n\",\n \" elif np.isclose(np.linalg.norm(com), c_1):\\n\",\n \" gmsh.model.addPhysicalGroup(domain[0], [domain[1]], inner_tag)\\n\",\n \" inner_tag += 1\\n\",\n \" other_surfaces.append(domain)\\n\",\n \" # Identify the outer circles by their center of mass\\n\",\n \" elif np.isclose(np.linalg.norm(com), c_2):\\n\",\n \" gmsh.model.addPhysicalGroup(domain[0], [domain[1]], outer_tag)\\n\",\n \" outer_tag += 1\\n\",\n \" other_surfaces.append(domain)\\n\",\n \" # Add marker for the vacuum\\n\",\n \" gmsh.model.addPhysicalGroup(2, background_surfaces, tag=0)\\n\",\n \" # Create mesh resolution that is fine around the wires and\\n\",\n \" # iron cylinder, coarser the further away you get\\n\",\n \" gmsh.model.mesh.field.add(\\\"Distance\\\", 1)\\n\",\n \" edges = gmsh.model.getBoundary(other_surfaces, oriented=False)\\n\",\n \" gmsh.model.mesh.field.setNumbers(1, \\\"EdgesList\\\", [e[1] for e in edges])\\n\",\n \" gmsh.model.mesh.field.add(\\\"Threshold\\\", 2)\\n\",\n \" gmsh.model.mesh.field.setNumber(2, \\\"IField\\\", 1)\\n\",\n \" gmsh.model.mesh.field.setNumber(2, \\\"LcMin\\\", r / 3)\\n\",\n \" gmsh.model.mesh.field.setNumber(2, \\\"LcMax\\\", 6 * r)\\n\",\n \" gmsh.model.mesh.field.setNumber(2, \\\"DistMin\\\", 4 * r)\\n\",\n \" gmsh.model.mesh.field.setNumber(2, \\\"DistMax\\\", 10 * r)\\n\",\n \" gmsh.model.mesh.field.add(\\\"Min\\\", 5)\\n\",\n \" gmsh.model.mesh.field.setNumbers(5, \\\"FieldsList\\\", [2])\\n\",\n \" gmsh.model.mesh.field.setAsBackgroundMesh(5)\\n\",\n \" # Generate mesh\\n\",\n \" gmsh.option.setNumber(\\\"Mesh.Algorithm\\\", 7)\\n\",\n \" gmsh.model.mesh.generate(gdim)\\n\",\n \" gmsh.model.mesh.optimize(\\\"Netgen\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"As in [the Navier-Stokes tutorial](../chapter2/ns_code2) we load the mesh directly into DOLFINx, without writing it to file.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"tags\": []\n },\n \"outputs\": [],\n \"source\": [\n \"mesh_data = gmshio.model_to_mesh(gmsh.model, mesh_comm, model_rank, gdim=2)\\n\",\n \"mesh = mesh_data.mesh\\n\",\n \"assert mesh_data.cell_tags is not None\\n\",\n \"ct = mesh_data.cell_tags\\n\",\n \"gmsh.finalize()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"To inspect the mesh, we use Paraview, and obtain the following mesh\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"tags\": []\n },\n \"outputs\": [],\n \"source\": [\n \"with XDMFFile(MPI.COMM_WORLD, \\\"mt.xdmf\\\", \\\"w\\\") as xdmf:\\n\",\n \" xdmf.write_mesh(mesh)\\n\",\n \" xdmf.write_meshtags(ct, mesh.geometry)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We can also visualize the subdommains using pyvista\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"lines_to_next_cell\": 2,\n \"tags\": []\n },\n \"outputs\": [],\n \"source\": [\n \"plotter = pyvista.Plotter()\\n\",\n \"tdim = mesh.topology.dim\\n\",\n \"mesh.topology.create_connectivity(tdim, tdim)\\n\",\n \"grid = pyvista.UnstructuredGrid(*vtk_mesh(mesh, tdim))\\n\",\n \"num_local_cells = mesh.topology.index_map(tdim).size_local\\n\",\n \"grid.cell_data[\\\"Marker\\\"] = ct.values[ct.indices < num_local_cells]\\n\",\n \"grid.set_active_scalars(\\\"Marker\\\")\\n\",\n \"actor = plotter.add_mesh(grid, show_edges=True)\\n\",\n \"plotter.view_xy()\\n\",\n \"if not pyvista.OFF_SCREEN:\\n\",\n \" plotter.show()\\n\",\n \"else:\\n\",\n \" cell_tag_fig = plotter.screenshot(\\\"cell_tags.png\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"Next, we define the discontinous functions for the permeability $\\\\mu$ and current $J_z$ using the `MeshTags` as in [Defining material parameters through subdomains](./subdomains)\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"tags\": []\n },\n \"outputs\": [],\n \"source\": [\n \"Q = functionspace(mesh, (\\\"DG\\\", 0))\\n\",\n \"material_tags = np.unique(ct.values)\\n\",\n \"mu = Function(Q)\\n\",\n \"J = Function(Q)\\n\",\n \"# As we only set some values in J, initialize all as 0\\n\",\n \"J.x.array[:] = 0\\n\",\n \"for tag in material_tags:\\n\",\n \" cells = ct.find(tag)\\n\",\n \" # Set values for mu\\n\",\n \" if tag == 0:\\n\",\n \" mu_ = 4 * np.pi * 1e-7 # Vacuum\\n\",\n \" elif tag == 1:\\n\",\n \" mu_ = 1e-5 # Iron (This should really be 6.3e-3)\\n\",\n \" else:\\n\",\n \" mu_ = 1.26e-6 # Copper\\n\",\n \" mu.x.array[cells] = np.full_like(cells, mu_, dtype=default_scalar_type)\\n\",\n \" if tag in range(2, 2 + N):\\n\",\n \" J.x.array[cells] = np.full_like(cells, 1, dtype=default_scalar_type)\\n\",\n \" elif tag in range(2 + N, 2 * N + 2):\\n\",\n \" J.x.array[cells] = np.full_like(cells, -1, dtype=default_scalar_type)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"In the code above, we have used a somewhat less extreme value for the magnetic permability of iron. This is to make the solution a little more interesting. It would otherwise be completely dominated by the field in the iron cylinder.\\n\",\n \"\\n\",\n \"We can now define the weak problem\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"tags\": []\n },\n \"outputs\": [],\n \"source\": [\n \"V = functionspace(mesh, (\\\"Lagrange\\\", 1))\\n\",\n \"tdim = mesh.topology.dim\\n\",\n \"facets = locate_entities_boundary(mesh, tdim - 1, lambda x: np.full(x.shape[1], True))\\n\",\n \"dofs = locate_dofs_topological(V, tdim - 1, facets)\\n\",\n \"bc = dirichletbc(default_scalar_type(0), dofs, V)\\n\",\n \"\\n\",\n \"u = TrialFunction(V)\\n\",\n \"v = TestFunction(V)\\n\",\n \"a = (1 / mu) * dot(grad(u), grad(v)) * dx\\n\",\n \"L = J * v * dx\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We are now ready to solve the linear problem\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"tags\": []\n },\n \"outputs\": [],\n \"source\": [\n \"A_z = Function(V)\\n\",\n \"problem = LinearProblem(a, L, u=A_z, bcs=[bc], petsc_options_prefix=\\\"em_\\\")\\n\",\n \"problem.solve()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"As we have computed the magnetic potential, we can now compute the magnetic field, by setting `B=curl(A_z)`. Note that as we have chosen a function space of first order piecewise linear function to describe our potential, the curl of a function in this space is a discontinous zeroth order function (a function of cell-wise constants). We use `dolfinx.fem.Expression` to interpolate the curl into `W`.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"tags\": []\n },\n \"outputs\": [],\n \"source\": [\n \"W = functionspace(mesh, (\\\"DG\\\", 0, (mesh.geometry.dim,)))\\n\",\n \"B = Function(W)\\n\",\n \"B_expr = Expression(as_vector((A_z.dx(1), -A_z.dx(0))), W.element.interpolation_points)\\n\",\n \"B.interpolate(B_expr)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"Note that we used `ufl.as_vector` to interpret the `Python`-tuple `(A_z.dx(1), -A_z.dx(0))` as a vector in the unified form language (UFL).\\n\",\n \"\\n\",\n \"We now plot the magnetic potential $A_z$ and the magnetic field $B$. We start by creating a new plotter\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"tags\": []\n },\n \"outputs\": [],\n \"source\": [\n \"plotter = pyvista.Plotter()\\n\",\n \"\\n\",\n \"Az_grid = pyvista.UnstructuredGrid(*vtk_mesh(V))\\n\",\n \"Az_grid.point_data[\\\"A_z\\\"] = A_z.x.array\\n\",\n \"Az_grid.set_active_scalars(\\\"A_z\\\")\\n\",\n \"warp = Az_grid.warp_by_scalar(\\\"A_z\\\", factor=1e7)\\n\",\n \"actor = plotter.add_mesh(warp, show_edges=True)\\n\",\n \"if not pyvista.OFF_SCREEN:\\n\",\n \" plotter.show()\\n\",\n \"else:\\n\",\n \" Az_fig = plotter.screenshot(\\\"Az.png\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Visualizing the magnetic field\\n\",\n \"As the magnetic field is a piecewise constant vector field, we need create a custom plotting function.\\n\",\n \"We start by computing the midpoints of each cell, which is where we would like to visualize the cell-wise constant vector.\\n\",\n \"Next, we take the data from the function `B`, and shape it to become a 3D vector.\\n\",\n \"We connect the vector field with the midpoint by using `pyvista.PolyData`.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"tags\": []\n },\n \"outputs\": [],\n \"source\": [\n \"plotter = pyvista.Plotter()\\n\",\n \"plotter.set_position([0, 0, 5])\\n\",\n \"\\n\",\n \"# We include ghosts cells as we access all degrees of freedom (including ghosts) on each process\\n\",\n \"top_imap = mesh.topology.index_map(mesh.topology.dim)\\n\",\n \"num_cells = top_imap.size_local + top_imap.num_ghosts\\n\",\n \"mesh.topology.create_connectivity(mesh.topology.dim, mesh.topology.dim)\\n\",\n \"midpoints = compute_midpoints(\\n\",\n \" mesh, mesh.topology.dim, np.arange(num_cells, dtype=np.int32)\\n\",\n \")\\n\",\n \"\\n\",\n \"num_dofs = W.dofmap.index_map.size_local + W.dofmap.index_map.num_ghosts\\n\",\n \"assert num_cells == num_dofs\\n\",\n \"values = np.zeros((num_dofs, 3), dtype=np.float64)\\n\",\n \"values[:, : mesh.geometry.dim] = B.x.array.real.reshape(num_dofs, W.dofmap.index_map_bs)\\n\",\n \"cloud = pyvista.PolyData(midpoints)\\n\",\n \"cloud[\\\"B\\\"] = values\\n\",\n \"glyphs = cloud.glyph(\\\"B\\\", factor=2e6)\\n\",\n \"actor = plotter.add_mesh(grid, style=\\\"wireframe\\\", color=\\\"k\\\")\\n\",\n \"actor2 = plotter.add_mesh(glyphs)\\n\",\n \"\\n\",\n \"if not pyvista.OFF_SCREEN:\\n\",\n \" plotter.show()\\n\",\n \"else:\\n\",\n \" B_fig = plotter.screenshot(\\\"B.png\\\")\"\n ]\n }\n ],\n \"metadata\": {\n \"jupytext\": {\n \"formats\": \"ipynb,py:light\"\n },\n \"kernelspec\": {\n \"display_name\": \"Python 3 (ipykernel)\",\n \"language\": \"python\",\n \"name\": \"python3\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 3\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython3\",\n \"version\": \"3.10.12\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 4\n}\n" }, { "path": "chapter3/em.py", "content": "# ---\n# jupyter:\n# jupytext:\n# formats: ipynb,py:light\n# text_representation:\n# extension: .py\n# format_name: light\n# format_version: '1.5'\n# jupytext_version: 1.18.1\n# kernelspec:\n# display_name: Python 3 (ipykernel)\n# language: python\n# name: python3\n# ---\n\n# # Electromagnetics example\n#\n# Theoretical introduction by: Hans Petter Langtangen and Anders Logg\n#\n# Implementation by: Jørgen S. Dokken\n#\n# In this example, we will consider an iron cylinder with copper wires wound around the cylinder, as shown below\n#\n# ![Cross section of wires](wire.png)\n#\n# Through the copper wires a static current of $J=1A$ is flowing.\n# We would like to compute the magnetic field $B$ in the iron cylinder, the copper wires, and the surrounding vaccum.\n#\n# We start by simplifying the problem to a 2D problem.\n# We can do this by assuming that the cylinder extends far along the z-axis and\n# as a consequence the field is virtually independent of the z-coordinate.\n# Next, we consider Maxwell's equation to derive a Poisson equation for the magnetic field (or rather its potential)\n#\n# $$\n# \\nabla \\cdot D = \\rho,\n# $$\n# $$\n# \\nabla \\cdot B = 0,\n# $$\n# $$\n# \\nabla \\times E = -\\frac{\\partial B}{\\partial t},\n# $$\n# $$\n# \\nabla \\times H = \\frac{\\partial D}{\\partial t}+ J.\n# $$\n#\n# Here, $D$ is the displacement field, $B$ is the magnetic field, $E$ is the electric field, and $H$ is the magnetizing field.\n# In addition to Maxwell's equation, we need a constitutive relation between $B$ and $H$,\n#\n# $$\n# B =\\mu H,\n# $$\n#\n# which holds for an isotropic linear magnetic medium.\n# Here, $\\mu$ is the magnetic permability of the material.\n# Now, since $B$ is solenodial (divergence free) according to Maxwell's equations, we known that $B$ must be the curl of some vector field $A$. This field is called the magnetic vector potential. Since the problem is static and thus $\\frac{\\partial D}{\\partial t}=0$, it follows that\n#\n# $$\n# J = \\nabla \\times H = \\nabla \\times(\\mu^{-1} B)=\\nabla \\times (\\mu^{-1}\\nabla \\times A ) = -\\nabla \\cdot (\\mu^{-1}\\nabla A).\n# $$\n#\n# In the last step, we have expanded the second derivatives and used the gauge freedom of $A$ to simplify the equations to a simple vector-valued Poisson equation for the magnetic vector potential; if $B=\\nabla \\times A$, then $B=\\nabla \\times (A+\\nabla \\phi)$ for any scalar field $\\phi$ (the gauge function).\n# For the current problem, we thus need to solve the following 2D Poisson problem for the $z$-component $A_z$ of the magnetic vector potential\n#\n# $$\n# - \\nabla \\cdot (\\mu^{-1} \\nabla A_z) = J_z \\qquad \\text{in } \\mathbb{R}^2,\\\\\n# $$\n# $$\n# \\lim_{\\vert(x,y)\\vert\\to \\infty}A_z = 0.\n# $$\n#\n# Since we cannot solve the problem on an infinite domain, we will truncate the domain using a large disk, and set $A_z=0$ on the boundary. The current $J_z$ is set to $+1$A in the interior set of the circles (copper-wire cross sections) and to $-1$ A in the exterior set of circles in the cross section figure.\n# Once the magnetic field vector potential has been computed, we can compute the magnetic field $B=B(x,y)$ by\n#\n# $$\n# B(x,y)=\\left(\\frac{\\partial A_z}{\\partial y}, - \\frac{\\partial A_z}{\\partial x} \\right).\n# $$\n#\n# The weak formulation is easily obtained by multiplication of a test function $v$, followed by integration by parts, where all boundary integrals vanish due to the Dirichlet condition, we obtain $a(A_z,v)=L(v)$ with\n#\n# $$\n# a(A_z, v)=\\int_\\Omega \\mu^{-1}\\nabla A_z \\cdot \\nabla v ~\\mathrm{d}x,\n# $$\n# $$\n# L(v)=\\int_\\Omega J_z v~\\mathrm{d} x.\n# $$\n\n# ## Meshing a complex structure with subdomains\n#\n# We create the domain visualized in the cross section figure above using gmsh. Note that we are using the `gmsh.model.occ.fragment` commands to ensure that the boundaries of the wires are resolved in the mesh.\n\n# +\nfrom dolfinx import default_scalar_type\nfrom dolfinx.fem import (\n dirichletbc,\n Expression,\n Function,\n functionspace,\n locate_dofs_topological,\n)\nfrom dolfinx.fem.petsc import LinearProblem\nfrom dolfinx.io import XDMFFile\nfrom dolfinx.io import gmsh as gmshio\nfrom dolfinx.mesh import compute_midpoints, locate_entities_boundary\nfrom dolfinx.plot import vtk_mesh\n\nfrom ufl import TestFunction, TrialFunction, as_vector, dot, dx, grad\nfrom mpi4py import MPI\n\nimport gmsh\nimport numpy as np\nimport pyvista\n\nrank = MPI.COMM_WORLD.rank\n\ngmsh.initialize()\nr = 0.1 # Radius of copper wires\nR = 5 # Radius of domain\na = 1 # Radius of inner iron cylinder\nb = 1.2 # Radius of outer iron cylinder\nN = 8 # Number of windings\nc_1 = 0.8 # Radius of inner copper wires\nc_2 = 1.4 # Radius of outer copper wires\ngdim = 2 # Geometric dimension of the mesh\nmodel_rank = 0\nmesh_comm = MPI.COMM_WORLD\nif mesh_comm.rank == model_rank:\n # Define geometry for iron cylinder\n outer_iron = gmsh.model.occ.addCircle(0, 0, 0, b)\n inner_iron = gmsh.model.occ.addCircle(0, 0, 0, a)\n gmsh.model.occ.addCurveLoop([outer_iron], 5)\n gmsh.model.occ.addCurveLoop([inner_iron], 6)\n iron = gmsh.model.occ.addPlaneSurface([5, 6])\n gmsh.model.occ.synchronize()\n\n # Define geometry for background\n background = gmsh.model.occ.addDisk(0, 0, 0, R, R)\n gmsh.model.occ.synchronize()\n\n # Define the copper-wires inside iron cylinder\n angles_N = [i * 2 * np.pi / N for i in range(N)]\n wires_N = [\n (2, gmsh.model.occ.addDisk(c_1 * np.cos(v), c_1 * np.sin(v), 0, r, r))\n for v in angles_N\n ]\n\n # Define the copper-wires outside the iron cylinder\n angles_S = [(i + 0.5) * 2 * np.pi / N for i in range(N)]\n wires_S = [\n (2, gmsh.model.occ.addDisk(c_2 * np.cos(v), c_2 * np.sin(v), 0, r, r))\n for v in angles_S\n ]\n gmsh.model.occ.synchronize()\n # Resolve all boundaries of the different wires in the background domain\n all_surfaces = [(2, iron)]\n all_surfaces.extend(wires_S)\n all_surfaces.extend(wires_N)\n whole_domain = gmsh.model.occ.fragment([(2, background)], all_surfaces)\n gmsh.model.occ.synchronize()\n # Create physical markers for the different wires.\n # We use the following markers:\n # - Vacuum: 0\n # - Iron cylinder: 1\n # - Inner copper wires: $[2,3,\\dots,N+1]$\n # - Outer copper wires: $[N+2,\\dots, 2\\cdot N+1]\n inner_tag = 2\n outer_tag = 2 + N\n background_surfaces = []\n other_surfaces = []\n for domain in whole_domain[0]:\n com = gmsh.model.occ.getCenterOfMass(domain[0], domain[1])\n mass = gmsh.model.occ.getMass(domain[0], domain[1])\n # Identify iron circle by its mass\n if np.isclose(mass, np.pi * (b**2 - a**2)):\n gmsh.model.addPhysicalGroup(domain[0], [domain[1]], tag=1)\n other_surfaces.append(domain)\n # Identify the background circle by its center of mass\n elif np.allclose(com, [0, 0, 0]):\n background_surfaces.append(domain[1])\n\n # Identify the inner circles by their center of mass\n elif np.isclose(np.linalg.norm(com), c_1):\n gmsh.model.addPhysicalGroup(domain[0], [domain[1]], inner_tag)\n inner_tag += 1\n other_surfaces.append(domain)\n # Identify the outer circles by their center of mass\n elif np.isclose(np.linalg.norm(com), c_2):\n gmsh.model.addPhysicalGroup(domain[0], [domain[1]], outer_tag)\n outer_tag += 1\n other_surfaces.append(domain)\n # Add marker for the vacuum\n gmsh.model.addPhysicalGroup(2, background_surfaces, tag=0)\n # Create mesh resolution that is fine around the wires and\n # iron cylinder, coarser the further away you get\n gmsh.model.mesh.field.add(\"Distance\", 1)\n edges = gmsh.model.getBoundary(other_surfaces, oriented=False)\n gmsh.model.mesh.field.setNumbers(1, \"EdgesList\", [e[1] for e in edges])\n gmsh.model.mesh.field.add(\"Threshold\", 2)\n gmsh.model.mesh.field.setNumber(2, \"IField\", 1)\n gmsh.model.mesh.field.setNumber(2, \"LcMin\", r / 3)\n gmsh.model.mesh.field.setNumber(2, \"LcMax\", 6 * r)\n gmsh.model.mesh.field.setNumber(2, \"DistMin\", 4 * r)\n gmsh.model.mesh.field.setNumber(2, \"DistMax\", 10 * r)\n gmsh.model.mesh.field.add(\"Min\", 5)\n gmsh.model.mesh.field.setNumbers(5, \"FieldsList\", [2])\n gmsh.model.mesh.field.setAsBackgroundMesh(5)\n # Generate mesh\n gmsh.option.setNumber(\"Mesh.Algorithm\", 7)\n gmsh.model.mesh.generate(gdim)\n gmsh.model.mesh.optimize(\"Netgen\")\n# -\n\n# As in [the Navier-Stokes tutorial](../chapter2/ns_code2) we load the mesh directly into DOLFINx, without writing it to file.\n\nmesh_data = gmshio.model_to_mesh(gmsh.model, mesh_comm, model_rank, gdim=2)\nmesh = mesh_data.mesh\nassert mesh_data.cell_tags is not None\nct = mesh_data.cell_tags\ngmsh.finalize()\n\n# To inspect the mesh, we use Paraview, and obtain the following mesh\n\nwith XDMFFile(MPI.COMM_WORLD, \"mt.xdmf\", \"w\") as xdmf:\n xdmf.write_mesh(mesh)\n xdmf.write_meshtags(ct, mesh.geometry)\n\n# We can also visualize the subdommains using pyvista\n\nplotter = pyvista.Plotter()\ntdim = mesh.topology.dim\nmesh.topology.create_connectivity(tdim, tdim)\ngrid = pyvista.UnstructuredGrid(*vtk_mesh(mesh, tdim))\nnum_local_cells = mesh.topology.index_map(tdim).size_local\ngrid.cell_data[\"Marker\"] = ct.values[ct.indices < num_local_cells]\ngrid.set_active_scalars(\"Marker\")\nactor = plotter.add_mesh(grid, show_edges=True)\nplotter.view_xy()\nif not pyvista.OFF_SCREEN:\n plotter.show()\nelse:\n cell_tag_fig = plotter.screenshot(\"cell_tags.png\")\n\n\n# Next, we define the discontinous functions for the permeability $\\mu$ and current $J_z$ using the `MeshTags` as in [Defining material parameters through subdomains](./subdomains)\n#\n\nQ = functionspace(mesh, (\"DG\", 0))\nmaterial_tags = np.unique(ct.values)\nmu = Function(Q)\nJ = Function(Q)\n# As we only set some values in J, initialize all as 0\nJ.x.array[:] = 0\nfor tag in material_tags:\n cells = ct.find(tag)\n # Set values for mu\n if tag == 0:\n mu_ = 4 * np.pi * 1e-7 # Vacuum\n elif tag == 1:\n mu_ = 1e-5 # Iron (This should really be 6.3e-3)\n else:\n mu_ = 1.26e-6 # Copper\n mu.x.array[cells] = np.full_like(cells, mu_, dtype=default_scalar_type)\n if tag in range(2, 2 + N):\n J.x.array[cells] = np.full_like(cells, 1, dtype=default_scalar_type)\n elif tag in range(2 + N, 2 * N + 2):\n J.x.array[cells] = np.full_like(cells, -1, dtype=default_scalar_type)\n\n# In the code above, we have used a somewhat less extreme value for the magnetic permability of iron. This is to make the solution a little more interesting. It would otherwise be completely dominated by the field in the iron cylinder.\n#\n# We can now define the weak problem\n\n# +\nV = functionspace(mesh, (\"Lagrange\", 1))\ntdim = mesh.topology.dim\nfacets = locate_entities_boundary(mesh, tdim - 1, lambda x: np.full(x.shape[1], True))\ndofs = locate_dofs_topological(V, tdim - 1, facets)\nbc = dirichletbc(default_scalar_type(0), dofs, V)\n\nu = TrialFunction(V)\nv = TestFunction(V)\na = (1 / mu) * dot(grad(u), grad(v)) * dx\nL = J * v * dx\n# -\n\n# We are now ready to solve the linear problem\n\nA_z = Function(V)\nproblem = LinearProblem(a, L, u=A_z, bcs=[bc], petsc_options_prefix=\"em_\")\nproblem.solve()\n\n# As we have computed the magnetic potential, we can now compute the magnetic field, by setting `B=curl(A_z)`. Note that as we have chosen a function space of first order piecewise linear function to describe our potential, the curl of a function in this space is a discontinous zeroth order function (a function of cell-wise constants). We use `dolfinx.fem.Expression` to interpolate the curl into `W`.\n\nW = functionspace(mesh, (\"DG\", 0, (mesh.geometry.dim,)))\nB = Function(W)\nB_expr = Expression(as_vector((A_z.dx(1), -A_z.dx(0))), W.element.interpolation_points)\nB.interpolate(B_expr)\n\n# Note that we used `ufl.as_vector` to interpret the `Python`-tuple `(A_z.dx(1), -A_z.dx(0))` as a vector in the unified form language (UFL).\n#\n# We now plot the magnetic potential $A_z$ and the magnetic field $B$. We start by creating a new plotter\n\n# +\nplotter = pyvista.Plotter()\n\nAz_grid = pyvista.UnstructuredGrid(*vtk_mesh(V))\nAz_grid.point_data[\"A_z\"] = A_z.x.array\nAz_grid.set_active_scalars(\"A_z\")\nwarp = Az_grid.warp_by_scalar(\"A_z\", factor=1e7)\nactor = plotter.add_mesh(warp, show_edges=True)\nif not pyvista.OFF_SCREEN:\n plotter.show()\nelse:\n Az_fig = plotter.screenshot(\"Az.png\")\n# -\n\n# ## Visualizing the magnetic field\n# As the magnetic field is a piecewise constant vector field, we need create a custom plotting function.\n# We start by computing the midpoints of each cell, which is where we would like to visualize the cell-wise constant vector.\n# Next, we take the data from the function `B`, and shape it to become a 3D vector.\n# We connect the vector field with the midpoint by using `pyvista.PolyData`.\n\n# +\nplotter = pyvista.Plotter()\nplotter.set_position([0, 0, 5])\n\n# We include ghosts cells as we access all degrees of freedom (including ghosts) on each process\ntop_imap = mesh.topology.index_map(mesh.topology.dim)\nnum_cells = top_imap.size_local + top_imap.num_ghosts\nmesh.topology.create_connectivity(mesh.topology.dim, mesh.topology.dim)\nmidpoints = compute_midpoints(\n mesh, mesh.topology.dim, np.arange(num_cells, dtype=np.int32)\n)\n\nnum_dofs = W.dofmap.index_map.size_local + W.dofmap.index_map.num_ghosts\nassert num_cells == num_dofs\nvalues = np.zeros((num_dofs, 3), dtype=np.float64)\nvalues[:, : mesh.geometry.dim] = B.x.array.real.reshape(num_dofs, W.dofmap.index_map_bs)\ncloud = pyvista.PolyData(midpoints)\ncloud[\"B\"] = values\nglyphs = cloud.glyph(\"B\", factor=2e6)\nactor = plotter.add_mesh(grid, style=\"wireframe\", color=\"k\")\nactor2 = plotter.add_mesh(glyphs)\n\nif not pyvista.OFF_SCREEN:\n plotter.show()\nelse:\n B_fig = plotter.screenshot(\"B.png\")\n" }, { "path": "chapter3/multiple_dirichlet.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Setting multiple Dirichlet condition\\n\",\n \"\\n\",\n \"In the previous section, we used a single function for $u_D$ to set Dirichlet conditions on two parts of the boundary. However, it is often more practical to use multiple functions, one for each subdomain of the boundary. We consider a similar example to [the previous example](./neumann_dirichlet_code) and redefine it to consist of two Dirichlet boundary conditions\\n\",\n \"\\n\",\n \"$$\\n\",\n \"-\\\\nabla^2 u =f \\\\quad \\\\text{in } \\\\Omega,\\n\",\n \"$$\\n\",\n \"\\n\",\n \"$$\\n\",\n \"u=u_L \\\\quad \\\\text{on } \\\\Lambda_D^L\\n\",\n \"$$\\n\",\n \"\\n\",\n \"$$\\n\",\n \"u=u_R \\\\quad \\\\text{on } \\\\Lambda_D^R\\n\",\n \"$$\\n\",\n \"\\n\",\n \"$$\\n\",\n \"-\\\\frac{\\\\partial u}{\\\\partial n} = g \\\\quad \\\\text{on } \\\\Lambda_N.\\n\",\n \"$$\\n\",\n \"Here, $\\\\Lambda_D^L$ is the left boundary $x=0$, while $\\\\Lambda_D^R$ is the right boundary $x=1$.\\n\",\n \"We note that $u_L(y)=1+2y^2$, $u_R(y)=2+2y^2$ and $g(y)=-4y$ using the same analytical example as in the previous section.\\n\",\n \"\\n\",\n \"We start by defining the mesh, function space and variational formulation as in the previous exercise\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"from dolfinx import default_scalar_type\\n\",\n \"from dolfinx.fem import (\\n\",\n \" Constant,\\n\",\n \" Function,\\n\",\n \" functionspace,\\n\",\n \" assemble_scalar,\\n\",\n \" dirichletbc,\\n\",\n \" form,\\n\",\n \" locate_dofs_geometrical,\\n\",\n \")\\n\",\n \"from dolfinx.fem.petsc import LinearProblem\\n\",\n \"from dolfinx.mesh import create_unit_square\\n\",\n \"from dolfinx.plot import vtk_mesh\\n\",\n \"\\n\",\n \"from mpi4py import MPI\\n\",\n \"from ufl import SpatialCoordinate, TestFunction, TrialFunction, dot, dx, ds, grad\\n\",\n \"\\n\",\n \"import numpy as np\\n\",\n \"import pyvista\\n\",\n \"\\n\",\n \"\\n\",\n \"def u_exact(x):\\n\",\n \" return 1 + x[0] ** 2 + 2 * x[1] ** 2\\n\",\n \"\\n\",\n \"\\n\",\n \"mesh = create_unit_square(MPI.COMM_WORLD, 10, 10)\\n\",\n \"V = functionspace(mesh, (\\\"Lagrange\\\", 1))\\n\",\n \"u = TrialFunction(V)\\n\",\n \"v = TestFunction(V)\\n\",\n \"a = dot(grad(u), grad(v)) * dx\\n\",\n \"x = SpatialCoordinate(mesh)\\n\",\n \"g = -4 * x[1]\\n\",\n \"f = Constant(mesh, default_scalar_type(-6))\\n\",\n \"L = f * v * dx - g * v * ds\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We next mark the two boundaries separately, starting with the left boundary\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"dofs_L = locate_dofs_geometrical(V, lambda x: np.isclose(x[0], 0))\\n\",\n \"u_L = Function(V)\\n\",\n \"u_L.interpolate(lambda x: 1 + 2 * x[1] ** 2)\\n\",\n \"bc_L = dirichletbc(u_L, dofs_L)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"Note that we have used `lambda`-functions to compactly define the functions returning the subdomain evaluation and function evaluation. We can use a similar procedure for the right boundary condition, and gather both boundary conditions in a vector `bcs`.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"dofs_R = locate_dofs_geometrical(V, lambda x: np.isclose(x[0], 1))\\n\",\n \"u_R = Function(V)\\n\",\n \"u_R.interpolate(lambda x: 2 + 2 * x[1] ** 2)\\n\",\n \"bc_R = dirichletbc(u_R, dofs_R)\\n\",\n \"bcs = [bc_R, bc_L]\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We are now ready to again solve the problem, and check the $L^2$ and max error at the mesh vertices.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"problem = LinearProblem(\\n\",\n \" a,\\n\",\n \" L,\\n\",\n \" bcs=bcs,\\n\",\n \" petsc_options={\\\"ksp_type\\\": \\\"preonly\\\", \\\"pc_type\\\": \\\"lu\\\"},\\n\",\n \" petsc_options_prefix=\\\"multiple_dirichlet_\\\",\\n\",\n \")\\n\",\n \"uh = problem.solve()\\n\",\n \"\\n\",\n \"V2 = functionspace(mesh, (\\\"Lagrange\\\", 2))\\n\",\n \"uex = Function(V2)\\n\",\n \"uex.interpolate(u_exact)\\n\",\n \"error_L2 = assemble_scalar(form((uh - uex) ** 2 * dx))\\n\",\n \"error_L2 = np.sqrt(MPI.COMM_WORLD.allreduce(error_L2, op=MPI.SUM))\\n\",\n \"\\n\",\n \"u_vertex_values = uh.x.array\\n\",\n \"uex_1 = Function(V)\\n\",\n \"uex_1.interpolate(uex)\\n\",\n \"u_ex_vertex_values = uex_1.x.array\\n\",\n \"error_max = np.max(np.abs(u_vertex_values - u_ex_vertex_values))\\n\",\n \"error_max = MPI.COMM_WORLD.allreduce(error_max, op=MPI.MAX)\\n\",\n \"print(f\\\"Error_L2 : {error_L2:.2e}\\\")\\n\",\n \"print(f\\\"Error_max : {error_max:.2e}\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Visualization\\n\",\n \"To visualize the solution, run the script with in a Jupyter notebook with `off_screen=False` or as a python script with `off_screen=True`.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"pyvista_cells, cell_types, geometry = vtk_mesh(V)\\n\",\n \"grid = pyvista.UnstructuredGrid(pyvista_cells, cell_types, geometry)\\n\",\n \"grid.point_data[\\\"u\\\"] = uh.x.array\\n\",\n \"grid.set_active_scalars(\\\"u\\\")\\n\",\n \"\\n\",\n \"plotter = pyvista.Plotter()\\n\",\n \"plotter.add_text(\\\"uh\\\", position=\\\"upper_edge\\\", font_size=14, color=\\\"black\\\")\\n\",\n \"plotter.add_mesh(grid, show_edges=True)\\n\",\n \"plotter.view_xy()\\n\",\n \"if not pyvista.OFF_SCREEN:\\n\",\n \" plotter.show()\\n\",\n \"else:\\n\",\n \" figure = plotter.screenshot(\\\"multiple_dirichlet.png\\\")\"\n ]\n }\n ],\n \"metadata\": {\n \"jupytext\": {\n \"formats\": \"ipynb,py:light\"\n },\n \"kernelspec\": {\n \"display_name\": \"Python 3 (ipykernel)\",\n \"language\": \"python\",\n \"name\": \"python3\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 3\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython3\",\n \"version\": \"3.10.12\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 4\n}\n" }, { "path": "chapter3/multiple_dirichlet.py", "content": "# ---\n# jupyter:\n# jupytext:\n# formats: ipynb,py:light\n# text_representation:\n# extension: .py\n# format_name: light\n# format_version: '1.5'\n# jupytext_version: 1.18.1\n# kernelspec:\n# display_name: Python 3 (ipykernel)\n# language: python\n# name: python3\n# ---\n\n# # Setting multiple Dirichlet condition\n#\n# In the previous section, we used a single function for $u_D$ to set Dirichlet conditions on two parts of the boundary. However, it is often more practical to use multiple functions, one for each subdomain of the boundary. We consider a similar example to [the previous example](./neumann_dirichlet_code) and redefine it to consist of two Dirichlet boundary conditions\n#\n# $$\n# -\\nabla^2 u =f \\quad \\text{in } \\Omega,\n# $$\n#\n# $$\n# u=u_L \\quad \\text{on } \\Lambda_D^L\n# $$\n#\n# $$\n# u=u_R \\quad \\text{on } \\Lambda_D^R\n# $$\n#\n# $$\n# -\\frac{\\partial u}{\\partial n} = g \\quad \\text{on } \\Lambda_N.\n# $$\n# Here, $\\Lambda_D^L$ is the left boundary $x=0$, while $\\Lambda_D^R$ is the right boundary $x=1$.\n# We note that $u_L(y)=1+2y^2$, $u_R(y)=2+2y^2$ and $g(y)=-4y$ using the same analytical example as in the previous section.\n#\n# We start by defining the mesh, function space and variational formulation as in the previous exercise\n\n# +\nfrom dolfinx import default_scalar_type\nfrom dolfinx.fem import (\n Constant,\n Function,\n functionspace,\n assemble_scalar,\n dirichletbc,\n form,\n locate_dofs_geometrical,\n)\nfrom dolfinx.fem.petsc import LinearProblem\nfrom dolfinx.mesh import create_unit_square\nfrom dolfinx.plot import vtk_mesh\n\nfrom mpi4py import MPI\nfrom ufl import SpatialCoordinate, TestFunction, TrialFunction, dot, dx, ds, grad\n\nimport numpy as np\nimport pyvista\n\n\ndef u_exact(x):\n return 1 + x[0] ** 2 + 2 * x[1] ** 2\n\n\nmesh = create_unit_square(MPI.COMM_WORLD, 10, 10)\nV = functionspace(mesh, (\"Lagrange\", 1))\nu = TrialFunction(V)\nv = TestFunction(V)\na = dot(grad(u), grad(v)) * dx\nx = SpatialCoordinate(mesh)\ng = -4 * x[1]\nf = Constant(mesh, default_scalar_type(-6))\nL = f * v * dx - g * v * ds\n# -\n\n# We next mark the two boundaries separately, starting with the left boundary\n\ndofs_L = locate_dofs_geometrical(V, lambda x: np.isclose(x[0], 0))\nu_L = Function(V)\nu_L.interpolate(lambda x: 1 + 2 * x[1] ** 2)\nbc_L = dirichletbc(u_L, dofs_L)\n\n# Note that we have used `lambda`-functions to compactly define the functions returning the subdomain evaluation and function evaluation. We can use a similar procedure for the right boundary condition, and gather both boundary conditions in a vector `bcs`.\n\ndofs_R = locate_dofs_geometrical(V, lambda x: np.isclose(x[0], 1))\nu_R = Function(V)\nu_R.interpolate(lambda x: 2 + 2 * x[1] ** 2)\nbc_R = dirichletbc(u_R, dofs_R)\nbcs = [bc_R, bc_L]\n\n# We are now ready to again solve the problem, and check the $L^2$ and max error at the mesh vertices.\n\n# +\nproblem = LinearProblem(\n a,\n L,\n bcs=bcs,\n petsc_options={\"ksp_type\": \"preonly\", \"pc_type\": \"lu\"},\n petsc_options_prefix=\"multiple_dirichlet_\",\n)\nuh = problem.solve()\n\nV2 = functionspace(mesh, (\"Lagrange\", 2))\nuex = Function(V2)\nuex.interpolate(u_exact)\nerror_L2 = assemble_scalar(form((uh - uex) ** 2 * dx))\nerror_L2 = np.sqrt(MPI.COMM_WORLD.allreduce(error_L2, op=MPI.SUM))\n\nu_vertex_values = uh.x.array\nuex_1 = Function(V)\nuex_1.interpolate(uex)\nu_ex_vertex_values = uex_1.x.array\nerror_max = np.max(np.abs(u_vertex_values - u_ex_vertex_values))\nerror_max = MPI.COMM_WORLD.allreduce(error_max, op=MPI.MAX)\nprint(f\"Error_L2 : {error_L2:.2e}\")\nprint(f\"Error_max : {error_max:.2e}\")\n# -\n\n# ## Visualization\n# To visualize the solution, run the script with in a Jupyter notebook with `off_screen=False` or as a python script with `off_screen=True`.\n\n# +\npyvista_cells, cell_types, geometry = vtk_mesh(V)\ngrid = pyvista.UnstructuredGrid(pyvista_cells, cell_types, geometry)\ngrid.point_data[\"u\"] = uh.x.array\ngrid.set_active_scalars(\"u\")\n\nplotter = pyvista.Plotter()\nplotter.add_text(\"uh\", position=\"upper_edge\", font_size=14, color=\"black\")\nplotter.add_mesh(grid, show_edges=True)\nplotter.view_xy()\nif not pyvista.OFF_SCREEN:\n plotter.show()\nelse:\n figure = plotter.screenshot(\"multiple_dirichlet.png\")\n" }, { "path": "chapter3/neumann_dirichlet_code.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {\n \"tags\": []\n },\n \"source\": [\n \"# Combining Dirichlet and Neumann conditions\\n\",\n \"Author: Jørgen S. Dokken\\n\",\n \"\\n\",\n \"Let's return to the Poisson problem from the [Fundamentals chapter](./../chapter1/fundamentals.md)\\n\",\n \"and see how to extend the mathematics and the implementation to handle Dirichlet condition\\n\",\n \"in combination with a Neumann condition.\\n\",\n \"The domain is still the unit square, but now we set the Dirichlet condition $u=u_D$ at the left and right sides,\\n\",\n \"while the Neumann condition\\n\",\n \"\\n\",\n \"$$\\n\",\n \"-\\\\frac{\\\\partial u}{\\\\partial n}=g\\n\",\n \"$$\\n\",\n \"\\n\",\n \"is applied to the remaining sides $y=0$ and $y=1$.\\n\",\n \"\\n\",\n \"## The PDE problem\\n\",\n \"Let $\\\\Lambda_D$ and $\\\\Lambda_N$ denote parts of the boundary $\\\\partial \\\\Omega$\\n\",\n \"where the Dirichlet and Neumann conditions apply, respectively.\\n\",\n \"The complete boundary-value problem can be written as\\n\",\n \"\\n\",\n \"$$\\n\",\n \"-\\\\nabla^2 u =f \\\\qquad \\\\text{in } \\\\Omega,\\n\",\n \"$$\\n\",\n \"$$\\n\",\n \"u=u_D \\\\qquad\\\\text{on } \\\\Lambda_D,\\n\",\n \"$$\\n\",\n \"$$\\n\",\n \"-\\\\frac{\\\\partial u}{\\\\partial n}=g \\\\qquad \\\\text{on }\\\\Lambda_N\\n\",\n \"$$\\n\",\n \"\\n\",\n \"Again, we choose $u=1+x^2+2y^2$ as the exact solution and adjust $f, g,$ and $u_D$ accordingly\\n\",\n \"\\n\",\n \"$$\\n\",\n \"f(x,y)=-6,\\n\",\n \"$$\\n\",\n \"$$\\n\",\n \"g(x,y)=\\\\begin{cases}\\n\",\n \"0, & y=0,\\\\\\\\\\n\",\n \"-4, & y=1,\\n\",\n \"\\\\end{cases}\\n\",\n \"$$\\n\",\n \"$$\\n\",\n \"u_D(x,y)=1+x^2+2y^2.\\n\",\n \"$$\\n\",\n \"\\n\",\n \"For the ease of programming, we define $g$ as a function over the whole domain $\\\\Omega$ such that\\n\",\n \"$g$ takes on the correct values at $y=0$ and $y=1$. One possible extension is\\n\",\n \"\\n\",\n \"$$\\n\",\n \" g(x,y)=-4y.\\n\",\n \"$$\\n\",\n \"\\n\",\n \"## The variational formulation\\n\",\n \"The first task is to derive the variational formulation.\\n\",\n \"This time we cannot omit the boundary term arising from integration by parts,\\n\",\n \"because $v$ is only zero on $\\\\Lambda_D$. We have\\n\",\n \"\\n\",\n \"$$\\n\",\n \"-\\\\int_\\\\Omega (\\\\nabla^2u)v~\\\\mathrm{d} x =\\n\",\n \"\\\\int_\\\\Omega \\\\nabla u \\\\cdot \\\\nabla v ~\\\\mathrm{d} x - \\\\int_{\\\\partial\\\\Omega}\\\\frac{\\\\partial u}{\\\\partial n}v~\\\\mathrm{d}s,\\n\",\n \"$$\\n\",\n \"\\n\",\n \"and since $v=0$ on $\\\\Lambda_D$,\\n\",\n \"\\n\",\n \"$$\\n\",\n \"- \\\\int_{\\\\partial\\\\Omega}\\\\frac{\\\\partial u}{\\\\partial n}v~\\\\mathrm{d}s=\\n\",\n \"- \\\\int_{\\\\Lambda_N}\\\\frac{\\\\partial u}{\\\\partial n}v~\\\\mathrm{d}s =\\\\int_{\\\\Lambda_N} gv~\\\\mathrm{d}s,\\n\",\n \"$$\\n\",\n \"\\n\",\n \"by applying the boundary condition on $\\\\Lambda_N$.\\n\",\n \"The resulting weak from reads\\n\",\n \"\\n\",\n \"$$\\n\",\n \"\\\\int_\\\\Omega \\\\nabla u \\\\cdot \\\\nabla v~\\\\mathrm{d} x =\\n\",\n \"\\\\int_\\\\Omega fv~\\\\mathrm{d} x - \\\\int_{\\\\Lambda_N}gv~\\\\mathrm{d}s.\\n\",\n \"$$\\n\",\n \"Expressing this equation in the standard notation $a(u,v)=L(v)$ is straight-forward with\\n\",\n \"\\n\",\n \"$$\\n\",\n \" a(u,v) = \\\\int_{\\\\Omega} \\\\nabla u \\\\cdot \\\\nabla v ~\\\\mathrm{d} x,\\\\\\\\\\n\",\n \"$$\\n\",\n \"$$\\n\",\n \"L(v) = \\\\int_{\\\\Omega} fv ~\\\\mathrm{d} x - \\\\int_{\\\\Lambda_N} gv~\\\\mathrm{d} s.\\n\",\n \"$$\\n\",\n \"\\n\",\n \"## Implementation\\n\",\n \"As in the previous example, we define our mesh,function space and bilinear form $a(u,v)$.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"from dolfinx import default_scalar_type\\n\",\n \"from dolfinx.fem import (\\n\",\n \" Constant,\\n\",\n \" Function,\\n\",\n \" functionspace,\\n\",\n \" assemble_scalar,\\n\",\n \" dirichletbc,\\n\",\n \" form,\\n\",\n \" locate_dofs_geometrical,\\n\",\n \")\\n\",\n \"from dolfinx.fem.petsc import LinearProblem\\n\",\n \"from dolfinx.mesh import create_unit_square\\n\",\n \"from dolfinx.plot import vtk_mesh\\n\",\n \"\\n\",\n \"from mpi4py import MPI\\n\",\n \"from ufl import SpatialCoordinate, TestFunction, TrialFunction, dot, ds, dx, grad\\n\",\n \"\\n\",\n \"import numpy as np\\n\",\n \"import pyvista\\n\",\n \"\\n\",\n \"mesh = create_unit_square(MPI.COMM_WORLD, 10, 10)\\n\",\n \"V = functionspace(mesh, (\\\"Lagrange\\\", 1))\\n\",\n \"u = TrialFunction(V)\\n\",\n \"v = TestFunction(V)\\n\",\n \"a = dot(grad(u), grad(v)) * dx\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"Now we get to the Neumann and Dirichlet boundary condition. As previously, we use a Python-function to define the boundary where we should have a Dirichlet condition. Then, with this function, we locate degrees of freedom that fulfill this condition.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"def u_exact(x):\\n\",\n \" return 1 + x[0] ** 2 + 2 * x[1] ** 2\\n\",\n \"\\n\",\n \"\\n\",\n \"def boundary_D(x):\\n\",\n \" return np.logical_or(np.isclose(x[0], 0), np.isclose(x[0], 1))\\n\",\n \"\\n\",\n \"\\n\",\n \"dofs_D = locate_dofs_geometrical(V, boundary_D)\\n\",\n \"u_bc = Function(V)\\n\",\n \"u_bc.interpolate(u_exact)\\n\",\n \"bc = dirichletbc(u_bc, dofs_D)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"The next step is to define the Neumann condition. We first define $g$ uses `UFL`s `SpatialCoordinate`-function, and then in turn create a boundary integration measure `ds`. As the test function $v$ is zero on the boundary integrals over the Dirichlet boundary disappears, and we can integrate `g*v*ds` over the entire boundary.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"x = SpatialCoordinate(mesh)\\n\",\n \"g = -4 * x[1]\\n\",\n \"f = Constant(mesh, default_scalar_type(-6))\\n\",\n \"L = f * v * dx - g * v * ds\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We can now assemble and solve the linear system of equations\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"problem = LinearProblem(\\n\",\n \" a,\\n\",\n \" L,\\n\",\n \" bcs=[bc],\\n\",\n \" petsc_options={\\\"ksp_type\\\": \\\"preonly\\\", \\\"pc_type\\\": \\\"lu\\\"},\\n\",\n \" petsc_options_prefix=\\\"neumann_dirichlet_\\\",\\n\",\n \")\\n\",\n \"uh = problem.solve()\\n\",\n \"\\n\",\n \"V2 = functionspace(mesh, (\\\"Lagrange\\\", 2))\\n\",\n \"uex = Function(V2)\\n\",\n \"uex.interpolate(u_exact)\\n\",\n \"error_L2 = assemble_scalar(form((uh - uex) ** 2 * dx))\\n\",\n \"error_L2 = np.sqrt(MPI.COMM_WORLD.allreduce(error_L2, op=MPI.SUM))\\n\",\n \"\\n\",\n \"u_vertex_values = uh.x.array\\n\",\n \"uex_1 = Function(V)\\n\",\n \"uex_1.interpolate(uex)\\n\",\n \"u_ex_vertex_values = uex_1.x.array\\n\",\n \"error_max = np.max(np.abs(u_vertex_values - u_ex_vertex_values))\\n\",\n \"error_max = MPI.COMM_WORLD.allreduce(error_max, op=MPI.MAX)\\n\",\n \"print(f\\\"Error_L2 : {error_L2:.2e}\\\")\\n\",\n \"print(f\\\"Error_max : {error_max:.2e}\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Visualization\\n\",\n \"To look at the actual solution, run the script as a python script with `off_screen=True` or as a Jupyter notebook with `off_screen=False`\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"tags\": []\n },\n \"outputs\": [],\n \"source\": [\n \"pyvista_cells, cell_types, geometry = vtk_mesh(V)\\n\",\n \"grid = pyvista.UnstructuredGrid(pyvista_cells, cell_types, geometry)\\n\",\n \"grid.point_data[\\\"u\\\"] = uh.x.array\\n\",\n \"grid.set_active_scalars(\\\"u\\\")\\n\",\n \"\\n\",\n \"plotter = pyvista.Plotter()\\n\",\n \"plotter.add_text(\\\"uh\\\", position=\\\"upper_edge\\\", font_size=14, color=\\\"black\\\")\\n\",\n \"plotter.add_mesh(grid, show_edges=True)\\n\",\n \"plotter.view_xy()\\n\",\n \"\\n\",\n \"if not pyvista.OFF_SCREEN:\\n\",\n \" plotter.show()\\n\",\n \"else:\\n\",\n \" figure = plotter.screenshot(\\\"neumann_dirichlet.png\\\")\"\n ]\n }\n ],\n \"metadata\": {\n \"jupytext\": {\n \"formats\": \"ipynb,py:light\"\n },\n \"kernelspec\": {\n \"display_name\": \"Python 3 (ipykernel)\",\n \"language\": \"python\",\n \"name\": \"python3\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 3\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython3\",\n \"version\": \"3.10.12\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 4\n}\n" }, { "path": "chapter3/neumann_dirichlet_code.py", "content": "# ---\n# jupyter:\n# jupytext:\n# formats: ipynb,py:light\n# text_representation:\n# extension: .py\n# format_name: light\n# format_version: '1.5'\n# jupytext_version: 1.18.1\n# kernelspec:\n# display_name: Python 3 (ipykernel)\n# language: python\n# name: python3\n# ---\n\n# # Combining Dirichlet and Neumann conditions\n# Author: Jørgen S. Dokken\n#\n# Let's return to the Poisson problem from the [Fundamentals chapter](./../chapter1/fundamentals.md)\n# and see how to extend the mathematics and the implementation to handle Dirichlet condition\n# in combination with a Neumann condition.\n# The domain is still the unit square, but now we set the Dirichlet condition $u=u_D$ at the left and right sides,\n# while the Neumann condition\n#\n# $$\n# -\\frac{\\partial u}{\\partial n}=g\n# $$\n#\n# is applied to the remaining sides $y=0$ and $y=1$.\n#\n# ## The PDE problem\n# Let $\\Lambda_D$ and $\\Lambda_N$ denote parts of the boundary $\\partial \\Omega$\n# where the Dirichlet and Neumann conditions apply, respectively.\n# The complete boundary-value problem can be written as\n#\n# $$\n# -\\nabla^2 u =f \\qquad \\text{in } \\Omega,\n# $$\n# $$\n# u=u_D \\qquad\\text{on } \\Lambda_D,\n# $$\n# $$\n# -\\frac{\\partial u}{\\partial n}=g \\qquad \\text{on }\\Lambda_N\n# $$\n#\n# Again, we choose $u=1+x^2+2y^2$ as the exact solution and adjust $f, g,$ and $u_D$ accordingly\n#\n# $$\n# f(x,y)=-6,\n# $$\n# $$\n# g(x,y)=\\begin{cases}\n# 0, & y=0,\\\\\n# -4, & y=1,\n# \\end{cases}\n# $$\n# $$\n# u_D(x,y)=1+x^2+2y^2.\n# $$\n#\n# For the ease of programming, we define $g$ as a function over the whole domain $\\Omega$ such that\n# $g$ takes on the correct values at $y=0$ and $y=1$. One possible extension is\n#\n# $$\n# g(x,y)=-4y.\n# $$\n#\n# ## The variational formulation\n# The first task is to derive the variational formulation.\n# This time we cannot omit the boundary term arising from integration by parts,\n# because $v$ is only zero on $\\Lambda_D$. We have\n#\n# $$\n# -\\int_\\Omega (\\nabla^2u)v~\\mathrm{d} x =\n# \\int_\\Omega \\nabla u \\cdot \\nabla v ~\\mathrm{d} x - \\int_{\\partial\\Omega}\\frac{\\partial u}{\\partial n}v~\\mathrm{d}s,\n# $$\n#\n# and since $v=0$ on $\\Lambda_D$,\n#\n# $$\n# - \\int_{\\partial\\Omega}\\frac{\\partial u}{\\partial n}v~\\mathrm{d}s=\n# - \\int_{\\Lambda_N}\\frac{\\partial u}{\\partial n}v~\\mathrm{d}s =\\int_{\\Lambda_N} gv~\\mathrm{d}s,\n# $$\n#\n# by applying the boundary condition on $\\Lambda_N$.\n# The resulting weak from reads\n#\n# $$\n# \\int_\\Omega \\nabla u \\cdot \\nabla v~\\mathrm{d} x =\n# \\int_\\Omega fv~\\mathrm{d} x - \\int_{\\Lambda_N}gv~\\mathrm{d}s.\n# $$\n# Expressing this equation in the standard notation $a(u,v)=L(v)$ is straight-forward with\n#\n# $$\n# a(u,v) = \\int_{\\Omega} \\nabla u \\cdot \\nabla v ~\\mathrm{d} x,\\\\\n# $$\n# $$\n# L(v) = \\int_{\\Omega} fv ~\\mathrm{d} x - \\int_{\\Lambda_N} gv~\\mathrm{d} s.\n# $$\n#\n# ## Implementation\n# As in the previous example, we define our mesh,function space and bilinear form $a(u,v)$.\n\n# +\nfrom dolfinx import default_scalar_type\nfrom dolfinx.fem import (\n Constant,\n Function,\n functionspace,\n assemble_scalar,\n dirichletbc,\n form,\n locate_dofs_geometrical,\n)\nfrom dolfinx.fem.petsc import LinearProblem\nfrom dolfinx.mesh import create_unit_square\nfrom dolfinx.plot import vtk_mesh\n\nfrom mpi4py import MPI\nfrom ufl import SpatialCoordinate, TestFunction, TrialFunction, dot, ds, dx, grad\n\nimport numpy as np\nimport pyvista\n\nmesh = create_unit_square(MPI.COMM_WORLD, 10, 10)\nV = functionspace(mesh, (\"Lagrange\", 1))\nu = TrialFunction(V)\nv = TestFunction(V)\na = dot(grad(u), grad(v)) * dx\n# -\n\n# Now we get to the Neumann and Dirichlet boundary condition. As previously, we use a Python-function to define the boundary where we should have a Dirichlet condition. Then, with this function, we locate degrees of freedom that fulfill this condition.\n\n\n# +\ndef u_exact(x):\n return 1 + x[0] ** 2 + 2 * x[1] ** 2\n\n\ndef boundary_D(x):\n return np.logical_or(np.isclose(x[0], 0), np.isclose(x[0], 1))\n\n\ndofs_D = locate_dofs_geometrical(V, boundary_D)\nu_bc = Function(V)\nu_bc.interpolate(u_exact)\nbc = dirichletbc(u_bc, dofs_D)\n# -\n\n# The next step is to define the Neumann condition. We first define $g$ uses `UFL`s `SpatialCoordinate`-function, and then in turn create a boundary integration measure `ds`. As the test function $v$ is zero on the boundary integrals over the Dirichlet boundary disappears, and we can integrate `g*v*ds` over the entire boundary.\n\nx = SpatialCoordinate(mesh)\ng = -4 * x[1]\nf = Constant(mesh, default_scalar_type(-6))\nL = f * v * dx - g * v * ds\n\n# We can now assemble and solve the linear system of equations\n\n# +\nproblem = LinearProblem(\n a,\n L,\n bcs=[bc],\n petsc_options={\"ksp_type\": \"preonly\", \"pc_type\": \"lu\"},\n petsc_options_prefix=\"neumann_dirichlet_\",\n)\nuh = problem.solve()\n\nV2 = functionspace(mesh, (\"Lagrange\", 2))\nuex = Function(V2)\nuex.interpolate(u_exact)\nerror_L2 = assemble_scalar(form((uh - uex) ** 2 * dx))\nerror_L2 = np.sqrt(MPI.COMM_WORLD.allreduce(error_L2, op=MPI.SUM))\n\nu_vertex_values = uh.x.array\nuex_1 = Function(V)\nuex_1.interpolate(uex)\nu_ex_vertex_values = uex_1.x.array\nerror_max = np.max(np.abs(u_vertex_values - u_ex_vertex_values))\nerror_max = MPI.COMM_WORLD.allreduce(error_max, op=MPI.MAX)\nprint(f\"Error_L2 : {error_L2:.2e}\")\nprint(f\"Error_max : {error_max:.2e}\")\n# -\n\n# ## Visualization\n# To look at the actual solution, run the script as a python script with `off_screen=True` or as a Jupyter notebook with `off_screen=False`\n\n# +\npyvista_cells, cell_types, geometry = vtk_mesh(V)\ngrid = pyvista.UnstructuredGrid(pyvista_cells, cell_types, geometry)\ngrid.point_data[\"u\"] = uh.x.array\ngrid.set_active_scalars(\"u\")\n\nplotter = pyvista.Plotter()\nplotter.add_text(\"uh\", position=\"upper_edge\", font_size=14, color=\"black\")\nplotter.add_mesh(grid, show_edges=True)\nplotter.view_xy()\n\nif not pyvista.OFF_SCREEN:\n plotter.show()\nelse:\n figure = plotter.screenshot(\"neumann_dirichlet.png\")\n" }, { "path": "chapter3/robin_neumann_dirichlet.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Setting multiple Dirichlet, Neumann, and Robin conditions\\n\",\n \"Author: Hans Petter Langtangen and Anders Logg\\n\",\n \"\\n\",\n \"We consider the variable coefficient example from [the previous section](subdomains.ipynb). In this section we will cover how to apply a mixture of Dirichlet, Neumann and Robin type boundary conditions for this type of problem.\\n\",\n \"\\n\",\n \"We divide our boundary into three distinct sections:\\n\",\n \"- $\\\\Gamma_D$ for Dirichlet conditions:\\n\",\n \"$u=u_D^i \\\\text{ on } \\\\Gamma_D^i, \\\\dots$ where $\\\\Gamma_D=\\\\Gamma_D^0\\\\cup \\\\Gamma_D^1 \\\\cup \\\\dots$.\\n\",\n \"- $\\\\Gamma_N$ for Neumann conditions: $-\\\\kappa \\\\frac{\\\\partial u}{\\\\partial n}=g_j \\\\text{ on } \\\\Gamma_N^j$ where $\\\\Gamma_N=\\\\Gamma_N^0\\\\cup \\\\Gamma_N^1 \\\\cup \\\\dots$.\\n\",\n \"- $\\\\Gamma_R$ for Robin conditions: $-\\\\kappa \\\\frac{\\\\partial u}{\\\\partial n}=r(u-s)$\\n\",\n \"\\n\",\n \"where $r$ and $s$ are specified functions. The Robin condition is most often used to model heat transfer to the surroundings and arises naturally from Newton's cooling law.\\n\",\n \"In that case, $r$ is a heat transfer coefficient, and $s$ is the temperature of the surroundings.\\n\",\n \"Both can be space and time-dependent. The Robin conditions apply at some parts $\\\\Gamma_R^0,\\\\Gamma_R^1,\\\\dots$, of the boundary:\\n\",\n \"\\n\",\n \"$$\\n\",\n \" -\\\\kappa \\\\frac{\\\\partial u}{\\\\partial n}=r_k(u-s_k) \\\\text{ on } \\\\Gamma_R^k\\n\",\n \"$$\\n\",\n \"\\n\",\n \"\\n\",\n \"## The PDE problem and variational formulation\\n\",\n \"We can summarize the PDE problem as\\n\",\n \"\\n\",\n \"$$\\n\",\n \"-\\\\nabla (\\\\kappa \\\\nabla u) = f \\\\qquad \\\\text{in } \\\\Omega,\\n\",\n \"$$\\n\",\n \"$$\\n\",\n \"u=u_D^i \\\\qquad \\\\text{on } \\\\Gamma_D^i,\\n\",\n \"$$\\n\",\n \"$$\\n\",\n \"-\\\\kappa \\\\frac{\\\\partial u}{\\\\partial n}=g_j \\\\quad\\\\text{on } \\\\Gamma_N^j,\\n\",\n \"$$\\n\",\n \"$$\\n\",\n \"-\\\\kappa \\\\frac{\\\\partial u}{\\\\partial n}=r_k(u-s_k)\\\\quad \\\\text{ on } \\\\Gamma_R^k,\\n\",\n \"$$\\n\",\n \"\\n\",\n \"As usual, we multiply by a test function and integrate by parts.\\n\",\n \"\\n\",\n \"$$\\n\",\n \"-\\\\int_{\\\\Omega}\\\\nabla \\\\cdot (\\\\kappa \\\\nabla u)v ~\\\\mathrm{d} x = \\\\int_{\\\\Omega}\\\\kappa \\\\nabla u\\\\cdot \\\\nabla v~\\\\mathrm{d} x - \\\\int_{\\\\partial\\\\Omega}\\\\kappa \\\\frac{\\\\partial u}{\\\\partial n} v ~\\\\mathrm{d} s.\\n\",\n \"$$\\n\",\n \"\\n\",\n \"On the Dirichlet part ($\\\\Gamma_D^i$), the boundary integral vanishes since $v=0$. On the remaining part of the boundary, we split the boundary into contributions from the Neumann parts ($\\\\Gamma_N^i$) and Robin parts ($\\\\Gamma_R^i$).\\n\",\n \"Inserting the boundary conditions, we obtain\\n\",\n \"\\n\",\n \"$$\\n\",\n \"-\\\\int_{\\\\partial\\\\Omega}\\\\kappa\\\\frac{\\\\partial u }{\\\\partial n }v~\\\\mathrm{d} s=\\\\sum_i \\\\int_{\\\\Gamma_N^i} g_iv~\\\\mathrm{d} s + \\\\sum_i\\\\int_{\\\\Gamma_R^i}r_i(u-s_i)v~\\\\mathrm{d}s.\\n\",\n \"$$\\n\",\n \"\\n\",\n \"Thus we have the following variational problem\\n\",\n \"\\n\",\n \"$$\\n\",\n \"F(u, v)=\\\\int_\\\\Omega \\\\kappa \\\\nabla u \\\\cdot \\\\nabla v~\\\\mathrm{d} x + \\\\sum_i\\\\int_{\\\\Gamma_N^i}g_i v~\\\\mathrm{d}s +\\\\sum_i\\\\int_{\\\\Gamma_R^i}r_i(u-s_i)v~\\\\mathrm{d}s - \\\\int_\\\\Omega fv~\\\\mathrm{d} x = 0.\\n\",\n \"$$\\n\",\n \"\\n\",\n \"We have been used to writing the variational formulation as $a(u,v)=L(v)$, which requires that we identify the integrals dependent on the trial function $u$ and collect these in $a(u,v)$, while the remaining terms form $L(v)$. We note that the Robin condition has a contribution to both $a(u,v)$ and $L(v)$.\\n\",\n \"We then have\\n\",\n \"\\n\",\n \"$$\\n\",\n \"a(u,v)= \\\\int_{\\\\Omega} \\\\kappa \\\\nabla u \\\\cdot \\\\nabla v ~\\\\mathrm{d} x + \\\\sum_i \\\\int_{\\\\Gamma_R^i}r_i u v~\\\\mathrm{d} s,\\n\",\n \"$$\\n\",\n \"$$\\n\",\n \"L(v) = \\\\int_{\\\\Omega} fv~\\\\mathrm{d} x - \\\\sum_i \\\\int_{\\\\Gamma_N^i}g_i v~\\\\mathrm{d} s + \\\\sum_i \\\\int_{\\\\Gamma_R^i}r_i s_i v ~\\\\mathrm{d}s.\\n\",\n \"$$\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Implementation\\n\",\n \"Author: Jørgen S. Dokken\\n\",\n \"\\n\",\n \"We start by defining the domain $\\\\Omega$ as the unit square $[0,1]\\\\times[0,1]$.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"from dolfinx import default_scalar_type\\n\",\n \"from dolfinx.fem import (\\n\",\n \" Constant,\\n\",\n \" Function,\\n\",\n \" functionspace,\\n\",\n \" assemble_scalar,\\n\",\n \" dirichletbc,\\n\",\n \" form,\\n\",\n \" locate_dofs_topological,\\n\",\n \")\\n\",\n \"from dolfinx.fem.petsc import LinearProblem\\n\",\n \"from dolfinx.io import XDMFFile\\n\",\n \"from dolfinx.mesh import create_unit_square, locate_entities, meshtags\\n\",\n \"from dolfinx.plot import vtk_mesh\\n\",\n \"\\n\",\n \"from mpi4py import MPI\\n\",\n \"from ufl import (\\n\",\n \" FacetNormal,\\n\",\n \" Measure,\\n\",\n \" SpatialCoordinate,\\n\",\n \" TestFunction,\\n\",\n \" TrialFunction,\\n\",\n \" div,\\n\",\n \" dot,\\n\",\n \" dx,\\n\",\n \" grad,\\n\",\n \" inner,\\n\",\n \" lhs,\\n\",\n \" rhs,\\n\",\n \")\\n\",\n \"\\n\",\n \"import numpy as np\\n\",\n \"import pyvista\\n\",\n \"\\n\",\n \"mesh = create_unit_square(MPI.COMM_WORLD, 10, 10)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"In this section, we will solve the Poisson problem for the manufactured solution $u_{ex} = 1+x^2+2y^2$, which yields $\\\\kappa=1$, $f=-6$. The next step is to define the parameters of the boundary condition, and where we should apply them. In this example, we will apply the following\\n\",\n \"\\n\",\n \"$$\\n\",\n \"u = u_D \\\\qquad \\\\text{for } x=0,1\\n\",\n \"$$\\n\",\n \"$$\\n\",\n \"-\\\\kappa \\\\frac{\\\\partial u}{\\\\partial n} = r(u-s) \\\\quad \\\\text{for } y=0\\n\",\n \"$$\\n\",\n \"$$\\n\",\n \"-\\\\kappa \\\\frac{\\\\partial u}{\\\\partial n} =g_0 \\\\quad\\\\text{for } y = 1\\n\",\n \"$$\\n\",\n \"\\n\",\n \"To reproduce the analytical solution, we have that\\n\",\n \"\\n\",\n \"$$\\n\",\n \" u_D=u_{ex}=1+x^2+2y^2\\n\",\n \"$$\\n\",\n \"$$\\n\",\n \" g_0=-\\\\left.\\\\frac{\\\\partial u_{ex}}{\\\\partial y}\\\\right\\\\vert_{y=1}=-4y\\\\vert_{y=1}=-4\\n\",\n \"$$\\n\",\n \"\\n\",\n \"The Robin condition can be specified in many ways. As\\n\",\n \"$-\\\\left.\\\\frac{\\\\partial u_{ex}}{\\\\partial n}\\\\right\\\\vert_{y=0}=\\\\left.\\\\frac{\\\\partial u_{ex}}{\\\\partial y}\\\\right\\\\vert_{y=0}=4y=0,$\\n\",\n \"we can specify $r\\\\neq 0$ arbitrarily and $s=u_{ex}$. We choose $r=1000$.\\n\",\n \"We can now create all the necessary variable definitions and the traditional part of the variational form.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"def u_ex(x):\\n\",\n \" return 1 + x[0] ** 2 + 2 * x[1] ** 2\\n\",\n \"\\n\",\n \"\\n\",\n \"x = SpatialCoordinate(mesh)\\n\",\n \"# Define physical parameters and boundary condtions\\n\",\n \"s = u_ex(x)\\n\",\n \"f = -div(grad(u_ex(x)))\\n\",\n \"n = FacetNormal(mesh)\\n\",\n \"g = -dot(n, grad(u_ex(x)))\\n\",\n \"kappa = Constant(mesh, default_scalar_type(1))\\n\",\n \"r = Constant(mesh, default_scalar_type(1000))\\n\",\n \"# Define function space and standard part of variational form\\n\",\n \"V = functionspace(mesh, (\\\"Lagrange\\\", 1))\\n\",\n \"u, v = TrialFunction(V), TestFunction(V)\\n\",\n \"F = kappa * inner(grad(u), grad(v)) * dx - inner(f, v) * dx\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We start by identifying the facets contained in each boundary and create a custom integration measure `ds`.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"boundaries = [\\n\",\n \" (1, lambda x: np.isclose(x[0], 0)),\\n\",\n \" (2, lambda x: np.isclose(x[0], 1)),\\n\",\n \" (3, lambda x: np.isclose(x[1], 0)),\\n\",\n \" (4, lambda x: np.isclose(x[1], 1)),\\n\",\n \"]\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We now loop through all the boundary conditions and create `MeshTags` identifying the facets for each boundary condition.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"facet_indices, facet_markers = [], []\\n\",\n \"fdim = mesh.topology.dim - 1\\n\",\n \"for marker, locator in boundaries:\\n\",\n \" facets = locate_entities(mesh, fdim, locator)\\n\",\n \" facet_indices.append(facets)\\n\",\n \" facet_markers.append(np.full_like(facets, marker))\\n\",\n \"facet_indices = np.hstack(facet_indices).astype(np.int32)\\n\",\n \"facet_markers = np.hstack(facet_markers).astype(np.int32)\\n\",\n \"sorted_facets = np.argsort(facet_indices)\\n\",\n \"facet_tag = meshtags(\\n\",\n \" mesh, fdim, facet_indices[sorted_facets], facet_markers[sorted_facets]\\n\",\n \")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Debugging boundary condition\\n\",\n \"To debug boundary conditions, the easiest thing to do is to visualize the boundary in Paraview by writing the `MeshTags` to file. We can then inspect individual boundaries using the `Threshold`-filter.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"mesh.topology.create_connectivity(mesh.topology.dim - 1, mesh.topology.dim)\\n\",\n \"with XDMFFile(mesh.comm, \\\"facet_tags.xdmf\\\", \\\"w\\\") as xdmf:\\n\",\n \" xdmf.write_mesh(mesh)\\n\",\n \" xdmf.write_meshtags(facet_tag, mesh.geometry)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"Now we can create a custom integration measure `ds`, which can be used to restrict integration. If we integrate over `ds(1)`, we only integrate over facets marked with value 1 in the corresponding `facet_tag`.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"outputs\": [],\n \"source\": [\n \"ds = Measure(\\\"ds\\\", domain=mesh, subdomain_data=facet_tag)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"We can now create a general boundary condition class.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"class BoundaryCondition:\\n\",\n \" def __init__(self, type, marker, values):\\n\",\n \" self._type = type\\n\",\n \" if type == \\\"Dirichlet\\\":\\n\",\n \" u_D = Function(V)\\n\",\n \" u_D.interpolate(values)\\n\",\n \" facets = facet_tag.find(marker)\\n\",\n \" dofs = locate_dofs_topological(V, fdim, facets)\\n\",\n \" self._bc = dirichletbc(u_D, dofs)\\n\",\n \" elif type == \\\"Neumann\\\":\\n\",\n \" self._bc = inner(values, v) * ds(marker)\\n\",\n \" elif type == \\\"Robin\\\":\\n\",\n \" self._bc = values[0] * inner(u - values[1], v) * ds(marker)\\n\",\n \" else:\\n\",\n \" raise TypeError(\\\"Unknown boundary condition: {0:s}\\\".format(type))\\n\",\n \"\\n\",\n \" @property\\n\",\n \" def bc(self):\\n\",\n \" return self._bc\\n\",\n \"\\n\",\n \" @property\\n\",\n \" def type(self):\\n\",\n \" return self._type\\n\",\n \"\\n\",\n \"\\n\",\n \"# Define the Dirichlet condition\\n\",\n \"boundary_conditions = [\\n\",\n \" BoundaryCondition(\\\"Dirichlet\\\", 1, u_ex),\\n\",\n \" BoundaryCondition(\\\"Dirichlet\\\", 2, u_ex),\\n\",\n \" BoundaryCondition(\\\"Robin\\\", 3, (r, s)),\\n\",\n \" BoundaryCondition(\\\"Neumann\\\", 4, g),\\n\",\n \"]\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We can now loop through the boundary condition and append them to `L(v)` or the list of Dirichlet boundary conditions\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"bcs = []\\n\",\n \"for condition in boundary_conditions:\\n\",\n \" if condition.type == \\\"Dirichlet\\\":\\n\",\n \" bcs.append(condition.bc)\\n\",\n \" else:\\n\",\n \" F += condition.bc\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We can now create the bilinear form $a$ and linear form $L$ by using the `ufl`-functions `lhs` and `rhs`\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"# Solve linear variational problem\\n\",\n \"a = lhs(F)\\n\",\n \"L = rhs(F)\\n\",\n \"problem = LinearProblem(\\n\",\n \" a,\\n\",\n \" L,\\n\",\n \" bcs=bcs,\\n\",\n \" petsc_options={\\\"ksp_type\\\": \\\"preonly\\\", \\\"pc_type\\\": \\\"lu\\\"},\\n\",\n \" petsc_options_prefix=\\\"robin_neumann_dirichlet_\\\",\\n\",\n \")\\n\",\n \"uh = problem.solve()\\n\",\n \"\\n\",\n \"# Visualize solution\\n\",\n \"pyvista_cells, cell_types, geometry = vtk_mesh(V)\\n\",\n \"grid = pyvista.UnstructuredGrid(pyvista_cells, cell_types, geometry)\\n\",\n \"grid.point_data[\\\"u\\\"] = uh.x.array\\n\",\n \"grid.set_active_scalars(\\\"u\\\")\\n\",\n \"\\n\",\n \"plotter = pyvista.Plotter()\\n\",\n \"plotter.add_text(\\\"uh\\\", position=\\\"upper_edge\\\", font_size=14, color=\\\"black\\\")\\n\",\n \"plotter.add_mesh(grid, show_edges=True)\\n\",\n \"plotter.view_xy()\\n\",\n \"if not pyvista.OFF_SCREEN:\\n\",\n \" plotter.show()\\n\",\n \"else:\\n\",\n \" figure = plotter.screenshot(\\\"robin_neumann_dirichlet.png\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Verification\\n\",\n \"As for the previous problems, we compute the error of our computed solution and compare it with the analytical solution.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"# Compute L2 error and error at nodes\\n\",\n \"V_ex = functionspace(mesh, (\\\"Lagrange\\\", 2))\\n\",\n \"u_exact = Function(V_ex)\\n\",\n \"u_exact.interpolate(u_ex)\\n\",\n \"error_L2 = np.sqrt(\\n\",\n \" mesh.comm.allreduce(assemble_scalar(form((uh - u_exact) ** 2 * dx)), op=MPI.SUM)\\n\",\n \")\\n\",\n \"\\n\",\n \"u_vertex_values = uh.x.array\\n\",\n \"uex_1 = Function(V)\\n\",\n \"uex_1.interpolate(u_ex)\\n\",\n \"u_ex_vertex_values = uex_1.x.array\\n\",\n \"error_max = np.max(np.abs(u_vertex_values - u_ex_vertex_values))\\n\",\n \"error_max = mesh.comm.allreduce(error_max, op=MPI.MAX)\\n\",\n \"print(f\\\"Error_L2 : {error_L2:.2e}\\\")\\n\",\n \"print(f\\\"Error_max : {error_max:.2e}\\\")\"\n ]\n }\n ],\n \"metadata\": {\n \"jupytext\": {\n \"formats\": \"ipynb,py:light\"\n },\n \"kernelspec\": {\n \"display_name\": \"Python 3 (ipykernel)\",\n \"language\": \"python\",\n \"name\": \"python3\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 3\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython3\",\n \"version\": \"3.10.12\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 4\n}\n" }, { "path": "chapter3/robin_neumann_dirichlet.py", "content": "# ---\n# jupyter:\n# jupytext:\n# formats: ipynb,py:light\n# text_representation:\n# extension: .py\n# format_name: light\n# format_version: '1.5'\n# jupytext_version: 1.18.1\n# kernelspec:\n# display_name: Python 3 (ipykernel)\n# language: python\n# name: python3\n# ---\n\n# # Setting multiple Dirichlet, Neumann, and Robin conditions\n# Author: Hans Petter Langtangen and Anders Logg\n#\n# We consider the variable coefficient example from [the previous section](subdomains.ipynb). In this section we will cover how to apply a mixture of Dirichlet, Neumann and Robin type boundary conditions for this type of problem.\n#\n# We divide our boundary into three distinct sections:\n# - $\\Gamma_D$ for Dirichlet conditions:\n# $u=u_D^i \\text{ on } \\Gamma_D^i, \\dots$ where $\\Gamma_D=\\Gamma_D^0\\cup \\Gamma_D^1 \\cup \\dots$.\n# - $\\Gamma_N$ for Neumann conditions: $-\\kappa \\frac{\\partial u}{\\partial n}=g_j \\text{ on } \\Gamma_N^j$ where $\\Gamma_N=\\Gamma_N^0\\cup \\Gamma_N^1 \\cup \\dots$.\n# - $\\Gamma_R$ for Robin conditions: $-\\kappa \\frac{\\partial u}{\\partial n}=r(u-s)$\n#\n# where $r$ and $s$ are specified functions. The Robin condition is most often used to model heat transfer to the surroundings and arises naturally from Newton's cooling law.\n# In that case, $r$ is a heat transfer coefficient, and $s$ is the temperature of the surroundings.\n# Both can be space and time-dependent. The Robin conditions apply at some parts $\\Gamma_R^0,\\Gamma_R^1,\\dots$, of the boundary:\n#\n# $$\n# -\\kappa \\frac{\\partial u}{\\partial n}=r_k(u-s_k) \\text{ on } \\Gamma_R^k\n# $$\n#\n#\n# ## The PDE problem and variational formulation\n# We can summarize the PDE problem as\n#\n# $$\n# -\\nabla (\\kappa \\nabla u) = f \\qquad \\text{in } \\Omega,\n# $$\n# $$\n# u=u_D^i \\qquad \\text{on } \\Gamma_D^i,\n# $$\n# $$\n# -\\kappa \\frac{\\partial u}{\\partial n}=g_j \\quad\\text{on } \\Gamma_N^j,\n# $$\n# $$\n# -\\kappa \\frac{\\partial u}{\\partial n}=r_k(u-s_k)\\quad \\text{ on } \\Gamma_R^k,\n# $$\n#\n# As usual, we multiply by a test function and integrate by parts.\n#\n# $$\n# -\\int_{\\Omega}\\nabla \\cdot (\\kappa \\nabla u)v ~\\mathrm{d} x = \\int_{\\Omega}\\kappa \\nabla u\\cdot \\nabla v~\\mathrm{d} x - \\int_{\\partial\\Omega}\\kappa \\frac{\\partial u}{\\partial n} v ~\\mathrm{d} s.\n# $$\n#\n# On the Dirichlet part ($\\Gamma_D^i$), the boundary integral vanishes since $v=0$. On the remaining part of the boundary, we split the boundary into contributions from the Neumann parts ($\\Gamma_N^i$) and Robin parts ($\\Gamma_R^i$).\n# Inserting the boundary conditions, we obtain\n#\n# $$\n# -\\int_{\\partial\\Omega}\\kappa\\frac{\\partial u }{\\partial n }v~\\mathrm{d} s=\\sum_i \\int_{\\Gamma_N^i} g_iv~\\mathrm{d} s + \\sum_i\\int_{\\Gamma_R^i}r_i(u-s_i)v~\\mathrm{d}s.\n# $$\n#\n# Thus we have the following variational problem\n#\n# $$\n# F(u, v)=\\int_\\Omega \\kappa \\nabla u \\cdot \\nabla v~\\mathrm{d} x + \\sum_i\\int_{\\Gamma_N^i}g_i v~\\mathrm{d}s +\\sum_i\\int_{\\Gamma_R^i}r_i(u-s_i)v~\\mathrm{d}s - \\int_\\Omega fv~\\mathrm{d} x = 0.\n# $$\n#\n# We have been used to writing the variational formulation as $a(u,v)=L(v)$, which requires that we identify the integrals dependent on the trial function $u$ and collect these in $a(u,v)$, while the remaining terms form $L(v)$. We note that the Robin condition has a contribution to both $a(u,v)$ and $L(v)$.\n# We then have\n#\n# $$\n# a(u,v)= \\int_{\\Omega} \\kappa \\nabla u \\cdot \\nabla v ~\\mathrm{d} x + \\sum_i \\int_{\\Gamma_R^i}r_i u v~\\mathrm{d} s,\n# $$\n# $$\n# L(v) = \\int_{\\Omega} fv~\\mathrm{d} x - \\sum_i \\int_{\\Gamma_N^i}g_i v~\\mathrm{d} s + \\sum_i \\int_{\\Gamma_R^i}r_i s_i v ~\\mathrm{d}s.\n# $$\n\n# ## Implementation\n# Author: Jørgen S. Dokken\n#\n# We start by defining the domain $\\Omega$ as the unit square $[0,1]\\times[0,1]$.\n\n# +\nfrom dolfinx import default_scalar_type\nfrom dolfinx.fem import (\n Constant,\n Function,\n functionspace,\n assemble_scalar,\n dirichletbc,\n form,\n locate_dofs_topological,\n)\nfrom dolfinx.fem.petsc import LinearProblem\nfrom dolfinx.io import XDMFFile\nfrom dolfinx.mesh import create_unit_square, locate_entities, meshtags\nfrom dolfinx.plot import vtk_mesh\n\nfrom mpi4py import MPI\nfrom ufl import (\n FacetNormal,\n Measure,\n SpatialCoordinate,\n TestFunction,\n TrialFunction,\n div,\n dot,\n dx,\n grad,\n inner,\n lhs,\n rhs,\n)\n\nimport numpy as np\nimport pyvista\n\nmesh = create_unit_square(MPI.COMM_WORLD, 10, 10)\n# -\n\n# In this section, we will solve the Poisson problem for the manufactured solution $u_{ex} = 1+x^2+2y^2$, which yields $\\kappa=1$, $f=-6$. The next step is to define the parameters of the boundary condition, and where we should apply them. In this example, we will apply the following\n#\n# $$\n# u = u_D \\qquad \\text{for } x=0,1\n# $$\n# $$\n# -\\kappa \\frac{\\partial u}{\\partial n} = r(u-s) \\quad \\text{for } y=0\n# $$\n# $$\n# -\\kappa \\frac{\\partial u}{\\partial n} =g_0 \\quad\\text{for } y = 1\n# $$\n#\n# To reproduce the analytical solution, we have that\n#\n# $$\n# u_D=u_{ex}=1+x^2+2y^2\n# $$\n# $$\n# g_0=-\\left.\\frac{\\partial u_{ex}}{\\partial y}\\right\\vert_{y=1}=-4y\\vert_{y=1}=-4\n# $$\n#\n# The Robin condition can be specified in many ways. As\n# $-\\left.\\frac{\\partial u_{ex}}{\\partial n}\\right\\vert_{y=0}=\\left.\\frac{\\partial u_{ex}}{\\partial y}\\right\\vert_{y=0}=4y=0,$\n# we can specify $r\\neq 0$ arbitrarily and $s=u_{ex}$. We choose $r=1000$.\n# We can now create all the necessary variable definitions and the traditional part of the variational form.\n\n\n# +\ndef u_ex(x):\n return 1 + x[0] ** 2 + 2 * x[1] ** 2\n\n\nx = SpatialCoordinate(mesh)\n# Define physical parameters and boundary condtions\ns = u_ex(x)\nf = -div(grad(u_ex(x)))\nn = FacetNormal(mesh)\ng = -dot(n, grad(u_ex(x)))\nkappa = Constant(mesh, default_scalar_type(1))\nr = Constant(mesh, default_scalar_type(1000))\n# Define function space and standard part of variational form\nV = functionspace(mesh, (\"Lagrange\", 1))\nu, v = TrialFunction(V), TestFunction(V)\nF = kappa * inner(grad(u), grad(v)) * dx - inner(f, v) * dx\n# -\n\n# We start by identifying the facets contained in each boundary and create a custom integration measure `ds`.\n\nboundaries = [\n (1, lambda x: np.isclose(x[0], 0)),\n (2, lambda x: np.isclose(x[0], 1)),\n (3, lambda x: np.isclose(x[1], 0)),\n (4, lambda x: np.isclose(x[1], 1)),\n]\n\n# We now loop through all the boundary conditions and create `MeshTags` identifying the facets for each boundary condition.\n\nfacet_indices, facet_markers = [], []\nfdim = mesh.topology.dim - 1\nfor marker, locator in boundaries:\n facets = locate_entities(mesh, fdim, locator)\n facet_indices.append(facets)\n facet_markers.append(np.full_like(facets, marker))\nfacet_indices = np.hstack(facet_indices).astype(np.int32)\nfacet_markers = np.hstack(facet_markers).astype(np.int32)\nsorted_facets = np.argsort(facet_indices)\nfacet_tag = meshtags(\n mesh, fdim, facet_indices[sorted_facets], facet_markers[sorted_facets]\n)\n\n# ## Debugging boundary condition\n# To debug boundary conditions, the easiest thing to do is to visualize the boundary in Paraview by writing the `MeshTags` to file. We can then inspect individual boundaries using the `Threshold`-filter.\n\nmesh.topology.create_connectivity(mesh.topology.dim - 1, mesh.topology.dim)\nwith XDMFFile(mesh.comm, \"facet_tags.xdmf\", \"w\") as xdmf:\n xdmf.write_mesh(mesh)\n xdmf.write_meshtags(facet_tag, mesh.geometry)\n\n# Now we can create a custom integration measure `ds`, which can be used to restrict integration. If we integrate over `ds(1)`, we only integrate over facets marked with value 1 in the corresponding `facet_tag`.\n\nds = Measure(\"ds\", domain=mesh, subdomain_data=facet_tag)\n\n\n# We can now create a general boundary condition class.\n\n\n# +\nclass BoundaryCondition:\n def __init__(self, type, marker, values):\n self._type = type\n if type == \"Dirichlet\":\n u_D = Function(V)\n u_D.interpolate(values)\n facets = facet_tag.find(marker)\n dofs = locate_dofs_topological(V, fdim, facets)\n self._bc = dirichletbc(u_D, dofs)\n elif type == \"Neumann\":\n self._bc = inner(values, v) * ds(marker)\n elif type == \"Robin\":\n self._bc = values[0] * inner(u - values[1], v) * ds(marker)\n else:\n raise TypeError(\"Unknown boundary condition: {0:s}\".format(type))\n\n @property\n def bc(self):\n return self._bc\n\n @property\n def type(self):\n return self._type\n\n\n# Define the Dirichlet condition\nboundary_conditions = [\n BoundaryCondition(\"Dirichlet\", 1, u_ex),\n BoundaryCondition(\"Dirichlet\", 2, u_ex),\n BoundaryCondition(\"Robin\", 3, (r, s)),\n BoundaryCondition(\"Neumann\", 4, g),\n]\n# -\n\n# We can now loop through the boundary condition and append them to `L(v)` or the list of Dirichlet boundary conditions\n\nbcs = []\nfor condition in boundary_conditions:\n if condition.type == \"Dirichlet\":\n bcs.append(condition.bc)\n else:\n F += condition.bc\n\n# We can now create the bilinear form $a$ and linear form $L$ by using the `ufl`-functions `lhs` and `rhs`\n\n# +\n# Solve linear variational problem\na = lhs(F)\nL = rhs(F)\nproblem = LinearProblem(\n a,\n L,\n bcs=bcs,\n petsc_options={\"ksp_type\": \"preonly\", \"pc_type\": \"lu\"},\n petsc_options_prefix=\"robin_neumann_dirichlet_\",\n)\nuh = problem.solve()\n\n# Visualize solution\npyvista_cells, cell_types, geometry = vtk_mesh(V)\ngrid = pyvista.UnstructuredGrid(pyvista_cells, cell_types, geometry)\ngrid.point_data[\"u\"] = uh.x.array\ngrid.set_active_scalars(\"u\")\n\nplotter = pyvista.Plotter()\nplotter.add_text(\"uh\", position=\"upper_edge\", font_size=14, color=\"black\")\nplotter.add_mesh(grid, show_edges=True)\nplotter.view_xy()\nif not pyvista.OFF_SCREEN:\n plotter.show()\nelse:\n figure = plotter.screenshot(\"robin_neumann_dirichlet.png\")\n# -\n\n# ## Verification\n# As for the previous problems, we compute the error of our computed solution and compare it with the analytical solution.\n\n# +\n# Compute L2 error and error at nodes\nV_ex = functionspace(mesh, (\"Lagrange\", 2))\nu_exact = Function(V_ex)\nu_exact.interpolate(u_ex)\nerror_L2 = np.sqrt(\n mesh.comm.allreduce(assemble_scalar(form((uh - u_exact) ** 2 * dx)), op=MPI.SUM)\n)\n\nu_vertex_values = uh.x.array\nuex_1 = Function(V)\nuex_1.interpolate(u_ex)\nu_ex_vertex_values = uex_1.x.array\nerror_max = np.max(np.abs(u_vertex_values - u_ex_vertex_values))\nerror_max = mesh.comm.allreduce(error_max, op=MPI.MAX)\nprint(f\"Error_L2 : {error_L2:.2e}\")\nprint(f\"Error_max : {error_max:.2e}\")\n" }, { "path": "chapter3/subdomains.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Defining subdomains for different materials\\n\",\n \"\\n\",\n \"Author: Jørgen S. Dokken\\n\",\n \"\\n\",\n \"Solving PDEs in domains made up of different materials is a frequently encountered task. In FEniCSx, we handle these problems by defining a Discontinous cell-wise constant function.\\n\",\n \"Such a function can be created over any mesh in the following way\\n\",\n \"\\n\",\n \"## Subdomains on built-in meshes\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"lines_to_end_of_cell_marker\": 2\n },\n \"outputs\": [],\n \"source\": [\n \"from dolfinx import default_scalar_type\\n\",\n \"from dolfinx.fem import (\\n\",\n \" Constant,\\n\",\n \" dirichletbc,\\n\",\n \" Function,\\n\",\n \" functionspace,\\n\",\n \" assemble_scalar,\\n\",\n \" form,\\n\",\n \" locate_dofs_geometrical,\\n\",\n \" locate_dofs_topological,\\n\",\n \")\\n\",\n \"from dolfinx.fem.petsc import LinearProblem\\n\",\n \"from dolfinx.io import XDMFFile, gmsh as gmshio\\n\",\n \"from dolfinx.mesh import create_unit_square, locate_entities\\n\",\n \"from dolfinx.plot import vtk_mesh\\n\",\n \"\\n\",\n \"from ufl import SpatialCoordinate, TestFunction, TrialFunction, dx, grad, inner\\n\",\n \"\\n\",\n \"from mpi4py import MPI\\n\",\n \"\\n\",\n \"import meshio\\n\",\n \"import gmsh\\n\",\n \"import numpy as np\\n\",\n \"import pyvista\\n\",\n \"\\n\",\n \"mesh = create_unit_square(MPI.COMM_WORLD, 10, 10)\\n\",\n \"Q = functionspace(mesh, (\\\"DG\\\", 0))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"We will use a simple example with two materials in two dimensions to demonstrate the idea. The whole domain will be $\\\\Omega=[0,1]\\\\times[0,1]$, which consists of two subdomains\\n\",\n \"$\\\\Omega_0=[0,1]\\\\times [0,1/2]$ and $\\\\Omega_1=[0,1]\\\\times[1/2, 1]$. We start by creating two python functions, where each returns `True` if the input coordinate is inside its domain.\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"def Omega_0(x):\\n\",\n \" return x[1] <= 0.5\\n\",\n \"\\n\",\n \"\\n\",\n \"def Omega_1(x):\\n\",\n \" return x[1] >= 0.5\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"Note that both functions use a $\\\\leq$ or $\\\\geq$, as FEniCSx will evaluate each cell at all of the vertices, and thus has to return `True` for all vertices aligned with the interface to be marked properly.\\n\",\n \"\\n\",\n \"We will solve a variable-coefficient extension of the Poisson equation\\n\",\n \"\\n\",\n \"$$\\n\",\n \"-\\\\nabla \\\\cdot [\\\\kappa (x,y)\\\\nabla u(x, y)]= 1 \\\\qquad \\\\text{in } \\\\Omega,\\n\",\n \"$$\\n\",\n \"\\n\",\n \"$$\\n\",\n \"u=u_D=1 \\\\qquad \\\\text{on } \\\\partial\\\\Omega_D=[0,y], y\\\\in[0,1]\\n\",\n \"$$\\n\",\n \"\\n\",\n \"$$\\n\",\n \"-\\\\frac{\\\\partial u}{\\\\partial n}=0 \\\\qquad \\\\text{on } \\\\partial\\\\Omega\\\\setminus \\\\partial\\\\Omega_D\\n\",\n \"$$\\n\",\n \"\\n\",\n \"Our next step is to define $\\\\kappa$\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"kappa = Function(Q)\\n\",\n \"cells_0 = locate_entities(mesh, mesh.topology.dim, Omega_0)\\n\",\n \"cells_1 = locate_entities(mesh, mesh.topology.dim, Omega_1)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"In the previous code block, we found which cells (triangular elements) satisfy the condition for being in $\\\\Omega_0, \\\\Omega_1$. As the $DG-0$ function contains only one degree of freedom per cell, there is a one to one mapping between the cell indicies and the degrees of freedom. We let $\\\\kappa=\\\\begin{cases}\\n\",\n \"1 &\\\\text{if } x\\\\in\\\\Omega_0\\\\\\\\\\n\",\n \"0.1& \\\\text{if } x\\\\in\\\\Omega_1\\\\\\\\\\n\",\n \"\\\\end{cases}$\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"kappa.x.array[cells_0] = np.full_like(cells_0, 1, dtype=default_scalar_type)\\n\",\n \"kappa.x.array[cells_1] = np.full_like(cells_1, 0.1, dtype=default_scalar_type)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We are now ready to define our variational formulation and Dirichlet boundary condition after using integration by parts\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"V = functionspace(mesh, (\\\"Lagrange\\\", 1))\\n\",\n \"u, v = TrialFunction(V), TestFunction(V)\\n\",\n \"a = inner(kappa * grad(u), grad(v)) * dx\\n\",\n \"x = SpatialCoordinate(mesh)\\n\",\n \"L = Constant(mesh, default_scalar_type(1)) * v * dx\\n\",\n \"dofs = locate_dofs_geometrical(V, lambda x: np.isclose(x[0], 0))\\n\",\n \"bcs = [dirichletbc(default_scalar_type(1), dofs, V)]\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We can now solve and visualize the solution of the problem\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"tags\": []\n },\n \"outputs\": [],\n \"source\": [\n \"problem = LinearProblem(\\n\",\n \" a,\\n\",\n \" L,\\n\",\n \" bcs=bcs,\\n\",\n \" petsc_options={\\\"ksp_type\\\": \\\"preonly\\\", \\\"pc_type\\\": \\\"lu\\\"},\\n\",\n \" petsc_options_prefix=\\\"subdomains_structured_\\\",\\n\",\n \")\\n\",\n \"uh = problem.solve()\\n\",\n \"\\n\",\n \"# Filter out ghosted cells\\n\",\n \"tdim = mesh.topology.dim\\n\",\n \"num_cells_local = mesh.topology.index_map(tdim).size_local\\n\",\n \"marker = np.zeros(num_cells_local, dtype=np.int32)\\n\",\n \"cells_0 = cells_0[cells_0 < num_cells_local]\\n\",\n \"cells_1 = cells_1[cells_1 < num_cells_local]\\n\",\n \"marker[cells_0] = 1\\n\",\n \"marker[cells_1] = 2\\n\",\n \"mesh.topology.create_connectivity(tdim, tdim)\\n\",\n \"topology, cell_types, x = vtk_mesh(\\n\",\n \" mesh, tdim, np.arange(num_cells_local, dtype=np.int32)\\n\",\n \")\\n\",\n \"\\n\",\n \"p = pyvista.Plotter(window_size=[800, 800])\\n\",\n \"grid = pyvista.UnstructuredGrid(topology, cell_types, x)\\n\",\n \"grid.cell_data[\\\"Marker\\\"] = marker\\n\",\n \"grid.set_active_scalars(\\\"Marker\\\")\\n\",\n \"p.add_mesh(grid, show_edges=True)\\n\",\n \"if pyvista.OFF_SCREEN:\\n\",\n \" figure = p.screenshot(\\\"subdomains_structured.png\\\")\\n\",\n \"p.show()\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"outputs\": [],\n \"source\": [\n \"p2 = pyvista.Plotter(window_size=[800, 800])\\n\",\n \"grid_uh = pyvista.UnstructuredGrid(*vtk_mesh(V))\\n\",\n \"grid_uh.point_data[\\\"u\\\"] = uh.x.array.real\\n\",\n \"grid_uh.set_active_scalars(\\\"u\\\")\\n\",\n \"p2.add_mesh(grid_uh, show_edges=True)\\n\",\n \"if not pyvista.OFF_SCREEN:\\n\",\n \" p2.show()\\n\",\n \"else:\\n\",\n \" figure = p2.screenshot(\\\"subdomains_structured2.png\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We clearly observe different behavior in the two regions, which both have the same Dirichlet boundary condition on the left side, where $x=0$.\\n\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"## Interpolation with Python-function\\n\",\n \"\\n\",\n \"As we saw in the first approach, in many cases, we can use the geometrical coordinates to determine which coefficient we should use. Using the unstructured mesh from the previous example, we illustrate an alternative approach using interpolation:\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"def eval_kappa(x):\\n\",\n \" values = np.zeros(x.shape[1], dtype=default_scalar_type)\\n\",\n \" # Create a boolean array indicating which dofs (corresponding to cell centers)\\n\",\n \" # that are in each domain\\n\",\n \" top_coords = x[1] > 0.5\\n\",\n \" bottom_coords = x[1] < 0.5\\n\",\n \" values[top_coords] = np.full(sum(top_coords), 0.1)\\n\",\n \" values[bottom_coords] = np.full(sum(bottom_coords), 1)\\n\",\n \" return values\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"kappa2 = Function(Q)\\n\",\n \"kappa2.interpolate(eval_kappa)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We verify this by assembling the error between this new function and the old one\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"# Difference in kappa's\\n\",\n \"error = mesh.comm.allreduce(assemble_scalar(form((kappa - kappa2) ** 2 * dx)))\\n\",\n \"print(error)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Subdomains defined from external mesh data\\n\",\n \"\\n\",\n \"Let us now consider the same problem, but using GMSH to generate the mesh and subdomains. We will then in turn show how to use this data to generate discontinuous functions in DOLFINx.\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"gmsh.initialize()\\n\",\n \"proc = MPI.COMM_WORLD.rank\\n\",\n \"top_marker = 2\\n\",\n \"bottom_marker = 1\\n\",\n \"left_marker = 1\\n\",\n \"if proc == 0:\\n\",\n \" # We create one rectangle for each subdomain\\n\",\n \" gmsh.model.occ.addRectangle(0, 0, 0, 1, 0.5, tag=1)\\n\",\n \" gmsh.model.occ.addRectangle(0, 0.5, 0, 1, 0.5, tag=2)\\n\",\n \" # We fuse the two rectangles and keep the interface between them\\n\",\n \" gmsh.model.occ.fragment([(2, 1)], [(2, 2)])\\n\",\n \" gmsh.model.occ.synchronize()\\n\",\n \"\\n\",\n \" # Mark the top (2) and bottom (1) rectangle\\n\",\n \" top, bottom = None, None\\n\",\n \" for surface in gmsh.model.getEntities(dim=2):\\n\",\n \" com = gmsh.model.occ.getCenterOfMass(surface[0], surface[1])\\n\",\n \" if np.allclose(com, [0.5, 0.25, 0]):\\n\",\n \" bottom = surface[1]\\n\",\n \" else:\\n\",\n \" top = surface[1]\\n\",\n \" gmsh.model.addPhysicalGroup(2, [bottom], bottom_marker)\\n\",\n \" gmsh.model.addPhysicalGroup(2, [top], top_marker)\\n\",\n \" # Tag the left boundary\\n\",\n \" left = []\\n\",\n \" for line in gmsh.model.getEntities(dim=1):\\n\",\n \" com = gmsh.model.occ.getCenterOfMass(line[0], line[1])\\n\",\n \" if np.isclose(com[0], 0):\\n\",\n \" left.append(line[1])\\n\",\n \" gmsh.model.addPhysicalGroup(1, left, left_marker)\\n\",\n \" gmsh.model.mesh.generate(2)\\n\",\n \" gmsh.write(\\\"mesh.msh\\\")\\n\",\n \"gmsh.finalize()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Read in MSH files with DOLFINx\\n\",\n \"\\n\",\n \"You can read in MSH files with DOLFINx, which will read them in on a single process, and then distribute them over the available ranks in the MPI communicator.\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"mesh_data = gmshio.read_from_msh(\\\"mesh.msh\\\", MPI.COMM_WORLD, gdim=2)\\n\",\n \"mesh = mesh_data.mesh\\n\",\n \"assert mesh_data.cell_tags is not None\\n\",\n \"cell_markers = mesh_data.cell_tags\\n\",\n \"assert mesh_data.facet_tags is not None\\n\",\n \"facet_markers = mesh_data.facet_tags\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"## Convert msh-files to XDMF using meshio\\n\",\n \"\\n\",\n \"We will use `meshio` to read in the `msh` file, and convert it to a more suitable IO format. Meshio requires `h5py`, and can be installed on linux with the following commands:\\n\",\n \"\\n\",\n \"```{code}\\n\",\n \"export HDF5_MPI=\\\"ON\\\"\\n\",\n \"export CC=mpicc\\n\",\n \"export HDF5_DIR=\\\"/usr/lib/x86_64-linux-gnu/hdf5/mpich/\\\"\\n\",\n \"pip3 install --no-cache-dir --no-binary=h5py h5py meshio\\n\",\n \"```\\n\",\n \"\\n\",\n \"We start by creating a convenience function for extracting data for a single cell type, and creating a new `meshio.Mesh`.\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"def create_mesh(mesh, cell_type, prune_z=False):\\n\",\n \" cells = mesh.get_cells_type(cell_type)\\n\",\n \" cell_data = mesh.get_cell_data(\\\"gmsh:physical\\\", cell_type)\\n\",\n \" points = mesh.points[:, :2] if prune_z else mesh.points\\n\",\n \" out_mesh = meshio.Mesh(\\n\",\n \" points=points,\\n\",\n \" cells={cell_type: cells},\\n\",\n \" cell_data={\\\"name_to_read\\\": [cell_data.astype(np.int32)]},\\n\",\n \" )\\n\",\n \" return out_mesh\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"This function returns a meshio mesh, including physical markers for the given type. The `prune_z` argument is for cases where we want to use two dimensional meshes. The last coordinate in the mesh (as it is generated in a 3D space) has to be removed for DOLFINx to consider this as a two dimensional geometry.\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"if proc == 0:\\n\",\n \" # Read in mesh\\n\",\n \" msh = meshio.read(\\\"mesh.msh\\\")\\n\",\n \"\\n\",\n \" # Create and save one file for the mesh, and one file for the facets\\n\",\n \" triangle_mesh = create_mesh(msh, \\\"triangle\\\", prune_z=True)\\n\",\n \" line_mesh = create_mesh(msh, \\\"line\\\", prune_z=True)\\n\",\n \" meshio.write(\\\"mesh.xdmf\\\", triangle_mesh, compression=None)\\n\",\n \" meshio.write(\\\"mt.xdmf\\\", line_mesh, compression=None)\\n\",\n \"MPI.COMM_WORLD.barrier()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We have now written the mesh and the cell markers to one file, and the facet markers in a separate file. We can now read this data in DOLFINx using `XDMFFile.read_mesh` and `XDMFFile.read_meshtags`. The `dolfinx.MeshTags` stores the index of the entity, along with the value of the marker in two one dimensional arrays.\\n\",\n \"\\n\",\n \"Note that we have generated and written the mesh on only one processor. However, the `xdmf`-format supports parallel IO, and we can thus read the mesh in parallel.\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"with XDMFFile(MPI.COMM_WORLD, \\\"mesh.xdmf\\\", \\\"r\\\") as xdmf:\\n\",\n \" mesh = xdmf.read_mesh(name=\\\"Grid\\\")\\n\",\n \" ct = xdmf.read_meshtags(mesh, name=\\\"Grid\\\")\\n\",\n \"mesh.topology.create_connectivity(mesh.topology.dim, mesh.topology.dim - 1)\\n\",\n \"with XDMFFile(MPI.COMM_WORLD, \\\"mt.xdmf\\\", \\\"r\\\") as xdmf:\\n\",\n \" ft = xdmf.read_meshtags(mesh, name=\\\"Grid\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We have now read in the mesh and corresponding cell and facet data. We can now create our discontinuous function `kappa` as follows\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"Q = functionspace(mesh, (\\\"DG\\\", 0))\\n\",\n \"kappa = Function(Q)\\n\",\n \"bottom_cells = ct.find(bottom_marker)\\n\",\n \"kappa.x.array[bottom_cells] = np.full_like(bottom_cells, 1, dtype=default_scalar_type)\\n\",\n \"top_cells = ct.find(top_marker)\\n\",\n \"kappa.x.array[top_cells] = np.full_like(top_cells, 0.1, dtype=default_scalar_type)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We can also efficiently use the facet data `ft` to create the Dirichlet boundary condition\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"V = functionspace(mesh, (\\\"Lagrange\\\", 1))\\n\",\n \"u_bc = Function(V)\\n\",\n \"left_facets = ft.find(left_marker)\\n\",\n \"mesh.topology.create_connectivity(mesh.topology.dim - 1, mesh.topology.dim)\\n\",\n \"left_dofs = locate_dofs_topological(V, mesh.topology.dim - 1, left_facets)\\n\",\n \"bcs = [dirichletbc(default_scalar_type(1), left_dofs, V)]\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We can now solve the problem in a similar fashion as above\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"lines_to_next_cell\": 0\n },\n \"outputs\": [],\n \"source\": [\n \"u, v = TrialFunction(V), TestFunction(V)\\n\",\n \"a = inner(kappa * grad(u), grad(v)) * dx\\n\",\n \"x = SpatialCoordinate(mesh)\\n\",\n \"L = Constant(mesh, default_scalar_type(1)) * v * dx\\n\",\n \"\\n\",\n \"problem = LinearProblem(\\n\",\n \" a,\\n\",\n \" L,\\n\",\n \" bcs=bcs,\\n\",\n \" petsc_options={\\\"ksp_type\\\": \\\"preonly\\\", \\\"pc_type\\\": \\\"lu\\\"},\\n\",\n \" petsc_options_prefix=\\\"subdomains_unstructured_\\\",\\n\",\n \")\\n\",\n \"uh = problem.solve()\\n\",\n \"\\n\",\n \"# As the dolfinx.MeshTag contains a value for every cell in the\\n\",\n \"# geometry, we can attach it directly to the grid\\n\",\n \"\\n\",\n \"tdim = mesh.topology.dim\\n\",\n \"mesh.topology.create_connectivity(tdim, tdim)\\n\",\n \"topology, cell_types, x = vtk_mesh(mesh, tdim)\\n\",\n \"grid = pyvista.UnstructuredGrid(topology, cell_types, x)\\n\",\n \"num_local_cells = mesh.topology.index_map(tdim).size_local\\n\",\n \"grid.cell_data[\\\"Marker\\\"] = ct.values[ct.indices < num_local_cells]\\n\",\n \"grid.set_active_scalars(\\\"Marker\\\")\\n\",\n \"\\n\",\n \"p = pyvista.Plotter(window_size=[800, 800])\\n\",\n \"p.add_mesh(grid, show_edges=True)\\n\",\n \"if not pyvista.OFF_SCREEN:\\n\",\n \" p.show()\\n\",\n \"else:\\n\",\n \" figure = p.screenshot(\\\"subdomains_unstructured.png\\\")\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"grid_uh = pyvista.UnstructuredGrid(*vtk_mesh(V))\\n\",\n \"grid_uh.point_data[\\\"u\\\"] = uh.x.array.real\\n\",\n \"grid_uh.set_active_scalars(\\\"u\\\")\\n\",\n \"p2 = pyvista.Plotter(window_size=[800, 800])\\n\",\n \"p2.add_mesh(grid_uh, show_edges=True)\\n\",\n \"if not pyvista.OFF_SCREEN:\\n\",\n \" p2.show()\\n\",\n \"else:\\n\",\n \" p2.screenshot(\\\"unstructured_u.png\\\")\"\n ]\n }\n ],\n \"metadata\": {\n \"jupytext\": {\n \"formats\": \"ipynb,py:light\"\n },\n \"kernelspec\": {\n \"display_name\": \"Python 3 (ipykernel)\",\n \"language\": \"python\",\n \"name\": \"python3\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 3\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython3\",\n \"version\": \"3.10.12\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 4\n}\n" }, { "path": "chapter3/subdomains.py", "content": "# ---\n# jupyter:\n# jupytext:\n# formats: ipynb,py:light\n# text_representation:\n# extension: .py\n# format_name: light\n# format_version: '1.5'\n# jupytext_version: 1.18.1\n# kernelspec:\n# display_name: Python 3 (ipykernel)\n# language: python\n# name: python3\n# ---\n\n# # Defining subdomains for different materials\n#\n# Author: Jørgen S. Dokken\n#\n# Solving PDEs in domains made up of different materials is a frequently encountered task. In FEniCSx, we handle these problems by defining a Discontinous cell-wise constant function.\n# Such a function can be created over any mesh in the following way\n#\n# ## Subdomains on built-in meshes\n#\n\n# +\nfrom dolfinx import default_scalar_type\nfrom dolfinx.fem import (\n Constant,\n dirichletbc,\n Function,\n functionspace,\n assemble_scalar,\n form,\n locate_dofs_geometrical,\n locate_dofs_topological,\n)\nfrom dolfinx.fem.petsc import LinearProblem\nfrom dolfinx.io import XDMFFile, gmsh as gmshio\nfrom dolfinx.mesh import create_unit_square, locate_entities\nfrom dolfinx.plot import vtk_mesh\n\nfrom ufl import SpatialCoordinate, TestFunction, TrialFunction, dx, grad, inner\n\nfrom mpi4py import MPI\n\nimport meshio\nimport gmsh\nimport numpy as np\nimport pyvista\n\nmesh = create_unit_square(MPI.COMM_WORLD, 10, 10)\nQ = functionspace(mesh, (\"DG\", 0))\n\n\n# -\n\n# We will use a simple example with two materials in two dimensions to demonstrate the idea. The whole domain will be $\\Omega=[0,1]\\times[0,1]$, which consists of two subdomains\n# $\\Omega_0=[0,1]\\times [0,1/2]$ and $\\Omega_1=[0,1]\\times[1/2, 1]$. We start by creating two python functions, where each returns `True` if the input coordinate is inside its domain.\n#\n\n\n# +\ndef Omega_0(x):\n return x[1] <= 0.5\n\n\ndef Omega_1(x):\n return x[1] >= 0.5\n\n\n# -\n\n# Note that both functions use a $\\leq$ or $\\geq$, as FEniCSx will evaluate each cell at all of the vertices, and thus has to return `True` for all vertices aligned with the interface to be marked properly.\n#\n# We will solve a variable-coefficient extension of the Poisson equation\n#\n# $$\n# -\\nabla \\cdot [\\kappa (x,y)\\nabla u(x, y)]= 1 \\qquad \\text{in } \\Omega,\n# $$\n#\n# $$\n# u=u_D=1 \\qquad \\text{on } \\partial\\Omega_D=[0,y], y\\in[0,1]\n# $$\n#\n# $$\n# -\\frac{\\partial u}{\\partial n}=0 \\qquad \\text{on } \\partial\\Omega\\setminus \\partial\\Omega_D\n# $$\n#\n# Our next step is to define $\\kappa$\n#\n\nkappa = Function(Q)\ncells_0 = locate_entities(mesh, mesh.topology.dim, Omega_0)\ncells_1 = locate_entities(mesh, mesh.topology.dim, Omega_1)\n\n# In the previous code block, we found which cells (triangular elements) satisfy the condition for being in $\\Omega_0, \\Omega_1$. As the $DG-0$ function contains only one degree of freedom per cell, there is a one to one mapping between the cell indicies and the degrees of freedom. We let $\\kappa=\\begin{cases}\n# 1 &\\text{if } x\\in\\Omega_0\\\\\n# 0.1& \\text{if } x\\in\\Omega_1\\\\\n# \\end{cases}$\n#\n\nkappa.x.array[cells_0] = np.full_like(cells_0, 1, dtype=default_scalar_type)\nkappa.x.array[cells_1] = np.full_like(cells_1, 0.1, dtype=default_scalar_type)\n\n# We are now ready to define our variational formulation and Dirichlet boundary condition after using integration by parts\n#\n\nV = functionspace(mesh, (\"Lagrange\", 1))\nu, v = TrialFunction(V), TestFunction(V)\na = inner(kappa * grad(u), grad(v)) * dx\nx = SpatialCoordinate(mesh)\nL = Constant(mesh, default_scalar_type(1)) * v * dx\ndofs = locate_dofs_geometrical(V, lambda x: np.isclose(x[0], 0))\nbcs = [dirichletbc(default_scalar_type(1), dofs, V)]\n\n# We can now solve and visualize the solution of the problem\n#\n\n# +\nproblem = LinearProblem(\n a,\n L,\n bcs=bcs,\n petsc_options={\"ksp_type\": \"preonly\", \"pc_type\": \"lu\"},\n petsc_options_prefix=\"subdomains_structured_\",\n)\nuh = problem.solve()\n\n# Filter out ghosted cells\ntdim = mesh.topology.dim\nnum_cells_local = mesh.topology.index_map(tdim).size_local\nmarker = np.zeros(num_cells_local, dtype=np.int32)\ncells_0 = cells_0[cells_0 < num_cells_local]\ncells_1 = cells_1[cells_1 < num_cells_local]\nmarker[cells_0] = 1\nmarker[cells_1] = 2\nmesh.topology.create_connectivity(tdim, tdim)\ntopology, cell_types, x = vtk_mesh(\n mesh, tdim, np.arange(num_cells_local, dtype=np.int32)\n)\n\np = pyvista.Plotter(window_size=[800, 800])\ngrid = pyvista.UnstructuredGrid(topology, cell_types, x)\ngrid.cell_data[\"Marker\"] = marker\ngrid.set_active_scalars(\"Marker\")\np.add_mesh(grid, show_edges=True)\nif pyvista.OFF_SCREEN:\n figure = p.screenshot(\"subdomains_structured.png\")\np.show()\n# -\n\np2 = pyvista.Plotter(window_size=[800, 800])\ngrid_uh = pyvista.UnstructuredGrid(*vtk_mesh(V))\ngrid_uh.point_data[\"u\"] = uh.x.array.real\ngrid_uh.set_active_scalars(\"u\")\np2.add_mesh(grid_uh, show_edges=True)\nif not pyvista.OFF_SCREEN:\n p2.show()\nelse:\n figure = p2.screenshot(\"subdomains_structured2.png\")\n\n\n# We clearly observe different behavior in the two regions, which both have the same Dirichlet boundary condition on the left side, where $x=0$.\n#\n\n# ## Interpolation with Python-function\n#\n# As we saw in the first approach, in many cases, we can use the geometrical coordinates to determine which coefficient we should use. Using the unstructured mesh from the previous example, we illustrate an alternative approach using interpolation:\n#\n\n\ndef eval_kappa(x):\n values = np.zeros(x.shape[1], dtype=default_scalar_type)\n # Create a boolean array indicating which dofs (corresponding to cell centers)\n # that are in each domain\n top_coords = x[1] > 0.5\n bottom_coords = x[1] < 0.5\n values[top_coords] = np.full(sum(top_coords), 0.1)\n values[bottom_coords] = np.full(sum(bottom_coords), 1)\n return values\n\n\nkappa2 = Function(Q)\nkappa2.interpolate(eval_kappa)\n\n# We verify this by assembling the error between this new function and the old one\n#\n\n# Difference in kappa's\nerror = mesh.comm.allreduce(assemble_scalar(form((kappa - kappa2) ** 2 * dx)))\nprint(error)\n\n# ## Subdomains defined from external mesh data\n#\n# Let us now consider the same problem, but using GMSH to generate the mesh and subdomains. We will then in turn show how to use this data to generate discontinuous functions in DOLFINx.\n#\n\ngmsh.initialize()\nproc = MPI.COMM_WORLD.rank\ntop_marker = 2\nbottom_marker = 1\nleft_marker = 1\nif proc == 0:\n # We create one rectangle for each subdomain\n gmsh.model.occ.addRectangle(0, 0, 0, 1, 0.5, tag=1)\n gmsh.model.occ.addRectangle(0, 0.5, 0, 1, 0.5, tag=2)\n # We fuse the two rectangles and keep the interface between them\n gmsh.model.occ.fragment([(2, 1)], [(2, 2)])\n gmsh.model.occ.synchronize()\n\n # Mark the top (2) and bottom (1) rectangle\n top, bottom = None, None\n for surface in gmsh.model.getEntities(dim=2):\n com = gmsh.model.occ.getCenterOfMass(surface[0], surface[1])\n if np.allclose(com, [0.5, 0.25, 0]):\n bottom = surface[1]\n else:\n top = surface[1]\n gmsh.model.addPhysicalGroup(2, [bottom], bottom_marker)\n gmsh.model.addPhysicalGroup(2, [top], top_marker)\n # Tag the left boundary\n left = []\n for line in gmsh.model.getEntities(dim=1):\n com = gmsh.model.occ.getCenterOfMass(line[0], line[1])\n if np.isclose(com[0], 0):\n left.append(line[1])\n gmsh.model.addPhysicalGroup(1, left, left_marker)\n gmsh.model.mesh.generate(2)\n gmsh.write(\"mesh.msh\")\ngmsh.finalize()\n\n# ## Read in MSH files with DOLFINx\n#\n# You can read in MSH files with DOLFINx, which will read them in on a single process, and then distribute them over the available ranks in the MPI communicator.\n#\n\nmesh_data = gmshio.read_from_msh(\"mesh.msh\", MPI.COMM_WORLD, gdim=2)\nmesh = mesh_data.mesh\nassert mesh_data.cell_tags is not None\ncell_markers = mesh_data.cell_tags\nassert mesh_data.facet_tags is not None\nfacet_markers = mesh_data.facet_tags\n\n# ## Convert msh-files to XDMF using meshio\n#\n# We will use `meshio` to read in the `msh` file, and convert it to a more suitable IO format. Meshio requires `h5py`, and can be installed on linux with the following commands:\n#\n# ```{code}\n# export HDF5_MPI=\"ON\"\n# export CC=mpicc\n# export HDF5_DIR=\"/usr/lib/x86_64-linux-gnu/hdf5/mpich/\"\n# pip3 install --no-cache-dir --no-binary=h5py h5py meshio\n# ```\n#\n# We start by creating a convenience function for extracting data for a single cell type, and creating a new `meshio.Mesh`.\n#\n\n\ndef create_mesh(mesh, cell_type, prune_z=False):\n cells = mesh.get_cells_type(cell_type)\n cell_data = mesh.get_cell_data(\"gmsh:physical\", cell_type)\n points = mesh.points[:, :2] if prune_z else mesh.points\n out_mesh = meshio.Mesh(\n points=points,\n cells={cell_type: cells},\n cell_data={\"name_to_read\": [cell_data.astype(np.int32)]},\n )\n return out_mesh\n\n\n# This function returns a meshio mesh, including physical markers for the given type. The `prune_z` argument is for cases where we want to use two dimensional meshes. The last coordinate in the mesh (as it is generated in a 3D space) has to be removed for DOLFINx to consider this as a two dimensional geometry.\n#\n\nif proc == 0:\n # Read in mesh\n msh = meshio.read(\"mesh.msh\")\n\n # Create and save one file for the mesh, and one file for the facets\n triangle_mesh = create_mesh(msh, \"triangle\", prune_z=True)\n line_mesh = create_mesh(msh, \"line\", prune_z=True)\n meshio.write(\"mesh.xdmf\", triangle_mesh, compression=None)\n meshio.write(\"mt.xdmf\", line_mesh, compression=None)\nMPI.COMM_WORLD.barrier()\n\n# We have now written the mesh and the cell markers to one file, and the facet markers in a separate file. We can now read this data in DOLFINx using `XDMFFile.read_mesh` and `XDMFFile.read_meshtags`. The `dolfinx.MeshTags` stores the index of the entity, along with the value of the marker in two one dimensional arrays.\n#\n# Note that we have generated and written the mesh on only one processor. However, the `xdmf`-format supports parallel IO, and we can thus read the mesh in parallel.\n#\n\nwith XDMFFile(MPI.COMM_WORLD, \"mesh.xdmf\", \"r\") as xdmf:\n mesh = xdmf.read_mesh(name=\"Grid\")\n ct = xdmf.read_meshtags(mesh, name=\"Grid\")\nmesh.topology.create_connectivity(mesh.topology.dim, mesh.topology.dim - 1)\nwith XDMFFile(MPI.COMM_WORLD, \"mt.xdmf\", \"r\") as xdmf:\n ft = xdmf.read_meshtags(mesh, name=\"Grid\")\n\n# We have now read in the mesh and corresponding cell and facet data. We can now create our discontinuous function `kappa` as follows\n#\n\nQ = functionspace(mesh, (\"DG\", 0))\nkappa = Function(Q)\nbottom_cells = ct.find(bottom_marker)\nkappa.x.array[bottom_cells] = np.full_like(bottom_cells, 1, dtype=default_scalar_type)\ntop_cells = ct.find(top_marker)\nkappa.x.array[top_cells] = np.full_like(top_cells, 0.1, dtype=default_scalar_type)\n\n# We can also efficiently use the facet data `ft` to create the Dirichlet boundary condition\n#\n\nV = functionspace(mesh, (\"Lagrange\", 1))\nu_bc = Function(V)\nleft_facets = ft.find(left_marker)\nmesh.topology.create_connectivity(mesh.topology.dim - 1, mesh.topology.dim)\nleft_dofs = locate_dofs_topological(V, mesh.topology.dim - 1, left_facets)\nbcs = [dirichletbc(default_scalar_type(1), left_dofs, V)]\n\n# We can now solve the problem in a similar fashion as above\n#\n\n# +\nu, v = TrialFunction(V), TestFunction(V)\na = inner(kappa * grad(u), grad(v)) * dx\nx = SpatialCoordinate(mesh)\nL = Constant(mesh, default_scalar_type(1)) * v * dx\n\nproblem = LinearProblem(\n a,\n L,\n bcs=bcs,\n petsc_options={\"ksp_type\": \"preonly\", \"pc_type\": \"lu\"},\n petsc_options_prefix=\"subdomains_unstructured_\",\n)\nuh = problem.solve()\n\n# As the dolfinx.MeshTag contains a value for every cell in the\n# geometry, we can attach it directly to the grid\n\ntdim = mesh.topology.dim\nmesh.topology.create_connectivity(tdim, tdim)\ntopology, cell_types, x = vtk_mesh(mesh, tdim)\ngrid = pyvista.UnstructuredGrid(topology, cell_types, x)\nnum_local_cells = mesh.topology.index_map(tdim).size_local\ngrid.cell_data[\"Marker\"] = ct.values[ct.indices < num_local_cells]\ngrid.set_active_scalars(\"Marker\")\n\np = pyvista.Plotter(window_size=[800, 800])\np.add_mesh(grid, show_edges=True)\nif not pyvista.OFF_SCREEN:\n p.show()\nelse:\n figure = p.screenshot(\"subdomains_unstructured.png\")\n# -\ngrid_uh = pyvista.UnstructuredGrid(*vtk_mesh(V))\ngrid_uh.point_data[\"u\"] = uh.x.array.real\ngrid_uh.set_active_scalars(\"u\")\np2 = pyvista.Plotter(window_size=[800, 800])\np2.add_mesh(grid_uh, show_edges=True)\nif not pyvista.OFF_SCREEN:\n p2.show()\nelse:\n p2.screenshot(\"unstructured_u.png\")\n" }, { "path": "chapter3/wire.ipe", "content": "\n\n\n\n\n\n\n0 0 m\n-1 0.333 l\n-1 -0.333 l\nh\n\n\n\n\n0 0 m\n-1 0.333 l\n-1 -0.333 l\nh\n\n\n\n\n0 0 m\n-1 0.333 l\n-0.8 0 l\n-1 -0.333 l\nh\n\n\n\n\n0 0 m\n-1 0.333 l\n-0.8 0 l\n-1 -0.333 l\nh\n\n\n\n\n0.6 0 0 0.6 0 0 e\n0.4 0 0 0.4 0 0 e\n\n\n\n\n0.6 0 0 0.6 0 0 e\n\n\n\n\n\n0.5 0 0 0.5 0 0 e\n\n\n0.6 0 0 0.6 0 0 e\n0.4 0 0 0.4 0 0 e\n\n\n\n\n\n-0.6 -0.6 m\n0.6 -0.6 l\n0.6 0.6 l\n-0.6 0.6 l\nh\n-0.4 -0.4 m\n0.4 -0.4 l\n0.4 0.4 l\n-0.4 0.4 l\nh\n\n\n\n\n-0.6 -0.6 m\n0.6 -0.6 l\n0.6 0.6 l\n-0.6 0.6 l\nh\n\n\n\n\n\n-0.5 -0.5 m\n0.5 -0.5 l\n0.5 0.5 l\n-0.5 0.5 l\nh\n\n\n-0.6 -0.6 m\n0.6 -0.6 l\n0.6 0.6 l\n-0.6 0.6 l\nh\n-0.4 -0.4 m\n0.4 -0.4 l\n0.4 0.4 l\n-0.4 0.4 l\nh\n\n\n\n\n\n\n-0.43 -0.57 m\n0.57 0.43 l\n0.43 0.57 l\n-0.57 -0.43 l\nh\n\n\n-0.43 0.57 m\n0.57 -0.43 l\n0.43 -0.57 l\n-0.57 0.43 l\nh\n\n\n\n\n\n0 0 m\n-1 0.333 l\n-1 -0.333 l\nh\n\n\n\n\n0 0 m\n-1 0.333 l\n-0.8 0 l\n-1 -0.333 l\nh\n\n\n\n\n0 0 m\n-1 0.333 l\n-0.8 0 l\n-1 -0.333 l\nh\n\n\n\n\n-1 0.333 m\n0 0 l\n-1 -0.333 l\n\n\n\n\n0 0 m\n-1 0.333 l\n-1 -0.333 l\nh\n-1 0 m\n-2 0.333 l\n-2 -0.333 l\nh\n\n\n\n\n0 0 m\n-1 0.333 l\n-1 -0.333 l\nh\n-1 0 m\n-2 0.333 l\n-2 -0.333 l\nh\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n102.45 0 0 102.45 304 576 e\n124.258 0 0 124.258 304 576 e\n\n\n12 0 0 12 304 720 e\n\n\n12 0 0 12 304 720 e\n\n\n12 0 0 12 304 720 e\n\n\n12 0 0 12 304 720 e\n\n\n12 0 0 12 304 720 e\n\n\n12 0 0 12 304 720 e\n\n\n12 0 0 12 304 720 e\n\n\n12 0 0 12 304 720 e\n\n\n12 0 0 12 304 720 e\n\n\n12 0 0 12 304 720 e\n\n\n12 0 0 12 304 720 e\n\n\n12 0 0 12 304 720 e\n\n\n12 0 0 12 304 720 e\n\n\n12 0 0 12 304 720 e\n\n\n12 0 0 12 304 720 e\n\n\n12 0 0 12 304 720 e\n\n\n\n" }, { "path": "chapter4/compiler_parameters.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"markdown\",\n \"id\": \"0\",\n \"metadata\": {},\n \"source\": [\n \"# JIT options and visualization using Pandas\\n\",\n \"Author: Jørgen S. Dokken\\n\",\n \"\\n\",\n \"In this chapter, we will explore how to optimize and inspect the integration kernels used in DOLFINx.\\n\",\n \"As we have seen in the previous demos, DOLFINx uses the [Unified form language](https://github.com/FEniCS/ufl/) to describe variational problems.\\n\",\n \"\\n\",\n \"These descriptions have to be translated into code for assembling the right and left hand side of the discrete variational problem.\\n\",\n \"\\n\",\n \"DOLFINx uses [ffcx](https://github.com/FEniCS/ffcx/) to generate efficient C code assembling the element matrices.\\n\",\n \"This C code is in turn compiled using [CFFI](https://cffi.readthedocs.io/en/latest/), and we can specify a variety of compile options.\\n\",\n \"\\n\",\n \"We start by specifying the current directory as the location to place the generated C files, we obtain the current directory using pathlib\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"1\",\n \"metadata\": {\n \"lines_to_end_of_cell_marker\": 2\n },\n \"outputs\": [],\n \"source\": [\n \"import pandas as pd\\n\",\n \"import seaborn\\n\",\n \"import time\\n\",\n \"\\n\",\n \"from ufl import TestFunction, TrialFunction, dx, inner\\n\",\n \"from dolfinx.mesh import create_unit_cube\\n\",\n \"from dolfinx.fem.petsc import assemble_matrix\\n\",\n \"from dolfinx.fem import functionspace, form\\n\",\n \"\\n\",\n \"from mpi4py import MPI\\n\",\n \"from pathlib import Path\\n\",\n \"from typing import Dict\\n\",\n \"\\n\",\n \"cache_dir = f\\\"{str(Path.cwd())}/.cache\\\"\\n\",\n \"print(f\\\"Directory to put C files in: {cache_dir}\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"2\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"Next we generate a general function to assemble the mass matrix for a unit cube. Note that we use `dolfinx.fem.form` to compile the variational form.\\n\",\n \"For codes using `dolfinx.fem.petsc.LinearProblem`, you can supply `jit_options` as a keyword argument.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"3\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"def compile_form(space: str, degree: int, jit_options: Dict):\\n\",\n \" N = 10\\n\",\n \" mesh = create_unit_cube(MPI.COMM_WORLD, N, N, N)\\n\",\n \" V = functionspace(mesh, (space, degree))\\n\",\n \" u = TrialFunction(V)\\n\",\n \" v = TestFunction(V)\\n\",\n \" a = inner(u, v) * dx\\n\",\n \" a_compiled = form(a, jit_options=jit_options)\\n\",\n \" start = time.perf_counter()\\n\",\n \" assemble_matrix(a_compiled)\\n\",\n \" end = time.perf_counter()\\n\",\n \" return end - start\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"4\",\n \"metadata\": {},\n \"source\": [\n \"We start by considering the different levels of optimization that the C compiler can use on the optimized code.\\n\",\n \"A list of optimization options and explanations can be found [here](https://gcc.gnu.org/onlinedocs/gcc/Optimize-Options.html)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"5\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"optimization_options = [\\\"-O1\\\", \\\"-O2\\\", \\\"-O3\\\", \\\"-Ofast\\\"]\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"6\",\n \"metadata\": {},\n \"source\": [\n \"The next option we can choose is if we want to compile the code with `-march=native` or not.\\n\",\n \"This option enables instructions for the local machine, and can give different results on different systems.\\n\",\n \"More information can be found [here](https://gcc.gnu.org/onlinedocs/gcc/AArch64-Options.html#g_t-march-and--mcpu-Feature-Modifiers)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"7\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"march_native = [True, False]\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"8\",\n \"metadata\": {},\n \"source\": [\n \"We choose a subset of finite element spaces, varying the order of the space to look at the effects it has on the assembly time with different compile options.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"9\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"results = {\\\"Space\\\": [], \\\"Degree\\\": [], \\\"Options\\\": [], \\\"Time\\\": []}\\n\",\n \"for space in [\\\"N1curl\\\", \\\"Lagrange\\\", \\\"RT\\\"]:\\n\",\n \" for degree in [1, 2, 3]:\\n\",\n \" for native in march_native:\\n\",\n \" for option in optimization_options:\\n\",\n \" if native:\\n\",\n \" cffi_options = [option, \\\"-march=native\\\"]\\n\",\n \" else:\\n\",\n \" cffi_options = [option]\\n\",\n \" jit_options = {\\n\",\n \" \\\"cffi_extra_compile_args\\\": cffi_options,\\n\",\n \" \\\"cache_dir\\\": cache_dir,\\n\",\n \" \\\"cffi_libraries\\\": [\\\"m\\\"],\\n\",\n \" }\\n\",\n \" runtime = compile_form(space, degree, jit_options=jit_options)\\n\",\n \" results[\\\"Space\\\"].append(space)\\n\",\n \" results[\\\"Degree\\\"].append(str(degree))\\n\",\n \" results[\\\"Options\\\"].append(\\\"\\\\n\\\".join(cffi_options))\\n\",\n \" results[\\\"Time\\\"].append(runtime)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"10\",\n \"metadata\": {},\n \"source\": [\n \"We have now stored all the results to a dictionary. To visualize it, we use pandas and its Dataframe class.\\n\",\n \"To instpect the data in a Jupyter notebook, call the code below.\\n\",\n \"If you are running this code in a script, you can use `print(results_df)` instead.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"11\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"results_df = pd.DataFrame.from_dict(results)\\n\",\n \"results_df\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"12\",\n \"metadata\": {},\n \"source\": [\n \"Next, we inspect the impact of the compiler option on each type of finite element family.\\n\",\n \"To achieve this, we add an extra column to the dataframe, which combines the space and degree of the finite element.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"13\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"seaborn.set(style=\\\"ticks\\\")\\n\",\n \"seaborn.set(font_scale=1.2)\\n\",\n \"seaborn.set_style(\\\"darkgrid\\\")\\n\",\n \"results_df[\\\"Element\\\"] = results_df[\\\"Space\\\"] + \\\" \\\" + results_df[\\\"Degree\\\"]\\n\",\n \"elements = sorted(set(results_df[\\\"Element\\\"]))\\n\",\n \"for element in elements:\\n\",\n \" df_e = results_df[results_df[\\\"Element\\\"] == element]\\n\",\n \" g = seaborn.catplot(x=\\\"Options\\\", y=\\\"Time\\\", kind=\\\"bar\\\", data=df_e, col=\\\"Element\\\")\\n\",\n \" g.fig.set_size_inches(16, 4)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"14\",\n \"metadata\": {},\n \"source\": [\n \"We observe that the compile time increases when increasing the degree of the function space,\\n\",\n \"and that we get most speedup by using \\\"-O3\\\" or \\\"-Ofast\\\" combined with \\\"-march=native\\\".\"\n ]\n }\n ],\n \"metadata\": {\n \"jupytext\": {\n \"formats\": \"ipynb,py:light\"\n },\n \"kernelspec\": {\n \"display_name\": \"Python 3 (ipykernel)\",\n \"language\": \"python\",\n \"name\": \"python3\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 3\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython3\",\n \"version\": \"3.10.12\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 5\n}\n" }, { "path": "chapter4/compiler_parameters.py", "content": "# ---\n# jupyter:\n# jupytext:\n# formats: ipynb,py:light\n# text_representation:\n# extension: .py\n# format_name: light\n# format_version: '1.5'\n# jupytext_version: 1.18.1\n# kernelspec:\n# display_name: Python 3 (ipykernel)\n# language: python\n# name: python3\n# ---\n\n# # JIT options and visualization using Pandas\n# Author: Jørgen S. Dokken\n#\n# In this chapter, we will explore how to optimize and inspect the integration kernels used in DOLFINx.\n# As we have seen in the previous demos, DOLFINx uses the [Unified form language](https://github.com/FEniCS/ufl/) to describe variational problems.\n#\n# These descriptions have to be translated into code for assembling the right and left hand side of the discrete variational problem.\n#\n# DOLFINx uses [ffcx](https://github.com/FEniCS/ffcx/) to generate efficient C code assembling the element matrices.\n# This C code is in turn compiled using [CFFI](https://cffi.readthedocs.io/en/latest/), and we can specify a variety of compile options.\n#\n# We start by specifying the current directory as the location to place the generated C files, we obtain the current directory using pathlib\n\n# +\nimport pandas as pd\nimport seaborn\nimport time\n\nfrom ufl import TestFunction, TrialFunction, dx, inner\nfrom dolfinx.mesh import create_unit_cube\nfrom dolfinx.fem.petsc import assemble_matrix\nfrom dolfinx.fem import functionspace, form\n\nfrom mpi4py import MPI\nfrom pathlib import Path\nfrom typing import Dict\n\ncache_dir = f\"{str(Path.cwd())}/.cache\"\nprint(f\"Directory to put C files in: {cache_dir}\")\n\n\n# -\n\n# Next we generate a general function to assemble the mass matrix for a unit cube. Note that we use `dolfinx.fem.form` to compile the variational form.\n# For codes using `dolfinx.fem.petsc.LinearProblem`, you can supply `jit_options` as a keyword argument.\n\n\ndef compile_form(space: str, degree: int, jit_options: Dict):\n N = 10\n mesh = create_unit_cube(MPI.COMM_WORLD, N, N, N)\n V = functionspace(mesh, (space, degree))\n u = TrialFunction(V)\n v = TestFunction(V)\n a = inner(u, v) * dx\n a_compiled = form(a, jit_options=jit_options)\n start = time.perf_counter()\n assemble_matrix(a_compiled)\n end = time.perf_counter()\n return end - start\n\n\n# We start by considering the different levels of optimization that the C compiler can use on the optimized code.\n# A list of optimization options and explanations can be found [here](https://gcc.gnu.org/onlinedocs/gcc/Optimize-Options.html)\n\noptimization_options = [\"-O1\", \"-O2\", \"-O3\", \"-Ofast\"]\n\n# The next option we can choose is if we want to compile the code with `-march=native` or not.\n# This option enables instructions for the local machine, and can give different results on different systems.\n# More information can be found [here](https://gcc.gnu.org/onlinedocs/gcc/AArch64-Options.html#g_t-march-and--mcpu-Feature-Modifiers)\n\nmarch_native = [True, False]\n\n# We choose a subset of finite element spaces, varying the order of the space to look at the effects it has on the assembly time with different compile options.\n\nresults = {\"Space\": [], \"Degree\": [], \"Options\": [], \"Time\": []}\nfor space in [\"N1curl\", \"Lagrange\", \"RT\"]:\n for degree in [1, 2, 3]:\n for native in march_native:\n for option in optimization_options:\n if native:\n cffi_options = [option, \"-march=native\"]\n else:\n cffi_options = [option]\n jit_options = {\n \"cffi_extra_compile_args\": cffi_options,\n \"cache_dir\": cache_dir,\n \"cffi_libraries\": [\"m\"],\n }\n runtime = compile_form(space, degree, jit_options=jit_options)\n results[\"Space\"].append(space)\n results[\"Degree\"].append(str(degree))\n results[\"Options\"].append(\"\\n\".join(cffi_options))\n results[\"Time\"].append(runtime)\n\n# We have now stored all the results to a dictionary. To visualize it, we use pandas and its Dataframe class.\n# To instpect the data in a Jupyter notebook, call the code below.\n# If you are running this code in a script, you can use `print(results_df)` instead.\n\nresults_df = pd.DataFrame.from_dict(results)\nresults_df\n\n# Next, we inspect the impact of the compiler option on each type of finite element family.\n# To achieve this, we add an extra column to the dataframe, which combines the space and degree of the finite element.\n\nseaborn.set(style=\"ticks\")\nseaborn.set(font_scale=1.2)\nseaborn.set_style(\"darkgrid\")\nresults_df[\"Element\"] = results_df[\"Space\"] + \" \" + results_df[\"Degree\"]\nelements = sorted(set(results_df[\"Element\"]))\nfor element in elements:\n df_e = results_df[results_df[\"Element\"] == element]\n g = seaborn.catplot(x=\"Options\", y=\"Time\", kind=\"bar\", data=df_e, col=\"Element\")\n g.fig.set_size_inches(16, 4)\n\n# We observe that the compile time increases when increasing the degree of the function space,\n# and that we get most speedup by using \"-O3\" or \"-Ofast\" combined with \"-march=native\".\n" }, { "path": "chapter4/convergence.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Error control: Computing convergence rates\\n\",\n \"Author: Jørgen S. Dokken, Hans Petter Langtangen, Anders Logg\\n\",\n \"\\n\",\n \"For any numerical method one of the most central questions is its *convergence rate*: How fast does the error go to zero when the resolution is increased (mesh size decreased).\\n\",\n \"\\n\",\n \"For the finite element method, this usually corresponds to proving, theoretically or imperically, that the error $e=u_e-u_h$ is bounded by the mesh size $h$ to some power $r$, that is $\\\\vert\\\\vert e \\\\vert\\\\vert\\\\leq Ch^r$ for some mesh independent constant $C$. The number $r$ is called the *convergence rate* of the method. Note that the different norms like the $L^2$-norm $\\\\vert\\\\vert e\\\\vert\\\\vert$ or the $H_0^1$-norm have different convergence rates.\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Computing error norms\\n\",\n \"We start by creating a manufactured problem, using the same problem as in [the solver configuration](./solvers.ipynb).\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"from dolfinx import default_scalar_type\\n\",\n \"from dolfinx.fem import (\\n\",\n \" Expression,\\n\",\n \" Function,\\n\",\n \" functionspace,\\n\",\n \" assemble_scalar,\\n\",\n \" dirichletbc,\\n\",\n \" form,\\n\",\n \" locate_dofs_topological,\\n\",\n \")\\n\",\n \"from dolfinx.fem.petsc import LinearProblem\\n\",\n \"from dolfinx.mesh import create_unit_square, locate_entities_boundary\\n\",\n \"\\n\",\n \"from mpi4py import MPI\\n\",\n \"from ufl import (\\n\",\n \" SpatialCoordinate,\\n\",\n \" TestFunction,\\n\",\n \" TrialFunction,\\n\",\n \" div,\\n\",\n \" dot,\\n\",\n \" dx,\\n\",\n \" grad,\\n\",\n \" inner,\\n\",\n \")\\n\",\n \"\\n\",\n \"import ufl\\n\",\n \"import numpy as np\\n\",\n \"\\n\",\n \"\\n\",\n \"def u_ex(mod):\\n\",\n \" return lambda x: mod.cos(2 * mod.pi * x[0]) * mod.cos(2 * mod.pi * x[1])\\n\",\n \"\\n\",\n \"\\n\",\n \"u_numpy = u_ex(np)\\n\",\n \"u_ufl = u_ex(ufl)\\n\",\n \"\\n\",\n \"\\n\",\n \"def solve_poisson(N=10, degree=1):\\n\",\n \" mesh = create_unit_square(MPI.COMM_WORLD, N, N)\\n\",\n \" x = SpatialCoordinate(mesh)\\n\",\n \" f = -div(grad(u_ufl(x)))\\n\",\n \" V = functionspace(mesh, (\\\"Lagrange\\\", degree))\\n\",\n \" u = TrialFunction(V)\\n\",\n \" v = TestFunction(V)\\n\",\n \" a = inner(grad(u), grad(v)) * dx\\n\",\n \" L = f * v * dx\\n\",\n \" u_bc = Function(V)\\n\",\n \" u_bc.interpolate(u_numpy)\\n\",\n \" facets = locate_entities_boundary(\\n\",\n \" mesh, mesh.topology.dim - 1, lambda x: np.full(x.shape[1], True)\\n\",\n \" )\\n\",\n \" dofs = locate_dofs_topological(V, mesh.topology.dim - 1, facets)\\n\",\n \" bcs = [dirichletbc(u_bc, dofs)]\\n\",\n \" default_problem = LinearProblem(\\n\",\n \" a,\\n\",\n \" L,\\n\",\n \" bcs=bcs,\\n\",\n \" petsc_options={\\\"ksp_type\\\": \\\"preonly\\\", \\\"pc_type\\\": \\\"lu\\\"},\\n\",\n \" petsc_options_prefix=\\\"poisson_convergence_\\\",\\n\",\n \" )\\n\",\n \" return default_problem.solve(), u_ufl(x)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"Now, we can compute the error between the analyical solution `u_ex=u_ufl(x)` and the approximated solution `uh`. A natural choice might seem to compute `(u_ex-uh)**2*ufl.dx`.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"uh, u_ex = solve_poisson(10)\\n\",\n \"comm = uh.function_space.mesh.comm\\n\",\n \"error = form((uh - u_ex) ** 2 * ufl.dx)\\n\",\n \"E = np.sqrt(comm.allreduce(assemble_scalar(error), MPI.SUM))\\n\",\n \"if comm.rank == 0:\\n\",\n \" print(f\\\"L2-error: {E:.2e}\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"Sometimes it is of interest to compute the error fo the gradient field, $\\\\vert\\\\vert \\\\nabla(u_e-u_h)\\\\vert\\\\vert$, often referred to as the $H_0^1$-norm of the error, this can be expressed as\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"outputs\": [],\n \"source\": [\n \"eh = uh - u_ex\\n\",\n \"error_H10 = form(dot(grad(eh), grad(eh)) * dx)\\n\",\n \"E_H10 = np.sqrt(comm.allreduce(assemble_scalar(error_H10), op=MPI.SUM))\\n\",\n \"if comm.rank == 0:\\n\",\n \" print(f\\\"H01-error: {E_H10:.2e}\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"### Reliable error norm computation\\n\",\n \"However, as this gets expanded to `u_ex**2 + uh**2 - 2*u_ex*uh`. If the error is small, (and the solution itself is of moderate size), this calculation will correspond to subtract two positive numbers `u_ex**2 + uh**2`$\\\\sim 1$ and `2*u_ex*u`$\\\\sim 1$ yielding a small number, prone to round-off errors.\\n\",\n \"\\n\",\n \"To avoid this issue, we interpolate the approximate and exact solution into a higher order function space. Then we subtract the degrees of freedom from the interpolated functions to create a new error function. Then, finally, we assemble/integrate the square difference and take the square root to get the L2 norm.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"def error_L2(uh, u_ex, degree_raise=3):\\n\",\n \" # Create higher order function space\\n\",\n \" degree = uh.function_space.ufl_element().degree\\n\",\n \" family = uh.function_space.ufl_element().family_name\\n\",\n \" mesh = uh.function_space.mesh\\n\",\n \" W = functionspace(mesh, (family, degree + degree_raise))\\n\",\n \" # Interpolate approximate solution\\n\",\n \" u_W = Function(W)\\n\",\n \" u_W.interpolate(uh)\\n\",\n \"\\n\",\n \" # Interpolate exact solution, special handling if exact solution\\n\",\n \" # is a ufl expression or a python lambda function\\n\",\n \" u_ex_W = Function(W)\\n\",\n \" if isinstance(u_ex, ufl.core.expr.Expr):\\n\",\n \" u_expr = Expression(u_ex, W.element.interpolation_points)\\n\",\n \" u_ex_W.interpolate(u_expr)\\n\",\n \" else:\\n\",\n \" u_ex_W.interpolate(u_ex)\\n\",\n \"\\n\",\n \" # Compute the error in the higher order function space\\n\",\n \" e_W = Function(W)\\n\",\n \" e_W.x.array[:] = u_W.x.array - u_ex_W.x.array\\n\",\n \"\\n\",\n \" # Integrate the error\\n\",\n \" error = form(ufl.inner(e_W, e_W) * ufl.dx)\\n\",\n \" error_local = assemble_scalar(error)\\n\",\n \" error_global = mesh.comm.allreduce(error_local, op=MPI.SUM)\\n\",\n \" return np.sqrt(error_global)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Computing convergence rates\\n\",\n \"Let us consider a sequence of mesh resolutions $h_0>h_1>h_2$, where $h_i=\\\\frac{1}{N_i}$ we compute the errors for a range of $N_i$s\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"Ns = [4, 8, 16, 32, 64]\\n\",\n \"Es = np.zeros(len(Ns), dtype=default_scalar_type)\\n\",\n \"hs = np.zeros(len(Ns), dtype=np.float64)\\n\",\n \"for i, N in enumerate(Ns):\\n\",\n \" uh, u_ex = solve_poisson(N, degree=1)\\n\",\n \" comm = uh.function_space.mesh.comm\\n\",\n \" # One can send in either u_numpy or u_ex\\n\",\n \" # For L2 error estimations it is reccommended to send in u_numpy\\n\",\n \" # as no JIT compilation is required\\n\",\n \" Es[i] = error_L2(uh, u_numpy)\\n\",\n \" hs[i] = 1.0 / Ns[i]\\n\",\n \" if comm.rank == 0:\\n\",\n \" print(f\\\"h: {hs[i]:.2e} Error: {Es[i]:.2e}\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"If we assume that $E_i$ is of the form $E_i=Ch_i^r$, with unknown constants $C$ and $r$, we can compare two consecutive experiments, $E_{i-1}= Ch_{i-1}^r$ and $E_i=Ch_i^r$, and solve for $r$:\\n\",\n \"```{math}\\n\",\n \"r=\\\\frac{\\\\ln(E_i/E_{i-1})}{\\\\ln(h_i/h_{i-1})}\\n\",\n \"```\\n\",\n \"The $r$ values should approach the expected convergence rate (which is typically the polynomial degree + 1 for the $L^2$-error.) as $i$ increases. This can be written compactly using `numpy`.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"rates = np.log(Es[1:] / Es[:-1]) / np.log(hs[1:] / hs[:-1])\\n\",\n \"if comm.rank == 0:\\n\",\n \" print(f\\\"Rates: {rates}\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We also do a similar study for different orders of polynomial spaces to verify our previous claim.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"outputs\": [],\n \"source\": [\n \"degrees = [1, 2, 3, 4]\\n\",\n \"for degree in degrees:\\n\",\n \" Es = np.zeros(len(Ns), dtype=default_scalar_type)\\n\",\n \" hs = np.zeros(len(Ns), dtype=np.float64)\\n\",\n \" for i, N in enumerate(Ns):\\n\",\n \" uh, u_ex = solve_poisson(N, degree=degree)\\n\",\n \" comm = uh.function_space.mesh.comm\\n\",\n \" Es[i] = error_L2(uh, u_numpy, degree_raise=3)\\n\",\n \" hs[i] = 1.0 / Ns[i]\\n\",\n \" if comm.rank == 0:\\n\",\n \" print(f\\\"h: {hs[i]:.2e} Error: {Es[i]:.2e}\\\")\\n\",\n \" rates = np.log(Es[1:] / Es[:-1]) / np.log(hs[1:] / hs[:-1])\\n\",\n \" if comm.rank == 0:\\n\",\n \" print(f\\\"Polynomial degree {degree:d}, Rates {rates}\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"### Infinity norm estimates\\n\",\n \"We start by creating a function to compute the infinity norm, the max difference between the approximate and exact solution.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"def error_infinity(u_h, u_ex):\\n\",\n \" # Interpolate exact solution, special handling if exact solution\\n\",\n \" # is a ufl expression or a python lambda function\\n\",\n \" comm = u_h.function_space.mesh.comm\\n\",\n \" u_ex_V = Function(u_h.function_space)\\n\",\n \" if isinstance(u_ex, ufl.core.expr.Expr):\\n\",\n \" u_expr = Expression(u_ex, u_h.function_space.element.interpolation_points)\\n\",\n \" u_ex_V.interpolate(u_expr)\\n\",\n \" else:\\n\",\n \" u_ex_V.interpolate(u_ex)\\n\",\n \" # Compute infinity norm, furst local to process, then gather the max\\n\",\n \" # value over all processes\\n\",\n \" error_max_local = np.max(np.abs(u_h.x.array - u_ex_V.x.array))\\n\",\n \" error_max = comm.allreduce(error_max_local, op=MPI.MAX)\\n\",\n \" return error_max\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"Running this for various polynomial degrees yields:\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"for degree in degrees:\\n\",\n \" Es = np.zeros(len(Ns), dtype=default_scalar_type)\\n\",\n \" hs = np.zeros(len(Ns), dtype=np.float64)\\n\",\n \" for i, N in enumerate(Ns):\\n\",\n \" uh, u_ex = solve_poisson(N, degree=degree)\\n\",\n \" comm = uh.function_space.mesh.comm\\n\",\n \" Es[i] = error_infinity(uh, u_numpy)\\n\",\n \" hs[i] = 1.0 / Ns[i]\\n\",\n \" if comm.rank == 0:\\n\",\n \" print(f\\\"h: {hs[i]:.2e} Error: {Es[i]:.2e}\\\")\\n\",\n \" rates = np.log(Es[1:] / Es[:-1]) / np.log(hs[1:] / hs[:-1])\\n\",\n \" if comm.rank == 0:\\n\",\n \" print(f\\\"Polynomial degree {degree:d}, Rates {rates}\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We observe super convergence for second order polynomials, yielding a fourth order convergence.\"\n ]\n }\n ],\n \"metadata\": {\n \"jupytext\": {\n \"formats\": \"ipynb,py:light\"\n },\n \"kernelspec\": {\n \"display_name\": \"Python 3 (ipykernel)\",\n \"language\": \"python\",\n \"name\": \"python3\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 3\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython3\",\n \"version\": \"3.10.12\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 4\n}\n" }, { "path": "chapter4/convergence.py", "content": "# ---\n# jupyter:\n# jupytext:\n# formats: ipynb,py:light\n# text_representation:\n# extension: .py\n# format_name: light\n# format_version: '1.5'\n# jupytext_version: 1.18.1\n# kernelspec:\n# display_name: Python 3 (ipykernel)\n# language: python\n# name: python3\n# ---\n\n# # Error control: Computing convergence rates\n# Author: Jørgen S. Dokken, Hans Petter Langtangen, Anders Logg\n#\n# For any numerical method one of the most central questions is its *convergence rate*: How fast does the error go to zero when the resolution is increased (mesh size decreased).\n#\n# For the finite element method, this usually corresponds to proving, theoretically or imperically, that the error $e=u_e-u_h$ is bounded by the mesh size $h$ to some power $r$, that is $\\vert\\vert e \\vert\\vert\\leq Ch^r$ for some mesh independent constant $C$. The number $r$ is called the *convergence rate* of the method. Note that the different norms like the $L^2$-norm $\\vert\\vert e\\vert\\vert$ or the $H_0^1$-norm have different convergence rates.\n\n# ## Computing error norms\n# We start by creating a manufactured problem, using the same problem as in [the solver configuration](./solvers.ipynb).\n#\n\n# +\nfrom dolfinx import default_scalar_type\nfrom dolfinx.fem import (\n Expression,\n Function,\n functionspace,\n assemble_scalar,\n dirichletbc,\n form,\n locate_dofs_topological,\n)\nfrom dolfinx.fem.petsc import LinearProblem\nfrom dolfinx.mesh import create_unit_square, locate_entities_boundary\n\nfrom mpi4py import MPI\nfrom ufl import (\n SpatialCoordinate,\n TestFunction,\n TrialFunction,\n div,\n dot,\n dx,\n grad,\n inner,\n)\n\nimport ufl\nimport numpy as np\n\n\ndef u_ex(mod):\n return lambda x: mod.cos(2 * mod.pi * x[0]) * mod.cos(2 * mod.pi * x[1])\n\n\nu_numpy = u_ex(np)\nu_ufl = u_ex(ufl)\n\n\ndef solve_poisson(N=10, degree=1):\n mesh = create_unit_square(MPI.COMM_WORLD, N, N)\n x = SpatialCoordinate(mesh)\n f = -div(grad(u_ufl(x)))\n V = functionspace(mesh, (\"Lagrange\", degree))\n u = TrialFunction(V)\n v = TestFunction(V)\n a = inner(grad(u), grad(v)) * dx\n L = f * v * dx\n u_bc = Function(V)\n u_bc.interpolate(u_numpy)\n facets = locate_entities_boundary(\n mesh, mesh.topology.dim - 1, lambda x: np.full(x.shape[1], True)\n )\n dofs = locate_dofs_topological(V, mesh.topology.dim - 1, facets)\n bcs = [dirichletbc(u_bc, dofs)]\n default_problem = LinearProblem(\n a,\n L,\n bcs=bcs,\n petsc_options={\"ksp_type\": \"preonly\", \"pc_type\": \"lu\"},\n petsc_options_prefix=\"poisson_convergence_\",\n )\n return default_problem.solve(), u_ufl(x)\n\n\n# -\n\n# Now, we can compute the error between the analyical solution `u_ex=u_ufl(x)` and the approximated solution `uh`. A natural choice might seem to compute `(u_ex-uh)**2*ufl.dx`.\n\nuh, u_ex = solve_poisson(10)\ncomm = uh.function_space.mesh.comm\nerror = form((uh - u_ex) ** 2 * ufl.dx)\nE = np.sqrt(comm.allreduce(assemble_scalar(error), MPI.SUM))\nif comm.rank == 0:\n print(f\"L2-error: {E:.2e}\")\n\n# Sometimes it is of interest to compute the error fo the gradient field, $\\vert\\vert \\nabla(u_e-u_h)\\vert\\vert$, often referred to as the $H_0^1$-norm of the error, this can be expressed as\n\neh = uh - u_ex\nerror_H10 = form(dot(grad(eh), grad(eh)) * dx)\nE_H10 = np.sqrt(comm.allreduce(assemble_scalar(error_H10), op=MPI.SUM))\nif comm.rank == 0:\n print(f\"H01-error: {E_H10:.2e}\")\n\n\n# ### Reliable error norm computation\n# However, as this gets expanded to `u_ex**2 + uh**2 - 2*u_ex*uh`. If the error is small, (and the solution itself is of moderate size), this calculation will correspond to subtract two positive numbers `u_ex**2 + uh**2`$\\sim 1$ and `2*u_ex*u`$\\sim 1$ yielding a small number, prone to round-off errors.\n#\n# To avoid this issue, we interpolate the approximate and exact solution into a higher order function space. Then we subtract the degrees of freedom from the interpolated functions to create a new error function. Then, finally, we assemble/integrate the square difference and take the square root to get the L2 norm.\n\n\ndef error_L2(uh, u_ex, degree_raise=3):\n # Create higher order function space\n degree = uh.function_space.ufl_element().degree\n family = uh.function_space.ufl_element().family_name\n mesh = uh.function_space.mesh\n W = functionspace(mesh, (family, degree + degree_raise))\n # Interpolate approximate solution\n u_W = Function(W)\n u_W.interpolate(uh)\n\n # Interpolate exact solution, special handling if exact solution\n # is a ufl expression or a python lambda function\n u_ex_W = Function(W)\n if isinstance(u_ex, ufl.core.expr.Expr):\n u_expr = Expression(u_ex, W.element.interpolation_points)\n u_ex_W.interpolate(u_expr)\n else:\n u_ex_W.interpolate(u_ex)\n\n # Compute the error in the higher order function space\n e_W = Function(W)\n e_W.x.array[:] = u_W.x.array - u_ex_W.x.array\n\n # Integrate the error\n error = form(ufl.inner(e_W, e_W) * ufl.dx)\n error_local = assemble_scalar(error)\n error_global = mesh.comm.allreduce(error_local, op=MPI.SUM)\n return np.sqrt(error_global)\n\n\n# ## Computing convergence rates\n# Let us consider a sequence of mesh resolutions $h_0>h_1>h_2$, where $h_i=\\frac{1}{N_i}$ we compute the errors for a range of $N_i$s\n\nNs = [4, 8, 16, 32, 64]\nEs = np.zeros(len(Ns), dtype=default_scalar_type)\nhs = np.zeros(len(Ns), dtype=np.float64)\nfor i, N in enumerate(Ns):\n uh, u_ex = solve_poisson(N, degree=1)\n comm = uh.function_space.mesh.comm\n # One can send in either u_numpy or u_ex\n # For L2 error estimations it is reccommended to send in u_numpy\n # as no JIT compilation is required\n Es[i] = error_L2(uh, u_numpy)\n hs[i] = 1.0 / Ns[i]\n if comm.rank == 0:\n print(f\"h: {hs[i]:.2e} Error: {Es[i]:.2e}\")\n\n# If we assume that $E_i$ is of the form $E_i=Ch_i^r$, with unknown constants $C$ and $r$, we can compare two consecutive experiments, $E_{i-1}= Ch_{i-1}^r$ and $E_i=Ch_i^r$, and solve for $r$:\n# ```{math}\n# r=\\frac{\\ln(E_i/E_{i-1})}{\\ln(h_i/h_{i-1})}\n# ```\n# The $r$ values should approach the expected convergence rate (which is typically the polynomial degree + 1 for the $L^2$-error.) as $i$ increases. This can be written compactly using `numpy`.\n\nrates = np.log(Es[1:] / Es[:-1]) / np.log(hs[1:] / hs[:-1])\nif comm.rank == 0:\n print(f\"Rates: {rates}\")\n\n# We also do a similar study for different orders of polynomial spaces to verify our previous claim.\n\ndegrees = [1, 2, 3, 4]\nfor degree in degrees:\n Es = np.zeros(len(Ns), dtype=default_scalar_type)\n hs = np.zeros(len(Ns), dtype=np.float64)\n for i, N in enumerate(Ns):\n uh, u_ex = solve_poisson(N, degree=degree)\n comm = uh.function_space.mesh.comm\n Es[i] = error_L2(uh, u_numpy, degree_raise=3)\n hs[i] = 1.0 / Ns[i]\n if comm.rank == 0:\n print(f\"h: {hs[i]:.2e} Error: {Es[i]:.2e}\")\n rates = np.log(Es[1:] / Es[:-1]) / np.log(hs[1:] / hs[:-1])\n if comm.rank == 0:\n print(f\"Polynomial degree {degree:d}, Rates {rates}\")\n\n\n# ### Infinity norm estimates\n# We start by creating a function to compute the infinity norm, the max difference between the approximate and exact solution.\n\n\ndef error_infinity(u_h, u_ex):\n # Interpolate exact solution, special handling if exact solution\n # is a ufl expression or a python lambda function\n comm = u_h.function_space.mesh.comm\n u_ex_V = Function(u_h.function_space)\n if isinstance(u_ex, ufl.core.expr.Expr):\n u_expr = Expression(u_ex, u_h.function_space.element.interpolation_points)\n u_ex_V.interpolate(u_expr)\n else:\n u_ex_V.interpolate(u_ex)\n # Compute infinity norm, furst local to process, then gather the max\n # value over all processes\n error_max_local = np.max(np.abs(u_h.x.array - u_ex_V.x.array))\n error_max = comm.allreduce(error_max_local, op=MPI.MAX)\n return error_max\n\n\n# Running this for various polynomial degrees yields:\n\nfor degree in degrees:\n Es = np.zeros(len(Ns), dtype=default_scalar_type)\n hs = np.zeros(len(Ns), dtype=np.float64)\n for i, N in enumerate(Ns):\n uh, u_ex = solve_poisson(N, degree=degree)\n comm = uh.function_space.mesh.comm\n Es[i] = error_infinity(uh, u_numpy)\n hs[i] = 1.0 / Ns[i]\n if comm.rank == 0:\n print(f\"h: {hs[i]:.2e} Error: {Es[i]:.2e}\")\n rates = np.log(Es[1:] / Es[:-1]) / np.log(hs[1:] / hs[:-1])\n if comm.rank == 0:\n print(f\"Polynomial degree {degree:d}, Rates {rates}\")\n\n# We observe super convergence for second order polynomials, yielding a fourth order convergence.\n" }, { "path": "chapter4/mixed_poisson.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"markdown\",\n \"id\": \"0\",\n \"metadata\": {},\n \"source\": [\n \"# Mixed Poisson with a Schur complement pre-conditioner\\n\",\n \"This example demonstrates how to use PETSC fieldsplits with custom preconditions in DOLFINx.\\n\",\n \"This example is heavily insipired by the [FEniCSx PCTools example](https://rafinex-external-rifle.gitlab.io/fenicsx-pctools/demo/demo_mixed-poisson.html)\\n\",\n \"which was presented in {cite}`mp-rehor2025pctools`.\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"1\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"We start with the mixed formulation of the Poisson equation, which is given by\\n\",\n \"\\\\begin{align}\\n\",\n \"\\\\sigma - \\\\nabla u &= 0&&\\\\text{in } \\\\Omega,\\\\\\\\\\n\",\n \"\\\\nabla \\\\cdot \\\\sigma &= -f&&\\\\text{in } \\\\Omega,\\\\\\\\\\n\",\n \"u &= u_D &&\\\\text{on } \\\\Gamma_D,\\\\\\\\\\n\",\n \"\\\\sigma \\\\cdot n &= g &&\\\\text{on } \\\\Gamma_N,\\n\",\n \"\\\\end{align}\\n\",\n \"\\n\",\n \"As in previous examples, we pick a manufactured solution to ensure that we can verify\\n\",\n \"the correctness of our implementation.\\n\",\n \"The manufactured solution is given by\\n\",\n \"\\\\begin{align}\\n\",\n \"u_{ex}(x, y) &= \\\\sin(\\\\pi x) + y^2.\\n\",\n \"\\\\end{align}\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"2\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"def u_ex(mod, x):\\n\",\n \" return mod.sin(mod.pi * x[0]) + x[1] ** 2\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"3\",\n \"metadata\": {},\n \"source\": [\n \"We choose to solve the problem on a unit square,\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"4\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"from mpi4py import MPI\\n\",\n \"from petsc4py import PETSc\\n\",\n \"import dolfinx\\n\",\n \"\\n\",\n \"N = 400\\n\",\n \"mesh = dolfinx.mesh.create_unit_square(MPI.COMM_WORLD, N, N)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"5\",\n \"metadata\": {},\n \"source\": [\n \"where $\\\\Gamma_D = \\\\{(x, 0) \\\\vert x \\\\in [0, 1]\\\\}\\\\cup\\\\{ (x, 1) \\\\vert x \\\\in [0, 1]\\\\}$\\n\",\n \"and $\\\\Gamma_N = \\\\{(0, y) \\\\vert y \\\\in [0, 1]\\\\}\\\\cup\\\\{(1, y) \\\\vert y \\\\in [0, 1]\\\\}$.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"6\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"import numpy as np\\n\",\n \"\\n\",\n \"\\n\",\n \"def Gamma_D(x):\\n\",\n \" return (\\n\",\n \" np.isclose(x[1], 0)\\n\",\n \" | np.isclose(x[1], 1)\\n\",\n \" | np.isclose(x[0], 0)\\n\",\n \" | np.isclose(x[0], 1)\\n\",\n \" )\\n\",\n \"\\n\",\n \"\\n\",\n \"def Gamma_N(x):\\n\",\n \" return np.full_like(\\n\",\n \" x[0], 0, dtype=bool\\n\",\n \" ) # np.isclose(x[0], 0) | np.isclose(x[0], 1)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"7\",\n \"metadata\": {},\n \"source\": [\n \"We define the function space for the vector-valued flux $p\\\\in Q$ as the zeroth order discontinuous Lagrange space,\\n\",\n \"while the scalar potential $u \\\\in V$ is defined in first order\\n\",\n \"[Brezzi-Douglas-Marini space](https://defelement.org/elements/brezzi-douglas-marini.html).\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"8\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"V = dolfinx.fem.functionspace(mesh, (\\\"DG\\\", 0))\\n\",\n \"Q = dolfinx.fem.functionspace(mesh, (\\\"BDM\\\", 1))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"9\",\n \"metadata\": {},\n \"source\": [\n \"We define a `ufl.MixedFunctionSpace` to automatically handle the block structure of the problem\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"10\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"import ufl\\n\",\n \"\\n\",\n \"W = ufl.MixedFunctionSpace(*[Q, V])\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"11\",\n \"metadata\": {},\n \"source\": [\n \"Next, we have to define the bilinear and linear forms.\\n\",\n \"We do this as usual, by introducing a test functions $v\\\\in V$ and $\\\\tau\\\\in Q$ and a trial function $u\\\\in V$ and $q\\\\in Q$,\\n\",\n \"and integrate the first equation by parts.\\n\",\n \"\\n\",\n \"\\\\begin{align}\\n\",\n \"\\\\int_\\\\Omega \\\\sigma \\\\cdot \\\\tau - \\\\nabla u \\\\cdot \\\\tau ~\\\\mathrm{d} x &=\\n\",\n \"\\\\int_\\\\Omega \\\\sigma \\\\cdot \\\\tau + u \\\\nabla \\\\cdot \\\\tau ~\\\\mathrm{d} x\\n\",\n \"- \\\\sum_{f_i\\\\in \\\\mathit{Fi}}\\\\int_{f_i}\\\\left[u\\\\right] \\\\tau \\\\cdot \\\\mathbf{n}_i~\\\\mathrm{d}s\\n\",\n \"- \\\\int_{\\\\partial\\\\Omega} u \\\\tau\\\\cdot \\\\mathbf{n}~\\\\mathrm{d}s,\\\\\\\\\\n\",\n \"&=\\\\int_\\\\Omega \\\\sigma \\\\cdot \\\\tau + u \\\\nabla \\\\cdot \\\\tau ~\\\\mathrm{d} x\\n\",\n \"- \\\\int_{\\\\Gamma_D} u_D \\\\tau\\\\cdot \\\\mathbf{n}~\\\\mathrm{d}s,\\\\\\\\\\n\",\n \"\\\\end{align}\\n\",\n \"\\n\",\n \"where $f_i$ is an interior facet of the mesh, $\\\\mathbf{n}_i$ is an outwards pointing normal of one of the two\\n\",\n \"adjacent elements. We will enforce the boundary conditions strongly by using a\\n\",\n \"`dolfinx.fem.dirichletbc` on both $\\\\Gamma_N$, which makes its integral dissapear, while we enforced the Dirichlet boundary condition\\n\",\n \"on $\\\\Gamma_D$ weakly._\\n\",\n \"We enforce the continuity of $u$ weakly by removing the jump term.\\n\",\n \"Thus we end up with:\\n\",\n \"\\n\",\n \"Find $u\\\\in V_{u_D}, \\\\sigma \\\\in Q_{g}$ such that\\n\",\n \"\\n\",\n \"\\\\begin{align}\\n\",\n \"\\\\begin{split}\\n\",\n \"\\\\int_\\\\Omega \\\\sigma \\\\cdot \\\\tau + u \\\\nabla \\\\cdot \\\\tau ~\\\\mathrm{d} x&= \\\\int_{\\\\Gamma_D} u_D \\\\tau\\\\cdot \\\\mathbf{n}~\\\\mathrm{d}s,\\\\\\\\\\\\\\\\\\n\",\n \"\\\\int_\\\\Omega \\\\nabla \\\\cdot \\\\sigma v ~\\\\mathrm{d} x&=-\\\\int_\\\\Omega f v ~\\\\mathrm{d} x\\n\",\n \"\\\\end{split}\\\\qquad \\\\forall v \\\\in V_{0}, \\\\tau \\\\in Q_{0}\\n\",\n \"\\\\end{align}\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"12\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"outputs\": [],\n \"source\": [\n \"u_D = dolfinx.fem.Function(V)\\n\",\n \"u_D.interpolate(lambda x: u_ex(np, x))\\n\",\n \"mesh.topology.create_connectivity(mesh.topology.dim, mesh.topology.dim)\\n\",\n \"gamma_d_facets = dolfinx.mesh.locate_entities_boundary(\\n\",\n \" mesh, mesh.topology.dim - 1, Gamma_D\\n\",\n \")\\n\",\n \"tag = 3\\n\",\n \"ft = dolfinx.mesh.meshtags(\\n\",\n \" mesh,\\n\",\n \" mesh.topology.dim - 1,\\n\",\n \" gamma_d_facets,\\n\",\n \" np.full(gamma_d_facets.shape[0], tag, dtype=np.int32),\\n\",\n \")\\n\",\n \"dGammaD = ufl.Measure(\\\"ds\\\", domain=mesh, subdomain_data=ft, subdomain_id=tag)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"13\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"sigma, u = ufl.TrialFunctions(W)\\n\",\n \"tau, v = ufl.TestFunctions(W)\\n\",\n \"n = ufl.FacetNormal(mesh)\\n\",\n \"a = ufl.inner(sigma, tau) * ufl.dx\\n\",\n \"a += u * ufl.div(tau) * ufl.dx\\n\",\n \"a += ufl.inner(ufl.div(sigma), v) * ufl.dx\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"14\",\n \"metadata\": {},\n \"source\": [\n \"This can be split into a saddle point problem, with discretized matrices $A$ and $B$ and discretized\\n\",\n \"right-hand side $\\\\mathbf{b}$.\\n\",\n \"\\\\begin{align}\\n\",\n \"\\\\begin{pmatrix}\\n\",\n \"A & B^T\\\\\\\\\\n\",\n \"B & 0\\n\",\n \"\\\\end{pmatrix}\\n\",\n \"\\\\begin{pmatrix}\\n\",\n \"u_h\\\\\\\\\\n\",\n \"\\\\sigma_h\\n\",\n \"\\\\end{pmatrix}\\n\",\n \"= \\\\begin{pmatrix}\\n\",\n \"b_0\\\\\\\\\\n\",\n \"b_1\\n\",\n \"\\\\end{pmatrix}\\n\",\n \"\\\\end{align}\\n\",\n \"We can extract the block structure of the bilinear form using `ufl.extract_blocks`, which returns a nested list of bilinear forms.\\n\",\n \"You can also build this nested list by hand if you want to, but it is usually more error-prone.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"15\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"a_blocked = ufl.extract_blocks(a)\\n\",\n \"\\n\",\n \"x = ufl.SpatialCoordinate(mesh)\\n\",\n \"u_exact = u_ex(ufl, x)\\n\",\n \"sigma_exact = ufl.grad(u_exact)\\n\",\n \"f = -ufl.div(sigma_exact)\\n\",\n \"L = ufl.inner(u_D, ufl.dot(tau, n)) * dGammaD - ufl.inner(f, v) * ufl.dx\\n\",\n \"L_blocked = ufl.extract_blocks(L)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"16\",\n \"metadata\": {},\n \"source\": [\n \"Next we create the Dirichlet boundary condition for $\\\\sigma$.\\n\",\n \"As we are using manufactured solutions for this problem, we could manually derive the explicit expression\\n\",\n \"for $\\\\sigma$ on the boundary $\\\\Gamma_N$.\\n\",\n \"However, in general this is not possible (especially for curved boundaries), and we have to use a more generic approach.\\n\",\n \"For this we will use the `dolfinx.fem.Expression` class to interpolate the expression into the function space $Q$.\\n\",\n \"This is done by evaluating the expression at the physical interpolation points of the mesh.\\n\",\n \"A convenience function for this is provided in the `interpolate_facet_expression` function below.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"17\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"import numpy.typing as npt\\n\",\n \"import basix.ufl\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"18\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"def interpolate_facet_expression(\\n\",\n \" Q: dolfinx.fem.FunctionSpace,\\n\",\n \" expr: ufl.core.expr.Expr,\\n\",\n \" facets: npt.NDArray[np.int32],\\n\",\n \") -> dolfinx.fem.Function:\\n\",\n \" \\\"\\\"\\\"\\n\",\n \" Interpolate a UFL-expression into a function space, only for the degrees of freedom assoicated with facets.\\n\",\n \" \\\"\\\"\\\"\\n\",\n \" domain = Q.mesh\\n\",\n \" Q_el = Q.element\\n\",\n \" fdim = domain.topology.dim - 1\\n\",\n \"\\n\",\n \" # Get coordinate element for facets of cell\\n\",\n \" c_el = domain.ufl_domain().ufl_coordinate_element()\\n\",\n \" facet_types = basix.cell.subentity_types(domain.basix_cell())[fdim]\\n\",\n \" unique_facet_types = np.unique(facet_types)\\n\",\n \" assert len(unique_facet_types) == 1, (\\n\",\n \" \\\"All facets must have the same type for interpolation.\\\"\\n\",\n \" )\\n\",\n \" facet_type = facet_types[0]\\n\",\n \" x_type = domain.geometry.x.dtype\\n\",\n \" facet_cmap = basix.ufl.element(\\n\",\n \" \\\"Lagrange\\\", facet_type, c_el.degree, shape=(domain.geometry.dim,), dtype=x_type\\n\",\n \" )\\n\",\n \" if np.issubdtype(x_type, np.float32):\\n\",\n \" facet_cel = dolfinx.cpp.fem.CoordinateElement_float32(\\n\",\n \" facet_cmap.basix_element._e\\n\",\n \" )\\n\",\n \" elif np.issubdtype(x_type, np.float64):\\n\",\n \" facet_cel = dolfinx.cpp.fem.CoordinateElement_float64(\\n\",\n \" facet_cmap.basix_element._e\\n\",\n \" )\\n\",\n \" else:\\n\",\n \" raise TypeError(\\n\",\n \" f\\\"Unsupported coordinate element type: {x_type}. \\\"\\n\",\n \" \\\"Only float32 and float64 are supported.\\\"\\n\",\n \" )\\n\",\n \" # Pull back interpolation points from reference coordinate element to facet reference element\\n\",\n \" ref_top = c_el.reference_topology\\n\",\n \" ref_geom = c_el.reference_geometry\\n\",\n \" reference_facet_points = None\\n\",\n \" interpolation_points = Q_el.basix_element.x\\n\",\n \" for i, points in enumerate(interpolation_points[fdim]):\\n\",\n \" geom = ref_geom[ref_top[fdim][i]]\\n\",\n \" ref_points = facet_cel.pull_back(points, geom)\\n\",\n \" # Assert that interpolation points are all equal on all facets\\n\",\n \" if reference_facet_points is None:\\n\",\n \" reference_facet_points = ref_points\\n\",\n \" else:\\n\",\n \" assert np.allclose(reference_facet_points, ref_points)\\n\",\n \"\\n\",\n \" assert isinstance(reference_facet_points, np.ndarray)\\n\",\n \"\\n\",\n \" # Create expression for BC\\n\",\n \" bndry_expr = dolfinx.fem.Expression(expr, reference_facet_points)\\n\",\n \"\\n\",\n \" # Compute number of interpolation points per sub entity\\n\",\n \" points_per_entity = [sum(ip.shape[0] for ip in ips) for ips in interpolation_points]\\n\",\n \" offsets = np.zeros(domain.topology.dim + 2, dtype=np.int32)\\n\",\n \" offsets[1:] = np.cumsum(points_per_entity[: domain.topology.dim + 1])\\n\",\n \" values_per_entity = np.zeros(\\n\",\n \" (offsets[-1], domain.geometry.dim), dtype=dolfinx.default_scalar_type\\n\",\n \" )\\n\",\n \"\\n\",\n \" # Map facet indices to (cell, local_facet) pairs\\n\",\n \" boundary_entities = dolfinx.fem.compute_integration_domains(\\n\",\n \" dolfinx.fem.IntegralType.exterior_facet, domain.topology, facets\\n\",\n \" )\\n\",\n \"\\n\",\n \" # Compute and insert the correct values for the interpolation points on the facets\\n\",\n \" entities = boundary_entities.reshape(-1, 2)\\n\",\n \" values = np.zeros(entities.shape[0] * offsets[-1] * domain.geometry.dim)\\n\",\n \" for i, entity in enumerate(entities):\\n\",\n \" insert_pos = offsets[fdim] + reference_facet_points.shape[0] * entity[1]\\n\",\n \" normal_on_facet = bndry_expr.eval(domain, entity.reshape(1, 2))\\n\",\n \" values_per_entity[insert_pos : insert_pos + reference_facet_points.shape[0]] = (\\n\",\n \" normal_on_facet.reshape(-1, domain.geometry.dim)\\n\",\n \" )\\n\",\n \" values[\\n\",\n \" i * offsets[-1] * domain.geometry.dim : (i + 1)\\n\",\n \" * offsets[-1]\\n\",\n \" * domain.geometry.dim\\n\",\n \" ] = values_per_entity.reshape(-1)\\n\",\n \" # Use lower-level interpolation that takes in the function evaluated at the physical\\n\",\n \" # interpolation points of the mesh.\\n\",\n \" qh = dolfinx.fem.Function(Q)\\n\",\n \" qh._cpp_object.interpolate(\\n\",\n \" values.reshape(-1, domain.geometry.dim).T.copy(), boundary_entities[::2].copy()\\n\",\n \" )\\n\",\n \" qh.x.scatter_forward()\\n\",\n \" return qh\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"19\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"sigma_facets = dolfinx.mesh.locate_entities_boundary(\\n\",\n \" mesh, mesh.topology.dim - 1, Gamma_N\\n\",\n \")\\n\",\n \"n = ufl.FacetNormal(mesh)\\n\",\n \"g = ufl.dot(ufl.grad(u_exact), n)\\n\",\n \"sigma_bc = interpolate_facet_expression(Q, g, sigma_facets)\\n\",\n \"bc_sigma = dolfinx.fem.dirichletbc(\\n\",\n \" sigma_bc,\\n\",\n \" dolfinx.fem.locate_dofs_topological(Q, mesh.topology.dim - 1, sigma_facets),\\n\",\n \")\\n\",\n \"assert len(sigma_facets) == 0\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"20\",\n \"metadata\": {},\n \"source\": [\n \"Now that we have created the bilinear and linear form, and the boundary conditions,\\n\",\n \"we turn to solving the problem. For this we use the `dolfinx.fem.petsc.LinearProblem` class.\\n\",\n \"As opposed to the previous examples, we now have an explicit block structure, which we would like to\\n\",\n \"exploit when solving the problem. However, first we will solve the problem without any preconditioner\\n\",\n \"to have a baseline performance.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"21\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"import dolfinx.fem.petsc\\n\",\n \"\\n\",\n \"problem = dolfinx.fem.petsc.LinearProblem(\\n\",\n \" a_blocked,\\n\",\n \" L_blocked,\\n\",\n \" bcs=[bc_sigma],\\n\",\n \" petsc_options={\\n\",\n \" \\\"ksp_type\\\": \\\"preonly\\\",\\n\",\n \" \\\"pc_type\\\": \\\"lu\\\",\\n\",\n \" \\\"pc_factor_mat_solver_type\\\": \\\"mumps\\\",\\n\",\n \" \\\"ksp_error_if_not_converged\\\": True,\\n\",\n \" \\\"mat_mumps_icntl_24\\\": 1,\\n\",\n \" \\\"mat_mumps_icntl_25\\\": 0,\\n\",\n \" },\\n\",\n \" kind=\\\"mpi\\\",\\n\",\n \" petsc_options_prefix=\\\"mixed_poisson_direct\\\",\\n\",\n \")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"22\",\n \"metadata\": {},\n \"source\": [\n \"Note that we have specified `kind=\\\"mpi\\\"` in the initialization of the `LinearProblem`.\\n\",\n \"This is to inform DOLFINx that we wan to preserve the block structure of the problem when assembling.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"23\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"import time\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"24\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"start = time.perf_counter()\\n\",\n \"(sigma_h, u_h) = problem.solve()\\n\",\n \"end = time.perf_counter()\\n\",\n \"print(f\\\"Direct solver took {end - start:.2f} seconds.\\\")\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"25\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"L2_u = dolfinx.fem.form(ufl.inner(u_h - u_exact, u_h - u_exact) * ufl.dx)\\n\",\n \"Hdiv_sigma = dolfinx.fem.form(\\n\",\n \" ufl.inner(\\n\",\n \" ufl.div(sigma_h) - ufl.div(ufl.grad(u_exact)),\\n\",\n \" ufl.div(sigma_h) - ufl.div(ufl.grad(u_exact)),\\n\",\n \" )\\n\",\n \" * ufl.dx\\n\",\n \")\\n\",\n \"local_u_error = dolfinx.fem.assemble_scalar(L2_u)\\n\",\n \"local_sigma_error = dolfinx.fem.assemble_scalar(Hdiv_sigma)\\n\",\n \"u_error = np.sqrt(mesh.comm.allreduce(local_u_error, op=MPI.SUM))\\n\",\n \"sigma_error = np.sqrt(mesh.comm.allreduce(local_sigma_error, op=MPI.SUM))\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"26\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"print(f\\\"Direct solver, L2(u): {u_error:.2e}, H(div)(sigma): {sigma_error:.2e}\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"27\",\n \"metadata\": {},\n \"source\": [\n \"## Iterative solver with Schur complement preconditioner\\n\",\n \"As mentioned earlier, there are more efficient ways of solving this problem, than using a direct solver.\\n\",\n \"Especially with the saddle point structure of the problem, we can use a Schur complement preconditioner.\\n\",\n \"As described in [FEniCSx PCTools: Mixed Poisson](https://rafinex-external-rifle.gitlab.io/fenicsx-pctools/demo/demo_mixed-poisson.html),\\n\",\n \"Instead of wrapping the matrices in a custom wrapper, we can use `dolfinx.fem.petsc.LinearProblem` to solve the problem.\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"28\",\n \"metadata\": {},\n \"source\": [\n \"We start by defining the $S$ matrix in the Schur complement (see the aforementioned link for details on the variational formulation).\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"29\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"alpha = dolfinx.fem.Constant(mesh, 4.0)\\n\",\n \"gamma = dolfinx.fem.Constant(mesh, 9.0)\\n\",\n \"h = ufl.CellDiameter(mesh)\\n\",\n \"s = -(\\n\",\n \" ufl.inner(ufl.grad(u), ufl.grad(v)) * ufl.dx\\n\",\n \" - ufl.inner(ufl.avg(ufl.grad(v)), ufl.jump(u, n)) * ufl.dS\\n\",\n \" - ufl.inner(ufl.jump(u, n), ufl.avg(ufl.grad(v))) * ufl.dS\\n\",\n \" + (alpha / ufl.avg(h)) * ufl.inner(ufl.jump(u, n), ufl.jump(v, n)) * ufl.dS\\n\",\n \" - ufl.inner(ufl.grad(u), v * n) * dGammaD\\n\",\n \" - ufl.inner(u * n, ufl.grad(v)) * dGammaD\\n\",\n \" + (gamma / h) * u * v * dGammaD\\n\",\n \")\\n\",\n \"\\n\",\n \"S = dolfinx.fem.petsc.assemble_matrix(dolfinx.fem.form(s))\\n\",\n \"S.assemble()\\n\",\n \"\\n\",\n \"\\n\",\n \"class SchurInv:\\n\",\n \" def setUp(self, pc):\\n\",\n \" self.ksp = PETSc.KSP().create(mesh.comm)\\n\",\n \" self.ksp.setOptionsPrefix(pc.getOptionsPrefix() + \\\"SchurInv_\\\")\\n\",\n \" self.ksp.setOperators(S)\\n\",\n \" self.ksp.setTolerances(atol=1e-10, rtol=1e-10)\\n\",\n \" self.ksp.setFromOptions()\\n\",\n \"\\n\",\n \" def apply(self, pc, x, y):\\n\",\n \" self.ksp.solve(x, y)\\n\",\n \"\\n\",\n \" def __del__(self):\\n\",\n \" self.ksp.destroy()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"30\",\n \"metadata\": {},\n \"source\": [\n \"Next we can create the linear problem instance with all the required options\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"31\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"u_it = dolfinx.fem.Function(V, name=\\\"u_it\\\")\\n\",\n \"sigma_it = dolfinx.fem.Function(Q, name=\\\"sigma_it\\\")\\n\",\n \"petsc_options = {\\n\",\n \" \\\"ksp_error_if_not_converged\\\": True,\\n\",\n \" \\\"ksp_type\\\": \\\"gmres\\\",\\n\",\n \" \\\"ksp_rtol\\\": 1e-10,\\n\",\n \" \\\"ksp_atol\\\": 1e-10,\\n\",\n \" \\\"pc_type\\\": \\\"fieldsplit\\\",\\n\",\n \" \\\"pc_fieldsplit_type\\\": \\\"schur\\\",\\n\",\n \" \\\"pc_fieldsplit_schur_fact_type\\\": \\\"upper\\\",\\n\",\n \" \\\"pc_fieldsplit_schur_precondition\\\": \\\"user\\\",\\n\",\n \" f\\\"fieldsplit_{sigma_it.name}_0_ksp_type\\\": \\\"preonly\\\",\\n\",\n \" f\\\"fieldsplit_{sigma_it.name}_0_pc_type\\\": \\\"bjacobi\\\",\\n\",\n \" f\\\"fieldsplit_{u_it.name}_1_ksp_type\\\": \\\"preonly\\\",\\n\",\n \" f\\\"fieldsplit_{u_it.name}_1_pc_type\\\": \\\"python\\\",\\n\",\n \" f\\\"fieldsplit_{u_it.name}_1_pc_python_type\\\": __name__ + \\\".SchurInv\\\",\\n\",\n \" f\\\"fieldsplit_{u_it.name}_1_SchurInv_ksp_type\\\": \\\"preonly\\\",\\n\",\n \" f\\\"fieldsplit_{u_it.name}_1_SchurInv_pc_type\\\": \\\"hypre\\\",\\n\",\n \"}\\n\",\n \"w_it = (sigma_it, u_it)\\n\",\n \"problem = dolfinx.fem.petsc.LinearProblem(\\n\",\n \" a_blocked,\\n\",\n \" L_blocked,\\n\",\n \" u=w_it,\\n\",\n \" bcs=[bc_sigma],\\n\",\n \" petsc_options=petsc_options,\\n\",\n \" petsc_options_prefix=\\\"mp_\\\",\\n\",\n \" kind=\\\"nest\\\",\\n\",\n \")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"3b3b48be\",\n \"metadata\": {},\n \"source\": [\n \"```{admonition} NEST matrices\\n\",\n \"Note that instead of using `kind=\\\"mpi\\\"` we use `kind=\\\"nest\\\"` to indicate that we want to use a nested matrix structure\\n\",\n \"and employ the power of [PETSc fieldsplit](https://petsc.org/release/manual/ksp/#solving-block-matrices-with-pcfieldsplit).\\n\",\n \"```\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"32\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"start_it = time.perf_counter()\\n\",\n \"problem.solve()\\n\",\n \"end_it = time.perf_counter()\\n\",\n \"print(\\n\",\n \" f\\\"Iterative solver took {end_it - start_it:.2f} seconds\\\"\\n\",\n \" + f\\\" in {problem.solver.getIterationNumber()} iterations\\\"\\n\",\n \")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"33\",\n \"metadata\": {},\n \"source\": [\n \"We compute the error norms for the iterative solution\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"34\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"L2_u_it = dolfinx.fem.form(ufl.inner(u_it - u_exact, u_it - u_exact) * ufl.dx)\\n\",\n \"Hdiv_sigma_it = dolfinx.fem.form(\\n\",\n \" ufl.inner(\\n\",\n \" ufl.div(sigma_it) - ufl.div(ufl.grad(u_exact)),\\n\",\n \" ufl.div(sigma_it) - ufl.div(ufl.grad(u_exact)),\\n\",\n \" )\\n\",\n \" * ufl.dx\\n\",\n \")\\n\",\n \"local_u_error_it = dolfinx.fem.assemble_scalar(L2_u_it)\\n\",\n \"local_sigma_error_it = dolfinx.fem.assemble_scalar(Hdiv_sigma_it)\\n\",\n \"u_error_it = np.sqrt(mesh.comm.allreduce(local_u_error_it, op=MPI.SUM))\\n\",\n \"sigma_error_it = np.sqrt(mesh.comm.allreduce(local_sigma_error_it, op=MPI.SUM))\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"35\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"print(f\\\"Iterative solver, L2(u): {u_error_it:.2e}, H(div)(sigma): {sigma_error_it:.2e}\\\")\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"36\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"outputs\": [],\n \"source\": [\n \"np.testing.assert_allclose(u_h.x.array, u_it.x.array, rtol=1e-7, atol=1e-7)\\n\",\n \"np.testing.assert_allclose(sigma_h.x.array, sigma_it.x.array, rtol=1e-7, atol=1e-7)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"37\",\n \"metadata\": {},\n \"source\": [\n \"```{bibliography}\\n\",\n \" :filter: cited\\n\",\n \" :labelprefix:\\n\",\n \" :keyprefix: mp-\\n\",\n \"```\"\n ]\n }\n ],\n \"metadata\": {\n \"jupytext\": {\n \"cell_metadata_filter\": \"-all\",\n \"formats\": \"ipynb,py:light\",\n \"main_language\": \"python\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 5\n}\n" }, { "path": "chapter4/mixed_poisson.py", "content": "# ---\n# jupyter:\n# jupytext:\n# cell_metadata_filter: -all\n# formats: ipynb,py:light\n# text_representation:\n# extension: .py\n# format_name: light\n# format_version: '1.5'\n# jupytext_version: 1.18.1\n# ---\n\n# # Mixed Poisson with a Schur complement pre-conditioner\n# This example demonstrates how to use PETSC fieldsplits with custom preconditions in DOLFINx.\n# This example is heavily insipired by the [FEniCSx PCTools example](https://rafinex-external-rifle.gitlab.io/fenicsx-pctools/demo/demo_mixed-poisson.html)\n# which was presented in {cite}`mp-rehor2025pctools`.\n\n# We start with the mixed formulation of the Poisson equation, which is given by\n# \\begin{align}\n# \\sigma - \\nabla u &= 0&&\\text{in } \\Omega,\\\\\n# \\nabla \\cdot \\sigma &= -f&&\\text{in } \\Omega,\\\\\n# u &= u_D &&\\text{on } \\Gamma_D,\\\\\n# \\sigma \\cdot n &= g &&\\text{on } \\Gamma_N,\n# \\end{align}\n#\n# As in previous examples, we pick a manufactured solution to ensure that we can verify\n# the correctness of our implementation.\n# The manufactured solution is given by\n# \\begin{align}\n# u_{ex}(x, y) &= \\sin(\\pi x) + y^2.\n# \\end{align}\n\n\ndef u_ex(mod, x):\n return mod.sin(mod.pi * x[0]) + x[1] ** 2\n\n\n# We choose to solve the problem on a unit square,\n\n# +\nfrom mpi4py import MPI\nfrom petsc4py import PETSc\nimport dolfinx\n\nN = 400\nmesh = dolfinx.mesh.create_unit_square(MPI.COMM_WORLD, N, N)\n# -\n\n# where $\\Gamma_D = \\{(x, 0) \\vert x \\in [0, 1]\\}\\cup\\{ (x, 1) \\vert x \\in [0, 1]\\}$\n# and $\\Gamma_N = \\{(0, y) \\vert y \\in [0, 1]\\}\\cup\\{(1, y) \\vert y \\in [0, 1]\\}$.\n\n# +\nimport numpy as np\n\n\ndef Gamma_D(x):\n return (\n np.isclose(x[1], 0)\n | np.isclose(x[1], 1)\n | np.isclose(x[0], 0)\n | np.isclose(x[0], 1)\n )\n\n\ndef Gamma_N(x):\n return np.full_like(\n x[0], 0, dtype=bool\n ) # np.isclose(x[0], 0) | np.isclose(x[0], 1)\n\n\n# -\n\n# We define the function space for the vector-valued flux $p\\in Q$ as the zeroth order discontinuous Lagrange space,\n# while the scalar potential $u \\in V$ is defined in first order\n# [Brezzi-Douglas-Marini space](https://defelement.org/elements/brezzi-douglas-marini.html).\n\nV = dolfinx.fem.functionspace(mesh, (\"DG\", 0))\nQ = dolfinx.fem.functionspace(mesh, (\"BDM\", 1))\n\n# We define a `ufl.MixedFunctionSpace` to automatically handle the block structure of the problem\n\n# +\nimport ufl\n\nW = ufl.MixedFunctionSpace(*[Q, V])\n# -\n\n# Next, we have to define the bilinear and linear forms.\n# We do this as usual, by introducing a test functions $v\\in V$ and $\\tau\\in Q$ and a trial function $u\\in V$ and $q\\in Q$,\n# and integrate the first equation by parts.\n#\n# \\begin{align}\n# \\int_\\Omega \\sigma \\cdot \\tau - \\nabla u \\cdot \\tau ~\\mathrm{d} x &=\n# \\int_\\Omega \\sigma \\cdot \\tau + u \\nabla \\cdot \\tau ~\\mathrm{d} x\n# - \\sum_{f_i\\in \\mathit{Fi}}\\int_{f_i}\\left[u\\right] \\tau \\cdot \\mathbf{n}_i~\\mathrm{d}s\n# - \\int_{\\partial\\Omega} u \\tau\\cdot \\mathbf{n}~\\mathrm{d}s,\\\\\n# &=\\int_\\Omega \\sigma \\cdot \\tau + u \\nabla \\cdot \\tau ~\\mathrm{d} x\n# - \\int_{\\Gamma_D} u_D \\tau\\cdot \\mathbf{n}~\\mathrm{d}s,\\\\\n# \\end{align}\n#\n# where $f_i$ is an interior facet of the mesh, $\\mathbf{n}_i$ is an outwards pointing normal of one of the two\n# adjacent elements. We will enforce the boundary conditions strongly by using a\n# `dolfinx.fem.dirichletbc` on both $\\Gamma_N$, which makes its integral dissapear, while we enforced the Dirichlet boundary condition\n# on $\\Gamma_D$ weakly._\n# We enforce the continuity of $u$ weakly by removing the jump term.\n# Thus we end up with:\n#\n# Find $u\\in V_{u_D}, \\sigma \\in Q_{g}$ such that\n#\n# \\begin{align}\n# \\begin{split}\n# \\int_\\Omega \\sigma \\cdot \\tau + u \\nabla \\cdot \\tau ~\\mathrm{d} x&= \\int_{\\Gamma_D} u_D \\tau\\cdot \\mathbf{n}~\\mathrm{d}s,\\\\\\\\\n# \\int_\\Omega \\nabla \\cdot \\sigma v ~\\mathrm{d} x&=-\\int_\\Omega f v ~\\mathrm{d} x\n# \\end{split}\\qquad \\forall v \\in V_{0}, \\tau \\in Q_{0}\n# \\end{align}\n\nu_D = dolfinx.fem.Function(V)\nu_D.interpolate(lambda x: u_ex(np, x))\nmesh.topology.create_connectivity(mesh.topology.dim, mesh.topology.dim)\ngamma_d_facets = dolfinx.mesh.locate_entities_boundary(\n mesh, mesh.topology.dim - 1, Gamma_D\n)\ntag = 3\nft = dolfinx.mesh.meshtags(\n mesh,\n mesh.topology.dim - 1,\n gamma_d_facets,\n np.full(gamma_d_facets.shape[0], tag, dtype=np.int32),\n)\ndGammaD = ufl.Measure(\"ds\", domain=mesh, subdomain_data=ft, subdomain_id=tag)\n\n\nsigma, u = ufl.TrialFunctions(W)\ntau, v = ufl.TestFunctions(W)\nn = ufl.FacetNormal(mesh)\na = ufl.inner(sigma, tau) * ufl.dx\na += u * ufl.div(tau) * ufl.dx\na += ufl.inner(ufl.div(sigma), v) * ufl.dx\n\n# This can be split into a saddle point problem, with discretized matrices $A$ and $B$ and discretized\n# right-hand side $\\mathbf{b}$.\n# \\begin{align}\n# \\begin{pmatrix}\n# A & B^T\\\\\n# B & 0\n# \\end{pmatrix}\n# \\begin{pmatrix}\n# u_h\\\\\n# \\sigma_h\n# \\end{pmatrix}\n# = \\begin{pmatrix}\n# b_0\\\\\n# b_1\n# \\end{pmatrix}\n# \\end{align}\n# We can extract the block structure of the bilinear form using `ufl.extract_blocks`, which returns a nested list of bilinear forms.\n# You can also build this nested list by hand if you want to, but it is usually more error-prone.\n\n# +\na_blocked = ufl.extract_blocks(a)\n\nx = ufl.SpatialCoordinate(mesh)\nu_exact = u_ex(ufl, x)\nsigma_exact = ufl.grad(u_exact)\nf = -ufl.div(sigma_exact)\nL = ufl.inner(u_D, ufl.dot(tau, n)) * dGammaD - ufl.inner(f, v) * ufl.dx\nL_blocked = ufl.extract_blocks(L)\n# -\n\n# Next we create the Dirichlet boundary condition for $\\sigma$.\n# As we are using manufactured solutions for this problem, we could manually derive the explicit expression\n# for $\\sigma$ on the boundary $\\Gamma_N$.\n# However, in general this is not possible (especially for curved boundaries), and we have to use a more generic approach.\n# For this we will use the `dolfinx.fem.Expression` class to interpolate the expression into the function space $Q$.\n# This is done by evaluating the expression at the physical interpolation points of the mesh.\n# A convenience function for this is provided in the `interpolate_facet_expression` function below.\n\nimport numpy.typing as npt\nimport basix.ufl\n\n\ndef interpolate_facet_expression(\n Q: dolfinx.fem.FunctionSpace,\n expr: ufl.core.expr.Expr,\n facets: npt.NDArray[np.int32],\n) -> dolfinx.fem.Function:\n \"\"\"\n Interpolate a UFL-expression into a function space, only for the degrees of freedom assoicated with facets.\n \"\"\"\n domain = Q.mesh\n Q_el = Q.element\n fdim = domain.topology.dim - 1\n\n # Get coordinate element for facets of cell\n c_el = domain.ufl_domain().ufl_coordinate_element()\n facet_types = basix.cell.subentity_types(domain.basix_cell())[fdim]\n unique_facet_types = np.unique(facet_types)\n assert len(unique_facet_types) == 1, (\n \"All facets must have the same type for interpolation.\"\n )\n facet_type = facet_types[0]\n x_type = domain.geometry.x.dtype\n facet_cmap = basix.ufl.element(\n \"Lagrange\", facet_type, c_el.degree, shape=(domain.geometry.dim,), dtype=x_type\n )\n if np.issubdtype(x_type, np.float32):\n facet_cel = dolfinx.cpp.fem.CoordinateElement_float32(\n facet_cmap.basix_element._e\n )\n elif np.issubdtype(x_type, np.float64):\n facet_cel = dolfinx.cpp.fem.CoordinateElement_float64(\n facet_cmap.basix_element._e\n )\n else:\n raise TypeError(\n f\"Unsupported coordinate element type: {x_type}. \"\n \"Only float32 and float64 are supported.\"\n )\n # Pull back interpolation points from reference coordinate element to facet reference element\n ref_top = c_el.reference_topology\n ref_geom = c_el.reference_geometry\n reference_facet_points = None\n interpolation_points = Q_el.basix_element.x\n for i, points in enumerate(interpolation_points[fdim]):\n geom = ref_geom[ref_top[fdim][i]]\n ref_points = facet_cel.pull_back(points, geom)\n # Assert that interpolation points are all equal on all facets\n if reference_facet_points is None:\n reference_facet_points = ref_points\n else:\n assert np.allclose(reference_facet_points, ref_points)\n\n assert isinstance(reference_facet_points, np.ndarray)\n\n # Create expression for BC\n bndry_expr = dolfinx.fem.Expression(expr, reference_facet_points)\n\n # Compute number of interpolation points per sub entity\n points_per_entity = [sum(ip.shape[0] for ip in ips) for ips in interpolation_points]\n offsets = np.zeros(domain.topology.dim + 2, dtype=np.int32)\n offsets[1:] = np.cumsum(points_per_entity[: domain.topology.dim + 1])\n values_per_entity = np.zeros(\n (offsets[-1], domain.geometry.dim), dtype=dolfinx.default_scalar_type\n )\n\n # Map facet indices to (cell, local_facet) pairs\n boundary_entities = dolfinx.fem.compute_integration_domains(\n dolfinx.fem.IntegralType.exterior_facet, domain.topology, facets\n )\n\n # Compute and insert the correct values for the interpolation points on the facets\n entities = boundary_entities.reshape(-1, 2)\n values = np.zeros(entities.shape[0] * offsets[-1] * domain.geometry.dim)\n for i, entity in enumerate(entities):\n insert_pos = offsets[fdim] + reference_facet_points.shape[0] * entity[1]\n normal_on_facet = bndry_expr.eval(domain, entity.reshape(1, 2))\n values_per_entity[insert_pos : insert_pos + reference_facet_points.shape[0]] = (\n normal_on_facet.reshape(-1, domain.geometry.dim)\n )\n values[\n i * offsets[-1] * domain.geometry.dim : (i + 1)\n * offsets[-1]\n * domain.geometry.dim\n ] = values_per_entity.reshape(-1)\n # Use lower-level interpolation that takes in the function evaluated at the physical\n # interpolation points of the mesh.\n qh = dolfinx.fem.Function(Q)\n qh._cpp_object.interpolate(\n values.reshape(-1, domain.geometry.dim).T.copy(), boundary_entities[::2].copy()\n )\n qh.x.scatter_forward()\n return qh\n\n\nsigma_facets = dolfinx.mesh.locate_entities_boundary(\n mesh, mesh.topology.dim - 1, Gamma_N\n)\nn = ufl.FacetNormal(mesh)\ng = ufl.dot(ufl.grad(u_exact), n)\nsigma_bc = interpolate_facet_expression(Q, g, sigma_facets)\nbc_sigma = dolfinx.fem.dirichletbc(\n sigma_bc,\n dolfinx.fem.locate_dofs_topological(Q, mesh.topology.dim - 1, sigma_facets),\n)\nassert len(sigma_facets) == 0\n\n# Now that we have created the bilinear and linear form, and the boundary conditions,\n# we turn to solving the problem. For this we use the `dolfinx.fem.petsc.LinearProblem` class.\n# As opposed to the previous examples, we now have an explicit block structure, which we would like to\n# exploit when solving the problem. However, first we will solve the problem without any preconditioner\n# to have a baseline performance.\n\n# +\nimport dolfinx.fem.petsc\n\nproblem = dolfinx.fem.petsc.LinearProblem(\n a_blocked,\n L_blocked,\n bcs=[bc_sigma],\n petsc_options={\n \"ksp_type\": \"preonly\",\n \"pc_type\": \"lu\",\n \"pc_factor_mat_solver_type\": \"mumps\",\n \"ksp_error_if_not_converged\": True,\n \"mat_mumps_icntl_24\": 1,\n \"mat_mumps_icntl_25\": 0,\n },\n kind=\"mpi\",\n petsc_options_prefix=\"mixed_poisson_direct\",\n)\n# -\n\n# Note that we have specified `kind=\"mpi\"` in the initialization of the `LinearProblem`.\n# This is to inform DOLFINx that we wan to preserve the block structure of the problem when assembling.\n\nimport time\n\nstart = time.perf_counter()\n(sigma_h, u_h) = problem.solve()\nend = time.perf_counter()\nprint(f\"Direct solver took {end - start:.2f} seconds.\")\n\nL2_u = dolfinx.fem.form(ufl.inner(u_h - u_exact, u_h - u_exact) * ufl.dx)\nHdiv_sigma = dolfinx.fem.form(\n ufl.inner(\n ufl.div(sigma_h) - ufl.div(ufl.grad(u_exact)),\n ufl.div(sigma_h) - ufl.div(ufl.grad(u_exact)),\n )\n * ufl.dx\n)\nlocal_u_error = dolfinx.fem.assemble_scalar(L2_u)\nlocal_sigma_error = dolfinx.fem.assemble_scalar(Hdiv_sigma)\nu_error = np.sqrt(mesh.comm.allreduce(local_u_error, op=MPI.SUM))\nsigma_error = np.sqrt(mesh.comm.allreduce(local_sigma_error, op=MPI.SUM))\n\nprint(f\"Direct solver, L2(u): {u_error:.2e}, H(div)(sigma): {sigma_error:.2e}\")\n\n# ## Iterative solver with Schur complement preconditioner\n# As mentioned earlier, there are more efficient ways of solving this problem, than using a direct solver.\n# Especially with the saddle point structure of the problem, we can use a Schur complement preconditioner.\n# As described in [FEniCSx PCTools: Mixed Poisson](https://rafinex-external-rifle.gitlab.io/fenicsx-pctools/demo/demo_mixed-poisson.html),\n# Instead of wrapping the matrices in a custom wrapper, we can use `dolfinx.fem.petsc.LinearProblem` to solve the problem.\n\n# We start by defining the $S$ matrix in the Schur complement (see the aforementioned link for details on the variational formulation).\n\n# +\nalpha = dolfinx.fem.Constant(mesh, 4.0)\ngamma = dolfinx.fem.Constant(mesh, 9.0)\nh = ufl.CellDiameter(mesh)\ns = -(\n ufl.inner(ufl.grad(u), ufl.grad(v)) * ufl.dx\n - ufl.inner(ufl.avg(ufl.grad(v)), ufl.jump(u, n)) * ufl.dS\n - ufl.inner(ufl.jump(u, n), ufl.avg(ufl.grad(v))) * ufl.dS\n + (alpha / ufl.avg(h)) * ufl.inner(ufl.jump(u, n), ufl.jump(v, n)) * ufl.dS\n - ufl.inner(ufl.grad(u), v * n) * dGammaD\n - ufl.inner(u * n, ufl.grad(v)) * dGammaD\n + (gamma / h) * u * v * dGammaD\n)\n\nS = dolfinx.fem.petsc.assemble_matrix(dolfinx.fem.form(s))\nS.assemble()\n\n\nclass SchurInv:\n def setUp(self, pc):\n self.ksp = PETSc.KSP().create(mesh.comm)\n self.ksp.setOptionsPrefix(pc.getOptionsPrefix() + \"SchurInv_\")\n self.ksp.setOperators(S)\n self.ksp.setTolerances(atol=1e-10, rtol=1e-10)\n self.ksp.setFromOptions()\n\n def apply(self, pc, x, y):\n self.ksp.solve(x, y)\n\n def __del__(self):\n self.ksp.destroy()\n\n\n# -\n\n# Next we can create the linear problem instance with all the required options\n\nu_it = dolfinx.fem.Function(V, name=\"u_it\")\nsigma_it = dolfinx.fem.Function(Q, name=\"sigma_it\")\npetsc_options = {\n \"ksp_error_if_not_converged\": True,\n \"ksp_type\": \"gmres\",\n \"ksp_rtol\": 1e-10,\n \"ksp_atol\": 1e-10,\n \"pc_type\": \"fieldsplit\",\n \"pc_fieldsplit_type\": \"schur\",\n \"pc_fieldsplit_schur_fact_type\": \"upper\",\n \"pc_fieldsplit_schur_precondition\": \"user\",\n f\"fieldsplit_{sigma_it.name}_0_ksp_type\": \"preonly\",\n f\"fieldsplit_{sigma_it.name}_0_pc_type\": \"bjacobi\",\n f\"fieldsplit_{u_it.name}_1_ksp_type\": \"preonly\",\n f\"fieldsplit_{u_it.name}_1_pc_type\": \"python\",\n f\"fieldsplit_{u_it.name}_1_pc_python_type\": __name__ + \".SchurInv\",\n f\"fieldsplit_{u_it.name}_1_SchurInv_ksp_type\": \"preonly\",\n f\"fieldsplit_{u_it.name}_1_SchurInv_pc_type\": \"hypre\",\n}\nw_it = (sigma_it, u_it)\nproblem = dolfinx.fem.petsc.LinearProblem(\n a_blocked,\n L_blocked,\n u=w_it,\n bcs=[bc_sigma],\n petsc_options=petsc_options,\n petsc_options_prefix=\"mp_\",\n kind=\"nest\",\n)\n\n# ```{admonition} NEST matrices\n# Note that instead of using `kind=\"mpi\"` we use `kind=\"nest\"` to indicate that we want to use a nested matrix structure\n# and employ the power of [PETSc fieldsplit](https://petsc.org/release/manual/ksp/#solving-block-matrices-with-pcfieldsplit).\n# ```\n\nstart_it = time.perf_counter()\nproblem.solve()\nend_it = time.perf_counter()\nprint(\n f\"Iterative solver took {end_it - start_it:.2f} seconds\"\n + f\" in {problem.solver.getIterationNumber()} iterations\"\n)\n\n# We compute the error norms for the iterative solution\n\nL2_u_it = dolfinx.fem.form(ufl.inner(u_it - u_exact, u_it - u_exact) * ufl.dx)\nHdiv_sigma_it = dolfinx.fem.form(\n ufl.inner(\n ufl.div(sigma_it) - ufl.div(ufl.grad(u_exact)),\n ufl.div(sigma_it) - ufl.div(ufl.grad(u_exact)),\n )\n * ufl.dx\n)\nlocal_u_error_it = dolfinx.fem.assemble_scalar(L2_u_it)\nlocal_sigma_error_it = dolfinx.fem.assemble_scalar(Hdiv_sigma_it)\nu_error_it = np.sqrt(mesh.comm.allreduce(local_u_error_it, op=MPI.SUM))\nsigma_error_it = np.sqrt(mesh.comm.allreduce(local_sigma_error_it, op=MPI.SUM))\n\nprint(f\"Iterative solver, L2(u): {u_error_it:.2e}, H(div)(sigma): {sigma_error_it:.2e}\")\n\nnp.testing.assert_allclose(u_h.x.array, u_it.x.array, rtol=1e-7, atol=1e-7)\nnp.testing.assert_allclose(sigma_h.x.array, sigma_it.x.array, rtol=1e-7, atol=1e-7)\n\n\n# ```{bibliography}\n# :filter: cited\n# :labelprefix:\n# :keyprefix: mp-\n# ```\n" }, { "path": "chapter4/newton-solver.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"markdown\",\n \"id\": \"0\",\n \"metadata\": {},\n \"source\": [\n \"# Custom Newton solvers\\n\",\n \"Author: Jørgen S. Dokken\\n\",\n \"\\n\",\n \"Newtons method, as used in the [non-linear Poisson](./../chapter2/nonlinpoisson_code) problem, is a way of solving a non-linear equation as a sequence of linear equations.\\n\",\n \"\\n\",\n \"Given a function $F:\\\\mathbb{R}^M\\\\mapsto \\\\mathbb{R}^M$, we have that $u_k, u_{k+1}\\\\in \\\\mathbb{R}^M$ is related as:\\n\",\n \"\\n\",\n \"$$u_{k+1} = u_{k} - J_F(u_k)^{-1} F(u_k)$$\\n\",\n \"\\n\",\n \"where $J_F$ is the Jacobian matrix of $F$.\\n\",\n \"\\n\",\n \"We can rewrite this equation as $\\\\delta u_k = u_{k+1} - u_{k}$,\\n\",\n \"\\n\",\n \"$$\\n\",\n \"J_F(u_k)\\\\delta u_k = - F(u_k)\\n\",\n \"$$\\n\",\n \"\\n\",\n \"and\\n\",\n \"\\n\",\n \"$$\\n\",\n \"u_{k+1} = u_k + \\\\delta u_k.\\n\",\n \"$$\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"1\",\n \"metadata\": {},\n \"source\": [\n \"## Problem specification\\n\",\n \"We start by importing all packages needed to solve the problem.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"2\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"outputs\": [],\n \"source\": [\n \"import dolfinx\\n\",\n \"import dolfinx.fem.petsc\\n\",\n \"import matplotlib.pyplot as plt\\n\",\n \"import numpy as np\\n\",\n \"import pyvista\\n\",\n \"import ufl\\n\",\n \"from mpi4py import MPI\\n\",\n \"from petsc4py import PETSc\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"3\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"We will consider the following non-linear problem:\\n\",\n \"\\n\",\n \"$$ u^2 - 2 u = x^2 + 4x + 3 \\\\text{ in } [0,1] $$\\n\",\n \"For this problem, we have two solutions, $u=-x-1$, $u=x+3$.\\n\",\n \"We define these roots as python functions, and create an appropriate spacing for plotting these soultions.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"4\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"def root_0(x):\\n\",\n \" return 3 + x[0]\\n\",\n \"\\n\",\n \"\\n\",\n \"def root_1(x):\\n\",\n \" return -1 - x[0]\\n\",\n \"\\n\",\n \"\\n\",\n \"N = 10\\n\",\n \"roots = [root_0, root_1]\\n\",\n \"x_spacing = np.linspace(0, 1, N)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"5\",\n \"metadata\": {},\n \"source\": [\n \"We will start with an initial guess for this problem, $u_0 = 0$.\\n\",\n \"Next, we define the mesh, and the appropriate function space and function `uh` to hold the approximate solution.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"6\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"mesh = dolfinx.mesh.create_unit_interval(MPI.COMM_WORLD, N)\\n\",\n \"V = dolfinx.fem.functionspace(mesh, (\\\"Lagrange\\\", 1))\\n\",\n \"uh = dolfinx.fem.Function(V)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"7\",\n \"metadata\": {},\n \"source\": [\n \"## Definition of residual and Jacobian\\n\",\n \"Next, we define the variational form, by multiplying by a test function and integrating over the domain $[0,1]$\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"8\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"v = ufl.TestFunction(V)\\n\",\n \"x = ufl.SpatialCoordinate(mesh)\\n\",\n \"F = uh**2 * v * ufl.dx - 2 * uh * v * ufl.dx - (x[0] ** 2 + 4 * x[0] + 3) * v * ufl.dx\\n\",\n \"residual = dolfinx.fem.form(F)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"9\",\n \"metadata\": {},\n \"source\": [\n \"Next, we can define the jacobian $J_F$, by using `ufl.derivative`.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"10\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"J = ufl.derivative(F, uh)\\n\",\n \"jacobian = dolfinx.fem.form(J)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"11\",\n \"metadata\": {},\n \"source\": [\n \"As we will solve this problem in an iterative fashion, we would like to create the sparse matrix and vector containing the residual only once.\\n\",\n \"## Setup of iteration-independent structures\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"12\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"A = dolfinx.fem.petsc.create_matrix(jacobian)\\n\",\n \"L = dolfinx.fem.petsc.create_vector(dolfinx.fem.extract_function_spaces(residual))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"13\",\n \"metadata\": {},\n \"source\": [\n \"Next, we create the linear solver and the vector to hold `du`.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"0d5ce361\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"solver = PETSc.KSP().create(mesh.comm)\\n\",\n \"solver.setType(\\\"preonly\\\")\\n\",\n \"solver.getPC().setType(\\\"lu\\\")\\n\",\n \"solver.getPC().setFactorSolverType(\\\"mumps\\\")\\n\",\n \"solver.setErrorIfNotConverged(True)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"14\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"solver.setOperators(A)\\n\",\n \"du = dolfinx.fem.Function(V)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"15\",\n \"metadata\": {},\n \"source\": [\n \"We would like to monitor the evolution of `uh` for each iteration. Therefore, we get the dof coordinates, and sort them in increasing order.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"16\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"i = 0\\n\",\n \"coords = V.tabulate_dof_coordinates()[:, 0]\\n\",\n \"sort_order = np.argsort(coords)\\n\",\n \"max_iterations = 25\\n\",\n \"solutions = np.zeros((max_iterations + 1, len(coords)))\\n\",\n \"solutions[0] = uh.x.array[sort_order]\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"17\",\n \"metadata\": {},\n \"source\": [\n \"We are now ready to solve the linear problem.\\n\",\n \"At each iteration, we reassemble the Jacobian and residual, and use the norm of the magnitude of the update (`dx`) as a termination criteria.\\n\",\n \"## The Newton iterations\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"18\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"i = 0\\n\",\n \"while i < max_iterations:\\n\",\n \" # Assemble Jacobian and residual\\n\",\n \" with L.localForm() as loc_L:\\n\",\n \" loc_L.set(0)\\n\",\n \" A.zeroEntries()\\n\",\n \" dolfinx.fem.petsc.assemble_matrix(A, jacobian)\\n\",\n \" A.assemble()\\n\",\n \" dolfinx.fem.petsc.assemble_vector(L, residual)\\n\",\n \" L.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)\\n\",\n \"\\n\",\n \" # Scale residual by -1\\n\",\n \" L.scale(-1)\\n\",\n \" L.ghostUpdate(addv=PETSc.InsertMode.INSERT_VALUES, mode=PETSc.ScatterMode.FORWARD)\\n\",\n \"\\n\",\n \" # Solve linear problem\\n\",\n \" solver.solve(L, du.x.petsc_vec)\\n\",\n \" du.x.scatter_forward()\\n\",\n \" # Update u_{i+1} = u_i + delta u_i\\n\",\n \" uh.x.array[:] += du.x.array\\n\",\n \" i += 1\\n\",\n \"\\n\",\n \" # Compute norm of update\\n\",\n \" correction_norm = du.x.petsc_vec.norm(0)\\n\",\n \" PETSc.Sys.Print(f\\\"Iteration {i}: Correction norm {correction_norm}\\\")\\n\",\n \" if correction_norm < 1e-10:\\n\",\n \" break\\n\",\n \" solutions[i, :] = uh.x.array[sort_order]\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"19\",\n \"metadata\": {},\n \"source\": [\n \"We now compute the magnitude of the residual.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"20\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"dolfinx.fem.petsc.assemble_vector(L, residual)\\n\",\n \"PETSc.Sys.Print(f\\\"Final residual {L.norm(0)}\\\")\\n\",\n \"A.destroy()\\n\",\n \"L.destroy()\\n\",\n \"solver.destroy()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"21\",\n \"metadata\": {},\n \"source\": [\n \"## Visualization of Newton iterations\\n\",\n \"We next look at the evolution of the solution and the error of the solution when compared to the two exact roots of the problem.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"22\",\n \"metadata\": {\n \"lines_to_end_of_cell_marker\": 2\n },\n \"outputs\": [],\n \"source\": [\n \"# Plot solution for each of the iterations\\n\",\n \"fig = plt.figure(figsize=(15, 8))\\n\",\n \"for j, solution in enumerate(solutions[:i]):\\n\",\n \" plt.plot(coords[sort_order], solution, label=f\\\"Iteration {j}\\\")\\n\",\n \"\\n\",\n \"# Plot each of the roots of the problem, and compare the approximate solution with each of them\\n\",\n \"args = (\\\"--go\\\",)\\n\",\n \"for j, root in enumerate(roots):\\n\",\n \" u_ex = root(x)\\n\",\n \" L2_error = dolfinx.fem.form(ufl.inner(uh - u_ex, uh - u_ex) * ufl.dx)\\n\",\n \" global_L2 = mesh.comm.allreduce(dolfinx.fem.assemble_scalar(L2_error), op=MPI.SUM)\\n\",\n \" PETSc.Sys.Print(f\\\"L2-error (root {j}) {np.sqrt(global_L2)}\\\")\\n\",\n \"\\n\",\n \" kwargs = {} if j == 0 else {\\\"label\\\": \\\"u_exact\\\"}\\n\",\n \" plt.plot(x_spacing, root(x_spacing.reshape(1, -1)), *args, **kwargs)\\n\",\n \"plt.grid()\\n\",\n \"plt.legend()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"23\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"# Newton's method with DirichletBC\\n\",\n \"In the previous example, we did not consider handling of Dirichlet boundary conditions.\\n\",\n \"For this example, we will consider the [non-linear Poisson](./../chapter2/nonlinpoisson)-problem.\\n\",\n \"We start by defining the mesh, the analytical solution and the forcing term $f$.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"24\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"def q(u):\\n\",\n \" return 1 + u**2\\n\",\n \"\\n\",\n \"\\n\",\n \"domain = dolfinx.mesh.create_unit_square(MPI.COMM_WORLD, 10, 10)\\n\",\n \"x = ufl.SpatialCoordinate(domain)\\n\",\n \"u_ufl = 1 + x[0] + 2 * x[1]\\n\",\n \"f = -ufl.div(q(u_ufl) * ufl.grad(u_ufl))\\n\",\n \"\\n\",\n \"\\n\",\n \"def u_exact(x):\\n\",\n \" return eval(str(u_ufl))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"25\",\n \"metadata\": {},\n \"source\": [\n \"Next, we define the boundary condition `bc`, the residual `F` and the Jacobian `J`.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"26\",\n \"metadata\": {\n \"lines_to_next_cell\": 0\n },\n \"outputs\": [],\n \"source\": [\n \"V = dolfinx.fem.functionspace(domain, (\\\"Lagrange\\\", 1))\\n\",\n \"u_D = dolfinx.fem.Function(V)\\n\",\n \"u_D.interpolate(u_exact)\\n\",\n \"fdim = domain.topology.dim - 1\\n\",\n \"domain.topology.create_connectivity(fdim, fdim + 1)\\n\",\n \"boundary_facets = dolfinx.mesh.exterior_facet_indices(domain.topology)\\n\",\n \"bc = dolfinx.fem.dirichletbc(\\n\",\n \" u_D, dolfinx.fem.locate_dofs_topological(V, fdim, boundary_facets)\\n\",\n \")\\n\",\n \"\\n\",\n \"uh = dolfinx.fem.Function(V)\\n\",\n \"v = ufl.TestFunction(V)\\n\",\n \"F = q(uh) * ufl.dot(ufl.grad(uh), ufl.grad(v)) * ufl.dx - f * v * ufl.dx\\n\",\n \"J = ufl.derivative(F, uh)\\n\",\n \"residual = dolfinx.fem.form(F)\\n\",\n \"jacobian = dolfinx.fem.form(J)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"27\",\n \"metadata\": {},\n \"source\": [\n \"Next, we define the matrix `A`, right hand side vector `L` and the correction function `du`\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"28\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"du = dolfinx.fem.Function(V)\\n\",\n \"A = dolfinx.fem.petsc.create_matrix(jacobian)\\n\",\n \"L = dolfinx.fem.petsc.create_vector(dolfinx.fem.extract_function_spaces(residual))\\n\",\n \"solver = PETSc.KSP().create(mesh.comm)\\n\",\n \"solver.setOperators(A)\\n\",\n \"solver.setType(\\\"preonly\\\")\\n\",\n \"solver.getPC().setType(\\\"lu\\\")\\n\",\n \"solver.getPC().setFactorSolverType(\\\"mumps\\\")\\n\",\n \"solver.setErrorIfNotConverged(True)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"29\",\n \"metadata\": {\n \"lines_to_next_cell\": 2\n },\n \"source\": [\n \"Since this problem has strong Dirichlet conditions, we need to apply lifting to the right hand side of our Newton problem.\\n\",\n \"We previously had that we wanted to solve the system:\\n\",\n \"\\n\",\n \"$$\\n\",\n \"\\\\begin{align}\\n\",\n \"J_F(u_k)\\\\delta u_k &= - F(u_k)\\\\\\\\\\n\",\n \"u_{k+1} &= u_k + \\\\delta u_k\\n\",\n \"\\\\end{align}\\n\",\n \"$$\\n\",\n \"\\n\",\n \"we want $u_{k+1}\\\\vert_{bc}= u_D$. However, we do not know if $u_k\\\\vert_{bc}=u_D$.\\n\",\n \"Therefore, we want to apply the following boundary condition for our correction $\\\\delta u_k$\\n\",\n \"\\n\",\n \"$$\\n\",\n \"\\\\delta u_k\\\\vert_{bc} = u_D-u_k\\\\vert_{bc}\\n\",\n \"$$\\n\",\n \"\\n\",\n \"We therefore arrive at the following Newton scheme\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"30\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"i = 0\\n\",\n \"error = dolfinx.fem.form(\\n\",\n \" ufl.inner(uh - u_ufl, uh - u_ufl) * ufl.dx(metadata={\\\"quadrature_degree\\\": 4})\\n\",\n \")\\n\",\n \"L2_error = []\\n\",\n \"du_norm = []\\n\",\n \"while i < max_iterations:\\n\",\n \" # Assemble Jacobian and residual\\n\",\n \" with L.localForm() as loc_L:\\n\",\n \" loc_L.set(0)\\n\",\n \" A.zeroEntries()\\n\",\n \" dolfinx.fem.petsc.assemble_matrix(A, jacobian, bcs=[bc])\\n\",\n \" A.assemble()\\n\",\n \" dolfinx.fem.petsc.assemble_vector(L, residual)\\n\",\n \"\\n\",\n \" # Compute b - alpha * J(u_D-u_(i-1))\\n\",\n \" dolfinx.fem.petsc.apply_lifting(\\n\",\n \" L, [jacobian], [[bc]], x0=[uh.x.petsc_vec], alpha=-1.0\\n\",\n \" )\\n\",\n \" L.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)\\n\",\n \"\\n\",\n \" # Set du|_bc = - (u_{i-1}-u_D)\\n\",\n \" dolfinx.fem.petsc.set_bc(L, [bc], uh.x.petsc_vec, -1.0)\\n\",\n \" L.ghostUpdate(addv=PETSc.InsertMode.INSERT_VALUES, mode=PETSc.ScatterMode.FORWARD)\\n\",\n \"\\n\",\n \" # Compute negative residual\\n\",\n \" L.scale(-1)\\n\",\n \"\\n\",\n \" # Solve linear problem\\n\",\n \" solver.solve(L, du.x.petsc_vec)\\n\",\n \" du.x.scatter_forward()\\n\",\n \"\\n\",\n \" # Update u_{i+1} = u_i + delta u_i\\n\",\n \" uh.x.array[:] += du.x.array\\n\",\n \" i += 1\\n\",\n \"\\n\",\n \" # Compute norm of update\\n\",\n \" correction_norm = du.x.petsc_vec.norm(0)\\n\",\n \"\\n\",\n \" # Compute L2 error comparing to the analytical solution\\n\",\n \" L2_error.append(\\n\",\n \" np.sqrt(mesh.comm.allreduce(dolfinx.fem.assemble_scalar(error), op=MPI.SUM))\\n\",\n \" )\\n\",\n \" du_norm.append(correction_norm)\\n\",\n \" PETSc.Sys.Print(\\n\",\n \" f\\\"Iteration {i}: Correction norm {correction_norm}, L2 error: {L2_error[-1]}\\\",\\n\",\n \" flush=True,\\n\",\n \" )\\n\",\n \" if correction_norm < 1e-10:\\n\",\n \" break\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"31\",\n \"metadata\": {},\n \"source\": [\n \"We plot the $L^2$-error and the residual norm ($\\\\delta u$) per iteration\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"32\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"fig = plt.figure(figsize=(15, 8))\\n\",\n \"plt.subplot(121)\\n\",\n \"plt.plot(np.arange(i), L2_error)\\n\",\n \"plt.title(r\\\"$L^2(\\\\Omega)$-error of $u_h$\\\")\\n\",\n \"ax = plt.gca()\\n\",\n \"ax.set_yscale(\\\"log\\\")\\n\",\n \"plt.xlabel(\\\"Iterations\\\")\\n\",\n \"plt.ylabel(r\\\"$L^2$-error\\\")\\n\",\n \"plt.grid()\\n\",\n \"plt.subplot(122)\\n\",\n \"plt.title(r\\\"Residual of $\\\\vert\\\\vert\\\\delta u_i\\\\vert\\\\vert$\\\")\\n\",\n \"plt.plot(np.arange(i), du_norm)\\n\",\n \"ax = plt.gca()\\n\",\n \"ax.set_yscale(\\\"log\\\")\\n\",\n \"plt.xlabel(\\\"Iterations\\\")\\n\",\n \"plt.ylabel(r\\\"$\\\\vert\\\\vert \\\\delta u\\\\vert\\\\vert$\\\")\\n\",\n \"plt.grid()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"33\",\n \"metadata\": {},\n \"source\": [\n \"We compute the max error and plot the solution\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"34\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"error_max = domain.comm.allreduce(np.max(np.abs(uh.x.array - u_D.x.array)), op=MPI.MAX)\\n\",\n \"PETSc.Sys.Print(f\\\"Error_max: {error_max:.2e}\\\")\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"35\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"u_topology, u_cell_types, u_geometry = dolfinx.plot.vtk_mesh(V)\\n\",\n \"u_grid = pyvista.UnstructuredGrid(u_topology, u_cell_types, u_geometry)\\n\",\n \"u_grid.point_data[\\\"u\\\"] = uh.x.array.real\\n\",\n \"u_grid.set_active_scalars(\\\"u\\\")\\n\",\n \"u_plotter = pyvista.Plotter()\\n\",\n \"u_plotter.add_mesh(u_grid, show_edges=True)\\n\",\n \"u_plotter.view_xy()\\n\",\n \"if not pyvista.OFF_SCREEN:\\n\",\n \" u_plotter.show()\"\n ]\n }\n ],\n \"metadata\": {\n \"jupytext\": {\n \"formats\": \"ipynb,py:light\"\n },\n \"kernelspec\": {\n \"display_name\": \"Python 3 (ipykernel)\",\n \"language\": \"python\",\n \"name\": \"python3\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 5\n}\n" }, { "path": "chapter4/newton-solver.py", "content": "# ---\n# jupyter:\n# jupytext:\n# formats: ipynb,py:light\n# text_representation:\n# extension: .py\n# format_name: light\n# format_version: '1.5'\n# jupytext_version: 1.19.1\n# kernelspec:\n# display_name: Python 3 (ipykernel)\n# language: python\n# name: python3\n# ---\n\n# # Custom Newton solvers\n# Author: Jørgen S. Dokken\n#\n# Newtons method, as used in the [non-linear Poisson](./../chapter2/nonlinpoisson_code) problem, is a way of solving a non-linear equation as a sequence of linear equations.\n#\n# Given a function $F:\\mathbb{R}^M\\mapsto \\mathbb{R}^M$, we have that $u_k, u_{k+1}\\in \\mathbb{R}^M$ is related as:\n#\n# $$u_{k+1} = u_{k} - J_F(u_k)^{-1} F(u_k)$$\n#\n# where $J_F$ is the Jacobian matrix of $F$.\n#\n# We can rewrite this equation as $\\delta u_k = u_{k+1} - u_{k}$,\n#\n# $$\n# J_F(u_k)\\delta u_k = - F(u_k)\n# $$\n#\n# and\n#\n# $$\n# u_{k+1} = u_k + \\delta u_k.\n# $$\n\n# ## Problem specification\n# We start by importing all packages needed to solve the problem.\n\nimport dolfinx\nimport dolfinx.fem.petsc\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport pyvista\nimport ufl\nfrom mpi4py import MPI\nfrom petsc4py import PETSc\n\n\n# We will consider the following non-linear problem:\n#\n# $$ u^2 - 2 u = x^2 + 4x + 3 \\text{ in } [0,1] $$\n# For this problem, we have two solutions, $u=-x-1$, $u=x+3$.\n# We define these roots as python functions, and create an appropriate spacing for plotting these soultions.\n\n\n# +\ndef root_0(x):\n return 3 + x[0]\n\n\ndef root_1(x):\n return -1 - x[0]\n\n\nN = 10\nroots = [root_0, root_1]\nx_spacing = np.linspace(0, 1, N)\n# -\n\n# We will start with an initial guess for this problem, $u_0 = 0$.\n# Next, we define the mesh, and the appropriate function space and function `uh` to hold the approximate solution.\n\nmesh = dolfinx.mesh.create_unit_interval(MPI.COMM_WORLD, N)\nV = dolfinx.fem.functionspace(mesh, (\"Lagrange\", 1))\nuh = dolfinx.fem.Function(V)\n\n# ## Definition of residual and Jacobian\n# Next, we define the variational form, by multiplying by a test function and integrating over the domain $[0,1]$\n\nv = ufl.TestFunction(V)\nx = ufl.SpatialCoordinate(mesh)\nF = uh**2 * v * ufl.dx - 2 * uh * v * ufl.dx - (x[0] ** 2 + 4 * x[0] + 3) * v * ufl.dx\nresidual = dolfinx.fem.form(F)\n\n# Next, we can define the jacobian $J_F$, by using `ufl.derivative`.\n\nJ = ufl.derivative(F, uh)\njacobian = dolfinx.fem.form(J)\n\n# As we will solve this problem in an iterative fashion, we would like to create the sparse matrix and vector containing the residual only once.\n# ## Setup of iteration-independent structures\n\nA = dolfinx.fem.petsc.create_matrix(jacobian)\nL = dolfinx.fem.petsc.create_vector(dolfinx.fem.extract_function_spaces(residual))\n\n# Next, we create the linear solver and the vector to hold `du`.\n\nsolver = PETSc.KSP().create(mesh.comm)\nsolver.setType(\"preonly\")\nsolver.getPC().setType(\"lu\")\nsolver.getPC().setFactorSolverType(\"mumps\")\nsolver.setErrorIfNotConverged(True)\n\nsolver.setOperators(A)\ndu = dolfinx.fem.Function(V)\n\n# We would like to monitor the evolution of `uh` for each iteration. Therefore, we get the dof coordinates, and sort them in increasing order.\n\ni = 0\ncoords = V.tabulate_dof_coordinates()[:, 0]\nsort_order = np.argsort(coords)\nmax_iterations = 25\nsolutions = np.zeros((max_iterations + 1, len(coords)))\nsolutions[0] = uh.x.array[sort_order]\n\n# We are now ready to solve the linear problem.\n# At each iteration, we reassemble the Jacobian and residual, and use the norm of the magnitude of the update (`dx`) as a termination criteria.\n# ## The Newton iterations\n\ni = 0\nwhile i < max_iterations:\n # Assemble Jacobian and residual\n with L.localForm() as loc_L:\n loc_L.set(0)\n A.zeroEntries()\n dolfinx.fem.petsc.assemble_matrix(A, jacobian)\n A.assemble()\n dolfinx.fem.petsc.assemble_vector(L, residual)\n L.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)\n\n # Scale residual by -1\n L.scale(-1)\n L.ghostUpdate(addv=PETSc.InsertMode.INSERT_VALUES, mode=PETSc.ScatterMode.FORWARD)\n\n # Solve linear problem\n solver.solve(L, du.x.petsc_vec)\n du.x.scatter_forward()\n # Update u_{i+1} = u_i + delta u_i\n uh.x.array[:] += du.x.array\n i += 1\n\n # Compute norm of update\n correction_norm = du.x.petsc_vec.norm(0)\n PETSc.Sys.Print(f\"Iteration {i}: Correction norm {correction_norm}\")\n if correction_norm < 1e-10:\n break\n solutions[i, :] = uh.x.array[sort_order]\n\n# We now compute the magnitude of the residual.\n\ndolfinx.fem.petsc.assemble_vector(L, residual)\nPETSc.Sys.Print(f\"Final residual {L.norm(0)}\")\nA.destroy()\nL.destroy()\nsolver.destroy()\n\n# ## Visualization of Newton iterations\n# We next look at the evolution of the solution and the error of the solution when compared to the two exact roots of the problem.\n\n# +\n# Plot solution for each of the iterations\nfig = plt.figure(figsize=(15, 8))\nfor j, solution in enumerate(solutions[:i]):\n plt.plot(coords[sort_order], solution, label=f\"Iteration {j}\")\n\n# Plot each of the roots of the problem, and compare the approximate solution with each of them\nargs = (\"--go\",)\nfor j, root in enumerate(roots):\n u_ex = root(x)\n L2_error = dolfinx.fem.form(ufl.inner(uh - u_ex, uh - u_ex) * ufl.dx)\n global_L2 = mesh.comm.allreduce(dolfinx.fem.assemble_scalar(L2_error), op=MPI.SUM)\n PETSc.Sys.Print(f\"L2-error (root {j}) {np.sqrt(global_L2)}\")\n\n kwargs = {} if j == 0 else {\"label\": \"u_exact\"}\n plt.plot(x_spacing, root(x_spacing.reshape(1, -1)), *args, **kwargs)\nplt.grid()\nplt.legend()\n\n\n# -\n\n# # Newton's method with DirichletBC\n# In the previous example, we did not consider handling of Dirichlet boundary conditions.\n# For this example, we will consider the [non-linear Poisson](./../chapter2/nonlinpoisson)-problem.\n# We start by defining the mesh, the analytical solution and the forcing term $f$.\n\n\n# +\ndef q(u):\n return 1 + u**2\n\n\ndomain = dolfinx.mesh.create_unit_square(MPI.COMM_WORLD, 10, 10)\nx = ufl.SpatialCoordinate(domain)\nu_ufl = 1 + x[0] + 2 * x[1]\nf = -ufl.div(q(u_ufl) * ufl.grad(u_ufl))\n\n\ndef u_exact(x):\n return eval(str(u_ufl))\n\n\n# -\n\n# Next, we define the boundary condition `bc`, the residual `F` and the Jacobian `J`.\n\n# +\nV = dolfinx.fem.functionspace(domain, (\"Lagrange\", 1))\nu_D = dolfinx.fem.Function(V)\nu_D.interpolate(u_exact)\nfdim = domain.topology.dim - 1\ndomain.topology.create_connectivity(fdim, fdim + 1)\nboundary_facets = dolfinx.mesh.exterior_facet_indices(domain.topology)\nbc = dolfinx.fem.dirichletbc(\n u_D, dolfinx.fem.locate_dofs_topological(V, fdim, boundary_facets)\n)\n\nuh = dolfinx.fem.Function(V)\nv = ufl.TestFunction(V)\nF = q(uh) * ufl.dot(ufl.grad(uh), ufl.grad(v)) * ufl.dx - f * v * ufl.dx\nJ = ufl.derivative(F, uh)\nresidual = dolfinx.fem.form(F)\njacobian = dolfinx.fem.form(J)\n# -\n# Next, we define the matrix `A`, right hand side vector `L` and the correction function `du`\n\ndu = dolfinx.fem.Function(V)\nA = dolfinx.fem.petsc.create_matrix(jacobian)\nL = dolfinx.fem.petsc.create_vector(dolfinx.fem.extract_function_spaces(residual))\nsolver = PETSc.KSP().create(mesh.comm)\nsolver.setOperators(A)\nsolver.setType(\"preonly\")\nsolver.getPC().setType(\"lu\")\nsolver.getPC().setFactorSolverType(\"mumps\")\nsolver.setErrorIfNotConverged(True)\n\n# Since this problem has strong Dirichlet conditions, we need to apply lifting to the right hand side of our Newton problem.\n# We previously had that we wanted to solve the system:\n#\n# $$\n# \\begin{align}\n# J_F(u_k)\\delta u_k &= - F(u_k)\\\\\n# u_{k+1} &= u_k + \\delta u_k\n# \\end{align}\n# $$\n#\n# we want $u_{k+1}\\vert_{bc}= u_D$. However, we do not know if $u_k\\vert_{bc}=u_D$.\n# Therefore, we want to apply the following boundary condition for our correction $\\delta u_k$\n#\n# $$\n# \\delta u_k\\vert_{bc} = u_D-u_k\\vert_{bc}\n# $$\n#\n# We therefore arrive at the following Newton scheme\n\n\ni = 0\nerror = dolfinx.fem.form(\n ufl.inner(uh - u_ufl, uh - u_ufl) * ufl.dx(metadata={\"quadrature_degree\": 4})\n)\nL2_error = []\ndu_norm = []\nwhile i < max_iterations:\n # Assemble Jacobian and residual\n with L.localForm() as loc_L:\n loc_L.set(0)\n A.zeroEntries()\n dolfinx.fem.petsc.assemble_matrix(A, jacobian, bcs=[bc])\n A.assemble()\n dolfinx.fem.petsc.assemble_vector(L, residual)\n\n # Compute b - alpha * J(u_D-u_(i-1))\n dolfinx.fem.petsc.apply_lifting(\n L, [jacobian], [[bc]], x0=[uh.x.petsc_vec], alpha=-1.0\n )\n L.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)\n\n # Set du|_bc = - (u_{i-1}-u_D)\n dolfinx.fem.petsc.set_bc(L, [bc], uh.x.petsc_vec, -1.0)\n L.ghostUpdate(addv=PETSc.InsertMode.INSERT_VALUES, mode=PETSc.ScatterMode.FORWARD)\n\n # Compute negative residual\n L.scale(-1)\n\n # Solve linear problem\n solver.solve(L, du.x.petsc_vec)\n du.x.scatter_forward()\n\n # Update u_{i+1} = u_i + delta u_i\n uh.x.array[:] += du.x.array\n i += 1\n\n # Compute norm of update\n correction_norm = du.x.petsc_vec.norm(0)\n\n # Compute L2 error comparing to the analytical solution\n L2_error.append(\n np.sqrt(mesh.comm.allreduce(dolfinx.fem.assemble_scalar(error), op=MPI.SUM))\n )\n du_norm.append(correction_norm)\n PETSc.Sys.Print(\n f\"Iteration {i}: Correction norm {correction_norm}, L2 error: {L2_error[-1]}\",\n flush=True,\n )\n if correction_norm < 1e-10:\n break\n\n# We plot the $L^2$-error and the residual norm ($\\delta u$) per iteration\n\nfig = plt.figure(figsize=(15, 8))\nplt.subplot(121)\nplt.plot(np.arange(i), L2_error)\nplt.title(r\"$L^2(\\Omega)$-error of $u_h$\")\nax = plt.gca()\nax.set_yscale(\"log\")\nplt.xlabel(\"Iterations\")\nplt.ylabel(r\"$L^2$-error\")\nplt.grid()\nplt.subplot(122)\nplt.title(r\"Residual of $\\vert\\vert\\delta u_i\\vert\\vert$\")\nplt.plot(np.arange(i), du_norm)\nax = plt.gca()\nax.set_yscale(\"log\")\nplt.xlabel(\"Iterations\")\nplt.ylabel(r\"$\\vert\\vert \\delta u\\vert\\vert$\")\nplt.grid()\n\n# We compute the max error and plot the solution\n\nerror_max = domain.comm.allreduce(np.max(np.abs(uh.x.array - u_D.x.array)), op=MPI.MAX)\nPETSc.Sys.Print(f\"Error_max: {error_max:.2e}\")\n\nu_topology, u_cell_types, u_geometry = dolfinx.plot.vtk_mesh(V)\nu_grid = pyvista.UnstructuredGrid(u_topology, u_cell_types, u_geometry)\nu_grid.point_data[\"u\"] = uh.x.array.real\nu_grid.set_active_scalars(\"u\")\nu_plotter = pyvista.Plotter()\nu_plotter.add_mesh(u_grid, show_edges=True)\nu_plotter.view_xy()\nif not pyvista.OFF_SCREEN:\n u_plotter.show()\n" }, { "path": "chapter4/solvers.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Solver configuration\\n\",\n \"Author: Jørgen S. Dokken\\n\",\n \"\\n\",\n \"In this section, we will go through how to specify what linear algebra solver we would like to use to solve our PDEs, as well as how to verify the implementation by considering convergence rates.\\n\",\n \"\\n\",\n \"$$\\n\",\n \"-\\\\Delta u = f \\\\text{ in } \\\\Omega\\n\",\n \"$$\\n\",\n \"$$\\n\",\n \"u = u_D \\\\text{ on } \\\\partial \\\\Omega.\\n\",\n \"$$\\n\",\n \"Using the manufactured solution $u_D=\\\\cos(2\\\\pi x)\\\\cos(2\\\\pi y)$, we obtain $f=8\\\\pi^2\\\\cos(2\\\\pi x)\\\\cos(2\\\\pi y)$.\\n\",\n \"We start by creating a generic module for evaluating the analytical solution at any point $x$.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"from dolfinx.fem import dirichletbc, functionspace, Function, locate_dofs_topological\\n\",\n \"from dolfinx.fem.petsc import LinearProblem\\n\",\n \"from dolfinx.mesh import create_unit_square, locate_entities_boundary\\n\",\n \"\\n\",\n \"from mpi4py import MPI\\n\",\n \"from petsc4py import PETSc\\n\",\n \"from ufl import SpatialCoordinate, TestFunction, TrialFunction, div, dx, inner, grad\\n\",\n \"\\n\",\n \"import numpy\\n\",\n \"import ufl\\n\",\n \"\\n\",\n \"\\n\",\n \"def u_ex(mod):\\n\",\n \" return lambda x: mod.cos(2 * mod.pi * x[0]) * mod.cos(2 * mod.pi * x[1])\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"Note that the return type of `u_ex` is a `lambda` function. Thus, we can create two different lambda functions, one using `numpy` (which will be used for interpolation) and one using `ufl` (which will be used for defining the source term)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"u_numpy = u_ex(numpy)\\n\",\n \"u_ufl = u_ex(ufl)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We start by using ufl to define our source term, using `ufl.SpatialCoordinate` as input to `u_ufl`.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"mesh = create_unit_square(MPI.COMM_WORLD, 30, 30)\\n\",\n \"x = SpatialCoordinate(mesh)\\n\",\n \"f = -div(grad(u_ufl(x)))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"Next, we define our linear variational problem\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"V = functionspace(mesh, (\\\"Lagrange\\\", 1))\\n\",\n \"u = TrialFunction(V)\\n\",\n \"v = TestFunction(V)\\n\",\n \"a = inner(grad(u), grad(v)) * dx\\n\",\n \"L = f * v * dx\\n\",\n \"u_bc = Function(V)\\n\",\n \"u_bc.interpolate(u_numpy)\\n\",\n \"facets = locate_entities_boundary(\\n\",\n \" mesh, mesh.topology.dim - 1, lambda x: numpy.full(x.shape[1], True)\\n\",\n \")\\n\",\n \"dofs = locate_dofs_topological(V, mesh.topology.dim - 1, facets)\\n\",\n \"bcs = [dirichletbc(u_bc, dofs)]\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We start by solving the problem with an LU factorization, a direct solver method (similar to Gaussian elimination).\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"default_problem = LinearProblem(\\n\",\n \" a,\\n\",\n \" L,\\n\",\n \" bcs=bcs,\\n\",\n \" petsc_options={\\\"ksp_type\\\": \\\"preonly\\\", \\\"pc_type\\\": \\\"lu\\\"},\\n\",\n \" petsc_options_prefix=\\\"poisson_l\\\",\\n\",\n \")\\n\",\n \"uh = default_problem.solve()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We now look at the solver process by inspecting the `PETSc`-solver. As the view-options in PETSc are not adjusted for notebooks (`solver.view()` will print output to the terminal if used in a `.py` file), we write the solver output to file and read it in and print the output.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"lu_solver = default_problem.solver\\n\",\n \"viewer = PETSc.Viewer().createASCII(\\\"lu_output.txt\\\")\\n\",\n \"lu_solver.view(viewer)\\n\",\n \"solver_output = open(\\\"lu_output.txt\\\", \\\"r\\\")\\n\",\n \"for line in solver_output.readlines():\\n\",\n \" print(line)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"This is a very robust and simple method, and is the recommended method up to a few thousand unknowns and can be efficiently used for many 2D and smaller 3D problems. However, sparse LU decomposition quickly becomes slow, as for a $N\\\\times N$-matrix the number of floating point operations scales as $\\\\sim (2/3)N^3$.\\n\",\n \"\\n\",\n \"For large problems, we instead need to use an iterative method which is faster and requires less memory.\\n\",\n \"## Choosing a linear solver and preconditioner\\n\",\n \"As the Poisson equation results in a symmetric, positive definite system matrix, the optimal Krylov solver is the conjugate gradient (Lagrange) method. The default preconditioner is the incomplete LU factorization (ILU), which is a popular and robust overall preconditioner. We can change the preconditioner by setting `\\\"pc_type\\\"` to some of the other preconditioners in petsc, which you can find at [PETSc KSP solvers](https://petsc.org/release/manual/ksp/#tab-kspdefaults) and [PETSc preconditioners](https://petsc.org/release/manual/ksp/#tab-pcdefaults).\\n\",\n \"You can set any option in `PETSc` through the `petsc_options` input, such as the absolute tolerance (`\\\"ksp_atol\\\"`), relative tolerance (`\\\"ksp_rtol\\\"`) and maximum number of iterations (`\\\"ksp_max_it\\\"`).\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"cg_problem = LinearProblem(\\n\",\n \" a,\\n\",\n \" L,\\n\",\n \" bcs=bcs,\\n\",\n \" petsc_options={\\n\",\n \" \\\"ksp_type\\\": \\\"cg\\\",\\n\",\n \" \\\"ksp_rtol\\\": 1e-6,\\n\",\n \" \\\"ksp_atol\\\": 1e-10,\\n\",\n \" \\\"ksp_max_it\\\": 1000,\\n\",\n \" },\\n\",\n \" petsc_options_prefix=\\\"poisson_cg_\\\",\\n\",\n \")\\n\",\n \"uh = cg_problem.solve()\\n\",\n \"cg_solver = cg_problem.solver\\n\",\n \"viewer = PETSc.Viewer().createASCII(\\\"cg_output.txt\\\")\\n\",\n \"cg_solver.view(viewer)\\n\",\n \"solver_output = open(\\\"cg_output.txt\\\", \\\"r\\\")\\n\",\n \"for line in solver_output.readlines():\\n\",\n \" print(line)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"For non-symmetric problems, a Krylov solver for non-symmetric systems, such as GMRES is better.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"gmres_problem = LinearProblem(\\n\",\n \" a,\\n\",\n \" L,\\n\",\n \" bcs=bcs,\\n\",\n \" petsc_options={\\n\",\n \" \\\"ksp_type\\\": \\\"gmres\\\",\\n\",\n \" \\\"ksp_rtol\\\": 1e-6,\\n\",\n \" \\\"ksp_atol\\\": 1e-10,\\n\",\n \" \\\"ksp_max_it\\\": 1000,\\n\",\n \" \\\"pc_type\\\": \\\"none\\\",\\n\",\n \" },\\n\",\n \" petsc_options_prefix=\\\"poisson_gmres_\\\",\\n\",\n \")\\n\",\n \"uh = gmres_problem.solve()\\n\",\n \"gmres_solver = gmres_problem.solver\\n\",\n \"viewer = PETSc.Viewer().createASCII(\\\"gmres_output.txt\\\")\\n\",\n \"gmres_solver.view(viewer)\\n\",\n \"solver_output = open(\\\"gmres_output.txt\\\", \\\"r\\\")\\n\",\n \"for line in solver_output.readlines():\\n\",\n \" print(line)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"```{admonition} A remark regarding verification using iterative solvers\\n\",\n \"When we consider manufactured solutions where we expect the resulting error to be of machine precision, it gets complicated when we use iterative methods. The problem is to keep the error due to the iterative solution smaller than the tolerance used in the iterative test. For linear elements and small meshes, a tolerance of between $10^{-11}$ and $10^{-12}$ works well in the case of Krylov solvers too.\\n\",\n \"```\"\n ]\n }\n ],\n \"metadata\": {\n \"jupytext\": {\n \"formats\": \"ipynb,py:light\"\n },\n \"kernelspec\": {\n \"display_name\": \"Python 3 (ipykernel)\",\n \"language\": \"python\",\n \"name\": \"python3\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 3\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython3\",\n \"version\": \"3.10.12\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 4\n}\n" }, { "path": "chapter4/solvers.py", "content": "# ---\n# jupyter:\n# jupytext:\n# formats: ipynb,py:light\n# text_representation:\n# extension: .py\n# format_name: light\n# format_version: '1.5'\n# jupytext_version: 1.18.1\n# kernelspec:\n# display_name: Python 3 (ipykernel)\n# language: python\n# name: python3\n# ---\n\n# # Solver configuration\n# Author: Jørgen S. Dokken\n#\n# In this section, we will go through how to specify what linear algebra solver we would like to use to solve our PDEs, as well as how to verify the implementation by considering convergence rates.\n#\n# $$\n# -\\Delta u = f \\text{ in } \\Omega\n# $$\n# $$\n# u = u_D \\text{ on } \\partial \\Omega.\n# $$\n# Using the manufactured solution $u_D=\\cos(2\\pi x)\\cos(2\\pi y)$, we obtain $f=8\\pi^2\\cos(2\\pi x)\\cos(2\\pi y)$.\n# We start by creating a generic module for evaluating the analytical solution at any point $x$.\n\n# +\nfrom dolfinx.fem import dirichletbc, functionspace, Function, locate_dofs_topological\nfrom dolfinx.fem.petsc import LinearProblem\nfrom dolfinx.mesh import create_unit_square, locate_entities_boundary\n\nfrom mpi4py import MPI\nfrom petsc4py import PETSc\nfrom ufl import SpatialCoordinate, TestFunction, TrialFunction, div, dx, inner, grad\n\nimport numpy\nimport ufl\n\n\ndef u_ex(mod):\n return lambda x: mod.cos(2 * mod.pi * x[0]) * mod.cos(2 * mod.pi * x[1])\n\n\n# -\n\n# Note that the return type of `u_ex` is a `lambda` function. Thus, we can create two different lambda functions, one using `numpy` (which will be used for interpolation) and one using `ufl` (which will be used for defining the source term)\n\nu_numpy = u_ex(numpy)\nu_ufl = u_ex(ufl)\n\n# We start by using ufl to define our source term, using `ufl.SpatialCoordinate` as input to `u_ufl`.\n\nmesh = create_unit_square(MPI.COMM_WORLD, 30, 30)\nx = SpatialCoordinate(mesh)\nf = -div(grad(u_ufl(x)))\n\n# Next, we define our linear variational problem\n\nV = functionspace(mesh, (\"Lagrange\", 1))\nu = TrialFunction(V)\nv = TestFunction(V)\na = inner(grad(u), grad(v)) * dx\nL = f * v * dx\nu_bc = Function(V)\nu_bc.interpolate(u_numpy)\nfacets = locate_entities_boundary(\n mesh, mesh.topology.dim - 1, lambda x: numpy.full(x.shape[1], True)\n)\ndofs = locate_dofs_topological(V, mesh.topology.dim - 1, facets)\nbcs = [dirichletbc(u_bc, dofs)]\n\n# We start by solving the problem with an LU factorization, a direct solver method (similar to Gaussian elimination).\n\ndefault_problem = LinearProblem(\n a,\n L,\n bcs=bcs,\n petsc_options={\"ksp_type\": \"preonly\", \"pc_type\": \"lu\"},\n petsc_options_prefix=\"poisson_l\",\n)\nuh = default_problem.solve()\n\n# We now look at the solver process by inspecting the `PETSc`-solver. As the view-options in PETSc are not adjusted for notebooks (`solver.view()` will print output to the terminal if used in a `.py` file), we write the solver output to file and read it in and print the output.\n\nlu_solver = default_problem.solver\nviewer = PETSc.Viewer().createASCII(\"lu_output.txt\")\nlu_solver.view(viewer)\nsolver_output = open(\"lu_output.txt\", \"r\")\nfor line in solver_output.readlines():\n print(line)\n\n# This is a very robust and simple method, and is the recommended method up to a few thousand unknowns and can be efficiently used for many 2D and smaller 3D problems. However, sparse LU decomposition quickly becomes slow, as for a $N\\times N$-matrix the number of floating point operations scales as $\\sim (2/3)N^3$.\n#\n# For large problems, we instead need to use an iterative method which is faster and requires less memory.\n# ## Choosing a linear solver and preconditioner\n# As the Poisson equation results in a symmetric, positive definite system matrix, the optimal Krylov solver is the conjugate gradient (Lagrange) method. The default preconditioner is the incomplete LU factorization (ILU), which is a popular and robust overall preconditioner. We can change the preconditioner by setting `\"pc_type\"` to some of the other preconditioners in petsc, which you can find at [PETSc KSP solvers](https://petsc.org/release/manual/ksp/#tab-kspdefaults) and [PETSc preconditioners](https://petsc.org/release/manual/ksp/#tab-pcdefaults).\n# You can set any option in `PETSc` through the `petsc_options` input, such as the absolute tolerance (`\"ksp_atol\"`), relative tolerance (`\"ksp_rtol\"`) and maximum number of iterations (`\"ksp_max_it\"`).\n\ncg_problem = LinearProblem(\n a,\n L,\n bcs=bcs,\n petsc_options={\n \"ksp_type\": \"cg\",\n \"ksp_rtol\": 1e-6,\n \"ksp_atol\": 1e-10,\n \"ksp_max_it\": 1000,\n },\n petsc_options_prefix=\"poisson_cg_\",\n)\nuh = cg_problem.solve()\ncg_solver = cg_problem.solver\nviewer = PETSc.Viewer().createASCII(\"cg_output.txt\")\ncg_solver.view(viewer)\nsolver_output = open(\"cg_output.txt\", \"r\")\nfor line in solver_output.readlines():\n print(line)\n\n# For non-symmetric problems, a Krylov solver for non-symmetric systems, such as GMRES is better.\n\ngmres_problem = LinearProblem(\n a,\n L,\n bcs=bcs,\n petsc_options={\n \"ksp_type\": \"gmres\",\n \"ksp_rtol\": 1e-6,\n \"ksp_atol\": 1e-10,\n \"ksp_max_it\": 1000,\n \"pc_type\": \"none\",\n },\n petsc_options_prefix=\"poisson_gmres_\",\n)\nuh = gmres_problem.solve()\ngmres_solver = gmres_problem.solver\nviewer = PETSc.Viewer().createASCII(\"gmres_output.txt\")\ngmres_solver.view(viewer)\nsolver_output = open(\"gmres_output.txt\", \"r\")\nfor line in solver_output.readlines():\n print(line)\n\n# ```{admonition} A remark regarding verification using iterative solvers\n# When we consider manufactured solutions where we expect the resulting error to be of machine precision, it gets complicated when we use iterative methods. The problem is to keep the error due to the iterative solution smaller than the tolerance used in the iterative test. For linear elements and small meshes, a tolerance of between $10^{-11}$ and $10^{-12}$ works well in the case of Krylov solvers too.\n# ```\n" }, { "path": "docker/Dockerfile", "content": "# Execute from root of repo as: docker buildx build --platform=linux/arm64,linux/amd64 -f docker/Dockerfile . --progress=plain\n\nFROM ghcr.io/fenics/dolfinx/lab:stable\n\nARG TARGETPLATFORM\n\n\nENV DEB_PYTHON_INSTALL_LAYOUT=deb_system\nENV HDF5_MPI=\"ON\"\nENV HDF5_DIR=\"/usr/local\"\n\nWORKDIR /tmp/\n# Requirements for pyvista (gl1 and render1) and jupyterlab (nodejs and curl)\nRUN apt-get update && apt-get install -y libgl1-mesa-dev mesa-utils curl\n# RUN curl -sL https://deb.nodesource.com/setup_18.x -o nodesource_setup.sh && \\\n# bash nodesource_setup.sh && \\\n# apt -y install nodejs\n\n# Install netgen from source\nRUN apt-get update && \\\n apt-get -y install python3 python3-tk libpython3-dev libxmu-dev tk-dev tcl-dev cmake git g++ libglu1-mesa-dev liblapacke-dev libocct-data-exchange-dev libocct-draw-dev occt-misc libtbb-dev libxi-dev\nWORKDIR /ngsuite\nRUN git clone --branch=v6.2.2505 --single-branch https://github.com/NGSolve/netgen.git netgen-src\nWORKDIR /ngsuite/netgen-src\nRUN git submodule update --init --recursive\nWORKDIR /ngsuite\nRUN mkdir netgen-build\nRUN mkdir netgen-install\nRUN cmake -DCMAKE=\"-cxx-flags=-flax-vector-conversions\" -DCMAKE_INSTALL_PREFIX=/ngsuite/netgen-install /ngsuite/netgen-src\nRUN make\nRUN make install\nENV NETGENDIR=/ngsuite/netgen-install/bin\nENV PATH=$NETGENDIR:$PATH\nENV PYTHONPATH=$NETGENDIR/../lib/python3.12/site-packages:${PYTHONPATH}:${PYTHONPATH}:$PATH\n\n# Install vtk\nRUN python3 -m pip install vtk\n\nADD pyproject.toml /tmp/pyproject.toml\nWORKDIR /tmp\n# As we install netgen from source we don't need the netgen deps here\nRUN python3 -m pip install --no-cache-dir --no-binary=h5py -v .\nRUN python3 -m pip cache purge\nRUN python3 -m pip install ngsPETSc --no-deps\n\nENV PYVISTA_OFF_SCREEN=\"false\"\nENV PYVISTA_JUPYTER_BACKEND=\"trame\"\nENV LIBGL_ALWAYS_SOFTWARE=1\n\nENV PYVISTA_OFF_SCREEN=\"false\"\nENV PYVISTA_JUPYTER_BACKEND=\"trame\"\nENV LIBGL_ALWAYS_SOFTWARE=1\n\nENTRYPOINT [\"jupyter\", \"lab\", \"--ip\", \"0.0.0.0\", \"--no-browser\", \"--allow-root\"]\n" }, { "path": "fem.md", "content": "# An overview of the FEniCS Project\n\nThe FEniCS project is a research and software project aimed at creating mathematical methods and software for solving partial differential equations (PDEs). This includes creating intuitive, efficient and flexible software. The project was initiated in 2003, and is developed in a collaboration among researchers from a number of universities and research institutes around the world. For the latest updates and more information about the FEniCS project, visit the [FEniCS](https://fenicsproject.org) webpage.\n\nThe latest version of the FEniCS project, FEniCSx, consists of several building blocks, namely [DOLFINx](https://github.com/FEniCS/dolfinx), [UFL](https://github.com/FEniCS/ufl), [FFCx](https://github.com/FEniCS/ffcx), and [Basix](https://github.com/FEniCS/basix/). We will now go through the main objectives of each of these building blocks. DOLFINx is the high performance C++ backend of FEniCSx, where structures such as meshes, function spaces and functions are implemented.\nAdditionally, DOLFINx also contains compute intensive functions such as finite element assembly and mesh refinement algorithms. It also provides an interface to linear algebra solvers and data-structures, such as [PETSc](https://www.mcs.anl.gov/petsc/). UFL is a high-level form language for describing variational formulations with a high-level mathematical syntax. FFCx is the form compiler of FEniCSx; given variational formulations written with UFL, it generates efficient C code. Basix is the finite element backend of FEniCSx, responsible for generating finite element basis functions.\n\n# What you will learn\n\nThe goal of this tutorial is to demonstrate how to apply the finite element to solve PDEs using FEniCS. Through a series of examples, we will demonstrate how to:\n\n- Solve linear PDEs (such as the Poisson equation),\n- Solve time-dependent PDEs (such as the heat equation),\n- Solve non-linear PDEs,\n- Solve systems of time-dependent non-linear PDEs.\n\nImportant topics include: how to set boundary conditions of various types (Dirichlet, Neumann, Robin), how to create meshes, how to define variable coefficients, how to interact with linear and non-linear solvers, and how to post-process and visualize solutions.\n\n# How to use this tutorial\n\nMost of the mathematical part of the examples will be kept at a simple level, such that we can keep the focus on the functionality and syntax of FEniCSx. Therefore, we will mostly use the Poisson equation and the time-dependent diffusion equation as model problems. We will use adjusted input data, such that the solution of the problem can be exactly reproduced on uniform, structured meshes with the finite element method. This greatly simplifies the verification of the implementations.\nOccasionally we will consider a more physically relevant example to remind the reader that there are no big leaps from solving simple model problems to challenging real-world problems when using FEniCSx.\n\n## Interactive tutorials\n\nAs this book has been published as a Jupyter Book, we provide interactive notebooks that can be run in the browser. To start such a notebook click the ![Binder symbol](binder.png)-symbol in the top right corner of the relevant tutorial.\n\n## Obtaining the software\n\nIf you would like to work with DOLFINx outside of the binder-notebooks, you need to install the FEniCS software.\nThe recommended way of installing DOLFINx for new users is by using Docker.\nDocker is a software that uses _containers_ to supply software across different kinds of operating systems (Linux, Mac, Windows). The first step is to install docker, following the instructions at their [webpage](https://docs.docker.com/get-started/).\n\nAll notebooks can be converted to python files using [nbconvert](https://nbconvert.readthedocs.io/en/latest/).\n\n### Tutorial compatible docker images\n\nThe tutorial uses several dependencies for meshing, plotting and timings. A compatible `JupyterLab` image is available in the [Github Packages](https://github.com/jorgensd/dolfinx-tutorial/pkgs/container/dolfinx-tutorial).\n\nTo use the notebooks in this tutorial with DOLFINx on your own computer, you should use the docker image obtained using the following command:\n\n```bash\n docker run --init -p 8888:8888 -v \"$(pwd)\":/root/shared ghcr.io/jorgensd/dolfinx-tutorial:release\n```\n\nThis image can also be used as a normal docker container by adding:\n\n```bash\n docker run --ti -v \"$(pwd)\":/root/shared --entrypoint=\"/bin/bash\" ghcr.io/jorgensd/dolfinx-tutorial:release\n```\n\nThe tutorials can also be exported as an IPython notebook or PDF by clicking the ![Download](save.png)-symbol in the top right corner of the relevant tutorial. The notebook can in turn be used with a Python kernel which has DOLFINx.\n\n### Official images\n\nThe FEniCS project supplies pre-built docker images at [https://hub.docker.com/r/dolfinx/dolfinx](https://hub.docker.com/r/dolfinx/dolfinx).\nThe [Dockerfile](https://github.com/FEniCS/dolfinx/blob/main/docker/Dockerfile)\nprovides a definitive build recipe. As the DOLFINx docker images are hosted at Docker-hub, one can directly access the image using:\n\n```bash\ndocker run dolfinx/dolfinx:stable\n```\n\nThere are several ways of customizing a docker container, such as mounting volumes/sharing folder, setting a working directory, sharing graphical interfaces etc. See `docker run --help` for an extensive list.\n\nOnce you have installed DOLFINx, either by using docker or installing from source, you can test the installation by running `python3 -c \"import dolfinx\"`.\nIf all goes well, no error-messages should appear.\n\n## Installation from source\n\nThe software is quite complex, and building the software and all the dependencies from source can be a daunting task. The list of dependencies can be found at [docs.fenicsproject.org/dolfinx/main/python/installation.html](https://docs.fenicsproject.org/dolfinx/main/python/installation.html).\n\n## Introduction to Python for beginners\n\nIf you are a beginner in Python, we suggest reading {cite}`fem-Langtangen2016` by Hans Petter Langtangen, which will give you a gentle introduction to the Python programming language. Note that DOLFINx, being a state of the art finite element solver, only supports Python 3, as Python 2 reached its end of life January 1st, 2020. To automatically transfer Python 2 scripts to Python 3, it is suggested to use the [2to3](https://docs.python.org/3/library/2to3.html)-package, which provides automated translation of the code.\n\n## Introduction to the finite element method\n\nIn the last decade, several sets of lecture notes on finite element methods have been made open access. See for instance:\n\n- [Numerical methods for partial differential equations](https://hplgit.github.io/num-methods-for-PDEs/doc/web/index.html), by Hans Petter Langtangen\n- [Finite elements - analysis and implementation](https://finite-element.github.io/), by David A. Ham and Colin J. Cotter\n- [Finite element analysis for coupled problems](https://drive.google.com/file/d/1o0DY1RWoXd-gOISqyRzJoDHUHvSMvSg3/view?usp=sharing), by David Kamensky.\n- [DefElement: an encyclopedia of finite element definitions](https://defelement.com/), by Matthew W. Scroggs.\n\nMany good textbooks on the finite element method have been written, and we refer to the original FEniCS tutorial for references to these, see Chapter 1.6.2 of The FEniCS tutorial {cite}`fem-FenicsTutorial`.\n\n## References\n\n```{bibliography}\n :filter: cited\n :labelprefix:\n :keyprefix: fem-\n```\n" }, { "path": "index.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"markdown\",\n \"id\": \"0\",\n \"metadata\": {},\n \"source\": [\n \"# The FEniCSx tutorial\\n\",\n \"Author: Jørgen S. Dokken\\n\",\n \"\\n\",\n \"These webpages give a concise overview of the functionality of [DOLFINx](https://github.com/FEniCS/dolfinx/), including a gentle introduction to the finite element method.\\n\",\n \"This webpage started as an adaptation of the FEniCS tutorial {cite}`FenicsTutorial`, but has evolved into a larger subset of tutorials.\\n\",\n \"\\n\",\n \"DOLFINx can be used as either C++ or Python software, but this tutorial will focus on Python programming, as it is the simplest and most effective approach for beginners. After having gone through this tutorial, the reader should familiarize themselves with the DOLFINx [documentation](https://docs.fenicsproject.org/dolfinx/main/python/), which includes the API and numerous demos.\\n\",\n \"\\n\",\n \"Comments and corrections to this webpage should be submitted to the issue tracker by going to the relevant page in the tutorial, then click the ![git](git.png)-symbol in the top right corner and \\\"open issue\\\".\\n\",\n \"\\n\",\n \"**Interactive tutorials**\\n\",\n \"\\n\",\n \"As this book has been published as a Jupyter Book, we provide interactive notebooks that can be run in the browser. To start such a notebook click the ![Binder symbol](binder.png)-symbol in the top right corner of the relevant tutorial.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"1\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"import dolfinx\\n\",\n \"\\n\",\n \"print(\\n\",\n \" f\\\"DOLFINx version: {dolfinx.__version__} based on GIT commit: {dolfinx.git_commit_hash} of https://github.com/FEniCS/dolfinx/\\\"\\n\",\n \")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"fcc98c0e\",\n \"metadata\": {},\n \"source\": [\n \"## Clickable API links\\n\",\n \"For the modules imported in the tutorial, you can click any function or class to go to the respective API:\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"c2f329cb\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": [\n \"from dolfinx.fem import functionspace # Click `functionspace`\\n\",\n \"from ufl import div, grad # Click `div` and `grad`\\n\",\n \"import numpy as np # Click `numpy`\\n\",\n \"\\n\",\n \"print(functionspace, div, grad, np)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"id\": \"2\",\n \"metadata\": {},\n \"source\": [\n \"**References**\\n\",\n \"```{bibliography}\\n\",\n \":filter: docname in docnames\\n\",\n \"```\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"id\": \"3\",\n \"metadata\": {},\n \"outputs\": [],\n \"source\": []\n }\n ],\n \"metadata\": {\n \"kernelspec\": {\n \"display_name\": \"Python 3 (ipykernel)\",\n \"language\": \"python\",\n \"name\": \"python3\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 3\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython3\",\n \"version\": \"3.9.7\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 5\n}\n" }, { "path": "jupyter_book.code-workspace", "content": "{\n\t\"folders\": [\n\t\t{\n\t\t\t\"path\": \".\"\n\t\t}\n\t}" }, { "path": "pyproject.toml", "content": "[build-system]\nrequires = [\"setuptools>=64.4.0\", \"wheel\", \"pip>=22.3\", \"poetry-core\"]\nbuild-backend = \"setuptools.build_meta\"\n\n[project]\nname = \"DOLFINx_Tutorial\"\nversion = \"0.11.0.dev0\"\ndependencies = [\n \"jupyter-book<2.0\",\n \"meshio\",\n \"h5py\",\n \"seaborn\",\n \"pandas\",\n \"tqdm\",\n \"pyvista[all]>=0.45.0\",\n \"fenics-dolfinx>=0.10.0\",\n \"sphinx-codeautolink\",\n \"trame_jupyter_extension\",\n \"jupytext\",\n \"vtk>=9.5.0\",\n]\n\n[project.optional-dependencies]\ndev = [\"pdbpp\", \"ipython\", \"ruff\", \"pre-commit\"]\nnetgen = [\n \"ngsPETSc@git+https://github.com/NGSolve/ngsPETSc/@main\",\n] # Has to be optional as we cant get netgen on linux/arm64\n\n[tool.setuptools]\npackages = []\n\n\n[tool.jupytext]\nformats = \"ipynb,py:percent\"\n\n\n[tool.ruff.lint.isort]\nknown-first-party = [\"basix\", \"dolfinx\", \"ffcx\", \"ufl\"]\nknown-third-party = [\"gmsh\", \"numpy\", \"pytest\"]\nsection-order = [\n \"future\",\n \"standard-library\",\n \"mpi\",\n \"third-party\",\n \"first-party\",\n \"local-folder\",\n]\n\n[tool.ruff.lint.isort.sections]\n\"mpi\" = [\"mpi4py\", \"petsc4py\"]\n" }, { "path": "references.bib", "content": "@book{Langtangen_Mardal_FEM_2019,\n author = {Langtangen, Hans Petter\n and Mardal, Kent-Andre},\n title = {{Introduction to Numerical Methods for Variational Problems}},\n year = {2019},\n publisher = {Springer International Publishing},\n address = {Cham},\n isbn = {978-3-030-23788-2},\n doi = {10.1007/978-3-030-23788-2_1}\n}\n\n@article{ufl2014,\n author = {Aln\\ae{}s, Martin S. and Logg, Anders and \\O{}lgaard, Kristian B. and Rognes, Marie E. and Wells, Garth N.},\n title = {{Unified Form Language: A Domain-Specific Language for Weak Formulations of Partial Differential Equations}},\n year = {2014},\n issue_date = {February 2014},\n publisher = {Association for Computing Machinery},\n address = {New York, NY, USA},\n volume = {40},\n number = {2},\n issn = {0098-3500},\n doi = {10.1145/2566630},\n journal = {ACM Trans. Math. Softw.},\n articleno = {9},\n numpages = {37}\n}\n\n@article{chorin1968numerical,\n author = {Chorin, Alexandre Joel},\n doi = {10.1090/S0025-5718-1968-0242392-2},\n journal = {Mathematics of Computation},\n number = {104},\n pages = {745--762},\n title = {{Numerical solution of the Navier-Stokes equations}},\n volume = {22},\n year = {1968}\n}\n\n\n@article{Temam1969,\n author = {Temam, Roger},\n doi = {10.1007/BF00247696},\n journal = {Archive for Rational Mechanics and Analysis},\n number = {2},\n pages = {135--153},\n publisher = {Springer},\n timestamp = {2019.08.05},\n title = {{Sur l'approximation de la solution des {\\'e}quations de Navier-Stokes par la m{\\'e}thode des pas fractionnaires (I)}},\n volume = {32},\n year = {1969}\n}\n\n\n\n@article{goda1979multistep,\n abstract = {A numerical algorithm for solving two- or three-dimensional incompressible viscous Navier-Stokes equations is presented. The technique presented here is based on a simple variant of the Chorin method and is related to the MAC method. Auxiliary velocity fields are introduced, which are calculated by the use of a fractional-step procedure for the convective and diffusive part of the solution. For the pressure resolution, a triple sweep is used to obtain the fluid pressure. By these fractional techniques, the three-dimensional equations are separated into only one-dimensional forms. Thus, this saves more computation time and makes algorithm simple. Some numerical computations are made on flows within square and cubic cavities, and some comparisons are made in regard to boundary effects in three-dimensional flows. Further, some discussions are made on primary and secondary eddies generated in a cubic cavity, and comparisons with those in a square cavity are also made. It was found that boundary effects mainly locate near a side wall, but these are not negligibly small in a central region in a cubic cavity.},\n author = {Goda, Katuhiko},\n doi = {10.1016/0021-9991(79)90088-3},\n issn = {0021-9991},\n journal = {Journal of computational physics},\n number = {1},\n pages = {76--95},\n publisher = {Elsevier},\n title = {{A multistep technique with implicit difference schemes for calculating two- or three-dimensional cavity flows}},\n volume = {30},\n year = {1979}\n}\n\n\n\n@book{QuarteroniSaccoSaleri2010,\n title = {Numerical mathematics},\n publisher = {Springer Science \\& Business Media},\n year = {2010},\n author = {Quarteroni, Alfio and Sacco, Riccardo and Saleri, Fausto},\n volume = {37},\n owner = {andre},\n timestamp = {2019.06.24},\n doi = {10.1007/b98885}\n}\n\n@article{Guermond1999,\n author = {Guermond, Jean-Luc},\n journal = {ESAIM: Mathematical Modelling and Numerical Analysis},\n title = {{Un r{\\'e}sultat de convergence d'ordre deux en temps pour l'approximation des {\\'e}quations de Navier--Stokes par une technique de projection incr{\\'e}mentale}},\n year = {1999},\n number = {1},\n pages = {169--189},\n volume = {33},\n publisher = {EDP Sciences},\n timestamp = {2020.04.19},\n doi = {10.1051/m2an:1999101}\n}\n\n\n@book{Langtangen2016,\n author = {Langtangen, Hans Petter},\n title = {{A Primer on Scientific Programming with Python}},\n year = {2016},\n publisher = {Springer Berlin Heidelberg},\n address = {Berlin, Heidelberg},\n pages = {1--49},\n isbn = {978-3-662-49887-3},\n doi = {10.1007/978-3-662-49887-3}\n}\n\n@book{FenicsTutorial,\n author = {Langtangen, Hans Petter and Logg, Anders},\n title = {{Solving PDEs in Python: The FEniCS Tutorial I}},\n year = {2016},\n publisher = {Springer International Publishing},\n address = {Cham},\n pages = {3--10},\n isbn = {978-3-319-52462-7},\n doi = {10.1007/978-3-319-52462-7}\n}\n\n@inbook{Langtangen2016scaling,\n author = {Langtangen, Hans Petter and Pedersen, Geir K.},\n title = {Advanced partial differential equation models},\n booktitle = {Scaling of Differential Equations},\n year = {2016},\n publisher = {Springer International Publishing},\n address = {Cham},\n pages = {99--134},\n isbn = {978-3-319-32726-6},\n doi = {10.1007/978-3-319-32726-6_4}\n}\n\n\n@article{Nitsche1971,\n author = {Nitsche, J.},\n title = {{\\\"U}ber ein Variationsprinzip zur L{\\\"o}sung von Dirichlet-Problemen bei Verwendung von Teilr{\\\"a}umen, die keinen Randbedingungen unterworfen sind},\n journal = {Abhandlungen aus dem Mathematischen Seminar der Universit{\\\"a}t Hamburg},\n year = {1971},\n month = {Jul},\n day = {01},\n volume = {36},\n number = {1},\n pages = {9-15},\n issn = {1865-8784},\n doi = {10.1007/BF02995904}\n}\n\n\n@misc{rehor2025pctools,\n author = {Řehoř, Martin and Hale, Jack S.},\n doi = {10.5334/jors.494},\n journal = {Journal of Open Research Software},\n keyword = {en},\n month = {Sep},\n title = {FEniCSx-pctools: Tools for PETSc Block Linear Algebra Preconditioning in FEniCSx},\n year = {2025}\n}\n" }, { "path": "tox.ini", "content": "[flake8]\nmax-line-length = 120\nignore = W503" } ]