Full Code of phys-sim-book/solid-sim-tutorial for AI

Repository: phys-sim-book/solid-sim-tutorial
Branch: main
Commit: f463ddc4cb51
Files: 141
Total size: 2.8 MB

Directory structure:
gitextract_jxhlhlxe/

├── .gitignore
├── .python-version
├── 0_getting_started/
│   ├── readme.md
│   ├── simulator0.py
│   ├── simulator1.py
│   ├── simulator2.py
│   └── square_mesh.py
├── 10_mpm_elasticity/
│   ├── readme.md
│   └── simulator.py
├── 11_mpm_plastic/
│   ├── readme.md
│   ├── simulator.py
│   └── simulator_viscoplastic.py
├── 11_mpm_sand/
│   ├── readme.md
│   └── simulator.py
├── 12_pbd_cloth/
│   ├── cloth_data.json
│   ├── constraints/
│   │   ├── __init__.py
│   │   ├── bending.py
│   │   └── stretching.py
│   ├── imgui.ini
│   ├── main.py
│   ├── utils/
│   │   ├── __init__.py
│   │   ├── cloth_utils.py
│   │   ├── constraint_finder.py
│   │   └── mesh_loader.py
│   └── xpbd_base.py
├── 13_pbd_mesh/
│   ├── constraints/
│   │   ├── __init__.py
│   │   ├── edge.py
│   │   └── volume.py
│   ├── dragon_data.json
│   ├── imgui.ini
│   ├── main.py
│   ├── skinning/
│   │   ├── __init__.py
│   │   ├── hash_grid.py
│   │   └── skinning.py
│   └── xpbd_base.py
├── 14_pbf/
│   ├── constraints/
│   │   ├── __init__.py
│   │   ├── density.py
│   │   ├── viscosity.py
│   │   └── vorticity.py
│   ├── main.py
│   ├── scene.py
│   └── sph_base.py
├── 1_mass_spring/
│   ├── InertiaEnergy.py
│   ├── MassSpringEnergy.py
│   ├── readme.md
│   ├── simulator.py
│   ├── square_mesh.py
│   ├── time_integrator.py
│   └── utils.py
├── 2_dirichlet/
│   ├── GravityEnergy.py
│   ├── InertiaEnergy.py
│   ├── MassSpringEnergy.py
│   ├── readme.md
│   ├── simulator.py
│   ├── square_mesh.py
│   ├── time_integrator.py
│   └── utils.py
├── 3_contact/
│   ├── BarrierEnergy.py
│   ├── GravityEnergy.py
│   ├── InertiaEnergy.py
│   ├── MassSpringEnergy.py
│   ├── readme.md
│   ├── simulator.py
│   ├── square_mesh.py
│   ├── time_integrator.py
│   └── utils.py
├── 4_friction/
│   ├── BarrierEnergy.py
│   ├── FrictionEnergy.py
│   ├── GravityEnergy.py
│   ├── InertiaEnergy.py
│   ├── MassSpringEnergy.py
│   ├── readme.md
│   ├── simulator.py
│   ├── square_mesh.py
│   ├── time_integrator.py
│   └── utils.py
├── 5_mov_dirichlet/
│   ├── BarrierEnergy.py
│   ├── FrictionEnergy.py
│   ├── GravityEnergy.py
│   ├── InertiaEnergy.py
│   ├── MassSpringEnergy.py
│   ├── SpringEnergy.py
│   ├── readme.md
│   ├── simulator.py
│   ├── square_mesh.py
│   ├── time_integrator.py
│   └── utils.py
├── 6_inv_free/
│   ├── BarrierEnergy.py
│   ├── FrictionEnergy.py
│   ├── GravityEnergy.py
│   ├── InertiaEnergy.py
│   ├── NeoHookeanEnergy.py
│   ├── SpringEnergy.py
│   ├── readme.md
│   ├── simulator.py
│   ├── square_mesh.py
│   ├── time_integrator.py
│   └── utils.py
├── 7_self_contact/
│   ├── BarrierEnergy.py
│   ├── FrictionEnergy.py
│   ├── GravityEnergy.py
│   ├── InertiaEnergy.py
│   ├── NeoHookeanEnergy.py
│   ├── SpringEnergy.py
│   ├── distance/
│   │   ├── CCD.py
│   │   ├── PointEdgeDistance.py
│   │   ├── PointLineDistance.py
│   │   └── PointPointDistance.py
│   ├── readme.md
│   ├── simulator.py
│   ├── square_mesh.py
│   ├── time_integrator.py
│   └── utils.py
├── 8_self_friction/
│   ├── BarrierEnergy.py
│   ├── FrictionEnergy.py
│   ├── GravityEnergy.py
│   ├── InertiaEnergy.py
│   ├── NeoHookeanEnergy.py
│   ├── SpringEnergy.py
│   ├── distance/
│   │   ├── CCD.py
│   │   ├── PointEdgeDistance.py
│   │   ├── PointLineDistance.py
│   │   └── PointPointDistance.py
│   ├── readme.md
│   ├── simulator.py
│   ├── square_mesh.py
│   ├── time_integrator.py
│   └── utils.py
├── 9_reduced_DOF/
│   ├── BarrierEnergy.py
│   ├── GravityEnergy.py
│   ├── InertiaEnergy.py
│   ├── NeoHookeanEnergy.py
│   ├── linear.py
│   ├── readme.md
│   ├── simulator.py
│   ├── square_mesh.py
│   ├── time_integrator.py
│   └── utils.py
├── LICENSE
├── pyproject.toml
└── readme.md

================================================
FILE CONTENTS
================================================

================================================
FILE: .gitignore
================================================
.DS_Store
test.py
*.pyc
output/

================================================
FILE: .python-version
================================================
3.13


================================================
FILE: 0_getting_started/readme.md
================================================
# Getting Started with Solid Simulation

Free fall, spring, and mass-spring solid simulation with symplectic Euler time integration.

## Dependencies
```
pip install numpy scipy pygame
```

## Run
Free falling particle:
```
python simulator0.py
```
Hanging particle:
```
python simulator1.py
```
Hanging square:
```
python simulator2.py
```

================================================
FILE: 0_getting_started/simulator0.py
================================================
# Free Fall Simulation

import math
import numpy as np  # for vector data structure and computations
import pygame       # for visualization

# simulation setup
x = np.array([0.0, 0.0])    # position of particle
v = np.array([0.0, 0.0])    # velocity of particle
g = np.array([0.0, -10.0])  # gravitational acceleration
h = 0.01                    # time step size in seconds

# visualization/rendering setup
pygame.init()
render_FPS = 100                    # number of frames to render per second
resolution = np.array([900, 900])   # visualization window size in pixels
offset = resolution / 2             # offset between window coordinates and simulated coordinates
scale = 200                         # scale between window coordinates and simulated coordinates
def screen_projection(x):           # convert simulated coordinates to window coordinates
    return [offset[0] + scale * x[0], resolution[1] - (offset[1] + scale * x[1])]
screen = pygame.display.set_mode(resolution)    # initialize visualizer

time_step = 0   # the number of the current time step
running = True  # flag indicating whether the simulation is still running
while running:
    # run until the user asks to quit
    for event in pygame.event.get():
        if event.type == pygame.QUIT:
            running = False
    
    # update the frame to display according to render_FPS
    if time_step % int(math.ceil((1.0 / render_FPS) / h)) == 0:
        # fill the background with white color, display simulation time at the top,
        # render a floor at y=-1, and render the particle as a circle:
        screen.fill((255, 255, 255))    
        pygame.display.set_caption('Current time: ' + f'{time_step * h: .2f}s')
        pygame.draw.aaline(screen, (0, 0, 255), screen_projection([-2, -1]), screen_projection([2, -1]))
        pygame.draw.circle(screen, (0, 0, 255), screen_projection(x), 0.1 * scale)
        pygame.display.flip()   # flip the display
        pygame.time.wait(int(1000.0 / render_FPS))  # wait to render the next frame

    # pause the simulation when the particle touches on the ground 
    if x[1] <= -1:
        input()
        break

    # step forward the simulation by updating particle velocity and position
    v += h * g
    x += h * v

    time_step += 1  # update time step counter

================================================
FILE: 0_getting_started/simulator1.py
================================================
# Spring Simulation

import math
import numpy as np  # for vector data structure and computations
import pygame       # for visualization

# simulation setup
m = 1000                    # mass of particle
x = np.array([0.4, 0.0])    # position of particle
v = np.array([0.0, 0.0])    # velocity of particle
g = np.array([0.0, -10.0])  # gravitational acceleration
spring_rest_len = 0.3       # rest length of the spring ###
spring_stiffness = 1e5      # stiffness of the spring ###
h = 0.01                    # time step size in seconds

# visualization/rendering setup
pygame.init()
render_FPS = 100                    # number of frames to render per second
resolution = np.array([900, 900])   # visualization window size in pixels
offset = resolution / 2             # offset between window coordinates and simulated coordinates
scale = 200                         # scale between window coordinates and simulated coordinates
def screen_projection(x):           # convert simulated coordinates to window coordinates
    return [offset[0] + scale * x[0], resolution[1] - (offset[1] + scale * x[1])]
screen = pygame.display.set_mode(resolution)    # initialize visualizer

time_step = 0   # the number of the current time step
running = True  # flag indicating whether the simulation is still running
while running:
    # run until the user asks to quit
    for event in pygame.event.get():
        if event.type == pygame.QUIT:
            running = False
    
    # update the frame to display according to render_FPS
    if time_step % int(math.ceil((1.0 / render_FPS) / h)) == 0:
        # fill the background with white color, display simulation time at the top,
        # draw the spring segment, and render the particle as a circle:
        screen.fill((255, 255, 255))    
        pygame.display.set_caption('Current time: ' + f'{time_step * h: .2f}s')
        pygame.draw.aaline(screen, (0, 0, 255), screen_projection([0, 0]), screen_projection(x)) ###
        pygame.draw.circle(screen, (0, 0, 255), screen_projection(x), 0.1 * scale)
        pygame.display.flip()   # flip the display
        pygame.time.wait(int(1000.0 / render_FPS))  # wait to render the next frame

    # step forward the simulation by updating particle velocity and position
    spring_cur_len = math.sqrt(x[0] * x[0] + x[1] * x[1]) ###
    spring_displacement = spring_cur_len - spring_rest_len ###
    spring_force = -spring_stiffness * spring_displacement * (x / spring_cur_len) ###
    v += h * (g + spring_force / m)
    x += h * v

    time_step += 1  # update time step counter

================================================
FILE: 0_getting_started/simulator2.py
================================================
# Mass-Spring Solid Simulation

import math
import numpy as np  # for vector data structure and computations
import pygame       # for visualization
import square_mesh  # for generating a square mesh

# simulation setup
side_length = 1             # side length of the square
n_seg = 4                   # number of springs per side of the square
m = 1000                    # mass of each particle
[x, e] = square_mesh.generate(side_length, n_seg)   # array of particle positions and springs   ###
v = np.array([[0.0, 0.0]] * len(x))     # velocity array of particles ###
g = np.array([0.0, -10.0])  # gravitational acceleration
spring_rest_len = []        # rest length array of the springs ###
for i in range(0, len(e)):  # calculate the rest length of each spring
    spring_vec = x[e[i][0]] - x[e[i][1]]    # the vector connecting two ends of spring i
    spring_rest_len.append(math.sqrt(spring_vec[0] * spring_vec[0] + spring_vec[1] * spring_vec[1]))
spring_stiffness = 1e6      # stiffness of the spring
h = 0.01                    # time step size in seconds

# visualization/rendering setup
pygame.init()
render_FPS = 100                    # number of frames to render per second
resolution = np.array([900, 900])   # visualization window size in pixels
offset = resolution / 2             # offset between window coordinates and simulated coordinates
scale = 200                         # scale between window coordinates and simulated coordinates
def screen_projection(x):           # convert simulated coordinates to window coordinates
    return [offset[0] + scale * x[0], resolution[1] - (offset[1] + scale * x[1])]
screen = pygame.display.set_mode(resolution)    # initialize visualizer

time_step = 0   # the number of the current time step
running = True  # flag indicating whether the simulation is still running
while running:
    # run until the user asks to quit
    for event in pygame.event.get():
        if event.type == pygame.QUIT:
            running = False
    
    # update the frame to display according to render_FPS
    if time_step % int(math.ceil((1.0 / render_FPS) / h)) == 0:
        # fill the background with white color, display simulation time at the top,
        # draw each spring segment, and render each particle as a circle:
        screen.fill((255, 255, 255))    
        pygame.display.set_caption('Current time: ' + f'{time_step * h: .2f}s')
        for i in range(0, len(e)):  ###
            pygame.draw.aaline(screen, (0, 0, 255), screen_projection(x[e[i][0]]), screen_projection(x[e[i][1]]))
        for i in range(0, len(x)):  ###
            pygame.draw.circle(screen, (0, 0, 255), screen_projection(x[i]), 0.02 * scale)
        pygame.display.flip()   # flip the display
        pygame.time.wait(int(1000.0 / render_FPS))  # wait to render the next frame

    # step forward the simulation by updating particle velocity and position ###
    for i in range(0, len(e)):
        # calculate elasticity force of spring i:
        spring_vec = x[e[i][0]] - x[e[i][1]]
        spring_cur_len = math.sqrt(spring_vec[0] * spring_vec[0] + spring_vec[1] * spring_vec[1])
        spring_displacement = spring_cur_len - spring_rest_len[i]
        spring_force = -spring_stiffness * spring_displacement * (spring_vec / spring_cur_len)
        # update the velocity of the two ends of spring i
        v[e[i][0]] += h * (g + spring_force / m)
        v[e[i][1]] += h * (g - spring_force / m)
    # fix the top left and top right corner by setting velocity to 0:
    v[n_seg] = v[(n_seg + 1) * (n_seg + 1) - 1] = np.array([0, 0])
    # update the position of each particle:
    for i in range(0, len(x)):
        x[i] += h * v[i]

    time_step += 1  # update time step counter

================================================
FILE: 0_getting_started/square_mesh.py
================================================
import numpy as np

def generate(side_length, n_seg):
    # sample nodes uniformly on a square
    x = np.array([[0.0, 0.0]] * ((n_seg + 1) ** 2))
    step = side_length / n_seg
    for i in range(0, n_seg + 1):
        for j in range(0, n_seg + 1):
            x[i * (n_seg + 1) + j] = [-side_length / 2 + i * step, -side_length / 2 + j * step]
    
    # connect the nodes with edges
    e = []
    # horizontal edges
    for i in range(0, n_seg):
        for j in range(0, n_seg + 1):
            e.append([i * (n_seg + 1) + j, (i + 1) * (n_seg + 1) + j])
    # vertical edges
    for i in range(0, n_seg + 1):
        for j in range(0, n_seg):
            e.append([i * (n_seg + 1) + j, i * (n_seg + 1) + j + 1])
    # diagonals
    for i in range(0, n_seg):
        for j in range(0, n_seg):
            e.append([i * (n_seg + 1) + j, (i + 1) * (n_seg + 1) + j + 1])
            e.append([(i + 1) * (n_seg + 1) + j, i * (n_seg + 1) + j + 1])

    return [x, e]

================================================
FILE: 10_mpm_elasticity/readme.md
================================================
# Two Colliding Elastic Blocks in 2D

Two elastic blocks with opposite initial velocities collide head-on using the simplest PIC transfer scheme, illustrating a minimal yet complete implementation of the Material Point Method (MPM).

![results](results.gif)

## Dependencies
```
pip install numpy taichi
```

## Run
```
python simulator.py
```

================================================
FILE: 10_mpm_elasticity/simulator.py
================================================
# Material Point Method Simulation
import numpy as np  # numpy for linear algebra
import taichi as ti # taichi for fast and parallelized computation 

ti.init(arch=ti.cpu)

# ANCHOR: property_def
# simulation setup
grid_size = 128 # background Eulerian grid's resolution, in 2D is [128, 128]
dx = 1.0 / grid_size # the domain size is [1m, 1m] in 2D, so dx for each cell is (1/128)m
dt = 2e-4 # time step size in second
ppc = 8 # average particles per cell

density = 1000 # mass density, unit: kg / m^3
E, nu = 1e4, 0.3 # block's Young's modulus and Poisson's ratio
mu, lam = E / (2 * (1 + nu)), E * nu / ((1 + nu) * (1 - 2 * nu)) # Lame parameters
# ANCHOR_END: property_def

# ANCHOR: setting
# uniformly sampling material particles
def uniform_grid(x0, y0, x1, y1, dx):
    xx, yy = np.meshgrid(np.arange(x0, x1 + dx, dx), np.arange(y0, y1 + dx, dx))
    return np.column_stack((xx.ravel(), yy.ravel()))

box1_samples = uniform_grid(0.2, 0.4, 0.4, 0.6, dx / np.sqrt(ppc))
box1_velocities = np.tile(np.array([10.0, 0]), (len(box1_samples), 1))
box2_samples = uniform_grid(0.6, 0.4, 0.8, 0.6, dx / np.sqrt(ppc))
box2_velocities = np.tile(np.array([-10.0, 0]), (len(box1_samples), 1))
all_samples = np.concatenate([box1_samples, box2_samples], axis=0)
all_velocities = np.concatenate([box1_velocities, box2_velocities], axis=0)
# ANCHOR_END: setting

# ANCHOR: data_def
# material particles data
N_particles = len(all_samples)
x = ti.Vector.field(2, float, N_particles) # the position of particles
x.from_numpy(all_samples)
v = ti.Vector.field(2, float, N_particles) # the velocity of particles
v.from_numpy(all_velocities)
vol = ti.field(float, N_particles)         # the volume of particle
vol.fill(0.2 * 0.4 / N_particles) # get the volume of each particle as V_rest / N_particles
m = ti.field(float, N_particles)           # the mass of particle
m.fill(vol[0] * density)
F = ti.Matrix.field(2, 2, float, N_particles)  # the deformation gradient of particles
F.from_numpy(np.tile(np.eye(2), (N_particles, 1, 1)))

# grid data
grid_m = ti.field(float, (grid_size, grid_size))
grid_v = ti.Vector.field(2, float, (grid_size, grid_size))
# ANCHOR_END: data_def

# ANCHOR: reset_grid
def reset_grid():
    # after each transfer, the grid is reset
    grid_m.fill(0)
    grid_v.fill(0)
# ANCHOR_END: reset_grid

################################
# Stvk Hencky Elasticity
################################
# ANCHOR: stvk
@ti.func
def StVK_Hencky_PK1_2D(F):
    U, sig, V = ti.svd(F)
    inv_sig = sig.inverse()
    e = ti.Matrix([[ti.log(sig[0, 0]), 0], [0, ti.log(sig[1, 1])]])
    return U @ (2 * mu * inv_sig @ e + lam * e.trace() * inv_sig) @ V.transpose()
# ANCHOR_END: stvk

# Particle-to-Grid (P2G) Transfers
# ANCHOR: p2g
@ti.kernel
def particle_to_grid_transfer():
    for p in range(N_particles):
        base = (x[p] / dx - 0.5).cast(int)
        fx = x[p] / dx - base.cast(float)
        # quadratic B-spline interpolating functions (Section 26.2)
        w = [0.5 * (1.5 - fx) ** 2, 0.75 - (fx - 1) ** 2, 0.5 * (fx - 0.5) ** 2]
        # gradient of the interpolating function (Section 26.2)
        dw_dx = [fx - 1.5, 2 * (1.0 - fx), fx - 0.5]

        P = StVK_Hencky_PK1_2D(F[p])
        for i in ti.static(range(3)):
            for j in ti.static(range(3)):
                offset = ti.Vector([i, j])
                weight = w[i][0] * w[j][1]
                grad_weight = ti.Vector([(1. / dx) * dw_dx[i][0] * w[j][1], 
                                          w[i][0] * (1. / dx) * dw_dx[j][1]])

                grid_m[base + offset] += weight * m[p] # mass transfer
                grid_v[base + offset] += weight * m[p] * v[p] # momentum Transfer, PIC formulation
                # internal force (stress) transfer
                fi = -vol[p] * P @ F[p].transpose() @ grad_weight
                grid_v[base + offset] += dt * fi
# ANCHOR_END: p2g

# Grid Update
# ANCHOR: grid_update
@ti.kernel
def update_grid():
    for i, j in grid_m:
        if grid_m[i, j] > 0:
            grid_v[i, j] = grid_v[i, j] / grid_m[i, j] # extract updated nodal velocity from transferred nodal momentum

            # Dirichlet BC near the bounding box
            if i <= 3 or i > grid_size - 3 or j <= 2 or j > grid_size - 3:
                grid_v[i, j] = 0
# ANCHOR_END: grid_update


# Grid-to-Particle (G2P) Transfers
# ANCHOR: g2p
@ti.kernel
def grid_to_particle_transfer():
    for p in range(N_particles):
        base = (x[p] / dx - 0.5).cast(int)
        fx = x[p] / dx - base.cast(float)
        # quadratic B-spline interpolating functions (Section 26.2)
        w = [0.5 * (1.5 - fx) ** 2, 0.75 - (fx - 1) ** 2, 0.5 * (fx - 0.5) ** 2]
        # gradient of the interpolating function (Section 26.2)
        dw_dx = [fx - 1.5, 2 * (1.0 - fx), fx - 0.5]

        new_v = ti.Vector.zero(float, 2)
        v_grad_wT = ti.Matrix.zero(float, 2, 2)
        for i in ti.static(range(3)):
            for j in ti.static(range(3)):
                offset = ti.Vector([i, j])
                weight = w[i][0] * w[j][1]
                grad_weight = ti.Vector([(1. / dx) * dw_dx[i][0] * w[j][1], 
                                          w[i][0] * (1. / dx) * dw_dx[j][1]])

                new_v += weight * grid_v[base + offset]
                v_grad_wT += grid_v[base + offset].outer_product(grad_weight)

        v[p] = new_v
        F[p] = (ti.Matrix.identity(float, 2) + dt * v_grad_wT) @ F[p]
# ANCHOR_END: g2p

# Deformation Gradient and Particle State Update
# ANCHOR: particle_update
@ti.kernel
def update_particle_state():
    for p in range(N_particles):
        x[p] += dt * v[p] # advection
# ANCHOR_END: particle_update

# ANCHOR: time_step
def step():
    # a single time step of the Material Point Method (MPM) simulation
    reset_grid()
    particle_to_grid_transfer()
    update_grid()
    grid_to_particle_transfer()
    update_particle_state()
# ANCHOR_END: time_step

################################
# Main 
################################
gui = ti.GUI("2D MPM Elasticity", res = 512, background_color = 0xFFFFFF)
while True:
    for s in range(50): step()

    gui.circles(x.to_numpy()[:len(box1_samples)], radius=3, color=0xFF0000)
    gui.circles(x.to_numpy()[len(box1_samples):], radius=3, color=0x0000FF)
    gui.show()

================================================
FILE: 11_mpm_plastic/readme.md
================================================
# 2D Sand with a Sphere Collider

A 2D sand block falling onto a static red sphere collider simulated by Material Point Method (MPM). The sand undergoes irreversible deformation and splashing upon impact, demonstrating granular flow and frictional boundary response.

![results](results.gif)

## Dependencies
```
pip install numpy taichi
```

## Run
```
python simulator.py
```

================================================
FILE: 11_mpm_plastic/simulator.py
================================================
# Material Point Method Simulation
import numpy as np  # numpy for linear algebra
import taichi as ti # taichi for fast and parallelized computation 

ti.init(arch=ti.cpu)

# sampling material particles with poisson-disk sampling
################################
# Poisson Disk Sampling Tool
################################
def poisson_disk_sampling(radius, domain_size, k=30):
    """Bridson's algorithm for Poisson-disk sampling in 2D"""
    cell_size = radius / np.sqrt(2)
    grid_shape = (int(domain_size[0] / cell_size) + 1, int(domain_size[1] / cell_size) + 1)
    grid = -np.ones(grid_shape, dtype=int)
    samples = []
    active_list = []

    def in_domain(p):
        return 0 <= p[0] < domain_size[0] and 0 <= p[1] < domain_size[1]

    def get_cell_coords(p):
        return int(p[0] / cell_size), int(p[1] / cell_size)

    def get_nearby_samples(p):
        i, j = get_cell_coords(p)
        neighbors = []
        for di in [-2, -1, 0, 1, 2]:
            for dj in [-2, -1, 0, 1, 2]:
                ni, nj = i + di, j + dj
                if 0 <= ni < grid.shape[0] and 0 <= nj < grid.shape[1]:
                    idx = grid[ni, nj]
                    if idx != -1:
                        neighbors.append(samples[idx])
        return neighbors

    # Start with a random point
    first_point = np.array([np.random.uniform(0, domain_size[0]), np.random.uniform(0, domain_size[1])])
    samples.append(first_point)
    active_list.append(0)
    grid[get_cell_coords(first_point)] = 0

    while active_list:
        idx = np.random.choice(active_list)
        base_point = samples[idx]
        found = False
        for _ in range(k):
            angle = np.random.uniform(0, 2 * np.pi)
            r = np.random.uniform(radius, 2 * radius)
            new_point = base_point + r * np.array([np.cos(angle), np.sin(angle)])
            if in_domain(new_point):
                neighbors = get_nearby_samples(new_point)
                if all(np.linalg.norm(new_point - n) >= radius for n in neighbors):
                    samples.append(new_point)
                    active_list.append(len(samples) - 1)
                    grid[get_cell_coords(new_point)] = len(samples) - 1
                    found = True
        if not found:
            active_list.remove(idx)
    return np.array(samples)

# ANCHOR: data_def
# simulation setup
grid_size = 128 # background Eulerian grid's resolution, in 2D is [128, 128]
dx = 1.0 / grid_size # the domain size is [1m, 1m] in 2D, so dx for each cell is (1/128)m
dt = 2e-4 # time step size in second
ppc = 8 # average particle-per-cell

density = 400 # kg / m^3
E, nu = 3.537e5, 0.3 # sand's Young's modulus and Poisson's ratio
mu, lam = E / (2 * (1 + nu)), E * nu / ((1 + nu) * (1 - 2 * nu)) # sand's Lame parameters
sdf_friction = 0.5 # frictional coefficient of SDF boundary condition
friction_angle_in_degrees = 25.0 # Drucker Prager friction angle
# ANCHOR: D_def
D = (1./4.) * dx * dx # constant D for Quadratic B-spline used for APIC
# ANCHOR_END: D_def

# sampling material particles with poisson-disk sampling
poisson_samples = poisson_disk_sampling(dx / np.sqrt(ppc), [0.2, 0.4]) # simulating a [30cm, 50cm] sand block

# material particles data
N_particles = len(poisson_samples)
x = ti.Vector.field(2, float, N_particles) # the position of particles
x.from_numpy(np.array(poisson_samples) + [0.4, 0.55])
v = ti.Vector.field(2, float, N_particles) # the velocity of particles
vol = ti.field(float, N_particles)         # the volume of particle
vol.fill(0.2 * 0.4 / N_particles) # get the volume of each particle as V_rest / N_particles
m = ti.field(float, N_particles)           # the mass of particle
m.fill(vol[0] * density)
F = ti.Matrix.field(2, 2, float, N_particles)  # the deformation gradient of particles
F.from_numpy(np.tile(np.eye(2), (N_particles, 1, 1)))
C = ti.Matrix.field(2, 2, float, N_particles)  # the affine-matrix of particles

diff_log_J = ti.field(float, N_particles) # tracks changes in the log of the volume gained during extension

# grid data
grid_m = ti.field(float, (grid_size, grid_size))
grid_v = ti.Vector.field(2, float, (grid_size, grid_size))
# ANCHOR_END: data_def

# ANCHOR: reset_grid
def reset_grid():
    # after each transfer, the grid is reset
    grid_m.fill(0)
    grid_v.fill(0)
# ANCHOR_END: reset_grid

################################
# Stvk Hencky Elasticity
################################
# ANCHOR: stvk
@ti.func
def StVK_Hencky_PK1_2D(F):
    U, sig, V = ti.svd(F)
    inv_sig = sig.inverse()
    e = ti.Matrix([[ti.log(sig[0, 0]), 0], [0, ti.log(sig[1, 1])]])
    return U @ (2 * mu * inv_sig @ e + lam * e.trace() * inv_sig) @ V.transpose()
# ANCHOR_END: stvk

################################
# Drucker Prager plasticity
################################
# ANCHOR: drucker_prager
@ti.func
def Drucker_Prager_return_mapping(F, diff_log_J):
    dim = ti.static(F.n)
    sin_phi = ti.sin(friction_angle_in_degrees/ 180.0 * ti.math.pi)
    friction_alpha = ti.sqrt(2.0 / 3.0) * 2.0 * sin_phi / (3.0 - sin_phi)
    U, sig_diag, V = ti.svd(F)
    sig = ti.Vector([ti.max(sig_diag[i,i], 0.05) for i in ti.static(range(dim))])
    epsilon = ti.log(sig)
    epsilon += diff_log_J / dim # volume correction treatment
    trace_epsilon = epsilon.sum()
    shifted_trace = trace_epsilon
    if shifted_trace >= 0:
        for d in ti.static(range(dim)):
            epsilon[d] = 0.
    else:
        epsilon_hat = epsilon - (trace_epsilon / dim)
        epsilon_hat_norm = ti.sqrt(epsilon_hat.dot(epsilon_hat)+1e-8)
        delta_gamma = epsilon_hat_norm + (dim * lam + 2. * mu) / (2. * mu) * (shifted_trace) * friction_alpha
        epsilon -= (ti.max(delta_gamma, 0) / epsilon_hat_norm) * epsilon_hat
    sig_out = ti.exp(epsilon)
    for d in ti.static(range(dim)):
        sig_diag[d,d] = sig_out[d]
    return U @ sig_diag @ V.transpose()
# ANCHOR_END: drucker_prager

# Particle-to-Grid (P2G) Transfers
# ANCHOR: p2g
@ti.kernel
def particle_to_grid_transfer():
    for p in range(N_particles):
        base = (x[p] / dx - 0.5).cast(int)
        fx = x[p] / dx - base.cast(float)
        # quadratic B-spline interpolating functions
        w = [0.5 * (1.5 - fx) ** 2, 0.75 - (fx - 1) ** 2, 0.5 * (fx - 0.5) ** 2]
        # gradient of the interpolating function
        dw_dx = [fx - 1.5, 2 * (1.0 - fx), fx - 0.5]

        P = StVK_Hencky_PK1_2D(F[p])
        for i in ti.static(range(3)):
            for j in ti.static(range(3)):
                offset = ti.Vector([i, j])
                dpos = (offset.cast(float) - fx) * dx
                weight = w[i][0] * w[j][1]
                grad_weight = ti.Vector([(1. / dx) * dw_dx[i][0] * w[j][1], 
                                          w[i][0] * (1. / dx) * dw_dx[j][1]])

                grid_m[base + offset] += weight * m[p] # mass transfer
                # ANCHOR: apic_p2g
                grid_v[base + offset] += weight * m[p] * (v[p] + C[p] @ dpos) # momentum Transfer, APIC formulation
                # ANCHOR_END: apic_p2g
                # internal force (stress) transfer
                fi = -vol[p] * P @ F[p].transpose() @ grad_weight
                grid_v[base + offset] += dt * fi
# ANCHOR_END: p2g

# Grid Update
@ti.kernel
def update_grid():
    for i, j in grid_m:
        if grid_m[i, j] > 0:
            grid_v[i, j] = grid_v[i, j] / grid_m[i, j] # extract updated nodal velocity from transferred nodal momentum
            grid_v[i, j] += ti.Vector([0, -9.81]) * dt # gravity

            # Dirichlet BC near the bounding box
            if i <= 3 or i > grid_size - 3 or j <= 2 or j > grid_size - 3:
                grid_v[i, j] = 0
            
            x_i = (ti.Vector([i, j])) * dx # position of the grid-node
            
            # ANCHOR: sphere_sdf
            # a sphere SDF as boundary condition
            sphere_center = ti.Vector([0.5, 0.5])
            sphere_radius = 0.05 + dx # add a dx-gap to avoid penetration
            if (x_i - sphere_center).norm() < sphere_radius:
                normal = (x_i - sphere_center).normalized()
                diff_vel = -grid_v[i, j]
                dotnv = normal.dot(diff_vel)
                dotnv_frac = dotnv * (1.0 - sdf_friction)
                grid_v[i, j] += diff_vel * sdf_friction + normal * dotnv_frac
            # ANCHOR_END: sphere_sdf


# Grid-to-Particle (G2P) Transfers
# ANCHOR: g2p
@ti.kernel
def grid_to_particle_transfer():
    for p in range(N_particles):
        base = (x[p] / dx - 0.5).cast(int)
        fx = x[p] / dx - base.cast(float)
        # quadratic B-spline interpolating functions 
        w = [0.5 * (1.5 - fx) ** 2, 0.75 - (fx - 1) ** 2, 0.5 * (fx - 0.5) ** 2]
        # gradient of the interpolating function
        dw_dx = [fx - 1.5, 2 * (1.0 - fx), fx - 0.5]

        new_v = ti.Vector.zero(float, 2)
        new_C = ti.Matrix.zero(float, 2, 2)
        v_grad_wT = ti.Matrix.zero(float, 2, 2)
        for i in ti.static(range(3)):
            for j in ti.static(range(3)):
                offset = ti.Vector([i, j])
                dpos = (offset.cast(float) - fx) * dx
                weight = w[i][0] * w[j][1]
                grad_weight = ti.Vector([(1. / dx) * dw_dx[i][0] * w[j][1], 
                                          w[i][0] * (1. / dx) * dw_dx[j][1]])

                new_v += weight * grid_v[base + offset]
                # ANCHOR: apic_g2p
                new_C += weight * grid_v[base + offset].outer_product(dpos) / D
                # ANCHOR_END: apic_g2p
                v_grad_wT += grid_v[base + offset].outer_product(grad_weight)

        v[p] = new_v
        C[p] = new_C
        # note the updated F here is the trial elastic deformation gradient
        F[p] = (ti.Matrix.identity(float, 2) + dt * v_grad_wT) @ F[p]
# ANCHOR_END: g2p

# Deformation Gradient and Particle State Update
# ANCHOR: particle_update
@ti.kernel
def update_particle_state():
    for p in range(N_particles):
        # trial elastic deformation gradient
        F_tr = F[p]
        # apply return mapping to correct the trial elastic state, projecting the stress induced by F_tr
        # back onto the yield surface, following the direction specified by the plastic flow rule.
        new_F = Drucker_Prager_return_mapping(F_tr, diff_log_J[p])
        # track the volume change incurred by return mapping to correct volume, following https://dl.acm.org/doi/10.1145/3072959.3073651 sec 4.3.4
        diff_log_J[p] += -ti.log(new_F.determinant()) + ti.log(F_tr.determinant()) 
        F[p] = new_F
        # advection
        x[p] += dt * v[p]
# ANCHOR_END: particle_update

# ANCHOR: time_step
def step():
    # a single time step of the Material Point Method (MPM) simulation
    reset_grid()
    particle_to_grid_transfer()
    update_grid()
    grid_to_particle_transfer()
    update_particle_state()
# ANCHOR_END: time_step

################################
# Main 
################################
gui = ti.GUI("2D MPM Sand", res = 512, background_color = 0xFFFFFF)
frame = 0
while True:
    for s in range(50):
        step()

    gui.circles(np.array([[0.5, 0.5]]), radius = 0.05 * gui.res[0], color = 0xFF0000)
    gui.circles(x.to_numpy(), radius = 1.5, color = 0xD6B588)
    gui.show()

================================================
FILE: 11_mpm_plastic/simulator_viscoplastic.py
================================================
import taichi as ti
import os
import shutil

ti.init(arch=ti.cuda)

quality = 6
n_particles, n_grid = 9000 * quality**2, 256 * quality
dx, inv_dx = 1 / n_grid, float(n_grid)
dt = 1e-4 / quality
p_vol, p_rho = (dx * 0.5) ** 2, 1
p_mass = p_vol * p_rho

nu = 0.2
E_toothpaste = 350.0
mu_toothpaste = E_toothpaste / (2 * (1 + nu))
lam_toothpaste = E_toothpaste * nu / ((1 + nu) * (1 - 2 * nu))

yield_stress_bar1 = 1.2
plastic_viscosity_bar1 = 0.7
yield_stress_bar2 = 1.0
plastic_viscosity_bar2 = 0.3

ground_friction = 0.3

x = ti.Vector.field(2, dtype=float, shape=n_particles)
v = ti.Vector.field(2, dtype=float, shape=n_particles)
C = ti.Matrix.field(2, 2, dtype=float, shape=n_particles)
F = ti.Matrix.field(2, 2, dtype=float, shape=n_particles)
color_stripe = ti.field(dtype=int, shape=n_particles)
yield_stress = ti.field(dtype=float, shape=n_particles)
plastic_viscosity = ti.field(dtype=float, shape=n_particles)
grid_v = ti.Vector.field(2, dtype=float, shape=(n_grid, n_grid))
grid_m = ti.field(dtype=float, shape=(n_grid, n_grid))
gravity = ti.Vector.field(2, dtype=float, shape=())


@ti.func
def viscoplastic_return_mapping_stvk_2d(F_trial, mu_p, lam_p, yield_stress_p, plastic_viscosity_p, dt_val):
    """Viscoplastic return mapping for toothpaste-like material (2D version)"""
    U, sig, V = ti.svd(F_trial)
    
    # Clamp singular values to prevent numerical issues
    sig0 = ti.max(sig[0, 0], 0.01)
    sig1 = ti.max(sig[1, 1], 0.01)
    
    # Compute logarithmic strain (2D)
    epsilon = ti.Vector([ti.log(sig0), ti.log(sig1)])
    trace_epsilon = epsilon[0] + epsilon[1]
    
    # Deviatoric strain
    epsilon_hat = epsilon - ti.Vector([trace_epsilon / 2.0, trace_epsilon / 2.0])
    
    # Deviatoric stress in logarithmic strain space
    s_trial = 2.0 * mu_p * epsilon_hat
    s_trial_norm = s_trial.norm()
    
    # Yield condition - using similar factor to 3D version (sqrt(2/3) ≈ 0.816)
    # For 2D, we use sqrt(2/3) to match the 3D implementation's scaling
    y = s_trial_norm - ti.sqrt(2.0 / 3.0) * yield_stress_p
    
    F_elastic_res = F_trial
    if y > 0:
        # Compute b_trial for mu_hat (2D adaptation)
        b_trial = sig0 * sig0 + sig1 * sig1
        mu_hat = mu_p * b_trial / 2.0
        
        # Rate-dependent plastic flow
        denom = 1.0 + plastic_viscosity_p / (2.0 * mu_hat * dt_val)
        s_new_norm = s_trial_norm - y / denom
        s_scale = s_new_norm / s_trial_norm if s_trial_norm > 1e-10 else 1.0
        s_new = s_scale * s_trial
        
        # Reconstruct strain
        epsilon_new = 1.0 / (2.0 * mu_p) * s_new + ti.Vector([trace_epsilon / 2.0, trace_epsilon / 2.0])
        
        # Reconstruct deformation gradient
        sig_elastic = ti.Matrix([[ti.exp(epsilon_new[0]), 0.0], [0.0, ti.exp(epsilon_new[1])]])
        F_elastic_res = U @ sig_elastic @ V.transpose()
    
    return F_elastic_res


@ti.func
def stvk_stress_2d(F_elastic, U, V, sig, mu_p, lam_p):
    """Saint Venant-Kirchhoff stress model (2D version)"""
    sig0 = ti.max(sig[0, 0], 0.01)
    sig1 = ti.max(sig[1, 1], 0.01)
    
    epsilon = ti.Vector([ti.log(sig0), ti.log(sig1)])
    log_sig_sum = ti.log(sig0) + ti.log(sig1)
    
    ONE = ti.Vector([1.0, 1.0])
    tau = 2.0 * mu_p * epsilon + lam_p * log_sig_sum * ONE
    
    tau_mat = ti.Matrix([[tau[0], 0.0], [0.0, tau[1]]])
    return U @ tau_mat @ V.transpose() @ F_elastic.transpose()


@ti.kernel
def p2g2p():
    for i, j in grid_m:
        grid_v[i, j] = [0, 0]
        grid_m[i, j] = 0
    for p in x:
        if x[p][0] < 0.0 or x[p][1] < 0.0 or x[p][0] > 1.0 or x[p][1] > 1.0:
            continue
        
        base = (x[p] * inv_dx - 0.5).cast(int)
        fx = x[p] * inv_dx - base.cast(float)
        w = [0.5 * (1.5 - fx) ** 2, 0.75 - (fx - 1) ** 2, 0.5 * (fx - 0.5) ** 2]
        F_trial = (ti.Matrix.identity(float, 2) + dt * C[p]) @ F[p]
        
        F[p] = viscoplastic_return_mapping_stvk_2d(
            F_trial, mu_toothpaste, lam_toothpaste, yield_stress[p], plastic_viscosity[p], dt
        )
        
        U, sig, V = ti.svd(F[p])
        stress = stvk_stress_2d(F[p], U, V, sig, mu_toothpaste, lam_toothpaste)
        
        stress = (-dt * p_vol * 4 * inv_dx * inv_dx) * stress
        affine = stress + p_mass * C[p]
        for i, j in ti.static(ti.ndrange(3, 3)):
            offset = ti.Vector([i, j])
            dpos = (offset.cast(float) - fx) * dx
            weight = w[i][0] * w[j][1]
            grid_v[base + offset] += weight * (p_mass * v[p] + affine @ dpos)
            grid_m[base + offset] += weight * p_mass
    for i, j in grid_m:
        if grid_m[i, j] > 0:
            grid_v[i, j] = (1 / grid_m[i, j]) * grid_v[i, j]
            grid_v[i, j] += dt * gravity[None] * 30
            
            if i < 3 and grid_v[i, j][0] < 0:
                grid_v[i, j][0] = 0
            if i > n_grid - 3 and grid_v[i, j][0] > 0:
                grid_v[i, j][0] = 0
            
            if j < 3 and grid_v[i, j][1] < 0:
                v_y = grid_v[i, j][1]
                v_x = grid_v[i, j][0]
                normal_component = ti.abs(v_y)
                tangential_velocity = v_x
                
                if ti.abs(tangential_velocity) > 1e-20:
                    friction_reduction = ground_friction * normal_component
                    v_x_new = tangential_velocity
                    if v_x_new > 0:
                        v_x_new = ti.max(0.0, v_x_new - friction_reduction)
                    else:
                        v_x_new = ti.min(0.0, v_x_new + friction_reduction)
                    grid_v[i, j] = ti.Vector([v_x_new, 0.0])
                else:
                    grid_v[i, j] = ti.Vector([0.0, 0.0])
            
            if j > n_grid - 3 and grid_v[i, j][1] > 0:
                grid_v[i, j][1] = 0
    for p in x:
        if x[p][0] < 0.0 or x[p][1] < 0.0 or x[p][0] > 1.0 or x[p][1] > 1.0:
            continue
        
        base = (x[p] * inv_dx - 0.5).cast(int)
        fx = x[p] * inv_dx - base.cast(float)
        w = [0.5 * (1.5 - fx) ** 2, 0.75 - (fx - 1.0) ** 2, 0.5 * (fx - 0.5) ** 2]
        new_v = ti.Vector.zero(float, 2)
        new_C = ti.Matrix.zero(float, 2, 2)
        for i, j in ti.static(ti.ndrange(3, 3)):
            dpos = ti.Vector([i, j]).cast(float) - fx
            g_v = grid_v[base + ti.Vector([i, j])]
            weight = w[i][0] * w[j][1]
            new_v += weight * g_v
            new_C += 4 * inv_dx * weight * g_v.outer_product(dpos)
        v[p], C[p] = new_v, new_C
        x[p] += dt * v[p]


@ti.kernel
def reset():
    bar_width = 0.015
    bar_height = 0.85
    bar_bottom_y = 0.1
    bar_spacing = 0.2
    bar1_center_x = 0.5 - bar_spacing / 2.0
    bar2_center_x = 0.5 + bar_spacing / 2.0
    
    particles_per_bar = n_particles // 2
    
    for i in range(n_particles):
        bar_center_x = 0.5
        bar_yield_stress = 1.2
        bar_plastic_viscosity = 0.7
        
        if i < particles_per_bar:
            bar_center_x = bar1_center_x
            bar_yield_stress = 1.2
            bar_plastic_viscosity = 0.7
        else:
            bar_center_x = bar2_center_x
            bar_yield_stress = 1.0
            bar_plastic_viscosity = 0.3
        
        x_pos = ti.random() * bar_width + (bar_center_x - bar_width / 2.0)
        y_pos = ti.random() * bar_height + bar_bottom_y
        
        x[i] = [x_pos, y_pos]
        
        stripe_width = bar_width / 3.0
        relative_x = x_pos - (bar_center_x - bar_width / 2.0)
        if relative_x < stripe_width:
            color_stripe[i] = 0
        elif relative_x < 2.0 * stripe_width:
            color_stripe[i] = 1
        else:
            color_stripe[i] = 2
        
        v[i] = [0, 0]
        F[i] = ti.Matrix([[1, 0], [0, 1]])
        C[i] = ti.Matrix.zero(float, 2, 2)
        yield_stress[i] = bar_yield_stress
        plastic_viscosity[i] = bar_plastic_viscosity


gui = ti.GUI("Taichi MLS-MPM-128", res=1024, background_color=0xEEEEF0, show_gui=False)
reset()
gravity[None] = [0, -1]

save_duration = 10.0
output_fps = 30
frame_time = 3e-2
total_frames_to_run = int(save_duration / frame_time)
frames_per_save = int(1.0 / (output_fps * frame_time))
total_saves = int(save_duration * output_fps)
save_counter = 0

output_dir = os.path.join(os.path.dirname(__file__), "frames")
os.makedirs(output_dir, exist_ok=True)
video_output = os.path.join(os.path.dirname(__file__), "viscoplastic_animation.mp4")


for frame in range(total_frames_to_run):
    substeps_per_frame = int(2e-3 // dt)
    
    for s in range(substeps_per_frame):
        p2g2p()
    
    if frame % frames_per_save == 0 and save_counter < total_saves:
        gui.circles(
            x.to_numpy(),
            radius=1.5,
            palette=[0x3366FF, 0xFFFF00, 0xFF3333],
            palette_indices=color_stripe,
        )
        filename = os.path.join(output_dir, f'{save_counter:06d}.png')
        gui.show(filename)
        save_counter += 1
        if save_counter % 30 == 0:
            print(f"Saved {save_counter}/{total_saves} frames...")

print(f"Frame saving complete! Saved {save_counter} frames.")

ffmpeg_path = shutil.which("ffmpeg")
if ffmpeg_path is not None and save_counter > 0:
    pattern = os.path.join(output_dir, "%06d.png")
    cmd = (
        f'"{ffmpeg_path}" -y -framerate {output_fps} -i "{pattern}" '
        f'-c:v libx264 -pix_fmt yuv420p -crf 18 "{video_output}"'
    )
    print("Running:", cmd)
    os.system(cmd)
    print(f"Video written to: {video_output}")
else:
    print("ffmpeg not found or no frames saved. To build video manually, run:")
    print(
        f"ffmpeg -y -framerate {output_fps} -i '{os.path.join(output_dir, '%06d.png')}' "
        f"-c:v libx264 -pix_fmt yuv420p -crf 18 '{video_output}'"
    )

================================================
FILE: 11_mpm_sand/readme.md
================================================
# 2D Sand with a Sphere Collider

A 2D sand block falling onto a static red sphere collider simulated by Material Point Method (MPM). The sand undergoes irreversible deformation and splashing upon impact, demonstrating granular flow and frictional boundary response.

![results](results.gif)

## Dependencies
```
pip install numpy taichi
```

## Run
```
python simulator.py
```

================================================
FILE: 11_mpm_sand/simulator.py
================================================
# Material Point Method Simulation
import numpy as np  # numpy for linear algebra
import taichi as ti # taichi for fast and parallelized computation 

ti.init(arch=ti.cpu)

# sampling material particles with poisson-disk sampling
################################
# Poisson Disk Sampling Tool
################################
def poisson_disk_sampling(radius, domain_size, k=30):
    """Bridson's algorithm for Poisson-disk sampling in 2D"""
    cell_size = radius / np.sqrt(2)
    grid_shape = (int(domain_size[0] / cell_size) + 1, int(domain_size[1] / cell_size) + 1)
    grid = -np.ones(grid_shape, dtype=int)
    samples = []
    active_list = []

    def in_domain(p):
        return 0 <= p[0] < domain_size[0] and 0 <= p[1] < domain_size[1]

    def get_cell_coords(p):
        return int(p[0] / cell_size), int(p[1] / cell_size)

    def get_nearby_samples(p):
        i, j = get_cell_coords(p)
        neighbors = []
        for di in [-2, -1, 0, 1, 2]:
            for dj in [-2, -1, 0, 1, 2]:
                ni, nj = i + di, j + dj
                if 0 <= ni < grid.shape[0] and 0 <= nj < grid.shape[1]:
                    idx = grid[ni, nj]
                    if idx != -1:
                        neighbors.append(samples[idx])
        return neighbors

    # Start with a random point
    first_point = np.array([np.random.uniform(0, domain_size[0]), np.random.uniform(0, domain_size[1])])
    samples.append(first_point)
    active_list.append(0)
    grid[get_cell_coords(first_point)] = 0

    while active_list:
        idx = np.random.choice(active_list)
        base_point = samples[idx]
        found = False
        for _ in range(k):
            angle = np.random.uniform(0, 2 * np.pi)
            r = np.random.uniform(radius, 2 * radius)
            new_point = base_point + r * np.array([np.cos(angle), np.sin(angle)])
            if in_domain(new_point):
                neighbors = get_nearby_samples(new_point)
                if all(np.linalg.norm(new_point - n) >= radius for n in neighbors):
                    samples.append(new_point)
                    active_list.append(len(samples) - 1)
                    grid[get_cell_coords(new_point)] = len(samples) - 1
                    found = True
        if not found:
            active_list.remove(idx)
    return np.array(samples)

# ANCHOR: data_def
# simulation setup
grid_size = 128 # background Eulerian grid's resolution, in 2D is [128, 128]
dx = 1.0 / grid_size # the domain size is [1m, 1m] in 2D, so dx for each cell is (1/128)m
dt = 2e-4 # time step size in second
ppc = 8 # average particle-per-cell

density = 400 # kg / m^3
E, nu = 3.537e5, 0.3 # sand's Young's modulus and Poisson's ratio
mu, lam = E / (2 * (1 + nu)), E * nu / ((1 + nu) * (1 - 2 * nu)) # sand's Lame parameters
sdf_friction = 0.5 # frictional coefficient of SDF boundary condition
friction_angle_in_degrees = 25.0 # Drucker Prager friction angle
# ANCHOR: D_def
D = (1./4.) * dx * dx # constant D for Quadratic B-spline used for APIC
# ANCHOR_END: D_def

# sampling material particles with poisson-disk sampling
poisson_samples = poisson_disk_sampling(dx / np.sqrt(ppc), [0.2, 0.4]) # simulating a [30cm, 50cm] sand block

# material particles data
N_particles = len(poisson_samples)
x = ti.Vector.field(2, float, N_particles) # the position of particles
x.from_numpy(np.array(poisson_samples) + [0.4, 0.55])
v = ti.Vector.field(2, float, N_particles) # the velocity of particles
vol = ti.field(float, N_particles)         # the volume of particle
vol.fill(0.2 * 0.4 / N_particles) # get the volume of each particle as V_rest / N_particles
m = ti.field(float, N_particles)           # the mass of particle
m.fill(vol[0] * density)
F = ti.Matrix.field(2, 2, float, N_particles)  # the deformation gradient of particles
F.from_numpy(np.tile(np.eye(2), (N_particles, 1, 1)))
C = ti.Matrix.field(2, 2, float, N_particles)  # the affine-matrix of particles

diff_log_J = ti.field(float, N_particles) # tracks changes in the log of the volume gained during extension

# grid data
grid_m = ti.field(float, (grid_size, grid_size))
grid_v = ti.Vector.field(2, float, (grid_size, grid_size))
# ANCHOR_END: data_def

# ANCHOR: reset_grid
def reset_grid():
    # after each transfer, the grid is reset
    grid_m.fill(0)
    grid_v.fill(0)
# ANCHOR_END: reset_grid

################################
# Stvk Hencky Elasticity
################################
# ANCHOR: stvk
@ti.func
def StVK_Hencky_PK1_2D(F):
    U, sig, V = ti.svd(F)
    inv_sig = sig.inverse()
    e = ti.Matrix([[ti.log(sig[0, 0]), 0], [0, ti.log(sig[1, 1])]])
    return U @ (2 * mu * inv_sig @ e + lam * e.trace() * inv_sig) @ V.transpose()
# ANCHOR_END: stvk

################################
# Drucker Prager plasticity
################################
# ANCHOR: drucker_prager
@ti.func
def Drucker_Prager_return_mapping(F, diff_log_J):
    dim = ti.static(F.n)
    sin_phi = ti.sin(friction_angle_in_degrees/ 180.0 * ti.math.pi)
    friction_alpha = ti.sqrt(2.0 / 3.0) * 2.0 * sin_phi / (3.0 - sin_phi)
    U, sig_diag, V = ti.svd(F)
    sig = ti.Vector([ti.max(sig_diag[i,i], 0.05) for i in ti.static(range(dim))])
    epsilon = ti.log(sig)
    epsilon += diff_log_J / dim # volume correction treatment
    trace_epsilon = epsilon.sum()
    shifted_trace = trace_epsilon
    if shifted_trace >= 0:
        for d in ti.static(range(dim)):
            epsilon[d] = 0.
    else:
        epsilon_hat = epsilon - (trace_epsilon / dim)
        epsilon_hat_norm = ti.sqrt(epsilon_hat.dot(epsilon_hat)+1e-8)
        delta_gamma = epsilon_hat_norm + (dim * lam + 2. * mu) / (2. * mu) * (shifted_trace) * friction_alpha
        epsilon -= (ti.max(delta_gamma, 0) / epsilon_hat_norm) * epsilon_hat
    sig_out = ti.exp(epsilon)
    for d in ti.static(range(dim)):
        sig_diag[d,d] = sig_out[d]
    return U @ sig_diag @ V.transpose()
# ANCHOR_END: drucker_prager

# Particle-to-Grid (P2G) Transfers
# ANCHOR: p2g
@ti.kernel
def particle_to_grid_transfer():
    for p in range(N_particles):
        base = (x[p] / dx - 0.5).cast(int)
        fx = x[p] / dx - base.cast(float)
        # quadratic B-spline interpolating functions
        w = [0.5 * (1.5 - fx) ** 2, 0.75 - (fx - 1) ** 2, 0.5 * (fx - 0.5) ** 2]
        # gradient of the interpolating function
        dw_dx = [fx - 1.5, 2 * (1.0 - fx), fx - 0.5]

        P = StVK_Hencky_PK1_2D(F[p])
        for i in ti.static(range(3)):
            for j in ti.static(range(3)):
                offset = ti.Vector([i, j])
                dpos = (offset.cast(float) - fx) * dx
                weight = w[i][0] * w[j][1]
                grad_weight = ti.Vector([(1. / dx) * dw_dx[i][0] * w[j][1], 
                                          w[i][0] * (1. / dx) * dw_dx[j][1]])

                grid_m[base + offset] += weight * m[p] # mass transfer
                # ANCHOR: apic_p2g
                grid_v[base + offset] += weight * m[p] * (v[p] + C[p] @ dpos) # momentum Transfer, APIC formulation
                # ANCHOR_END: apic_p2g
                # internal force (stress) transfer
                fi = -vol[p] * P @ F[p].transpose() @ grad_weight
                grid_v[base + offset] += dt * fi
# ANCHOR_END: p2g

# Grid Update
@ti.kernel
def update_grid():
    for i, j in grid_m:
        if grid_m[i, j] > 0:
            grid_v[i, j] = grid_v[i, j] / grid_m[i, j] # extract updated nodal velocity from transferred nodal momentum
            grid_v[i, j] += ti.Vector([0, -9.81]) * dt # gravity

            # Dirichlet BC near the bounding box
            if i <= 3 or i > grid_size - 3 or j <= 2 or j > grid_size - 3:
                grid_v[i, j] = 0
            
            x_i = (ti.Vector([i, j])) * dx # position of the grid-node
            
            # ANCHOR: sphere_sdf
            # a sphere SDF as boundary condition
            sphere_center = ti.Vector([0.5, 0.5])
            sphere_radius = 0.05 + dx # add a dx-gap to avoid penetration
            if (x_i - sphere_center).norm() < sphere_radius:
                normal = (x_i - sphere_center).normalized()
                diff_vel = -grid_v[i, j]
                dotnv = normal.dot(diff_vel)
                dotnv_frac = dotnv * (1.0 - sdf_friction)
                grid_v[i, j] += diff_vel * sdf_friction + normal * dotnv_frac
            # ANCHOR_END: sphere_sdf


# Grid-to-Particle (G2P) Transfers
# ANCHOR: g2p
@ti.kernel
def grid_to_particle_transfer():
    for p in range(N_particles):
        base = (x[p] / dx - 0.5).cast(int)
        fx = x[p] / dx - base.cast(float)
        # quadratic B-spline interpolating functions 
        w = [0.5 * (1.5 - fx) ** 2, 0.75 - (fx - 1) ** 2, 0.5 * (fx - 0.5) ** 2]
        # gradient of the interpolating function
        dw_dx = [fx - 1.5, 2 * (1.0 - fx), fx - 0.5]

        new_v = ti.Vector.zero(float, 2)
        new_C = ti.Matrix.zero(float, 2, 2)
        v_grad_wT = ti.Matrix.zero(float, 2, 2)
        for i in ti.static(range(3)):
            for j in ti.static(range(3)):
                offset = ti.Vector([i, j])
                dpos = (offset.cast(float) - fx) * dx
                weight = w[i][0] * w[j][1]
                grad_weight = ti.Vector([(1. / dx) * dw_dx[i][0] * w[j][1], 
                                          w[i][0] * (1. / dx) * dw_dx[j][1]])

                new_v += weight * grid_v[base + offset]
                # ANCHOR: apic_g2p
                new_C += weight * grid_v[base + offset].outer_product(dpos) / D
                # ANCHOR_END: apic_g2p
                v_grad_wT += grid_v[base + offset].outer_product(grad_weight)

        v[p] = new_v
        C[p] = new_C
        # note the updated F here is the trial elastic deformation gradient
        F[p] = (ti.Matrix.identity(float, 2) + dt * v_grad_wT) @ F[p]
# ANCHOR_END: g2p

# Deformation Gradient and Particle State Update
# ANCHOR: particle_update
@ti.kernel
def update_particle_state():
    for p in range(N_particles):
        # trial elastic deformation gradient
        F_tr = F[p]
        # apply return mapping to correct the trial elastic state, projecting the stress induced by F_tr
        # back onto the yield surface, following the direction specified by the plastic flow rule.
        new_F = Drucker_Prager_return_mapping(F_tr, diff_log_J[p])
        # track the volume change incurred by return mapping to correct volume, following https://dl.acm.org/doi/10.1145/3072959.3073651 sec 4.3.4
        diff_log_J[p] += -ti.log(new_F.determinant()) + ti.log(F_tr.determinant()) 
        F[p] = new_F
        # advection
        x[p] += dt * v[p]
# ANCHOR_END: particle_update

# ANCHOR: time_step
def step():
    # a single time step of the Material Point Method (MPM) simulation
    reset_grid()
    particle_to_grid_transfer()
    update_grid()
    grid_to_particle_transfer()
    update_particle_state()
# ANCHOR_END: time_step

################################
# Main 
################################
gui = ti.GUI("2D MPM Sand", res = 512, background_color = 0xFFFFFF)
frame = 0
while True:
    for s in range(50):
        step()

    gui.circles(np.array([[0.5, 0.5]]), radius = 0.05 * gui.res[0], color = 0xFF0000)
    gui.circles(x.to_numpy(), radius = 1.5, color = 0xD6B588)
    gui.show()

================================================
FILE: 12_pbd_cloth/cloth_data.json
================================================
				{
					"name" : "cloth",
					"vertices" : [
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						1230, 1661, 3150,  1232, 1663, 3149,  1234, 1665, 3148,  3147, 203, 1667,  1238, 1669, 3146,  
						1240, 1671, 3145,  1242, 1673, 3144,  1244, 1675, 3143,  1246, 1677, 3142,  1248, 1679, 3141,  
						1251, 1682, 3140,  1253, 1684, 3139,  1255, 1686, 3138,  3137, 214, 1688,  1259, 1690, 3136,  
						1261, 1692, 3135,  3134, 217, 1694,  1265, 1696, 3133,  1267, 1698, 3132,  3131, 220, 1700,  
						1271, 1702, 3130,  1273, 1704, 3129,  1275, 1706, 3128,  3127, 224, 1708,  1279, 1710, 3126,  
						1281, 1712, 3125,  1283, 1714, 3124,  1285, 1716, 3123,  1287, 1718, 3122,  1289, 1720, 3121,  
						1721, 1290, 3120,  3119, 661, 1723,  1719, 1724, 3118,  1725, 1288, 3117,  1726, 1727, 3116,  
						1717, 1728, 3115,  1729, 1286, 3114,  1730, 1731, 3113,  1715, 1732, 3112,  1733, 1284, 3111,  
						3110, 664, 1735,  1713, 1736, 3109,  1737, 1282, 3108,  1738, 1739, 3107,  1711, 1740, 3106,  
						1741, 1280, 3105,  1742, 1743, 3104,  1709, 1744, 3103,  1745, 1278, 3102,  1746, 1747, 3101,  
						1707, 1748, 3100,  1749, 1276, 3099,  1750, 1751, 3098,  1705, 1752, 3097,  1753, 1274, 3096,  
						1754, 1755, 3095,  1703, 1756, 3094,  3093, 642, 1272,  3092, 670, 1759,  1701, 1760, 3091,  
						1761, 1270, 3090,  1762, 1763, 3089,  1699, 1764, 3088,  1765, 1268, 3087,  1766, 1767, 3086,  
						1697, 1768, 3085,  1769, 1266, 3084,  1770, 1771, 3083,  1695, 1772, 3082,  1773, 1264, 3081,  
						1774, 1775, 3080,  3079, 633, 1776,  1777, 1262, 3078,  1778, 1779, 3077,  1691, 1780, 3076,  
						1781, 1260, 3075,  1782, 1783, 3074,  1689, 1784, 3073,  3072, 628, 1258,  3071, 677, 1787,  
						3070, 627, 1788,  1789, 1256, 3069,  1790, 1791, 3068,  1685, 1792, 3067,  1793, 1254, 3066,  
						1794, 1795, 3065,  1683, 1796, 3064,  1797, 1252, 3063,  1798, 1799, 3062,  1681, 1800, 3061,  
						1801, 1249, 3060,  1802, 1803, 3059,  1678, 1804, 3058,  1805, 1247, 3057,  1806, 1807, 3056,  
						1676, 1808, 3055,  1809, 1245, 3054,  1810, 1811, 3053,  1674, 1812, 3052,  1813, 1243, 3051,  
						1814, 1815, 3050,  1672, 1816, 3049,  1817, 1241, 3048,  1818, 1819, 3047,  1670, 1820, 3046,  
						1821, 1239, 3045,  1822, 1823, 3044,  1668, 1824, 3043,  1825, 1237, 3042,  1826, 1827, 3041,  
						1666, 1828, 3040,  1829, 1235, 3039,  1830, 1831, 3038,  1664, 1832, 3037,  1833, 1233, 3036,  
						1834, 1835, 3035,  1662, 1836, 3034,  1837, 1231, 3033,  1838, 1839, 3032,  1660, 1840, 3031,  
						1841, 1229, 3030,  1842, 1843, 3029,  3028, 598, 1844,  1845, 1227, 3027,  1846, 1847, 3026,  
						1656, 1848, 3025,  1849, 1225, 3024,  1850, 1851, 3023,  1654, 1852, 3022,  1853, 1223, 3021,  
						1854, 1855, 3020,  1652, 1856, 3019,  1857, 1221, 3018,  1858, 1859, 3017,  1650, 1860, 3016,  
						1861, 1219, 3015,  1862, 1863, 3014,  1648, 1864, 3013,  3012, 587, 1217,  3011, 697, 1867,  
						1646, 1868, 3010,  1869, 1215, 3009,  1870, 1871, 3008,  1644, 1872, 3007,  1873, 1213, 3006,  
						1874, 1875, 3005,  1642, 1876, 3004,  1877, 1211, 3003,  1878, 1879, 3002,  1640, 1880, 3001,  
						1881, 1208, 3000,  1882, 1883, 2999,  1637, 1884, 2998,  1885, 1206, 2997,  2996, 702, 1887,  
						2995, 575, 1888,  1889, 1204, 2994,  1890, 1891, 2993,  1633, 1892, 2992,  1893, 1202, 2991,  
						1894, 1895, 2990,  1631, 1896, 2989,  1897, 1200, 2988,  1898, 1899, 2987,  1629, 1900, 2986,  
						1901, 1198, 2985,  1902, 1903, 2984,  1627, 1904, 2983,  1905, 1196, 2982,  1906, 1907, 2981,  
						2980, 565, 1908,  1909, 1194, 2979,  1910, 1911, 2978,  1623, 1912, 2977,  1913, 1192, 2976,  
						1914, 1915, 2975,  1621, 1916, 2974,  1917, 1190, 2973,  1918, 1919, 2972,  1619, 1920, 2971,  
						1921, 1188, 2970,  1922, 1923, 2969,  1617, 1924, 2968,  1925, 1186, 2967,  1926, 1927, 2966,  
						1615, 1928, 2965,  1929, 1184, 2964,  1930, 1931, 2963,  2962, 553, 1932,  1933, 1182, 2961,  
						1934, 1935, 2960,  1611, 1936, 2959,  1937, 1180, 2958,  1938, 1939, 2957,  1609, 1940, 2956,  
						1941, 1178, 2955,  1942, 1943, 2954,  1607, 1944, 2953,  2952, 546, 1176,  2951, 717, 1947,  
						1605, 1948, 2950,  1949, 1174, 2949,  1950, 1951, 2948,  1603, 1952, 2947,  1953, 1172, 2946,  
						1954, 1955, 2945,  1601, 1956, 2944,  1957, 1170, 2943,  1958, 1959, 2942,  1599, 1960, 2941,  
						1961, 1167, 2940,  1962, 1963, 2939,  1596, 1964, 2938,  2937, 535, 1165,  2936, 722, 1967,  
						1594, 1968, 2935,  1969, 1163, 2934,  1970, 1971, 2933,  1592, 1972, 2932,  1973, 1161, 2931,  
						2930, 724, 1975,  1590, 1976, 2929,  1977, 1159, 2928,  1978, 1979, 2927,  1588, 1980, 2926,  
						2925, 527, 1157,  2924, 726, 1983,  2923, 526, 1984,  1985, 1155, 2922,  1986, 1987, 2921,  
						1584, 1988, 2920,  2919, 523, 1153,  1990, 1991, 2918,  1582, 1992, 2917,  1993, 1151, 2916,  
						1994, 1995, 2915,  2914, 520, 1996,  1997, 1149, 2913,  1998, 1999, 2912,  1578, 2000, 2911,  
						2001, 1147, 2910,  2002, 2003, 2909,  1576, 2004, 2908,  2005, 1145, 2907,  2006, 2007, 2906,  
						1574, 2008, 2905,  2009, 1143, 2904,  2010, 2011, 2903,  2902, 512, 2012,  2013, 1141, 2901,  
						2014, 2015, 2900,  1570, 2016, 2899,  2017, 1139, 2898,  2018, 2019, 2897,  1568, 2020, 2896,  
						2021, 1137, 2895,  2022, 2023, 2894,  1566, 2024, 2893,  2892, 505, 1135,  2026, 2027, 2891,  
						1564, 2028, 2890,  2889, 503, 1133,  2888, 738, 2031,  1562, 2032, 2887,  2033, 1131, 2886,  
						2034, 2035, 2885,  1560, 2036, 2884,  2037, 1129, 2883,  2038, 2039, 2882,  1558, 2040, 2881,  
						2041, 1126, 2880,  2042, 2043, 2879,  1555, 2044, 2878,  2045, 1124, 2877,  2046, 2047, 2876,  
						1553, 2048, 2875,  2874, 492, 1122,  2873, 743, 2051,  2872, 491, 2052,  2871, 490, 1120,  
						2054, 2055, 2870,  1549, 2056, 2869,  2057, 1118, 2868,  2058, 2059, 2867,  1547, 2060, 2866,  
						2061, 1116, 2865,  2062, 2063, 2864,  1545, 2064, 2863,  2862, 484, 1114,  2861, 747, 2067,  
						2860, 483, 2068,  2069, 1112, 2859,  2070, 2071, 2858,  1541, 2072, 2857,  2073, 1110, 2856,  
						2855, 749, 2075,  2854, 479, 2076,  2077, 1108, 2853,  2078, 2079, 2852,  1537, 2080, 2851,  
						2081, 1106, 2850,  2082, 2083, 2849,  1535, 2084, 2848,  2085, 1104, 2847,  2846, 752, 2087,  
						2845, 473, 2088,  2844, 472, 1102,  2090, 2091, 2843,  1531, 2092, 2842,  2093, 1100, 2841,  
						2840, 754, 2095,  1529, 2096, 2839,  2097, 1098, 2838,  2098, 2099, 2837,  1527, 2100, 2836,  
						2835, 466, 1096,  2102, 2103, 2834,  1525, 2104, 2833,  2105, 1094, 2832,  2106, 2107, 2831,  
						2830, 463, 2108,  2829, 462, 1092,  2110, 2111, 2828,  2827, 461, 2112,  2113, 1090, 2826,  
						2114, 2115, 2825,  1519, 2116, 2824,  2117, 1088, 2823,  2118, 2119, 2822,  1517, 2120, 2821,  
						2820, 455, 1085,  2819, 761, 2123,  2818, 454, 2124,  2817, 453, 1083,  2126, 2127, 2816,  
						1512, 2128, 2815,  2814, 451, 1081,  2813, 763, 2131,  2812, 450, 2132,  2133, 1079, 2811,  
						2134, 2135, 2810,  1508, 2136, 2809,  2137, 1077, 2808,  2807, 765, 2139,  1506, 2140, 2806,  
						2141, 1075, 2805,  2142, 2143, 2804,  1504, 2144, 2803,  2802, 443, 1073,  2801, 767, 2147,  
						1502, 2148, 2800,  2149, 1071, 2799,  2150, 2151, 2798,  1500, 2152, 2797,  2796, 439, 1069,  
						2795, 769, 2155,  1498, 2156, 2794,  2157, 1067, 2793,  2158, 2159, 2792,  2791, 436, 2160,  
						2790, 435, 1065,  2162, 2163, 2789,  1494, 2164, 2788,  2787, 433, 1063,  2166, 2167, 2786,  
						1492, 2168, 2785,  2169, 1061, 2784,  2170, 2171, 2783,  2782, 430, 2172,  2781, 429, 1059,  
						2780, 774, 2175,  1488, 2176, 2779,  2177, 1057, 2778,  2777, 775, 2179,  2776, 426, 2180,  
						2181, 1055, 2775,  2182, 2183, 2774,  1484, 2184, 2773,  2772, 423, 1053,  2771, 777, 2187,  
						2770, 422, 2188,  2189, 1051, 2769,  2190, 2191, 2768,  1480, 2192, 2767,  2193, 1049, 2766,  
						2194, 2195, 2765,  1478, 2196, 2764,  2197, 1047, 2763,  2198, 2199, 2762,  2761, 416, 2200,  
						2201, 1044, 2760,  2759, 781, 2203,  1473, 2204, 2758,  2205, 1042, 2757,  2206, 2207, 2756,  
						1471, 2208, 2755,  2209, 1040, 2754,  2210, 2211, 2753,  1469, 2212, 2752,  2213, 1038, 2751,  
						2214, 2215, 2750,  1467, 2216, 2749,  2748, 406, 1036,  2218, 2219, 2747,  1465, 2220, 2746,  
						2221, 1034, 2745,  2222, 2223, 2744,  1463, 2224, 2743,  2225, 1032, 2742,  2226, 2227, 2741,  
						1461, 2228, 2740,  2229, 1030, 2739,  2230, 2231, 2738,  1459, 2232, 2737,  2233, 1028, 2736,  
						2234, 2235, 2735,  2734, 397, 2236,  2237, 1026, 2733,  2238, 2239, 2732,  1455, 2240, 2731,  
						2241, 1024, 2730,  2242, 2243, 2729,  1453, 2244, 2728,  2245, 1022, 2727,  2726, 792, 2247,  
						1451, 2248, 2725,  2249, 1020, 2724,  2250, 2251, 2723,  1449, 2252, 2722,  2253, 1018, 2721,  
						2254, 2255, 2720,  1447, 2256, 2719,  2718, 386, 1016,  2258, 2259, 2717,  2716, 385, 2260,  
						2261, 1014, 2715,  2262, 2263, 2714,  1443, 2264, 2713,  2265, 1012, 2712,  2266, 2267, 2711,  
						1441, 2268, 2710,  2269, 1010, 2709,  2708, 798, 2271,  2707, 379, 2272,  2273, 1008, 2706,  
						2274, 2275, 2705,  1437, 2276, 2704,  2277, 1006, 2703,  2278, 2279, 2702,  1435, 2280, 2701,  
						2281, 1003, 2700,  2282, 2283, 2699,  1432, 2284, 2698,  2285, 1001, 2697,  2286, 2287, 2696,  
						1430, 2288, 2695,  2289, 999, 2694,  2290, 2291, 2693,  1428, 2292, 2692,  2293, 997, 2691,  
						2294, 2295, 2690,  1426, 2296, 2689,  2297, 995, 2688,  2298, 2299, 2687,  1424, 2300, 2686,  
						2301, 993, 2685,  2302, 2303, 2684,  1422, 2304, 2683,  2305, 991, 2682,  2306, 2307, 2681,  
						2680, 360, 2308,  2309, 989, 2679,  2310, 2311, 2678,  1418, 2312, 2677,  2313, 987, 2676,  
						2314, 2315, 2675,  1416, 2316, 2674,  2317, 985, 2673,  2318, 2319, 2672,  1414, 2320, 2671,  
						2321, 983, 2670,  2322, 2323, 2669,  1412, 2324, 2668,  2325, 981, 2667,  2326, 2327, 2666,  
						1410, 2328, 2665,  2329, 979, 2664,  2330, 2331, 2663,  1408, 2332, 2662,  2661, 347, 977,  
						2660, 814, 2335,  1406, 2336, 2659,  2337, 975, 2658,  2338, 2339, 2657,  1404, 2340, 2656,  
						2341, 973, 2655,  2342, 2343, 2654,  1402, 2344, 2653,  2345, 971, 2652,  2346, 2347, 2651,  
						1400, 2348, 2650,  2649, 339, 969,  2648, 818, 2351,  1398, 2352, 2647,  2353, 967, 2646,  
						2354, 2355, 2645,  1396, 2356, 2644,  2357, 965, 2643,  2358, 2359, 2642,  1394, 2360, 2641,  
						2361, 962, 2640,  2362, 2363, 2639,  1391, 2364, 2638,  2365, 960, 2637,  2366, 2367, 2636,  
						1389, 2368, 2635,  2369, 958, 2634,  2370, 2371, 2633,  1387, 2372, 2632,  2373, 956, 2631,  
						2374, 2375, 2630,  1385, 2376, 2629,  2377, 954, 2628,  2378, 2379, 2627,  1383, 2380, 2626,  
						2381, 952, 2625,  2382, 2383, 2624,  1381, 2384, 2623,  2385, 950, 2622,  2386, 2387, 2621,  
						1379, 2388, 2620,  2389, 948, 2619,  2390, 2391, 2618,  1377, 2392, 2617,  2393, 946, 2616,  
						2394, 2395, 2615,  1375, 2396, 2614,  2397, 944, 2613,  2398, 2399, 2612,  1373, 2400, 2611,  
						2401, 942, 2610,  2402, 2403, 2609,  1371, 2404, 2608,  2405, 940, 2607,  2406, 2407, 2606,  
						1369, 2408, 2605,  2409, 938, 2604,  2410, 2411, 2603,  1367, 2412, 2602,  2413, 936, 2601,  
						2600, 834, 2415,  1365, 2416, 2599,  2417, 934, 2598,  2418, 2419, 2597,  1363, 2420, 2596,  
						2421, 932, 2595,  2422, 2423, 2594,  1361, 2424, 2593,  2425, 930, 2592,  2426, 2427, 2591,  
						1359, 2428, 2590,  2429, 928, 2589,  2430, 2431, 2588,  1357, 2432, 2587,  2433, 926, 2586,  
						2434, 2435, 2585,  1355, 2436, 2584,  2437, 924, 2583,  2438, 2439, 2582,  1353, 2440, 2581,  
						2441, 921, 2580,  2442, 2443, 2579,  1350, 2444, 2578,  2445, 918, 2577,  2446, 2447, 2576,  
						1347, 2448, 2575,  2449, 915, 2574,  2450, 2451, 2573,  1344, 2452, 2572,  2453, 912, 2571,  
						2570, 844, 2455,  2569, 281, 2456,  2457, 909, 2568,  2458, 2459, 2567,  1338, 2460, 2566,  
						2461, 906, 2565,  2462, 2463, 2564,  1335, 2464, 2563,  2465, 903, 2562,  2466, 2467, 2561,  
						1332, 2468, 2560,  2469, 900, 2559,  2470, 2471, 2558,  1329, 2472, 2557,  2473, 897, 2556,  
						2474, 2475, 2555,  1326, 2476, 2554,  2553, 264, 894,  2552, 850, 2479,  1323, 2480, 2551,  
						2550, 261, 891,  2482, 2483, 2549,  1320, 2484, 2548,  2547, 258, 888,  2486, 2487, 2546,  
						1317, 2488, 2545,  2544, 255, 885,  2490, 2491, 2543,  2542, 254, 2492,  2493, 882, 2541,  
						2494, 2495, 2540,  1311, 2496, 2539,  2497, 879, 2538,  2498, 2499, 2537,  1308, 2500, 2536,  
						2535, 246, 876,  2502, 2503, 2534,  1305, 2504, 2533,  2505, 873, 2532,  2506, 2507, 2531,  
						2530, 242, 2508,  2509, 870, 2529,  2510, 2511, 2528,  1299, 2512, 2527,  2526, 237, 867,  
						2514, 2515, 2525,  1296, 2516, 2524,  2523, 234, 864,  2518, 2519, 2522,  2521, 233, 2520,  
						2521, 860, 2518,  232, 2521, 2518,  862, 1293, 2521,  2522, 231, 861,  1292, 861, 0,  
						232, 2522, 1292,  2523, 21, 1291,  231, 2523, 1291,  2519, 2517, 2523,  2524, 859, 2514,  
						865, 2514, 235,  2, 2524, 865,  1293, 2515, 233,  1295, 1293, 1,  235, 2525, 1295,  
						863, 867, 22,  233, 2526, 863,  2515, 2513, 2526,  2527, 858, 2510,  868, 2510, 238,  
						3, 2527, 868,  2528, 236, 1296,  1298, 1296, 2,  238, 2528, 1298,  2529, 23, 866,  
						2511, 866, 236,  858, 2529, 2511,  2506, 2508, 857,  241, 2530, 2506,  871, 1302, 2530,  
						2531, 239, 1299,  1301, 1299, 3,  1301, 2506, 2531,  2532, 24, 869,  2507, 869, 239,  
						857, 2532, 2507,  2533, 856, 2502,  874, 2502, 244,  5, 2533, 874,  1302, 2503, 242,  
						1304, 1302, 4,  244, 2534, 1304,  872, 876, 25,  242, 2535, 872,  2503, 2501, 2535,  
						2536, 855, 2498,  877, 2498, 247,  6, 2536, 877,  2537, 245, 1305,  1307, 1305, 5,  
						247, 2537, 1307,  2538, 26, 875,  2499, 875, 245,  855, 2538, 2499,  2539, 854, 2494,  
						880, 2494, 250,  7, 2539, 880,  1308, 2495, 248,  6, 2540, 1308,  250, 2540, 1310,  
						2541, 27, 878,  2495, 878, 248,  854, 2541, 2495,  2542, 853, 2490,  883, 2490, 253,  
						8, 2542, 883,  2543, 251, 1311,  1313, 1311, 7,  253, 2543, 1313,  881, 885, 28,  
						251, 2544, 881,  2491, 2489, 2544,  2545, 852, 2486,  886, 2486, 256,  9, 2545, 886,  
						1314, 2487, 254,  1316, 1314, 8,  256, 2546, 1316,  884, 888, 29,  254, 2547, 884,  
						2487, 2485, 2547,  2482, 2484, 851,  889, 2482, 259,  889, 1320, 2548,  2549, 257, 1317,  
						1319, 1317, 9,  259, 2549, 1319,  887, 891, 30,  2483, 887, 257,  851, 2550, 2483,  
						2551, 850, 2478,  892, 2478, 262,  892, 1323, 2551,  1320, 2479, 260,  10, 2552, 1320,  
						1322, 2478, 2552,  2553, 31, 890,  2479, 890, 260,  2479, 2477, 2553,  2554, 849, 2474,  
						895, 2474, 265,  12, 2554, 895,  2555, 263, 1323,  1325, 1323, 11,  265, 2555, 1325,  
						2556, 32, 893,  2475, 893, 263,  849, 2556, 2475,  2557, 848, 2470,  898, 2470, 268,  
						13, 2557, 898,  2558, 266, 1326,  1328, 1326, 12,  268, 2558, 1328,  2559, 33, 896,  
						2471, 896, 266,  848, 2559, 2471,  2560, 847, 2466,  901, 2466, 271,  14, 2560, 901,  
						2561, 269, 1329,  1331, 1329, 13,  271, 2561, 1331,  2562, 34, 899,  2467, 899, 269,  
						847, 2562, 2467,  2563, 846, 2462,  904, 2462, 274,  15, 2563, 904,  2564, 272, 1332,  
						1334, 1332, 14,  274, 2564, 1334,  2565, 35, 902,  2463, 902, 272,  846, 2565, 2463,  
						2566, 845, 2458,  907, 2458, 277,  16, 2566, 907,  2567, 275, 1335,  1337, 1335, 15,  
						277, 2567, 1337,  905, 909, 36,  2459, 905, 275,  845, 2568, 2459,  2454, 2456, 844,  
						910, 2454, 280,  910, 1341, 2569,  2570, 278, 1338,  16, 2570, 1338,  1340, 2454, 2570,  
						2571, 37, 908,  2455, 908, 278,  2455, 2453, 2571,  2572, 843, 2450,  913, 2450, 283,  
						18, 2572, 913,  2573, 281, 1341,  1343, 1341, 17,  283, 2573, 1343,  2574, 38, 911,  
						2451, 911, 281,  843, 2574, 2451,  2575, 842, 2446,  916, 2446, 286,  19, 2575, 916,  
						2576, 284, 1344,  1346, 1344, 18,  286, 2576, 1346,  2577, 39, 914,  2447, 914, 284,  
						842, 2577, 2447,  2578, 841, 2442,  919, 2442, 289,  919, 1350, 2578,  2579, 287, 1347,  
						1349, 1347, 19,  289, 2579, 1349,  2580, 40, 917,  2443, 917, 287,  841, 2580, 2443,  
						2581, 840, 2438,  1294, 2438, 234,  22, 2581, 1294,  2582, 292, 922,  864, 922, 21,  
						234, 2582, 864,  2583, 42, 1352,  2439, 1352, 292,  840, 2583, 2439,  2584, 839, 2434,  
						1297, 2434, 237,  23, 2584, 1297,  2585, 293, 1353,  867, 1353, 22,  237, 2585, 867,  
						2586, 43, 923,  2435, 923, 293,  839, 2586, 2435,  2587, 838, 2430,  1300, 2430, 240,  
						24, 2587, 1300,  2588, 295, 1355,  870, 1355, 23,  240, 2588, 870,  2589, 44, 925,  
						2431, 925, 295,  838, 2589, 2431,  2590, 837, 2426,  1303, 2426, 243,  25, 2590, 1303,  
						2591, 297, 1357,  873, 1357, 24,  243, 2591, 873,  2592, 45, 927,  2427, 927, 297,  
						837, 2592, 2427,  2593, 836, 2422,  1306, 2422, 246,  26, 2593, 1306,  2594, 299, 1359,  
						876, 1359, 25,  246, 2594, 876,  2595, 46, 929,  2423, 929, 299,  836, 2595, 2423,  
						2596, 835, 2418,  1309, 2418, 249,  27, 2596, 1309,  2597, 301, 1361,  879, 1361, 26,  
						249, 2597, 879,  2598, 47, 931,  2419, 931, 301,  835, 2598, 2419,  2414, 2416, 834,  
						1312, 2414, 252,  28, 2599, 1312,  2600, 303, 1363,  882, 1363, 27,  252, 2600, 882,  
						933, 936, 48,  2415, 933, 303,  2415, 2413, 2601,  2602, 833, 2410,  1315, 2410, 255,  
						29, 2602, 1315,  2603, 305, 1365,  885, 1365, 28,  255, 2603, 885,  2604, 49, 935,  
						2411, 935, 305,  833, 2604, 2411,  2605, 832, 2406,  1318, 2406, 258,  30, 2605, 1318,  
						2606, 307, 1367,  888, 1367, 29,  258, 2606, 888,  2607, 50, 937,  2407, 937, 307,  
						832, 2607, 2407,  2608, 831, 2402,  1321, 2402, 261,  31, 2608, 1321,  2609, 309, 1369,  
						891, 1369, 30,  261, 2609, 891,  2610, 51, 939,  2403, 939, 309,  831, 2610, 2403,  
						2611, 830, 2398,  1324, 2398, 264,  32, 2611, 1324,  2612, 311, 1371,  894, 1371, 31,  
						264, 2612, 894,  2613, 52, 941,  2399, 941, 311,  830, 2613, 2399,  2614, 829, 2394,  
						1327, 2394, 267,  33, 2614, 1327,  2615, 313, 1373,  897, 1373, 32,  267, 2615, 897,  
						2616, 53, 943,  2395, 943, 313,  829, 2616, 2395,  2617, 828, 2390,  1330, 2390, 270,  
						1330, 1377, 2617,  2618, 315, 1375,  900, 1375, 33,  270, 2618, 900,  2619, 54, 945,  
						2391, 945, 315,  828, 2619, 2391,  2620, 827, 2386,  1333, 2386, 273,  35, 2620, 1333,  
						1377, 2387, 317,  903, 1377, 34,  273, 2621, 903,  2622, 55, 947,  317, 2622, 947,  
						827, 2622, 2387,  2623, 826, 2382,  1336, 2382, 276,  36, 2623, 1336,  2624, 319, 1379,  
						906, 1379, 35,  276, 2624, 906,  2625, 56, 949,  2383, 949, 319,  826, 2625, 2383,  
						2626, 825, 2378,  1339, 2378, 279,  37, 2626, 1339,  1381, 2379, 321,  36, 2627, 1381,  
						279, 2627, 909,  2628, 57, 951,  2379, 951, 321,  825, 2628, 2379,  2629, 824, 2374,  
						1342, 2374, 282,  38, 2629, 1342,  2630, 323, 1383,  912, 1383, 37,  282, 2630, 912,  
						2631, 58, 953,  2375, 953, 323,  824, 2631, 2375,  2632, 823, 2370,  1345, 2370, 285,  
						39, 2632, 1345,  2633, 325, 1385,  915, 1385, 38,  285, 2633, 915,  2634, 59, 955,  
						2371, 955, 325,  823, 2634, 2371,  2635, 822, 2366,  1348, 2366, 288,  40, 2635, 1348,  
						2636, 327, 1387,  918, 1387, 39,  288, 2636, 918,  2637, 60, 957,  2367, 957, 327,  
						822, 2637, 2367,  2638, 821, 2362,  1351, 2362, 291,  41, 2638, 1351,  2639, 329, 1389,  
						921, 1389, 40,  291, 2639, 921,  2640, 61, 959,  2363, 959, 329,  821, 2640, 2363,  
						2641, 820, 2358,  1354, 2358, 294,  43, 2641, 1354,  2642, 333, 963,  924, 963, 42,  
						294, 2642, 924,  2643, 63, 1393,  2359, 1393, 333,  820, 2643, 2359,  2644, 819, 2354,  
						1356, 2354, 296,  1356, 1396, 2644,  2645, 334, 1394,  926, 1394, 43,  296, 2645, 926,  
						2646, 64, 964,  2355, 964, 334,  819, 2646, 2355,  2647, 818, 2350,  1358, 2350, 298,  
						45, 2647, 1358,  1396, 2351, 336,  44, 2648, 1396,  298, 2648, 928,  966, 969, 65,  
						336, 2649, 966,  2351, 2349, 2649,  2650, 817, 2346,  1360, 2346, 300,  46, 2650, 1360,  
						2651, 338, 1398,  930, 1398, 45,  300, 2651, 930,  2652, 66, 968,  2347, 968, 338,  
						817, 2652, 2347,  2653, 816, 2342,  1362, 2342, 302,  47, 2653, 1362,  2654, 340, 1400,  
						932, 1400, 46,  302, 2654, 932,  2655, 67, 970,  2343, 970, 340,  816, 2655, 2343,  
						2656, 815, 2338,  1364, 2338, 304,  48, 2656, 1364,  2657, 342, 1402,  934, 1402, 47,  
						304, 2657, 934,  2658, 68, 972,  2339, 972, 342,  815, 2658, 2339,  2659, 814, 2334,  
						1366, 2334, 306,  49, 2659, 1366,  1404, 2335, 344,  48, 2660, 1404,  306, 2660, 936,  
						974, 977, 69,  344, 2661, 974,  2335, 2333, 2661,  2662, 813, 2330,  1368, 2330, 308,  
						50, 2662, 1368,  2663, 346, 1406,  938, 1406, 49,  308, 2663, 938,  2664, 70, 976,  
						2331, 976, 346,  813, 2664, 2331,  2665, 812, 2326,  1370, 2326, 310,  51, 2665, 1370,  
						2666, 348, 1408,  940, 1408, 50,  310, 2666, 940,  978, 981, 71,  2327, 978, 348,  
						812, 2667, 2327,  2668, 811, 2322,  1372, 2322, 312,  52, 2668, 1372,  2669, 350, 1410,  
						942, 1410, 51,  312, 2669, 942,  2670, 72, 980,  2323, 980, 350,  811, 2670, 2323,  
						2671, 810, 2318,  1374, 2318, 314,  53, 2671, 1374,  2672, 352, 1412,  944, 1412, 52,  
						314, 2672, 944,  2673, 73, 982,  2319, 982, 352,  810, 2673, 2319,  2674, 809, 2314,  
						1376, 2314, 316,  54, 2674, 1376,  2675, 354, 1414,  946, 1414, 53,  316, 2675, 946,  
						2676, 74, 984,  2315, 984, 354,  809, 2676, 2315,  2677, 808, 2310,  1378, 2310, 318,  
						55, 2677, 1378,  2678, 356, 1416,  948, 1416, 54,  318, 2678, 948,  986, 989, 75,  
						356, 2679, 986,  808, 2679, 2311,  2680, 807, 2306,  1380, 2306, 320,  1380, 1420, 2680,  
						1418, 2307, 358,  55, 2681, 1418,  320, 2681, 950,  2682, 76, 988,  2307, 988, 358,  
						807, 2682, 2307,  2683, 806, 2302,  1382, 2302, 322,  57, 2683, 1382,  2684, 360, 1420,  
						952, 1420, 56,  322, 2684, 952,  2685, 77, 990,  2303, 990, 360,  806, 2685, 2303,  
						2686, 805, 2298,  1384, 2298, 324,  58, 2686, 1384,  2687, 362, 1422,  954, 1422, 57,  
						324, 2687, 954,  2688, 78, 992,  2299, 992, 362,  805, 2688, 2299,  2689, 804, 2294,  
						1386, 2294, 326,  59, 2689, 1386,  2690, 364, 1424,  956, 1424, 58,  326, 2690, 956,  
						2691, 79, 994,  2295, 994, 364,  804, 2691, 2295,  2692, 803, 2290,  1388, 2290, 328,  
						60, 2692, 1388,  2693, 366, 1426,  958, 1426, 59,  328, 2693, 958,  2694, 80, 996,  
						2291, 996, 366,  803, 2694, 2291,  2695, 802, 2286,  1390, 2286, 330,  61, 2695, 1390,  
						2696, 368, 1428,  960, 1428, 60,  330, 2696, 960,  2697, 81, 998,  2287, 998, 368,  
						802, 2697, 2287,  2698, 801, 2282,  1392, 2282, 332,  62, 2698, 1392,  2699, 370, 1430,  
						962, 1430, 61,  332, 2699, 962,  2700, 82, 1000,  2283, 1000, 370,  801, 2700, 2283,  
						2701, 800, 2278,  1395, 2278, 335,  64, 2701, 1395,  2702, 374, 1004,  965, 1004, 63,  
						335, 2702, 965,  2703, 84, 1434,  2279, 1434, 374,  800, 2703, 2279,  2704, 799, 2274,  
						1397, 2274, 337,  65, 2704, 1397,  2705, 375, 1435,  967, 1435, 64,  337, 2705, 967,  
						2706, 85, 1005,  2275, 1005, 375,  799, 2706, 2275,  2707, 798, 2270,  339, 2707, 2270,  
						66, 2707, 1399,  2708, 377, 1437,  969, 1437, 65,  969, 2270, 2708,  2709, 86, 1007,  
						2271, 1007, 377,  798, 2709, 2271,  2710, 797, 2266,  1401, 2266, 341,  67, 2710, 1401,  
						2711, 379, 1439,  971, 1439, 66,  341, 2711, 971,  2712, 87, 1009,  2267, 1009, 379,  
						797, 2712, 2267,  2713, 796, 2262,  1403, 2262, 343,  68, 2713, 1403,  2714, 381, 1441,  
						973, 1441, 67,  343, 2714, 973,  2715, 88, 1011,  2263, 1011, 381,  796, 2715, 2263,  
						2716, 795, 2258,  1405, 2258, 345,  1405, 1445, 2716,  1443, 2259, 383,  68, 2717, 1443,  
						345, 2717, 975,  1013, 1016, 89,  383, 2718, 1013,  2259, 2257, 2718,  2719, 794, 2254,  
						1407, 2254, 347,  70, 2719, 1407,  2720, 385, 1445,  977, 1445, 69,  347, 2720, 977,  
						1015, 1018, 90,  2255, 1015, 385,  794, 2721, 2255,  2722, 793, 2250,  1409, 2250, 349,  
						71, 2722, 1409,  2723, 387, 1447,  979, 1447, 70,  349, 2723, 979,  2724, 91, 1017,  
						2251, 1017, 387,  793, 2724, 2251,  2725, 792, 2246,  1411, 2246, 351,  72, 2725, 1411,  
						2726, 389, 1449,  71, 2726, 1449,  351, 2726, 981,  2727, 92, 1019,  2247, 1019, 389,  
						792, 2727, 2247,  2728, 791, 2242,  1413, 2242, 353,  73, 2728, 1413,  2729, 391, 1451,  
						983, 1451, 72,  353, 2729, 983,  1021, 1024, 93,  391, 2730, 1021,  791, 2730, 2243,  
						2731, 790, 2238,  1415, 2238, 355,  74, 2731, 1415,  2732, 393, 1453,  985, 1453, 73,  
						355, 2732, 985,  2733, 94, 1023,  2239, 1023, 393,  790, 2733, 2239,  2734, 789, 2234,  
						1417, 2234, 357,  1417, 1457, 2734,  1455, 2235, 395,  74, 2735, 1455,  357, 2735, 987,  
						2736, 95, 1025,  395, 2736, 1025,  789, 2736, 2235,  2737, 788, 2230,  1419, 2230, 359,  
						76, 2737, 1419,  2738, 397, 1457,  75, 2738, 1457,  359, 2738, 989,  2739, 96, 1027,  
						2231, 1027, 397,  788, 2739, 2231,  2740, 787, 2226,  1421, 2226, 361,  77, 2740, 1421,  
						2741, 399, 1459,  991, 1459, 76,  361, 2741, 991,  2742, 97, 1029,  2227, 1029, 399,  
						787, 2742, 2227,  2743, 786, 2222,  1423, 2222, 363,  78, 2743, 1423,  2744, 401, 1461,  
						993, 1461, 77,  363, 2744, 993,  2745, 98, 1031,  2223, 1031, 401,  786, 2745, 2223,  
						2746, 785, 2218,  1425, 2218, 365,  79, 2746, 1425,  2747, 403, 1463,  995, 1463, 78,  
						365, 2747, 995,  2748, 99, 1033,  2219, 1033, 403,  785, 2748, 2219,  2749, 784, 2214,  
						1427, 2214, 367,  80, 2749, 1427,  2750, 405, 1465,  997, 1465, 79,  367, 2750, 997,  
						2751, 100, 1035,  2215, 1035, 405,  784, 2751, 2215,  2752, 783, 2210,  1429, 2210, 369,  
						81, 2752, 1429,  2753, 407, 1467,  999, 1467, 80,  369, 2753, 999,  2754, 101, 1037,  
						2211, 1037, 407,  783, 2754, 2211,  2755, 782, 2206,  1431, 2206, 371,  82, 2755, 1431,  
						2756, 409, 1469,  1001, 1469, 81,  371, 2756, 1001,  2757, 102, 1039,  2207, 1039, 409,  
						782, 2757, 2207,  2758, 781, 2202,  1433, 2202, 373,  83, 2758, 1433,  2759, 411, 1471,  
						1003, 1471, 82,  1003, 2202, 2759,  1041, 1044, 103,  411, 2760, 1041,  2203, 2201, 2760,  
						2198, 2200, 780,  376, 2761, 2198,  85, 2761, 1436,  2762, 415, 1045,  1006, 1045, 84,  
						1006, 2198, 2762,  1475, 1047, 105,  2199, 1475, 415,  780, 2763, 2199,  2764, 779, 2194,  
						1438, 2194, 378,  86, 2764, 1438,  2765, 416, 1476,  1008, 1476, 85,  378, 2765, 1008,  
						1046, 1049, 106,  2195, 1046, 416,  779, 2766, 2195,  2767, 778, 2190,  1440, 2190, 380,  
						87, 2767, 1440,  2768, 418, 1478,  1010, 1478, 86,  380, 2768, 1010,  2769, 107, 1048,  
						2191, 1048, 418,  778, 2769, 2191,  2186, 2188, 777,  382, 2770, 2186,  88, 2770, 1442,  
						2771, 420, 1480,  1012, 1480, 87,  1012, 2186, 2771,  1050, 1053, 108,  420, 2772, 1050,  
						2187, 2185, 2772,  2773, 776, 2182,  1444, 2182, 384,  89, 2773, 1444,  2774, 422, 1482,  
						1014, 1482, 88,  384, 2774, 1014,  1052, 1055, 109,  2183, 1052, 422,  776, 2775, 2183,  
						2178, 2180, 775,  386, 2776, 2178,  1446, 1486, 2776,  1484, 2179, 424,  89, 2777, 1484,  
						1016, 2178, 2777,  1054, 1057, 110,  424, 2778, 1054,  2179, 2177, 2778,  2779, 774, 2174,  
						1448, 2174, 388,  91, 2779, 1448,  2780, 426, 1486,  1018, 1486, 90,  388, 2780, 1018,  
						1056, 1059, 111,  2175, 1056, 426,  2175, 2173, 2781,  2782, 773, 2170,  1450, 2170, 390,  
						1450, 1490, 2782,  2783, 428, 1488,  1020, 1488, 91,  390, 2783, 1020,  2784, 112, 1058,  
						2171, 1058, 428,  773, 2784, 2171,  2785, 772, 2166,  1452, 2166, 392,  93, 2785, 1452,  
						1490, 2167, 430,  1022, 1490, 92,  392, 2786, 1022,  1060, 1063, 113,  430, 2787, 1060,  
						772, 2787, 2167,  2788, 771, 2162,  1454, 2162, 394,  94, 2788, 1454,  2789, 432, 1492,  
						93, 2789, 1492,  394, 2789, 1024,  2790, 114, 1062,  2163, 1062, 432,  2163, 2161, 2790,  
						2791, 770, 2158,  1456, 2158, 396,  1456, 1496, 2791,  2792, 434, 1494,  1026, 1494, 94,  
						396, 2792, 1026,  2793, 115, 1064,  2159, 1064, 434,  770, 2793, 2159,  2154, 2156, 769,  
						1458, 2154, 398,  96, 2794, 1458,  1496, 2155, 436,  95, 2795, 1496,  1028, 2154, 2795,  
						2796, 116, 1066,  436, 2796, 1066,  2155, 2153, 2796,  2797, 768, 2150,  1460, 2150, 400,  
						97, 2797, 1460,  2798, 438, 1498,  1030, 1498, 96,  400, 2798, 1030,  2799, 117, 1068,  
						2151, 1068, 438,  768, 2799, 2151,  2800, 767, 2146,  1462, 2146, 402,  98, 2800, 1462,  
						2801, 440, 1500,  1032, 1500, 97,  1032, 2146, 2801,  1070, 1073, 118,  2147, 1070, 440,  
						2147, 2145, 2802,  2803, 766, 2142,  1464, 2142, 404,  99, 2803, 1464,  2804, 442, 1502,  
						1034, 1502, 98,  404, 2804, 1034,  2805, 119, 1072,  2143, 1072, 442,  766, 2805, 2143,  
						2806, 765, 2138,  406, 2806, 2138,  1466, 1506, 2806,  2807, 444, 1504,  99, 2807, 1504,  
						1036, 2138, 2807,  2808, 120, 1074,  2139, 1074, 444,  765, 2808, 2139,  2809, 764, 2134,  
						1468, 2134, 408,  101, 2809, 1468,  2810, 446, 1506,  1038, 1506, 100,  408, 2810, 1038,  
						2811, 121, 1076,  2135, 1076, 446,  764, 2811, 2135,  2130, 2132, 763,  410, 2812, 2130,  
						102, 2812, 1470,  2813, 448, 1508,  1040, 1508, 101,  1040, 2130, 2813,  1078, 1081, 122,  
						448, 2814, 1078,  2131, 2129, 2814,  2815, 762, 2126,  1472, 2126, 412,  103, 2815, 1472,  
						2816, 450, 1510,  1042, 1510, 102,  412, 2816, 1042,  1080, 1083, 123,  2127, 1080, 450,  
						762, 2817, 2127,  2122, 2124, 761,  414, 2818, 2122,  1474, 1514, 2818,  1512, 2123, 452,  
						103, 2819, 1512,  1044, 2122, 2819,  1082, 1085, 124,  452, 2820, 1082,  2123, 2121, 2820,  
						2821, 760, 2118,  1477, 2118, 417,  106, 2821, 1477,  2822, 456, 1086,  1047, 1086, 105,  
						417, 2822, 1047,  2823, 126, 1516,  2119, 1516, 456,  760, 2823, 2119,  2824, 759, 2114,  
						1479, 2114, 419,  107, 2824, 1479,  2825, 457, 1517,  1049, 1517, 106,  419, 2825, 1049,  
						2826, 127, 1087,  2115, 1087, 457,  759, 2826, 2115,  2110, 2112, 758,  1481, 2110, 421,  
						108, 2827, 1481,  1519, 2111, 459,  107, 2828, 1519,  421, 2828, 1051,  1089, 1092, 128,  
						459, 2829, 1089,  2111, 2109, 2829,  2106, 2108, 757,  423, 2830, 2106,  1483, 1523, 2830,  
						1521, 2107, 461,  108, 2831, 1521,  1053, 2106, 2831,  2832, 129, 1091,  2107, 1091, 461,  
						757, 2832, 2107,  2833, 756, 2102,  1485, 2102, 425,  110, 2833, 1485,  2834, 463, 1523,  
						1055, 1523, 109,  425, 2834, 1055,  1093, 1096, 130,  2103, 1093, 463,  2103, 2101, 2835,  
						2836, 755, 2098,  1487, 2098, 427,  111, 2836, 1487,  2837, 465, 1525,  1057, 1525, 110,  
						427, 2837, 1057,  2838, 131, 1095,  2099, 1095, 465,  755, 2838, 2099,  2094, 2096, 754,  
						429, 2839, 2094,  112, 2839, 1489,  1527, 2095, 467,  111, 2840, 1527,  1059, 2094, 2840,  
						2841, 132, 1097,  467, 2841, 1097,  2095, 2093, 2841,  2842, 753, 2090,  1491, 2090, 431,  
						113, 2842, 1491,  2843, 469, 1529,  1061, 1529, 112,  431, 2843, 1061,  2844, 133, 1099,  
						2091, 1099, 469,  753, 2844, 2091,  2845, 752, 2086,  433, 2845, 2086,  114, 2845, 1493,  
						2846, 471, 1531,  113, 2846, 1531,  1063, 2086, 2846,  2847, 134, 1101,  2087, 1101, 471,  
						2087, 2085, 2847,  2848, 751, 2082,  1495, 2082, 435,  1495, 1535, 2848,  2849, 473, 1533,  
						1065, 1533, 114,  435, 2849, 1065,  1103, 1106, 135,  473, 2850, 1103,  751, 2850, 2083,  
						2851, 750, 2078,  1497, 2078, 437,  116, 2851, 1497,  2852, 475, 1535,  1067, 1535, 115,  
						437, 2852, 1067,  2853, 136, 1105,  475, 2853, 1105,  750, 2853, 2079,  2854, 749, 2074,  
						1499, 2074, 439,  117, 2854, 1499,  2855, 477, 1537,  1069, 1537, 116,  439, 2855, 1069,  
						2856, 137, 1107,  2075, 1107, 477,  749, 2856, 2075,  2857, 748, 2070,  1501, 2070, 441,  
						118, 2857, 1501,  2858, 479, 1539,  1071, 1539, 117,  441, 2858, 1071,  1109, 1112, 138,  
						479, 2859, 1109,  748, 2859, 2071,  2066, 2068, 747,  443, 2860, 2066,  1503, 1543, 2860,  
						1541, 2067, 481,  118, 2861, 1541,  1073, 2066, 2861,  2862, 139, 1111,  2067, 1111, 481,  
						2067, 2065, 2862,  2863, 746, 2062,  1505, 2062, 445,  120, 2863, 1505,  2864, 483, 1543,  
						1075, 1543, 119,  445, 2864, 1075,  2865, 140, 1113,  2063, 1113, 483,  746, 2865, 2063,  
						2866, 745, 2058,  1507, 2058, 447,  121, 2866, 1507,  2867, 485, 1545,  1077, 1545, 120,  
						447, 2867, 1077,  2868, 141, 1115,  2059, 1115, 485,  745, 2868, 2059,  2054, 2056, 744,  
						1509, 205