A Moving Least Squares Material Point Method implementation in Taichi language. Originally created by @yuanming-hu with 128 lines of code.

taichi_zoo_MLS-MPM-128.py

import taichi as ti

ti.init(arch=ti.gpu)  # Try to run on GPU

quality = 1  # Use a larger value for higher-res simulations
n_particles, n_grid = 9000 * quality**2, 128 * 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
E, nu = 5e3, 0.2  # Young's modulus and Poisson's ratio
mu_0, lambda_0 = E / (2 * (1 + nu)), E * nu / (
    (1 + nu) * (1 - 2 * nu))  # Lame parameters

x = ti.Vector.field(2, dtype=float, shape=n_particles)  # position
v = ti.Vector.field(2, dtype=float, shape=n_particles)  # velocity
C = ti.Matrix.field(2, 2, dtype=float,
                    shape=n_particles)  # affine velocity field
F = ti.Matrix.field(2, 2, dtype=float,
                    shape=n_particles)  # deformation gradient
material = ti.field(dtype=int, shape=n_particles)  # material id
Jp = ti.field(dtype=float, shape=n_particles)  # plastic deformation
grid_v = ti.Vector.field(2, dtype=float,
                         shape=(n_grid, n_grid))  # grid node momentum/velocity
grid_m = ti.field(dtype=float, shape=(n_grid, n_grid))  # grid node mass
gravity = ti.Vector.field(2, dtype=float, shape=())
attractor_strength = ti.field(dtype=float, shape=())
attractor_pos = ti.Vector.field(2, dtype=float, shape=())


@ti.kernel
def substep():
    for i, j in grid_m:
        grid_v[i, j] = [0, 0]
        grid_m[i, j] = 0
    for p in x:  # Particle state update and scatter to grid (P2G)
        base = (x[p] * inv_dx - 0.5).cast(int)
        fx = x[p] * inv_dx - base.cast(float)
        # Quadratic kernels  [http://mpm.graphics   Eqn. 123, with x=fx, fx-1,fx-2]
        w = [0.5 * (1.5 - fx)**2, 0.75 - (fx - 1)**2, 0.5 * (fx - 0.5)**2]
        F[p] = (ti.Matrix.identity(float, 2) +
                dt * C[p]) @ F[p]  # deformation gradient update
        h = max(0.1, min(5, ti.exp(10 * (1.0 - Jp[p])))
                )  # Hardening coefficient: snow gets harder when compressed
        if material[p] == 1:  # jelly, make it softer
            h = 0.3
        mu, la = mu_0 * h, lambda_0 * h
        if material[p] == 0:  # liquid
            mu = 0.0
        U, sig, V = ti.svd(F[p])
        J = 1.0
        for d in ti.static(range(2)):
            new_sig = sig[d, d]
            if material[p] == 2:  # Snow
                new_sig = min(max(sig[d, d], 1 - 2.5e-2),
                              1 + 4.5e-3)  # Plasticity
            Jp[p] *= sig[d, d] / new_sig
            sig[d, d] = new_sig
            J *= new_sig
        if material[
                p] == 0:  # Reset deformation gradient to avoid numerical instability
            F[p] = ti.Matrix.identity(float, 2) * ti.sqrt(J)
        elif material[p] == 2:
            F[p] = U @ sig @ V.transpose(
            )  # Reconstruct elastic deformation gradient after plasticity
        stress = 2 * mu * (F[p] - U @ V.transpose()) @ F[p].transpose(
        ) + ti.Matrix.identity(float, 2) * la * J * (J - 1)
        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)):  # Loop over 3x3 grid node neighborhood
            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:  # No need for epsilon here
            grid_v[i,
                   j] = (1 / grid_m[i, j]) * grid_v[i,
                                                    j]  # Momentum to velocity
            grid_v[i, j] += dt * gravity[None] * 30  # gravity
            dist = attractor_pos[None] - dx * ti.Vector([i, j])
            grid_v[i, j] += dist / (
                0.01 + dist.norm()) * attractor_strength[None] * dt * 100
            if i < 3 and grid_v[i, j][0] < 0:
                grid_v[i, j][0] = 0  # Boundary conditions
            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: grid_v[i, j][1] = 0
            if j > n_grid - 3 and grid_v[i, j][1] > 0: grid_v[i, j][1] = 0
    for p in x:  # grid to particle (G2P)
        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)):  # loop over 3x3 grid node neighborhood
            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]  # advection


@ti.kernel
def reset():
    group_size = n_particles // 3
    for i in range(n_particles):
        x[i] = [
            ti.random() * 0.2 + 0.3 + 0.10 * (i // group_size),
            ti.random() * 0.2 + 0.05 + 0.32 * (i // group_size)
        ]
        material[i] = i // group_size  # 0: fluid 1: jelly 2: snow
        v[i] = [0, 0]
        F[i] = ti.Matrix([[1, 0], [0, 1]])
        Jp[i] = 1
        C[i] = ti.Matrix.zero(float, 2, 2)


print(
    "[Hint] Use WSAD/arrow keys to control gravity. Use left/right mouse bottons to attract/repel. Press R to reset."
)
gui = ti.GUI("Taichi MLS-MPM-128", res=512, background_color=0x112F41)
reset()
gravity[None] = [0, -1]

for frame in range(20000):
    if gui.get_event(ti.GUI.PRESS):
        if gui.event.key == 'r': reset()
        elif gui.event.key in [ti.GUI.ESCAPE, ti.GUI.EXIT]: break
    if gui.event is not None: gravity[None] = [0, 0]  # if had any event
    if gui.is_pressed(ti.GUI.LEFT, 'a'): gravity[None][0] = -1
    if gui.is_pressed(ti.GUI.RIGHT, 'd'): gravity[None][0] = 1
    if gui.is_pressed(ti.GUI.UP, 'w'): gravity[None][1] = 1
    if gui.is_pressed(ti.GUI.DOWN, 's'): gravity[None][1] = -1
    mouse = gui.get_cursor_pos()
    attractor_pos[None] = [mouse[0], mouse[1]]
    attractor_strength[None] = 0
    if gui.is_pressed(ti.GUI.LMB):
        attractor_strength[None] = 1
    if gui.is_pressed(ti.GUI.RMB):
        attractor_strength[None] = -1
    for s in range(int(2e-3 // dt)):
        substep()
    gui.clear(0x112F41)
    gui.circles(x.to_numpy(),
                radius=1,
                palette=[0x068587, 0xED553B, 0xEEEEF0],
                palette_indices=material)
    gui.show()