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mpm128.py
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mpm128.py
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import taichi as ti
import numpy as np
ti.init(arch=ti.cuda) # 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(2, dt=ti.f32, shape=n_particles) # position
v = ti.Vector(2, dt=ti.f32, shape=n_particles) # velocity
C = ti.Matrix(2, 2, dt=ti.f32, shape=n_particles) # affine velocity field
F = ti.Matrix(2, 2, dt=ti.f32, shape=n_particles) # deformation gradient
material = ti.var(dt=ti.i32, shape=n_particles) # material id
Jp = ti.var(dt=ti.f32, shape=n_particles) # plastic deformation
grid_v = ti.Vector(2, dt=ti.f32, shape=(n_grid, n_grid)) # grid node momentum/velocity
grid_m = ti.var(dt=ti.f32, shape=(n_grid, n_grid)) # grid node mass
gravity = ti.Vector(2, dt=ti.f32, shape=())
attractor_strength = ti.var(dt=ti.f32, shape=())
attractor_pos = ti.Vector(2, dt=ti.f32, 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 * ti.sqr(1.5 - fx), 0.75 - ti.sqr(fx - 1), 0.5 * ti.sqr(fx - 0.5)]
F[p] = (ti.Matrix.identity(ti.f32, 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(ti.f32, 2) * ti.sqrt(J)
elif material[p] == 2:
F[p] = U @ sig @ V.T() # Reconstruct elastic deformation gradient after plasticity
stress = 2 * mu * (F[p] - U @ V.T()) @ F[p].T() + ti.Matrix.identity(ti.f32, 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 * ti.sqr(1.5 - fx), 0.75 - ti.sqr(fx - 1.0), 0.5 * ti.sqr(fx - 0.5)]
new_v = ti.Vector.zero(ti.f32, 2)
new_C = ti.Matrix.zero(ti.f32, 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 * ti.outer_product(g_v, 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, -1]
F[i] = ti.Matrix([[1, 0], [0, 1]])
Jp[i] = 1
C[i] = ti.Matrix.zero(ti.f32, 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()
for frame in range(20000):
gravity[None] = [0, 0]
while gui.has_key_event():
e = gui.get_key_event()
if e.type == ti.GUI.RELEASE: continue
elif e.key == 'r': reset()
elif e.key == ti.GUI.ESCAPE: exit(0)
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()
gui.circle((mouse[0], mouse[1]), color=0x336699, radius=15)
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()
colors = np.array([0x068587, 0xED553B, 0xEEEEF0], dtype=np.uint32)
gui.circles(x.to_numpy(), radius=1.5, color=colors[material.to_numpy()])
gui.show() # Change to gui.show(f'{frame:06d}.png') to write images to disk