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2 * nu)) # Lame parameters 12 | x = ti.Vector.field(2, dtype=float, shape=n_particles) # position 13 | v = ti.Vector.field(2, dtype=float, shape=n_particles) # velocity 14 | C = ti.Matrix.field(2, 2, dtype=float, shape=n_particles) # affine velocity field 15 | F = ti.Matrix.field(2, 2, dtype=float, shape=n_particles) # deformation gradient 16 | material = ti.field(dtype=int, shape=n_particles) # material id 17 | Jp = ti.field(dtype=float, shape=n_particles) # plastic deformation 18 | grid_v = ti.Vector.field(2, dtype=float, shape=(n_grid, n_grid)) # grid node momentum/velocity 19 | grid_m = ti.field(dtype=float, shape=(n_grid, n_grid)) # grid node mass 20 | 21 | @ti.kernel 22 | def substep(): 23 | for i, j in grid_m: 24 | grid_v[i, j] = [0, 0] 25 | grid_m[i, j] = 0 26 | for p in x: # Particle state update and scatter to grid (P2G) 27 | base = (x[p] * inv_dx).cast(int) 28 | fx = x[p] * inv_dx - base.cast(float) 29 | # Cubic kernels http://mpm.graphics Eqn. 122 with x=fx+1, fx, abs(fx-1), abs(fx-2) 30 | w = [1./6.*(2. - (fx + 1))**3, 0.5*fx**3 - fx**2 + 2./3., 0.5*(-(fx - 1.))**3 - (-(fx - 1.))**2 + 2./3., 1./6.*(2. + (fx - 2.))**3] 31 | F[p] = (ti.Matrix.identity(float, 2) + dt * C[p]) @ F[p] # deformation gradient update 32 | h = ti.exp(10 * (1.0 - Jp[p])) # Hardening coefficient: snow gets harder when compressed 33 | if material[p] == 1: # jelly, make it softer 34 | h = 0.3 35 | mu, la = mu_0 * h, lambda_0 * h 36 | if material[p] == 0: # liquid 37 | mu = 0.0 38 | U, sig, V = ti.svd(F[p]) 39 | J = 1.0 40 | for d in ti.static(range(2)): 41 | new_sig = sig[d, d] 42 | if material[p] == 2: # Snow 43 | new_sig = min(max(sig[d, d], 1 - 2.5e-2), 1 + 4.5e-3) # Plasticity 44 | Jp[p] *= sig[d, d] / new_sig 45 | sig[d, d] = new_sig 46 | J *= new_sig 47 | if material[p] == 0: # Reset deformation gradient to avoid numerical instability 48 | F[p] = ti.Matrix.identity(float, 2) * ti.sqrt(J) 49 | elif material[p] == 2: 50 | F[p] = U @ sig @ V.transpose() # Reconstruct elastic deformation gradient after plasticity 51 | stress = 2 * mu * (F[p] - U @ V.transpose()) @ F[p].transpose() + ti.Matrix.identity(float, 2) * la * J * (J - 1) 52 | stress = (-dt * p_vol * 3 * inv_dx * inv_dx) * stress 53 | affine = stress + p_mass * C[p] 54 | for i, j in ti.static(ti.ndrange(4, 4)): # Loop over 4x4 grid node neighborhood 55 | offset = ti.Vector([i, j]) - 1 56 | dpos = (offset.cast(float) - fx) * dx 57 | weight = w[i][0] * w[j][1] 58 | grid_v[base + offset] += weight * (p_mass * v[p] + affine @ dpos) 59 | grid_m[base + offset] += weight * p_mass 60 | for i, j in grid_m: 61 | if grid_m[i, j] > 0: # No need for epsilon here 62 | grid_v[i, j] = (1 / grid_m[i, j]) * grid_v[i, j] # Momentum to velocity 63 | grid_v[i, j][1] -= dt * 50 # gravity 64 | if i < 3 and grid_v[i, j][0] < 0: grid_v[i, j][0] = 0 # Boundary conditions 65 | if i > n_grid - 3 and grid_v[i, j][0] > 0: grid_v[i, j][0] = 0 66 | if j < 3 and grid_v[i, j][1] < 0: grid_v[i, j][1] = 0 67 | if j > n_grid - 3 and grid_v[i, j][1] > 0: grid_v[i, j][1] = 0 68 | for p in x: # grid to particle (G2P) 69 | base = (x[p] * inv_dx).cast(int) 70 | fx = x[p] * inv_dx - base.cast(float) 71 | w = [1./6.*(2. - (fx + 1))**3, 0.5*fx**3 - fx**2 + 2./3., 0.5*(-(fx - 1.))**3 - (-(fx - 1.))**2 + 2./3., 1./6.*(2. + (fx - 2.))**3] 72 | new_v = ti.Vector.zero(float, 2) 73 | new_C = ti.Matrix.zero(float, 2, 2) 74 | for i, j in ti.static(ti.ndrange(4, 4)): # loop over 4x4 grid node neighborhood 75 | dpos = ti.Vector([i, j]).cast(float) - fx - 1 76 | g_v = grid_v[base + ti.Vector([i, j]) - 1] 77 | weight = w[i][0] * w[j][1] 78 | new_v += weight * g_v 79 | new_C += 3 * inv_dx * weight * g_v.outer_product(dpos) 80 | v[p], C[p] = new_v, new_C 81 | x[p] += dt * v[p] # advection 82 | 83 | group_size = n_particles // 3 84 | @ti.kernel 85 | def initialize(): 86 | for i in range(n_particles): 87 | x[i] = [ti.random() * 0.2 + 0.3 + 0.10 * (i // group_size), ti.random() * 0.2 + 0.05 + 0.32 * (i // group_size)] 88 | material[i] = i // group_size # 0: fluid 1: jelly 2: snow 89 | v[i] = ti.Matrix([0, 0]) 90 | F[i] = ti.Matrix([[1, 0], [0, 1]]) 91 | Jp[i] = 1 92 | initialize() 93 | gui = ti.GUI("Taichi MLS-MPM-99", res=512, background_color=0x112F41) 94 | while not gui.get_event(ti.GUI.ESCAPE, ti.GUI.EXIT): 95 | for s in range(int(2e-3 // dt)): 96 | substep() 97 | colors = np.array([0x068587, 0xED553B, 0xEEEEF0], dtype=np.uint32) 98 | gui.circles(x.to_numpy(), radius=1.5, color=colors[material.to_numpy()]) 99 | gui.show() # Change to gui.show(f'{frame:06d}.png') to write images to disk 100 | --------------------------------------------------------------------------------