——闲聊物理引擎,可微编程,SIGGRAPH生产力难题,与Taichi编程语言目录序MaterialPointMethod(物资点法)MovingLeastSquaresM"> ——闲聊物理引擎,可微编程,SIGGRAPH生产力难题,与Taichi编程语言目录序MaterialPointMethod(物资点法)MovingLeastSquaresM" />

代码果冻99行代码的《冰雪奇缘》

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——闲聊物理引擎,可微编程,SIGGRAPH生产力难题,与Taichi编程语言

目录

  • Material Point Method(物资点法)
  • Moving Least Squares Material Point Method(移动最小二乘物资点法)
  • Differentiable MLS-MPM (ChainQueen可微物理引擎)
  • “硬核”的盘算机图形学
  • Taichi编程语言
  • Differentiable Programming (可微编程) 与 Taichi
  • import taichi as ti:伪装成Python的Taichi语言
  • 总结
  • 终于下决心一口吻把四颗智齿拔了,这两天随同着牙疼只能吃土豆泥和蒸蛋,写编译器之类的体力活是干不了了。登录知乎发明自己居然曾经开了个专栏,无奈一直沉迷于写代码,一篇文章都没写,实在有些惭愧。

    最近正好看了《冰雪奇缘2》,就顺便聊聊动画电影里面持续介质,比如雪的模仿吧。实际上,由于最近图形技巧的发展,只须要99行代码就可以写一个简略的持续介质模仿器,模仿三种相互作用的不同资料(水,果冻,雪),并且在你的笔记本上就能运行:

    14万个水,果冻,和雪“粒子”https://www.zhihu.com/video/1195682534812659712

    代码:

    import taichi as ti import numpy as np 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 = 0.1e4, 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 @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 = 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][1] -= dt * 50 # gravity 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 group_size = n_particles // 3 @ti.kernel def initialize(): 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] = ti.Matrix([0, 0]) F[i] = ti.Matrix([[1, 0], [0, 1]]) Jp[i] = 1 initialize() gui = ti.GUI("Taichi MLS-MPM-99", res=512, background_color=0x112F41) for frame in range(20000): 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