switched to NH

Author: liminchen

9_reduced_DOF/MassSpringEnergy.py

@@ -0,0 +1,187 @@
+# ANCHOR: helper_func
+import utils
+import numpy as np
+import math
+
+def polar_svd(F):
+    [U, s, VT] = np.linalg.svd(F)
+    if np.linalg.det(U) < 0:
+        U[:, 1] = -U[:, 1]
+        s[1] = -s[1]
+    if np.linalg.det(VT) < 0:
+        VT[1, :] = -VT[1, :]
+        s[1] = -s[1]
+    return [U, s, VT]
+
+def dPsi_div_dsigma(s, mu, lam):
+    ln_sigma_prod = math.log(s[0] * s[1])
+    inv0 = 1.0 / s[0]
+    dPsi_dsigma_0 = mu * (s[0] - inv0) + lam * inv0 * ln_sigma_prod
+    inv1 = 1.0 / s[1]
+    dPsi_dsigma_1 = mu * (s[1] - inv1) + lam * inv1 * ln_sigma_prod
+    return [dPsi_dsigma_0, dPsi_dsigma_1]
+
+def d2Psi_div_dsigma2(s, mu, lam):
+    ln_sigma_prod = math.log(s[0] * s[1])
+    inv2_0 = 1 / (s[0] * s[0])
+    d2Psi_dsigma2_00 = mu * (1 + inv2_0) - lam * inv2_0 * (ln_sigma_prod - 1)
+    inv2_1 = 1 / (s[1] * s[1])
+    d2Psi_dsigma2_11 = mu * (1 + inv2_1) - lam * inv2_1 * (ln_sigma_prod - 1)
+    d2Psi_dsigma2_01 = lam / (s[0] * s[1])
+    return [[d2Psi_dsigma2_00, d2Psi_dsigma2_01], [d2Psi_dsigma2_01, d2Psi_dsigma2_11]]
+
+def B_left_coef(s, mu, lam):
+    sigma_prod = s[0] * s[1]
+    return (mu + (mu - lam * math.log(sigma_prod)) / sigma_prod) / 2
+
+def Psi(F, mu, lam):
+    J = np.linalg.det(F)
+    lnJ = math.log(J)
+    return mu / 2 * (np.trace(np.transpose(F).dot(F)) - 2) - mu * lnJ + lam / 2 * lnJ * lnJ
+
+def dPsi_div_dF(F, mu, lam):
+    FinvT = np.transpose(np.linalg.inv(F))
+    return mu * (F - FinvT) + lam * math.log(np.linalg.det(F)) * FinvT
+
+def d2Psi_div_dF2(F, mu, lam, project_PSD = True):
+    [U, sigma, VT] = polar_svd(F)
+
+    Psi_sigma_sigma = np.array(d2Psi_div_dsigma2(sigma, mu, lam))
+    if project_PSD:
+        Psi_sigma_sigma = utils.make_PSD(Psi_sigma_sigma)
+
+    B_left = B_left_coef(sigma, mu, lam)
+    Psi_sigma = dPsi_div_dsigma(sigma, mu, lam)
+    B_right = (Psi_sigma[0] + Psi_sigma[1]) / (2 * max(sigma[0] + sigma[1], 1e-6))
+    B = np.array([[B_left + B_right, B_left - B_right], [B_left - B_right, B_left + B_right]])
+    if project_PSD:
+        B = utils.make_PSD(B)
+
+    M = np.array([[0, 0, 0, 0]] * 4)
+    M[0, 0] = Psi_sigma_sigma[0, 0]
+    M[0, 3] = Psi_sigma_sigma[0, 1]
+    M[1, 1] = B[0, 0]
+    M[1, 2] = B[0, 1]
+    M[2, 1] = B[1, 0]
+    M[2, 2] = B[1, 1]
+    M[3, 0] = Psi_sigma_sigma[1, 0]
+    M[3, 3] = Psi_sigma_sigma[1, 1]
+
+    dP_div_dF = np.array([[0, 0, 0, 0]] * 4)
+    for j in range(0, 2):
+        for i in range(0, 2):
+            ij = j * 2 + i
+            for s in range(0, 2):
+                for r in range(0, 2):
+                    rs = s * 2 + r
+                    dP_div_dF[ij, rs] = M[0, 0] * U[i, 0] * VT[0, j] * U[r, 0] * VT[0, s] \
+                        + M[0, 3] * U[i, 0] * VT[0, j] * U[r, 1] * VT[1, s] \
+                        + M[1, 1] * U[i, 1] * VT[0, j] * U[r, 1] * VT[0, s] \
+                        + M[1, 2] * U[i, 1] * VT[0, j] * U[r, 0] * VT[1, s] \
+                        + M[2, 1] * U[i, 0] * VT[1, j] * U[r, 1] * VT[0, s] \
+                        + M[2, 2] * U[i, 0] * VT[1, j] * U[r, 0] * VT[1, s] \
+                        + M[3, 0] * U[i, 1] * VT[1, j] * U[r, 0] * VT[0, s] \
+                        + M[3, 3] * U[i, 1] * VT[1, j] * U[r, 1] * VT[1, s]
+    return dP_div_dF
+# ANCHOR_END: helper_func
+
+# ANCHOR: stress_deriv
+def deformation_grad(x, elemVInd, IB):
+    F = [x[elemVInd[1]] - x[elemVInd[0]], x[elemVInd[2]] - x[elemVInd[0]]]
+    return np.transpose(F).dot(IB)
+
+def dPsi_div_dx(P, IB):  # applying chain-rule, dPsi_div_dx = dPsi_div_dF * dF_div_dx
+    dPsi_dx_2 = P[0, 0] * IB[0, 0] + P[0, 1] * IB[0, 1]
+    dPsi_dx_3 = P[1, 0] * IB[0, 0] + P[1, 1] * IB[0, 1]
+    dPsi_dx_4 = P[0, 0] * IB[1, 0] + P[0, 1] * IB[1, 1]
+    dPsi_dx_5 = P[1, 0] * IB[1, 0] + P[1, 1] * IB[1, 1]
+    return [np.array([-dPsi_dx_2 - dPsi_dx_4, -dPsi_dx_3 - dPsi_dx_5]), np.array([dPsi_dx_2, dPsi_dx_3]), np.array([dPsi_dx_4, dPsi_dx_5])]
+
+def d2Psi_div_dx2(dP_div_dF, IB):  # applying chain-rule, d2Psi_div_dx2 = dF_div_dx^T * d2Psi_div_dF2 * dF_div_dx (note that d2F_div_dx2 = 0)
+    intermediate = np.array([[0.0, 0.0, 0.0, 0.0]] * 6)
+    for colI in range(0, 4):
+        _000 = dP_div_dF[0, colI] * IB[0, 0]
+        _010 = dP_div_dF[0, colI] * IB[1, 0]
+        _101 = dP_div_dF[2, colI] * IB[0, 1]
+        _111 = dP_div_dF[2, colI] * IB[1, 1]
+        _200 = dP_div_dF[1, colI] * IB[0, 0]
+        _210 = dP_div_dF[1, colI] * IB[1, 0]
+        _301 = dP_div_dF[3, colI] * IB[0, 1]
+        _311 = dP_div_dF[3, colI] * IB[1, 1]
+        intermediate[2, colI] = _000 + _101
+        intermediate[3, colI] = _200 + _301
+        intermediate[4, colI] = _010 + _111
+        intermediate[5, colI] = _210 + _311
+        intermediate[0, colI] = -intermediate[2, colI] - intermediate[4, colI]
+        intermediate[1, colI] = -intermediate[3, colI] - intermediate[5, colI]
+    result = np.array([[0.0, 0.0, 0.0, 0.0, 0.0, 0.0]] * 6)
+    for colI in range(0, 6):
+        _000 = intermediate[colI, 0] * IB[0, 0]
+        _010 = intermediate[colI, 0] * IB[1, 0]
+        _101 = intermediate[colI, 2] * IB[0, 1]
+        _111 = intermediate[colI, 2] * IB[1, 1]
+        _200 = intermediate[colI, 1] * IB[0, 0]
+        _210 = intermediate[colI, 1] * IB[1, 0]
+        _301 = intermediate[colI, 3] * IB[0, 1]
+        _311 = intermediate[colI, 3] * IB[1, 1]
+        result[2, colI] = _000 + _101
+        result[3, colI] = _200 + _301
+        result[4, colI] = _010 + _111
+        result[5, colI] = _210 + _311
+        result[0, colI] = -_000 - _101 - _010 - _111
+        result[1, colI] = -_200 - _301 - _210 - _311
+    return result
+# ANCHOR_END: stress_deriv
+
+# ANCHOR: val_grad_hess
+def val(x, e, vol, IB, mu, lam):
+    sum = 0.0
+    for i in range(0, len(e)):
+        F = deformation_grad(x, e[i], IB[i])
+        sum += vol[i] * Psi(F, mu[i], lam[i])
+    return sum
+
+def grad(x, e, vol, IB, mu, lam):
+    g = np.array([[0.0, 0.0]] * len(x))
+    for i in range(0, len(e)):
+        F = deformation_grad(x, e[i], IB[i])
+        P = vol[i] * dPsi_div_dF(F, mu[i], lam[i])
+        g_local = dPsi_div_dx(P, IB[i])
+        for j in range(0, 3):
+            g[e[i][j]] += g_local[j]
+    return g
+
+def hess(x, e, vol, IB, mu, lam, project_PSD = True):
+    IJV = [[0] * (len(e) * 36), [0] * (len(e) * 36), np.array([0.0] * (len(e) * 36))]
+    for i in range(0, len(e)):
+        F = deformation_grad(x, e[i], IB[i])
+        dP_div_dF = vol[i] * d2Psi_div_dF2(F, mu[i], lam[i], project_PSD)
+        local_hess = d2Psi_div_dx2(dP_div_dF, IB[i])
+        for xI in range(0, 3):
+            for xJ in range(0, 3):
+                for dI in range(0, 2):
+                    for dJ in range(0, 2):
+                        ind = i * 36 + (xI * 3 + xJ) * 4 + dI * 2 + dJ
+                        IJV[0][ind] = e[i][xI] * 2 + dI
+                        IJV[1][ind] = e[i][xJ] * 2 + dJ
+                        IJV[2][ind] = local_hess[xI * 2 + dI, xJ * 2 + dJ]
+    return IJV
+# ANCHOR_END: val_grad_hess
+
+# ANCHOR: filter_line_search
+def init_step_size(x, e, p):
+    alpha = 1
+    for i in range(0, len(e)):
+        x21 = x[e[i][1]] - x[e[i][0]]
+        x31 = x[e[i][2]] - x[e[i][0]]
+        p21 = p[e[i][1]] - p[e[i][0]]
+        p31 = p[e[i][2]] - p[e[i][0]]
+        detT = np.linalg.det(np.transpose([x21, x31]))
+        a = np.linalg.det(np.transpose([p21, p31])) / detT
+        b = (np.linalg.det(np.transpose([x21, p31])) + np.linalg.det(np.transpose([p21, x31]))) / detT
+        c = 0.9  # solve for alpha that first brings the new volume to 0.1x the old volume for slackness
+        critical_alpha = utils.smallest_positive_real_root_quad(a, b, c)
+        if critical_alpha > 0:
+            alpha = min(alpha, critical_alpha)
+    return alpha
+# ANCHOR_END: filter_line_search

9_reduced_DOF/NeoHookeanEnergy.py

@@ -10,32 +10,38 @@
 
 # simulation setup
 side_len = 1
-rho = 1000      # density of square
-k = 2e5         # spring stiffness
-n_seg = 4       # num of segments per side of the square
-h = 0.005        # time step size in s
+rho = 2000      # density of square
+E = 1e4         # Young's modulus
+nu = 0.4        # Poisson's ratio
+n_seg = 10       # num of segments per side of the square
+h = 0.04        # time step size in s
 DBC = []        # no nodes need to be fixed
 y_ground = -1   # height of the planar ground
 
 # initialize simulation
 [x, e] = square_mesh.generate(side_len, n_seg)  # node positions and edge node indices
 v = np.array([[0.0, 0.0]] * len(x))             # velocity
 m = [rho * side_len * side_len / ((n_seg + 1) * (n_seg + 1))] * len(x)  # calculate node mass evenly
-# rest length squared
-l2 = []
+# ANCHOR: elem_precomp
+# rest shape basis, volume, and lame parameters
+vol = [0.0] * len(e)
+IB = [np.array([[0.0, 0.0]] * 2)] * len(e)
 for i in range(0, len(e)):
-    diff = x[e[i][0]] - x[e[i][1]]
-    l2.append(diff.dot(diff))
-k = [k] * len(e)    # spring stiffness
+    TB = [x[e[i][1]] - x[e[i][0]], x[e[i][2]] - x[e[i][0]]]
+    vol[i] = np.linalg.det(np.transpose(TB)) / 2
+    IB[i] = np.linalg.inv(np.transpose(TB))
+mu_lame = [0.5 * E / (1 + nu)] * len(e)
+lam = [E * nu / ((1 + nu) * (1 - 2 * nu))] * len(e)
+# ANCHOR_END: elem_precomp
 # identify whether a node is Dirichlet
 is_DBC = [False] * len(x)
 for i in DBC:
     is_DBC[i] = True
 # ANCHOR: contact_area
 contact_area = [side_len / n_seg] * len(x)     # perimeter split to each node
 # ANCHOR_END: contact_area
-# compute reduced basis using 0: polynomial functions or 1: modal reduction
-reduced_basis = utils.compute_reduced_basis(x, e, l2, k, is_DBC, method=0, order=3)
+# compute reduced basis using 0: no reduction; 1: polynomial functions; 2: modal reduction
+reduced_basis = utils.compute_reduced_basis(x, e, vol, IB, mu_lame, lam, method=1, order=2)
 
 # simulation with visualization
 resolution = np.array([900, 900])
@@ -45,7 +51,7 @@ def screen_projection(x):
     return [offset[0] + scale * x[0], resolution[1] - (offset[1] + scale * x[1])]
 
 time_step = 0
-square_mesh.write_to_file(time_step, x, n_seg)
+square_mesh.write_to_file(time_step, x, e)
 screen = pygame.display.set_mode(resolution)
 running = True
 while running:
@@ -60,16 +66,20 @@ def screen_projection(x):
     screen.fill((255, 255, 255))
     pygame.draw.aaline(screen, (0, 0, 255), screen_projection([-2, y_ground]), screen_projection([2, y_ground]))   # ground
     for eI in e:
+        # ANCHOR: draw_tri
         pygame.draw.aaline(screen, (0, 0, 255), screen_projection(x[eI[0]]), screen_projection(x[eI[1]]))
+        pygame.draw.aaline(screen, (0, 0, 255), screen_projection(x[eI[1]]), screen_projection(x[eI[2]]))
+        pygame.draw.aaline(screen, (0, 0, 255), screen_projection(x[eI[2]]), screen_projection(x[eI[0]]))
+        # ANCHOR_END: draw_tri
     for xI in x:
         pygame.draw.circle(screen, (0, 0, 255), screen_projection(xI), 0.1 * side_len / n_seg * scale)
 
     pygame.display.flip()   # flip the display
 
     # step forward simulation and wait for screen refresh
-    [x, v] = time_integrator.step_forward(x, e, v, m, l2, k, y_ground, contact_area, is_DBC, reduced_basis, h, 1e-2)
+    [x, v] = time_integrator.step_forward(x, e, v, m, vol, IB, mu_lame, lam, y_ground, contact_area, is_DBC, reduced_basis, h, 1e-2)
     time_step += 1
     pygame.time.wait(int(h * 1000))
-    square_mesh.write_to_file(time_step, x, n_seg)
+    square_mesh.write_to_file(time_step, x, e)
 
 pygame.quit()

9_reduced_DOF/simulator.py

@@ -9,25 +9,23 @@ def generate(side_length, n_seg):
         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
+    # ANCHOR: tri_vert_ind
+    # connect the nodes with triangle elements
     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])
+            # triangulate each cell following a symmetric pattern:
+            if (i % 2)^(j % 2):
+                e.append([i * (n_seg + 1) + j, (i + 1) * (n_seg + 1) + j, i * (n_seg + 1) + j + 1])
+                e.append([(i + 1) * (n_seg + 1) + j, (i + 1) * (n_seg + 1) + j + 1, i * (n_seg + 1) + j + 1])
+            else:
+                e.append([i * (n_seg + 1) + j, (i + 1) * (n_seg + 1) + j, (i + 1) * (n_seg + 1) + j + 1])
+                e.append([i * (n_seg + 1) + j, (i + 1) * (n_seg + 1) + j + 1, i * (n_seg + 1) + j + 1])
+    # ANCHOR_END: tri_vert_ind
 
     return [x, e]
 
-def write_to_file(frameNum, x, n_seg):
+def write_to_file(frameNum, x, e):
     # Check if 'output' directory exists; if not, create it
     if not os.path.exists('output'):
         os.makedirs('output')
@@ -39,8 +37,6 @@ def write_to_file(frameNum, x, n_seg):
         for row in x:
             f.write(f"v {float(row[0]):.6f} {float(row[1]):.6f} 0.0\n") 
         # write vertex indices for each triangle
-        for i in range(0, n_seg):
-            for j in range(0, n_seg):
-                #NOTE: each cell is exported as 2 triangles for rendering
-                f.write(f"f {i * (n_seg+1) + j + 1} {(i+1) * (n_seg+1) + j + 1} {(i+1) * (n_seg+1) + j+1 + 1}\n")
-                f.write(f"f {i * (n_seg+1) + j + 1} {(i+1) * (n_seg+1) + j+1 + 1} {i * (n_seg+1) + j+1 + 1}\n")
+        for row in e:
+            #NOTE: vertex indices start from 1 in obj file format
+            f.write(f"f {row[0] + 1} {row[1] + 1} {row[2] + 1}\n")