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implicit_mass_spring_system.py
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implicit_mass_spring_system.py
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# https://www.cs.cmu.edu/~baraff/papers/sig98.pdf
import argparse
import numpy as np
import taichi as ti
import time
@ti.data_oriented
class Cloth:
def __init__(self, N):
self.N = N
self.NF = 2 * N**2 # number of faces
self.NV = (N + 1)**2 # number of vertices
self.NE = 2 * N * (N + 1) + 2 * N * N # numbser of edges
self.initPos = ti.Vector.field(2, ti.f32, self.NV)
self.pos = ti.Vector.field(2, ti.f32, self.NV)
self.vel = ti.Vector.field(2, ti.f32, self.NV)
self.force = ti.Vector.field(2, ti.f32, self.NV)
self.mass = ti.field(ti.f32, self.NV)
self.spring = ti.Vector.field(2, ti.i32, self.NE)
self.rest_len = ti.field(ti.f32, self.NE)
self.ks = 1000.0 # spring stiffness
self.kf = 1.0e5 # Attachment point stiffness
self.Jx = ti.Matrix.field(2, 2, ti.f32, self.NE) # Force Jacobian
self.Jf = ti.Matrix.field(2, 2, ti.f32, 2) # Attachment Jacobian
self.init_pos()
self.init_edges()
# For sparse matrix solver， PPT: P45
max_num_triplets = 10000
self.MBuilder = ti.linalg.SparseMatrixBuilder(2 * self.NV, 2 * self.NV,
max_num_triplets)
self.init_mass_sp(self.MBuilder)
self.M = self.MBuilder.build()
self.KBuilder = ti.linalg.SparseMatrixBuilder(2 * self.NV, 2 * self.NV,
max_num_triplets)
# For conjugate gradient method, PPT: P106
self.x = ti.Vector.field(2, ti.f32, self.NV)
self.Ax = ti.Vector.field(2, ti.f32, self.NV)
self.b = ti.Vector.field(2, ti.f32, self.NV)
self.r = ti.Vector.field(2, ti.f32, self.NV)
self.d = ti.Vector.field(2, ti.f32, self.NV)
self.Ad = ti.Vector.field(2, ti.f32, self.NV)
@ti.kernel
def init_pos(self):
for i, j in ti.ndrange(self.N + 1, self.N + 1):
k = i * (self.N + 1) + j
self.initPos[k] = ti.Vector([i, j]) / self.N * 0.5 + ti.Vector(
[0.25, 0.25])
self.pos[k] = self.initPos[k]
self.vel[k] = ti.Vector([0, 0])
self.mass[k] = 1.0
@ti.kernel
def init_edges(self):
pos, spring, N, rest_len = ti.static(self.pos, self.spring, self.N,
self.rest_len)
for i, j in ti.ndrange(N + 1, N):
idx, idx1 = i * N + j, i * (N + 1) + j
spring[idx] = ti.Vector([idx1, idx1 + 1])
start = N * (N + 1)
for i, j in ti.ndrange(N, N + 1):
idx, idx1, idx2 = start + i + j * N, i * (N + 1) + j, i * (
N + 1) + j + N + 1
spring[idx] = ti.Vector([idx1, idx2])
start = 2 * N * (N + 1)
for i, j in ti.ndrange(N, N):
idx, idx1, idx2 = start + i * N + j, i * (N + 1) + j, (i + 1) * (
N + 1) + j + 1
spring[idx] = ti.Vector([idx1, idx2])
start = 2 * N * (N + 1) + N * N
for i, j in ti.ndrange(N, N):
idx, idx1, idx2 = start + i * N + j, i * (N + 1) + j + 1, (
i + 1) * (N + 1) + j
spring[idx] = ti.Vector([idx1, idx2])
for i in range(self.NE):
idx1, idx2 = spring[i]
rest_len[i] = (pos[idx1] - pos[idx2]).norm()
@ti.kernel
def init_mass_sp(self, M: ti.types.sparse_matrix_builder()):
for i in range(self.NV):
M[2 * i + 0, 2 * i + 0] += self.mass[i]
M[2 * i + 1, 2 * i + 1] += self.mass[i]
@ti.func
def clear_force(self):
for i in self.force:
self.force[i] = ti.Vector([0.0, 0.0])
@ti.kernel
def compute_force(self):
self.clear_force()
gravity = ti.Vector([0.0, -2.0])
for i in self.force:
self.force[i] += gravity * self.mass[i]
for i in self.spring:
idx1, idx2 = self.spring[i][0], self.spring[i][1]
pos1, pos2 = self.pos[idx1], self.pos[idx2]
dis = pos1 - pos2
# Hook's law
force = self.ks * (dis.norm() -
self.rest_len[i]) * dis.normalized()
self.force[idx1] -= force
self.force[idx2] += force
# Attachment constraint force
self.force[self.N] += self.kf * (self.initPos[self.N] -
self.pos[self.N])
self.force[self.NV - 1] += self.kf * (self.initPos[self.NV - 1] -
self.pos[self.NV - 1])
@ti.kernel
def compute_force_Jacobians(self):
for i in self.spring:
idx1, idx2 = self.spring[i][0], self.spring[i][1]
pos1, pos2 = self.pos[idx1], self.pos[idx2]
dx = pos1 - pos2
I = ti.Matrix([[1.0, 0.0], [0.0, 1.0]])
dxtdx = ti.Matrix([[dx[0] * dx[0], dx[0] * dx[1]],
[dx[1] * dx[0], dx[1] * dx[1]]])
l = dx.norm()
if l != 0.0:
l = 1.0 / l
self.Jx[i] = (I - self.rest_len[i] * l *
(I - dxtdx * l**2)) * self.ks
# Attachment constraint force Jacobian
self.Jf[0] = ti.Matrix([[-self.kf, 0], [0, -self.kf]])
self.Jf[1] = ti.Matrix([[-self.kf, 0], [0, -self.kf]])
@ti.kernel
def assemble_K(self, K: ti.types.sparse_matrix_builder()):
for i in self.spring:
idx1, idx2 = self.spring[i][0], self.spring[i][1]
for m, n in ti.static(ti.ndrange(2, 2)):
K[2 * idx1 + m, 2 * idx1 + n] -= self.Jx[i][m, n]
K[2 * idx1 + m, 2 * idx2 + n] += self.Jx[i][m, n]
K[2 * idx2 + m, 2 * idx1 + n] += self.Jx[i][m, n]
K[2 * idx2 + m, 2 * idx2 + n] -= self.Jx[i][m, n]
for m, n in ti.static(ti.ndrange(2, 2)):
K[2 * self.N + m, 2 * self.N + n] += self.Jf[0][m, n]
K[2 * (self.NV - 1) + m, 2 * (self.NV - 1) + n] += self.Jf[1][m, n]
@ti.kernel
def directUpdatePosVel(self, h: ti.f32, v_next: ti.ext_arr()):
for i in self.pos:
self.vel[i] = ti.Vector([v_next[2 * i], v_next[2 * i + 1]])
self.pos[i] += h * self.vel[i]
def update_direct(self, h):
self.compute_force()
self.compute_force_Jacobians()
# Assemble global system
self.assemble_K(self.KBuilder)
K = self.KBuilder.build()
A = self.M - h**2 * K
solver = ti.linalg.SparseSolver(solver_type="LLT")
solver.analyze_pattern(A)
solver.factorize(A)
vel = self.vel.to_numpy().reshape(2 * self.NV)
force = self.force.to_numpy().reshape(2 * self.NV)
b = h * force + self.M @ vel
v_next = solver.solve(b)
# flag = solver.info()
# print("solver flag: ", flag)
self.directUpdatePosVel(h, v_next)
@ti.kernel
def cgUpdatePosVel(self, h: ti.f32):
for i in self.pos:
self.vel[i] = self.x[i]
self.pos[i] += h * self.vel[i]
@ti.kernel
def compute_RHS(self, h: ti.f32):
#rhs = b = h * force + M @ v
for i in range(self.NV):
self.b[i] = h * self.force[i] + self.mass[i] * self.vel[i]
@ti.func
def dot(self, v1, v2):
result = 0.0
for i in range(self.NV):
result += v1[i][0] * v2[i][0]
result += v1[i][1] * v2[i][1]
return result
@ti.func
def A_mult_x(self, h, dst, src):
coeff = -h**2
for i in range(self.NV):
dst[i] = self.mass[i] * src[i]
for i in range(self.NE):
idx1, idx2 = self.spring[i][0], self.spring[i][1]
temp = self.Jx[i] @ (src[idx1] - src[idx2])
dst[idx1] -= coeff * temp
dst[idx2] += coeff * temp
# Attachment constraint
Attachment1, Attachment2 = self.N, self.NV - 1
dst[Attachment1] -= coeff * self.kf * src[Attachment1]
dst[Attachment2] -= coeff * self.kf * src[Attachment2]
# conjugate gradient solving
# https://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf
@ti.kernel
def before_ite(self) -> ti.f32:
for i in range(self.NV):
self.x[i] = ti.Vector([0.0, 0.0])
self.A_mult_x(h, self.Ax, self.x) # Ax = A @ x
for i in range(self.NV): # r = b - A @ x
self.r[i] = self.b[i] - self.Ax[i]
for i in range(self.NV): # d = r
self.d[i] = self.r[i]
delta_new = self.dot(self.r, self.r)
return delta_new
@ti.kernel
def run_iteration(self, delta_new: ti.f32) -> ti.f32:
self.A_mult_x(h, self.Ad, self.d) # Ad = A @ d
alpha = delta_new / self.dot(self.d,
self.Ad) # alpha = (r^T * r) / dot(d, Ad)
for i in range(self.NV):
self.x[i] += alpha * self.d[i] # x^{i+1} = x^{i} + alpha * d
self.r[i] -= alpha * self.Ad[i] # r^{i+1} = r^{i} + alpha * Ad
delta_old = delta_new
delta_new = self.dot(self.r, self.r)
beta = delta_new / delta_old
for i in range(self.NV):
self.d[i] = self.r[i] + beta * self.d[
i] #p^{i+1} = r^{i+1} + beta * p^{i}
return delta_new
def cg(self, h: ti.f32):
delta_new = self.before_ite()
ite, iteMax = 0, 2 * self.NV
while ite < iteMax and delta_new > 1.0e-6:
delta_new = self.run_iteration(delta_new)
ite += 1
def update_cg(self, h):
self.compute_force()
self.compute_force_Jacobians()
self.compute_RHS(h)
self.cg(h)
self.cgUpdatePosVel(h)
def display(self, gui, radius=5, color=0xffffff):
springs, pos = self.spring.to_numpy(), self.pos.to_numpy()
line_Begin = np.zeros(shape=(springs.shape[0], 2))
line_End = np.zeros(shape=(springs.shape[0], 2))
for i in range(springs.shape[0]):
idx1, idx2 = springs[i][0], springs[i][1]
line_Begin[i], line_End[i] = pos[idx1], pos[idx2]
gui.lines(line_Begin, line_End, radius=2, color=0x0000ff)
gui.circles(self.pos.to_numpy(), radius, color)
if __name__ == "__main__":
ti.init(arch=ti.cpu)
cloth = Cloth(N=5)
parser = argparse.ArgumentParser()
parser.add_argument('-cg',
'--use_cg',
action='store_true',
help='Solve Ax=b with conjugate gradient method (CG).')
args, unknowns = parser.parse_known_args()
use_cg = args.use_cg
gui = ti.GUI('Implicit Mass Spring System', res=(500, 500))
pause = False
h, max_step = 0.01, 3
while gui.running:
for e in gui.get_events():
if e.key == gui.ESCAPE:
gui.running = False
elif e.key == gui.SPACE:
pause = not pause
if not pause:
for i in range(max_step):
if use_cg:
cloth.update_cg(h)
else:
cloth.update_direct(h)
cloth.display(gui)
gui.show()