In [1]:
from math import sqrt
import matplotlib.pyplot as plt
import torch
from torchfem import Truss
from torchfem.materials import IsotropicElasticity1D
plt.rcParams["text.usetex"] = True
torch.set_default_dtype(torch.double)
In [2]:
p = 3
Cxx = 1.0
# Plot the SIMP relation
rho = torch.linspace(0, 1, 20)
plt.figure(figsize=(5, 2.5))
plt.plot(rho, Cxx * rho**1.0, label="$p=1$")
plt.plot(rho, Cxx * rho**2.0, label="$p=2$")
plt.plot(rho, Cxx * rho**3.0, label="$p=3$")
plt.xlabel(r"$\textrm{Normalized area } \rho_j$")
plt.ylabel(r"$\textrm{Relative stiffness } E(\rho_j) / E$")
plt.xlim([0, 1])
plt.ylim([0, Cxx])
plt.legend()
plt.grid()
plt.savefig("../figures/simp_truss.svg", transparent=True, bbox_inches="tight")
Sample truss¶
In [3]:
n1 = torch.linspace(0.0, 4.0, 5)
n2 = torch.linspace(0.0, 1.0, 2)
n1, n2 = torch.stack(torch.meshgrid(n1, n2, indexing="xy"))
nodes = torch.stack([n1.ravel(), n2.ravel()], dim=1)
elements = torch.tensor(
[
[0, 1],
[1, 2],
[2, 3],
[3, 4],
[5, 6],
[6, 7],
[7, 8],
[8, 9],
[1, 5],
[0, 6],
[2, 6],
[1, 7],
[3, 7],
[2, 8],
[4, 8],
[3, 9],
[1, 6],
[2, 7],
[3, 8],
[4, 9],
]
)
material = IsotropicElasticity1D(1.0)
truss_sample = Truss(nodes, elements, material)
truss_sample.forces[4, 1] = -0.1
truss_sample.constraints[0, 0] = True
truss_sample.constraints[0, 1] = True
truss_sample.constraints[5, 0] = True
truss_sample.areas[:] = 10.0
The optimization¶
In [4]:
def newton(grad, mu_init, max_iter=10, tol=0.0001):
# Newton's method to find root of grad (few iterations, if succesfull)
mu = torch.tensor([mu_init], requires_grad=True)
i = 0
while torch.abs(grad(mu)) > tol:
if i > max_iter:
raise Exception(f"Newton solver did not converge in {max_iter} iterations.")
gradgrad = torch.autograd.grad(grad(mu).sum(), mu)[0]
mu.data -= grad(mu) / gradgrad
mu.data = torch.max(mu.data, torch.tensor([0.0]))
i += 1
return mu.data
def bisection(grad, a, b, max_iter=50, tol=1e-10):
# Bisection method always finds a root, even with highly non-linear grad
i = 0
while (b - a) > tol:
c = (a + b) / 2.0
if i > max_iter:
raise Exception(f"Bisection did not converge in {max_iter} iterations.")
if grad(a) * grad(c) > 0:
a = c
else:
b = c
i += 1
return c
def compute_lengths(truss):
start_nodes = truss.nodes[truss.elements[:, 0]]
end_nodes = truss.nodes[truss.elements[:, 1]]
dx = end_nodes - start_nodes
return torch.linalg.norm(dx, dim=-1)
def optimize(truss, a_0, a_min, a_max, V_0, iter=10, s=0.7, p=1.0):
k0 = truss.k0() / truss.areas[:, None, None]
a = [a_0]
L = []
l = compute_lengths(truss)
# Check if there is a feasible solution before starting iteration
if torch.inner(a_min, l) > V_0:
raise Exception("x_min is not compatible with V_0.")
# Iterate solutions
for k in range(iter):
# Solve the truss problem at point a_k
truss.areas = a[k] * (a[k] / a_max) ** p
u_k, f_k, _, _, _ = truss.solve()
# Get strain energy of all truss elements for the given displacement
disp = u_k[truss.elements].reshape(-1, 4)
w_k = 0.5 * torch.einsum("...i,...ij,...j", disp, k0, disp)
sensitivity = -p * (a[k] / a_max) ** (p - 1) * 2.0 * w_k
# Compute lower asymptote
if k <= 1:
L.append(a[k] - s * (a_max - a_min))
else:
L_k = torch.zeros_like(L[k - 1])
osci = (a[k] - a[k - 1]) * (a[k - 1] - a[k - 2]) < 0.0
L_k[osci] = a[k][osci] - s * (a[k - 1][osci] - L[k - 1][osci])
L_k[~osci] = a[k][~osci] - 1 / sqrt(s) * (a[k - 1][~osci] - L[k - 1][~osci])
L.append(L_k)
# Compute lower move limit in this step
a_min_k = torch.max(a_min, 0.9 * L[k] + 0.1 * a[k])
# Analytical solution
def x_star(mu):
a_hat = L[k] + torch.sqrt((-sensitivity * (L[k] - a[k]) ** 2) / (mu * l))
return torch.clamp(a_hat, a_min_k, a_max)
# Analytical gradient
def grad(mu):
return torch.dot(x_star(mu), l) - V_0
# Solve dual problem
# mu_star = newton(grad, 1.0)
mu_star = bisection(grad, 1e-10, 100.0)
# Evaluation
compliance = torch.inner(f_k.ravel(), u_k.ravel())
print(f"Iteration k={k} - Compliance: {compliance:.5f}")
# Compute current optimal point with dual solution
a.append(x_star(mu_star))
return a
Topology optimization of the sample truss¶
In [5]:
a_0 = 5.0 * torch.ones((len(truss_sample.elements)))
a_min = 1.0 * torch.ones_like(a_0)
a_max = 20.0 * torch.ones_like(a_0)
# Compute volume restriction
l = compute_lengths(truss_sample)
V0 = 0.565 * torch.inner(a_max, l)
a_opt = optimize(truss_sample, a_0, a_min, a_max, V0, iter=10, p=2)
u, f, sigma, _, _ = truss_sample.solve()
truss_sample.plot(
u=u, element_property=sigma, thickness_threshold=2.0, node_labels=False
)
plt.savefig(
"../figures/truss_sample_topology_optimized.svg",
transparent=True,
bbox_inches="tight",
)
plt.show()
plt.plot(range(len(a_opt)), torch.stack(a_opt))
plt.xlabel("Iteration")
plt.ylabel("Design variables")
plt.xlim([0, len(a_opt)])
plt.grid()
plt.show()
Iteration k=0 - Compliance: 1.60058 Iteration k=1 - Compliance: 0.04992 Iteration k=2 - Compliance: 0.04823 Iteration k=3 - Compliance: 0.03525 Iteration k=4 - Compliance: 0.02889 Iteration k=5 - Compliance: 0.02768 Iteration k=6 - Compliance: 0.02768 Iteration k=7 - Compliance: 0.02768 Iteration k=8 - Compliance: 0.02768 Iteration k=9 - Compliance: 0.02768
Bridge¶
In [6]:
# Dimensions
A = 17
B = 2
# Nodes
n1 = torch.linspace(0.0, 5.0, A)
n2 = torch.linspace(0.0, 0.5, B)
n1, n2 = torch.stack(torch.meshgrid(n1, n2, indexing="xy"))
nodes = torch.stack([n1.ravel(), n2.ravel()], dim=1)
# Elements
elements = []
for i in range(A - 1):
for j in range(B):
elements.append([i + j * A, i + 1 + j * A])
for i in range(A):
for j in range(B - 1):
elements.append([i + j * A, i + A + j * A])
for i in range(A - 1):
for j in range(B - 1):
elements.append([i + j * A, i + 1 + A + j * A])
elements.append([i + 1 + j * A, i + A + j * A])
elements = torch.tensor(elements)
# Material
material = IsotropicElasticity1D(500.0)
# Truss
bridge = Truss(nodes.clone(), elements, material)
# Forces at bottom edge
bridge.forces[1 : A - 1, 1] = -0.1
# Constraints by the supports
bridge.constraints[0, 0] = True
bridge.constraints[0, 1] = True
bridge.constraints[A - 1, 1] = True
# Plot
bridge.plot(node_labels=False)
In [7]:
a_0 = 10.0 * torch.ones((len(elements)))
a_min = 1.0 * torch.ones_like(a_0)
a_max = 20.0 * torch.ones_like(a_0)
l = compute_lengths(bridge)
V0 = 0.43 * torch.inner(a_max, l)
a_opt = optimize(bridge, a_0, a_min, a_max, V0, iter=30, p=2)
bridge.plot(thickness_threshold=2.0, node_labels=False)
plt.savefig(
"../figures/bridge_topology_optimized.svg", transparent=True, bbox_inches="tight"
)
plt.show()
Iteration k=0 - Compliance: 0.01812 Iteration k=1 - Compliance: 0.02468 Iteration k=2 - Compliance: 0.41196 Iteration k=3 - Compliance: 3.54525 Iteration k=4 - Compliance: 0.48119 Iteration k=5 - Compliance: 0.05083 Iteration k=6 - Compliance: 0.04977 Iteration k=7 - Compliance: 0.07433 Iteration k=8 - Compliance: 0.01339 Iteration k=9 - Compliance: 0.02662 Iteration k=10 - Compliance: 0.04031 Iteration k=11 - Compliance: 0.02164 Iteration k=12 - Compliance: 0.02333 Iteration k=13 - Compliance: 0.01066 Iteration k=14 - Compliance: 0.00909 Iteration k=15 - Compliance: 0.00758 Iteration k=16 - Compliance: 0.00875 Iteration k=17 - Compliance: 0.00767 Iteration k=18 - Compliance: 0.00843 Iteration k=19 - Compliance: 0.00745 Iteration k=20 - Compliance: 0.00688 Iteration k=21 - Compliance: 0.00620 Iteration k=22 - Compliance: 0.00534 Iteration k=23 - Compliance: 0.00493 Iteration k=24 - Compliance: 0.00471 Iteration k=25 - Compliance: 0.00459 Iteration k=26 - Compliance: 0.00446 Iteration k=27 - Compliance: 0.00436 Iteration k=28 - Compliance: 0.00434 Iteration k=29 - Compliance: 0.00431
