In [1]:
from math import sqrt
import matplotlib.pyplot as plt
import torch
from torchfem import Planar
from torchfem.elements import Quad1
from torchfem.materials import IsotropicElasticityPlaneStress
# Set plot to use TeX for rendering equations.
plt.rcParams["text.usetex"] = True
plt.rcParams["font.size"] = 14
torch.set_default_dtype(torch.float64)
In [2]:
def get_cantilever(size, Lx, Ly, d=1.0, E=100, nu=0.3, etype=Quad1()):
# Material
material = IsotropicElasticityPlaneStress(E, nu)
# Dimensions
Nx = int(Lx / size)
Ny = int(Ly / size)
# Create nodes
n1 = torch.linspace(0.0, Lx, Nx + 1)
n2 = torch.linspace(0.0, Ly, Ny + 1)
n1, n2 = torch.stack(torch.meshgrid(n1, n2, indexing="xy"))
nodes = torch.stack([n1.ravel(), n2.ravel()], dim=1)
# Create elements connecting nodes
elements = []
for j in range(Ny):
for i in range(Nx):
if isinstance(etype, Quad1):
# Quad elements
n0 = i + j * (Nx + 1)
elements.append([n0, n0 + 1, n0 + Nx + 2, n0 + Nx + 1])
else:
# Tria elements
n0 = i + j * (Nx + 1)
elements.append([n0, n0 + 1, n0 + Nx + 2])
elements.append([n0 + Nx + 2, n0 + Nx + 1, n0])
elements = torch.tensor(elements)
cantilever = Planar(nodes, elements, material)
# Load at tip
cantilever.forces[(int((Ny + 1) / 2) + 1) * (Nx + 1) - 1, 1] = -1.0
# Constrained displacement at left end
for i in range(Ny + 1):
cantilever.constraints[i * (Nx + 1), :] = True
# Default
cantilever.thickness[:] = d
return cantilever
Define the cantilever problem¶
In [3]:
cantilever = get_cantilever(0.75, 28.0, 14.0)
cantilever.plot()
Optimization with MMA¶
In [ ]:
def bisection(f, a, b, max_iter=50, tol=1e-10):
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 f(a) * f(c) > 0:
a = c
else:
b = c
i += 1
return c
def compute_areas(truss):
areas = torch.zeros((truss.n_elem))
nodes = truss.nodes[truss.elements, :]
for w, q in zip(truss.etype.iweights, truss.etype.ipoints):
J = truss.etype.B(q) @ nodes
detJ = torch.linalg.det(J)
areas[:] += w * detJ
return areas
def optimize_mma(fem, rho_0, rho_min, rho_max, V_0, iter=15, s=0.9, p=1.0, r_sens=0.0):
k0 = torch.einsum("i,ijk->ijk", 1.0 / fem.thickness, fem.k0())
rho = [rho_0]
L = []
areas = compute_areas(fem)
# Precompute filter weights
if r_sens > 0.0:
ecenters = torch.stack([torch.mean(fem.nodes[e], dim=0) for e in fem.elements])
dist = torch.cdist(ecenters, ecenters)
H = r_sens - dist
H[dist > r_sens] = 0.0
# Check if there is a feasible solution before starting iteration
if torch.inner(rho_min, areas) > V_0:
raise Exception("rho_min is not compatible with V_0.")
# Iterate solutions
for k in range(iter):
# Solve the problem at rho_k
fem.thickness = rho[k] ** p
u_k, f_k, _, _, _ = fem.solve()
# Compute sensitivity
disp = u_k[fem.elements, :].reshape(fem.n_elem, -1)
w_k = 0.5 * torch.einsum("...i,...ij,...j", disp, k0, disp)
sensitivity = -p * rho[k] ** (p - 1) * 2.0 * w_k
# Filter sensitivities (if r_sens provided)
if r_sens > 0.0:
# Note: We assume that each cell has the same area here.
sensitivity = H @ (rho[k] * sensitivity) / H.sum(dim=0) / rho[k]
# Compute lower asymptote
if k <= 1:
L.append(rho[k] - s * (rho_max - rho_min))
else:
L_k = torch.zeros_like(L[k - 1])
osci = (rho[k] - rho[k - 1]) * (rho[k - 1] - rho[k - 2]) < 0.0
L_k[osci] = rho[k][osci] - s * (rho[k - 1][osci] - L[k - 1][osci])
L_k[~osci] = rho[k][~osci] - 1 / sqrt(s) * (
rho[k - 1][~osci] - L[k - 1][~osci]
)
L.append(L_k)
# Compute lower move limit in this step
rho_min_k = torch.max(rho_min, 0.9 * L[k] + 0.1 * rho[k])
# Analytical solution
def rho_star(mu):
rho_hat = L[k] + torch.sqrt(
(-sensitivity * (L[k] - rho[k]) ** 2) / (mu * areas)
)
return torch.clamp(rho_hat, rho_min_k, rho_max)
# Analytical gradient
def grad(mu):
return torch.dot(rho_star(mu), areas) - V_0
# Solve dual problem
mu_star = bisection(grad, 0.0, 100.0)
# Evaluation
compliance = torch.inner(f_k.ravel(), u_k.ravel())
print(f"Iteration k={k:3d} - Compliance: {compliance:.5f}")
# Compute current optimal point with dual solution
rho.append(rho_star(mu_star))
return rho
Variable thickness optimization (p=1)¶
In [5]:
rho_0 = 0.5 * torch.ones(len(cantilever.elements))
rho_min = 0.1 * torch.ones_like(rho_0)
rho_max = 1.0 * torch.ones_like(rho_0)
areas = compute_areas(cantilever)
V0 = 0.5 * torch.inner(rho_max, areas)
rho_opt = optimize_mma(cantilever, rho_0, rho_min, rho_max, V0, iter=25)
cantilever.plot(element_property=rho_opt[-1], cmap="gray_r")
plt.savefig(
"../figures/cantilever_fem_optimized.svg", transparent=True, bbox_inches="tight"
)
Iteration k= 0 - Compliance: 0.78348 Iteration k= 1 - Compliance: 0.58562 Iteration k= 2 - Compliance: 0.58086 Iteration k= 3 - Compliance: 0.56605 Iteration k= 4 - Compliance: 0.56055 Iteration k= 5 - Compliance: 0.55596 Iteration k= 6 - Compliance: 0.55339 Iteration k= 7 - Compliance: 0.55251 Iteration k= 8 - Compliance: 0.55184 Iteration k= 9 - Compliance: 0.55158 Iteration k= 10 - Compliance: 0.55134 Iteration k= 11 - Compliance: 0.55126 Iteration k= 12 - Compliance: 0.55118 Iteration k= 13 - Compliance: 0.55113 Iteration k= 14 - Compliance: 0.55110 Iteration k= 15 - Compliance: 0.55107 Iteration k= 16 - Compliance: 0.55105 Iteration k= 17 - Compliance: 0.55104 Iteration k= 18 - Compliance: 0.55103 Iteration k= 19 - Compliance: 0.55102 Iteration k= 20 - Compliance: 0.55102 Iteration k= 21 - Compliance: 0.55102 Iteration k= 22 - Compliance: 0.55101 Iteration k= 23 - Compliance: 0.55101 Iteration k= 24 - Compliance: 0.55101
SIMP approach for binarization¶
In [6]:
p = 3
Cxx = 1.0
# Plot the SIMP relation
rho = torch.linspace(0, 1, 20)
plt.figure(figsize=(5, 3))
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"$\textrm{Normalized density } \rho$")
plt.ylabel(r"$\textrm{Relative stiffness } C_{xx}(\rho)/C_{xx}$")
plt.xlim([0, 1])
plt.ylim([0, Cxx])
plt.legend()
plt.grid()
plt.savefig("../figures/simp.svg", transparent=True, bbox_inches="tight")
In [7]:
rho_opt = optimize_mma(cantilever, rho_0, rho_min, rho_max, V0, p=3, iter=50)
cantilever.plot(element_property=rho_opt[-1], cmap="gray_r")
plt.savefig(
"../figures/cantilever_fem_optimized_binary.svg",
transparent=True,
bbox_inches="tight",
)
Iteration k= 0 - Compliance: 3.13393 Iteration k= 1 - Compliance: 3.51266 Iteration k= 2 - Compliance: 5.28577 Iteration k= 3 - Compliance: 2.73529 Iteration k= 4 - Compliance: 1.46228 Iteration k= 5 - Compliance: 1.06049 Iteration k= 6 - Compliance: 0.88839 Iteration k= 7 - Compliance: 0.83167 Iteration k= 8 - Compliance: 0.81168 Iteration k= 9 - Compliance: 0.80092 Iteration k= 10 - Compliance: 0.79512 Iteration k= 11 - Compliance: 0.79241 Iteration k= 12 - Compliance: 0.79032 Iteration k= 13 - Compliance: 0.78968 Iteration k= 14 - Compliance: 0.78866 Iteration k= 15 - Compliance: 0.78766 Iteration k= 16 - Compliance: 0.78605 Iteration k= 17 - Compliance: 0.78527 Iteration k= 18 - Compliance: 0.78517 Iteration k= 19 - Compliance: 0.78512 Iteration k= 20 - Compliance: 0.78508 Iteration k= 21 - Compliance: 0.78499 Iteration k= 22 - Compliance: 0.78473 Iteration k= 23 - Compliance: 0.78424 Iteration k= 24 - Compliance: 0.78423 Iteration k= 25 - Compliance: 0.78422 Iteration k= 26 - Compliance: 0.78421 Iteration k= 27 - Compliance: 0.78421 Iteration k= 28 - Compliance: 0.78420 Iteration k= 29 - Compliance: 0.78420 Iteration k= 30 - Compliance: 0.78419 Iteration k= 31 - Compliance: 0.78417 Iteration k= 32 - Compliance: 0.78410 Iteration k= 33 - Compliance: 0.78375 Iteration k= 34 - Compliance: 0.78295 Iteration k= 35 - Compliance: 0.78295 Iteration k= 36 - Compliance: 0.78295 Iteration k= 37 - Compliance: 0.78295 Iteration k= 38 - Compliance: 0.78295 Iteration k= 39 - Compliance: 0.78295 Iteration k= 40 - Compliance: 0.78295 Iteration k= 41 - Compliance: 0.78295 Iteration k= 42 - Compliance: 0.78295 Iteration k= 43 - Compliance: 0.78295 Iteration k= 44 - Compliance: 0.78295 Iteration k= 45 - Compliance: 0.78295 Iteration k= 46 - Compliance: 0.78295 Iteration k= 47 - Compliance: 0.78295 Iteration k= 48 - Compliance: 0.78295 Iteration k= 49 - Compliance: 0.78295
In [8]:
d_opt = optimize_mma(cantilever, rho_0, rho_min, rho_max, V0, p=3, iter=50, r_sens=1.0)
cantilever.plot(element_property=d_opt[-1], cmap="gray_r")
plt.savefig(
"../figures/cantilever_fem_optimized_binary_filtered.svg",
transparent=True,
bbox_inches="tight",
)
Iteration k= 0 - Compliance: 3.13393 Iteration k= 1 - Compliance: 3.63990 Iteration k= 2 - Compliance: 5.51123 Iteration k= 3 - Compliance: 3.02951 Iteration k= 4 - Compliance: 1.61524 Iteration k= 5 - Compliance: 1.14307 Iteration k= 6 - Compliance: 0.97882 Iteration k= 7 - Compliance: 0.85315 Iteration k= 8 - Compliance: 0.79233 Iteration k= 9 - Compliance: 0.77423 Iteration k= 10 - Compliance: 0.76343 Iteration k= 11 - Compliance: 0.75818 Iteration k= 12 - Compliance: 0.75506 Iteration k= 13 - Compliance: 0.75250 Iteration k= 14 - Compliance: 0.75184 Iteration k= 15 - Compliance: 0.75030 Iteration k= 16 - Compliance: 0.75043 Iteration k= 17 - Compliance: 0.74935 Iteration k= 18 - Compliance: 0.74931 Iteration k= 19 - Compliance: 0.74875 Iteration k= 20 - Compliance: 0.74887 Iteration k= 21 - Compliance: 0.74829 Iteration k= 22 - Compliance: 0.74844 Iteration k= 23 - Compliance: 0.74796 Iteration k= 24 - Compliance: 0.74792 Iteration k= 25 - Compliance: 0.74781 Iteration k= 26 - Compliance: 0.74759 Iteration k= 27 - Compliance: 0.74731 Iteration k= 28 - Compliance: 0.74723 Iteration k= 29 - Compliance: 0.74705 Iteration k= 30 - Compliance: 0.74709 Iteration k= 31 - Compliance: 0.74692 Iteration k= 32 - Compliance: 0.74750 Iteration k= 33 - Compliance: 0.74679 Iteration k= 34 - Compliance: 0.74795 Iteration k= 35 - Compliance: 0.74708 Iteration k= 36 - Compliance: 0.74771 Iteration k= 37 - Compliance: 0.74799 Iteration k= 38 - Compliance: 0.74796 Iteration k= 39 - Compliance: 0.74838 Iteration k= 40 - Compliance: 0.74867 Iteration k= 41 - Compliance: 0.74920 Iteration k= 42 - Compliance: 0.75140 Iteration k= 43 - Compliance: 0.75105 Iteration k= 44 - Compliance: 0.75460 Iteration k= 45 - Compliance: 0.75011 Iteration k= 46 - Compliance: 0.75283 Iteration k= 47 - Compliance: 0.74907 Iteration k= 48 - Compliance: 0.75040 Iteration k= 49 - Compliance: 0.74783
In [9]:
cantilever_fine = get_cantilever(0.5, 28.0, 14.0)
rho_0_f = 0.5 * torch.ones(len(cantilever_fine.elements))
rho_min_f = 0.1 * torch.ones_like(rho_0_f)
rho_max_f = 1.0 * torch.ones_like(rho_0_f)
areas_f = compute_areas(cantilever_fine)
V0 = 0.5 * torch.inner(rho_max_f, areas_f)
rho_opt = optimize_mma(cantilever_fine, rho_0_f, rho_min_f, rho_max_f, V0, p=3, iter=50)
cantilever_fine.plot(element_property=rho_opt[-1], cmap="gray_r")
plt.savefig(
"../figures/cantilever_fem_optimized_binary_fine.svg",
transparent=True,
bbox_inches="tight",
)
Iteration k= 0 - Compliance: 3.15949 Iteration k= 1 - Compliance: 3.48000 Iteration k= 2 - Compliance: 5.39400 Iteration k= 3 - Compliance: 3.02577 Iteration k= 4 - Compliance: 1.54312 Iteration k= 5 - Compliance: 1.14941 Iteration k= 6 - Compliance: 0.90429 Iteration k= 7 - Compliance: 0.82058 Iteration k= 8 - Compliance: 0.79029 Iteration k= 9 - Compliance: 0.77547 Iteration k= 10 - Compliance: 0.76657 Iteration k= 11 - Compliance: 0.76302 Iteration k= 12 - Compliance: 0.76191 Iteration k= 13 - Compliance: 0.76099 Iteration k= 14 - Compliance: 0.76025 Iteration k= 15 - Compliance: 0.75940 Iteration k= 16 - Compliance: 0.75908 Iteration k= 17 - Compliance: 0.75874 Iteration k= 18 - Compliance: 0.75858 Iteration k= 19 - Compliance: 0.75848 Iteration k= 20 - Compliance: 0.75839 Iteration k= 21 - Compliance: 0.75812 Iteration k= 22 - Compliance: 0.75785 Iteration k= 23 - Compliance: 0.75747 Iteration k= 24 - Compliance: 0.75744 Iteration k= 25 - Compliance: 0.75744 Iteration k= 26 - Compliance: 0.75744 Iteration k= 27 - Compliance: 0.75744 Iteration k= 28 - Compliance: 0.75744 Iteration k= 29 - Compliance: 0.75744 Iteration k= 30 - Compliance: 0.75744 Iteration k= 31 - Compliance: 0.75744 Iteration k= 32 - Compliance: 0.75744 Iteration k= 33 - Compliance: 0.75744 Iteration k= 34 - Compliance: 0.75744 Iteration k= 35 - Compliance: 0.75744 Iteration k= 36 - Compliance: 0.75744 Iteration k= 37 - Compliance: 0.75744 Iteration k= 38 - Compliance: 0.75744 Iteration k= 39 - Compliance: 0.75744 Iteration k= 40 - Compliance: 0.75744 Iteration k= 41 - Compliance: 0.75744 Iteration k= 42 - Compliance: 0.75744 Iteration k= 43 - Compliance: 0.75744 Iteration k= 44 - Compliance: 0.75744 Iteration k= 45 - Compliance: 0.75744 Iteration k= 46 - Compliance: 0.75744 Iteration k= 47 - Compliance: 0.75744 Iteration k= 48 - Compliance: 0.75744 Iteration k= 49 - Compliance: 0.75744
In [10]:
rho_opt = optimize_mma(
cantilever_fine, rho_0_f, rho_min_f, rho_max_f, V0, p=3, iter=50, r_sens=1.0
)
cantilever_fine.plot(element_property=rho_opt[-1], cmap="gray_r")
plt.savefig(
"../figures/cantilever_fem_optimized_binary_fine_filtered.svg",
transparent=True,
bbox_inches="tight",
)
Iteration k= 0 - Compliance: 3.15949 Iteration k= 1 - Compliance: 3.58879 Iteration k= 2 - Compliance: 5.59235 Iteration k= 3 - Compliance: 3.19642 Iteration k= 4 - Compliance: 1.57245 Iteration k= 5 - Compliance: 1.17433 Iteration k= 6 - Compliance: 1.01420 Iteration k= 7 - Compliance: 0.86609 Iteration k= 8 - Compliance: 0.81336 Iteration k= 9 - Compliance: 0.78984 Iteration k= 10 - Compliance: 0.76872 Iteration k= 11 - Compliance: 0.76237 Iteration k= 12 - Compliance: 0.75780 Iteration k= 13 - Compliance: 0.75444 Iteration k= 14 - Compliance: 0.75189 Iteration k= 15 - Compliance: 0.75071 Iteration k= 16 - Compliance: 0.74914 Iteration k= 17 - Compliance: 0.74884 Iteration k= 18 - Compliance: 0.74813 Iteration k= 19 - Compliance: 0.74783 Iteration k= 20 - Compliance: 0.74778 Iteration k= 21 - Compliance: 0.74761 Iteration k= 22 - Compliance: 0.74766 Iteration k= 23 - Compliance: 0.74727 Iteration k= 24 - Compliance: 0.74784 Iteration k= 25 - Compliance: 0.74734 Iteration k= 26 - Compliance: 0.74747 Iteration k= 27 - Compliance: 0.74754 Iteration k= 28 - Compliance: 0.74716 Iteration k= 29 - Compliance: 0.74738 Iteration k= 30 - Compliance: 0.74722 Iteration k= 31 - Compliance: 0.74701 Iteration k= 32 - Compliance: 0.74739 Iteration k= 33 - Compliance: 0.74653 Iteration k= 34 - Compliance: 0.74761 Iteration k= 35 - Compliance: 0.74629 Iteration k= 36 - Compliance: 0.74807 Iteration k= 37 - Compliance: 0.74636 Iteration k= 38 - Compliance: 0.74721 Iteration k= 39 - Compliance: 0.74639 Iteration k= 40 - Compliance: 0.74686 Iteration k= 41 - Compliance: 0.74654 Iteration k= 42 - Compliance: 0.74641 Iteration k= 43 - Compliance: 0.74668 Iteration k= 44 - Compliance: 0.74615 Iteration k= 45 - Compliance: 0.74669 Iteration k= 46 - Compliance: 0.74618 Iteration k= 47 - Compliance: 0.74662 Iteration k= 48 - Compliance: 0.74629 Iteration k= 49 - Compliance: 0.74647
In [11]:
cantilever_extra_fine = get_cantilever(0.25, 28.0, 14.0)
rho_0_xf = 0.5 * torch.ones(len(cantilever_extra_fine.elements))
rho_min_xf = 0.1 * torch.ones_like(rho_0_xf)
rho_max_xf = 1.0 * torch.ones_like(rho_0_xf)
areas_xf = compute_areas(cantilever_extra_fine)
V0 = 0.5 * torch.inner(rho_max_xf, areas_xf)
rho_opt = optimize_mma(
cantilever_extra_fine, rho_0_xf, rho_min_xf, rho_max_xf, V0, p=3, iter=20
)
cantilever_extra_fine.plot(element_property=rho_opt[-1], cmap="gray_r")
plt.savefig(
"../figures/cantilever_fem_optimized_binary_extra_fine.svg",
transparent=True,
bbox_inches="tight",
)
Iteration k= 0 - Compliance: 3.19738 Iteration k= 1 - Compliance: 3.46626 Iteration k= 2 - Compliance: 5.41301 Iteration k= 3 - Compliance: 3.15927 Iteration k= 4 - Compliance: 1.66375 Iteration k= 5 - Compliance: 1.20163 Iteration k= 6 - Compliance: 0.95243 Iteration k= 7 - Compliance: 0.84727 Iteration k= 8 - Compliance: 0.80169 Iteration k= 9 - Compliance: 0.77944 Iteration k= 10 - Compliance: 0.76927 Iteration k= 11 - Compliance: 0.76396 Iteration k= 12 - Compliance: 0.76124 Iteration k= 13 - Compliance: 0.75965 Iteration k= 14 - Compliance: 0.75654 Iteration k= 15 - Compliance: 0.75515 Iteration k= 16 - Compliance: 0.75427 Iteration k= 17 - Compliance: 0.75316 Iteration k= 18 - Compliance: 0.75255 Iteration k= 19 - Compliance: 0.75118
In [12]:
rho_opt = optimize_mma(
cantilever_extra_fine,
rho_0_xf,
rho_min_xf,
rho_max_xf,
V0,
p=3,
iter=30,
r_sens=1.0,
)
plt.plot(torch.stack(rho_opt))
plt.show()
cantilever_extra_fine.plot(element_property=rho_opt[-1], cmap="gray_r")
plt.savefig(
"../figures/cantilever_fem_optimized_binary_extra_fine_filtered.svg",
transparent=True,
bbox_inches="tight",
)
Iteration k= 0 - Compliance: 3.19738 Iteration k= 1 - Compliance: 3.58924 Iteration k= 2 - Compliance: 5.65584 Iteration k= 3 - Compliance: 3.28497 Iteration k= 4 - Compliance: 1.67591 Iteration k= 5 - Compliance: 1.22087 Iteration k= 6 - Compliance: 1.05201 Iteration k= 7 - Compliance: 0.90092 Iteration k= 8 - Compliance: 0.83648 Iteration k= 9 - Compliance: 0.80396 Iteration k= 10 - Compliance: 0.78370 Iteration k= 11 - Compliance: 0.77625 Iteration k= 12 - Compliance: 0.77003 Iteration k= 13 - Compliance: 0.76624 Iteration k= 14 - Compliance: 0.76360 Iteration k= 15 - Compliance: 0.76140 Iteration k= 16 - Compliance: 0.76024 Iteration k= 17 - Compliance: 0.75867 Iteration k= 18 - Compliance: 0.75808 Iteration k= 19 - Compliance: 0.75715 Iteration k= 20 - Compliance: 0.75676 Iteration k= 21 - Compliance: 0.75635 Iteration k= 22 - Compliance: 0.75601 Iteration k= 23 - Compliance: 0.75591 Iteration k= 24 - Compliance: 0.75545 Iteration k= 25 - Compliance: 0.75568 Iteration k= 26 - Compliance: 0.75512 Iteration k= 27 - Compliance: 0.75531 Iteration k= 28 - Compliance: 0.75506 Iteration k= 29 - Compliance: 0.75503
Optimization with optimality conditions¶
In [ ]:
def optimize_oc(
fem, rho_0, rho_min, rho_max, V_0, iter=15, xi=0.5, m=0.2, p=1.0, r_sens=0.0
):
k0 = torch.einsum("i,ijk->ijk", 1.0 / fem.thickness, fem.k0())
rho = [rho_0]
areas = compute_areas(fem)
# Check if there is a feasible solution before starting iteration
if torch.inner(rho_min, areas) > V_0:
raise Exception("rho_min is not compatible with V_0.")
# Precompute filter weights
if r_sens > 0.0:
ecenters = torch.stack([torch.mean(fem.nodes[e], dim=0) for e in fem.elements])
dist = torch.cdist(ecenters, ecenters)
H = r_sens - dist
H[dist > r_sens] = 0.0
# Iterate solutions
for k in range(iter):
# Adjust thickness variables
fem.thickness = rho[k] ** p
# Compute solution
u_k, f_k, _, _, _ = fem.solve()
# Copy original rho
rho_k = rho[k].clone()
# Compute sensitivities
disp = u_k[fem.elements, :].reshape(fem.n_elem, -1)
w_k = 0.5 * torch.einsum("...i,...ij,...j", disp, k0, disp)
sensitivity = -p * rho[k] ** (p - 1) * 2.0 * w_k
# Filter sensitivities (if r_filt provided)
if r_sens > 0.0:
# Note: We assume that each cell has the same area here.
sensitivity = H @ (rho[k] * sensitivity) / H.sum(dim=0) / rho[k]
def make_step(mu):
# Assuming a certain value of mu, apply the iteration scheme to
G_k = -sensitivity / (mu * areas)
upper = torch.min(rho_max, (1 + m) * rho_k)
lower = torch.max(rho_min, (1 - m) * rho_k)
rho_trial = G_k**xi * rho_k
return torch.clamp(rho_trial, lower, upper)
def g(mu):
rho_k = make_step(mu)
return torch.dot(rho_k, areas) - V_0
mu = bisection(g, 1e-10, 1.0, tol=1e-15)
# Evaluation
compliance = torch.inner(f_k.ravel(), u_k.ravel())
print(f"Iteration k={k} - Compliance: {compliance:.5f}")
rho.append(make_step(mu))
return rho
In [14]:
rho_opt = optimize_oc(cantilever, rho_0, rho_min, rho_max, V0, p=3, iter=50, r_sens=1.0)
cantilever.plot(element_property=rho_opt[-1], cmap="gray_r")
Iteration k=0 - Compliance: 3.13393 Iteration k=1 - Compliance: 2.29492 Iteration k=2 - Compliance: 1.82952 Iteration k=3 - Compliance: 1.50663 Iteration k=4 - Compliance: 1.28084 Iteration k=5 - Compliance: 1.18956 Iteration k=6 - Compliance: 1.12292 Iteration k=7 - Compliance: 1.06040 Iteration k=8 - Compliance: 1.00956 Iteration k=9 - Compliance: 0.96913 Iteration k=10 - Compliance: 0.93354 Iteration k=11 - Compliance: 0.90106 Iteration k=12 - Compliance: 0.87155 Iteration k=13 - Compliance: 0.84559 Iteration k=14 - Compliance: 0.82147 Iteration k=15 - Compliance: 0.80151 Iteration k=16 - Compliance: 0.78556 Iteration k=17 - Compliance: 0.77340 Iteration k=18 - Compliance: 0.76446 Iteration k=19 - Compliance: 0.75858 Iteration k=20 - Compliance: 0.75476 Iteration k=21 - Compliance: 0.75265 Iteration k=22 - Compliance: 0.75130 Iteration k=23 - Compliance: 0.75047 Iteration k=24 - Compliance: 0.74984 Iteration k=25 - Compliance: 0.74935 Iteration k=26 - Compliance: 0.74898 Iteration k=27 - Compliance: 0.74869 Iteration k=28 - Compliance: 0.74844 Iteration k=29 - Compliance: 0.74823 Iteration k=30 - Compliance: 0.74808 Iteration k=31 - Compliance: 0.74797 Iteration k=32 - Compliance: 0.74788 Iteration k=33 - Compliance: 0.74781 Iteration k=34 - Compliance: 0.74776 Iteration k=35 - Compliance: 0.74773 Iteration k=36 - Compliance: 0.74771 Iteration k=37 - Compliance: 0.74769 Iteration k=38 - Compliance: 0.74768 Iteration k=39 - Compliance: 0.74767 Iteration k=40 - Compliance: 0.74766 Iteration k=41 - Compliance: 0.74765 Iteration k=42 - Compliance: 0.74764 Iteration k=43 - Compliance: 0.74765 Iteration k=44 - Compliance: 0.74766 Iteration k=45 - Compliance: 0.74769 Iteration k=46 - Compliance: 0.74772 Iteration k=47 - Compliance: 0.74775 Iteration k=48 - Compliance: 0.74778 Iteration k=49 - Compliance: 0.74781
