From 93f87ca94abc8b99ccba8e9f463d67bae5018be6 Mon Sep 17 00:00:00 2001 From: Sebastian Blauth Date: Wed, 11 Oct 2023 09:19:56 +0200 Subject: [PATCH 1/3] Start with documentation of topology optimization --- .../poisson_clover/config.ini | 59 ++++ .../poisson_clover/demo_poisson_clover.py | 285 ++++++++++++++++++ .../poisson_clover/img_poisson_clover.png | Bin 0 -> 61402 bytes .../poisson_clover/results/history.json | 1 + .../poisson_clover/results/history.txt | 131 ++++++++ .../demos/topology_optimization/index.rst | 11 + docs/source/user/index.rst | 5 +- 7 files changed, 490 insertions(+), 2 deletions(-) create mode 100755 demos/documented/topology_optimization/poisson_clover/config.ini create mode 100644 demos/documented/topology_optimization/poisson_clover/demo_poisson_clover.py create mode 100644 demos/documented/topology_optimization/poisson_clover/img_poisson_clover.png create mode 100644 demos/documented/topology_optimization/poisson_clover/results/history.json create mode 100644 demos/documented/topology_optimization/poisson_clover/results/history.txt create mode 100644 docs/source/user/demos/topology_optimization/index.rst diff --git a/demos/documented/topology_optimization/poisson_clover/config.ini b/demos/documented/topology_optimization/poisson_clover/config.ini new file mode 100755 index 00000000..8a87a267 --- /dev/null +++ b/demos/documented/topology_optimization/poisson_clover/config.ini @@ -0,0 +1,59 @@ +[StateSystem] +is_linear = True +newton_rtol = 1e-11 +newton_atol = 1e-13 +newton_iter = 50 +newton_damped = True +newton_inexact = False +newton_verbose = False +picard_iteration = False +picard_rtol = 1e-10 +picard_atol = 1e-12 +picard_iter = 10 +picard_verbose = False + +[OptimizationRoutine] +algorithm = bfgs +rtol = 1e-10 +atol = 0.0 +max_iter = 10 +gradient_method = direct +gradient_tol = 1e-9 +soft_exit = True + +[LineSearch] +initial_stepsize = 1.0 +#safeguard_stepsize = True +epsilon_armijo = 1e-4 +beta_armijo = 2 +;method = polynomial + +[AlgoLBFGS] +bfgs_memory_size = 5 +use_bfgs_scaling = True + +[AlgoCG] +cg_method = PR +cg_periodic_restart = False +cg_periodic_its = 5 +cg_relative_restart = False +cg_restart_tol = 0.5 + +[AlgoTNM] +inner_newton = cg +max_it_inner_newton = 100 +inner_newton_rtol = 1e-15 +inner_newton_atol = 0.0 + +[TopologyOptimization] +topological_derivative_is_identical = True + +[Output] +verbose = True +save_results = True +save_txt = True +save_state = False +save_adjoint = False +save_gradient = False +time_suffix = False + diff --git a/demos/documented/topology_optimization/poisson_clover/demo_poisson_clover.py b/demos/documented/topology_optimization/poisson_clover/demo_poisson_clover.py new file mode 100644 index 00000000..37381f62 --- /dev/null +++ b/demos/documented/topology_optimization/poisson_clover/demo_poisson_clover.py @@ -0,0 +1,285 @@ +# --- +# jupyter: +# jupytext: +# text_representation: +# extension: .py +# format_name: light +# format_version: '1.5' +# jupytext_version: 1.14.4 +# --- + +# ```{eval-rst} +# .. include:: ../../../global.rst +# ``` +# +# (demo_poisson_clover)= +# # Topology Optimization with a Poisson Equation +# +# ## Some remarks about topology optimization +# +# As topology optimization problems are very involved from a theoretical point of view, +# it is, at the moment, not possible to automatically derive topological derivatives. +# Therefore, cashocs cannot be used as "black-box" solver for topology optimization +# problems in general. +# +# Moreover, our framework for topology optimization of using a level-set function is +# quite flexible, but requires a lot of theoretical understanding. In the following, we +# briefly go over some theoretical foundations required for using cashocs' topology +# optimization features. We refer the reader, e.g., to CITE for an exhaustive treatment +# of these topics. +# +# ## Solving topology optimization problems with a level-set method +# +# The general form of a topology optimization problem is given by +# +# $$ +# \min_\Omega J(\Omega), +# $$ +# +# i.e., we want to minimize some cost functional {math}`J` depending on a geometry or +# set {math}`\Omega` which is contained in some hold-all domain {math}`\mathrm{D}` by +# varying the topology and shape of {math}`\Omega`. In contrast to shape optimization +# problems we have discussed previously, topology optimization considers topological +# changes of {math}`\Omega`, i.e., the addition or removal of material. +# +# One can represent the domain {math}`\Omega` with a continuous level-set function {math}`\Psi +# \colon \mathrm{D} \to \mathbb{R}` as follows: +# +# $$ +# \begin{align} +# \Psi(x) < 0 \quad &\Leftrightarrow \quad x \in \Omega,\\ +# \Psi(x) > 0 \quad &\Leftrightarrow \quad x \in \mathcal{D} \setminus \bar{\Omega},\\ +# \Psi(x) = 0 \quad &\Leftrightarrow \quad x \in \partial \Omega \setminus \partial \mathrm{D}. +# \end{align} +# $$ +# +# Assuming that the topological derivative of the problem exists, which we denote by +# {math}`DJ(\Omega)`, then the generalized topological derivative is given by +# +# $$ +# \mathcal{D}J(\Omega)(x) = +# \begin{cases} +# -DJ(\Omega)(x) \quad &\text{ if } x \in \Omega,\\ +# DJ(\Omega)(x) \quad &\text{ if } x \in \mathcal{D} \setminus \bar{\Omega}. +# \end{cases} +# $$ +# +# This is motivated by the fact that if there exists some constant {math}`c > 0` such +# that +# +# $$ +# \mathcal{D}J(\Omega)(x) = c\Psi(x) \quad \text{ for all } x \in \mathrm{D}, +# $$ +# +# then we have that {math}`DJ(\Omega) \geq 0` which is a necessary condition for +# {math}`\Omega` being a local minimizer of the problem. +# According to this, several optimization algorithms have been developed in the +# literature to solve topology optimization problems with level-set methods. We refer +# the reader to [Blauth and Sturm - Quasi-Newton methods for topology optimization +# using a level-set method](https://doi.org/10.1007/s00158-023-03653-2) for a detailed +# discussion of the algorithms implemented in cashocs. +# +# +# ## Problem Formulation +# +# In this demo, we investigate the basics of using cashocs for solving topology +# optimization problems with a level-set approach. This approachs goes back to +# [Amstutz and Andrä - A new algorithm for topology optimization using a level-set +# method](https://doi.org/10.1016/j.jcp.2005.12.015) and in cashocs we have implemented +# novel gradient-type and quasi-Newton methods for solving such problems from [Blauth +# and Sturm - Quasi-Newton methods for topology optimization using a level-set +# method](https://doi.org/10.1007/s00158-023-03653-2). +# +# In this demo, we consider the following model problem +# +# $$ +# \begin{align} +# &\min_{\Omega,u} J(\Omega, u) = \frac{1}{2} \int_\mathrm{D} \left( u - u_\mathrm{des} \right)^2 \text{ d}x\\ +# &\text{subject to} \qquad +# \begin{alignedat}[t]{2} +# - \Delta u + \alpha_\Omega u &= f_\Omega \quad &&\text{ in } \mathrm{D},\\ +# u &= 0 \quad &&\text{ on } \partial \mathrm{D}. +# \end{alignedat} +# \end{align} +# $$ +# +# Here, {math}`\alpha_\Omega` and {math}`f_\Omega` have jumps between {math}`\Omega` +# and {math}`\mathrm{D} \setminus \bar{\Omega}` and are given by +# {math}`\alpha_\Omega(x) = \chi_\Omega(x)\alpha_\mathrm{in} + +# \chi_{\Omega^c}(x) \alpha_\mathrm{out}` and {math}`f_\Omega(x) = \chi_\Omega(x) +# f_\mathrm{in} + \chi_{\Omega^c}(x) f_\mathrm{out}` with {math}`\alpha_\mathrm{in}, +# \alpha_\mathrm{out} > 0` and {math}`f_\mathrm{in}, f_\mathrm{out} \in \mathbb{R}`, and +# {math}`\Omega^c = \mathrm{D} \setminus \bar{\Omega}` is the complement of +# {math}`\Omega` in {math}`\mathrm{D}`. Moreover, {math}`u_\mathrm{des}` is a desired +# state, which in our case comes from the solution of the state equation for a desired +# shape, which we want to recover by the above problem. The generalized topological +# derivative for this problem is given by +# +# $$ +# \mathcal{D}J(\Omega)(x) = (\alpha_\mathrm{out} - \alpha_\mathrm{in}) u(x)p(x) +# - (f_\mathrm{out} - f_\mathrm{in})p(x), +# $$ +# where {math}`u` solves the state equation and {math}`p` solves the adjoint equation. +# +# ## Implementation +# +# The complete python code can be found in the file {download}`demo_poisson_clover.py +# `, +# and the corresponding config can be found in {download}`config.ini +# `. +# +# ### Initialization and setup +# +# As usual, we start by using a wildcard import for FEniCS and by importing cashocs + +# + +from fenics import * + +import cashocs + +# - + +# Next, as before, we can specify the verbosity of cashocs with the line + +cashocs.set_log_level(cashocs.LogLevel.INFO) + +# As with the other problem types, the solution algorithms of cashocs can be adapted +# with the help of configuration files, which is loaded with the {py:func}`load_config < +# cashocs.load_config>` function + +cfg = cashocs.load_config("config.ini") + +# In the next step, we define the mesh used for discretization of the hold-all domain +# {math}`\mathrm{D}`. To do so, we use the in-built {py:func}`cashocs.regular_box_mesh` +# function, which gives us a discretization of {math}`\mathrm{D} = (-2, 2)^2`. + +mesh, subdomains, boundaries, dx, ds, dS = cashocs.regular_box_mesh( + n=32, start_x=-2, start_y=-2, end_x=2, end_y=2, diagonal="crossed" +) + +# Now, we define the function spaces required for solving the state system, i.e., a +# finite element space consisting of piecewise linear Lagrange elements, and we define +# a finite element space consisting of piecewise constant elements, which is used to +# discretize the jumping coefficients {math}`\alpha_\Omega` and {math}`f_\Omega`. + +V = FunctionSpace(mesh, "CG", 1) +DG0 = FunctionSpace(mesh, "DG", 0) + +# Now, we define the level-set function used to represent our domain {math}`\Omega` and +# initialize {math}`\Omega` to the empty set by setting {math}`\Psi = 1` with the lines + +psi = Function(V) +psi.vector()[:] = 1.0 + +# We define the constants for the jumping coefficients as follows + +# + +f_1 = 10.0 +f_2 = 1.0 +f_diff = f_1 - f_2 + +alpha_1 = 10.0 +alpha_2 = 1.0 +alpha_diff = alpha_1 - alpha_2 + +alpha = Function(DG0) +f = Function(DG0) +# - +# + + +def create_desired_state(): + a = 4.0 / 5.0 + b = 2 + + f_expr = Expression( + "0.1 * ( " + + "(sqrt(pow(x[0] - a, 2) + b * pow(x[1], 2)) - 1)" + + "* (sqrt(pow(x[0] + a, 2) + b * pow(x[1], 2)) - 1)" + + "* (sqrt(b * pow(x[0], 2) + pow(x[1] - a, 2)) - 1)" + + "* (sqrt(b * pow(x[0], 2) + pow(x[1] + a, 2)) - 1)" + + "- 0.001)", + degree=1, + a=a, + b=b, + ) + psi_des = interpolate(f_expr, V) + + cashocs.interpolate_levelset_function_to_cells(psi_des, alpha_1, alpha_2, alpha) + cashocs.interpolate_levelset_function_to_cells(psi_des, f_1, f_2, f) + + y_des = Function(V) + v = TestFunction(V) + + F = dot(grad(y_des), grad(v)) * dx + alpha * y_des * v * dx - f * v * dx + bcs = cashocs.create_dirichlet_bcs(V, Constant(0.0), boundaries, [1, 2, 3, 4]) + + cashocs.newton_solve(F, y_des, bcs, is_linear=True) + + return y_des, psi_des + + +y_des, psi_des = create_desired_state() + +y = Function(V) +p = Function(V) +F = dot(grad(y), grad(p)) * dx + alpha * y * p * dx - f * p * dx +bcs = cashocs.create_dirichlet_bcs(V, Constant(0.0), boundaries, [1, 2, 3, 4]) +J = cashocs.IntegralFunctional(Constant(0.5) * pow(y - y_des, 2) * dx) + +# Now, we have to define the topological derivative of the problem, which we do with + +dJ_in = Constant(alpha_diff) * y * p - Constant(f_diff) * p +dJ_out = Constant(alpha_diff) * y * p - Constant(f_diff) * p + +# ::::{note} +# We remark that the generalized topological derivative for this problem is identical +# in {math}`\Omega` and {math}`\Omega^c`, which is usually not the case. For this +# reason, the special structure of the problem can be exploited with the following +# lines in the configuration file +# ```{code-block} ini +# :caption: config.ini +# [TopologyOptimization] +# topological_derivative_is_identical = True +# ``` +# :::: + + +def update_level_set(): + cashocs.interpolate_levelset_function_to_cells(psi, alpha_1, alpha_2, alpha) + cashocs.interpolate_levelset_function_to_cells(psi, f_1, f_2, f) + + +top = cashocs.TopologyOptimizationProblem( + F, bcs, J, y, p, psi, dJ_in, dJ_out, update_level_set, config=cfg +) +top.solve(algorithm="bfgs", rtol=0.0, atol=0.0, angle_tol=1.0, max_iter=100) + +# We visualize the results with the following code + +# + +import matplotlib.pyplot as plt + +plt.figure(figsize=(10, 5)) + +ax_mesh = plt.subplot(1, 2, 1) +top.plot_shape() +plt.title("Obtained shape") + +ax_u = plt.subplot(1, 2, 2) +psi.vector()[:] = psi_des.vector()[:] +top.plot_shape() +plt.title("Desired shape") + +plt.tight_layout() +# plt.savefig("./img_poisson_clover.png", dpi=150, bbox_inches="tight") +# - + +# and the result looks like this +# ![](/../../demos/documented/topology_optimization/poisson_clover/img_poisson_clover.png) +# +# :::{note} +# The method {py:meth}`plot_shape ` can +# be used to visualize the current shape based on the level-set function `psi`. 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cost function, rel. grad. norm, abs. grad. norm, angle, step size + + 50, 9.535e-08, 1.606e-05, 1.598e-05, 128.687, 2.000e+00 + 51, 9.252e-08, 8.358e-06, 8.313e-06, 120.722, 4.000e+00 + 52, 8.735e-08, 1.113e-05, 1.107e-05, 125.272, 1.000e+00 + 53, 8.666e-08, 7.606e-06, 7.565e-06, 122.146, 1.000e+00 + 54, 8.451e-08, 5.215e-06, 5.187e-06, 103.400, 1.000e+00 + 55, 7.902e-08, 4.687e-06, 4.662e-06, 100.363, 1.000e+00 + 56, 4.615e-08, 3.968e-05, 3.946e-05, 50.964, 1.000e+00 + 57, 4.547e-08, 3.956e-05, 3.935e-05, 51.670, 2.500e-01 + 58, 4.184e-08, 2.364e-05, 2.351e-05, 50.390, 1.000e+00 + 59, 3.896e-08, 1.812e-05, 1.802e-05, 50.537, 1.000e+00 + +iter, cost function, rel. grad. norm, abs. grad. norm, angle, step size + + 60, 3.393e-08, 1.596e-05, 1.588e-05, 55.182, 1.000e+00 + 61, 3.122e-08, 4.962e-06, 4.935e-06, 113.815, 3.125e-02 + 62, 3.094e-08, 4.319e-06, 4.296e-06, 56.404, 1.000e+00 + 63, 3.093e-08, 7.118e-06, 7.080e-06, 54.503, 2.500e-01 + 64, 3.056e-08, 3.672e-06, 3.652e-06, 57.683, 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norm, angle, step size + + 80, 1.239e-08, 1.814e-06, 1.805e-06, 80.754, 1.000e+00 + 81, 1.151e-08, 3.363e-06, 3.345e-06, 61.982, 1.000e+00 + 82, 1.088e-08, 4.780e-06, 4.755e-06, 47.399, 1.000e+00 + 83, 1.075e-08, 1.044e-05, 1.038e-05, 51.171, 2.500e-01 + 84, 1.050e-08, 5.721e-06, 5.690e-06, 49.199, 1.000e+00 + 85, 1.033e-08, 3.351e-06, 3.333e-06, 48.368, 1.000e+00 + 86, 9.986e-09, 2.432e-06, 2.419e-06, 48.626, 1.000e+00 + 87, 8.772e-09, 9.585e-06, 9.534e-06, 51.448, 1.000e+00 + 88, 8.531e-09, 4.881e-06, 4.854e-06, 49.094, 1.000e+00 + 89, 8.411e-09, 2.606e-06, 2.592e-06, 47.590, 1.000e+00 + +iter, cost function, rel. grad. norm, abs. grad. norm, angle, step size + + 90, 8.212e-09, 2.550e-06, 2.536e-06, 48.003, 1.000e+00 + 91, 8.209e-09, 3.063e-06, 3.047e-06, 98.480, 1.562e-02 + 92, 8.012e-09, 4.404e-06, 4.380e-06, 50.333, 5.000e-01 + 93, 7.916e-09, 2.329e-06, 2.317e-06, 48.532, 1.000e+00 + 94, 7.829e-09, 1.666e-06, 1.657e-06, 50.600, 1.000e+00 + 95, 7.552e-09, 3.511e-06, 3.492e-06, 48.171, 1.000e+00 + 96, 7.518e-09, 2.573e-06, 2.559e-06, 47.321, 1.000e+00 + 97, 7.380e-09, 2.901e-06, 2.885e-06, 47.979, 1.000e+00 + 98, 7.349e-09, 2.203e-06, 2.191e-06, 47.554, 1.000e+00 + 99, 7.198e-09, 2.770e-06, 2.755e-06, 47.725, 1.000e+00 diff --git a/docs/source/user/demos/topology_optimization/index.rst b/docs/source/user/demos/topology_optimization/index.rst new file mode 100644 index 00000000..a54c7dbb --- /dev/null +++ b/docs/source/user/demos/topology_optimization/index.rst @@ -0,0 +1,11 @@ +Topology Optimization Problems +============================== + +In this part of the tutorial, we investigate how topology optimization problems can +be treated with cashocs. + +.. toctree:: + :maxdepth: 1 + :caption: List of all topology optimization problems: + + demo_poisson_clover.md diff --git a/docs/source/user/index.rst b/docs/source/user/index.rst index eb9b7bab..566b5891 100644 --- a/docs/source/user/index.rst +++ b/docs/source/user/index.rst @@ -4,8 +4,8 @@ cashocs Tutorial ================ Welcome to the cashocs tutorial. In the following, we present several example -programs that showcase how cashocs can be used to solve optimal control and -shape optimization problems. +programs that showcase how cashocs can be used to solve optimal control, +shape optimization, and topology optimization problems. .. include:: ../../../README.rst :start-after: readme_start_disclaimer @@ -25,6 +25,7 @@ Shape Optimization and Optimal Control Software Date: Thu, 12 Oct 2023 13:12:44 +0200 Subject: [PATCH 2/3] Add demos for topology optimization --- README.rst | 3 + .../cantilever/config.ini | 56 ++++ .../cantilever/demo_cantilever.py | 317 ++++++++++++++++++ .../cantilever/img_cantilever.png | Bin 0 -> 53961 bytes .../cantilever/results/history.json | 1 + .../cantilever/results/history.txt | 40 +++ .../pipe_bend/config.ini | 58 ++++ .../pipe_bend/demo_pipe_bend.py | 276 +++++++++++++++ .../pipe_bend/img_pipe_bend.png | Bin 0 -> 28385 bytes .../pipe_bend/results/history.json | 1 + .../pipe_bend/results/history.txt | 65 ++++ .../poisson_clover/demo_poisson_clover.py | 159 +++++++-- .../user/demos/shape_optimization/index.rst | 2 +- .../demos/topology_optimization/index.rst | 11 +- 14 files changed, 957 insertions(+), 32 deletions(-) create mode 100644 demos/documented/topology_optimization/cantilever/config.ini create mode 100644 demos/documented/topology_optimization/cantilever/demo_cantilever.py create mode 100644 demos/documented/topology_optimization/cantilever/img_cantilever.png create mode 100644 demos/documented/topology_optimization/cantilever/results/history.json create mode 100644 demos/documented/topology_optimization/cantilever/results/history.txt create mode 100644 demos/documented/topology_optimization/pipe_bend/config.ini create mode 100644 demos/documented/topology_optimization/pipe_bend/demo_pipe_bend.py create mode 100644 demos/documented/topology_optimization/pipe_bend/img_pipe_bend.png create mode 100644 demos/documented/topology_optimization/pipe_bend/results/history.json create mode 100644 demos/documented/topology_optimization/pipe_bend/results/history.txt diff --git a/README.rst b/README.rst index 864d7dcd..34e48356 100755 --- a/README.rst +++ b/README.rst @@ -65,6 +65,9 @@ we can recommend the textbooks - Shape Optimization - `Delfour and Zolesio - Shapes and Geometries `_ - `Sokolowski and Zolesio - Introduction to Shape Optimization `_ +- Topology Optimization + - `Sokolowski and Novotny - Topological Derivatives in Shape Optimization `_ + - `Amstutz - An Introduction to the Topological Derivative `_ - FEniCS - `Logg, Mardal, and Wells - Automated Solution of Differential Equations by the Finite Element Method `_ - `The FEniCS demos `_ diff --git a/demos/documented/topology_optimization/cantilever/config.ini b/demos/documented/topology_optimization/cantilever/config.ini new file mode 100644 index 00000000..0f529226 --- /dev/null +++ b/demos/documented/topology_optimization/cantilever/config.ini @@ -0,0 +1,56 @@ +[StateSystem] +is_linear = True +newton_rtol = 1e-11 +newton_atol = 1e-13 +newton_iter = 50 +newton_damped = True +newton_inexact = False +newton_verbose = False +picard_iteration = False +picard_rtol = 1e-10 +picard_atol = 1e-12 +picard_iter = 10 +picard_verbose = False + +[OptimizationRoutine] +algorithm = bfgs +rtol = 1e-10 +atol = 0.0 +max_iter = 1000 +gradient_method = direct +gradient_tol = 1e-9 +soft_exit = True + +[LineSearch] +;method = polynomial +initial_stepsize = 1e-3 +safeguard_stepsize = False +epsilon_armijo = 1e-20 +beta_armijo = 2.0 + +[AlgoLBFGS] +bfgs_memory_size = 5 +use_bfgs_scaling = True +bfgs_periodic_restart = 3 + +[AlgoCG] +cg_method = PR +cg_periodic_restart = False +cg_periodic_its = 5 +cg_relative_restart = False +cg_restart_tol = 0.5 + +[AlgoTNM] +inner_newton = cg +max_it_inner_newton = 100 +inner_newton_rtol = 1e-15 +inner_newton_atol = 0.0 + +[Output] +verbose = True +save_results = True +save_txt = True +save_state = False +save_adjoint = False +save_gradient = False +time_suffix = False diff --git a/demos/documented/topology_optimization/cantilever/demo_cantilever.py b/demos/documented/topology_optimization/cantilever/demo_cantilever.py new file mode 100644 index 00000000..475d1de8 --- /dev/null +++ b/demos/documented/topology_optimization/cantilever/demo_cantilever.py @@ -0,0 +1,317 @@ +# --- +# jupyter: +# jupytext: +# text_representation: +# extension: .py +# format_name: light +# format_version: '1.5' +# jupytext_version: 1.14.4 +# --- + +# ```{eval-rst} +# .. include:: ../../../global.rst +# ``` +# +# (demo_cantilever)= +# # Topology Optimization with Linear Elasticity - Cantilever +# +# ## Problem Formulation +# +# In this demo, we consider the topology optimization of linear elastic structures, +# using the well-known cantilever example, which has been investigated, e.g., in +# [Blauth and Sturm - Quasi-Newton methods for topology optimization +# using a level-set method](https://doi.org/10.1007/s00158-023-03653-2) and +# [Amstutz and Andrä - A new algorithm for topology optimization using a level-set +# method](https://doi.org/10.1016/j.jcp.2005.12.015). Our goal is to minimize the +# compliance of a linear elastic material, which can be formulated via the topology +# optimization problem +# +# $$ +# \begin{align} +# &\min_{\Omega,u} J(\Omega,u) = \int_\mathrm{D} \alpha_\Omega \sigma(u):e(u) \text{d}x + \gamma \lvert \Omega \rvert \\ +# &\text{subject to} \qquad +# \begin{alignedat}{2} +# -\text{div}(\alpha_\Omega \sigma(u)) &= f \quad &&\text{ in } \mathrm{D},\\ +# u &= 0 \quad &&\text{ on } \Gamma_D,\\ +# \alpha_\Omega \sigma(u)n &= g \quad &&\text{ on } \Gamma_N. +# \end{alignedat} +# \end{align} +# $$ +# +# Here, {math}`u` is the deformation of a linear elastic material, {math}`\sigma(u) = +# 2 \mu e(u) + \lambda \text{tr}e(u)I` is Hooke's tensor, where {math}`e(u) = +# \frac{1}{2}\left( \nabla u + (\nabla u)^T \right)` is the symmetric gradient. The +# coefficients {math}`\mu` and {math}`\lambda` are the Lamé parameters which satisfy +# {math}`\mu \geq 0` and {math}`2\mu + d \lambda > 0`, where {math}`d` is the dimension +# of the problem. As in {ref}`demo_poisson_clover`, the coefficient +# {math}`\alpha_\Omega` is given by {math}`\alpha_\Omega(x) = +# \chi_\Omega(x)\alpha_\mathrm{in} + \chi_{\Omega^c}(x) \alpha_\mathrm{out}`, and it +# models the elasticity of the material. We note that the term +# {math}`\gamma \lvert \Omega\rvert` is used to penalize too large domains +# {math}`\Omega` so that not the trivial solution {math}`\Omega = \mathrm{D}` is found. +# On the Dirichlet boundary {math}`\Gamma_D`, the material is fixed, and at the +# Neumann boundary {math}`\Gamma_N` a load {math}`g` is applied. +# +# Note that the generalized topological derivative for this problem is given by +# +# $$ +# \mathcal{D}J(\Omega)(x) = +# \begin{cases} +# \alpha_\mathrm{in} \frac{r_\mathrm{in} - 1}{\kappa r_\mathrm{in} +1} +# \frac{\kappa + 1}{2}\left( 2 \sigma(u): e(u) + +# \frac{(r_\mathrm{in}-1)(\kappa - 2)}{\kappa + 2 r_\mathrm{in} - 1} +# \text{tr}\sigma(u)\text{tr}e(u) \right) + \gamma \quad &\text{ for } x \in \Omega,\\ +# -\alpha_\mathrm{out} \frac{r_\mathrm{out} - 1}{\kappa r_\mathrm{out} + 1} +# \frac{\kappa + 1}{2} \left( 2 \sigma(u):e(u) + +# \frac{(r_\mathrm{out} - 1)(\kappa - 2)}{\kappa + 2r_\mathrm{out} - 1} \text{tr} +# \sigma(u) \text{tr}e(u) \right) + \gamma \quad &\text{ for } x \in +# \Omega^c = \mathrm{D}\setminus \bar{\Omega}, +# \end{cases} +# $$ +# +# where {math}`r_\mathrm{in} = \frac{\alpha_\mathrm{out}}{\alpha_\mathrm{in}}`, +# {math}`r_\mathrm{out} = \frac{\alpha_\mathrm{in}}{\alpha_\mathrm{out}}`, and +# {math}`\kappa = \frac{\lambda + 3\mu}{\lambda + \mu}`. +# +# :::{note} +# In the following, we consider only two-dimensional problems. For this case of plane +# stress, the Lamé parameters are given by {math}`\mu = \frac{E}{2(1+\nu)}` and +# {math}`\lambda = \frac{2\mu \lambda^*}{\lambda^* + 2\mu}`, where {math}`\lambda^* = +# \frac{E\nu}{(1+\nu)(1-2\nu)}` and {math}`E` and {math}`\nu` denote the Young's +# modulus and Poisson's ratio of the material, respectively. +# ::: +# +# ## Implementation +# +# The complete python code can be found in the file {download}`demo_cantilever.py +# `, +# and the corresponding config can be found in {download}`config.ini +# `. +# +# ### Initialization and Setup +# +# As with all other demos, we start by importing FEniCS and cashocs. +# +# + + +from fenics import * + +import cashocs + +# - +# +# Next, we load the configuration file of the problem with the line + +cfg = cashocs.load_config("config.ini") + +# Following this, we define the computational domain, using the built-in +# {py:func}`cashocs.regular_mesh` function, so that our hold-all domain is given by +# {math}`\mathrm{D} = (0,2) \times (0,1)`. + +mesh, subdomains, boundaries, dx, ds, dS = cashocs.regular_mesh( + 32, length_x=2.0, diagonal="crossed" +) + +# Next up, we define the function spaces for the problems. The solution of the state +# (and adjoint) systems lives in a vector function space of piecewise linear Lagrange +# elements, which is defined with + +V = VectorFunctionSpace(mesh, "CG", 1) + +# We also define scalar piecewise linear Lagrange elements (the `CG1` space) and +# piecewise constant discontinuous Lagrange elements (the `DG0` space) in the following. +# The `CG1` space is needed to represent the level-set function and the `DG0` space +# is used to treat the jumping coefficients. + +CG1 = FunctionSpace(mesh, "CG", 1) +DG0 = FunctionSpace(mesh, "DG", 0) + +# Now, we define the level-set function which represents our geometry (see +# {ref}`the previous demo `), and we initialize it with +# {math}`\Psi = -1`, so that {math}`\Omega = \mathrm{D}` as initial guess. + +psi = Function(CG1) +psi.vector()[:] = -1.0 + +# ### Definition of the Material Parameters +# +# In the following lines, we define the physical and numerical parameters for the +# material and optimization problem. First, the penalty parameter {math}`\gamma` is +# defined as + +gamma = 100.0 + +# Next, we define the Lamé parameters for the material, using a Young's modulus of +# {math}`E = 1` and Poisson's ratio of {math}`\nu = 0.3` + +E = 1.0 +nu = 0.3 +mu = E / (2.0 * (1.0 + nu)) +lambd_star = E * nu / ((1.0 + nu) * (1.0 - 2.0 * nu)) +lambd = 2 * mu * lambd_star / (lambd_star + 2.0 * mu) + +# Finally, the parameters {math}`\alpha_\mathrm{in}` and {math}`\alpha_\mathrm{out}` are +# defined as + +alpha_in = 1.0 +alpha_out = 1e-3 +alpha = Function(DG0) + +# which models the presence of material in {math}`\Omega` as well as the absence +# thereof in {math}`\Omega^c`. As before, {math}`\alpha` is a jumping coefficient, +# so that we define a piecewise constant FEM function which will represent it later. +# +# As we also need to take care of the volume {math}`\lvert \Omega \rvert`, we define a +# indicator function for the domain {math}`\Omega`, which is, as the jumping coefficient +# represented by a piecewise constant function. + + +indicator_omega = Function(DG0) + +# :::{note} +# As cells cut by the level-set function +# will recieve averaged values in these piecewise constant functions, the integral +# of the indicator function even gives us the exact volume of {math}`\Omega`. +# ::: +# +# ### Definition of the State System +# +# In the following, we define two python functions which return Hooke's tensor and +# the symmetrized gradient + +# + + + +def eps(u): + return Constant(0.5) * (grad(u) + grad(u).T) + + +def sigma(u): + return Constant(2.0 * mu) * eps(u) + Constant(lambd) * tr(eps(u)) * Identity(2) + + +# - +# +# For the load applied to the system we use a unitary point load at (2, 0.5), i.e., in +# the middle of the outer rightmost boundary. To do so, a Dirac-Delta function can be +# defined via a FEniCS `UserExpression` as follows. + + +class Delta(UserExpression): + def __init__(self, *args, **kwargs): + super().__init__(*args, **kwargs) + + def eval(self, values, x): + if near(x[0], 2.0) and near(x[1], 0.5): + values[0] = 3.0 / mesh.hmax() + else: + values[0] = 0.0 + + def value_shape(self): + return () + + +# We use this definition to define the point load as follows + +delta = Delta(degree=2) +g = delta * Constant((0.0, -1.0)) + +# Next, we define the state and adjoint variables {math}`u` and {math}`v` + +u = Function(V) +v = Function(V) + +# as well as the state system, the linear elasticity equations + +F = alpha * inner(sigma(u), eps(v)) * dx - dot(g, v) * ds(2) + +# which is supplied with boundary conditions of the form + +bcs = cashocs.create_dirichlet_bcs(V, Constant((0.0, 0.0)), boundaries, 1) + +# ### Setup of the Cost Functional and Topological Derivative +# +# To define the topology optimization problem, we first define the cost functional of +# the problem, given by + +J = cashocs.IntegralFunctional( + alpha * inner(sigma(u), eps(u)) * dx + Constant(gamma) * indicator_omega * dx +) + +# :::{note} +# As stated before, the integration of the function `indiciator_omega` gives the exact +# volume of {math}`\Omega`, so that the regularization term can be written as above. +# ::: + +# Finally, as stated previously, we have to specify the generalized topological +# derivative of the problem, which has been derived above. This is done with the lines + +# + + +kappa = (lambd + 3.0 * mu) / (lambd + mu) +r_in = alpha_out / alpha_in +r_out = alpha_in / alpha_out + +dJ_in = ( + Constant(alpha_in * (r_in - 1.0) / (kappa * r_in + 1.0) * (kappa + 1.0) / 2.0) + * ( + Constant(2.0) * inner(sigma(u), eps(u)) + + Constant((r_in - 1.0) * (kappa - 2.0) / (kappa + 2 * r_in - 1.0)) + * tr(sigma(u)) + * tr(eps(u)) + ) +) + Constant(gamma) +dJ_out = ( + Constant(-alpha_out * (r_out - 1.0) / (kappa * r_out + 1.0) * (kappa + 1.0) / 2.0) + * ( + Constant(2.0) * inner(sigma(u), eps(u)) + + Constant((r_out - 1.0) * (kappa - 2.0) / (kappa + 2 * r_out - 1.0)) + * tr(sigma(u)) + * tr(eps(u)) + ) +) + Constant(gamma) + +# - +# +# As in {ref}`demo_poisson_clover`, we now only have to specify what needs to happen +# when the level-set function is updated, i.e., when the geometry changes. Of course, +# the jumping coefficient {math}`\alpha` needs to be updated with the +# {py:func}`cashocs.interpolate_levelset_function_to_cells` function, but so does the +# indicator function of the geometry. This is specified in the following function. + + +def update_level_set(): + cashocs.interpolate_levelset_function_to_cells(psi, alpha_in, alpha_out, alpha) + cashocs.interpolate_levelset_function_to_cells(psi, 1.0, 0.0, indicator_omega) + + +# :::{note} +# As we want `indicator_omega` to be the indicator function of {math}`\Omega`, the +# above call makes sure that it is {math}`1` inside {math}`\Omega` and {math}`0` outside +# of it, representing a proper indicator function. +# ::: +# +# ### Definition and Solution of the Topology Optimization Problem +# +# Now, all that is left to do is to define the +# {py:class}`TopologyOptimizationProblem ` and +# solve it with its {py:meth}`solve ` method. + +top = cashocs.TopologyOptimizationProblem( + F, bcs, J, u, v, psi, dJ_in, dJ_out, update_level_set, config=cfg +) +top.solve(algorithm="bfgs", rtol=0.0, atol=0.0, angle_tol=1.5, max_iter=250) + +# As before, we can visualize the result using the following code + +# + +import matplotlib.pyplot as plt + +top.plot_shape() +plt.title("Obtained Cantilever Design") +plt.tight_layout() +# plt.savefig("./img_cantilever.png", dpi=150, bbox_inches="tight") +# - +# +# and the result looks like this +# ![](/../../demos/documented/topology_optimization/cantilever/img_cantilever.png) diff --git a/demos/documented/topology_optimization/cantilever/img_cantilever.png b/demos/documented/topology_optimization/cantilever/img_cantilever.png new file mode 100644 index 0000000000000000000000000000000000000000..a2efd5c83a0af7441da50d0194db407b1fba41f3 GIT binary patch literal 53961 zcmc$`byQVb*gm?EqbMS{rCZ@32q>X+8GwO;NJ@t&-Q7n;Hb|&|AgLhIY&tg~sicUs zfPjF~n{IyZ+USYzj_)__A9sw)P+2VIUNPqz&-1+RG~l-4)nlX#q$m{X*!63&cTgze z92DvR`tTt*;}EDU1>eLR)II|A4Q3rE&!>a>Vx9T?Z73_5|{m=t06~xYcLj>#~ayN!gw9&jXiz|2T1HZa#Fx!@?_g69)T&rd!f ztL48aOZcLi{A?se#J2xMO{TejaO3d*k4K5VmS0QIZ*CukKa3*QK`V|-95qZ8WQ@XO zaHe?J+aojOL~%nfqyKHL$97{4X1ekLv#SF(SGlNaatv#Yq^Q{vQ;n)mpi8;Vook>} zlKe|KN&d|hfAUoA;&TkbmY7-!872)~C+dmz1lAw3HgGezsKi(%&CFmqZK&kpVfkPN zzYu1LM7;`kdq#*nK+an6;i^bN1{C1c=6&yrcUVzCFzYn6~%x48SC6{w!YG%PyDJDUevCU+2vb^#N6TUn?$b_ zQkB1bpI>O6XxBS^jbxh0WZ&L zvz*L1&IhGfjcj0F9E=3YGJ0zn_%H>g$pcI+@*dc5577{@%TN26JCa=`NTaFR~eC znKtXWHc(`hYd?MinFp(ZqTm%r^FjUHMOq!V@rd3E4-s;iZClB$nf&$fNc|k2J#Q*1 zDucb65rLq0p(4Qs?ZpH^*;^Ko%=UshMAx!)S? 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153.80255017368734, 153.8005988441504, 153.799518539605, 153.79290868826308], "gradient_norm": [1.0, 1.0504034110943057, 1.1086364591825344, 1.0334296036131743, 0.9756773557243431, 0.8969587795103783, 0.883465007537447, 0.8764339888075806, 1.1166582098283442, 0.8802108273956757, 1.3527710023399095, 0.8525642977354847, 0.8341363444218072, 0.6465491575498764, 0.4681788522393535, 0.3349867715843251, 0.19452904859918965, 0.12186497762760716, 0.09121079226504318, 0.06917660965920856, 0.06072170916692272, 0.04376877374272062], "stepsize": [1.0, 0.001, 0.002, 0.004, 0.001, 0.03125, 0.015625, 0.0078125, 0.000125, 0.00025, 0.5, 0.001, 1.0, 0.25, 0.001, 1.0, 1.0, 1.0, 0.001, 0.25, 0.125, 0.25], "MeshQuality": [], "angle": [116.1396902089962, 109.44338267687696, 95.58788664858685, 68.08416186350985, 61.16220743510479, 54.06642617028483, 51.83975160125166, 50.6645398689501, 51.41695973229307, 46.13259388791498, 49.424308346035666, 35.78489880614035, 28.936452343936867, 21.91286194790358, 16.271289689226855, 11.101104739392598, 6.441962000461975, 4.045708869251757, 3.0353523937632665, 2.306465793093987, 2.0267069400692987, 1.4644305967635298], "initial_gradient_norm": 117.40766682321009, "state_solves": 84, "adjoint_solves": 23, "iterations": 21} \ No newline at end of file diff --git a/demos/documented/topology_optimization/cantilever/results/history.txt b/demos/documented/topology_optimization/cantilever/results/history.txt new file mode 100644 index 00000000..2e9da1aa --- /dev/null +++ b/demos/documented/topology_optimization/cantilever/results/history.txt @@ -0,0 +1,40 @@ + +iter, cost function, rel. grad. norm, abs. grad. norm, angle, step size + + 0, 2.395e+02, 1.000e+00, 1.174e+02, 116.140, + + 1, 2.395e+02, 1.050e+00, 1.233e+02, 109.443, 1.000e-03 + 2, 2.395e+02, 1.109e+00, 1.302e+02, 95.588, 2.000e-03 + 3, 2.395e+02, 1.033e+00, 1.213e+02, 68.084, 4.000e-03 + 4, 2.355e+02, 9.757e-01, 1.146e+02, 61.162, 1.000e-03 + 5, 2.174e+02, 8.970e-01, 1.053e+02, 54.066, 3.125e-02 + 6, 2.089e+02, 8.835e-01, 1.037e+02, 51.840, 1.562e-02 + 7, 2.043e+02, 8.764e-01, 1.029e+02, 50.665, 7.812e-03 + 8, 2.029e+02, 1.117e+00, 1.311e+02, 51.417, 1.250e-04 + 9, 1.923e+02, 8.802e-01, 1.033e+02, 46.133, 2.500e-04 + +iter, cost function, rel. grad. norm, abs. grad. norm, angle, step size + + 10, 1.897e+02, 1.353e+00, 1.588e+02, 49.424, 5.000e-01 + 11, 1.739e+02, 8.526e-01, 1.001e+02, 35.785, 1.000e-03 + 12, 1.652e+02, 8.341e-01, 9.793e+01, 28.936, 1.000e+00 + 13, 1.603e+02, 6.465e-01, 7.591e+01, 21.913, 2.500e-01 + 14, 1.584e+02, 4.682e-01, 5.497e+01, 16.271, 1.000e-03 + 15, 1.552e+02, 3.350e-01, 3.933e+01, 11.101, 1.000e+00 + 16, 1.542e+02, 1.945e-01, 2.284e+01, 6.442, 1.000e+00 + 17, 1.538e+02, 1.219e-01, 1.431e+01, 4.046, 1.000e+00 + 18, 1.538e+02, 9.121e-02, 1.071e+01, 3.035, 1.000e-03 + 19, 1.538e+02, 6.918e-02, 8.122e+00, 2.306, 2.500e-01 + +iter, cost function, rel. grad. norm, abs. grad. norm, angle, step size + + 20, 1.538e+02, 6.072e-02, 7.129e+00, 2.027, 1.250e-01 + 21, 1.538e+02, 4.377e-02, 5.139e+00, 1.464, 2.500e-01 + +Optimization was successful. +Statistics: + total iterations: 21 + final objective value: 1.538e+02 + final gradient norm: 4.377e-02 + total number of state systems solved: 84 + total number of adjoint systems solved: 23 diff --git a/demos/documented/topology_optimization/pipe_bend/config.ini b/demos/documented/topology_optimization/pipe_bend/config.ini new file mode 100644 index 00000000..e3be4b4b --- /dev/null +++ b/demos/documented/topology_optimization/pipe_bend/config.ini @@ -0,0 +1,58 @@ +[StateSystem] +is_linear = True +newton_rtol = 1e-8 +newton_atol = 1e-13 +newton_iter = 20 +newton_damped = True +newton_inexact = False +newton_verbose = False +picard_iteration = False +picard_rtol = 1e-10 +picard_atol = 1e-12 +picard_iter = 10 +picard_verbose = False + +[OptimizationRoutine] +algorithm = bfgs +rtol = 1e-10 +atol = 0.0 +max_iter = 1000 +gradient_method = direct +gradient_tol = 1e-9 +soft_exit = True + +[LineSearch] +initial_stepsize = 1.0 +safeguard_stepsize = False +epsilon_armijo = 1e-4 +beta_armijo = 2 +;method = polynomial + +[AlgoLBFGS] +bfgs_memory_size = 5 +use_bfgs_scaling = True + +[AlgoCG] +cg_method = PR +cg_periodic_restart = False +cg_periodic_its = 5 +cg_relative_restart = False +cg_restart_tol = 0.5 + +[AlgoTNM] +inner_newton = cg +max_it_inner_newton = 100 +inner_newton_rtol = 1e-15 +inner_newton_atol = 0.0 + +[TopologyOptimization] +topological_derivative_is_identical = True + +[Output] +verbose = True +save_results = True +save_txt = True +save_state = False +save_adjoint = False +save_gradient = False +time_suffix = False diff --git a/demos/documented/topology_optimization/pipe_bend/demo_pipe_bend.py b/demos/documented/topology_optimization/pipe_bend/demo_pipe_bend.py new file mode 100644 index 00000000..f65351de --- /dev/null +++ b/demos/documented/topology_optimization/pipe_bend/demo_pipe_bend.py @@ -0,0 +1,276 @@ +# --- +# jupyter: +# jupytext: +# text_representation: +# extension: .py +# format_name: light +# format_version: '1.5' +# jupytext_version: 1.14.4 +# --- + +# ```{eval-rst} +# .. include:: ../../../global.rst +# ``` +# +# (demo_pipe_bend)= +# # Topology Optimization with Stokes Flow - Pipe Bend +# +# ## Problem Formulation +# +# In this demo, we consider the topology optimization with Stokes flow, where +# we aim at finding the optimal shape for a pipe bend. This problem has been +# investigated previously, e.g., in [Blauth and Sturm - Quasi-Newton methods for +# topology optimization using a level-set +# method](https://doi.org/10.1007/s00158-023-03653-2) and [Sa, Amigo, Novotny, Silva - +# Topological derivatives applied to fluid flow channel design optimization +# problems](https://doi.org/10.1007/s00158-016-1399-0). The problem can be written +# as follows +# +# $$ +# \begin{align} +# &\min_{\Omega,u} J(\Omega, u) = \int_\mathrm{D} \mu \nabla u : \nabla u + +# \alpha_\Omega u \cdot u \text{d}x + \frac{\lambda}{2}\left( \lvert \Omega \rvert - \text{vol}_\mathrm{des} \right)^2 \\ +# &\text{subject to} \qquad +# \begin{alignedat}{2} +# -\mu \Delta u + \nabla p + \alpha_\Omega u &= 0 \quad &&\text{ in } \mathrm{D},\\ +# \nabla \cdot u &= 0 \quad &&\text{ in } \mathrm{D},\\ +# u &= u_D \quad &&\text{ on } \partial \mathrm{D},\\ +# p &= 0 \quad &&\text{ at } x^*. +# \end{alignedat} +# \end{align} +# $$ +# +# Here, {math}`u` and {math}`p` denote the fluids velocity and pressure, respectively, +# {math}`\mu` is its viscosity. Moreover, +# {math}`\alpha` denotes the viscous resistance or inverse permeability of the material, +# which is used to distinguish between fluid (where {math}`\alpha` is low) and solid +# (where {math}`\alpha` is high). As in the previous demos, e.g. {ref}`demo_cantilever`, +# {math}`\alpha` represents a jumping coefficient between the considered materials, +# i.e., it is given by {math}`\alpha_\Omega(x) = +# \chi_\Omega(x)\alpha_\mathrm{in} + \chi_{\Omega^c}(x) \alpha_\mathrm{out}`, so that +# {math}`\Omega` models our fluid and {math}`\Omega^c` models the solid part. +# +# On the outer boundary of the hold-all domain, +# Dirichlet boundary conditions are specified, +# indicating, where the fluid enters and exits. Moreover, the goal of the optimization +# problem is to minimize the energy dissipation of the fluid while achieving a certain +# volume of the fluid region. For more details on this problem, we refer the reader +# to [Blauth and Sturm - Quasi-Newton methods for topology optimization using a +# level-set method](https://doi.org/10.1007/s00158-023-03653-2). +# +# The generalized topological derivative for this problem is given by +# +# $$ +# \mathcal{D}J(\Omega)(x) = \left(\alpha_\mathrm{in} - \alpha_\mathrm{out}\right) +# u(x)\cdot (u(x) + v(x)) +# + \lambda \left( \lvert \Omega \rvert - \text{vol}_\mathrm{des}\right). +# $$ +# +# :::{note} +# We do not specify the adjoint equation here, as this is derived automatically +# by cashocs. For more details, we refer to [Blauth and Sturm - Quasi-Newton methods +# for topology optimization using a level-set +# method](https://doi.org/10.1007/s00158-023-03653-2). +# ::: +# +# :::{attention} +# As in {ref}`demo_poisson_clover`, there is a minus sign missing in our paper +# [Blauth and Sturm - Quasi-Newton methods for topology optimization using a +# level-set method](https://doi.org/10.1007/s00158-023-03653-2). The topological +# derivative presented here is, in fact, correct. +# ::: +# +# ## Implementation +# +# The complete python code can be found in the file {download}`demo_pipe_bend.py +# `, +# and the corresponding config can be found in {download}`config.ini +# `. +# +# ### Initialization and Setup +# +# We start by importing cashocs and FEniCS into our script +# +# + + +from fenics import * +import numpy as np + +import cashocs + +# - +# +# Next, we load the configuration file for the problem and define the mesh with the +# lines + +cfg = cashocs.load_config("config.ini") +mesh, subdomains, boundaries, dx, ds, dS = cashocs.regular_mesh(25, diagonal="crossed") + +# Now, we define the finite element spaces for the functions. These are given by +# a Taylor-Hood space for velocity and pressure variables, a piecewise linear Lagrange +# space for the level-set function, and a piecewise constant discontinuous Lagrange +# space for the jumping coefficients. + +v_elem = VectorElement("CG", mesh.ufl_cell(), 2) +p_elem = FiniteElement("CG", mesh.ufl_cell(), 1) +V = FunctionSpace(mesh, v_elem * p_elem) +CG1 = FunctionSpace(mesh, "CG", 1) +DG0 = FunctionSpace(mesh, "DG", 0) + +# Additionally, we define a scalar real +# function space (`R`), which will be used to deal with the volume regularization +# of the problem, and a function `vol`, which represents the current volume of +# {math}`\Omega`, as follows. + +R = FunctionSpace(mesh, "R", 0) +vol = Function(R) + +# ### Definition of Physical Parameters and Jumping Coefficients +# +# Next, we define the physical parameters for the problem, including a viscosity of +# {math}`\mu = 0.01` as well as parameters for {math}`\alpha_\mathrm{in}` and +# {math}`\alpha_\mathrm{out}` + +mu = 1e-2 +alpha_in = 2.5 * mu / 1e2**2 +alpha_out = 2.5 * mu * 1e2**2 +alpha = Function(DG0) + +# As before, the jumping coefficient {math}`\alpha` is represented using a piecewise +# constant (on each element) function. +# +# Next, the desired volume and the regularization parameter {math}`\lambda` are defined, +# together with the indicator function of {math}`\Omega` (see {ref}`demo_cantilever`). + +vol_des = assemble(1 * dx) * 0.08 * np.pi +lambd = 1e4 +indicator_omega = Function(DG0) + +# Then, the level-set function is introduced and set to {math}`\Psi = -1`, so that we +# use {math}`\Omega = \mathrm{D}` as initial guess. + +psi = Function(CG1) +psi.vector()[:] = -1.0 + +# ### Definition of the State System +# +# In the following, we define the state and adjoint variables of our problem, +# in analogy to, e.g., {ref}`demo_shape_stokes`: + +up = Function(V) +u, p = split(up) +vq = Function(V) +v, q = split(vq) + +# and the weak form of the Stokes system is given by the lines + +F = ( + Constant(mu) * inner(grad(u), grad(v)) * dx + - p * div(v) * dx + - q * div(u) * dx + + alpha * dot(u, v) * dx +) + +# For the Dirichlet boundary conditions, we specify that we have an inflow at +# the upper left part of the boundary as well as an outflow on the lower right part, +# with the following expressions.Additionally, we specify the pressure at the point +# {math}`x^* = (0,0)` to obtain uniqueness of the pressure. +# Altogether, the boundary conditions are specified using the code +# +# + + +v_max = 1e0 +v_in = Expression( + ("(x[1] >= 0.7 && x[1] <= 0.9) ? v_max*(1 - 100*pow(x[1] - 0.8, 2)) : 0.0", "0.0"), + degree=2, + v_max=v_max, +) +v_out = Expression( + ("0.0", "(x[0] >= 0.7 && x[0] <= 0.9) ? -v_max*(1 - 100*pow(x[0] - 0.8, 2)) : 0.0"), + degree=2, + v_max=v_max, +) + + +def pressure_point(x, on_boundary): + return near(x[0], 0) and near(x[1], 0) + + +bcs = cashocs.create_dirichlet_bcs(V.sub(0), v_in, boundaries, 1) +bcs += cashocs.create_dirichlet_bcs(V.sub(0), v_out, boundaries, 3) +bcs += cashocs.create_dirichlet_bcs(V.sub(0), Constant((0.0, 0.0)), boundaries, [2, 4]) +bcs += [DirichletBC(V.sub(1), Constant(0), pressure_point, method="pointwise")] + +# - +# +# ### Cost Functional and Topological Derivative +# +# Now, we define the cost functional of our problem as well as its corresponding +# generalized topological derivative with the lines + +J = cashocs.IntegralFunctional( + Constant(mu) * inner(grad(u), grad(u)) * dx + + alpha * dot(u, u) * dx + + Constant(lambd / 2) * pow(vol - Constant(vol_des), 2) * dx +) +dJ_in = Constant(alpha_in - alpha_out) * (dot(u, v) + dot(u, u)) + Constant(lambd) * ( + vol - Constant(vol_des) +) +dJ_out = Constant(alpha_in - alpha_out) * (dot(u, v) + dot(u, u)) + Constant(lambd) * ( + vol - Constant(vol_des) +) + +# :::{note} +# Note, that the second term of the cost functional measures the discrepancy between +# the current volume `vol` of {math}`\Omega` and the desired volume. Due to the +# `update_level_set` function, which is defined below, the variable `vol` holds the +# correct value in each iteration. +# ::: + +# As in the previous demos, we have to specify the update routine of the level-set +# function `update_level_set`, which we do as follows: + + +def update_level_set(): + cashocs.interpolate_levelset_function_to_cells(psi, alpha_in, alpha_out, alpha) + cashocs.interpolate_levelset_function_to_cells(psi, 1.0, 0.0, indicator_omega) + vol.vector()[0] = assemble(indicator_omega * dx) + + +# That is, in the `update_level_set` function, first the jumping coefficient is updated +# with the {py:func}`cashocs.interpolate_levelset_function_to_cells` function. +# Then, the indicator function is updated and used to compute the current volume of the +# domain, which is written to the variable `vol`. +# +# ### Solution of the Topology Optimization Problem +# +# Finally, we can define the topology optimization problem and solve it via +# the lines + +top = cashocs.TopologyOptimizationProblem( + F, bcs, J, up, vq, psi, dJ_in, dJ_out, update_level_set, config=cfg +) +top.solve(algorithm="bfgs", rtol=0.0, atol=0.0, angle_tol=5.0, max_iter=500) + +# :::{note} +# For solving this problem, we choose a rather high tolerance (regarding the angle) +# of 5 degrees. This means, our optimization algorithm has not yet converged. We do +# this to ensure a low running time of the demo. If one wishes, they can run the demo +# with `angle_tol=1.0` and still recieve a result, however, it may take a while +# until this lower tolerance can be reached. +# ::: +# +# We visualize the result with the code +# +# + + +import matplotlib.pyplot as plt + +top.plot_shape() +plt.title("Obtained Pipe Bend Design") +plt.tight_layout() +# plt.savefig("./img_pipe_bend.png", dpi=150, bbox_inches="tight") +# - +# +# and the results looks as follows +# ![](/../../demos/documented/topology_optimization/pipe_bend/img_pipe_bend.png) diff --git a/demos/documented/topology_optimization/pipe_bend/img_pipe_bend.png b/demos/documented/topology_optimization/pipe_bend/img_pipe_bend.png new file mode 100644 index 0000000000000000000000000000000000000000..9a5101b0d7355c11b96251d940e4487ab1c0c560 GIT binary patch literal 28385 zcmb@u2|U#6`#(JCv}uuqlWd_>D(9pjWN9IjP_oB3l6A6#k;qbMp@^XzyO5p9mTgdo zB$RFJYxX^ctn*y=P`~f*|Gb{p|M@-t=llAe(R}8!+@Jft?(2HL-`92DURP8v?qud- z#$YfzaY`4|F&L&a`0vBk&2S{xK+p^RNIEF!I9#(ab+}_>XM#~Na_NuPp23Ei7tMB%09-z2ij{Nr~=`v~NrewfU z{j+3a)gM)2mB%EWeCLkrJM9aN@z@a*FtG{#fl24r-+;k%Z}Sv^FK6!C!a)z=-OvS? zsXt2C;ZKBoE)xdxo^7i(d~ssxK|Q7R|K=jxJJV{eg?qB_|5V+~LmgGysOA3b{MT(RGty$BJr7q%#}b-`210FAEdov0*a}~}hn%vAK4FkQ z^kDmmuP&2?K@%o=?8Ejom6hK%Y~hZI@s+w%yx2(2avX1U5XBy6ijkO7GctOumt&n8 zA>%T#JX1?O&mt2$-j);Zk|(cqK>z;XPN%+|#EEv@AbmS8{M~+bS6ZX?=c=k}CMIuW z>C5&jG_rf+HsP?S05w_Nws%&_pShY+bYr?!=W-UuZQMfx7f6Ajp~-0m1xW;I+UM$O zmDZJ;8SU-**_Mr;>G9Fg=aRKj>)6!Jyxy>JbA$8p_U+qOUIc4Da;vEh!PfioNM|_= zt9KBo22x{hOHaAXjX8*lrG{V$1Mc^jw{Oq3Y7;+x{oQ4E`cfBh_)SofN~n1LSVmPA zJj+3}kqjMXP~fO(VPTOU#H4dEmx*<`kB|PfQaCb&HcCylYJ2DW-Tz{i>yo4Ee8-)n zgoLkVk*;w^uEix7l|BxUae3!98^dQYGf+K@cRbhn&MLW9j;(spgTC3LO zwq%tbB!ZJ#nXRWS3`UNE9bKMR$Bb^(=6{YF#wM)VP~^Jg0Iy_ONjjK66#c9rMOTbY z%qNbf_$r)vVOMawY0^ zU-@vS%j8QTtF9sgczxx`=#!V@Rl^-7@&-rb=nF3C+tyZopW9xdN*6DU-;Fc5myrN2Zs+w$d!+-|090#$Wk zVPWL!>xU=uzVC5ep;5Eb+7hn`^SCchuhMt9R(EXT)bH-n($st-(dl$P$iOk!S90J_ z9*3`IkqK74I*z^7{Fltk7co0-;jxhDLP?QXl@rO@*Ah)H#0AL$|^PWeq&?8RYBrpr$J7K zV<=4D=*~L|3Xdj=>251OE9A7EaU2f}3W{$^y-t`N`nCuIY&99Mqc0)G_lG;hez4{# zwJpmcAgyptk2cnF?c>LfM-LzVdf*9nbmZ!6{NilAT4`r_6SZM5(ml<-M)G3n^_;n` z$?uf`G%$lQ`ZnPqZ;`aGqV5aIIBczbR$-V;@yfdz*`-^=p|58*ZQ6A7#0jmgB3D|4 z-0I&W_;D)UMa3-a_T$!ULOg|XhR62(2C46U=N;;B!ozCri`P=AE#*H3Qtr+S3^!a; z(}c!1ptl=u)s$$redsvRzK=G?sc(DsYi&fZnAr&Q(>1UYO&uM0AE`?)nIe;tlLh6> zJ@)VN;NLHP%ht}$jvH44UuBq@wlQ1K6|B`|m%oUhPei&-Sikcv)mRUVKZ^>XU~(fIWhzP8i$!yhb)lmlQff~#M;E3JR*Fk9V%L!T9#27#7Qp^<~1mjW;ZI zJ4-ln;>51S$$C}k>co_k6k?mVH?tYo#Oi4O_wT2{qY%M03sx?GryE~}k#rQ2ycVzO zGrYij(|QTq$?0BVxi8PClAWEseW~n;*Kul*6H0bcj$=cYL&d9$o8nYLH_Mn7I@9dl z*>v&F$!e$R4?9e>x40Wb)@qObz4TGkjdgx-|LEZI_W-rUNweX)H%4SJLQ_fAai$PFqt`Qf_5f?O{X#x^O4gsVVv$YMMbfgpr1PaMw$1 zQEluT98HTwy9|uGsfMTLyWETEgt~j81O5FlT&26DzvK-@xCnH*m3VIR6*7a*8nC&l zsL9Ir*l_pN1){9Aya4!uvU)YSae9w5yvDyacLsvZ7TphkZ(9&$d0x`ndn5kh3*MTN zN85QSZikJGSPo`W@%9kqx}o3T^B#R={;Yk>CiSTKPM6w!a&DBFkz}$|sJN9KSQWYk ze5oWdGW+)%wwxw~Elm`Vo2-m?Qo%+fJ5x-@hx_sh;ln!>p5XR~5ViK$2^C{6ss1pH-(kc-vIB8g_Q}8t^&c z8(GeLg^`0PX=HXxKldx2G$cr`|G zLG;Bz{pxnx7cGJ>&h!@7EKL^kWsPeWFTRx1JI|-q_lOg3D5rUCHf*__=*aZKJypPDnYUUp-sY^VE=`7mmBQTPG`#oCeF2P_8>bp{!k|~r zJSl-EbHpryd<|c6vKn70e@bf+qARn0X3KD z022!V(8r#calI^aR!7AsZ z64sZ^%K_ey!@^qS!~u#P#IOYaN<&rx(DT-oW$+n5+_f^5BFdM0w-pim{UI zq8flBgf__y`t}W_SMUZ184w#aQJ`+li@Ers34vKKqLES-x#unh>eZvs#R~)b>5=cL zm)yWDtj3so>g78m!i3n^KQJ(ol$_>$nhxkTNT@nk>>^AG7MF0R;yCV#Z_uAY3af5X zy-x*~yVMfvD0l7Y7588C#k`P3IPUY2_@PgNFLt=JMMn$x@XF}HvvhsHqhBIH5Hsrk zw5u)0hI-wm@E|T9X5GukNW)R73O;v=ugqMtlo_%F`toC`@vM3w9KaR(bUl8`lpO1Z zvTCm%0|I@XY^qQ5$HB0hGEAy8G&Ns-oz}QB*8Cc3OZXWhOV5*pW(X0U%c&uai+M0%&&(s)!=5q;Igb?U>Gd_Z?J=&K>{ z<@GSP6Kleym&ju7tM_o~L?W@FuQ^1gBbyKeu7*SWWFL4b@INbb1L}OeSnV0M+XYts zMf7T86x*_`JEVHYH%2PQ0D89k7X3)vx;?>dWpS9KeBzsza6|&oi9-&GZx1TJ`bk}2 zgkT!H#OXqa16R#x!}Y?w)VRA7rzJO%fYqJ*W5T54oz?X(6<8k=HLEOL(>6fK2=oI7 zIU2Ww)k@P(JaGQ8`N|?~wF+28W0u7=y6B%X)}t5ZJ1?{vbLDFP^;e*TRqJ>6JGWDc z%N!>=$LY=7oH=QOQGhPfO0f`Qi>Y4*XA;42Sk}k+Y~Jx_6oPFdb#Ij0#jc~7Mt)@^ zOMc%+w{Cv>fj$j^=0<+x`-(5A8+VsTK2JD0mF_)9G@3JwztsJ={>J&DkP6Gen((VHYX12{Ruy1MBXncTbYBHw;YWYS znY$UWA#6K#@Sf2z3n#uR_u0=mV`#UrID(0OWB1Wlk$3$gB!-_1o@h4=*^qp$j@|+8{}({8X%fR4T-;KBA9=ld0z*rx3q=8Z|3zz1JJB0!E18_I%}5 zo#?!u0X@scwL!eUW-~A z)>e9*F%{pmy6M=|&mvPF!+X;TpKFuYf9ibvu@cA(zI2xU5j)#2G>~pLqYqu61G6!L z*M0eRs(zmOD3wg*TbA88aBBMy8CW}7Yw zc%z$Xx+>G`MHwFZ*VxxzV@6MV=I3L*Y)j2>qtFxJtY?bL{|@Xsy(*?2Tzf6)uCf1J z8JmpG1kU9571;=kzSy#h`!KGx;0_soZ~9|k+Hm(nmO1*HW3BVox|R2n#^>5>=INPE z-MF){e7wophOXGSAm*}Y@ILtv6_;(Ei@95*DUfL#_(=2k!w#~Zm_EV9Yj?QNJ-?zDR zR8*8jIwv=`nV!BNTqQ5SsR~OBc-hVeKKB6ZspWLGtGcl5MQ)eWPe5pS^S*p$)3wa zJ@sSZ9<-DSz>|jvoL2KnyJFjQQJfDKmL!BV|H>biEpF4sItoT1*j|0OdFSkO1t0KSyz^Lk#eOl}q! 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In python code, this can be done as follows + +dJ_in = Constant(alpha_in - alpha_out) * y * p - Constant(f_in - f_out) * p +dJ_out = Constant(alpha_in - alpha_out) * y * p - Constant(f_in - f_out) * p -dJ_in = Constant(alpha_diff) * y * p - Constant(f_diff) * p -dJ_out = Constant(alpha_diff) * y * p - Constant(f_diff) * p +# :::{attention} +# There is a typo in our paper [Blauth and Sturm - Quasi-Newton methods for topology +# optimization using a level-set method](https://doi.org/10.1007/s00158-023-03653-2), +# i.e., in the paper a minus sign is missing. The topological derivative as stated +# here is, in fact, correct. +# ::: # ::::{note} # We remark that the generalized topological derivative for this problem is identical @@ -243,18 +309,46 @@ def create_desired_state(): # topological_derivative_is_identical = True # ``` # :::: +# +# Finally, we have to define what happens, when the geometry, i.e., the level-set +# function is updated. Of course, a change in the level-set function changes the +# geometry, so that the jumping coefficients {math}`\alpha` and {math}`f` have to be +# updated. This is, as before, done with the function +# {py:func}`cashocs.interpolate_levelset_function_to_cells`, which is called twice, once +# for updating `alpha` and once for `f`. def update_level_set(): - cashocs.interpolate_levelset_function_to_cells(psi, alpha_1, alpha_2, alpha) - cashocs.interpolate_levelset_function_to_cells(psi, f_1, f_2, f) + cashocs.interpolate_levelset_function_to_cells(psi, alpha_in, alpha_out, alpha) + cashocs.interpolate_levelset_function_to_cells(psi, f_in, f_out, f) + +# ### Definition and Solution of the Topology Optimization Problem +# +# Now, defining a topology optimization problem is nearly as easy as defining a shape +# optimization or optimal control problem, namely we instantiate a +# {py:class}`cashocs.TopologyOptimizationProblem` and then can call it's +# {py:meth}`solve ` method. top = cashocs.TopologyOptimizationProblem( F, bcs, J, y, p, psi, dJ_in, dJ_out, update_level_set, config=cfg ) top.solve(algorithm="bfgs", rtol=0.0, atol=0.0, angle_tol=1.0, max_iter=100) +# :::{note} +# Note, that there are several algorithms available for solving such topology +# optimization problems. Among them are the usual "sphere combination" method of +# [Amstutz and Andrä - A new algorithm for topology optimization using a level-set +# method](https://doi.org/10.1016/j.jcp.2005.12.015), which is invoked with +# `algorithm="sphere_combination"`, a convex combination approach, invoked with +# `algorithm="convex_combination"`, and all optimizaton algorithms available for shape +# optimization, i.e., gradient descent (`algorithm="gd"`), nonlinear CG +# (`algorithm="ncg"`) and L-BFGS (`algorithm="bfgs"`) methods. For a comparison of these +# optimization algorithms, we refer the reader to [Blauth +# and Sturm - Quasi-Newton methods for topology optimization using a level-set +# method](https://doi.org/10.1007/s00158-023-03653-2) +# ::: + # We visualize the results with the following code # + @@ -277,9 +371,14 @@ def update_level_set(): # and the result looks like this # ![](/../../demos/documented/topology_optimization/poisson_clover/img_poisson_clover.png) +# As we can see, the BFGS method is able to reconstruct the desired clover shape after +# only about 100 iterations. We encourage readers to try the other (established) methods +# to compare the novel BFGS approach to established techniques and see that the new +# methods are significantly more efficient. # # :::{note} # The method {py:meth}`plot_shape ` can # be used to visualize the current shape based on the level-set function `psi`. Note # that the cells inside {math}`\Omega` are colored in yellow and the ones outside are # colored blue. +# ::: diff --git a/docs/source/user/demos/shape_optimization/index.rst b/docs/source/user/demos/shape_optimization/index.rst index a1cd8bf6..05b8b152 100644 --- a/docs/source/user/demos/shape_optimization/index.rst +++ b/docs/source/user/demos/shape_optimization/index.rst @@ -6,7 +6,7 @@ be treated with cashocs. .. toctree:: :maxdepth: 1 - :caption: List of all shape optimization problems: + :caption: List of all shape optimization demos: demo_shape_poisson.md doc_config diff --git a/docs/source/user/demos/topology_optimization/index.rst b/docs/source/user/demos/topology_optimization/index.rst index a54c7dbb..4f048427 100644 --- a/docs/source/user/demos/topology_optimization/index.rst +++ b/docs/source/user/demos/topology_optimization/index.rst @@ -6,6 +6,15 @@ be treated with cashocs. .. toctree:: :maxdepth: 1 - :caption: List of all topology optimization problems: + :caption: List of all topology optimization demos: demo_poisson_clover.md + demo_cantilever.md + demo_pipe_bend.md + + +.. note:: + + As topology optimization problems are very involved from a theoretical point of view, it is, at the moment, not possible to automatically derive topological derivatives. Therefore, cashocs cannot be used as "black-box" solver for topology optimization problems in general. + + Moreover, our framework for topology optimization of using a level-set function is quite flexible, but requires a lot of theoretical understanding. In :ref:`demo_poisson_clover`, we briefly go over some theoretical foundations required for using cashocs' topology optimization features. We refer the reader, e.g., to `Sokolowski and Novotny - Topological Derivatives in Shape Optimization `_ for an exhaustive treatment of these topics. From 88e0de186af9a03333bbe5305513a3fd282ed5cc Mon Sep 17 00:00:00 2001 From: Sebastian Blauth Date: Thu, 12 Oct 2023 13:30:13 +0200 Subject: [PATCH 3/3] Update the demos --- .../cantilever/demo_cantilever.py | 19 +++++---- .../pipe_bend/demo_pipe_bend.py | 39 +++++++++++++------ .../poisson_clover/demo_poisson_clover.py | 32 ++++++++------- 3 files changed, 55 insertions(+), 35 deletions(-) diff --git a/demos/documented/topology_optimization/cantilever/demo_cantilever.py b/demos/documented/topology_optimization/cantilever/demo_cantilever.py index 475d1de8..a911a7f5 100644 --- a/demos/documented/topology_optimization/cantilever/demo_cantilever.py +++ b/demos/documented/topology_optimization/cantilever/demo_cantilever.py @@ -91,21 +91,21 @@ # ### Initialization and Setup # # As with all other demos, we start by importing FEniCS and cashocs. -# -# + +# + from fenics import * import cashocs # - -# + # Next, we load the configuration file of the problem with the line cfg = cashocs.load_config("config.ini") # Following this, we define the computational domain, using the built-in -# {py:func}`cashocs.regular_mesh` function, so that our hold-all domain is given by +# {py:func}`regular_mesh ` function, so that our hold-all domain +# is given by # {math}`\mathrm{D} = (0,2) \times (0,1)`. mesh, subdomains, boundaries, dx, ds, dS = cashocs.regular_mesh( @@ -179,9 +179,8 @@ # In the following, we define two python functions which return Hooke's tensor and # the symmetrized gradient -# + - +# + def eps(u): return Constant(0.5) * (grad(u) + grad(u).T) @@ -191,7 +190,7 @@ def sigma(u): # - -# + # For the load applied to the system we use a unitary point load at (2, 0.5), i.e., in # the middle of the outer rightmost boundary. To do so, a Dirac-Delta function can be # defined via a FEniCS `UserExpression` as follows. @@ -247,7 +246,6 @@ def value_shape(self): # derivative of the problem, which has been derived above. This is done with the lines # + - kappa = (lambd + 3.0 * mu) / (lambd + mu) r_in = alpha_out / alpha_in r_out = alpha_in / alpha_out @@ -272,11 +270,12 @@ def value_shape(self): ) + Constant(gamma) # - -# + # As in {ref}`demo_poisson_clover`, we now only have to specify what needs to happen # when the level-set function is updated, i.e., when the geometry changes. Of course, # the jumping coefficient {math}`\alpha` needs to be updated with the -# {py:func}`cashocs.interpolate_levelset_function_to_cells` function, but so does the +# {py:func}`interpolate_levelset_function_to_cells +# ` function, but so does the # indicator function of the geometry. This is specified in the following function. diff --git a/demos/documented/topology_optimization/pipe_bend/demo_pipe_bend.py b/demos/documented/topology_optimization/pipe_bend/demo_pipe_bend.py index f65351de..9f8693b3 100644 --- a/demos/documented/topology_optimization/pipe_bend/demo_pipe_bend.py +++ b/demos/documented/topology_optimization/pipe_bend/demo_pipe_bend.py @@ -90,16 +90,15 @@ # ### Initialization and Setup # # We start by importing cashocs and FEniCS into our script -# -# + +# + from fenics import * import numpy as np import cashocs # - -# + # Next, we load the configuration file for the problem and define the mesh with the # lines @@ -176,9 +175,8 @@ # with the following expressions.Additionally, we specify the pressure at the point # {math}`x^* = (0,0)` to obtain uniqueness of the pressure. # Altogether, the boundary conditions are specified using the code -# -# + +# + v_max = 1e0 v_in = Expression( ("(x[1] >= 0.7 && x[1] <= 0.9) ? v_max*(1 - 100*pow(x[1] - 0.8, 2)) : 0.0", "0.0"), @@ -200,9 +198,8 @@ def pressure_point(x, on_boundary): bcs += cashocs.create_dirichlet_bcs(V.sub(0), v_out, boundaries, 3) bcs += cashocs.create_dirichlet_bcs(V.sub(0), Constant((0.0, 0.0)), boundaries, [2, 4]) bcs += [DirichletBC(V.sub(1), Constant(0), pressure_point, method="pointwise")] - # - -# + # ### Cost Functional and Topological Derivative # # Now, we define the cost functional of our problem as well as its corresponding @@ -226,6 +223,18 @@ def pressure_point(x, on_boundary): # `update_level_set` function, which is defined below, the variable `vol` holds the # correct value in each iteration. # ::: +# +# ::::{note} +# As in {ref}`demo_poisson_clover`, the generalized topological derivative for this +# problem is identical in {math}`\Omega` and {math}`\Omega^c`, which is usually not the +# case. For this reason, the special structure of the problem can be exploited with the +# following lines in the configuration file +# ```{code-block} ini +# :caption: config.ini +# [TopologyOptimization] +# topological_derivative_is_identical = True +# ``` +# :::: # As in the previous demos, we have to specify the update routine of the level-set # function `update_level_set`, which we do as follows: @@ -238,7 +247,8 @@ def update_level_set(): # That is, in the `update_level_set` function, first the jumping coefficient is updated -# with the {py:func}`cashocs.interpolate_levelset_function_to_cells` function. +# with the {py:func}`interpolate_levelset_function_to_cells +# ` function. # Then, the indicator function is updated and used to compute the current volume of the # domain, which is written to the variable `vol`. # @@ -261,9 +271,8 @@ def update_level_set(): # ::: # # We visualize the result with the code -# -# + +# + import matplotlib.pyplot as plt top.plot_shape() @@ -271,6 +280,14 @@ def update_level_set(): plt.tight_layout() # plt.savefig("./img_pipe_bend.png", dpi=150, bbox_inches="tight") # - -# + # and the results looks as follows # ![](/../../demos/documented/topology_optimization/pipe_bend/img_pipe_bend.png) +# +# :::{note} +# Note that this design is not final due to the following: First, the tolerance for +# the optimization algorithm is chosen too large, as explained previously. Second, +# the chosen mesh is rather coarse and, thus, the discretization of the shape is rather +# coarse, too. These problems can be overcome by using a finer discretization and a +# lower tolerance. +# ::: diff --git a/demos/documented/topology_optimization/poisson_clover/demo_poisson_clover.py b/demos/documented/topology_optimization/poisson_clover/demo_poisson_clover.py index 9b582788..2923f6a7 100644 --- a/demos/documented/topology_optimization/poisson_clover/demo_poisson_clover.py +++ b/demos/documented/topology_optimization/poisson_clover/demo_poisson_clover.py @@ -108,6 +108,13 @@ # $$ # where {math}`u` solves the state equation and {math}`p` solves the adjoint equation. # +# :::{attention} +# There is a typo in our paper [Blauth and Sturm - Quasi-Newton methods for topology +# optimization using a level-set method](https://doi.org/10.1007/s00158-023-03653-2), +# i.e., in the paper a minus sign is missing. The topological derivative as stated +# here is, in fact, correct. +# ::: +# # ## Implementation # # The complete python code can be found in the file {download}`demo_poisson_clover.py @@ -131,13 +138,14 @@ cashocs.set_log_level(cashocs.LogLevel.INFO) # As with the other problem types, the solution algorithms of cashocs can be adapted -# with the help of configuration files, which is loaded with the {py:func}`load_config < -# cashocs.load_config>` function +# with the help of configuration files, which is loaded with the +# {py:func}`load_config ` function cfg = cashocs.load_config("config.ini") # In the next step, we define the mesh used for discretization of the hold-all domain -# {math}`\mathrm{D}`. To do so, we use the in-built {py:func}`cashocs.regular_box_mesh` +# {math}`\mathrm{D}`. To do so, we use the built-in +# {py:func}`regular_box_mesh ` # function, which gives us a discretization of {math}`\mathrm{D} = (-2, 2)^2`. mesh, subdomains, boundaries, dx, ds, dS = cashocs.regular_box_mesh( @@ -242,9 +250,10 @@ def create_desired_state(): # # Note, that this level-set function is taken from the [NGSolve Tutorials](https://docu.ngsolve.org/latest/i-tutorials/unit-7-optimization/04_Topological_Derivative_Levelset.html>) # -# Then, the values of the jumping coefficients {math}`\alpha` and {math\`f` are +# Then, the values of the jumping coefficients {math}`\alpha` and {math}`f` are # computed for this level-set function, using the -# {py:func}`cashocs.interpolate_levelset_function_to_cells`, which sets the piecewise +# {py:func}`interpolate_levelset_function_to_cells +# `, which sets the piecewise # constant functions `alpha` and `f` to the appropriate values according to the # level-set function: # :::{code-block} python @@ -291,13 +300,6 @@ def create_desired_state(): dJ_in = Constant(alpha_in - alpha_out) * y * p - Constant(f_in - f_out) * p dJ_out = Constant(alpha_in - alpha_out) * y * p - Constant(f_in - f_out) * p -# :::{attention} -# There is a typo in our paper [Blauth and Sturm - Quasi-Newton methods for topology -# optimization using a level-set method](https://doi.org/10.1007/s00158-023-03653-2), -# i.e., in the paper a minus sign is missing. The topological derivative as stated -# here is, in fact, correct. -# ::: - # ::::{note} # We remark that the generalized topological derivative for this problem is identical # in {math}`\Omega` and {math}`\Omega^c`, which is usually not the case. For this @@ -314,7 +316,8 @@ def create_desired_state(): # function is updated. Of course, a change in the level-set function changes the # geometry, so that the jumping coefficients {math}`\alpha` and {math}`f` have to be # updated. This is, as before, done with the function -# {py:func}`cashocs.interpolate_levelset_function_to_cells`, which is called twice, once +# {py:func}`interpolate_levelset_function_to_cells +# `, which is called twice, once # for updating `alpha` and once for `f`. @@ -327,7 +330,8 @@ def update_level_set(): # # Now, defining a topology optimization problem is nearly as easy as defining a shape # optimization or optimal control problem, namely we instantiate a -# {py:class}`cashocs.TopologyOptimizationProblem` and then can call it's +# {py:class}`TopologyOptimizationProblem ` and +# then can call it's # {py:meth}`solve ` method. top = cashocs.TopologyOptimizationProblem(