From 54014b06157a41188903f24f26e4d4abf777ef3a Mon Sep 17 00:00:00 2001 From: Justin O'Connor Date: Wed, 19 May 2021 15:52:53 -0600 Subject: [PATCH] step-79: a topology optimization problem. --- doc/doxygen/references.bib | 64 + doc/doxygen/tutorial/tutorial.h.in | 6 + examples/step-79/CMakeLists.txt | 39 + examples/step-79/doc/builds-on | 1 + examples/step-79/doc/intro.dox | 610 +++++++ examples/step-79/doc/kind | 1 + examples/step-79/doc/results.dox | 79 + examples/step-79/doc/tooltip | 1 + examples/step-79/step-79.cc | 2479 ++++++++++++++++++++++++++++ 9 files changed, 3280 insertions(+) create mode 100644 examples/step-79/CMakeLists.txt create mode 100644 examples/step-79/doc/builds-on create mode 100644 examples/step-79/doc/intro.dox create mode 100644 examples/step-79/doc/kind create mode 100644 examples/step-79/doc/results.dox create mode 100644 examples/step-79/doc/tooltip create mode 100644 examples/step-79/step-79.cc diff --git a/doc/doxygen/references.bib b/doc/doxygen/references.bib index fef5c412f1..30c44359bd 100644 --- a/doc/doxygen/references.bib +++ b/doc/doxygen/references.bib @@ -1173,6 +1173,70 @@ eprint = {http://dx.doi.org/10.1137/0917003} } +% ------------------------------------ +% Step 79 +% ------------------------------------ + +@book{Bendse2004, + doi = {10.1007/978-3-662-05086-6}, + url = {https://doi.org/10.1007/978-3-662-05086-6}, + year = {2004}, + publisher = {Springer Berlin Heidelberg}, + author = {Martin P. Bends{\o}e and Ole Sigmund}, + title = {Topology Optimization} +} + +@book{Nocedal2006, + doi = {10.1007/978-0-387-40065-5}, + url = {https://doi.org/10.1007/978-0-387-40065-5}, + year = {2006}, + publisher = {Springer New York}, + author = {Jorge Nocedal and Stephen Wright}, + title = {Numerical Optimization} +} + +@article{Waechter2005, + doi = {10.1007/s10107-004-0559-y}, + url = {https://doi.org/10.1007/s10107-004-0559-y}, + year = {2005}, + month = apr, + publisher = {Springer Science and Business Media {LLC}}, + volume = {106}, + number = {1}, + pages = {25--57}, + author = {Andreas W\"{a}chter and Lorenz T. Biegler}, + title = {On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming}, + journal = {Mathematical Programming} +} + +@article{Nocedal2009, + doi = {10.1137/060649513}, + url = {https://doi.org/10.1137/060649513}, + year = {2009}, + month = jan, + publisher = {Society for Industrial {\&} Applied Mathematics ({SIAM})}, + volume = {19}, + number = {4}, + pages = {1674--1693}, + author = {Jorge Nocedal and Andreas W\"{a}chter and Richard A. Waltz}, + title = {Adaptive Barrier Update Strategies for Nonlinear Interior Methods}, + journal = {{SIAM} Journal on Optimization} +} + +@article{Benson2002, + doi = {10.1023/a:1020533003783}, + url = {https://doi.org/10.1023/a:1020533003783}, + year = {2002}, + publisher = {Springer Science and Business Media {LLC}}, + volume = {23}, + number = {2}, + pages = {257--272}, + author = {Hande Y. Benson and Robert J. Vanderbei and David F. Shanno}, + journal = {Computational Optimization and Applications}, + title = {Interior-Point Methods for Nonconvex Nonlinear Programs: Filter Methods and Merit Functions} +} + + % ------------------------------------ % References used elsewhere % ------------------------------------ diff --git a/doc/doxygen/tutorial/tutorial.h.in b/doc/doxygen/tutorial/tutorial.h.in index 16bba66822..4b3f5689d5 100644 --- a/doc/doxygen/tutorial/tutorial.h.in +++ b/doc/doxygen/tutorial/tutorial.h.in @@ -653,6 +653,12 @@ * Solves the Black-Scholes equation for options pricing in 1-D. * * + * + * step-79 + * A topology optimization program for elastic media using the solid + * isotropic material penalization (SIMP) formulation. + * + * * * * diff --git a/examples/step-79/CMakeLists.txt b/examples/step-79/CMakeLists.txt new file mode 100644 index 0000000000..529b717745 --- /dev/null +++ b/examples/step-79/CMakeLists.txt @@ -0,0 +1,39 @@ +## +# CMake script for the step-79 tutorial program: +## + +# Set the name of the project and target: +SET(TARGET "step-79") + +# Declare all source files the target consists of. Here, this is only +# the one step-X.cc file, but as you expand your project you may wish +# to add other source files as well. If your project becomes much larger, +# you may want to either replace the following statement by something like +# FILE(GLOB_RECURSE TARGET_SRC "source/*.cc") +# FILE(GLOB_RECURSE TARGET_INC "include/*.h") +# SET(TARGET_SRC ${TARGET_SRC} ${TARGET_INC}) +# or switch altogether to the large project CMakeLists.txt file discussed +# in the "CMake in user projects" page accessible from the "User info" +# page of the documentation. +SET(TARGET_SRC + ${TARGET}.cc + ) + +# Usually, you will not need to modify anything beyond this point... + +CMAKE_MINIMUM_REQUIRED(VERSION 3.1.0) + +FIND_PACKAGE(deal.II 9.3.0 + HINTS ${deal.II_DIR} ${DEAL_II_DIR} ../ ../../ $ENV{DEAL_II_DIR} + ) +IF(NOT ${deal.II_FOUND}) + MESSAGE(FATAL_ERROR "\n" + "*** Could not locate a (sufficiently recent) version of deal.II. ***\n\n" + "You may want to either pass a flag -DDEAL_II_DIR=/path/to/deal.II to cmake\n" + "or set an environment variable \"DEAL_II_DIR\" that contains this path." + ) +ENDIF() + +DEAL_II_INITIALIZE_CACHED_VARIABLES() +PROJECT(${TARGET}) +DEAL_II_INVOKE_AUTOPILOT() diff --git a/examples/step-79/doc/builds-on b/examples/step-79/doc/builds-on new file mode 100644 index 0000000000..92c4fce962 --- /dev/null +++ b/examples/step-79/doc/builds-on @@ -0,0 +1 @@ +step-8 step-15 diff --git a/examples/step-79/doc/intro.dox b/examples/step-79/doc/intro.dox new file mode 100644 index 0000000000..df186bfbe0 --- /dev/null +++ b/examples/step-79/doc/intro.dox @@ -0,0 +1,610 @@ + +

Introduction

+ +Topology Optimization of Elastic Media is a technique used to optimize a +structure that is bearing some load. Ideally, we would like to minimize the +maximum stress placed on a structure by selecting a region $E$ where material is +placed. In other words, +@f[ + \text{Minimize}\| \boldsymbol{\sigma} (\mathbf{u}) \|_\infty +@f] +@f[ + \text{subject to } |E|\leq V_{\max}, +@f] +@f[ + \text{and } \nabla \cdot \boldsymbol{\sigma} + \mathbf{F} = \mathbf{0}. +@f] + +Here, $\boldsymbol{\sigma} = \mathbf{C} : \boldsymbol{\varepsilon}(\mathbf{u})$ is the stress +within the body that is caused by the external forces $\mathbf F$, where we have for simplicity assumed +that the material is linear-elastic and so $\mathbf{C}$ is the stress-strain tensor and +$\boldsymbol{\varepsilon}(\mathbf{u})=\frac{1}{2} (\nabla \mathbf{u} + (\nabla\mathbf{u})^T)$ is the +small-deformation strain as a function of the displacement $\mathbf{u}$ -- see +step-8 and step-17 for more on linear elasticity. In the formulation above, +$V_\text{max}$ is the maximal amount of material we are willing to provide to +build the object. The last of the constraints is the partial differential +equation that relates stress $\boldsymbol{\sigma}$ and forces $\mathbf F$ and is simply the +steady-state force balance. + +That said, the infinity norm above creates a problem: As a function of location +of material, this objective function is necessarily not differentiable, making +prospects of optimization rather bleak. So instead, a common approach in +topology optimization is to find an approximate solution by optimizing a related +problem: We would like to minimize the strain energy. This is a +measure of the potential energy stored in an object due to its deformation, but +also works as a measure of total deformation over the structure. + +@f[ + \text{Minimize } \int_E \frac{1}{2}\boldsymbol{\sigma} : \boldsymbol{\varepsilon} dV +@f] +@f[ + \text{subject to } \|E\| \leq V_{\max} +@f] +@f[ + \text{and } \nabla \cdot \boldsymbol{\sigma} + \mathbf{F} = \mathbf{0} +@f] + +The value of the objective function is calculated using a finite element method, +where the solution is the displacements. This is placed inside of a nonlinear +solver loop that solves for a vector denoting placement of material. + +

Solid Isotropic Material with Penalization

+ +In actual practice, we can only build objects in which the material is either +present, or not present, at any given point -- i.e., we would have an indicator +function $\rho_E(\mathbf{x})\in \{0,1\}$ that describes the material-filled +region and that we want to find through the optimization problem. In this case, +the optimization problem becomes combinatorial, and very expensive to solve. +Instead, we use an approach called Solid Isotropic Material with Penalization, +or SIMP. @cite Bendse2004 + +The SIMP method is based on an idea of allowing the material to exist in a +location with a density $\rho$ between 0 and 1. A density of 0 suggests the +material is not there, and it is not a part of the structure, while a density of +1 suggests the material is present. Values between 0 and 1 do not reflect a +design we can create in the real-world, but allow us to turn the combinatorial +problem into a continuous one. One then looks at density values $\rho$, +with the constraint that $0 < \rho_{\min} \leq \rho \leq 1$. The minimum value +$\rho_{\min}$, typically chosen to be around $10^{-3}$, avoids the possibility +of having an infinite strain energy, but is small enough to provide accurate +results. + +The straightforward application of the effect of this "density" on the +elasticity of the media would be to simply multiply the stiffness tensor $\mathbf{C}_0$ +of the medium by the given density, that is, $\mathbf{C} = \rho \mathbf{C}_0$. However, this +approach often gives optimal solutions where density values are far from both 0 +and 1. As one wants to find a real-world solution, meaning the material either +is present or it is not, a penalty is applied to these in-between values. A +simple and effective way to do this is to multiply the stiffness tensor by the +density raised to some integer power penalty parameter $p$, so that +$\mathbf{C} = \rho^p \mathbf{C}_0$. This makes density values farther away from 0 or 1 less +effective. It has been shown that using $p=3$ is sufficiently high to create +'black-and-white' solutions: that is, one gets optimal solutions in which +material is either present or not present at all points. + +More material should always provide a structure with a lower strain energy, and so the +inequality constraint can be viewed as an equality where the total volume used +is the maximum volume. + +Using this density idea also allows us to reframe the volume constraint on the +optimization problem. Use of SIMP then turns the optimization problem into the +following: + +@f[ + \text{Minimize } \int_\Omega \frac{1}{2}\boldsymbol{\sigma}(\rho) : \boldsymbol{\varepsilon}(\rho) d_\Omega +@f] +@f[ + \text{subject to } \int_\Omega \rho(x) d_\Omega= V_{\max}, +@f] +@f[ + 0<\rho_{\min}\leq \rho(x) \leq 1, +@f] +@f[ + + \nabla \cdot \boldsymbol{\sigma}(\rho) + \mathbf{F} = 0 \quad \text{on } \Omega +@f] +The final constraint, the balance of linear momentum (sometimes referred to as the elasticity equation), + gives a method for finding $\boldsymbol{\sigma}$ and $\boldsymbol{\varepsilon}$ given the density $\rho$. + +

Elasticity Equation

+The elasticity equation in the time independent limit reads +@f[ + \nabla \cdot \boldsymbol{\sigma} + \mathbf{F} = \mathbf{0} . +@f] +In the situations we will care about, we will assume that the medium has a linear material response +and in that case, we have that +@f[ + \boldsymbol{\sigma} = \mathbf{C} : \boldsymbol{\varepsilon} = \rho^p \mathbf{C}_0 : \boldsymbol{\varepsilon}(\mathbf{u}) + = \rho^p \mathbf{C}_0 : \left[\frac{1}{2} (\nabla \mathbf{u} + (\nabla \mathbf{u})^T) \right] . +@f] +In everything we will do below, we will always consider the displacement +field $\mathbf{u}$ as the only solution variable, rather than considering +$\mathbf{u}$ and $\boldsymbol{\sigma}$ as solution variables (as is done in mixed +formulations). + +Furthermore, we will make the assumption that the material is linear isotropic, +in which case the stress-strain tensor can be expressed in terms of the Lam\'{e} +parameters $\lambda,\mu$ such that +@f{align} + \boldsymbol{\sigma} &= \rho^p (\lambda \text{tr}(\boldsymbol{\varepsilon}) \mathbf{I} + 2 \mu \boldsymbol{\varepsilon}) , \\ + \sigma_{i,j} &= \rho^p (\lambda \varepsilon_{k,k} \delta_{i,j} + 2 \mu \varepsilon_{i,j}) . +@f} +See step-8 for how this transformation works. + +Integrating the objective function by parts gives +@f[ + \int_\Omega \boldsymbol{\sigma}(\rho) : (\nabla \mathbf{u} + (\nabla \mathbf{u}))^T d\Omega+ + \int_\Omega (\nabla \cdot \boldsymbol{\sigma}(\rho)) \cdot \mathbf{u} d\Omega= + \int_{\partial \Omega} \mathbf{t} \cdot \mathbf{u} d\partial\Omega , +@f] +into which the linear elasticity equation can then be substituted, giving +@f[ + \int_\Omega \boldsymbol{\sigma}(\rho) : (\nabla \mathbf{u} + (\nabla \mathbf{u})^T) d\Omega = + \int_\Omega \mathbf{F}\cdot \mathbf{u} d\Omega+ + \int_{\partial \Omega} \mathbf{t} \cdot \mathbf{u} d\partial\Omega . +@f] +Because we are assuming no body forces, this simplifies further to +@f[ + \int_\Omega \boldsymbol{\sigma}(\rho) : (\nabla \mathbf{u} + (\nabla \mathbf{u})^T) d\Omega + = \int_{\partial \Omega} \mathbf{t} \cdot \mathbf{u} d\partial\Omega, +@f] +which is the final form of the governing equation that we'll be considering +from this point forward. + +

Making the solution mesh-independent

+ +Typically, the solutions to topology optimization problems are +mesh-dependent, and as such the problem is ill-posed. This is because +fractal structures are often formed as the mesh is refined further. As the mesh gains +resolution, the optimal solution typically gains smaller and smaller structures. +There are a few competing workarounds to this issue, but the most popular for +first order optimization is the sensitivity filter, while second order +optimization methods tend to prefer use of a density filter. + +As the filters affect the gradient and Hessian of the strain energy (i.e., the +objective function), the choice of filter has an effect on the solution of the +problem. The density filter as part of a second order method works by +introducing an unfiltered density, which we refer to as $\varrho$, and then +requiring that the density be a convolution of the unfiltered density: +@f[ + \rho = H(\varrho). +@f] +Here, $H$ is an operator so that $\rho(\mathbf{x})$ is some kind of average of +the values of $\varrho$ in the area around $\mathbf{x}$ -- i.e., it is a smoothed +version of $\varrho$. + +This prevents checkerboarding; the radius of the filter allows the user to +define an effective minimal beam width for the optimal structures we seek to +find. + +
+ Checkerboarding occurring in an MBB Beam +
+ +

Complete Problem Formulation

+ +The minimization problem is now +@f[ + \min_{\rho,\varrho,\mathbf{u}} \int_{\partial\Omega} \mathbf{u} \cdot \mathbf{t} d\partial\Omega +@f] +@f[ + \text{subject to } \rho = H(\varrho) +@f] +@f[ + \int_\Omega \rho^p \left(\frac{\mu}{2}\left(\boldsymbol{\varepsilon}(\mathbf{v}): + \boldsymbol{\varepsilon}(\mathbf{u})) \right) + \lambda \left( \nabla \cdot \mathbf{u} \nabla + \cdot \mathbf{v} \right) \right) d\Omega = \int_{\partial \Omega} \mathbf{v} \cdot + \mathbf{t} d\partial\Omega +@f] +@f[ + \int_\Omega \rho d\Omega= V +@f] +@f[ + 0\leq \varrho \leq 1 +@f] + +The inequality constraints are dealt with by first introducing slack variables, +and second using log barriers to ensure that we obtain an interior-point +method. The penalty parameter is going to be $\alpha$, and the following slack +variables are +
    +
  1. $s_1$ - a slack variable corresponding to the lower bound
    2. +
  2. $s_2$ - a slack variable corresponding to the upper bound.
    3. +
+This now gives the following problem: +@f[ + \min_{\rho,\varrho,\mathbf{u}, s_1, s_2} \int_{\partial\Omega} \mathbf{u} \cdot + \mathbf{t} d\partial\Omega- \alpha \int_\Omega \left(\log(s_1) + \log(s_2)\right) d\Omega +@f] +@f[ + \text{subject to } \rho = H(\varrho) +@f] +@f[ + \int_\Omega \rho^p \left(\frac{\mu}{2}\left(\boldsymbol{\varepsilon}(\mathbf{v}): + \boldsymbol{\varepsilon}(\mathbf{u})) \right) + \lambda \left( \nabla \cdot \mathbf{u} \nabla + \cdot \mathbf{v} \right) \right) d\Omega = \int_{\partial \Omega} \mathbf{v} \cdot + \mathbf{t} d\partial\Omega +@f] +@f[ + \int_\Omega \rho d\Omega = V +@f] +@f[ + \varrho = s_1 +@f] +@f[ + 1-\varrho = s_2 +@f] + +With these variables in place, we can then follow the usual approach to solving +constrained optimization problems: We introduce a Lagrangian in which we combine +the objective function and the constraints by multiplying the constraints by +Lagrange multipliers. Specifically, we will use the following symbols for the +Lagrange multipliers for the various constraints: +
    +
  1. $\mathbf{y}_1 $ - a Lagrange multiplier corresponding to the + elasticity constraint
    2. +
  2. $y_2$ - a Lagrange multiplier corresponding to the convolution + filter constraint
    3. +
  3. $z_1$ - a Lagrange multiplier corresponding to the lower slack variable
    4. +
  4. $z_2$ - a Lagrange multiplier corresponding to the upper slack variable.
    5. +
+With these variables, the Lagrangian function reads as follows: + +@f{align}{ + \mathcal{L} =& \int_{\partial\Omega} \mathbf{u} \cdot \mathbf{t} d\partial\Omega + - \alpha \int_\Omega \left(\log(s_1) + \log(s_2)\right) d\Omega- \left(\int_\Omega + \rho^p \left(\frac{\mu}{2}\left(\boldsymbol{\varepsilon}(\mathbf{y}_1):\boldsymbol{\varepsilon}(\mathbf{u})) + \right) + \lambda \left( \nabla \cdot \mathbf{u} \nabla \cdot \mathbf{y}_1 + \right) d\Omega \right)- \int_{\partial \Omega} \mathbf{y}_1 \cdot \mathbf{t} d\partial\Omega \right) \\ + & -\int_\Omega y_2 (\rho - H(\varrho)) d\Omega - \int_\Omega z_1 (\varrho-s_1) d\Omega + - \int_\Omega z_2 (1 - s_2 -\varrho) d\Omega +@f} + +The solution of the optimization problem then needs to satisfy what are known as +the [Karush-Kuhn-Tucker (KKT) conditions](https://en.wikipedia.org/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions): +The derivatives of the Lagrangian with respect to all of its arguments need to be equal to zero, and because we have +inequality constraints, we also have "complementarity" conditions. Since we +here have an infinite-dimensional problem, these conditions all involve +directional derivatives of the Lagrangian with regard to certain test +functions -- in other words, all of these conditions have to be stated in weak +form as is typically the basis for finite element methods anyway. + +The barrier method allows us to initially weaken the "complementary slackness" +as required by the typical KKT conditions. Typically, we would require that +$s_i z_i = 0$, but the barrier formulations give KKT conditions where +$s_i z_i = \alpha$, where $\alpha$ is our barrier parameter. As part of the +barrier method, this parameter must be driven close to 0 to give a good +approximation of the original problem. + +In the following, let us state all of these conditions where +$d_{\{\bullet\}}$ is a test function that is naturally paired with variational +derivatives of the Lagrangian with respect to the $\{\bullet\}$ function. +For simplicity, we introduce $\Gamma$ to indicate the portion of the boundary +where forces are applied, and Neumann boundary conditions are used. + +
    +
  1. Stationarity: +@f[ + \int_\Omega - d_\rho y_2 + p\rho^{p-1}d_\rho \left[\lambda + (\nabla \cdot \mathbf{y}_1) (\nabla \cdot \mathbf{u}) + + \mu \boldsymbol{\varepsilon}(\mathbf{u}):\boldsymbol{\varepsilon}(\mathbf{y}_1)\right] d\Omega=0\;\; + \forall d_\rho +@f] +@f[ + \int_\Gamma \mathbf d_\mathbf{u} \cdot \mathbf{t} d\partial\Omega+ \int_\Omega p\rho^{p} + \left[\lambda (\nabla \cdot \mathbf d_\mathbf{u})( \nabla \cdot \mathbf{y}_1) + + \mu \boldsymbol{\varepsilon}(\mathbf d_\mathbf{u}):\boldsymbol{\varepsilon}(\mathbf{y}_1)\right] d\Omega=0\;\; + \forall \mathbf{d}_\mathbf{u} +@f] +@f[ + \int_\Omega -d_\varrho z_1 + d_\varrho z_2 + H(d_\varrho)y_2 d\Omega= 0\;\;\forall + d_\varrho + +@f] +
    2. +
  2. Primal Feasibility: +@f[ + \int_\Omega \rho^{p}\lambda (\nabla \cdot \mathbf d_{\mathbf{y}_1}) + (\nabla \cdot \mathbf{u}) + \rho^{p}\mu \boldsymbol{\varepsilon}(\mathbf + d_{\mathbf{y}_1}) : \boldsymbol{\varepsilon}(\mathbf{u}) d\Omega - \int_\Gamma \mathbf + d_{\mathbf{y}_1} \cdot \mathbf{t} d\partial\Omega =0 \;\;\forall \mathbf{d}_{\mathbf{y}_1} +@f] +@f[ + \int_\Omega d_{z_1}(\varrho - s_1) d\Omega = 0\;\;\forall d_{z_1} +@f] +@f[ + \int_\Omega d_{z_z}(1-\varrho-s_2) d\Omega = 0\;\;\forall d_{z_2} +@f] +@f[ + \int_\Omega d_{y_2}(\rho - H(\varrho)) d\Omega = 0\;\;\forall d_{y_2} +@f] +
    3. +
  3. Complementary Slackness: +@f[ + \int_\Omega d_{s_1}(s_1z_1 - \alpha) d\Omega = 0 \;\;\forall d_{s_1} ,\;\;\; + \alpha \to 0 +@f] +@f[ + \int_\Omega d_{s_2}(s_2z_2 - \alpha) d\Omega = 0 \;\;\forall d_{s_2} ,\;\;\; + \alpha \to 0 +@f] +
    4. +
  4. Dual Feasibility: +@f[ + s_{1,i},s_{2,i},z_{1,i},z_{2,i} \geq 0 \;\;\;\; \forall i +@f] +
    5. +
+ +

Solution procedure

+ +The optimality conditions above are, in addition to being convoluted, of a kind +that is not easy to solve: They are generally nonlinear, and some of the +relationships are also inequalities. We will address the nonlinearity using a +Newton method to compute search directions, and come back to how to deal with +the inequalities below when talking about step length procedures. + +Newton's method applied to the equations above results in the following system. +Here, variational derivatives with respect to the $\{\bullet\}$ variable are taken + in the $c_{\{\bullet\}}$ direction. This gives + +
    +
  1. Stationarity - these equations ensure we are at a critical point of the +objective function when constrained + +Equation 1 +@f{align}{ + &\int_\Omega-d_\rho c_{y_2} + p(p-1) \rho^{p-2} d_\rho c_\rho [\lambda \nabla + \cdot \mathbf{y}_1 \nabla \cdot \mathbf{u} + \mu \boldsymbol{\varepsilon}(\mathbf{u}) + \boldsymbol{\varepsilon}(\mathbf{y}_1)] + + p \rho^{p-1} d_\rho[\lambda \nabla \cdot + \mathbf{c}_{\mathbf{y}_1} \nabla \cdot \mathbf{u} + \mu \boldsymbol{\varepsilon} + (\mathbf{u}) \boldsymbol{\varepsilon}(\mathbf{c}_{\mathbf{y}_1})] + p \rho^{p-1} d_\rho + [\lambda \nabla \cdot {\mathbf{y}_1} \nabla \cdot \mathbf{c}_\mathbf{u} + + \mu \boldsymbol{\varepsilon}(\mathbf{c}_\mathbf{u}) \boldsymbol{\varepsilon}(\mathbf{y}_1)] d\Omega \\ + &= -\int_\Omega -d_\rho z_1 + d_\rho z_2 - d_\rho y_2 + p\rho^{p-1}d_\rho +[\lambda \nabla \cdot \mathbf{y}_1 \nabla \cdot \mathbf{u} + \mu \boldsymbol{\varepsilon} +(\mathbf{u})\boldsymbol{\varepsilon}(\mathbf{y}_1)] d\Omega +@f} + +Equation 2 +@f{align}{ + &\int_\Omega p \rho^{p-1} c_\rho [\lambda \nabla \cdot {\mathbf{y}_1} \nabla + \cdot \mathbf{d}_\mathbf{u} + \mu \boldsymbol{\varepsilon}(\mathbf{d}_\mathbf{u}) + \boldsymbol{\varepsilon}(\mathbf{y})] + \rho^{p} [\lambda \nabla \cdot + \mathbf{c}_{\mathbf{y}_1} \nabla \cdot \mathbf{d}_\mathbf{u} + \mu + \boldsymbol{\varepsilon}(\mathbf{d}_\mathbf{u})\boldsymbol{\varepsilon}(\mathbf{c}_{\mathbf{y}_1})] d\Omega \\ + &= -\int_\Gamma \mathbf{d}_\mathbf{u} \cdot \mathbf{t} -\int_\Omega \rho^{p} + [\lambda \nabla \cdot \mathbf{y} \nabla \cdot \mathbf{d}_\mathbf{u} + \mu + \boldsymbol{\varepsilon}(d_\mathbf{u})\boldsymbol{\varepsilon}(\mathbf{y}_1)] d\Omega +@f} + +Equation 3 +@f[ + \int_\Omega - d_\varrho c_{z_1} +d_\varrho c_{z_2} + H(d_\varrho)c_{y_2} d\Omega = + -\int_\Omega -d_\varrho z_1 + d_\varrho z_2 + H(d_\varrho)y_2 d\Omega +@f] +
    2. + +
  2. Primal Feasibility - these equations ensure the equality constraints +are met. + +Equation 4 +@f{align}{ + &\int_\Omega p \rho^{p-1} c_p[\lambda \nabla \cdot + \mathbf{d}_{\mathbf{y}_1} \nabla \cdot \mathbf{u} + \mu + \boldsymbol{\varepsilon}(\mathbf{u}) \boldsymbol{\varepsilon}(\mathbf{d}_{\mathbf{y}_1})] + + \rho^{p}[\lambda \nabla \cdot \mathbf{d}_{\mathbf{y}_1} \nabla \cdot + \mathbf{c}_\mathbf{u} + \mu \boldsymbol{\varepsilon}(\mathbf{c}_\mathbf{u}) + \boldsymbol{\varepsilon}(\mathbf{d}_{\mathbf{y}_1})] d\Omega \\ + &= -\int_\Omega \rho^{p}[\lambda \nabla \cdot \mathbf{d}_{\mathbf{y}_1} \nabla + \cdot \mathbf{u} + \mu \boldsymbol{\varepsilon}(\mathbf{u}) \boldsymbol{\varepsilon} + (\mathbf{d}_{\mathbf{y}_1})] + \int_\Gamma \mathbf{d}_{\mathbf{y}_1} + \cdot \mathbf{t} d\partial\Omega +@f} + +Equation 5 +@f[ + -\int_\Omega d_{z_1}(c_\varrho - c_{s_1}) d\Omega=\int_\Omega d_{z_1} (\varrho - s_1) d\Omega +@f] + +Equation 6 +@f[ + -\int_\Omega d_{z_2}(-c_\varrho-c_{s_2}) d\Omega= \int_\Omega d_{z_2} (1-\varrho-s_2) d\Omega +@f] + +Equation 7 +@f[ + -\int_\Omega d_{y_2}(c_\rho - H(c_\varrho)) d\Omega=\int_\Omega d_{y_2} + (\rho - H(\varrho)) d\Omega +@f] +
    3. + +
  3. Complementary Slackness - these equations essentially ensure the barrier +is met - in the final solution, we need $s^T z = 0$ + +Equation 8 +@f[ + \int_\Omega d_{s_1}(c_{s_1}z_1/s_1 + c_{z_1} ) d\Omega=-\int_\Omega d_{s_1} + (z_1 - \alpha/s_1) d\Omega ,\;\;\; \alpha \to 0 +@f] + +Equation 9 +@f[ + \int_\Omega d_{s_2} (c_{s_2}z_2/s_2 + c_{z_2} ) d\Omega=-\int_\Omega d_{s_2} + (z_2 - \alpha/s_2) d\Omega,\;\;\; \alpha \to 0 +@f] +
    4. + +
  4. Dual Feasibility - Multiplier on slacks and slack variables must be kept +greater than 0. (This is the only part not implemented in the +SANDTopOpt::assemble_system() function) +@f[ + s,z \geq 0 +@f] +
    5. +
+ + + +

Discretization

+We use a quadrilateral mesh with $Q_1$ elements to discretize the displacement and +displacement Lagrange multiplier. Piecewise constant $DGQ_0$ elements are used +to discretize the density, unfiltered density, density slack variables, and +multipliers for the slack variables and filter constraint. + +

Nonlinear Algorithm

+ +While most of the discussion above follows traditional and well-known approaches +to solving nonlinear optimization problems, it turns out that the problem is +actually quite difficult to solve in practice. In particular, it is quite +nonlinear and an important question is not just to find search directions +$c_{\{\bullet\}}$ as discussed above based on a Newton method, but that one needs to +spend quite a lot of attention to how far one wants to go in this direction. +This is often called "line search" and comes down to the question of how to +choose the step length $\alpha_k \in (0,1]$ so that we move from the current +iterate $\mathbf{x}_k$ to the next iterate $\mathbf{x}_{k+1}=\mathbf{x}_k+\alpha_k \mathbf{x}_k$ +in as efficient a way as possible. It is well understood that we need to eventually choose +$\alpha_k=1$ to realize the Newton's method's quadratic convergence; however, +in the early iterations, taking such a long step might actually make things +worse, either by leading to a point that has a worse objective function or at +which the constraints are satisfied less well than they are at $\mathbf{x}_k$. + +Very complex algorithms have been proposed to deal with this issue +@cite Nocedal2009 @cite Waechter2005. Here, we implement a watchdog-search +algorithm @cite Nocedal2006. When discussing this algorithm, we will use the +vector $\mathbf{x}$ to represent all primal variables - the filtered and +unfiltered densities, slack variables and displacement - and use the vector +$\mathbf{y}$ to represent all of the dual vectors. The (incremental) solution to the nonlinear +system of equations stated above will now be referred to as $\Delta \mathbf{x}$ and $\Delta +\mathbf{y}$ instead of $c_{\{\bullet\}}$. A merit function (explained in more detail later) +is referred to here as $\phi(\mathbf{x,\mathbf{y}})$. + +The watchdog algorithm applied to a subproblem with a given barrier parameter +works in the following way: First, the current iteration is saved as a +"watchdog" state, and the merit of the watchdog state is recorded. +A maximal feasible Newton step is then taken. If the merit sufficiently +decreased from the first step, this new step is accepted. If not, another +maximal feasible Newton step is taken, and the merit is again compared to the +watchdog merit. +If after some number (typically between 5 and 8) of Newton steps, the merit did +not adequately decrease, the algorithm takes a scaled Newton step from either +the watchdog state or the last iteration that guarantees +a sufficient decrease of the merit, and that step is accepted. Once a step is +accepted, the norm of the KKT error is measured, and if it is sufficiently +small, the barrier value is decreased. If it is not sufficiently small, the +last accepted step is taken to be the new watchdog step, and the process is +repeated. + + +Above, the "maximal feasible step" is a scaling of the Newton step in both the +primal and dual variables given by + +@f[ + \beta^\mathbf{y} = \min\{1,\max \beta \text{ such that }\left(\mathbf{z}_{k+i} + + \beta^\mathbf{z}_{k+i} \Delta \mathbf{z}_{k+i}\right)_j \geq \zeta + \mathbf{z}_{k+i,j} \forall j\} +@f] +@f[ + \beta^\mathbf{x} = \min\{1,\max \beta \text{ such that }\left(\mathbf{s}_{k+i} + + \beta^\mathbf{s}_{k+i} \Delta \mathbf{s}_{k+i}\right)_j \geq \zeta + \mathbf{s}_{k+i,j} \forall j\} +@f] + +Above, $\zeta$ is the "fraction to boundary" that is allowed on any step. +Because the derivatives become ill-conditioned near the boundary, this technique +stands in for a [trust region](https://en.wikipedia.org/wiki/Trust_region) and is +necessary to ensure good approximations in +the future. $\zeta$ is taken to be $\max\{0.8, 1-\alpha\}$, which allows +movement closer to the boundary as the barrier becomes smaller. In the future, +when implementing the LOQO algorithm for barrier reduction, this must be kept +to 0.8 as the barrier parameter can vary wildly. + +Separately, we need to deal with the log-barrier that we have used to enforce +the positivity constraint on the slack variables $s_1,s_2$: In the statement of +the final optimization problem we solve, we have added the term +@f[ + -\alpha \int_\Omega (\log(s_1) + \log(s_2)) d\Omega. +@f] +The question is how we should choose the penalty factor $\alpha$. As with all +penalty methods, we are in reality only interested in the limit as +$\alpha\to 0$, since this is then the problem we really wanted to solve, +subject to the positivity constraints on the slack variables. On the other hand, +we need to choose $\alpha$ large enough to make the problem solvable in +practice. Actual implementations therefore start with a larger value of +$\alpha$ and gradually decrease it as the outer iterations proceed. + +In the monotone method implemented here, the barrier parameter is updated +whenever some level of convergence is achieved at the current barrier parameter. +We use the $l_\infty$ norm of the KKT conditions to check for convergence at +each barrier size. The requirement is that +$\|KKT\|_{l_\infty} < c \cdot \alpha$ where $c$ is a constant over any +barrier size and $\alpha$ is the barrier parameter. This forces better +convergence in later iterations, and is the same requirement as is used in +[IPOPT](https://coin-or.github.io/Ipopt/) (an open source software package for +large-scale nonlinear optimization). + +Here, the barrier is reduced linearly at larger values, and superlinearly at +smaller values. At larger values, it is multiplied by a constant (around 0.6), +and at lower values the barrier value is replaced by the barrier value raised +to some exponent (around 1.2). This method has proven to be effective at keeping + the subproblem solvable at large barrier values, while still allowing + superlinear convergence at smaller barrier values. In practice, this looks like + the following: +@f[ + \alpha_{k+1} = \min\{\alpha_k^{1.2},0.6\alpha_k\} +@f] + +While taking large steps at reducing the barrier size when convergence is +reached is widely used, more recent research has shown that it is typically faster +to use algorithms that adaptively update barrier each iteration, i.e., methods in which +we use concrete criteria at the end of each iteration to determine what the +penalty parameter should be in the next iteration, rather than using reduction +factors that are independent of the current solution. That said, such methods +are also more complicated and we will not do this here. + +

Merit Function

+ +The algorithm outlined above makes use of a "merit function". Merit functions +are used to determine whether a step from $x_k$ to a proposed point $x_{k+1}$ is +beneficial. In unconstrained optimization problems, one can simply check this +with the objective function we try to minimize, and typically uses conditions such +as the [Wolfe and Goldstein conditions](https://en.wikipedia.org/wiki/Wolfe_conditions). + +In constrained optimization problems, the question is how to balance reduction +in the objective function against a possible increase in the violation of +constraints: A proposed step might make the objective function smaller but be +further away from the set of points that satisfy the constraints -- or the other +way around. This trade-off is typically resolved by using a merit function that +combines the two criteria. + +Here, we use an exact $l_1$ merit function to test the steps: +@f{align}{ + \phi(\mathbf{x},\mathbf{y}) =& \int_{\partial \Omega} \mathbf{u}\cdot + \mathbf{t} d\partial\Omega- \alpha \int_\Omega (\log(s_1) + \log(s_2)) + p \sum_i\left| + \int_\Omega y_{2,i}(H(\varrho) - \rho) d\Omega \right| \\ + & + p \sum_i\left| \int_{\partial \Omega} \mathbf{y}_{1,i}\cdot \mathbf{t} d\partial\Omega + - \int_\Omega \rho^p[\lambda \nabla \cdot \mathbf{u} \nabla \cdot \mathbf{y}_{1,i} + + \mu \boldsymbol{\varepsilon}{\mathbf{u}}\boldsymbol{\varepsilon}{\mathbf{y}_{1,i}}] d\Omega \right| + + p \sum_i\left| \int_\Omega z_{1,i}(s_1 - \varrho) d\Omega\right| + + p \sum_i\left| \int_\Omega z_{2,i}(1-\varrho - s_2) d\Omega\right| +@f} + +Here, $p$ is a penalty parameter. This merit function being exact means that +there exists some $p_0$ so that for any $p > p_0$, the merit function has its +minima at the same location as the original problem. This penalty parameter is +updated (by recommendation of Nocedal and Wright @cite Benson2002) as follows: +@f[ + p > \frac{\frac{1}{2} \mathbf{x}^T \cdot \mathbf{H} \cdot \mathbf{x} - \mathbf{x}^T \cdot \nabla f}{\|c_i\|_{l_\infty}, i \in \mathcal{E}} , +@f] +where $\mathbf{H}$ is the Hessian of the objective function, $\mathbf{x}$ is a vector of our +decision (primal) variables, $f$ is the objective function, and $c_i$ is the error on a +current equality constraint. + +Our use of this method is partially due to already having most of the necessary +parts calculated in finding the right hand side, but also the use of an exact +merit function ensures that it is minimized in the same location as the overall +problem. Recent research has shown that one can replace merit functions by what +are called "filter methods", and one should consider using these instead as they +prove to be more efficient. diff --git a/examples/step-79/doc/kind b/examples/step-79/doc/kind new file mode 100644 index 0000000000..56e049c91a --- /dev/null +++ b/examples/step-79/doc/kind @@ -0,0 +1 @@ +solids diff --git a/examples/step-79/doc/results.dox b/examples/step-79/doc/results.dox new file mode 100644 index 0000000000..1c023ed7c1 --- /dev/null +++ b/examples/step-79/doc/results.dox @@ -0,0 +1,79 @@ +

Results

+

Test Problem

+The algorithms used above are tested against a traditional topology optimization + problem called the Messerschmitt-Bolkow-Blohm Beam (MBB Beam). + +This problem considers the optimal 2-d structure that can be built on a +rectangle 6 units wide, and 1 unit tall. The bottom corners are fixed in place +in the $y$ direction using a zero Dirichlet boundary condition, and a downward +force is applied in the center of the top of the beam by enforcing a Neumann +boundary condition. The rest of the boundary is allowed to move, and has no +external force applied, which takes the form of a zero Neumann boundary +condition. In essence, we are asking the following question: How should we +design a bridge in a way so that if the bottom left and bottom right point of +the bridge are on rollers that allow these points to move horizontally but not +vertically, and so that the displacement in response to the vertical force in +the center is minimal. + +While the total volume of the domain is 6 units, 3 units of material are allowed for +the structure. Because of the symmetry of the problem, it could be posed on a +rectangle of width 3 and height 1 by cutting the original domain in half, and +using zero Dirichlet boundary conditions in the $x$ direction along the cut +edge. That said, symmetry of the solution is a good indicator that the program +is working as expected, so we solved the problem on the whole domain, +as shown below. @cite Bendse2004 + +
+ The MBB problem domain and boundary conditions +
+ + +Using the program discussed above, we find the minimum volume of the MBB Beam and the +individual components of the solution look as follows: + +
+
+ Filtered Density Solution +
+
+ Unfiltered Density Solution +
+
+ + +These pictures show that what we find here is in accordance with what one +typically sees in other publications on the topic @cite Bendse2004. Maybe more interestingly, the +result looks like a truss bridge (except that we apply the load at the top of +the trusses, rather than the bottom as in real truss bridges, akin to a "deck +truss" bridge), suggesting that the designs that have been used in bridge- +building for centuries are indeed based on ideas we can now show to be optimal +in some sense. + + +

Possibilities for extensions

+ +The results shown above took around 75 iterations to find, which is quite +concerning given the expense in solving the large linear systems in each +iteration. Looking at the evolution, it does look as though the convergence has +moments of happening quickly and moments of happening slowly. We believe this to +be due to both a lack of precision on when and how to decrease the boundary +values, as well as our choice of merit function being sub-optimal. In the future, +a LOQO barrier update replacing the monotone reduction, as well as a Markov +Filter in place of a merit function will decrease the number of necessary +iterations significantly. + +The barrier decrease is most sensitive in the middle of the convergence, which +is problematic, as it seems like we need it to decrease quickly, then slowly, +then quickly again. + +Secondly, the linear solver used here is just the sparse direct solver based on +the SparseDirectUMFPACK class. This works reasonably well on small problems, +but the formulation of the optimization problem detailed above has quite a large +number of variables and so the linear problem is not only large but also has a +lot of nonzero entries in many rows, even on meshes that overall are still +relatively coarse. As a consequence, the solver time dominates the +computations, and more sophisticated approaches at solving the linear system +are necessary. diff --git a/examples/step-79/doc/tooltip b/examples/step-79/doc/tooltip new file mode 100644 index 0000000000..93585fb813 --- /dev/null +++ b/examples/step-79/doc/tooltip @@ -0,0 +1 @@ +Topology optimization of elastic media. diff --git a/examples/step-79/step-79.cc b/examples/step-79/step-79.cc new file mode 100644 index 0000000000..d89e0ab170 --- /dev/null +++ b/examples/step-79/step-79.cc @@ -0,0 +1,2479 @@ +/* --------------------------------------------------------------------- + * + * Copyright (C) 2021 by the deal.II authors + * + * This file is part of the deal.II library. + * + * The deal.II library is free software; you can use it, redistribute + * it, and/or modify it under the terms of the GNU Lesser General + * Public License as published by the Free Software Foundation; either + * version 2.1 of the License, or (at your option) any later version. + * The full text of the license can be found in the file LICENSE.md at + * the top level directory of deal.II. + * + * --------------------------------------------------------------------- + + * + * Author: Justin O'Connor, Colorado State University, 2021. + */ + + +// @sect3{Preliminaries} + +#include +#include +#include +#include +#include + +#include +#include +#include +#include +#include +#include +#include + +#include +#include +#include + +#include +#include +#include + +#include +#include +#include +#include + +#include +#include +#include + + + +#include +#include +#include + +// Above are fairly common files to include. These also include the +// one for the sparse direct class SparseDirectUMFPACK. This is not +// the most efficient way to solve large linear problems, but it will +// do for now. +// +// As usual, we put everything into a common namespace. We then start +// by declaring a number of symbolic names for constants that will be +// used throughout this tutorial. Specifically, we have a *lot* of +// variables in this program (of course the density and the displacement, +// but also the unfiltered density and quite a number of Lagrange multipliers). +// It is easy to forget which of these variables is at which position in +// the solution vector, and trying to use numbers for these vector +// components is a prescription for bugs. Rather, we define static +// variables that can be used in all of these places and that have to +// be initialized only once. In practice, this will lead to some +// lengthy expressions, but they are more readable and less likely to +// be wrong. +// +// A similar issue arises with the ordering of blocks in the system +// matrix and in vectors. The matrices have $9\times 9$ blocks, and +// it's difficult to remember which is which. It is far easier to just +// use symbolic names for those as well. +// +// Finally, while we're at it, we introduce symbolic names also for +// the boundary indicators we will use, in the same spirit as was done +// in step-19. +// +// In all of these cases, we declare these variables as members in a +// namespace. In the case of the solution components, the concrete +// values of these variables depend on the space dimension, so we use +// [template +// variables](https://en.cppreference.com/w/cpp/language/variable_template) +// to make the value of the variable depend on a template argument in +// the same way as we often use template functions. +namespace SAND +{ + using namespace dealii; + + // This namespace keeps track of the first component in + // our finite element system that corresponds to each variable. + namespace SolutionComponents + { + template + constexpr unsigned int density = 0; + template + constexpr unsigned int displacement = 1; + template + constexpr unsigned int unfiltered_density = 1 + dim; + template + constexpr unsigned int displacement_multiplier = 2 + dim; + template + constexpr unsigned int unfiltered_density_multiplier = 2 + 2 * dim; + template + constexpr unsigned int density_lower_slack = 3 + 2 * dim; + template + constexpr unsigned int density_lower_slack_multiplier = 4 + 2 * dim; + template + constexpr unsigned int density_upper_slack = 5 + 2 * dim; + template + constexpr unsigned int density_upper_slack_multiplier = 6 + 2 * dim; + } // namespace SolutionComponents + + // This is the namespace which keeps track of which block + // corresponds to which variable. + namespace SolutionBlocks + { + constexpr unsigned int density = 0; + constexpr unsigned int displacement = 1; + constexpr unsigned int unfiltered_density = 2; + constexpr unsigned int displacement_multiplier = 3; + constexpr unsigned int unfiltered_density_multiplier = 4; + constexpr unsigned int density_lower_slack = 5; + constexpr unsigned int density_lower_slack_multiplier = 6; + constexpr unsigned int density_upper_slack = 7; + constexpr unsigned int density_upper_slack_multiplier = 8; + } // namespace SolutionBlocks + + namespace BoundaryIds + { + constexpr types::boundary_id down_force = 101; + constexpr types::boundary_id no_force = 102; + } // namespace BoundaryIds + + namespace ValueExtractors + { + template + const FEValuesExtractors::Scalar + densities(SolutionComponents::density); + template + const FEValuesExtractors::Vector + displacements(SolutionComponents::displacement); + template + const FEValuesExtractors::Scalar + unfiltered_densities(SolutionComponents::unfiltered_density); + template + const FEValuesExtractors::Vector displacement_multipliers( + SolutionComponents::displacement_multiplier); + template + const FEValuesExtractors::Scalar unfiltered_density_multipliers( + SolutionComponents::unfiltered_density_multiplier); + template + const FEValuesExtractors::Scalar + density_lower_slacks(SolutionComponents::density_lower_slack); + template + const FEValuesExtractors::Scalar density_lower_slack_multipliers( + SolutionComponents::density_lower_slack_multiplier); + template + const FEValuesExtractors::Scalar + density_upper_slacks(SolutionComponents::density_upper_slack); + template + const FEValuesExtractors::Scalar density_upper_slack_multipliers( + SolutionComponents::density_upper_slack_multiplier); + } // namespace ValueExtractors + + + // @sect3{The SANDTopOpt main class} + + // Next up is the main class for this problem. The majority of functions + // follow the usual naming schemes of tutorial programs, though there + // are a couple that have been broken out of what is usually called + // the `setup_system()` function because of their length, and there + // are also a number that deal with various aspects of the + // optimization algorithm. + // + // As an added bonus, the program writes the computed design as an STL + // file that one can, for example, send to a 3d printer. + template + class SANDTopOpt + { + public: + SANDTopOpt(); + + void run(); + + private: + void create_triangulation(); + + void setup_boundary_values(); + + void setup_block_system(); + + void setup_filter_matrix(); + + void assemble_system(); + + BlockVector solve(); + + std::pair + calculate_max_step_size(const BlockVector &state, + const BlockVector &step) const; + + BlockVector + calculate_test_rhs(const BlockVector &test_solution) const; + + double calculate_exact_merit(const BlockVector &test_solution); + + BlockVector find_max_step(); + + BlockVector compute_scaled_step(const BlockVector &state, + const BlockVector &step, + const double descent_requirement); + + bool check_convergence(const BlockVector &state); + + void output_results(const unsigned int j) const; + + void write_as_stl(); + + std::set::cell_iterator> + find_relevant_neighbors( + typename Triangulation::cell_iterator cell) const; + + + // Most of the member variables are also standard. There are, + // however, a number of variables that are specifically related + // to the optimization algorithm (such the various scalar + // factors below) as well as the filter matrix to ensure that + // the design remains smooth. + Triangulation triangulation; + FESystem fe; + DoFHandler dof_handler; + AffineConstraints constraints; + + std::map boundary_values; + + BlockSparsityPattern sparsity_pattern; + BlockSparseMatrix system_matrix; + + SparsityPattern filter_sparsity_pattern; + SparseMatrix filter_matrix; + + BlockVector system_rhs; + BlockVector nonlinear_solution; + + const double density_ratio; + const double density_penalty_exponent; + const double filter_r; + double penalty_multiplier; + double barrier_size; + + + TimerOutput timer; + }; + + + // @sect3{Constructor and set-up functions} + + + // We initialize a FESystem composed of 2$\times$dim `FE_Q(1)` elements + // for the displacement variable and its Lagrange multiplier, and 7 + // `FE_DGQ(0)` elements. These piecewise constant functions are + // for density-related variables: the density itself, the + // unfiltered density, the slack variables for the lower and upper + // bounds on the unfiltered density, and then Lagrange multipliers + // for the connection between filtered and unfiltered densities as + // well as for the inequality constraints. + // + // The order in which these elements appear is documented above. + template + SANDTopOpt::SANDTopOpt() + : fe(FE_DGQ(0), + 1, + (FESystem(FE_Q(1) ^ dim)), + 1, + FE_DGQ(0), + 1, + (FESystem(FE_Q(1) ^ dim)), + 1, + FE_DGQ(0), + 5) + , dof_handler(triangulation) + , density_ratio(.5) + , density_penalty_exponent(3) + , filter_r(.251) + , penalty_multiplier(1) + , timer(std::cout, TimerOutput::summary, TimerOutput::wall_times) + { + Assert(dim > 1, ExcNotImplemented()); + } + + + // The first step then is to create the triangulation that matches + // the problem description in the introduction -- a 6-by-1 + // rectangle (or a 6-by-1-by-1 box in 3d) where a force will be + // applied in the top center. This triangulation is then uniformly + // refined a number of times. + // + // In contrast to nearly the entire rest of this program, this + // function specifically assumes that we are in 2d and will + // require changes if we wanted to move to 3d simulations. We + // ensure that nobody tries to accidentally run in 3d without such + // modifications through an assertion at the top of the function. + template + void SANDTopOpt::create_triangulation() + { + Assert(dim == 2, ExcNotImplemented()); + GridGenerator::subdivided_hyper_rectangle(triangulation, + {6, 1}, + Point(0, 0), + Point(6, 1)); + + triangulation.refine_global(3); + + // The second step is to apply boundary indicators to parts of + // the boundary. The following code assigns boundary + // indicators to the bottom, top, left, and right boundaries + // of the box, respectively. The center region of the top + // boundary is given a separate boundary indicator: This is + // where we will apply the down force. + for (const auto &cell : triangulation.active_cell_iterators()) + { + for (const auto &face : cell->face_iterators()) + { + if (face->at_boundary()) + { + const auto center = face->center(); + if (std::fabs(center(1) - 1) < 1e-12) + { + if ((std::fabs(center(0) - 3) < .3)) + face->set_boundary_id(BoundaryIds::down_force); + else + face->set_boundary_id(BoundaryIds::no_force); + } + else + face->set_boundary_id(BoundaryIds::no_force); + } + } + } + } + + + // Next, determine the constraints due to boundary values. The + // bottom corners of the domain are kept in place in the $y$ + // direction -- the bottom left also in the $x$ direction. deal.II + // generally thinks of boundary values as attached to pieces of the + // boundary, i.e., faces, rather than individual vertices. Indeed, + // mathematically speaking, one can not assign boundary values to + // individual points for the infinite-dimensional partial + // differential equation. But, since we are trying to reproduce a + // widely used benchmark, we will do so anyway and keep in mind that + // we have a finite-dimensional problem for which imposing boundary + // conditions at a single node is valid. + template + void SANDTopOpt::setup_boundary_values() + { + boundary_values.clear(); + for (const auto &cell : dof_handler.active_cell_iterators()) + { + for (const auto &face : cell->face_iterators()) + { + if (face->at_boundary()) + { + const auto center = face->center(); + + // Check whether the current face is on the bottom + // boundary, and if it is whether one of its + // vertices might be the bottom left or bottom + // right vertex: + if (std::fabs(center(1) - 0) < 1e-12) + { + for (const auto vertex_number : cell->vertex_indices()) + { + const auto vert = cell->vertex(vertex_number); + + if (std::fabs(vert(0) - 0) < 1e-12 && + std::fabs(vert(1) - 0) < 1e-12) + { + types::global_dof_index x_displacement = + cell->vertex_dof_index(vertex_number, 0); + types::global_dof_index y_displacement = + cell->vertex_dof_index(vertex_number, 1); + types::global_dof_index x_displacement_multiplier = + cell->vertex_dof_index(vertex_number, 2); + types::global_dof_index y_displacement_multiplier = + cell->vertex_dof_index(vertex_number, 3); + + boundary_values[x_displacement] = 0; + boundary_values[y_displacement] = 0; + boundary_values[x_displacement_multiplier] = 0; + boundary_values[y_displacement_multiplier] = 0; + } + + else if (std::fabs(vert(0) - 6) < 1e-12 && + std::fabs(vert(1) - 0) < 1e-12) + { + types::global_dof_index y_displacement = + cell->vertex_dof_index(vertex_number, 1); + types::global_dof_index y_displacement_multiplier = + cell->vertex_dof_index(vertex_number, 3); + + boundary_values[y_displacement] = 0; + boundary_values[y_displacement_multiplier] = 0; + } + } + } + } + } + } + } + + // @sect3{Setting up block matrices and vectors} + + // The next function makes a giant 9-by-9 block matrix, and also + // sets up the necessary block vectors. The sparsity pattern for + // this matrix includes the sparsity pattern for the filter + // matrix. It also initializes any block vectors we will use. + // + // Setting up the blocks by themselves is not overly complicated + // and follows what is already done in programs such as step-22, + // for example. + template + void SANDTopOpt::setup_block_system() + { + std::vector block_component(9, 2); + block_component[0] = 0; + block_component[1] = 1; + const std::vector dofs_per_block = + DoFTools::count_dofs_per_fe_block(dof_handler, block_component); + + const types::global_dof_index n_p = dofs_per_block[0]; + const types::global_dof_index n_u = dofs_per_block[1]; + const std::vector::size_type> block_sizes = { + n_p, n_u, n_p, n_u, n_p, n_p, n_p, n_p, n_p}; + + BlockDynamicSparsityPattern dsp(9, 9); + for (unsigned int k = 0; k < 9; ++k) + for (unsigned int j = 0; j < 9; ++j) + dsp.block(j, k).reinit(block_sizes[j], block_sizes[k]); + dsp.collect_sizes(); + + + // The bulk of the function is in setting up which of these + // blocks will actually contain anything, i.e., which + // variables couple with which other variables. This is + // cumbersome but necessary to ensure that we don't just + // allocate a very large number of entries for our matrix that + // will then end up being zero. + // + // The concrete pattern you see below is something one + // probably has to draw once on a piece of paper, but follows + // in an otherwise relatively straightforward way from looking + // through the many terms of the bilinear form we will have to + // assemble in each nonlinear iteration. + // + // The use of the symbolic names defined in namespace + // `SolutionComponents` helps understand what each of the + // following terms corresponds to, but it also makes the + // expressions lengthy and unwieldy: An term such as + // `coupling[SolutionComponents::density_upper_slack_multiplier][SolutionComponents::density]` + // just doesn't read very well, and would either have to be + // split over several lines or run off the right edge of + // nearly every screen. As a consequence, we open a + // curly-brace enclosed code block in which we temporarily + // make the names in namespace `SolutionComponents` available + // without the namespace qualifier, by saying `using namespace + // SolutionComponents`. + Table<2, DoFTools::Coupling> coupling(2 * dim + 7, 2 * dim + 7); + { + using namespace SolutionComponents; + + coupling[density][density] = DoFTools::always; + + for (unsigned int i = 0; i < dim; ++i) + { + coupling[density][displacement + i] = DoFTools::always; + coupling[displacement + i][density] = DoFTools::always; + } + + for (unsigned int i = 0; i < dim; ++i) + { + coupling[density][displacement_multiplier + i] = + DoFTools::always; + coupling[displacement_multiplier + i][density] = + DoFTools::always; + } + + coupling[density][unfiltered_density_multiplier] = + DoFTools::always; + coupling[unfiltered_density_multiplier][density] = + DoFTools::always; + + /* Coupling for displacement */ + + for (unsigned int i = 0; i < dim; ++i) + { + for (unsigned int k = 0; k < dim; ++k) + { + coupling[displacement + i] + [displacement_multiplier + k] = DoFTools::always; + coupling[displacement_multiplier + k] + [displacement + i] = DoFTools::always; + } + } + + /* Coupling for slack variables */ + coupling[density_lower_slack][density_lower_slack] = + DoFTools::always; + coupling[density_lower_slack][density_upper_slack] = + DoFTools::always; + coupling[density_upper_slack][density_lower_slack] = + DoFTools::always; + + coupling[density_lower_slack_multiplier] + [density_lower_slack_multiplier] = DoFTools::always; + coupling[density_lower_slack_multiplier] + [density_upper_slack_multiplier] = DoFTools::always; + coupling[density_upper_slack_multiplier] + [density_lower_slack_multiplier] = DoFTools::always; + } + + // Before we can create the sparsity pattern, we also have to + // set up constraints. Since this program does not adaptively + // refine the mesh, the only constraint we have is one that + // couples all density variables to enforce the volume + // constraint. This will ultimately lead to a dense sub-block + // of the matrix, but there is little we can do about that. + const ComponentMask density_mask = + fe.component_mask(ValueExtractors::densities); + const IndexSet density_dofs = + DoFTools::extract_dofs(dof_handler, density_mask); + + types::global_dof_index last_density_dof = + density_dofs.nth_index_in_set(density_dofs.n_elements() - 1); + constraints.clear(); + constraints.add_line(last_density_dof); + for (unsigned int i = 0; i < density_dofs.n_elements() - 1; ++i) + constraints.add_entry(last_density_dof, + density_dofs.nth_index_in_set(i), + -1); + constraints.set_inhomogeneity(last_density_dof, 0); + + constraints.close(); + + // We can now finally create the sparsity pattern for the + // matrix, taking into account which variables couple with + // which other variables, and the constraints we have on the + // density. + DoFTools::make_sparsity_pattern(dof_handler, coupling, dsp, constraints); + + // The only part of the matrix we have not dealt with is the + // filter matrix and its transpose. These are non-local + // (integral) operators for which deal.II does not currently + // have functions. What we will ultimately need to do is go + // over all cells and couple the unfiltered density on this + // cell to all filtered densities of neighboring cells that + // are less than a threshold distance away, and the other way + // around; for the moment, we are only concerned with building + // the sparsity pattern that would correspond to this kind of + // matrix, so we perform the equivalent loop and where later + // on we would write into an entry of the matrix, we now + // simply add an entry to the sparsity matrix: + for (const auto &cell : dof_handler.active_cell_iterators()) + { + const unsigned int i = cell->active_cell_index(); + for (const auto &check_cell : find_relevant_neighbors(cell)) + { + const double distance = + cell->center().distance(check_cell->center()); + if (distance < filter_r) + { + dsp + .block(SolutionBlocks::unfiltered_density, + SolutionBlocks::unfiltered_density_multiplier) + .add(i, check_cell->active_cell_index()); + dsp + .block(SolutionBlocks::unfiltered_density_multiplier, + SolutionBlocks::unfiltered_density) + .add(i, check_cell->active_cell_index()); + } + } + } + + // Having so generated the "dynamic" sparsity pattern, we can + // finally copy it to the structure that is used to associate + // matrices with a sparsity pattern. Because the sparsity + // pattern is large and complex, we also output it into a file + // of its own for visualization purposes -- in other words, + // for "visual debugging". + sparsity_pattern.copy_from(dsp); + + std::ofstream out("sparsity.plt"); + sparsity_pattern.print_gnuplot(out); + + system_matrix.reinit(sparsity_pattern); + + + // What is left is to correctly size the various vectors and + // their blocks, as well as setting initial guesses for some + // of the components of the (nonlinear) solution vector. We + // here use the symbolic component names for individual blocks + // of the solution vector and, for brevity, use the same trick + // with `using namespace` as above: + nonlinear_solution.reinit(block_sizes); + system_rhs.reinit(block_sizes); + + { + using namespace SolutionBlocks; + nonlinear_solution.block(density).add(density_ratio); + nonlinear_solution.block(unfiltered_density).add(density_ratio); + nonlinear_solution.block(unfiltered_density_multiplier) + .add(density_ratio); + nonlinear_solution.block(density_lower_slack).add(density_ratio); + nonlinear_solution.block(density_lower_slack_multiplier).add(50); + nonlinear_solution.block(density_upper_slack).add(1 - density_ratio); + nonlinear_solution.block(density_upper_slack_multiplier).add(50); + } + } + + + // @sect3{Creating the filter matrix} + + // Next up, a function that is used once at the beginning of the + // program: It creates a matrix $H$ so that the filtered density + // vector equals $H$ times the unfiltered density. The creation + // of this matrix is non-trivial, and it is used in every + // iteration, and so rather than reforming it as we do with the + // Newton matrix, it is made only once and stored separately. + // + // The way this matrix is computed follows the outline used above + // already to form its sparsity pattern. We repeat this process here + // for the sparsity pattern of this separately formed matrix, and + // then actually build the matrix itself. You may want to check the + // definition of this matrix in the introduction to this program. + template + void SANDTopOpt::setup_filter_matrix() + { + // The sparsity pattern of the filter has already been determined + // and implemented in the setup_system() function. We copy the + // structure from the appropriate block and use it again here. + + filter_sparsity_pattern.copy_from( + sparsity_pattern.block(SolutionBlocks::unfiltered_density, + SolutionBlocks::unfiltered_density_multiplier)); + filter_matrix.reinit(filter_sparsity_pattern); + + // Having so built the sparsity pattern, now we re-do all of + // these loops to actually compute the necessary values of the + // matrix entries: + + for (const auto &cell : dof_handler.active_cell_iterators()) + { + const unsigned int i = cell->active_cell_index(); + for (const auto &check_cell : find_relevant_neighbors(cell)) + { + const double distance = + cell->center().distance(check_cell->center()); + if (distance < filter_r) + { + filter_matrix.add(i, + check_cell->active_cell_index(), + filter_r - distance); + // + } + } + } + + // The final step is to normalize the matrix so that for each + // row, the sum of entries equals one. + for (unsigned int i = 0; i < filter_matrix.m(); ++i) + { + double denominator = 0; + for (SparseMatrix::iterator iter = filter_matrix.begin(i); + iter != filter_matrix.end(i); + iter++) + denominator = denominator + iter->value(); + for (SparseMatrix::iterator iter = filter_matrix.begin(i); + iter != filter_matrix.end(i); + iter++) + iter->value() = iter->value() / denominator; + } + } + + // This function is used for building the filter matrix. We create a set of + // all the cell iterators within a certain radius of the cell that is input. + // These are the neighboring cells that will be relevant for the filter. + template + std::set::cell_iterator> + SANDTopOpt::find_relevant_neighbors( + typename Triangulation::cell_iterator cell) const + { + std::set neighbor_ids; + std::set::cell_iterator> cells_to_check; + + neighbor_ids.insert(cell->active_cell_index()); + cells_to_check.insert(cell); + + bool new_neighbors_found; + do + { + new_neighbors_found = false; + for (const auto &check_cell : + std::vector::cell_iterator>( + cells_to_check.begin(), cells_to_check.end())) + { + for (const auto n : check_cell->face_indices()) + { + if (!(check_cell->face(n)->at_boundary())) + { + const auto & neighbor = check_cell->neighbor(n); + const double distance = + cell->center().distance(neighbor->center()); + if ((distance < filter_r) && + !(neighbor_ids.count(neighbor->active_cell_index()))) + { + cells_to_check.insert(neighbor); + neighbor_ids.insert(neighbor->active_cell_index()); + new_neighbors_found = true; + } + } + } + } + } + while (new_neighbors_found); + return cells_to_check; + } + + // @sect3{Assembling the Newton matrix} + + // Whereas the setup_filter_matrix function built a matrix that is the same as + // long as the mesh does not change (which we don't do anyway in + // this program), the next function builds the matrix to be solved + // in each iteration. This is where the magic happens. The components + // of the system of linear equations describing Newton's method for + // finding the solution of the KKT conditions are implemented here. + // + // The top of the function is as in most of these functions and just + // sets up all sorts of variables necessary for the actual assembly, + // including a whole bunch of extractors. The entire set up should + // look familiar, though somewhat lengthier, if you've previously + // looked at step-22. + template + void SANDTopOpt::assemble_system() + { + TimerOutput::Scope t(timer, "assembly"); + + system_matrix = 0; + system_rhs = 0; + + + MappingQGeneric mapping(1); + QGauss quadrature_formula(fe.degree + 1); + QGauss face_quadrature_formula(fe.degree + 1); + FEValues fe_values(mapping, + fe, + quadrature_formula, + update_values | update_gradients | + update_quadrature_points | update_JxW_values); + FEFaceValues fe_face_values(mapping, + fe, + face_quadrature_formula, + update_values | update_quadrature_points | + update_normal_vectors | + update_JxW_values); + + const unsigned int dofs_per_cell = fe.dofs_per_cell; + const unsigned int n_q_points = quadrature_formula.size(); + + FullMatrix cell_matrix(dofs_per_cell, dofs_per_cell); + Vector dummy_cell_rhs(dofs_per_cell); + + std::vector local_dof_indices(dofs_per_cell); + + std::vector lambda_values(n_q_points); + std::vector mu_values(n_q_points); + const Functions::ConstantFunction lambda(1.); + const Functions::ConstantFunction mu(1.); + std::vector> rhs_values(n_q_points); + + // At this point, we apply the filter to the unfiltered + // density, and apply the adjoint (transpose) operation to the + // unfiltered density multiplier, both to the current best + // guess for the nonlinear solution. We use this later to tell + // us how far off our filtered density is from the filter + // applied to the unfiltered density. That is because while at + // the solution of the nonlinear problem, we have + // $\rho=H\sigma$, but at intermediate iterations, we in + // general have $\rho^k\neq H\sigma^k$ and the "residual" + // $\rho^k-H\sigma^k$ will then appear as the right hand side + // of one of the Newton update equations that we compute + // below. + BlockVector filtered_unfiltered_density_solution = + nonlinear_solution; + BlockVector filter_adjoint_unfiltered_density_multiplier_solution = + nonlinear_solution; + + filter_matrix.vmult(filtered_unfiltered_density_solution.block( + SolutionBlocks::unfiltered_density), + nonlinear_solution.block( + SolutionBlocks::unfiltered_density)); + filter_matrix.Tvmult( + filter_adjoint_unfiltered_density_multiplier_solution.block( + SolutionBlocks::unfiltered_density_multiplier), + nonlinear_solution.block(SolutionBlocks::unfiltered_density_multiplier)); + + + std::vector old_density_values(n_q_points); + std::vector> old_displacement_values(n_q_points); + std::vector old_displacement_divs(n_q_points); + std::vector> old_displacement_symmgrads(n_q_points); + std::vector> old_displacement_multiplier_values(n_q_points); + std::vector old_displacement_multiplier_divs(n_q_points); + std::vector> old_displacement_multiplier_symmgrads( + n_q_points); + std::vector old_lower_slack_multiplier_values(n_q_points); + std::vector old_upper_slack_multiplier_values(n_q_points); + std::vector old_lower_slack_values(n_q_points); + std::vector old_upper_slack_values(n_q_points); + std::vector old_unfiltered_density_values(n_q_points); + std::vector old_unfiltered_density_multiplier_values(n_q_points); + std::vector filtered_unfiltered_density_values(n_q_points); + std::vector filter_adjoint_unfiltered_density_multiplier_values( + n_q_points); + + using namespace ValueExtractors; + for (const auto &cell : dof_handler.active_cell_iterators()) + { + cell_matrix = 0; + + cell->get_dof_indices(local_dof_indices); + + fe_values.reinit(cell); + + lambda.value_list(fe_values.get_quadrature_points(), lambda_values); + mu.value_list(fe_values.get_quadrature_points(), mu_values); + + // As part of the construction of our system matrix, we need to + // retrieve values from our current guess at the solution. + // The following lines of code retrieve the needed values. + fe_values[densities].get_function_values(nonlinear_solution, + old_density_values); + fe_values[displacements].get_function_values( + nonlinear_solution, old_displacement_values); + fe_values[displacements].get_function_divergences( + nonlinear_solution, old_displacement_divs); + fe_values[displacements].get_function_symmetric_gradients( + nonlinear_solution, old_displacement_symmgrads); + fe_values[displacement_multipliers].get_function_values( + nonlinear_solution, old_displacement_multiplier_values); + fe_values[displacement_multipliers].get_function_divergences( + nonlinear_solution, old_displacement_multiplier_divs); + fe_values[displacement_multipliers] + .get_function_symmetric_gradients( + nonlinear_solution, old_displacement_multiplier_symmgrads); + fe_values[density_lower_slacks].get_function_values( + nonlinear_solution, old_lower_slack_values); + fe_values[density_lower_slack_multipliers].get_function_values( + nonlinear_solution, old_lower_slack_multiplier_values); + fe_values[density_upper_slacks].get_function_values( + nonlinear_solution, old_upper_slack_values); + fe_values[density_upper_slack_multipliers].get_function_values( + nonlinear_solution, old_upper_slack_multiplier_values); + fe_values[unfiltered_densities].get_function_values( + nonlinear_solution, old_unfiltered_density_values); + fe_values[unfiltered_density_multipliers].get_function_values( + nonlinear_solution, old_unfiltered_density_multiplier_values); + fe_values[unfiltered_densities].get_function_values( + filtered_unfiltered_density_solution, + filtered_unfiltered_density_values); + fe_values[unfiltered_density_multipliers].get_function_values( + filter_adjoint_unfiltered_density_multiplier_solution, + filter_adjoint_unfiltered_density_multiplier_values); + + for (const auto q_point : fe_values.quadrature_point_indices()) + { + // We need several more values corresponding to the test functions + // coming from the first derivatives taken from the Lagrangian, + // that is the $d_{\bullet}$ functions. These are calculated here: + for (const auto i : fe_values.dof_indices()) + { + const SymmetricTensor<2, dim> displacement_phi_i_symmgrad = + fe_values[displacements].symmetric_gradient(i, q_point); + const double displacement_phi_i_div = + fe_values[displacements].divergence(i, q_point); + + const SymmetricTensor<2, dim> + displacement_multiplier_phi_i_symmgrad = + fe_values[displacement_multipliers].symmetric_gradient( + i, q_point); + const double displacement_multiplier_phi_i_div = + fe_values[displacement_multipliers].divergence(i, + q_point); + + const double density_phi_i = + fe_values[densities].value(i, q_point); + const double unfiltered_density_phi_i = + fe_values[unfiltered_densities].value(i, q_point); + const double unfiltered_density_multiplier_phi_i = + fe_values[unfiltered_density_multipliers].value(i, + q_point); + + const double lower_slack_multiplier_phi_i = + fe_values[density_lower_slack_multipliers].value( + i, q_point); + + const double lower_slack_phi_i = + fe_values[density_lower_slacks].value(i, q_point); + + const double upper_slack_phi_i = + fe_values[density_upper_slacks].value(i, q_point); + + const double upper_slack_multiplier_phi_i = + fe_values[density_upper_slack_multipliers].value( + i, q_point); + + + for (const auto j : fe_values.dof_indices()) + { + // Finally, we need values that come from the second round + // of derivatives taken from the Lagrangian, + // the $c_{\bullet}$ functions. These are calculated here: + const SymmetricTensor<2, dim> displacement_phi_j_symmgrad = + fe_values[displacements].symmetric_gradient(j, + q_point); + const double displacement_phi_j_div = + fe_values[displacements].divergence(j, q_point); + + const SymmetricTensor<2, dim> + displacement_multiplier_phi_j_symmgrad = + fe_values[displacement_multipliers] + .symmetric_gradient(j, q_point); + const double displacement_multiplier_phi_j_div = + fe_values[displacement_multipliers].divergence( + j, q_point); + + const double density_phi_j = + fe_values[densities].value(j, q_point); + + const double unfiltered_density_phi_j = + fe_values[unfiltered_densities].value(j, q_point); + const double unfiltered_density_multiplier_phi_j = + fe_values[unfiltered_density_multipliers].value( + j, q_point); + + + const double lower_slack_phi_j = + fe_values[density_lower_slacks].value(j, q_point); + + const double upper_slack_phi_j = + fe_values[density_upper_slacks].value(j, q_point); + + const double lower_slack_multiplier_phi_j = + fe_values[density_lower_slack_multipliers].value( + j, q_point); + + const double upper_slack_multiplier_phi_j = + fe_values[density_upper_slack_multipliers].value( + j, q_point); + + // This is where the actual work starts. In + // the following, we will build all of the + // terms of the matrix -- they are numerous + // and not entirely self-explanatory, also + // depending on the previous solutions and its + // derivatives (which we have already + // evaluated above and put into the variables + // called `old_*`). To understand what each of + // these terms corresponds to, you will want + // to look at the explicit form of these terms + // in the introduction above. + // + // The right hand sides of the equations being + // driven to 0 give all the KKT conditions + // for finding a local minimum -- the descriptions of what + // each individual equation are given with the computations + // of the right hand side. + + /* Equation 1 */ + cell_matrix(i, j) += + fe_values.JxW(q_point) * + ( + + -density_phi_i * unfiltered_density_multiplier_phi_j + + + density_penalty_exponent * + (density_penalty_exponent - 1) * + std::pow(old_density_values[q_point], + density_penalty_exponent - 2) * + density_phi_i * density_phi_j * + (old_displacement_multiplier_divs[q_point] * + old_displacement_divs[q_point] * + lambda_values[q_point] + + 2 * mu_values[q_point] * + (old_displacement_symmgrads[q_point] * + old_displacement_multiplier_symmgrads[q_point])) + + + density_penalty_exponent * + std::pow(old_density_values[q_point], + density_penalty_exponent - 1) * + density_phi_i * + (displacement_multiplier_phi_j_div * + old_displacement_divs[q_point] * + lambda_values[q_point] + + 2 * mu_values[q_point] * + (old_displacement_symmgrads[q_point] * + displacement_multiplier_phi_j_symmgrad)) + + + density_penalty_exponent * + std::pow(old_density_values[q_point], + density_penalty_exponent - 1) * + density_phi_i * + (displacement_phi_j_div * + old_displacement_multiplier_divs[q_point] * + lambda_values[q_point] + + 2 * mu_values[q_point] * + (old_displacement_multiplier_symmgrads[q_point] * + displacement_phi_j_symmgrad))); + + /* Equation 2 */ + cell_matrix(i, j) += + fe_values.JxW(q_point) * + (density_penalty_exponent * + std::pow(old_density_values[q_point], + density_penalty_exponent - 1) * + density_phi_j * + (old_displacement_multiplier_divs[q_point] * + displacement_phi_i_div * lambda_values[q_point] + + 2 * mu_values[q_point] * + (old_displacement_multiplier_symmgrads[q_point] * + displacement_phi_i_symmgrad)) + + + std::pow(old_density_values[q_point], + density_penalty_exponent) * + (displacement_multiplier_phi_j_div * + displacement_phi_i_div * lambda_values[q_point] + + 2 * mu_values[q_point] * + (displacement_multiplier_phi_j_symmgrad * + displacement_phi_i_symmgrad)) + + ); + + /* Equation 3, which has to do with the filter and which is + * calculated elsewhere. */ + cell_matrix(i, j) += + fe_values.JxW(q_point) * + (-1 * unfiltered_density_phi_i * + lower_slack_multiplier_phi_j + + unfiltered_density_phi_i * upper_slack_multiplier_phi_j); + + + /* Equation 4: Primal feasibility */ + cell_matrix(i, j) += + fe_values.JxW(q_point) * + ( + + density_penalty_exponent * + std::pow(old_density_values[q_point], + density_penalty_exponent - 1) * + density_phi_j * + (old_displacement_divs[q_point] * + displacement_multiplier_phi_i_div * + lambda_values[q_point] + + 2 * mu_values[q_point] * + (old_displacement_symmgrads[q_point] * + displacement_multiplier_phi_i_symmgrad)) + + + std::pow(old_density_values[q_point], + density_penalty_exponent) * + (displacement_phi_j_div * + displacement_multiplier_phi_i_div * + lambda_values[q_point] + + 2 * mu_values[q_point] * + (displacement_phi_j_symmgrad * + displacement_multiplier_phi_i_symmgrad))); + + /* Equation 5: Primal feasibility */ + cell_matrix(i, j) += + -1 * fe_values.JxW(q_point) * + lower_slack_multiplier_phi_i * + (unfiltered_density_phi_j - lower_slack_phi_j); + + /* Equation 6: Primal feasibility */ + cell_matrix(i, j) += + -1 * fe_values.JxW(q_point) * + upper_slack_multiplier_phi_i * + (-1 * unfiltered_density_phi_j - upper_slack_phi_j); + + /* Equation 7: Primal feasibility - the part with the filter + * is added later */ + cell_matrix(i, j) += -1 * fe_values.JxW(q_point) * + unfiltered_density_multiplier_phi_i * + (density_phi_j); + + /* Equation 8: Complementary slackness */ + cell_matrix(i, j) += + fe_values.JxW(q_point) * + (lower_slack_phi_i * lower_slack_multiplier_phi_j + + + lower_slack_phi_i * lower_slack_phi_j * + old_lower_slack_multiplier_values[q_point] / + old_lower_slack_values[q_point]); + + /* Equation 9: Complementary slackness */ + cell_matrix(i, j) += + fe_values.JxW(q_point) * + (upper_slack_phi_i * upper_slack_multiplier_phi_j + + + + upper_slack_phi_i * upper_slack_phi_j * + old_upper_slack_multiplier_values[q_point] / + old_upper_slack_values[q_point]); + } + } + } + + // Now that we have everything assembled, all we have to + // do is deal with the effect of (Dirichlet) boundary + // conditions and other constraints. We incorporate the + // former locally with just the contributions from the + // current cell, and then let the AffineConstraint class + // deal with the latter while copying contributions from + // the current cell into the global linear system: + MatrixTools::local_apply_boundary_values(boundary_values, + local_dof_indices, + cell_matrix, + dummy_cell_rhs, + true); + + constraints.distribute_local_to_global(cell_matrix, + local_dof_indices, + system_matrix); + } + + // Having accumulated all of the terms that belong + // into the Newton matrix, we now also have to + // compute the terms for the right hand side + // (i.e., the negative residual). We already do this + // in another function, and so we call that here: + system_rhs = calculate_test_rhs(nonlinear_solution); + + // Here we use the filter matrix we have already + // constructed. We only need to integrate this filter applied + // to test functions, which are piecewise constant, and so the + // integration becomes a simple multiplication by the measure + // of the cell. Iterating over the pre-made filter matrix + // allows us to use the information about which cells are in + // or out of the filter without repeatedly checking neighbor + // cells again. + for (const auto &cell : dof_handler.active_cell_iterators()) + { + const unsigned int i = cell->active_cell_index(); + for (typename SparseMatrix::iterator iter = + filter_matrix.begin(i); + iter != filter_matrix.end(i); + ++iter) + { + const unsigned int j = iter->column(); + const double value = iter->value() * cell->measure(); + + system_matrix + .block(SolutionBlocks::unfiltered_density_multiplier, + SolutionBlocks::unfiltered_density) + .add(i, j, value); + system_matrix + .block(SolutionBlocks::unfiltered_density, + SolutionBlocks::unfiltered_density_multiplier) + .add(j, i, value); + } + } + } + + + // @sect3{Solving the Newton linear system} + + + // We will need to solve a linear system in each iteration. We use + // a direct solver, for now -- this is clearly not an efficient + // choice for a matrix that has so many non-zeroes, and it will + // not scale to anything interesting. For "real" applications, we + // will need an iterative solver but the complexity of the system + // means that an iterative solver algorithm will take a good deal + // of work. Because this is not the focus of the current program, + // we simply stick with the direct solver we have here -- the + // function follows the same structure as used in step-29. + template + BlockVector SANDTopOpt::solve() + { + TimerOutput::Scope t(timer, "solver"); + + BlockVector linear_solution; + linear_solution.reinit(nonlinear_solution); + + SparseDirectUMFPACK A_direct; + A_direct.initialize(system_matrix); + A_direct.vmult(linear_solution, system_rhs); + + constraints.distribute(linear_solution); + + return linear_solution; + } + + + // @sect3{Details of the optimization algorithm} + + // The next several functions deal with specific parts of the + // optimization algorithm, most notably with deciding whether the + // direction computed by solving the linearized (Newton) system is + // viable and, if so, how far we want to go in this direction. + + // @sect4{Computing step lengths} + + // We start with a function that does a binary search to figure + // out the maximum step that meets the dual feasibility -- that + // is, how far can we go so that $s>0$ and $z>0$. The function + // returns a pair of values, one each for the $s$ and $z$ slack + // variables. + template + std::pair SANDTopOpt::calculate_max_step_size( + const BlockVector &state, + const BlockVector &step) const + { + double fraction_to_boundary; + const double min_fraction_to_boundary = .8; + const double max_fraction_to_boundary = 1. - 1e-5; + + if (min_fraction_to_boundary < 1 - barrier_size) + { + if (1 - barrier_size < max_fraction_to_boundary) + fraction_to_boundary = 1 - barrier_size; + else + fraction_to_boundary = max_fraction_to_boundary; + } + else + fraction_to_boundary = min_fraction_to_boundary; + + double step_size_s_low = 0; + double step_size_z_low = 0; + double step_size_s_high = 1; + double step_size_z_high = 1; + double step_size_s, step_size_z; + + const int max_bisection_method_steps = 50; + for (unsigned int k = 0; k < max_bisection_method_steps; ++k) + { + step_size_s = (step_size_s_low + step_size_s_high) / 2; + step_size_z = (step_size_z_low + step_size_z_high) / 2; + + const BlockVector state_test_s = + (fraction_to_boundary * state) + (step_size_s * step); + + const BlockVector state_test_z = + (fraction_to_boundary * state) + (step_size_z * step); + + const bool accept_s = + (state_test_s.block(SolutionBlocks::density_lower_slack) + .is_non_negative()) && + (state_test_s.block(SolutionBlocks::density_upper_slack) + .is_non_negative()); + const bool accept_z = + (state_test_z.block(SolutionBlocks::density_lower_slack_multiplier) + .is_non_negative()) && + (state_test_z.block(SolutionBlocks::density_upper_slack_multiplier) + .is_non_negative()); + + if (accept_s) + step_size_s_low = step_size_s; + else + step_size_s_high = step_size_s; + + if (accept_z) + step_size_z_low = step_size_z; + else + step_size_z_high = step_size_z; + } + + return {step_size_s_low, step_size_z_low}; + } + + + // @sect4{Computing residuals} + + // The next function computes a right hand side vector linearized + // around a "test solution vector" that we can use to look at the + // magnitude of the KKT conditions. This is then used for testing + // the convergence before shrinking the barrier size, as well as in the + // calculation of the $l_1$ merit. + // + // The function is lengthy and complicated, but it is really just a + // copy of the right hand side part of what the `assemble_system()` + // function above did. + template + BlockVector SANDTopOpt::calculate_test_rhs( + const BlockVector &test_solution) const + { + // We first create a zero vector with size and blocking of system_rhs + BlockVector test_rhs; + test_rhs.reinit(system_rhs); + + MappingQGeneric mapping(1); + const QGauss quadrature_formula(fe.degree + 1); + const QGauss face_quadrature_formula(fe.degree + 1); + FEValues fe_values(mapping, + fe, + quadrature_formula, + update_values | update_gradients | + update_quadrature_points | update_JxW_values); + FEFaceValues fe_face_values(mapping, + fe, + face_quadrature_formula, + update_values | update_quadrature_points | + update_normal_vectors | + update_JxW_values); + + const unsigned int dofs_per_cell = fe.dofs_per_cell; + const unsigned int n_q_points = quadrature_formula.size(); + + Vector cell_rhs(dofs_per_cell); + FullMatrix dummy_cell_matrix(dofs_per_cell, dofs_per_cell); + + std::vector local_dof_indices(dofs_per_cell); + + std::vector lambda_values(n_q_points); + std::vector mu_values(n_q_points); + + const Functions::ConstantFunction lambda(1.), mu(1.); + std::vector> rhs_values(n_q_points); + + + BlockVector filtered_unfiltered_density_solution = test_solution; + BlockVector filter_adjoint_unfiltered_density_multiplier_solution = + test_solution; + filtered_unfiltered_density_solution.block( + SolutionBlocks::unfiltered_density) = 0; + filter_adjoint_unfiltered_density_multiplier_solution.block( + SolutionBlocks::unfiltered_density_multiplier) = 0; + + filter_matrix.vmult(filtered_unfiltered_density_solution.block( + SolutionBlocks::unfiltered_density), + test_solution.block( + SolutionBlocks::unfiltered_density)); + filter_matrix.Tvmult( + filter_adjoint_unfiltered_density_multiplier_solution.block( + SolutionBlocks::unfiltered_density_multiplier), + test_solution.block(SolutionBlocks::unfiltered_density_multiplier)); + + + std::vector old_density_values(n_q_points); + std::vector> old_displacement_values(n_q_points); + std::vector old_displacement_divs(n_q_points); + std::vector> old_displacement_symmgrads(n_q_points); + std::vector> old_displacement_multiplier_values(n_q_points); + std::vector old_displacement_multiplier_divs(n_q_points); + std::vector> old_displacement_multiplier_symmgrads( + n_q_points); + std::vector old_lower_slack_multiplier_values(n_q_points); + std::vector old_upper_slack_multiplier_values(n_q_points); + std::vector old_lower_slack_values(n_q_points); + std::vector old_upper_slack_values(n_q_points); + std::vector old_unfiltered_density_values(n_q_points); + std::vector old_unfiltered_density_multiplier_values(n_q_points); + std::vector filtered_unfiltered_density_values(n_q_points); + std::vector filter_adjoint_unfiltered_density_multiplier_values( + n_q_points); + + using namespace ValueExtractors; + for (const auto &cell : dof_handler.active_cell_iterators()) + { + cell_rhs = 0; + + cell->get_dof_indices(local_dof_indices); + + fe_values.reinit(cell); + + lambda.value_list(fe_values.get_quadrature_points(), lambda_values); + mu.value_list(fe_values.get_quadrature_points(), mu_values); + + fe_values[densities].get_function_values(test_solution, + old_density_values); + fe_values[displacements].get_function_values( + test_solution, old_displacement_values); + fe_values[displacements].get_function_divergences( + test_solution, old_displacement_divs); + fe_values[displacements].get_function_symmetric_gradients( + test_solution, old_displacement_symmgrads); + fe_values[displacement_multipliers].get_function_values( + test_solution, old_displacement_multiplier_values); + fe_values[displacement_multipliers].get_function_divergences( + test_solution, old_displacement_multiplier_divs); + fe_values[displacement_multipliers] + .get_function_symmetric_gradients( + test_solution, old_displacement_multiplier_symmgrads); + fe_values[density_lower_slacks].get_function_values( + test_solution, old_lower_slack_values); + fe_values[density_lower_slack_multipliers].get_function_values( + test_solution, old_lower_slack_multiplier_values); + fe_values[density_upper_slacks].get_function_values( + test_solution, old_upper_slack_values); + fe_values[density_upper_slack_multipliers].get_function_values( + test_solution, old_upper_slack_multiplier_values); + fe_values[unfiltered_densities].get_function_values( + test_solution, old_unfiltered_density_values); + fe_values[unfiltered_density_multipliers].get_function_values( + test_solution, old_unfiltered_density_multiplier_values); + fe_values[unfiltered_densities].get_function_values( + filtered_unfiltered_density_solution, + filtered_unfiltered_density_values); + fe_values[unfiltered_density_multipliers].get_function_values( + filter_adjoint_unfiltered_density_multiplier_solution, + filter_adjoint_unfiltered_density_multiplier_values); + + for (const auto q_point : fe_values.quadrature_point_indices()) + { + for (const auto i : fe_values.dof_indices()) + { + const SymmetricTensor<2, dim> displacement_phi_i_symmgrad = + fe_values[displacements].symmetric_gradient(i, q_point); + const double displacement_phi_i_div = + fe_values[displacements].divergence(i, q_point); + + const SymmetricTensor<2, dim> + displacement_multiplier_phi_i_symmgrad = + fe_values[displacement_multipliers].symmetric_gradient( + i, q_point); + const double displacement_multiplier_phi_i_div = + fe_values[displacement_multipliers].divergence(i, + q_point); + + + const double density_phi_i = + fe_values[densities].value(i, q_point); + const double unfiltered_density_phi_i = + fe_values[unfiltered_densities].value(i, q_point); + const double unfiltered_density_multiplier_phi_i = + fe_values[unfiltered_density_multipliers].value(i, + q_point); + + const double lower_slack_multiplier_phi_i = + fe_values[density_lower_slack_multipliers].value( + i, q_point); + + const double lower_slack_phi_i = + fe_values[density_lower_slacks].value(i, q_point); + + const double upper_slack_phi_i = + fe_values[density_upper_slacks].value(i, q_point); + + const double upper_slack_multiplier_phi_i = + fe_values[density_upper_slack_multipliers].value( + i, q_point); + + /* Equation 1: This equation, along with equations + * 2 and 3, are the variational derivatives of the + * Lagrangian with respect to the decision + * variables - the density, displacement, and + * unfiltered density. */ + cell_rhs(i) += + -1 * fe_values.JxW(q_point) * + (density_penalty_exponent * + std::pow(old_density_values[q_point], + density_penalty_exponent - 1) * + density_phi_i * + (old_displacement_multiplier_divs[q_point] * + old_displacement_divs[q_point] * + lambda_values[q_point] + + 2 * mu_values[q_point] * + (old_displacement_symmgrads[q_point] * + old_displacement_multiplier_symmgrads[q_point])) - + density_phi_i * + old_unfiltered_density_multiplier_values[q_point]); + + /* Equation 2; the boundary terms will be added further down + * below. */ + cell_rhs(i) += + -1 * fe_values.JxW(q_point) * + (std::pow(old_density_values[q_point], + density_penalty_exponent) * + (old_displacement_multiplier_divs[q_point] * + displacement_phi_i_div * lambda_values[q_point] + + 2 * mu_values[q_point] * + (old_displacement_multiplier_symmgrads[q_point] * + displacement_phi_i_symmgrad))); + + /* Equation 3 */ + cell_rhs(i) += + -1 * fe_values.JxW(q_point) * + (unfiltered_density_phi_i * + filter_adjoint_unfiltered_density_multiplier_values + [q_point] + + unfiltered_density_phi_i * + old_upper_slack_multiplier_values[q_point] + + -1 * unfiltered_density_phi_i * + old_lower_slack_multiplier_values[q_point]); + + + + /* Equation 4; boundary term will again be dealt + * with below. This equation being driven to 0 + * ensures that the elasticity equation is met as + * a constraint. */ + cell_rhs(i) += -1 * fe_values.JxW(q_point) * + (std::pow(old_density_values[q_point], + density_penalty_exponent) * + (old_displacement_divs[q_point] * + displacement_multiplier_phi_i_div * + lambda_values[q_point] + + 2 * mu_values[q_point] * + (displacement_multiplier_phi_i_symmgrad * + old_displacement_symmgrads[q_point]))); + + /* Equation 5: This equation sets the lower slack + * variable equal to the unfiltered density, + * giving a minimum density of 0. */ + cell_rhs(i) += fe_values.JxW(q_point) * + (lower_slack_multiplier_phi_i * + (old_unfiltered_density_values[q_point] - + old_lower_slack_values[q_point])); + + /* Equation 6: This equation sets the upper slack + * variable equal to one minus the unfiltered + * density. */ + cell_rhs(i) += fe_values.JxW(q_point) * + (upper_slack_multiplier_phi_i * + (1 - old_unfiltered_density_values[q_point] - + old_upper_slack_values[q_point])); + + /* Equation 7: This is the difference between the + * density and the filter applied to the + * unfiltered density. This being driven to 0 by + * the Newton steps ensures that the filter is + * applied correctly. */ + cell_rhs(i) += fe_values.JxW(q_point) * + (unfiltered_density_multiplier_phi_i * + (old_density_values[q_point] - + filtered_unfiltered_density_values[q_point])); + + /* Equation 8: This along with equation 9 give the + * requirement that $s*z = \alpha$ for the barrier + * size alpha, and gives complementary slackness + * from KKT conditions when $\alpha$ goes to 0. */ + cell_rhs(i) += + -1 * fe_values.JxW(q_point) * + (lower_slack_phi_i * + (old_lower_slack_multiplier_values[q_point] - + barrier_size / old_lower_slack_values[q_point])); + + /* Equation 9 */ + cell_rhs(i) += + -1 * fe_values.JxW(q_point) * + (upper_slack_phi_i * + (old_upper_slack_multiplier_values[q_point] - + barrier_size / old_upper_slack_values[q_point])); + } + } + + for (const auto &face : cell->face_iterators()) + { + if (face->at_boundary() && + face->boundary_id() == BoundaryIds::down_force) + { + fe_face_values.reinit(cell, face); + + for (const auto face_q_point : + fe_face_values.quadrature_point_indices()) + { + for (const auto i : fe_face_values.dof_indices()) + { + Tensor<1, dim> traction; + traction[1] = -1.; + + cell_rhs(i) += + -1 * + (traction * fe_face_values[displacements].value( + i, face_q_point)) * + fe_face_values.JxW(face_q_point); + + cell_rhs(i) += + (traction * + fe_face_values[displacement_multipliers].value( + i, face_q_point)) * + fe_face_values.JxW(face_q_point); + } + } + } + } + + MatrixTools::local_apply_boundary_values(boundary_values, + local_dof_indices, + dummy_cell_matrix, + cell_rhs, + true); + + constraints.distribute_local_to_global(cell_rhs, + local_dof_indices, + test_rhs); + } + + return test_rhs; + } + + + // @sect4{Computing the merit function} + + // The algorithm we use herein uses a "watchdog" strategy to + // determine where and how far to go from the current iterate. We + // base the watchdog strategy on an exact $l_1$ merit function. This + // function calculates the exact $l_1$ merit of a given, putative, + // next iterate. + // + // The merit function consists of the sum of the objective function + // (which is simply an integral of external forces (on the boundary + // of the domain) times the displacement values of a test solution + // (typically, the current solution plus some multiple of the Newton + // update), and the $l_1$ norms of the Lagrange multiplier + // components of residual vectors. The following code computes these + // parts in turn: + template + double SANDTopOpt::calculate_exact_merit( + const BlockVector &test_solution) + { + TimerOutput::Scope t(timer, "merit function"); + + // Start with computing the objective function: + double objective_function_merit = 0; + { + MappingQGeneric mapping(1); + const QGauss quadrature_formula(fe.degree + 1); + const QGauss face_quadrature_formula(fe.degree + 1); + FEValues fe_values(mapping, + fe, + quadrature_formula, + update_values | update_gradients | + update_quadrature_points | update_JxW_values); + FEFaceValues fe_face_values(mapping, + fe, + face_quadrature_formula, + update_values | + update_quadrature_points | + update_normal_vectors | + update_JxW_values); + + const unsigned int n_face_q_points = face_quadrature_formula.size(); + + std::vector> displacement_face_values(n_face_q_points); + + for (const auto &cell : dof_handler.active_cell_iterators()) + { + for (const auto &face : cell->face_iterators()) + { + if (face->at_boundary() && + face->boundary_id() == BoundaryIds::down_force) + { + fe_face_values.reinit(cell, face); + fe_face_values[ValueExtractors::displacements] + .get_function_values(test_solution, + displacement_face_values); + for (unsigned int face_q_point = 0; + face_q_point < n_face_q_points; + ++face_q_point) + { + Tensor<1, dim> traction; + traction[1] = -1.; + + objective_function_merit += + (traction * displacement_face_values[face_q_point]) * + fe_face_values.JxW(face_q_point); + } + } + } + } + } + + for (const auto &cell : triangulation.active_cell_iterators()) + { + objective_function_merit = + objective_function_merit - + barrier_size * cell->measure() * + std::log(test_solution.block( + SolutionBlocks::density_lower_slack)[cell->active_cell_index()]); + objective_function_merit = + objective_function_merit - + barrier_size * cell->measure() * + std::log(test_solution.block( + SolutionBlocks::density_upper_slack)[cell->active_cell_index()]); + } + + // Then compute the residual and take the $l_1$ norms of the + // components that correspond to Lagrange mulipliers. We add + // those to the objective function computed above, and return + // the sum at the bottom: + const BlockVector test_rhs = calculate_test_rhs(test_solution); + + const double elasticity_constraint_merit = + penalty_multiplier * + test_rhs.block(SolutionBlocks::displacement_multiplier).l1_norm(); + const double filter_constraint_merit = + penalty_multiplier * + test_rhs.block(SolutionBlocks::unfiltered_density_multiplier).l1_norm(); + const double lower_slack_merit = + penalty_multiplier * + test_rhs.block(SolutionBlocks::density_lower_slack_multiplier).l1_norm(); + const double upper_slack_merit = + penalty_multiplier * + test_rhs.block(SolutionBlocks::density_upper_slack_multiplier).l1_norm(); + + const double total_merit = + objective_function_merit + elasticity_constraint_merit + + filter_constraint_merit + lower_slack_merit + upper_slack_merit; + return total_merit; + } + + + + // @sect4{Finding a search direction} + + // Next up is the function that actually computes a search direction + // starting at the current state (passed as the first argument) and + // returns the resulting vector. To this end, the function first + // calls the functions that assemble the linear system that + // corresponds to the Newton system, and that solve it. + + // This function also updates the penalty multiplier in the merit + // function, and then returns the largest scaled feasible step. + // It uses the `calculate_max_step_sizes()` function to find the + // largest feasible step that satisfies $s>0$ and $z>0$. + + template + BlockVector SANDTopOpt::find_max_step() + { + assemble_system(); + BlockVector step = solve(); + + // Next we are going to update penalty_multiplier. In + // essence, a larger penalty multiplier makes us consider the + // constraints more. Looking at the Hessian and gradient with + // respect to the step we want to take with our decision + // variables, and comparing that to the norm of our constraint + // error gives us a way to ensure that our merit function is + // "exact" - that is, it has a minimum in the same location + // that the objective function does. As our merit function is + // exact for any penalty multiplier over some minimum value, + // we only keep the computed value if it increases the penalty + // multiplier. + + const std::vector decision_variables = { + SolutionBlocks::density, + SolutionBlocks::displacement, + SolutionBlocks::unfiltered_density, + SolutionBlocks::density_upper_slack, + SolutionBlocks::density_lower_slack}; + double hess_part = 0; + double grad_part = 0; + for (const unsigned int decision_variable_i : decision_variables) + { + for (const unsigned int decision_variable_j : decision_variables) + { + Vector temp_vector(step.block(decision_variable_i).size()); + system_matrix.block(decision_variable_i, decision_variable_j) + .vmult(temp_vector, step.block(decision_variable_j)); + hess_part += step.block(decision_variable_i) * temp_vector; + } + grad_part -= system_rhs.block(decision_variable_i) * + step.block(decision_variable_i); + } + + const std::vector equality_constraint_multipliers = { + SolutionBlocks::displacement_multiplier, + SolutionBlocks::unfiltered_density_multiplier, + SolutionBlocks::density_lower_slack_multiplier, + SolutionBlocks::density_upper_slack_multiplier}; + double constraint_norm = 0; + for (unsigned int multiplier_i : equality_constraint_multipliers) + constraint_norm += system_rhs.block(multiplier_i).linfty_norm(); + + + double test_penalty_multiplier; + if (hess_part > 0) + test_penalty_multiplier = + (grad_part + .5 * hess_part) / (.05 * constraint_norm); + else + test_penalty_multiplier = (grad_part) / (.05 * constraint_norm); + + penalty_multiplier = std::max(penalty_multiplier, test_penalty_multiplier); + + // Based on all of this, we can now compute step sizes for the + // primal and dual (Lagrange multiplier) variables. Once we + // have these, we scale the components of the solution vector, + // and that is what this function returns. + const std::pair max_step_sizes = + calculate_max_step_size(nonlinear_solution, step); + const double step_size_s = max_step_sizes.first; + const double step_size_z = max_step_sizes.second; + + step.block(SolutionBlocks::density) *= step_size_s; + step.block(SolutionBlocks::displacement) *= step_size_s; + step.block(SolutionBlocks::unfiltered_density) *= step_size_s; + step.block(SolutionBlocks::displacement_multiplier) *= step_size_z; + step.block(SolutionBlocks::unfiltered_density_multiplier) *= step_size_z; + step.block(SolutionBlocks::density_lower_slack) *= step_size_s; + step.block(SolutionBlocks::density_lower_slack_multiplier) *= step_size_z; + step.block(SolutionBlocks::density_upper_slack) *= step_size_s; + step.block(SolutionBlocks::density_upper_slack_multiplier) *= step_size_z; + + return step; + } + + + + // @sect4{Computing a scaled step} + + // The next function then implements a back-tracking algorithm for a + // line search. It keeps shrinking step size until it finds a step + // where the merit is decreased, and then returns the new location + // based on the current state vector, and the direction to go into, + // times the step length. + template + BlockVector + SANDTopOpt::compute_scaled_step(const BlockVector &state, + const BlockVector &max_step, + const double descent_requirement) + { + const double merit_derivative = + (calculate_exact_merit(state + 1e-4 * max_step) - + calculate_exact_merit(state)) / + 1e-4; + double step_size = 1; + unsigned int max_linesearch_iterations = 10; + for (unsigned int k = 0; k < max_linesearch_iterations; ++k) + { + if (calculate_exact_merit(state + step_size * max_step) < + calculate_exact_merit(state) + + step_size * descent_requirement * merit_derivative) + break; + else + step_size = step_size / 2; + } + return state + (step_size * max_step); + } + + + // @sect4{Checking for convergence} + + // The final auxiliary function in this block is the one that checks + // to see if the KKT conditions are sufficiently met so that the + // overall algorithm can lower the barrier size. It does so by + // computing the $l_1$ norm of the residual, which is what + // `calculate_test_rhs()` computes. + template + bool SANDTopOpt::check_convergence(const BlockVector &state) + { + const BlockVector test_rhs = calculate_test_rhs(state); + const double test_rhs_norm = test_rhs.l1_norm(); + + const double convergence_condition = 1e-2; + const double target_norm = convergence_condition * barrier_size; + + std::cout << " Checking convergence. Current rhs norm is " + << test_rhs_norm << ", target is " << target_norm << std::endl; + + return (test_rhs_norm < target_norm); + } + + + // @sect3{Postprocessing the solution} + + // The first of the postprocessing functions outputs information + // in a VTU file for visualization. It looks long, but it's really + // just the same as what was done in step-22, for example, just + // with (a lot) more solution variables: + template + void SANDTopOpt::output_results(const unsigned int iteration) const + { + std::vector solution_names(1, "density"); + std::vector + data_component_interpretation( + 1, DataComponentInterpretation::component_is_scalar); + for (unsigned int i = 0; i < dim; ++i) + { + solution_names.emplace_back("displacement"); + data_component_interpretation.push_back( + DataComponentInterpretation::component_is_part_of_vector); + } + solution_names.emplace_back("unfiltered_density"); + data_component_interpretation.push_back( + DataComponentInterpretation::component_is_scalar); + for (unsigned int i = 0; i < dim; ++i) + { + solution_names.emplace_back("displacement_multiplier"); + data_component_interpretation.push_back( + DataComponentInterpretation::component_is_part_of_vector); + } + solution_names.emplace_back("unfiltered_density_multiplier"); + data_component_interpretation.push_back( + DataComponentInterpretation::component_is_scalar); + solution_names.emplace_back("low_slack"); + data_component_interpretation.push_back( + DataComponentInterpretation::component_is_scalar); + solution_names.emplace_back("low_slack_multiplier"); + data_component_interpretation.push_back( + DataComponentInterpretation::component_is_scalar); + solution_names.emplace_back("high_slack"); + data_component_interpretation.push_back( + DataComponentInterpretation::component_is_scalar); + solution_names.emplace_back("high_slack_multiplier"); + data_component_interpretation.push_back( + DataComponentInterpretation::component_is_scalar); + + DataOut data_out; + data_out.attach_dof_handler(dof_handler); + data_out.add_data_vector(nonlinear_solution, + solution_names, + DataOut::type_dof_data, + data_component_interpretation); + data_out.build_patches(); + + std::ofstream output("solution" + std::to_string(iteration) + ".vtu"); + data_out.write_vtu(output); + } + + + // The second of these functions outputs the solution as an `.stl` + // file for 3d + // printing. [STL](https://en.wikipedia.org/wiki/STL_(file_format)) + // files are made up of triangles and normal vectors, and we will + // use it to show all of those cells with a density value larger + // than zero by first extruding the mesh from a $z$ value of zero + // to $z=0.25$, and then generating two triangles for each face of + // the cells with a sufficiently large density value. The triangle + // nodes must go counter-clockwise when looking from the outside, + // and the normal vectors must be unit vectors pointing outwards, + // which requires a few checks. + template + void SANDTopOpt::write_as_stl() + { + static_assert(dim == 2, + "This function is not implemented for anything " + "other than the 2d case."); + + std::ofstream stlfile; + stlfile.open("bridge.stl"); + + stlfile << "solid bridge\n" << std::scientific; + double height = .25; + + for (const auto &cell : dof_handler.active_cell_iterators()) + { + if (nonlinear_solution.block( + SolutionBlocks::density)[cell->active_cell_index()] > 0.5) + { + // We have now found a cell with a density value larger + // than zero. Let us start by writing out the bottom + // and top faces. Owing to the ordering issue mentioned + // above, we have to make sure that we understand + // whether a cell has a right- or left-handed + // coordinate system. We do this by interrogating the + // directions of the two edges starting at vertex 0 and + // whether they form a right-handed coordinate system. + const Tensor<1, dim> edge_directions[2] = {cell->vertex(1) - + cell->vertex(0), + cell->vertex(2) - + cell->vertex(0)}; + const Tensor<2, dim> edge_tensor( + {{edge_directions[0][0], edge_directions[0][1]}, + {edge_directions[1][0], edge_directions[1][1]}}); + const bool is_right_handed_cell = (determinant(edge_tensor) > 0); + + if (is_right_handed_cell) + { + /* Write one side at z = 0. */ + stlfile << " facet normal " << 0.000000e+00 << " " + << 0.000000e+00 << " " << -1.000000e+00 << "\n"; + stlfile << " outer loop\n"; + stlfile << " vertex " << cell->vertex(0)[0] << " " + << cell->vertex(0)[1] << " " << 0.000000e+00 << "\n"; + stlfile << " vertex " << cell->vertex(2)[0] << " " + << cell->vertex(2)[1] << " " << 0.000000e+00 << "\n"; + stlfile << " vertex " << cell->vertex(1)[0] << " " + << cell->vertex(1)[1] << " " << 0.000000e+00 << "\n"; + stlfile << " endloop\n"; + stlfile << " endfacet\n"; + stlfile << " facet normal " << 0.000000e+00 << " " + << 0.000000e+00 << " " << -1.000000e+00 << "\n"; + stlfile << " outer loop\n"; + stlfile << " vertex " << cell->vertex(1)[0] << " " + << cell->vertex(1)[1] << " " << 0.000000e+00 << "\n"; + stlfile << " vertex " << cell->vertex(2)[0] << " " + << cell->vertex(2)[1] << " " << 0.000000e+00 << "\n"; + stlfile << " vertex " << cell->vertex(3)[0] << " " + << cell->vertex(3)[1] << " " << 0.000000e+00 << "\n"; + stlfile << " endloop\n"; + stlfile << " endfacet\n"; + + /* Write one side at z = height. */ + stlfile << " facet normal " << 0.000000e+00 << " " + << 0.000000e+00 << " " << 1.000000e+00 << "\n"; + stlfile << " outer loop\n"; + stlfile << " vertex " << cell->vertex(0)[0] << " " + << cell->vertex(0)[1] << " " << height << "\n"; + stlfile << " vertex " << cell->vertex(1)[0] << " " + << cell->vertex(1)[1] << " " << height << "\n"; + stlfile << " vertex " << cell->vertex(2)[0] << " " + << cell->vertex(2)[1] << " " << height << "\n"; + stlfile << " endloop\n"; + stlfile << " endfacet\n"; + stlfile << " facet normal " << 0.000000e+00 << " " + << 0.000000e+00 << " " << 1.000000e+00 << "\n"; + stlfile << " outer loop\n"; + stlfile << " vertex " << cell->vertex(1)[0] << " " + << cell->vertex(1)[1] << " " << height << "\n"; + stlfile << " vertex " << cell->vertex(3)[0] << " " + << cell->vertex(3)[1] << " " << height << "\n"; + stlfile << " vertex " << cell->vertex(2)[0] << " " + << cell->vertex(2)[1] << " " << height << "\n"; + stlfile << " endloop\n"; + stlfile << " endfacet\n"; + } + else /* The cell has a left-handed set up */ + { + /* Write one side at z = 0. */ + stlfile << " facet normal " << 0.000000e+00 << " " + << 0.000000e+00 << " " << -1.000000e+00 << "\n"; + stlfile << " outer loop\n"; + stlfile << " vertex " << cell->vertex(0)[0] << " " + << cell->vertex(0)[1] << " " << 0.000000e+00 << "\n"; + stlfile << " vertex " << cell->vertex(1)[0] << " " + << cell->vertex(1)[1] << " " << 0.000000e+00 << "\n"; + stlfile << " vertex " << cell->vertex(2)[0] << " " + << cell->vertex(2)[1] << " " << 0.000000e+00 << "\n"; + stlfile << " endloop\n"; + stlfile << " endfacet\n"; + stlfile << " facet normal " << 0.000000e+00 << " " + << 0.000000e+00 << " " << -1.000000e+00 << "\n"; + stlfile << " outer loop\n"; + stlfile << " vertex " << cell->vertex(1)[0] << " " + << cell->vertex(1)[1] << " " << 0.000000e+00 << "\n"; + stlfile << " vertex " << cell->vertex(3)[0] << " " + << cell->vertex(3)[1] << " " << 0.000000e+00 << "\n"; + stlfile << " vertex " << cell->vertex(2)[0] << " " + << cell->vertex(2)[1] << " " << 0.000000e+00 << "\n"; + stlfile << " endloop\n"; + stlfile << " endfacet\n"; + + /* Write one side at z = height. */ + stlfile << " facet normal " << 0.000000e+00 << " " + << 0.000000e+00 << " " << 1.000000e+00 << "\n"; + stlfile << " outer loop\n"; + stlfile << " vertex " << cell->vertex(0)[0] << " " + << cell->vertex(0)[1] << " " << height << "\n"; + stlfile << " vertex " << cell->vertex(2)[0] << " " + << cell->vertex(2)[1] << " " << height << "\n"; + stlfile << " vertex " << cell->vertex(1)[0] << " " + << cell->vertex(1)[1] << " " << height << "\n"; + stlfile << " endloop\n"; + stlfile << " endfacet\n"; + stlfile << " facet normal " << 0.000000e+00 << " " + << 0.000000e+00 << " " << 1.000000e+00 << "\n"; + stlfile << " outer loop\n"; + stlfile << " vertex " << cell->vertex(1)[0] << " " + << cell->vertex(1)[1] << " " << height << "\n"; + stlfile << " vertex " << cell->vertex(2)[0] << " " + << cell->vertex(2)[1] << " " << height << "\n"; + stlfile << " vertex " << cell->vertex(3)[0] << " " + << cell->vertex(3)[1] << " " << height << "\n"; + stlfile << " endloop\n"; + stlfile << " endfacet\n"; + } + + // Next we need to deal with the four faces of the + // cell, extended into the $z$ direction. However, we + // only need to write these faces if either the face + // is on the domain boundary, or if it is the + // interface between a cell with density greater than + // 0.5, and a cell with a density less than 0.5. + for (unsigned int face_number = 0; + face_number < GeometryInfo::faces_per_cell; + ++face_number) + { + const typename DoFHandler::face_iterator face = + cell->face(face_number); + + if ((face->at_boundary()) || + (!face->at_boundary() && + (nonlinear_solution.block( + 0)[cell->neighbor(face_number)->active_cell_index()] < + 0.5))) + { + const Tensor<1, dim> normal_vector = + (face->center() - cell->center()); + const double normal_norm = normal_vector.norm(); + if ((face->vertex(0)[0] - face->vertex(0)[0]) * + (face->vertex(1)[1] - face->vertex(0)[1]) * + 0.000000e+00 + + (face->vertex(0)[1] - face->vertex(0)[1]) * (0 - 0) * + normal_vector[0] + + (height - 0) * + (face->vertex(1)[0] - face->vertex(0)[0]) * + normal_vector[1] - + (face->vertex(0)[0] - face->vertex(0)[0]) * (0 - 0) * + normal_vector[1] - + (face->vertex(0)[1] - face->vertex(0)[1]) * + (face->vertex(1)[0] - face->vertex(0)[0]) * + normal_vector[0] - + (height - 0) * + (face->vertex(1)[1] - face->vertex(0)[1]) * 0 > + 0) + { + stlfile << " facet normal " + << normal_vector[0] / normal_norm << " " + << normal_vector[1] / normal_norm << " " + << 0.000000e+00 << "\n"; + stlfile << " outer loop\n"; + stlfile << " vertex " << face->vertex(0)[0] + << " " << face->vertex(0)[1] << " " + << 0.000000e+00 << "\n"; + stlfile << " vertex " << face->vertex(0)[0] + << " " << face->vertex(0)[1] << " " << height + << "\n"; + stlfile << " vertex " << face->vertex(1)[0] + << " " << face->vertex(1)[1] << " " + << 0.000000e+00 << "\n"; + stlfile << " endloop\n"; + stlfile << " endfacet\n"; + stlfile << " facet normal " + << normal_vector[0] / normal_norm << " " + << normal_vector[1] / normal_norm << " " + << 0.000000e+00 << "\n"; + stlfile << " outer loop\n"; + stlfile << " vertex " << face->vertex(0)[0] + << " " << face->vertex(0)[1] << " " << height + << "\n"; + stlfile << " vertex " << face->vertex(1)[0] + << " " << face->vertex(1)[1] << " " << height + << "\n"; + stlfile << " vertex " << face->vertex(1)[0] + << " " << face->vertex(1)[1] << " " + << 0.000000e+00 << "\n"; + stlfile << " endloop\n"; + stlfile << " endfacet\n"; + } + else + { + stlfile << " facet normal " + << normal_vector[0] / normal_norm << " " + << normal_vector[1] / normal_norm << " " + << 0.000000e+00 << "\n"; + stlfile << " outer loop\n"; + stlfile << " vertex " << face->vertex(0)[0] + << " " << face->vertex(0)[1] << " " + << 0.000000e+00 << "\n"; + stlfile << " vertex " << face->vertex(1)[0] + << " " << face->vertex(1)[1] << " " + << 0.000000e+00 << "\n"; + stlfile << " vertex " << face->vertex(0)[0] + << " " << face->vertex(0)[1] << " " << height + << "\n"; + stlfile << " endloop\n"; + stlfile << " endfacet\n"; + stlfile << " facet normal " + << normal_vector[0] / normal_norm << " " + << normal_vector[1] / normal_norm << " " + << 0.000000e+00 << "\n"; + stlfile << " outer loop\n"; + stlfile << " vertex " << face->vertex(0)[0] + << " " << face->vertex(0)[1] << " " << height + << "\n"; + stlfile << " vertex " << face->vertex(1)[0] + << " " << face->vertex(1)[1] << " " + << 0.000000e+00 << "\n"; + stlfile << " vertex " << face->vertex(1)[0] + << " " << face->vertex(1)[1] << " " << height + << "\n"; + stlfile << " endloop\n"; + stlfile << " endfacet\n"; + } + } + } + } + } + stlfile << "endsolid bridge"; + } + + + + // @sect3{The run() function driving the overall algorithm} + + // This function finally provides the overall driver logic. It is, + // in the grand scheme of things, a rather complicated function + // primarily because the optimization algorithm is difficult: It + // isn't just about finding a Newton direction like in step-15 and + // then going a fixed distance in this direction any more, but + // instead about (i) determining what the optimal log-barrier + // penalty parameter should be in the current step, (ii) a + // complicated algorithm to determine how far we want to go, and + // other ingredients. Let us see how we can break this down into + // smaller chunks in the following documentation. + // + // The function starts out simple enough with first setting up the + // mesh, the DoFHandler, and then the various linear algebra objects + // necessary for the following: + template + void SANDTopOpt::run() + { + std::cout << "filter r is: " << filter_r << std::endl; + + { + TimerOutput::Scope t(timer, "setup"); + + create_triangulation(); + + dof_handler.distribute_dofs(fe); + DoFRenumbering::component_wise(dof_handler); + + setup_boundary_values(); + setup_block_system(); + setup_filter_matrix(); + } + + // We then set a number of parameters that affect the + // log-barrier and line search components of the optimization + // algorithm: + barrier_size = 25; + const double min_barrier_size = .0005; + + const unsigned int max_uphill_steps = 8; + const double descent_requirement = .0001; + + + // Now start the principal iteration. The overall algorithm + // works by using an outer loop in which we loop until either + // (i) the log-barrier parameter has become small enough, or (ii) + // we have reached convergence. In any case, we terminate if + // end up with too large a number of iterations. This overall + // structure is encoded as a `do { ... } while (...)` loop + // where the convergence condition is at the bottom. + unsigned int iteration_number = 0; + const unsigned int max_iterations = 10000; + + do + { + std::cout << "Starting outer step in iteration " << iteration_number + << " with barrier parameter " << barrier_size << std::endl; + + // Within this outer loop, we have an inner loop in which we + // try to find an update direction using the watchdog + // algorithm described in the introduction. + // + // The general idea of the watchdog algorithm itself is + // this: For a maximum of `max_uphill_steps` (i.e., a loop + // within the "inner loop" mentioned above) attempts, we use + // `find_max_step()` to compute a Newton update step, and + // add these up in the `nonlinear_solution` vector. In each of + // these attempts (starting from the place reached at the + // end of the previous attempt), we check whether we have + // reached a target value of the merit function described + // above. The target value is computed based on where this + // algorithm starts (the `nonlinear_solution` at the beginning of + // the watchdog loop, saves as `watchdog_state`) and the + // first proposed direction provided by `find_max_step()` in + // the first go-around of this loop (the `k==0` case). + do + { + std::cout << " Starting inner step in iteration " + << iteration_number + << " with merit function penalty multiplier " + << penalty_multiplier << std::endl; + + bool watchdog_step_found = false; + + const BlockVector watchdog_state = nonlinear_solution; + BlockVector first_step; + double target_merit = numbers::signaling_nan(); + double merit_derivative = numbers::signaling_nan(); + + for (unsigned int k = 0; k < max_uphill_steps; ++k) + { + ++iteration_number; + const BlockVector update_step = find_max_step(); + + if (k == 0) + { + first_step = update_step; + merit_derivative = + ((calculate_exact_merit(watchdog_state + + .0001 * first_step) - + calculate_exact_merit(watchdog_state)) / + .0001); + target_merit = calculate_exact_merit(watchdog_state) + + descent_requirement * merit_derivative; + } + + nonlinear_solution += update_step; + const double current_merit = + calculate_exact_merit(nonlinear_solution); + + std::cout << " current watchdog state merit is: " + << current_merit << "; target merit is " + << target_merit << std::endl; + + if (current_merit < target_merit) + { + watchdog_step_found = true; + std::cout << " found workable step after " << k + 1 + << " iterations" << std::endl; + break; + } + } + + + // The next part of the algorithm then depends on + // whether the watchdog loop above succeeded. If it + // did, then we are satisfied and no further action is + // necessary: We just stay where we are. If, however, + // we took the maximal number of unsuccessful steps in + // the loop above, then we need to do something else, + // and this is what the following code block does. + // + // Specifically, from the final (unsuccessful) state + // of the loop above, we seek one more update + // direction and take what is called a "stretch + // step". If that stretch state satisfies a condition + // involving the merit function, then we go there. On + // the other hand, if the stretch state is also + // unacceptable (as all of the watchdog steps above + // were), then we discard all of the watchdog steps + // taken above and start over again where we had + // started the watchdog iterations -- that place was + // stored in the `watchdog_state` variable above. More + // specifically, the conditions below first test + // whether we take a step from `watchdog_state` in + // direction `first_step`, or whether we can do one + // more update from the stretch state to find a new + // place. It is possible that neither of these is + // actually better than the state we started from at + // the beginning of the watchdog algorithm, but even + // if that is so, that place clearly was a difficult + // place to be in, and getting away to start the next + // iteration from another place might be a useful + // strategy to eventually converge. + // + // We keep repeating the watchdog steps above along + // with the logic below until this inner iteration is + // finally converged (or if we run up against the + // maximal number of iterations -- where we count the + // number of linear solves as iterations and increment + // the counter every time we call `find_max_step()` + // since that is where the linear solve actually + // happens). In any case, at the end of each of these + // inner iterations we also output the solution in a + // form suitable for visualization. + + if (watchdog_step_found == false) + { + ++iteration_number; + const BlockVector update_step = find_max_step(); + const BlockVector stretch_state = + compute_scaled_step(nonlinear_solution, + update_step, + descent_requirement); + + // If we did not get a successful watchdog step, + // we now need to decide between going back to + // where we started, or using the final state. We + // compare the merits of both of these locations, + // and then take a scaled step from whichever + // location is better. As the scaled step is + // guaranteed to lower the merit, we will end up + // keeping one of the two. + if ((calculate_exact_merit(nonlinear_solution) < + calculate_exact_merit(watchdog_state)) || + (calculate_exact_merit(stretch_state) < target_merit)) + { + std::cout << " Taking scaled step from end of watchdog" + << std::endl; + nonlinear_solution = stretch_state; + } + else + { + std::cout + << " Taking scaled step from beginning of watchdog" + << std::endl; + if (calculate_exact_merit(stretch_state) > + calculate_exact_merit(watchdog_state)) + { + nonlinear_solution = + compute_scaled_step(watchdog_state, + first_step, + descent_requirement); + } + else + { + ++iteration_number; + nonlinear_solution = stretch_state; + const BlockVector stretch_step = + find_max_step(); + nonlinear_solution = + compute_scaled_step(nonlinear_solution, + stretch_step, + descent_requirement); + } + } + } + + output_results(iteration_number); + } + while ((iteration_number < max_iterations) && + (check_convergence(nonlinear_solution) == false)); + + + // At the end of the outer loop, we have to update the + // barrier parameter, for which we use the following + // formula. The rest of the function is then simply about + // checking the outer loop convergence condition, and if + // we decide to terminate computations, about writing the + // final "design" as an STL file for use in 3d printing, + // and to output some timing information. + const double barrier_size_multiplier = .8; + const double barrier_size_exponent = 1.2; + + barrier_size = + std::max(std::min(barrier_size * barrier_size_multiplier, + std::pow(barrier_size, barrier_size_exponent)), + min_barrier_size); + + std::cout << std::endl; + } + while (((barrier_size > min_barrier_size) || + (check_convergence(nonlinear_solution) == false)) && + (iteration_number < max_iterations)); + + write_as_stl(); + timer.print_summary(); + } +} // namespace SAND + +// @sect3{The main function} + +// The remainder of the code, the `main()` function, is as usual: +int main() +{ + try + { + SAND::SANDTopOpt<2> elastic_problem_2d; + elastic_problem_2d.run(); + } + catch (std::exception &exc) + { + std::cerr << std::endl + << std::endl + << "----------------------------------------------------" + << std::endl; + std::cerr << "Exception on processing: " << std::endl + << exc.what() << std::endl + << "Aborting!" << std::endl + << "----------------------------------------------------" + << std::endl; + + return 1; + } + catch (...) + { + std::cerr << std::endl + << std::endl + << "----------------------------------------------------" + << std::endl; + std::cerr << "Unknown exception!" << std::endl + << "Aborting!" << std::endl + << "----------------------------------------------------" + << std::endl; + return 1; + } + return 0; +} -- 2.39.5