Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

FormOpt: A FEniCSx toolbox for level set-based shape optimization supporting parallel computing

Josué D. Díaz-Avalos, Antoine Laurain

TL;DR

This article presents the toolbox FormOpt for two- and three-dimensional shape optimization with parallel computing capabilities, built on the FEniCSx software framework, and adopts an optimize-then-discretize strategy based on the distributed shape derivative and its tensor representation.

Abstract

This article presents the toolbox FormOpt for two- and three-dimensional shape optimization with parallel computing capabilities, built on the FEniCSx software framework. We introduce fundamental concepts of shape sensitivity analysis and their numerical applications, mainly for educational purposes, while also emphasizing computational efficiency via parallelism for practitioners. We adopt an optimize-then-discretize strategy based on the distributed shape derivative and its tensor representation, following the approach of \cite{MR3843884} and extending it in several directions. The numerical shape modeling relies on a level set method, whose evolution is driven by a descent direction computed from the shape derivative. Geometric constraints are treated accurately through a Proximal-Perturbed Lagrangian approach. FormOpt leverages the powerful features of FEniCSx, particularly its support for weak formulations of partial differential equations, diverse finite element types, and scalable parallelism. The implementation supports three different parallel computing modes: data parallelism, task parallelism, and a mixed mode. Data parallelism exploits FEniCSx's mesh partitioning features, and we implement a task parallelism mode which is useful for problems governed by a set of partial differential equations with varying parameters. The mixed mode conveniently combines both strategies to achieve efficient utilization of computational resources.

FormOpt: A FEniCSx toolbox for level set-based shape optimization supporting parallel computing

TL;DR

This article presents the toolbox FormOpt for two- and three-dimensional shape optimization with parallel computing capabilities, built on the FEniCSx software framework, and adopts an optimize-then-discretize strategy based on the distributed shape derivative and its tensor representation.

Abstract

This article presents the toolbox FormOpt for two- and three-dimensional shape optimization with parallel computing capabilities, built on the FEniCSx software framework. We introduce fundamental concepts of shape sensitivity analysis and their numerical applications, mainly for educational purposes, while also emphasizing computational efficiency via parallelism for practitioners. We adopt an optimize-then-discretize strategy based on the distributed shape derivative and its tensor representation, following the approach of \cite{MR3843884} and extending it in several directions. The numerical shape modeling relies on a level set method, whose evolution is driven by a descent direction computed from the shape derivative. Geometric constraints are treated accurately through a Proximal-Perturbed Lagrangian approach. FormOpt leverages the powerful features of FEniCSx, particularly its support for weak formulations of partial differential equations, diverse finite element types, and scalable parallelism. The implementation supports three different parallel computing modes: data parallelism, task parallelism, and a mixed mode. Data parallelism exploits FEniCSx's mesh partitioning features, and we implement a task parallelism mode which is useful for problems governed by a set of partial differential equations with varying parameters. The mixed mode conveniently combines both strategies to achieve efficient utilization of computational resources.
Paper Structure (19 sections, 1 theorem, 99 equations, 15 figures, 1 algorithm)

This paper contains 19 sections, 1 theorem, 99 equations, 15 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model problem and shape derivatives
  3. Shape derivative for a model problem
  4. Main algorithm
  5. Numerical implementation
  6. Toolbox structure
  7. Model construction
  8. Parallelization
  9. Lagrangian method
  10. Velocity
  11. Transport equation
  12. Other implementation aspects
  13. Numerical results
  14. Inverse elasticity
  15. Compliance minimization
  16. ...and 4 more sections

Key Result

Proposition 1

Let $\Omega\in \mathds{P}$ and assume $\partial \Omega$ is $C^2$. Suppose that $dJ(\Omega)$ has the tensor representation ea:volume_form2. If $S_1^+\in W^{1,1}(\Omega,\mathds{R}^{d\times d})$ and $S_1^-\in W^{1,1}(\mathcal{D}\setminus\overline \Omega,\mathds{R}^{d\times d})$, then we obtain the so-c with $G := [(S_1^+-S_1^-)n]\cdot n,$ where $+$ and $-$ denote the traces on $\partial\Omega$ of th

Figures (15)

  • Figure 1: Geometric configuration for the inverse problem in elasticity.
  • Figure 2: Inverse elasticity, Example 1. Initial and recovered inclusion. A mesh with $12\,713$ triangles was employed. The green line represents the ground truth.
  • Figure 3: Inverse elasticity, Example 2. Initial and recovered inclusions. A mesh with $13\,391$ triangles was employed. The green line represents the ground truth.
  • Figure 4: Compliance minimization, Example 1. Initial guess (top-left) and optimized design (top-right) at iteration $i=50$. The resulting level set function $\phi^{i}$ approximates the distance function associated to $\Omega^{i}$ (bottom). The domain $\mathcal{D}$ was discretized with $20\,946$ triangles. $\Gamma_0$ appears in red and $\Gamma_1$ in blue.
  • Figure 5: Compliance minimization, Example 2. The three-dimensional cantilever at $i=0$ and $i=14$ (top); $i=30$ and $i=81$ (bottom). A mesh with $2\,592\,000$ tetrahedra was employed.
  • ...and 10 more figures

Theorems & Definitions (3)

  • Definition 1: Shape derivative
  • Definition 2: Tensor representation
  • Proposition 1: Hadamard form