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

TL;DR

FormOpt is a FEniCSx-based toolbox enabling efficient parallel shape optimization in 2D and 3D, integrating shape sensitivity analysis, level set methods, and geometric constraints for educational and practical applications.

Contribution

It introduces a flexible, parallel-capable shape optimization toolbox built on FEniCSx, extending existing methods with new parallel modes and detailed numerical implementation.

Findings

Supports three parallel modes: data, task, and mixed.

Efficiently handles geometric constraints with a Proximal-Perturbed Lagrangian.

Demonstrates scalability and effectiveness in shape optimization tasks.

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.