First and Second Order Shape Optimization based on Restricted Mesh Deformations

TL;DR

This paper introduces a restricted mesh deformation approach for shape optimization that prevents mesh deterioration, utilizing first and second order methods inspired by the Hadamard structure theorem, with practical implementation in FEniCS.

Contribution

It proposes a novel restriction on mesh deformations based on the Hadamard theorem, improving mesh quality during shape optimization without remeshing.

Findings

Mesh degeneracy can be avoided with the proposed restrictions.

First and second order methods are effective in maintaining mesh integrity.

The approach eliminates the need for remeshing in shape optimization.

Abstract

We consider shape optimization problems subject to elliptic partial differential equations. In the context of the finite element method, the geometry to be optimized is represented by the computational mesh, and the optimization proceeds by repeatedly updating the mesh node positions. It is well known that such a procedure eventually may lead to a deterioration of mesh quality, or even an invalidation of the mesh, when interior nodes penetrate neighboring cells. We examine this phenomenon, which can be traced back to the ineptness of the discretized objective when considered over the space of mesh node positions. As a remedy, we propose a restriction in the admissible mesh deformations, inspired by the Hadamard structure theorem. First and second order methods are considered in this setting. Numerical results show that mesh degeneracy can be overcome, avoiding the need for remeshing or other strategies. FEniCS code for the proposed methods is available on GitHub.