A Local Mesh Modification Strategy for Interface Problems with Application to Shape and Topology Optimization

TL;DR

This paper introduces a finite element method that locally modifies triangulations to accurately resolve interfaces, ensuring optimal convergence and condition numbers, demonstrated through an electric motor design optimization.

Contribution

The paper proposes a novel local mesh modification technique for interface problems that improves accuracy and stability, with applications to shape and topology optimization.

Findings

Achieves optimal order of convergence.

Ensures optimal condition number of the stiffness matrix.

Successfully applied to electric motor design with evolving interfaces.

Abstract

We present and analyze a new finite element method for solving interface problems on a triangular grid. The method locally modifies a given triangulation such that the interfaces are accurately resolved and the maximal angle condition holds. Therefore, optimal order of convergence can be shown. Moreover, an appropriate scaling of the basis functions yields an optimal condition number of the stiffness matrix. The method is applied to an optimal design problem for an electric motor where the interface between different materials is evolving in the course of the optimization procedure.