Higher-order moving mesh methods for PDE-constrained shape optimization

TL;DR

This paper introduces a higher-order moving mesh method for PDE-constrained shape optimization, enabling high-resolution, smooth shape representations and accurate solutions through advanced discretization techniques.

Contribution

It generalizes existing moving mesh methods to higher-order deformations compatible with high-order finite element discretizations.

Findings

Achieves high-accuracy solutions for PDE-constrained shape optimization

Allows arbitrarily smooth and high-resolution shape representations

Demonstrates effectiveness through numerical experiments

Abstract

We present a new approach to discretizing shape optimization problems that generalizes standard moving mesh methods to higher-order mesh deformations and that is naturally compatible with higher-order finite element discretizations of PDE-constraints. This shape optimization method is based on discretized deformation diffeomorphisms and allows for arbitrarily high resolution of shapes with arbitrary smoothness. Numerical experiments show that it allows the solution of PDE-constrained shape optimization problems to high accuracy.