On algorithms with good mesh properties for problems with moving boundaries based on the Harmonic Map Heat Flow and the DeTurck trick

TL;DR

This paper introduces a novel numerical approach using the DeTurck trick and harmonic map heat flow to improve mesh quality in simulations involving moving boundaries and evolving submanifolds.

Contribution

It presents a new reparametrization technique that enhances mesh properties for problems with moving boundaries, integrating the ALE-method for practical computations.

Findings

Meshes maintain high quality during evolution

The approach simplifies handling boundary movements

Compatible with existing ALE-method algorithms

Abstract

In this paper, we present a general approach to obtain numerical schemes with good mesh properties for problems with moving boundaries, that is for evolving submanifolds with boundaries. This includes moving domains and surfaces with boundaries. Our approach is based on a variant of the so-called the DeTurck trick. By reparametrizing the evolution of the submanifold via solutions to the harmonic map heat flow of manifolds with boundary, we obtain a new velocity field for the motion of the submanifold. Moving the vertices of the computational mesh according to this velocity field automatically leads to computational meshes of high quality both for the submanifold and its boundary. Using the ALE-method in [16], this idea can be easily built into algorithms for the computation of physical problems with moving boundaries.