A new Diffuse-interface approximation of the Willmore flow

TL;DR

This paper introduces a novel two-variable diffuse approximation for the Willmore flow that prevents intersecting phase boundaries and aligns better with sharp interface evolution, supported by theoretical and numerical validation.

Contribution

A new diffuse approximation with a simple penalization term is proposed, ensuring quasi-one dimensional phase fields and improved approximation of Willmore flow.

Findings

Gamma convergence of the energies is established.

Ground states are shown to be one-dimensional.

Numerical simulations demonstrate the effectiveness of the new approach.

Abstract

Standard diffuse approximations of the Willmore flow often lead to intersecting phase boundaries that in many cases do not correspond to the intended sharp interface evolution. Here we introduce a new two-variable diffuse approximation that includes a rather simple but efficient penalization of the deviation from a quasi-one dimensional structure of the phase fields. We justify the approximation property by a Gamma convergence result for the energies and a matched asymptotic expansion for the flow. Ground states of the energy are shown to be one-dimensional, in contrast to the presence of saddle solutions for the usual diffuse approximation. Finally we present numerical simulations that illustrate the approximation property and apply our new approach to problems where the usual approach leads to an undesired behavior.