A Cahn--Hilliard--Willmore phase field model for non-oriented interfaces

TL;DR

This paper introduces a new phase field model for non-oriented interfaces that uses a variational approach with a Cahn-Hilliard functional and Willmore stabilization, providing theoretical analysis and numerical validation.

Contribution

It replaces neural network methods with a standard variational approach for modeling non-oriented interfaces, including $mbda$-convergence analysis and numerical schemes.

Findings

The model approximates mean curvature flow in 2D and 3D.

The variational approach achieves results comparable to neural network methods.

Numerical simulations validate the effectiveness of the proposed flow.

Abstract

We investigate a new phase field model for representing non-oriented interfaces, approximating their area and simulating their area-minimizing flow. Our contribution is related to the approach proposed in arXiv:2105.09627 that involves ad hoc neural networks. We show here that, instead of neural networks, similar results can be obtained using a more standard variational approach that combines a Cahn-Hilliard-type functional involving an appropriate non-smooth potential and a Willmore-type stabilization energy. We give a $\Gamma$-convergence analysis of this phase field model in dimension $1$ and, for radially symmetric functions, in arbitrary dimension. We also propose a simple numerical scheme to approximate its $L^2$-gradient flow. We illustrate numerically that the new flow approximates fairly well the mean curvature flow of codimension $1$ or $2$ interfaces in dimensions $2$ and $3$.