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

2024/12/20 by Bretin, Elie, Chambolle, Antonin, Masnou, Simon · 1 citation
#35A15 #35A35 #53E10 #53E40 #65M32 #74N20 #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2412.15926

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 Γ-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 L²-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.

Cited by

• Curvature penalization of strongly anisotropic interfaces models and their phase-field approximation

Related

• An End-to-End Deep Learning Method for Solving Nonlocal Allen-Cahn and Cahn-Hilliard Phase-Field Models
• Volume Preserving Willmore Flow in a Generalized Cahn-Hilliard Flow
• Non-isothermal nonlocal phase-field models with a double-obstacle potential
• A phase field model for the optimization of the Willmore energy in the\n class of connected surfaces
• Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks