Phase-field approximations of the Willmore functional and flow

TL;DR

This paper explores phase-field methods to approximate the Willmore functional and its flow, deriving flow equations, analyzing their behavior through asymptotic expansions, and validating with numerical simulations in 2D and 3D.

Contribution

It introduces a new numerical scheme with proven local convergence for simulating Willmore flows and compares classical and Mugnai's approximation models.

Findings

Numerical schemes accurately capture flow behavior in smooth and singular cases.

Asymptotic analysis clarifies the relation between different approximation models.

Simulations demonstrate the effectiveness of the proposed methods in 2D and 3D.

Abstract

We discuss in this paper phase-field approximations of the Willmore functional and the associated L2-flow. After recollecting known results on the approximation of the Willmore energy and its L1-relaxation, we derive the expression of the flows associated with various approximations, and we show their behavior by formal arguments based on matched asymptotic expansions. We introduce an accurate numerical scheme, whose local convergence can be proved, to describe with more details the behavior of two flows, the classical and the flow associated with an approximation model due to Mugnai. We propose a series of numerical simulations in 2D and 3D to illustrate their behavior in both smooth and singular situations.