A modified phase field approximation for mean curvature flow with conservation of the volume

TL;DR

This paper introduces a modified phase field model for mean curvature flow that better preserves volume, even with external forces, improving upon the classical nonlocal Allen-Cahn equation.

Contribution

A new modified phase field equation is proposed that enhances volume conservation during mean curvature flow with external forcing.

Findings

Improved volume preservation properties demonstrated

Modified equation outperforms classical models in simulations

Applicable to flows with external forcing

Abstract

This paper is concerned with the motion of a time dependent hypersurface that evolves by mean curvature flow with a a volume constraint. Phase field approximation of this motion leads to the well known nonlocal Allen--Cahn equation. Here we propose a modified version of this equation, and we show that it has better volume preserving properties than the classical one, even in the presence of an additional forcing term.