A penalized Allen–Cahn equation for the mean curvature flow of thin structures

Abstract

This paper addresses the approximation of the mean curvature flow of thin structures for which classical phase field methods are not suitable. By thin structures, we mean surfaces that are not domain boundaries, typically higher codimension objects such as 1D curves in 3D—that is, filaments, or soap films spanning a boundary curve. To approximate the mean curvature flow of such surfaces, we consider a small thickening and we apply to the thickened set an evolution model that combines the classical Allen–Cahn equation with a penalty term that takes on larger values around the skeleton of the set. The novelty of our approach lies in the definition of this penalty term that guarantees a minimal thickness of the evolving set and prevents it from disappearing unexpectedly. We prove a few theoretical properties of our model, provide examples showing the connection with higher codimension mean curvature flow, and introduce a quasi-static numerical scheme with explicit integration of the penalty term. We illustrate the numerical efficiency of the model with accurate approximations of filament structures evolving by mean curvature flow, and we also illustrate its ability to find complex 3D approximations of (at least local) solutions to the Steiner problem or the Plateau problem.