A thickness-aware Allen-Cahn equation for the mean curvature flow of thin structures

This work 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 either structures of higher codimension, typically filaments, or surfaces (including non orientables surfaces) that are not boundaries of a set. We propose a novel approach which consists in plugging into the classical Allen--Cahn equation a penalization term localized around the skeleton of the evolving set. This ensures that a minimal thickness is preserved during the evolution process. The numerical efficacy of our approach is illustrated with accurate approximations of the evolution by mean curvature flow of filament structures. Furthermore, we show the seamless adaptability of our approach to compute numerical approximations of solutions to the Steiner and Plateau problems in three dimensions.