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.

翻译：本文针对经典相场方法不适用于薄结构平均曲率流逼近的问题展开研究。所谓薄结构，是指高余维结构（典型如丝状结构）或非集合边界的曲面（包括不可定向曲面）。我们提出一种新方法，通过在经典Allen-Cahn方程中引入位于演化集骨架周围的惩罚项，确保演化过程中保持最小厚度。数值实验表明，该方法能精确逼近丝状结构平均曲率流演化过程。此外，我们展示了该方法可无缝应用于三维空间中Steiner问题与Plateau问题解的数值逼近。