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, i.e. 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 solutions to the Steiner problem or the Plateau problem.

翻译：本文研究薄结构平均曲率流的近似计算问题，此类问题不适用于经典相场方法。所谓薄结构，是指非区域边界的曲面，通常为高余维对象，例如三维空间中的一维曲线（即细丝状结构）或跨越边界曲线的皂膜。为近似此类曲面的平均曲率流，我们考虑对其进行微小增厚处理，并对增厚后的集合应用演化模型。该模型结合了经典Allen-Cahn方程与惩罚项，该惩罚项在集合骨架附近取值较大。本方法的核心创新在于惩罚项的定义，该定义保证了演化集合的最小厚度并防止其意外消失。我们证明了该模型的若干理论性质，通过算例展示了其与高余维平均曲率流的关联，并提出了惩罚项显式积分的准静态数值格式。通过精确模拟细丝结构在平均曲率流下的演化过程，我们验证了模型的数值有效性，同时展示了该模型在求解Steiner问题与Plateau问题时获得复杂三维近似解的能力。