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

TL;DR

This work proposes a novel approach which consists in plugging into the classical Allen--Cahn equation a penalization term localized around the skeleton of the evolving set that ensures that a minimal thickness is preserved during the evolution process.

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, 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.

Figures (11)

• Figure 1: Phase field approximation beyond singularities of the mean curvature flow of a dumbbell.
• Figure 2: Numerical values of $S n^\sigma$ in $2D$ for various configurations and various values of $\sigma$; The first column of each line corresponds to the considered set ; On each line, columns $2$, $3$ and $4$ show the values of the scalar field $S n^\sigma$ for, respectively, $\sigma = 0.02$, $0.01$ and $0.005$.
• Figure 3: Numerical values of $S n^\sigma$ for $3D$ examples. Each line shows the considered set, then the values taken by the skeletal term on the planes $\{x_1 = 0\}$, $\{x_2 = 0\}$, and $\{x_3 = 0\}$, respectively.
• Figure 4: Evolution of a dumbbell: the solution $u^{n}$ plotted at different times $t$. The first and second lines correspond to the approximate mean curvature flow with or without, respectively, the additional skeletal term.
• Figure 5: Evolution of a circle: the solution $u^{n}$ plotted at different times $t$ and the squared mass $t \mapsto (\int_Q u(x,t) dx)^2$ along the iterations.

Theorems & Definitions (20)

• Lemma 2.1
• proof
• Lemma 2.2
• Remark 2.3
• Remark 2.4
• proof : Proof of Lemma \ref{['claim:minimal_distance']}
• Theorem 2.5
• proof
• Definition 3.1
• Remark 3.2