Numerical Simulations of Mean Curvature Flow in the Presence of a Nonconvex Anisotropy

Abstract

In this paper we present and discuss the results of some numerical simulations in order to investigate the mean curvature flow problem in the presence of a nonconvex anisotropy. Mathematically, nonconvexity of the anisotropy leads to the ill-posedness of the evolution problem, which becomes forward–backward parabolic. Simulations presented here refer to two different settings: curvature driven vertical motion of graphs (nonparametric setting) and motion in the normal direction by anisotropic mean curvature of surfaces (parametric setting). In the latter we first relax the problem via an Allen–Cahn type reaction-diffusion equation, in the context of Finsler geometry (diffused interface approximation). Our results suggest three main points. A nonconvex anisotropy and its convexification give rise, for both settings and the discretizations considered, to different evolutions. Wrinkled regions seem to appear only in correspondence to locally concave parts of the anisotropy. Moreover, locally convex regions (interior to the convexification of the so-called Frank diagram) seem to play an important role.

A concept of solution and numerical experiments for forward-backward diffusion equations

“…The idea is to show that the points where wrinkles originate depend on the parameter s in agreement with (6.5). From (6.5) and (6.7) it follows that the points Fig 35 we show similar simulations corresponding to the energy density (2.5) for which φ (p) = The graphs in Figs 29,30,31 show that, besides the wrinkle component, the difference U (t)−U (0) also contains a quite evident smooth component. The presence of this component can be understood by observing that the time required for wrinkle formation is very short but it is not zero.…”

“…Wrinkles appear only in Σ L (u). It was already observed in [31] that wrinkles appear only in the set Σ L (u). Our simulations fully confirm this observation.…”

“…This is to be expected because the formation of microstructures should correspond to a local minimization of the energy and therefore the evolution of u ε (·, t, u) in regions where microstructures have appeared (wrinkled regions) should be slower than in the remaining (unwrinkled) regions. In [31] it was already pointed out that numerical experiments reveal that microstructures develop only in Σ L (u). Our numerical experiments confirm this behaviour, see Section 6.3 below.…”

“…See also [7], and the recent paper [30], where some properties of the limit function are given. The effect of nonconvexity in the context of anisotropic geometric evolution of interfaces was considered in [38] and [31]. Regularizations by means of nonlocal operators of various kinds have been proposed and studied by several authors, see [16], [48], [10].…”

The $\Gamma $-limit and the related gradient flow for singular perturbation functionals of Perona-Malik type

“…For instance if, as we do in the following, we assume n = m = 1, Ω = I = ]α, β[, then one is led to the parabolic equation (1.2) u t = (φ (u x )) x , coupled with suitable boundary and initial conditions, which for smooth enough solutions u and provided that φ is smooth, takes the form u t = φ (u x )u xx ; such an equation has a forward-backward character depending on the sign of φ (u x ). A consequence of the ill-posedness of (1.2) is that, in spite of some interesting attempts, [33], [45], [37], [27], [26], [36], [31], there is not yet a satisfactory notion of weak solution which can adequately describe the rich phenomenology observed in numerical experiments [28], [41]. A natural approach in this situation is to associate with (1.2) a family of regular problems depending on a small parameter ε > 0 which are expected to approach (1.2) in the limit ε → 0 + , and try to define solutions to (1.2) as limits as ε → 0 + of solutions of the regularized problems, i.e.,…”