Kernels with prescribed surface tension & mobility for threshold dynamics schemes

Abstract

We show how to construct a convolution kernel that has a desired anisotropic surface tension and desired anisotropic mobility to be used in threshold dynamics schemes for simulating weighted motion by mean curvature of interfaces, including networks of them, in both two and three dimensions. Moreover, we discuss necessary and sufficient conditions for the positivity of the kernel which, in the case of two-phase flow, ensures that the resulting scheme respects a comparison principle and implies convergence to the viscosity solution of the level set formulation of the flow. In particular, we show, in a barrier-type statement, that the kernel cannot possibly be positive unless both the mobility and the surface tension satisfy necessary conditions in three dimensions, and give a complete characterization. Among other results is a threshold dynamics scheme that is guaranteed to dissipate a nonlocal approximation to the interfacial energy in the fully anisotropic, multiphase setting, using the new kernel construction.

Dewetting dynamics of anisotropic particles -- a level set numerical approach

“…However, as pointed out later in [5], these kernels do not correctly separate the effect of anisotropic energy and mobility, and thus are not applicable to multi-phase evolution such as the one considered in this paper. This can be observed from equations ( 7)- (8), where the effect of surface energy and mobility are inseparably combined in the normal velocity but the contact angle condition depends solely on the surface energy. For this reason, we will review here only kernels developed after this seminal work.…”

“…Notice that the sign of γ + γ is again important here, since when γ (θ) + γ(θ) is negative for some angles θ, equation ( 7) becomes backwards parabolic and thus ill-posed. When topology changes occur, such as merging and splitting of particles, equation ( 7) still holds for smooth parts of Γ away from singularities but it is not anymore possible to describe the evolution fully using simple formulas such as (7)- (8). For smooth weak anisotropies, one can give a precise mathematical definition in terms of functions of bounded variation in a similar manner to Definition 1.1 of multiphase mean curvature flow in [12].…”

“…The purpose of this article is to extend the thresholding method of [20] to the anisotropic setting, which is motivated by the numerical advantages of the thresholding method: unconditional stability, low computational cost and natural handling of topology changes. In this generalization, we apply the theory of anisotropic convolution kernels developed in a series of papers [3,5,8]. Since the numerical properties of these kernels were not yet systematically studied, we first provide a numerical analysis of the two-phase problem, i.e., without obstacle.…”

On motion by curvature of a network with a triple junction

“…But we are not aware of any work that addresses convergence of schemes (in a parametric setting) for evolving networks with triple junctions subject to (1.1) and (1.2). For completeness, let us mention that there are also interface capturing approaches that avoid the need P. Pozzi & B. Stinner to look after the mesh quality [7,23,28,5,17]. Such approaches comprise phase field models and level set methods, for overviews we refer to [11,4,13,27].…”

A Monotone, Second Order Accurate Scheme for Curvature Motion

“…It is also worth recalling that using different (namely, in this case, non-radially symmetric) kernels in threshold dynamics comes up in its extensions to anisotropic curvature flows [3,5,6,22]. In particular, barrier type theorems [5,6] show that any threshold dynamics scheme that is at all consistent (never mind second order) with certain anisotropic curvature flows in three dimensions cannot possibly be monotone.…”