Parametric approximation of surface clusters driven by isotropic and anisotropic surface energies

Abstract

We present a variational formulation for the evolution of surface clusters in R 3 by mean curvature flow, surface diffusion and their anisotropic variants. We introduce the triple junction line conditions that are induced by the considered gradient flows, and present weak formulations of these flows. In addition, we consider the case where a subset of the boundaries of these clusters are constrained to lie on an external boundary. These formulations lead to unconditionally stable, fully discrete, parametric finite element approximations. The resulting schemes have very good properties with respect to the distribution of mesh points and, if applicable, volume conservation. This is demonstrated by several numerical experiments, including isotropic double, triple and quadruple bubbles, as well as clusters evolving under anisotropic mean curvature flow and anisotropic surface diffusion, including computations for regularized crystalline surface energy densities.

Finite-element approximation of coupled surface and grain boundary motion with applications to thermal grooving and sintering

“…On adapting the results derived in Barrett, Garcke, and Nürnberg (2009), it is not difficult to show that (2.4) with (2.5a-d) and (2.8a-c) is a gradient flow of the total surface area…”

“…Two possible such setups can be seen in Figures 10 and 17 below. Generalizations to more complicated setups, including to clusters with quadruple junction points, where four triple junction lines meet, are straightforward, see Barrett, Garcke, and Nürnberg (2009). But for the sake of clarity and brevity we restrict ourselves to the simpler situation described above in the majority of this paper.…”

“…These flows, for general surface clusters with an arbitrary number of triple junction lines and quadruple junction points, were investigated in Barrett, Garcke, and Nürnberg (2009). However, in this paper we want to restrict our attention to the following motion, which has physical applications in thermal grooving and sintering; recall Section 1.…”

“…For a derivation of the conditions (2.15a-c) and (2.18a,b), with (2.19), we refer to Barrett, Garcke, and Nürnberg (2009), where one can also find the necessary techniques in order to show that (2.14) together with these conditions is a gradient flow of (2.10), i.e. that the total weighted surface area is monotonically decreasing in time.…”

“…As a consequence, while ς ± k uniquely determine k , the converse is not true. In the remainder of this section, we adopt the techniques in Barrett, Garcke, and Nürnberg (2009) in order to derive a weak formulation of the flow (2.14) with (2.5a), (2.15a-c) and (2.6), (2.18a,b). This weak formulation will form the basis of our finite element approximation, which we will introduce in Section 3.…”