A symmetrized parametric finite element method for simulating solid-state dewetting problems

Abstract

We propose an accurate, area/mass-conservative and energy-dissipative parametric finite element method for solving the sharp-interface continuum model of solid-state dewetting with arbitrary anisotropic surface energy. The model is governed by the anisotropic surface diffusion equation with suitable boundary conditions at the contact points, and could be used to describe the kinetic evolution of the film/vapor interface with contact line migration. The sharp-interface continuum model is firstly reformulated into a conservative form by introducing a symmetric positive definite matrix Zk(n) that depends on the Cahn–Hoffman ξ-vector and a stabilizing function k(n). A new symmetrized variational formulation with arbitrary anisotropic surface energy is built and then its area/mass conservation and energy dissipation properties are proved. By using piecewise linear elements in space, we propose the spatial semi-discretized scheme, and using backward Euler in time with an appropriate approximation of the unit normal vector, we further derive the fully discretized parametric finite element approximation. We show, theoretically, that the semi-discretized and fully discretized schemes are both area/mass-conservative and energy-dissipative. Finally, numerical experiments are reported to show the accuracy and efficiency of the numerical method as well as the anisotropic effects on the morphological evolution processes for solid-state dewetting of thin films.