An energy-stable parametric finite element method for simulating solid-state dewetting

TL;DR

This paper introduces an energy-stable, efficient parametric finite element method for simulating solid-state dewetting of thin films, accurately capturing interface evolution and contact line migration with proven energy dissipation.

Contribution

The paper develops a novel variational formulation and a semi-implicit discretization that ensures unconditional energy stability and second-order spatial accuracy for solid-state dewetting simulations.

Findings

Method is unconditionally energy-stable and second-order accurate in space.

Numerical results demonstrate high accuracy and efficiency.

The approach maintains mesh quality and captures long-time dynamics effectively.

Abstract

We propose an energy-stable parametric finite element method (ES-PFEM) for simulating solid-state dewetting of thin films in two dimensions via a sharp-interface model, which is governed by surface diffusion and contact line (point) migration together with proper boundary conditions. By reformulating the relaxed contact angle condition into a Robin-type boundary condition and then treating it as a natural boundary condition, we obtain a new variational formulation for the problem, in which the interface curve and its contact points are evolved simultaneously. Then, the variational problem is discretized in space by using piecewise linear elements. A full discretization is presented by adopting the backward Euler method in time, and the well-posedness and energy dissipation of the full discretization are established. The numerical method is semi-implicit (i.e., a linear system to be solved at each time step and thus efficient), unconditionally energy-stable with respect to the time step, and second-order in space measured by a manifold distance between two curves. In addition, it demonstrates equal mesh distribution when the solution reaches its equilibrium, i.e., long-time dynamics. Numerical results are reported to show accuracy and efficiency as well as some good properties of the proposed numerical method.