A structure-preserving parametric finite element method for solid-state dewetting on curved substrates

TL;DR

This paper develops a structure-preserving parametric finite element method for simulating solid-state dewetting on curved substrates, ensuring energy decay and area preservation with efficient numerical algorithms.

Contribution

It introduces a novel energy-stable, structure-preserving finite element approach with error estimates and exact area preservation techniques for curved substrates.

Findings

Method demonstrates robustness across various curved substrates.

Ensures energy decay and area preservation in simulations.

Efficient hybrid iterative solver for nonlinear systems.

Abstract

We consider a two-dimensional sharp-interface model for solid-state dewetting of thin films with anisotropic surface energies on curved substrates, where the film/vapor interface and substrate surface are represented by an evolving and a static curve, respectively. The model is governed by the anisotropic surface diffusion for the evolving curve, with appropriate boundary conditions at the contact points where the two curves meet. The continuum model obeys an energy decay law and preserves the enclosed area between the two curves. We introduce an arclength parameterization for the substrate curve, which plays a crucial role in a structure-preserving approximation as it straightens the curved substrate and tracks length changes between contact points. Based on this insight, we introduce a symmetrized weak formulation which leads to an unconditional energy stable parametric approximation in terms of the discrete energy. We also provide an error estimate of the enclosed area, which depends on the substrate profile and can be zero in the case of a flat substrate. Furthermore, we introduce a correction to the discrete normals to enable an exact area preservation for general curved substrates. The resulting nonlinear system is efficiently solved using a hybrid iterative algorithm which combines both Picard and Newton's methods. Numerical results are presented to show the robustness and good properties of the introduced method for simulating solid-state dewetting on various curved substrates.