A Parametric Finite Element Method for Solid-State Dewetting Problems in Three Dimensions

Abstract

We propose a parametric finite element method (PFEM) for efficiently solving the morphological evolution of solid-state dewetting of thin films on a flat rigid substrate in three dimensions (3D). The interface evolution of the dewetting problem in 3D is described by a sharp-interface model, which includes surface diffusion coupled with contact line migration. A variational formulation of the sharp-interface model is presented, and a PFEM is proposed for spatial discretization. For temporal discretization, at each time step, we first update the position of the contact line according to the relaxed contact angle condition; then, by using the position of the new contact line as the boundary condition, we solve a linear algebra system resulted from the discretization of PFEM to obtain the new interface surface for the next step. The well-posedness of the solution of the PFEM is also established. Extensive numerical results are reported to demonstrate the accuracy and efficiency of the proposed PFEM and to show the complexities of the dewetting morphology evolution observed in solid-state dewetting experiments.

A symmetrized parametric finite element method for anisotropic surface diffusion of closed curves via a Cahn-Hoffman $\boldsymbolξ$-vector formulation

“…In the existing literature, a Willmore regularization energy term is often added into the model to deal with the strongly anisotropic cases [14,3,19], but here we only use one unified scheme to tackle the two cases. In the future, we will further explore the high performance of the schemes, for the strongly anisotropic cases; and extend the new variational formulation to anisotropic surface diffusion of open/closed surfaces in three dimensions [21,39].…”

A structure-preserving parametric finite element method for surface diffusion

“…e h,τ (t = 0.2) order e h,τ (t = 0.5) order e h,τ (t = 2.0) order We test the convergence rate of the numerical method in (2.9) by carrying out simulations using different mesh sizes and time step sizes. To measure the difference between two different closed curves Γ 1 and Γ 2 , we adopt the manifold distance in [32]. Let Ω 1 and Ω 2 be the inner regions enclosed by Γ 1 and Γ 2 , respectively, then the manifold distance is given by the area of the symmetric difference region between Ω 1 and Ω 2 [32]:…”

“…To measure the difference between two different closed curves Γ 1 and Γ 2 , we adopt the manifold distance in [32]. Let Ω 1 and Ω 2 be the inner regions enclosed by Γ 1 and Γ 2 , respectively, then the manifold distance is given by the area of the symmetric difference region between Ω 1 and Ω 2 [32]:…”

“…However, the method is fully implicit and the mesh quality is not well preserved during time evolution. For more related works, we refer the readers to [4,18,12,3,32,31] and references therein.…”

“…We remark here that the SP-PFEM (2.9) in 2D and (3.11) in 3D can be straightforwardly extended to the anisotropic surface diffusion flow based on the works in [7,32,22], the volume-preserving mean curvature flows [19], and other curvature driven flows that preserve the volume. Of course, these extensions are required further investigation in terms of preserving the mesh quality, especially the asymptotic equal mesh distribution.…”