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Volume 41, Issue 4
An Energy-Stable Parametric Finite Element Method for Simulating Solid-State Dewetting Problems in Three Dimensions

Weizhu Bao & Quan Zhao

J. Comp. Math., 41 (2023), pp. 771-796.

Published online: 2023-04

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  • Abstract

We propose an accurate and energy-stable parametric finite element method for solving the sharp-interface continuum model of solid-state dewetting in three-dimensional space. The model describes the motion of the film\slash vapor interface with contact line migration and is governed by the surface diffusion equation with proper boundary conditions at the contact line. We present a weak formulation for the problem, in which the contact angle condition is weakly enforced. By using piecewise linear elements in space and backward Euler method in time, we then discretize the formulation to obtain a parametric finite element approximation, where the interface and its contact line are evolved simultaneously. The resulting numerical method is shown to be well-posed and unconditionally energy-stable. Furthermore, the numerical method is generalized to the case of anisotropic surface energies in the Riemannian metric form.  Numerical results are reported to show the convergence and efficiency of the proposed numerical method as well as the anisotropic effects on the morphological evolution of thin films in solid-state dewetting.

  • AMS Subject Headings

74H15, 74S05, 74M15, 65Z99

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-41-771, author = {Bao , Weizhu and Zhao , Quan}, title = {An Energy-Stable Parametric Finite Element Method for Simulating Solid-State Dewetting Problems in Three Dimensions}, journal = {Journal of Computational Mathematics}, year = {2023}, volume = {41}, number = {4}, pages = {771--796}, abstract = {

We propose an accurate and energy-stable parametric finite element method for solving the sharp-interface continuum model of solid-state dewetting in three-dimensional space. The model describes the motion of the film\slash vapor interface with contact line migration and is governed by the surface diffusion equation with proper boundary conditions at the contact line. We present a weak formulation for the problem, in which the contact angle condition is weakly enforced. By using piecewise linear elements in space and backward Euler method in time, we then discretize the formulation to obtain a parametric finite element approximation, where the interface and its contact line are evolved simultaneously. The resulting numerical method is shown to be well-posed and unconditionally energy-stable. Furthermore, the numerical method is generalized to the case of anisotropic surface energies in the Riemannian metric form.  Numerical results are reported to show the convergence and efficiency of the proposed numerical method as well as the anisotropic effects on the morphological evolution of thin films in solid-state dewetting.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2205-m2021-0237}, url = {http://global-sci.org/intro/article_detail/jcm/21639.html} }
TY - JOUR T1 - An Energy-Stable Parametric Finite Element Method for Simulating Solid-State Dewetting Problems in Three Dimensions AU - Bao , Weizhu AU - Zhao , Quan JO - Journal of Computational Mathematics VL - 4 SP - 771 EP - 796 PY - 2023 DA - 2023/04 SN - 41 DO - http://doi.org/10.4208/jcm.2205-m2021-0237 UR - https://global-sci.org/intro/article_detail/jcm/21639.html KW - Solid-state dewetting, Surface diffusion, Contact line migration, Contact angle, Parametric finite element method, Anisotropic surface energy. AB -

We propose an accurate and energy-stable parametric finite element method for solving the sharp-interface continuum model of solid-state dewetting in three-dimensional space. The model describes the motion of the film\slash vapor interface with contact line migration and is governed by the surface diffusion equation with proper boundary conditions at the contact line. We present a weak formulation for the problem, in which the contact angle condition is weakly enforced. By using piecewise linear elements in space and backward Euler method in time, we then discretize the formulation to obtain a parametric finite element approximation, where the interface and its contact line are evolved simultaneously. The resulting numerical method is shown to be well-posed and unconditionally energy-stable. Furthermore, the numerical method is generalized to the case of anisotropic surface energies in the Riemannian metric form.  Numerical results are reported to show the convergence and efficiency of the proposed numerical method as well as the anisotropic effects on the morphological evolution of thin films in solid-state dewetting.

Weizhu Bao & Quan Zhao. (2023). An Energy-Stable Parametric Finite Element Method for Simulating Solid-State Dewetting Problems in Three Dimensions. Journal of Computational Mathematics. 41 (4). 771-796. doi:10.4208/jcm.2205-m2021-0237
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