一种用于模拟三维固态脱湿问题的能量稳定参数有限元方法
An Energy-Stable Parametric Finite Element Method for Simulating Solid-state Dewetting Problems in Three Dimensions
AI总结：
提出一种三维固态脱湿锐界面模型的能量稳定参数有限元方法，通过同时演化界面与接触线并采用分段线性元和后向欧拉离散，证明了方法的适定性与无条件能量稳定性，并扩展至各向异性表面能情形。
摘要：
我们提出了一种精确且能量稳定的参数有限元方法，用于求解三维空间中固态脱湿的锐界面连续介质模型。该模型描述了具有接触线迁移的薄膜/蒸汽界面的运动，并由带有接触线处适当边界条件的表面扩散方程所控制。我们针对该问题提出了一种新的弱形式，其中界面及其接触线被同时演化。通过在空间上使用分段线性元和在时间上使用后向欧拉方法，我们将弱形式离散化，得到一个完全离散的参数有限元逼近。结果表明，所得的数值方法是适定的且无条件能量稳定的。此外，该数值方法还被扩展用于求解具有黎曼度量形式各向异性表面能的固态脱湿锐界面模型。报告的数值结果表明了所提出数值方法的收敛性和效率，以及各向异性对固态脱湿中薄膜形态演变的影响。
英文摘要：
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 new weak formulation for the problem, in which the interface and its contact line are evolved simultaneously. By using piecewise linear elements in space and backward Euler in time, we then discretize the weak formulation to obtain a fully discretized parametric finite element approximation. The resulting numerical method is shown to be well-posed and unconditionally energy-stable. Furthermore, the numerical method is extended for solving the sharp interface model of solid-state dewetting with anisotropic surface energies in the Riemmanian 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.
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