各向异性表面扩散的对称参数有限元法[j]。三个维度

W. Bao, Yifei Li
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引用次数: 1

摘要

. 对于具有一般各向异性表面能γ (n)的封闭表面在三维(3D)中的演化,其中n为单位向外法向量,通过引入依赖于稳定函数k (n)和Cahn-Hoffman ξ向量的对称正定表面能矩阵Z k (n),我们给出了具有弱或强各向异性表面能的各向异性表面扩散的新的对称变分公式。它保留了两个重要的结构:体积守恒和能量耗散。在此基础上,提出了一种保持结构的参数有限元法(SP-PFEM)对对称变分问题进行离散化,在离散化水平上保持了体积。在相对温和和简单的γ (n)条件下,我们证明SP-PFEM对于实际应用中产生的几乎所有各向异性γ (n)表面能都是无条件能量稳定的。大量的数值结果表明,所提出的SP-PFEM在求解三维各向异性表面扩散时的效率、精度和能量消耗都得到了验证。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A symmetrized parametric finite element method for anisotropic surface diffusion ii. three dimensions
. For the evolution of a closed surface under anisotropic surface diffusion with a general anisotropic surface energy γ ( n ) in three dimensions (3D), where n is the unit outward normal vector, by introducing a novel symmetric positive definite surface energy matrix Z k ( n ) depending on a stabilizing function k ( n ) and the Cahn-Hoffman ξ -vector, we present a new symmetrized variational formulation for anisotropic surface diffusion with weakly or strongly anisotropic surface energy, which preserves two important structures including volume conservation and energy dissipation. Then we propose a structural-preserving parametric finite element method (SP-PFEM) to discretize the symmetrized variational problem, which preserves the volume in the discretized level. Under a relatively mild and simple condition on γ ( n ), we show that SP-PFEM is unconditionally energy- stable for almost all anisotropic surface energies γ ( n ) arising in practical applications. Extensive numerical results are reported to demonstrate the efficiency and accuracy as well as energy dissipation of the proposed SP-PFEM for solving anisotropic surface diffusion in 3D.
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