强各向异性材料中正则化固态去湿问题的能量稳定参数有限元逼近
Energy-stable parametric finite element approximations for regularized solid-state dewetting in strongly anisotropic materials
AI总结：
本文针对强各向异性材料中的固态去湿问题，建立等价正则化尖锐界面模型，提出能量稳定且保面积守恒的参数有限元格式，并通过数值实验验证其精度、效率、网格质量和长时间计算适用性。
摘要：
在这项工作中，我们旨在为具有强表面能各向异性的尖锐界面模型发展能量稳定的参数有限元逼近，该模型由薄膜/蒸气界面能、基底能和正则化 Willmore 能组成的能量泛函的一阶变分导出。通过引入两个几何关系，我们创新性地建立了一个等价的正则化尖锐界面模型，并进一步为该等价模型构造了一种能量稳定的参数有限元算法。我们给出了数值格式能量稳定性的详细证明，填补了相关理论中的空白。此外，我们还发展了另一种保结构的参数有限元格式，能够同时保持面积守恒和能量稳定性。最后，我们给出了若干数值模拟，以展示所提数值方法的精度、效率以及一些保结构性质。更重要的是，大量数值模拟表明，我们的格式具有更好的网格质量，并且更适合长时间计算。
英文摘要：
In this work, we aim to develop energy-stable parametric finite element approximations for a sharp-interface model with strong surface energy anisotropy, which is derived from the first variation of an energy functional composed of film/vapor interfacial energy, substrate energy, and regularized Willmore energy. By introducing two geometric relations, we innovatively establish an equivalent regularized sharp-interface model and further construct an energy-stable parametric finite element algorithm for this equivalent model. We provide a detailed proof of the energy stability of the numerical scheme, addressing a gap in the relevant theory. Additionally, we develop another structure-preserving parametric finite element scheme that can preserve both area conservation and energy stability. Finally, we present several numerical simulations to show accuracy and efficiency as well as some structure-preserving properties of the proposed numerical methods. More importantly, extensive numerical simulations reveal that our schemes provide better mesh quality and are more suitable for long-term computations.
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