论文标题
具有各向异性表面能的几何流量的结构保存参数有限元法
A structure-preserving parametric finite element method for geometric flows with anisotropic surface energy
论文作者
论文摘要
We propose and analyze structure-preserving parametric finite element methods (SP-PFEM) for evolution of a closed curve under different geometric flows with arbitrary anisotropic surface energy $γ(\boldsymbol{n})$ for $\boldsymbol{n}\in \mathbb{S}^1$ representing the outward unit normal vector.通过引入一个新型的表面能量矩阵$ \ boldsymbol {g} _k（\ boldsymbol {n}）$，具体取决于$γ（\ boldsymbol {n}）$和cahn-hoffman $ \ boldsymbol $ \ boldsymbol配量$ k（\ boldsymbol配量$ k） \ Mathbb {S}^1 \ to \ Mathbb {r} $，它是对称正定矩阵和反对称矩阵的总和，我们获得了新的几何部分微分方程及其相应的变化曲线，用于在异位表面下封闭曲线的进化。基于新的弱公式，我们提出了一种针对各向异性表面扩散的参数有限元方法，并表明它是$γ（\ boldsymbol {n}）$的非常温和的条件下的区域保护和能量耗散。然后，将SP-PFEM扩展以模拟其他各向异性几何流动下的近距离曲线的演变，包括各向异性曲率流量和区域保存的各向异性曲率流。据报道，广泛的数值结果证明了所提出的SP-PFEM的效率和无条件的能量稳定性以及良好的网格质量特性，用于模拟各向异性几何流量。
We propose and analyze structure-preserving parametric finite element methods (SP-PFEM) for evolution of a closed curve under different geometric flows with arbitrary anisotropic surface energy $γ(\boldsymbol{n})$ for $\boldsymbol{n}\in \mathbb{S}^1$ representing the outward unit normal vector. By introducing a novel surface energy matrix $\boldsymbol{G}_k(\boldsymbol{n})$ depending on $γ(\boldsymbol{n})$ and the Cahn-Hoffman $\boldsymbolξ$-vector as well as a nonnegative stabilizing function $k(\boldsymbol{n}):\ \mathbb{S}^1\to \mathbb{R}$, which is a sum of a symmetric positive definite matrix and an anti-symmetric matrix, we obtain a new geometric partial differential equation and its corresponding variational formulation for the evolution of a closed curve under anisotropic surface diffusion. Based on the new weak formulation, we propose a parametric finite element method for the anisotropic surface diffusion and show that it is area conservation and energy dissipation under a very mild condition on $γ(\boldsymbol{n})$. The SP-PFEM is then extended to simulate evolution of a close curve under other anisotropic geometric flows including anisotropic curvature flow and area-conserved anisotropic curvature flow. Extensive numerical results are reported to demonstrate the efficiency and unconditional energy stability as well as good mesh quality property of the proposed SP-PFEM for simulating anisotropic geometric flows.
