参数各向异性Willmore流的嵌套变分时间离散化
A Nested Variational Time Discretization for Parametric Anisotropic Willmore Flow
AI总结：
本文提出了一种结合变分时间离散化与有限元空间离散化的参数各向异性Willmore流方法，通过嵌套优化问题解决能量和度量的各向异性问题，验证了该方案在大时间步下的鲁棒性。
摘要：
本文提出了一种参数各向异性Willmore流的变分时间离散化方法，结合分段线性有限元的空间离散化。在此方法中，能量和度量均为各向异性，确保Wulff形状在梯度流下保持不变。在每个时间步中需解决嵌套优化问题，其中外层变分问题反映时间离散化的真实Willmore流，并包含两个连续时间步之间的近似各向异性L²距离以及各向异性Willmore能量的全隐式近似。所需的各向异性均曲率通过内层全隐式变分方案近似得到。为解决嵌套优化问题，应用了针对关联Lagrangian的牛顿方法。计算结果验证了新方案的鲁棒性，特别是在大时间步情况下。
英文摘要：
A variational time discretization of anisotropic Willmore flow combined with a spatial discretization via piecewise affine finite elements is presented. Here, both the energy and the metric underlying the gradient flow are anisotropic, which in particular ensures that Wulff shapes are invariant up to scaling under the gradient flow. In each time step of the gradient flow a nested optimization problem has to be solved. Thereby, an outer variational problem reflects the time discretization of the actual Willmore flow and involves an approximate anisotropic $L^2$-distance between two consecutive time steps and a fully implicit approximation of the anisotropic Willmore energy. The anisotropic mean curvature needed to evaluate the energy integrand is replaced by the time discrete, approximate speed from an inner, fully implicit variational scheme for anisotropic mean curvature motion. To solve the nested optimization problem a Newton method for the associated Lagrangian is applied. Computational results for the evolution of curves underline the robustness of the new scheme, in particular with respect to large time steps.
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