Stable approximations for axisymmetric Willmore flow for closed and open surfaces
Stable approximations for axisymmetric Willmore flow for closed and open surfaces
复制标题
封闭和开放表面的轴对称威尔莫尔流的稳定近似
DOI：
10.1051/m2an/2021014
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
R. Nürnberg
中科院分区：
文献类型：
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作者：
J. Barrett;H. Garcke;R. Nürnberg
对于ℝ3中的超曲面，Willmore流被定义为经典Willmore能量的L2-梯度流：平均曲率平方的积分。这一几何演化规律在微分几何、图像重建和数学生物学中具有重要意义。本文对轴对称超曲面的Willmore流提出了新的数值逼近方法。对于半离散时间连续变量，我们证明了一个稳定性结果。我们既考虑闭合曲面，也考虑有边界的曲面。在后一种情况下，我们仔细地推导出适当边界条件的弱公式。此外，我们还考虑了经典Willmore能量的许多推广，特别是那些在生物膜研究中发挥作用的推广。在广义模型中，我们考虑了自发曲率和面积差弹性(ADE)效应、高斯曲率和线能量贡献。几个数值实验证明了我们开发的数值方法的有效性和稳健性。
For a hypersurface in ℝ3, Willmore flow is defined as the L2-gradient flow of the classical Willmore energy: the integral of the squared mean curvature. This geometric evolution law is of interest in differential geometry, image reconstruction and mathematical biology. In this paper, we propose novel numerical approximations for the Willmore flow of axisymmetric hypersurfaces. For the semidiscrete continuous-in-time variants we prove a stability result. We consider both closed surfaces, and surfaces with a boundary. In the latter case, we carefully derive weak formulations of suitable boundary conditions. Furthermore, we consider many generalizations of the classical Willmore energy, particularly those that play a role in the study of biomembranes. In the generalized models we include spontaneous curvature and area difference elasticity (ADE) effects, Gaussian curvature and line energy contributions. Several numerical experiments demonstrate the efficiency and robustness of our developed numerical methods.