耦合方向与曲率依赖的几何流的一种统一保结构框架

中文摘要 AI 辅助

我们为几何流开发了一种保结构的参数有限元框架，其能量密度耦合了单位法向与曲率。该类别包括具有方向依赖性刚度或自发曲率的弯曲能量。一个共同的完全离散公式处理二维中的闭合曲线和三维中的闭合曲面，并适用于$L^2$流、曲线或曲面扩散，以及面积或体积约束的$L^2$流。该公式耦合了几何与曲率的更新，使得它们的贡献满足离散能量不等式。它使用连续分段线性单元、表面能量矩阵以及曲率导数密度的质量集中投影。对于满足方向条件和曲率凸性的正密度，我们证明了无时间步限制的能量耗散。扩散流和约束流也精确保持封闭面积或体积。该分析允许非偶数各向异性以及方向与曲率的不可分离依赖。数值实验在流形距离上表现出近似二阶收敛，并确认了离散结构性质。各向异性Helfrich型能量下的形状弛豫说明了该框架在耦合方向与弯曲效应中的应用。

英文摘要

We develop a structure-preserving parametric finite element framework for geometric flows whose energy density couples the unit normal and the curvature. This class includes bending energies with orientation-dependent rigidity or spontaneous curvature. A common fully discrete formulation treats closed curves in two dimensions and closed surfaces in three dimensions, and accommodates the $L^2$ flow, curve or surface diffusion, and the area- or volume-constrained $L^2$ flow. The formulation couples the geometric and curvature updates so that their contributions satisfy a discrete energy inequality. It uses continuous piecewise linear elements, a surface energy matrix, and a mass-lumped projection of the curvature derivative of the density. For positive densities satisfying a directional condition and convexity in curvature, we prove energy dissipation without a time-step restriction. The diffusion and constrained flows also preserve the enclosed area or volume exactly. The analysis allows non-even anisotropies and nonseparable dependence on orientation and curvature. Numerical experiments exhibit approximately second-order convergence in the manifold distance and confirm the discrete structural properties. Shape relaxation under an anisotropic Helfrich-type energy illustrates the use of the framework for coupled directional and bending effects.

发表机构

• School of Mathematical Sciences and Institute of Natural Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院和自然科学研究院）

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