用于Willmore和Helfrich流参数近似的约束Onsager变分框架
A constrained Onsager variational framework for parametric approximations of Willmore and Helfrich flows
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中文总结 AI 辅助
该研究提出了一种用于Willmore和Helfrich流参数近似的约束Onsager变分框架，结合PDE约束与松弛-MDR约束，构建了满足能量耗散定律的半离散格式及线性隐式全离散格式，经数值实验验证了其有效性。
中文摘要 AI 辅助
我们开发了一种用于Willmore和Helfrich流参数有限元近似的约束Onsager变分框架。弱曲率关系作为PDE约束，以曲率向量形式表达弯曲能的一阶变分。Onsager原理将该变分与基于法向速度的耗散相结合，同时独立的松弛最小变形率（relaxed-MDR）约束选取切向速度。所得混合形式允许使用连续分段线性元素进行半离散化，当施加相应约束时，该半离散化满足精确的能量耗散定律，且能精确保持面积和体积。该框架可适应空间变化的自发曲率，适用于闭曲面以及在Navier或 clamped边界条件下的带固定边界的开曲面。我们还提出了一种线性隐式全离散格式，证明了其唯一可解性，并给出数值实验，验证了其收敛性、能量衰减、约束保持性及有效的网格重分布。
英文摘要
We develop a constrained Onsager variational framework for parametric finite element approximations of Willmore and Helfrich flows. The weak curvature relation is used as a PDE constraint to express the first variation of the bending energy in curvature-vector form. Onsager's principle then combines this variation with dissipation based on the normal velocity, while an independent relaxed minimal-deformation-rate (relaxed-MDR) constraint selects the tangential velocity. The resulting mixed formulation admits a semidiscretization by continuous piecewise linear elements that satisfies an exact energy-dissipation law and exactly preserves area and volume when the corresponding constraints are imposed. The framework accommodates spatially varying spontaneous curvature and applies to both closed surfaces and open surfaces with fixed boundaries under Navier or clamped conditions. We also propose a linearly implicit fully discrete scheme, prove its unique solvability, and present numerical experiments demonstrating convergence, energy decay, constraint preservation, and effective mesh redistribution.
