Abstract:We propose a structure-preserving parametric finite element method for the constrained Helfrich flow of closed curves and surfaces. The proposed method is based on a two-stage velocity-splitting strategy. In the first stage, the normal velocity is computed from a curvature evolution equation, with the volume and surface area constraints imposed softly in terms of the normal velocity. This step approximates the gradient-flow structure of the Helfrich flow, and its fully discrete parametric finite element approximation leads to a linear system and yields an unconditional energy dissipation estimate at the fully discrete level. In the second stage, the surface mesh is updated by combining the computed normal velocity with a BGN-type tangential velocity. A time-weighted interface normal and an area-correction multiplier are also used to enforce exact preservation of the enclosed volume and surface area. This correction step leads to a nonlinear system, which can be efficiently solved by Newton iteration. The resulting method simultaneously achieves energy decay, exact geometric conservation, and good mesh quality. Numerical experiments for two-dimensional curves and three-dimensional surfaces, including nonsmooth initial data and nonzero spontaneous curvature, are presented to demonstrate the accuracy, robustness, and structure-preserving properties of the proposed method.
From: Quan Zhao [view email]
[v1]
Mon, 15 Jun 2026 13:31:54 UTC (7,534 KB)
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