Geometric structure-preserving parametric finite element approximations for the constrained Helfrich flow
This work provides a robust numerical method for simulating constrained Helfrich flow, which is important for applications in biology and materials science, but the approach is an incremental improvement over existing methods.
The authors propose a structure-preserving parametric finite element method for the constrained Helfrich flow of curves and surfaces, achieving unconditional energy dissipation, exact volume and area conservation, and good mesh quality. Numerical experiments demonstrate accuracy and robustness for 2D and 3D cases.
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.