A unified structure-preserving framework for geometric flows with coupled orientation and curvature dependence

Abstract

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.