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 L2 flow, curve or surface diffusion, and the area- or volume-constrained L2 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.
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BibTeX
@misc{zhang2026unified,
title = {{A unified structure-preserving framework for geometric flows with coupled orientation and curvature dependence}},
author = {Yulin Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36419},
eprint = {2609.36419},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36419},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A unified structure-preserving framework for geometric flows with coupled orientation and curvature dependence
AU - Yulin Zhang
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.36419
UR - https://arxiv.org/abs/2609.36419
AB - 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.
ER -