{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.36419",
  "author_total": 1,
  "authors": [
    {
      "name": "Yulin Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2q0frbw04rd5t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.36419",
  "fulltext_url": null,
  "id": "2q0frbw04rd5t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.36419",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2206|Advanced Numerical Methods in Computational Mathematics"
    ]
  ],
  "openalex": "W7214961728",
  "pdf_url": "https://arxiv.org/pdf/2609.36419",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2q0frbw04rd5t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.36419",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.36419"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "A unified structure-preserving framework for geometric flows with coupled orientation and curvature dependence",
  "title_text": "A unified structure-preserving framework for geometric flows with coupled orientation and curvature dependence",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/2q0frbw04rd5t",
  "venue": "arXiv",
  "versions": []
}