{
  "abstract": "Volume quadrature can change nonlinear finite-element constraints while preserving their reference-state derivatives. We connect an explicit determinant defect to feasible-set geometry and singular mechanical response. For affine tensor elements of coordinate degree $p\\ge3$ with $n\\ge p+1$ Gauss points per coordinate, determinant volume is exact precisely when $2n\\ge3p$. Below that threshold we construct a boundary-fixed cubic defect at every order. The same directions yield a full-space cube-root residual--distance bound under explicit cell-support, coefficient and physical-norm assumptions, with mesh-uniform upper constants at fixed order. With all cell-pressure equations retained, the volume Jacobian gains rank at nearby feasible states despite agreement through second derivatives at rest; the cube-root exponent is sharp on each fixed mesh. A general localized-minimum theorem shows that the first reduced compatibility term contributes its weighted square to the leading energy in a joint small-load, large-bulk limit. Cubic and quadratic defects therefore produce sextic and quartic terms. The full cubic-element interior space has an exact normal form and sharp local error exponents. Curved quadratic tetrahedra supply the second-order contrast, a sparse rational witness and an exact four-Jacobian volume formula. Finite-strain tensor calculations illustrate normalized response separation, with explicit stationary-point, extreme-bulk and pressure-recovery qualifications. The constructive correction preserves exact cell volumes, so quadrature feasibility remains distinct from physical volume preservation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13446",
  "author_total": 4,
  "authors": [
    {
      "name": "Yanlin Liu"
    },
    {
      "name": "Chao Huang"
    },
    {
      "name": "Kaixiang Yao"
    },
    {
      "name": "Yao Shen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3p74y4vfrk70k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13446",
  "fulltext_url": null,
  "id": "3p74y4vfrk70k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13446",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.13446",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3p74y4vfrk70k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.13446",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13446"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "RUPA: Nonlinear volume consistency, constraint geometry and singular penalty limits in finite elements",
  "title_text": "RUPA: Nonlinear volume consistency, constraint geometry and singular penalty limits in finite elements",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/3p74y4vfrk70k",
  "venue": "arXiv",
  "versions": []
}