{
  "abstract": "We extend the scaled boundary cubature (SBC) scheme from planar regions to higher-dimensional regions described by oriented boundary patches. The resulting parametrization transforms integrals over compact regions in $\\mathbb{R}^d$ into sums of integrals over the boundary-patch parameter domains and a radial coordinate. In three dimensions, this yields a direct volume-integration rule for solids bounded by affine faces, triangular or tensor-product surface patches, B-spline patches, NURBS patches, and combinations of curved and affine boundary representations. For affine polytopes, recursive application of the scaled boundary map yields nested tensor-product rules over simplex sectors; in three dimensions, these reduce to tetrahedral-sector rules that apply equally to convex and nonconvex oriented polyhedra. We also develop transformations for weakly singular integrands. Placing the scaling center at a point singularity exposes the radial power cancelled by the Jacobian, while generalized radial scalings and Gauss--Jacobi quadrature handle fractional powers. A transverse scaled-boundary map provides the analogous construction for affine singular sets, with straight-line examples in three dimensions. Numerical examples verify polynomial exactness on affine polyhedra and a four-dimensional tesseract, rapid convergence on curved B-spline and NURBS solids, and the expected convergence improvements for point and line singularities. Near-boundary singularity tests also identify when additional patch subdivision or patch-parameter transformations are required.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.15563",
  "author_total": 2,
  "authors": [
    {
      "name": "Eric B. Chin"
    },
    {
      "name": "N. Sukumar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8chspsfqqfzb6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.15563",
  "fulltext_url": null,
  "id": "8chspsfqqfzb6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.15563",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2608.15563",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8chspsfqqfzb6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2608.15563",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.15563"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Scaled boundary cubature scheme in higher dimensions: integration over polytopes and curved regions",
  "title_text": "Scaled boundary cubature scheme in higher dimensions: integration over polytopes and curved regions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/8chspsfqqfzb6",
  "venue": "arXiv",
  "versions": []
}