Scaled boundary cubature scheme in higher dimensions: integration over polytopes and curved regions
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 ℝd 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.
Related
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BibTeX
@misc{chin2026scaled,
title = {{Scaled boundary cubature scheme in higher dimensions: integration over polytopes and curved regions}},
author = {Eric B. Chin and N. Sukumar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.15563},
eprint = {2608.15563},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.15563},
}
RIS
TY - UNPB
TI - Scaled boundary cubature scheme in higher dimensions: integration over polytopes and curved regions
AU - Eric B. Chin
AU - N. Sukumar
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2608.15563
UR - https://arxiv.org/abs/2608.15563
AB - 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 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.
ER -