高维下的比例边界求积方案：多面体与曲区域上的积分
Scaled boundary cubature scheme in higher dimensions: integration over polytopes and curved regions
AI总结：
本研究将比例边界求积方案扩展至高维，提出适用于多面体与曲区域的体积积分规则，通过数值示例验证了其多项式精确性、快速收敛性及对奇点的处理效果。
AI中文摘要：
我们将比例边界求积（SBC）方案从平面区域扩展到由定向边界片描述的高维区域。所得参数化将ℝ^d中紧区域上的积分转化为边界片参数域与径向坐标上的积分之和。在三维中，这产生了对由仿射面、三角形或张量积曲面片、B样条片、NURBS片以及曲与仿射边界表示组合所界定的实体的直接体积积分规则。对于仿射多面体，比例边界映射的递归应用产生了单纯形扇区上的嵌套张量积规则；在三维中，这些规则简化为四面体扇区规则，可同等应用于凸与非凸定向多面体。我们还开发了弱奇异被积函数的变换：将缩放中心置于点奇点处可揭示被雅可比抵消的径向幂，而广义径向缩放与高斯-雅可比求积则处理分数幂；横向比例边界映射为仿射奇异集提供了类似构造，三维中存在直线示例。数值示例验证了仿射多面体与四维超正方体上的多项式精确性、弯曲B样条与NURBS实体上的快速收敛，以及点与线奇点的预期收敛改进，近边界奇点测试还确定了何时需要额外的片细分或片参数变换。
英文摘要：
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 $\Re^d$ 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.
