论文标题
缩放边界立方体方案，用于具有仿射和弯曲边界的平面区域的数值集成
Scaled boundary cubature scheme for numerical integration over planar regions with affine and curved boundaries
论文作者
论文摘要
本文介绍了缩放边界立方体（SBC）方案，以精确有效地集成多边形和由参数曲线界定的二维区域。在二维域上，SBC方法可在$ m $曲线界定到$ m $区域（称为弯曲的三角形区域）上的集成区域上的集成，在该区域中，每个区域都由两个线段和一个曲线界定。边界曲线的正确（逆时针）方向，该方案适用于凸和非凸域。此外，对于恒星凸域，获得了一个张量产物的立方体规则，在域内具有正权重和积分点。如果集成剂是均匀的，我们表明这种新方法将减少为均匀的数值集成方案。但是，SBC方案更具用途，因为它同样适用于同质和非均匀函数。本文还介绍了几种与点奇点和近乎差异性平滑整合的方法。当使用这些方法时，实现了弱奇异函数的高效整合。与现有的集成方法相比，SBC方法应用于许多基准问题，这些问题揭示了其广泛的适用性和出色的性能（就每次立方体生成规则和准确性的时间和准确性而言）。
This paper introduces the scaled boundary cubature (SBC) scheme for accurate and efficient integration of functions over polygons and two-dimensional regions bounded by parametric curves. Over two-dimensional domains, the SBC method reduces integration over a region bounded by $m$ curves to integration over $m$ regions (referred to as curved triangular regions), where each region is bounded by two line segments and a curve. With proper (counterclockwise) orientation of the boundary curves, the scheme is applicable to convex and nonconvex domains. Additionally, for star-convex domains, a tensor-product cubature rule with positive weights and integration points in the interior of the domain is obtained. If the integrand is homogeneous, we show that this new method reduces to the homogeneous numerical integration scheme; however, the SBC scheme is more versatile since it is equally applicable to both homogeneous and non-homogeneous functions. This paper also introduces several methods for smoothing integrands with point singularities and near-singularities. When these methods are used, highly efficient integration of weakly singular functions is realized. The SBC method is applied to a number of benchmark problems, which reveal its broad applicability and superior performance (in terms of time to generate a rule and accuracy per cubature point) when compared to existing methods for integration.
