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.
翻译:本文介绍了用于精确和高效整合多边形和受参数曲线约束的二维区域功能的边框缩放(SBC)方案。在二维域中,SBC方法减少了以美元曲线覆盖的区域(称为曲线三角区域)的整合,以整合超过百万美元的区域(称为曲线三角区域),每个区域都受两条线段和一个曲线的约束。有了适当的(对数时)边界曲线方向,这个方案还适用于二次曲线和非二次曲线域。此外,对于恒星-convex域,还获得了具有正权重和内部集成点的强产孵化规则。如果正重和内部集成点为一体,我们则显示这种新方法将降低为单一数字整合办法;然而,SBC方案则更加灵活,因为它同样适用于单一和不均分的函数。本文还介绍了用点和近级域域平滑动的复合体域域。在使用这些方法时,将高效率的软体产品孵化法集成一个比准的直径法则,而其直径比直径直径法则可以实现。