仿射和曲线边界平面区域上数值积分的比例边界体积格式
Scaled boundary cubature scheme for numerical integration over planar regions with affine and curved boundaries
摘要：
本文介绍了一种用于多边形和二维参数曲线区域上函数积分的尺度边界体积（SBC）格式。在二维域上，SBC方法将由$m$曲线界定的区域上的积分减少为$m$区域（称为弯曲三角形区域）上的积分，其中每个区域由两个线段和一条曲线界定。只要边界曲线具有适当的（逆时针）方向，该格式就适用于凸和非凸区域。另外，对于星型凸区域，得到了一个权值为正且积分点在区域内部的张量积求积规则。如果被积函数是齐次的，我们表明，这种新的方法减少到齐次数值积分计划，然而，SBC计划是更通用的，因为它是同样适用于齐次和非齐次函数。本文还介绍了几种光滑具有点奇点和近奇点的被积函数的方法。当使用这些方法时，实现了弱奇异函数的高效积分。SBC方法被应用到一些基准问题，这揭示了其广泛的适用性和优越的性能（在时间上产生的规则和精度每立方点）相比，现有的方法进行整合。
英文摘要：
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.
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