Scaled boundary cubature scheme for numerical integration over polytopes and curved solids. Part I: Two-dimensional domains.

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. A forthcoming paper will cover integration over polyhedra and solids bounded by curved surfaces using the SBC scheme. 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 [1, 2]; 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 based on the generalized Duffy transformation [3] and a distance transformation described in Ma and Kamiya [4]. When these methods are combined, 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.