On New Triangle Quadrature Rules for the Locally Corrected Nyström Method Formulated on NURBS-Generated Bézier Surfaces in 3-D

Abstract

Nonuniform rational b-splines (NURBS) are the most widely used technique in today’s geometric computer-aided design systems for modeling surfaces. Combining the locally corrected Nystrom (LCN) method with NURBS requires formulating LCN on both quadrilateral and triangular Bezier surfaces, as a typical NURBS-generated Bezier mesh includes elements of both the types. While on quadrilateral elements the product of 1-D Gaussian quadrature rules can be applied to LCN effectively, Gaussian integration rules available for triangles cannot efficiently be applied to LCN for two reasons. First, they do not possess the same number of quadrature points as the number of functions in a complete set of polynomial basis at an arbitrary order. Second, they exacerbate the condition number of the resulting matrix equation at higher orders due to an increasing density of quadrature points near the edges and corners of triangles. In this paper, we construct a new set of quadrature rules for Bezier triangles (i.e., degenerate quadrilaterals) based on the Newton–Cotes (equidistant) quadrature rules and apply these rules to the LCN solution of the electric, magnetic, and combined field integral equations. Results show that the new family of quadrature rules overcomes both the aforementioned issues and can be applied to LCN effectively for orders from 0 to 9, inclusively.