On New Triangle Quadrature Rules for the Locally Corrected Nystrom Method Formulated on NURBS Generated Bezier Surfaces in 3D


This paper was essentially the last work I did as a PhD student. I say last, only because it was published latest, but in fact this work started very early on in my PhD. The goal was to show high-order convergence on some realistic scattering objects. Such goal proved to be very hard to achieve. In fact, as seen in the paper, HO converge was only achieved for smooth scatterers and objects with sharp edges prevent LCN from achieving HO behavior. In the final chapter of my PhD thesis, I suggest some approaches for doing so.

Let us see the 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 Nyström (LCN) method with NURBS requires formulating LCN on both quadrilateral and triangular Bézier surfaces, as a typical NURBS-generated Bézier 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 Bézier 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.

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