Evaluating Singular, Near-Singular, and Non-Singular Integrals on Curvilinear Elements

Abstract

Recent progress in developing accurate quadrature rules for singular integrals is potentially very useful in higher-order electromagnetic modeling. It is often not so clear, however, how methods developed for linear elements can be applied to curvilinear elements. To this end, a detailed implementation of schemes is presented for handling numerical quadrature for modeling curvilinear elements in both integral and differential equation formulations. It is shown that the introduction of a tangent element not only incorporates fundamental geometry information about the surface and is useful in describing bases defined on the element, but it also facilitates the handling of quadrature rules, especially for singular or near-singular integrals. Detailed treatment of the use of singularity cancelation methods on curvilinear elements is given, and it is shown how singularity cancelation quadrature rules derived on linear elements apply to curvilinear elements.