Adaptive Singularity Cancellation for Efficient Treatment of Near-Singular and Near-Hypersingular Integrals in Surface Integral Equation Formulations

Abstract

A proposed singularity cancellation technique for fully numerical evaluation of method of moments integrals in surface integral equation solutions produces reasonably accurate results with few quadrature points for singular and hypersingular integrals. However, for near-singular and near-hypersingular integrals, time-consuming computations need to be repeatedly performed over unnecessary regions outside the actual integration domain. For a more efficient treatment of these integrals, an adaptive singularity cancellation technique is proposed. As such, the source triangular domain is subdivided in a way that all sample points remain inside the desired integration domain and unnecessary computations are avoided. Second the accuracy of results in existing singularity cancellation transformations is greatly affected by variations in height of observation point above the plane of source domain. This drawback has been removed in the adaptive singularity cancellation transformations. Additionally, an optimum selection criterion for the distribution of quadrature samples is presented. The criterion enables run-time selection of optimum number of samples in different directions by consideration of the instantaneous geometry of the transformed integration domain.