On a Unified Procedure for the Numerical Evaluation of Singular and Quasi-Singular Double Integrals

Abstract

A boundary integral implementation generally involves the evaluation of singular and quasi-singular integrals. Dealing with them requires elaborated codes, which quite frequently results in prohibitive computational effort as the price of adequate numerical accuracy. In the frame of a research line in progress at PUC-Rio, a simple and accurate scheme has been developed for the treatment of general singular linear integrals. It is now being implemented for double integrals. This technique makes use of fixed abscissas, as taken from Gauss-Legendre quadrature, with different weight sets calculated for different singularities — a procedure that demands low computational effort. This feature allows the development of robust codes, in which integration demands low computational effort and yields high precision. The core of the present contribution is the introduction of an adequate scheme for the global-to-local transformation of the coordinates of a given singularity pole. This transformation may be used in connection with polar coordinates, as singular integrals are usually dealt with. In the paper, however, it is an essential feature of the unified integration procedure the authors are proposing. The technique is straightforward to implement, independently of the degree of singularity or quasi-singularity.