A generalisation of Telles’ method for evaluating weakly singular boundary element integrals

摘要

The accurate numerical integration of line integrals is of fundamental importance for the reliable implementation of the boundary element method. Usually, the regular integrals arising from a boundary element method implementation are evaluated using standard Gaussian quadrature. However, the singular integrals which arise are often evaluated in another way, sometimes using a different integration method with different nodes and weights. This paper presents a simple transformation to improve the accuracy of evaluating weakly singular integrals. The transformation is, in a sense, a generalisation of the popular method of Telles (Internat. J. Numer. Methods Eng. 24 (1987) 959–973) with the underlying idea being to utilise the same Gaussian quadrature points as used for evaluating nonsingular integrals in a typical boundary element method implementation. Comparison of the generalised method with existing coordinate transformation techniques shows that a more accurate evaluation of weakly singular integrals can be obtained. Based on observation of several integrals considered, guidelines are suggested for the best transformation order to use (i.e. the degree to which nodes should be clustered at the singular points). An analysis of the generalised approach is also presented where asymptotic estimates for the truncation error, after evaluating the transformed integrals using Gauss–Legendre quadrature, are determined. Further numerical experimentation shows that these are very good estimates.