Application of sigmoidal transformations to weakly singular and near-singular boundary element integrals

Accurate numerical determination of line integrals is fundamental to reliable implementation of the boundary element method. For a source point distant from a particular element, standard Gaussian quadrature is ade- quate, as well as being the technique of choice. However, when the integrals are weakly singular or nearly singular (source point near the element) this technique is no longer adequate. Here a co-ordinate transformation technique, based on sigmoidal transformations, is introduced to evaluate weakly singular and near-singular integrals. A sigmoidal transformation has the e ect of clustering the inte- gration points towards the endpoints of the interval of integration. The degree of clustering is governed by the order of the transformation. Comparison of this new method with existing co-ordinate transformation techniques shows that more accu- rate evaluation of these integrals can be obtained. Based on observations of several integrals considered, guide- lines are suggested for the order of the sigmoidal transformations