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 adequate, 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 effect of clustering the integration 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 accurate evaluation of these integrals can be obtained. Based on observations of several integrals considered, guidelines are suggested for the order of the sigmoidal transformations. Copyright © 1999 John Wiley & Sons, Ltd.