Semi-sigmoidal transformations for evaluating weakly singular boundary element integrals

Accurate numerical integration of line integrals is of fundamental importance to 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. Here, a co-ordinate transformation technique is introduced for evaluating weakly singular integrals which, after some initial manipulation of the integral, uses the same integration nodes and weights as those of the regular integrals. The transformation technique is based on newly defined semi-sigmoidal transformations, which cluster integration nodes only near the singular point. The semi-sigmoidal transformations are defined in terms of existing sigmoidal transformations and have the benefit of evaluating integrals more accurately than full sigmoidal transformations as the clustering is restricted to one end point of the interval. Comparison of this new method with existing coordinate transformation techniques shows that more accurate evaluation of weakly singular integrals can be obtained. Based on observation of several integrals considered, guidelines are suggested for the type of semi-sigmoidal transformation to use and the degree to which nodes should be clustered at the singular points. Copyright © 2000 John Wiley & Sons, Ltd.