The evaluation of the inverse distance singular integrand which occurs in the three-dimensional boundary element formulation of potential problems is treated using weighted Gaussian integration. Three methods are investigated. The first involves repeated use of a one-dimensional Gaussian formula in which the weights are inverse powers. The second involves the inverse distance in the local parameter plane to which the surface elements are transformed. Comparisons with some exact integrals developed over plane parallelograms show that neither of these first two methods performs particularly well, with the second being appropriate only for integration over unit squares.
A third method is developed using an approximation to the real space distance in the weighting. Such a method does take into account the local properties of the surface and gives integrals accurate to the precision of the computer being used for tests on a plane parallelogram. In a further test integrating over a curved element on a sphere, good accuracy is again achieved. Overall, however, in the type of integration test considered, none of these weighted Gaussian methods integrates as accurately as the expansion or transformation methods published elsewhere.