On non-linear transformations for the integration of weakly-singular and Cauchy Principal Value integrals

The mathematical foundations of the application of non-linear transformations to the numerical integration of weakly singular and Cauchy Principal Value (CPV) integrals are revised in this paper. This approach was Þrstly introduced to compute the singular kernels appearing in the Boundary Element Method (BEM) for 2-D (two-dimensional) problems. Here, the mathematical requirements for a consistent application of these methods both for the 2-D and 3-D cases and both for collocation points located in the interior of a typical boundary element and located on its boundary are established. From them, it is shown why some of the transformations proposed in previous papers work while others do not, and also why some of the latter ones work only for the particular examples presented in those papers. Finally, some non-linear transformations for 2-D and 3-D problems that fulÞll the mentioned mathematical requirements are here introduced, including a complete numerical study of their accuracy