A direct approach for boundary integral equations with high-order singularities

Boundary integral equations with extremely singular (i.e., more than hypersingular) kernels would be useful in several elds of applied mechanics, particularly when second- and third-order derivatives of the primary variable are required. However, their de nition and numerical treatment pose several problems. In this paper, it is shown how to obtain these boundary integral equations with still unnamed singularities and, moreover, how to e ciently and reliably compute all the singular integrals. This is done by extending in full generality the so-called direct approach. Only for de niteness, the method is presented for the analysis of the de ection of thin elastic plates. Numerical results concerning integrals with singularities up to order r−4 are presented to validate the proposed algorithm