An efficient method to evaluate hypersingular and supersingular integrals in boundary integral equations analysis

An efficient algorithm is employed to evaluated hyper and super singular integral equations encountered in boundary integral equations analysis of engineering problems. The algorithm is based on multiple subtractions and additions to separate singular and regular integral terms in the polar transformation domain, primarily established in Refs. (Guiggiani M, Krishnasamy G, Rudolphi TJ, Rizzo FJ. A general algorithm for the numerical solution of hypersingular boundary integral equations. Trans ASME 1992;59:604–614; Guiggiani M, Casalini P. Direct computation of Cauchy principal value integral in advanced boundary element. Int J Numer Meth Engng 1987;24:1711–1720. Guiggiani M, Gigante A. A general algorithm for multidimensional Cauchy principal value integrals in the boundary element method. J Appl Mech Trans ASME 1990;57:906–915). It can be proved that the regular terms have finite analytical solutions in the range of integration, and the singular terms will be replaced by special periodic kernels in the integral equations. The subtractions involve to multiple derivatives of analytical kernels and the additions require some manipulation to separate the remaining regular terms from singular ones. The regular terms are computed numerically. Three examples on numerical evaluation of singular boundary integrals are presented to show the efficiency and accuracy of the algorithm. In this respect, strongly singular and hypersingular integrals of potential flow problems are considered, followed by a supersingular integral which is extracted from the partial differentiation of a hypersingular integral with respect to the source point.