Two general algorithms for nearly singular integrals in two dimensional anisotropic boundary element method

This paper presents an extension of the previously published sinh transformation and semi-analytical method for the evaluation of nearly singular integrals. The extension involves applying the two methods to two dimensional (2D) general anisotropic boundary element method (BEM). The new feature of the present method is that the distance from the calculation point to parabolic elements is expressed as $$r^{2}=(\xi -\eta )^{2}g(\xi )+b^{2}$$r2=(ξ-η)2g(ξ)+b2, where $$g(\xi )$$g(ξ) is a well-behaved function, $$\eta \hbox { and }b$$ηandb stand for the position of the projection of the nearly singular point and the shortest distance from the calculation point to the integration element, respectively. As a result, the two methods can be employed in a straightforward fashion. The accuracy and the efficiency of the proposed methods are demonstrated with four benchmark test integrals that are commonly encountered in the application of anisotropic BEM. Comparisons between the two proposed methods are also presented in the paper.