Analytic formulations for calculating nearly singular integrals in two-dimensional BEM

There exist the nearly singular integrals in the boundary integral equations when a source point is close to an integration element but not on the element, such as the field problems with thin domains. In this paper, the analytic formulations are achieved to calculate the nearly weakly singular, strongly singular and hyper-singular integrals on the straight elements for the two-dimensional (2D) boundary element methods (BEM). The algorithm is performed after the BIE are discretized by a set of boundary elements. The singular factor, which is expressed by the minimum relative distance from the source point to the closer element, is separated from the nearly singular integrands by the use of integration by parts. Thus, it results in exact integrations of the nearly singular integrals for the straight elements, instead of the numerical integration. The analytic algorithm is also used to calculate nearly singular integrals on the curved element by subdividing it into several linear or sub-parametric elements only when the nearly singular integrals need to be determined. The approach can achieve high accuracy in cases of the curved elements without increasing other computational efforts. As an application, the technique is employed to analyze the 2D elasticity problems, including the thin-walled structures. Some numerical results demonstrate the accuracy and effectiveness of the algorithm.