Calculation of 2D singular and near singular integrals of boundary elements method based on the complex space CC

The solution of Poissons equation is essential in many branches of science and engineering such as fluid-mechanics, geosciences, and electrostatics. Solution of the two-dimensional Poissons equation is carried out by the boundary elements method (BEM) based on the dual reciprocity method (DRM) and the multiple reciprocity method (MRM). There are some singular and near singular integrals in the DRM and MRM which are critical in BEM solutions. In this work two integral algorithms (Exact and Numerical schemes) are proposed based on the complex space CC, and are applied to calculating the boundary integrals. In fact treating singularity and near singularity of the boundary integrals is the main aim of this paper. The potentials at the interior points very close to boundary are evaluated by the present scheme. The integrals are computed for constant, linear and higher order elements when the geometry of the boundary elements is straight. Finally numerical examples demonstrate that the new algorithms presented in the current paper can effectively handle singular and near singular integrals emerging in the boundary layer effect and the thin body problems in BEM.