Boundary Layer Effect in BEM with High Order Geometry Elements Using Transformation

Abstract

The accurate evaluation of nearly singular integrals is one of the major concerned problems in the boundary element method (BEM). Although the current methods have achieved great progress, it is often possible only for problems defined in the simplest geometrical domains when the nearly singular integrals need to be calculated. However, engineering processes occur mostly in complex geometrical domains, and always, involve nonlinearities of the unknown variables and its derivatives. Therefore, effective methods of dealing with nearly singular integrals for such practical problems are necessary and need to be further investigated. In this paper, a general strategy based on a nonlinear transformation is introduced and applied to evaluate the nearly singular integrals in two dimensional (2D) elasticity problems. The proposed nonlinear transformation method can figure out the rapid variations of nearly singular kernels and extremely high accuracy of numerical results can be achieved without increasing other computational efforts. The accuracy and efficiency of the method are demonstrated through three examples that are commonly encountered in the applications of the BEM.