A systematic derived sinh based method for singular and nearly singular boundary integrals

Abstract

When using boundary element analysis for thin walled structures, to ensure the computational accuracy, special considerations on the singular and nearly singular integrals are indispensible. The real issue about the singular and nearly singular integrals is how to remove the near singularity (for singular and nearly singular integrals) and singularity (for singular integrals only). The near singularity is caused by the location of the singular points or near singular points in the sub-regions which the integrals are performed. The performance of our method is as follows. For the singular integrals, a coordinate transformation at the singular point is employed to remove the singularities, by which the variables are also separated independently. Then, a derived sinh method is developed for removal of the near singularities. For the nearly singular integrals, a coordinate transformation at the nearest point is employed to separate the two variables. For each variable, sinh methods are constructed. Three numerical examples are proposed, considering a cube with coarse mesh, a plane structure and a simplified millstone of a vertical mill. It can be found that the results are in good agreement with the analytical solutions and FEM solutions.