A new method for evaluating singular integrals in stress analysis of solids by the direct boundary element method

Abstract

AbstractThe purpose of this paper is to report on a new and efficient method for the evaluation of singular integrals in stress analysis of elastic and elasto‐plastic solids, respectively, by the direct boundary element method (BEM). Triangle polar co‐ordinates are used to reduce the order of singularity of the boundary integrals by one degree and to carry out the integration over mappings of the boundary elements onto plane squares. The method was subsequently extended to the cubature of singular integrals over three‐dimensional internal cells as occur in applications of the BEM to three‐dimensional elasto‐plasticity. For this purpose so‐called tetrahedron polar co‐ordinates were introduced. Singular boundary integrals stretching over either linear, triangular, or quadratic quadilateral, isoparametric boundry elements and singular volume integrals extending over either linear, tetrahedral, or quadratic, hexahedral, isoparametric internal cells are treated. In case of higher order isoparametric boundary elements and internal cells, division into a number of subelements and subcells, respectively, is necessary. The analytical investigation is followed by a numerical study restricted to the use of quadratic, quadrilateral, isoparametric boundary elements. This is justified by the fact that such elements, as opposed to linear elements, yield singular boundary integrals which cannot be integrated analytically. The results of the numerical investigation demonstrate the potential of the developed concept.