A sinh transformation for evaluating two‐dimensional nearly singular boundary element integrals

A new transformation technique is introduced for evaluating the two-dimensional nearly singular integrals, which arise in the solution of Laplace's equation in three dimensions, using the boundary element method, when the source point is very close to the element of integration. The integrals are evaluated using (in a product fashion) a transformation which has recently been used to evaluate one-dimensional near singular integrals. This sinh transformation method automatically takes into account the position of the projection of the source point onto the element and also the distance b between the source point and the element. The method is straightforward to implement and, when it is compared with a number of existing techniques for evaluating two-dimensional near singular integrals, it is found that the sinh method is superior to the existing methods considered, both for potential integrals across the full range of b values considered (0 0.01. For smaller values of b, the use of the Lmethod is recommended for flux integrals. Copyright © 2006 John Wiley & Sons, Ltd.

[1]  Thomas J. Rudolphi,et al.  Stress intensity sensitivities via Hypersingular boundary integral equations , 1999 .

[2]  M. H. Aliabadi,et al.  Boundary element hyper‐singular formulation for elastoplastic contact problems , 2000 .

[3]  J. Watson,et al.  Effective numerical treatment of boundary integral equations: A formulation for three‐dimensional elastostatics , 1976 .

[4]  J. Telles.  A self-adaptive co-ordinate transformation for efficient numerical evaluation of general boundary element integrals , 1987 .

[5]  Yijun Liu.  ANALYSIS OF SHELL-LIKE STRUCTURES BY THE BOUNDARY ELEMENT METHOD BASED ON 3-D ELASTICITY: FORMULATION AND VERIFICATION , 1998 .

[6]  C. Brebbia,et al.  Boundary Element Techniques , 1984 .

[7]  Masatake Mori,et al.  Quadrature formulas obtained by variable transformation , 1973 .

[8]  Vladimir Sladek,et al.  Evaluation of 1/r integrals in BEM formulations for 3-D problems using coordinate multitransformations , 1997 .

[9]  David Elliott,et al.  A sinh transformation for evaluating nearly singular boundary element integrals , 2005 .

[10]  C. A. Brebbia,et al.  A New Coordinate Transformation Method for Singular and Nearly Singular Integrals over General Curved Boundary Elements , 1987 .

[11]  T. Cruse,et al.  Non-singular boundary integral equation implementation , 1993 .

[12]  Peter Rex Johnston,et al.  A comparison of transformation methods for evaluating two‐dimensional weakly singular integrals , 2003 .

[13]  Ken Hayami,et al.  A numerical quadrature for nearly singular boundary element integrals , 1994 .

[14]  Tatacipta Dirgantara,et al.  Crack Growth analysis of plates Loaded by bending and tension using dual boundary element method , 2000 .