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A sinh transformation for evaluating two‐dimensional nearly singular boundary element integrals
Author(s) -
Johnston Barbara M.,
Johnston Peter R.,
Elliott David
Publication year - 2006
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1816
Subject(s) - boundary element method , singular integral , mathematical analysis , order of integration (calculus) , mathematics , transformation (genetics) , boundary (topology) , hyperbolic function , position (finance) , element (criminal law) , laplace transform , slater integrals , projection (relational algebra) , singular point of a curve , integral equation , physics , finite element method , algorithm , biochemistry , chemistry , finance , gene , political science , law , economics , thermodynamics
Abstract 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< b ⩽10), and for flux integrals where b >0.01. For smaller values of b , the use of the L 1 ‐1/5 method is recommended for flux integrals. Copyright © 2006 John Wiley & Sons, Ltd.