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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 89
PROCEEDINGS OF THE SIXTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY
Edited by: M. Papadrakakis and B.H.V. Topping
Paper 136

Evaluation of Singular Integrals in the Two-Dimensional Symmetric Galerkin Boundary Element Method

W.F. Yuan

School of Civil & Environmental Engineering, Nanyang Technological University, Singapore

Full Bibliographic Reference for this paper
W.F. Yuan, "Evaluation of Singular Integrals in the Two-Dimensional Symmetric Galerkin Boundary Element Method", in M. Papadrakakis, B.H.V. Topping, (Editors), "Proceedings of the Sixth International Conference on Engineering Computational Technology", Civil-Comp Press, Stirlingshire, UK, Paper 136, 2008. doi:10.4203/ccp.89.136
Keywords: symmetric Galerkin boundary element method, artificial body force, singular integral.

Summary
The symmetric Galerkin boundary element method (SGBEM) rests on both displacement and traction integral equations, which are derived from four fundamental solutions, viz. Guu, Gpu, Gup and Gpp. However, the fundamental displacement Gpu and the fundamental traction Gpp will require strong and hypersingular double integrals in boundary integral equations [1]. To date no general effective interpretations and consequent computational methods have been proposed for evaluating hypersingular integrals with desirable confidence. This situation has hindered the progress of the SGBEM. In this manuscript, a new approach is proposed to carry out all the singular integrals. The proposed method introduces a set of artificial body forces to the domain considered to calculate strong singular integrals numerically, and then evaluates those hypersingular integrals indirectly based on the basic relationship between displacement and traction integral equations. In the corresponding implementation of SGBEM, singular integrals behave such as 1/r and 1/r2 are avoided completely.

The strategy of the proposed artificial body force method (ABFM) is summarised as follows:

  1. Assume all the boundary displacements are zero.
  2. Apply different artificial body forces to the elastic body and solve the boundary fictitious tractions due to each artificial body force case from the traction integral equation.
  3. Successively substitute the fictitious tractions into the traction integral equation and generate several equations in which the terms contain the strong singular integrals are involved.
  4. Solve the equations generated in the last step and further evaluate the strong singular double integrals.
  5. Evaluate hypersingular double integrals indirectly based on the relationship existing among Guu Gpu and Gpp.

As an inevitable result, the ABFM induces volume integrals in the numerical implementation of the SGBEM. In general, those volume integrals can be carried out by dividing the domain into many small areas. In this study, the artificial body force takes the form of a Dirac delta function, thus, the volume integrals are simply evaluated.

Besides the method proposed in the manuscript, a potential alternative approach is also suggested to calculate the fictitious boundary traction due to an artificial body force. The idea is to relate the domain modelled to a semi-infinite domain using Schwarz-Christoffel mapping and Rongved's solutions for the problem of concentrated force in the interior of a semi-infinite solid [2].

References
1
M. Bonnet, G. Maier, C. Polizzotto, "Symmetric Galerkin boundary element methods", Appl. Mech. Rev., 51, 669-704, 1998. doi:10.1115/1.3098983
2
L. Rongved, "Point force in the interior of a semi-infinite solid with fixed boundary", J. Appl. Mech. 22, 545-546, 1955.

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