Keywords: boundary elements, symmetric formulation, double integration, hypersingular kernels, complex variables.

The standard form of the boundary element method is based on the use of boundary integral equations obtained weighting domain integral equations by singular fundamental solutions, corresponding to point sources, and approximating the boundary variables by interpolation functions. The coefficient matrix of the boundary system is unsymmetrical, limiting the boundary element analysis of dynamic and coupling problems.

The Galerkin formulation represents a procedure for obtaining symmetric boundary models [1]. In this approach the boundary integral equations are generated by sources distributed on the boundary and interpolated by the same shape functions used for the boundary variables. The terms of the boundary system are derived from the double integrals of the product of two shape functions and a fundamental solution. In two-dimensional elasticity these kernels exhibit singularities up to O when the integration domains of the field and source distributions are totally or partially overlapped.

The main computational difficulties arising in the construction of symmetric boundary element models concern the computation of double boundary integrals. The integration is developed separately for the source and the field distributions, defining the extremes of one integration domain as a function of the location of the other integration variable. This process becomes intricate in the case of generically oriented integration domains, encouraging the use of numerical integration procedures. Difficulties related to the evaluation of the singular contributions require special techniques [1,2,4].

The present work proposes an analytical procedure based on a complex variables algebra [3] and a suitable integration rule which allows a concise development of the double integration. The integrands involved in the entries of the boundary system can be simply expressed in terms of a limited number of one-dimensional basic functions. The integration strategy also includes a process consisting of a suitable number of Gauss transformations which regularize the kernels. This process is supported by the use of hat shape functions defined over supports made of two contiguous boundary elements. The use of the integration by parts turns out to be convenient also for the evaluation of nonsingular double integrals. Indeed, it leads to a significant reduction of the number of the type of integrals. The proposed complex variables technique is presented referring to the symmetric boundary element analysis of plane elasticity problems defined on polygonal domains.

The paper is organized as follows. By referring to plane elasticity problems, the symmetric boundary element formulation is briefly recalled. Successively, the relevant quantities are described in complex variables. A presentation of the integration procedure follows, giving the integration rule and the basic functions involved. Some typical integration results are also given.

1

M. Bonnet, G. Maier, C. Polizzotto, "Symmetric Galerkin boundary element methods", Applied Mechanical Review, 51, 11, 1998. doi:10.1115/1.3098983

2

S. Holzer, "How to deal with hypersingular integrals in the symmetric BEM", Communications in Numerical Methods for Engineering, 9, 219-232, 1993. doi:10.1002/cnm.1640090306

3

S. Sirtori, G. Maier, G. Novati, S. Miccoli, "A Galerkin symmetric boundary-element method in elasticity: formulation and implementation", International Journal of Numerical Methods in Engineering, 35, 255-282, 1992. doi:10.1002/nme.1620350204

4

M. Mazza, M. Aristodemo, "Integration of hypersingular kernels in symmetric Bem analysis of Kirchhoff plates", Proceedings of the Sixth International Conference on Computational Structures Technology, Prague, Czech Republic, Civil-Comp Press, UK, 2002. doi:10.4203/ccp.76.19

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