Keywords: Kirchhoff plates, boundary elements, Galerkin approach, symmetric formulation, hypersingular kernels, integration.

The boundary element methods lead to a substantial reduction of the number of variables and represent displacement and stress fields with comparable accuracy. The standard forms of these methods evaluate the unknown boundary fields by imposing the boundary conditions at the collocation points [1]. The entries of the boundary system are obtained by single integration of kernels which become singular when the collocation point belongs to the integration domain. The resulting matrix is fully populated, unsymmetrical and non definite. This feature is undesirable in dynamical and coupling problems.

The Galerkin method represents a way of defining symmetric boundary element models [2,3,4]. This approach introduces further integral equations, associated with boundary displacement discontinuities, and involves the double boundary integration of the fundamental solutions weighted with the functions approximating the boundary variables and the source density distributions. In the analysis of Kirchhoff plates the weighted boundary equations correspond to force, couple and displacement discontinuity and rotation discontinuity sources.

In developing symmetrical boundary element models the main computational difficulties concern the double boundary integration of the kernels when the integration domains are overlapping or contiguous. In these situations the fundamental solutions exhibit singularities which range from the lower order () to hypersingularities ( and higher). Several strategies have been proposed to attack the double boundary integration of singular kernels. They are based on regularization procedures [5], the Hadamard finite part concept [6] or a limit approach [7,8,9]. The singularities arising from the Galerkin boundary element analysis of Kirchhoff plates represent a severe test for these techniques because of the order of singularity which attains O.

The present work deals with polygonal Kirchhoff plate problems, particularly with the analytical integration of the entries of boundary systems generated by a Galerkin approximation. A simple technique for the analytical evaluation of these terms is proposed. It is based on the use of the limit approach in combination with the assumption of suitable interpolation functions. According to the limit approach, the source point is first moved away from the boundary to carry out the inner integration and then is lead back on it to perform the outer integration. The hat shape functions are chosen in order to satisfy the consistency rules which guarantee symmetry and to regularize the singular kernels, vanishing at the ends of the integration domains where the integration process locates the singularity poles.

After a brief description of the boundary integral formulation of Kirchhoff plate problems, the paper reports the basic equations of the symmetrical boundary element model and discusses the selection of the interpolation functions. Then the double integration of singular contributions is analyzed describing the proposed procedure for both regular and corner points. Some results are also reported from the integration relative to the kernels exhibiting the highest singularity.

1

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M. Mazza, M. Aristodemo, "Finite elements derived from symmetrical boundary elements", Fifth International Conference on Computational Structures Technology, 6-8 september 2000, Leuven - Belgium.

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M. Aristodemo, A. Leone, M. Mazza, "Energy boundary elements for finite element analysis" Meccanica, 36, 4, 2001. doi:10.1023/A:1015096723832

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