A Hybrid Boundary Element Approach without Singular Boundary Integrals
A boundary element formulation for 3D-elastostatics and 3D-elastodynamics is presented which avoids singular boundary integrals. The proposed method is based on a generalized variational principle. A weighted superposition of static fundamental solutions
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Abstract. A boundary element formulation for 3D-elastostatics and 3D-elastodynamics is presented which avoids singular boundary integrals. The proposed method is based on a generalized variational principle. A weighted superposition of static fundamental solutions is used for the field approximation in the domain, whereas the displacement and stress field on the boundary are interpolated by well-known polynomial shape functions. By separating time- and space-dependence a symmetric equation of motion is derived with timeindependent mass and stiffness matrix. The domain integral over inertia terms, leading to the mass matrix, is analytically transformed to the boundary. Thus, a boundary only formulation is derived. Comparing numerical results with analytical solutions clearly shows that the obtained system of equations is well-suited for dynamic problems.
1 Introduction The combination of different discretization methods leads to reliable numerical solutions even for challenging tasks in engineering, because the advantages of the involved methods are combined. The finite element method (FEM) is widely used since powerful user-friendly computer codes are available. In order to efficiently couple the symmetric FEM with the boundary element method (BEM) a symmetric BEM formulation is needed - the hybrid displacement boundary element method (HDBEM) fulfills this requirement. In elastostatics as well as in elastodynamics the method is based on generalized variational principles (DeFigueiredo and Brebbia, 1989; Gaul and Fiedler, 1996). The independent field variables are the displacements in the domain as well as the displacements and the tractions on the boundary. The domain field is approximated by a weighted superposition of fundamental solutions, where the weighting factors are unknowns. The displacement and traction field on the boundary are interpolated by shape functions multiplied by nodal data. Using these approximations a symmetric equation system for unknown boundary displacements is obtained by integration over the discretized boundary. Besides regular, weakly and strongly singular integrals also hypersingu1ar integrals have to be numerically evaluated. Latter integrals appear when the load point of the weakly singular displacement fundamental solution coincides with the load point of the strongly singular traction fundamental solution. These integrals are indirectly calculated by considering a stress-free rigid body motion (DeFigueiredo and * Support by the Deutsche Forschungsgemeinschaft DFG of the Graduate Collegium 'Modelling and discretization methods for continua and fluids' at the University of Stuttgart is gratefully acknowledged.
V. Kompiš (ed.), Selected Topics in Boundary Integral Formulations for Solids and Fluids © Springer-Verlag Wien 2002
L. Gaul and F. Moser
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Brebbia, 1989; Dumont, 1999), similar to the rigid body motion technique used in collocation BEM. This technique leads to an overdetermined system of equations for the unknown elements of main diagonal submatrices. The overdetermin
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