Numerical Treatment of Stress Singularities With BEM
by
A.-M. Saendig
Mathematical Institute A/6, University of Stuttgart
Coauthors: Marius Bochniak (University of Stuttgart)

According to Cea's lemma, the accuracy of finite element and boundary element methods depends on the approximation properties of test and ansatz spaces which, in turn, depend on the regularity of the solution. In linear elastic structures the stresses can be unbounded due to geometrical singularities as corner and edges or due to structural singularities as generated by the heterogenity of materials. In such cases the regularity of the solution is limited in the sense that it can not be improved by increasing the regularity of given volume and boundary forces.
In order to improve the convergence properties of finite and boundary element methods it is necessary to have detailed information on the regularity of the solution and the influence of stress singularities on the orders of convergence. This can be done by means of local asymptotic expansions of the solution in the neighbourhood of singular points. Based on this knowledge it is possible to improve the numerical algorithms by choosing ansatz spaces with better approximation properties.
In this paper we investigate boundary element domain decomposition methods (BEM-DDM's) for the solution of elastic boundary value and transmission problems in non-smooth domains. First we apply the Aubin-Nitsche trick to obtain the order of convergence for the Galerkin discretisation of BEM-DDM's on quasi-uniform meshes [3]. It turns out that the maximal order of convergence in weak Sobolev norms coincides with the order of convergence for the approximation of stress intensity factors with Maz'ya/Plamenevsky formulae. Finally we propose in analogy to the dual singular function method [1] a modified BEM-DDM with improved convergence properties. The theoretical results are illustrated by numerical examples.

[1] H. Blum, M. Dobrowolski On finite element methods for elliptic equations on domains with corners, Computing 28 (1982) pp. 53-63.
[2] M. Bochniak Numerical treatment of stress singularities in transmission problems by BEM, In: W. Hackbusch, S. Sauter (eds.) Numerical Techniques for Composite Materials, Vieweg-Verlag, Braunschweig, to appear.
[3] M. Bochniak, A.-M. Sändig Computation of generalized stress intensity factors for bonded elastic structures, Math. Model. Numer. Anal., to appear 1999.

Date received: June 22, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadk-08.