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COMPUTATIONAL METHODS FOR ACOUSTICS PROBLEMS
Edited by: F. Magoulès
Boundary Integral Equations Methods in Acoustic Scattering
A. Bendali1,2 and M. Fares2
1Institute of Mathematics in Toulouse, INSA, Department of Mathematics, Toulouse, France
A. Bendali, M. Fares, "Boundary Integral Equations Methods in Acoustic Scattering", in F. Magoulès, (Editor), "Computational Methods for Acoustics Problems", Saxe-Coburg Publications, Stirlingshire, UK, Chapter 1, pp 1-36, 2008. doi:10.4203/csets.18.1
Keywords: acoustic scattering, Helmholtz equation, boundary integral equations, finite element method, coupling, domain decomposition method, cross-points.
The main subject of this contribution is to present some recent methods, specially designed to be implemented on parallel platforms, to deal with acoustic scattering problems involving a bounded zone filled by a heterogeneous medium. The main approach is to couple a finite element method for handling this zone, with a boundary integral equation specially adapted to treat the unbounded part of the computational domain. After giving a short review of some alternative methods, we focus on the methods based on this approach and give a framework which makes it possible to construct almost all the standard boundary integral equations. As is well-known, each instance of this kind of scattering problem can be solved by a manifold of such equations. This framework allows one to have a good insight into the advantages and the drawbacks of each of them. It is seen next that the above coupling gives rise to non standard linear systems, with a matrix being partly sparse and partly dense. Serious difficulties then arise when the solution of such systems has to be tackled on a parallel platform. It is shown how techniques from domain decomposition methods can be used to efficiently overcome these difficulties.
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