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Spectral volumetric integral equation methods for acoustic medium scattering in a 3D waveguide

Armin Lechleiter 1 Dinh Liem Nguyen 1
1 DeFI - Shape reconstruction and identification
Inria Saclay - Ile de France, CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
Abstract : The Lippmann–Schwinger integral equation describes the scattering of acoustic waves from an inhomogeneous medium. For scattering problems in free space, Vainikko proposed a fast spectral solution method exploiting the convolution structure of this equation's integral operator and the fast Fourier transform. Although the integral operator of the Lippmann–Schwinger integral equation for scattering in a planar three-dimensional waveguide is not a convolution, we show in this paper that the separable structure of the kernel allows to construct fast spectral collocation methods. The numerical analysis of this method requires smooth material parameters; for discontinuous materials there is no theoretical convergence statement. Therefore, we construct a Galerkin variant of Vainikko's method avoiding this drawback. For several distant scattering objects inside the three-dimensional waveguide this discretization technique would lead to a computational domain consisting of one large box containing all scatterers and hence many unnecessary unknowns. However, the integral equation can be reformulated as a coupled system with unknowns defined on the different parts of the scatterer. Discretizing this coupled system by a combined spectral/multipole approach yields an efficient method for waveguide scattering from multiple objects.
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Submitted on : Thursday, January 31, 2013 - 10:24:21 AM
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Armin Lechleiter, Dinh Liem Nguyen. Spectral volumetric integral equation methods for acoustic medium scattering in a 3D waveguide. IMA Journal of Numerical Analysis, Oxford University Press (OUP), 2012, 32 (3), pp.813-844. ⟨10.1093/imanum/drr036⟩. ⟨hal-00783010⟩



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