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An O(1) integration scheme for three-dimensional surface scattering problems

Bruno, Oscar P. and Geuzaine, Christophe A. (2007) An O(1) integration scheme for three-dimensional surface scattering problems. Journal of Computational and Applied Mathematics, 204 (2). pp. 463-476. ISSN 0377-0427. doi:10.1016/

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We present an accurate method of O(1)-complexity with respect to frequency (i.e., a method that, to achieve a prescribed error tolerance, requires a bounded computational cost for arbitrarily high frequencies) for the computation of singular oscillatory integrals arising in the boundary integral formulation of problems of acoustic scattering by surfaces in three-dimensional space. Like the two-dimensional counterpart of this algorithm, which we introduced recently and which is applicable to scattering by curves in the plane, the present method is based on a combination of two main elements: (1) a high-frequency ansatz for the unknown density in a boundary integral formulation of the problem, and (2) an extension of the ideas of the method of stationary phase to allow for O(1) (high-order-accurate) integration of oscillatory functions. The techniques we introduce to implement an efficient O(1) integrator in the present three-dimensional context differ significantly from those used in the earlier two-dimensional algorithm. In particular, in the present text, we introduce an efficient “canonical” (hybrid analytic-numerical) algorithm which, in addition to allowing for integration of oscillatory functions around both singular points and points of stationary phase, can handle the significant difficulty that arises as singular points and one or more stationary points approach each other within the two-dimensional scattering surface. We include numerical results illustrating the behavior of the integration algorithm on sound-soft spheres with diameters of up to 5000 wavelengths: in such cases, for a single integral, the algorithm yields accuracies of the order of three digits in computational times of less than two seconds. In a preliminary full scattering simulation we present, a solution with two digits of accuracy in the surface density was obtained in about three hours running time, in a single 1.5 GHz AMD Athlon processor, for a sphere of 500 wavelengths in diameter.

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Bruno, Oscar P.0000-0001-8369-3014
Additional Information:© 2006 Elsevier B.V. Received 16 October 2005; received in revised form 7 February 2006. Available online 10 July 2006.
Subject Keywords:wave scattering; boundary integral equations; high frequency methods
Issue or Number:2
Classification Code:MSC: 78A45; 78M25; 78M35
Record Number:CaltechAUTHORS:20100820-091539194
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19545
Deposited By: Tony Diaz
Deposited On:20 Aug 2010 20:25
Last Modified:08 Nov 2021 23:53

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