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Shadow boundary effects in hybrid numerical-asymptotic methods for high-frequency scattering

Published online by Cambridge University Press:  30 June 2015

D. P. HEWETT
D. P. HEWETT*
Affiliation:
Department of Mathematics and Statistics, University of Reading, UK
*
Current address: Mathematical Institute, University of Oxford, UK Email: [email protected]
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Abstract

The hybrid numerical-asymptotic (HNA) approach aims to reduce the computational cost of conventional numerical methods for high-frequency wave scattering problems by enriching the numerical approximation space with oscillatory basis functions, chosen based on partial knowledge of the high-frequency solution asymptotics. In this paper, we propose a new methodology for the treatment of shadow boundary effects in HNA boundary element methods, using the classical geometrical theory of diffraction phase functions combined with mesh refinement. We develop our methodology in the context of scattering by a class of sound-soft non-convex polygons, presenting a rigorous numerical analysis (supported by numerical results) which proves the effectiveness of our HNA approximation space at high frequencies. Our analysis is based on a study of certain approximation properties of the Fresnel integral and related functions, which govern the shadow boundary behaviour.

Keywords

high frequency scatteringdiffractionboundary element methodnumerical analysisfresnel integral
Type
Papers
Information
European Journal of Applied Mathematics , Volume 26 , Special Anniversay Issue 5: Celebrating 75 years , October 2015 , pp. 773 - 793
Copyright
Copyright © Cambridge University Press 2015 

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References

[1]Digital Library of Mathematical Functions. National Institute of Standards and Technology, from http://dlmf.nist.gov/, release date: 2010-05-07.Google Scholar
[2]Alazah, M., Chandler-Wilde, S. N. & La Porte, S. (2014) Computing Fresnel integrals via modified trapezium rules. Numer. Math. 128, 635–661.CrossRefGoogle Scholar
[3]Asheim, A. & Huybrechs, D. (2010) Local solutions to high-frequency 2D scattering problems. J. Comput. Phys. 229, 5357–5372.CrossRefGoogle Scholar
[4]Borovikov, V. A. & Kinber, B. Y. (1994) Geometrical Theory of Diffraction, Institution of Electrical Engineers, London.CrossRefGoogle Scholar
[5]Bowman, J. J., Senior, T. B. A. & Uslenghi, P. L. E. (1969) Electromagnetic and Acoustic Scattering by Simple Shapes, North-Holland, Amsterdam.Google Scholar
[6]Chandler-Wilde, S. N., Graham, I. G., Langdon, S. & Spence, E. A. (2012) Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering. Acta Numer. 21, 89–305.CrossRefGoogle Scholar
[7]Chandler-Wilde, S. N., Hewett, D. P., Langdon, S. & Twigger, A. (2015) A high frequency boundary element method for scattering by a class of nonconvex obstacles. Numer. Math. 129, 647–689.CrossRefGoogle Scholar
[8]Chandler-Wilde, S. N. & Langdon, S. (2007) A Galerkin boundary element method for high frequency scattering by convex polygons. SIAM J. Numer. Anal. 45, 610–640.CrossRefGoogle Scholar
[9]Dominguez, V., Graham, I. G. & Smyshlyaev, V. P. (2007) A hybrid numerical-asymptotic boundary integral method for high-frequency acoustic scattering. Numer. Math. 106, 471–510.CrossRefGoogle Scholar
[10]Ganesh, M. & Hawkins, S. C. (2011) A fully discrete Galerkin method for high frequency exterior acoustic scattering in three dimensions. J. Comput. Phys. 230, 104–125.CrossRefGoogle Scholar
[11]Groth, S. G., Hewett, D. P. & Langdon, S. (2015) Hybrid numerical-asymptotic approximation for high frequency scattering by penetrable convex polygons. IMA J. Appl. Math. 80, 324–353.CrossRefGoogle Scholar
[12]Hewett, D. P. (2015) Tangent ray diffraction and the Pekeris caret function. Wave Motion. 57, 257–267.CrossRefGoogle Scholar
[13]Hewett, D. P., Langdon, S. & Chandler-Wilde, S. N. (2014) A frequency-independent boundary element method for scattering by two-dimensional screens and apertures. IMA J. Numer. Anal. doi: 10.1093/imanum/dru043.Google Scholar
[14]Hewett, D. P., Langdon, S. & Melenk, J. M. (2011) A high frequency hp boundary element method for scattering by convex polygons, University of Reading Department of Mathematics and Statistics Preprint MPS-2011-118.Google Scholar
[15]Hewett, D. P., Langdon, S. & Melenk, J. M. (2013) A high frequency hp boundary element method for scattering by convex polygons. SIAM J. Numer. Anal. 51, 629–653.CrossRefGoogle Scholar
[16]Keller, J. B. (1962) Geometrical theory of diffraction. J. Opt. Soc. Am. 52, 116–130.CrossRefGoogle ScholarPubMed
[17]Kouyoumjian, R. G. & Pathak, P. H. (1974) A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface. Proc. IEEE 62, 1448–1461.CrossRefGoogle Scholar
[18]Oberhettinger, F. (1956) On asymptotic series for functions occuring in the theory of diffraction of waves by wedges. J. Math. Phys. 34, 245–255.CrossRefGoogle Scholar
[19]Ockendon, J. R. & Tew, R. H. (2012) Thin-layer solutions of the Helmholtz and related equations. SIAM Rev. 54 (1), 3–51.CrossRefGoogle Scholar
[20]Perrey-Debain, E., Lagrouche, O., Bettess, P. & Trevelyan, J. (2004) Plane-wave basis finite elements and boundary elements for three-dimensional wave scattering. Philos. Trans. R. Soc. Lond. Ser. A 362, 561–577.CrossRefGoogle ScholarPubMed
[21]Spence, E. A., Chandler-Wilde, S. N., Graham, I. G. & Smyshlyaev, V. P. (2011) A new frequency-uniform coercive boundary integral equation for acoustic scattering. Commun. Pure Appl. Math. 64, 1384–1415.CrossRefGoogle Scholar
[22]Stenger, F. (1993) Numerical Methods Based on Sinc and Analytic Functions, Springer-Verlag.CrossRefGoogle Scholar
[23]Tew, R. H., Chapman, S. J., King, J. R., Ockendon, J. R., Smith, B. J. & Zafarullah, I. (2000) Scalar wave diffraction by tangent rays. Wave Motion 32, 363–380.CrossRefGoogle Scholar