Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-28T01:49:38.520Z Has data issue: false hasContentIssue false
  • English
  • Français

An Efficient and Stable Spectral-Element Method for Acoustic Scattering by an Obstacle

Published online by Cambridge University Press:  28 May 2015

Jing An  and
Jie Shen
Jing An*
Affiliation:
School of Mathematics and Science, Xiamen University, Xiamen 361005, China
Jie Shen*
Affiliation:
School of Mathematics and Science, Xiamen University, Xiamen 361005, China Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA
*
Corresponding author. Email Address: [email protected]
Corresponding author. Email Address: [email protected]
Get access

Abstract

A spectral-element method is developed to solve the scattering problem for time-harmonic sound waves due to an obstacle in an homogeneous compressible fluid. The method is based on a boundary perturbation technique coupled with an efficient spectral-element solver. Extensive numerical results are presented, in order to show the accuracy and stability of the method.

Keywords

78A4565N3535J0541A58Spectral methodacoustic scatteringboundary perturbationHelmholtz equation
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 3 , Issue 3 , August 2013 , pp. 190 - 208
Copyright
Copyright © Global-Science Press 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bergmann, P.G., The wave equation in a medium with a variable index of refraction, J. Acoust. Soc. Am. 17, 329–333 (1946).Google Scholar
[2]Bruno, O.P. and Reitich, F., Solution of a boundary value problem for the Helmholtz equation via variation of the boundary into the complex domain, Proc. Roy. Soc. Edinburgh A – Mathematics 122, 317–340 (1992).Google Scholar
[3]Colton, D.L. and Kress, R., Inverse Acoustic and Electromagnetic Scattering Theory, Springer Verlag (1998).CrossRefGoogle Scholar
[4]Fang, Q., Nicholls, D.P and Shen, J., A stable, high-order method for three-dimensional, bounded-obstacle, acoustic scattering. J. Comput. Phys. 224, 1145–1169 (2007).Google Scholar
[5]He, Y., Nicholls, D.P. and Shen, J., An efficient and stable spectral method for electromagnetic scattering from a layered periodic structure, J. Comput. Phys. 231, 3007–3022 (2012).Google Scholar
[6]Martin, P.A., Acoustic scattering by inhomogeneous spheres, J. Acoust. Soc. Am. 111, 2013–2018 (2002).Google Scholar
[7]Nicholls, D.P. and Nigam, N., Exact non-reflecting boundary conditions on general domains, J. Comput. Phys. 194, 278–303 (2004).Google Scholar
[8]Nicholls, D.P. and Reitich, F., A new approach to analyticity of Dirichlet-Neumann operators, Proc. Roy. Soc. Edinburgh A – Mathematics 131, 1411–1434 (2001).Google Scholar
[9]Nicholls, D.P. and Reitich, F., Stability of high-order perturbative methods for the computation of Dirichlet-Neumann operators, J. Comput. Phys. 170, 276–298 (2001).Google Scholar
[10]Nicholls, D.P. and Reitich, F., Analytic continuation of Dirichlet-Neumann operators, Numer. Math. 94, 107–146 (2003).Google Scholar
[11]Nicholls, D.P. and Reitich, F., Shape deformations in rough-surface scattering: Cancellations, conditioning, and convergence, JOSA A 21, 590–605 (2004).CrossRefGoogle ScholarPubMed
[12]Nicholls, D.P. and Reitich, F., Shape deformations in rough-surface scattering: Improved algorithms, JOSA A 21, 606–621 (2004).Google Scholar
[13]Nicholls, D.P. and Shen, J., A stable high-order method for two-dimensional bounded-obstacle scattering, SIAM J. Sci. Comput. 28, 1398–1419 (2006).Google Scholar
[14]Nicholls, D.P and Shen, J., A rigorous numerical analysis of the transformed field expansion method, SIAM J. Numer. Anal. 47, 2708–2723 (2009).Google Scholar
[15]Rayleigh, Lord, On the dynamical theory of gratings, Soc. London Ser. A 79, 399–416 (1907).Google Scholar
[16]Reitich, F. and Tamma, K.K., State-of-the-art, trends, and directions in computational electromagnetics, Comput. Model. Engr. Sci. 5, 287–294 (2004).Google Scholar
[17]Rice, S.O., Reflection of electromagnetic waves from slightly rough surfaces, Comm. Pure Appl. Math. 4, 351–378 (1951).CrossRefGoogle Scholar
[18]Shen, J., Efficient Spectral-Galerkin method I. Direct solvers of second-and fourth-order equations using Legendre polynomials, SIAM J. Sci. Comput. 15, 1489–1508 (1994).Google Scholar
[19]Shen, J., Efficient Spectral-Galerkin method II. Direct solvers of second-and fourth-order equations using Chebyshev polynomials, SIAM J. Sci. Comput. 16, 74–87 (1995).Google Scholar
[20]Shen, J., Efficient Spectral-Galerkin methods III: Polar and cylindrical geometries, SIAM J. Sci. Comput. 18, 1583–1604 (1997).Google Scholar
[21]Shen, J., Efficient Spectral-Galerkin methods IV. Spherical geometries, SIAM J. Sci. Comput. 20, 1438–1455 (1999).CrossRefGoogle Scholar
[22]Shen, J. and Wang, L.-L., Analysis of a Spectral-Galerkin approximation to the Helmholtz equation in exterior domains, SIAM J. Numer. Anal. 45, 1954–1978 (2007).Google Scholar
[23]Warnick, K.F. and Chew, W.C., Numerical simulation methods for rough surface scattering, Waves in Random Media 11, 1–30 (2001).Google Scholar