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The Method of Fundamental Solutions for Solving Exterior Axisymmetric Helmholtz Problems with High Wave-Number

Published online by Cambridge University Press:  03 June 2015

Wen Chen ,
Ji Lin  and
C.S. Chen
Wen Chen*
Affiliation:
College of Engineering Mechanics, Hohai University, Nanjing 210098, Jiangsu, China
Ji Lin
Affiliation:
College of Engineering Mechanics, Hohai University, Nanjing 210098, Jiangsu, China
C.S. Chen
Affiliation:
Department of Mathematics, University of Southern Mississippi, Hattiesburg, MS 39406, USA
*
*Corresponding author. Email: [email protected]
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Abstract

In this paper, we investigate the method of fundamental solutions (MFS) for solving exterior Helmholtz problems with high wave-number in axisymmetric domains. Since the coefficient matrix in the linear system resulting from the MFS approximation has a block circulant structure, it can be solved by the matrix decomposition algorithm and fast Fourier transform for the fast computation of large-scale problems and meanwhile saving computer memory space. Several numerical examples are provided to demonstrate its applicability and efficacy in two and three dimensional domains.

Keywords

65M1078A48Method of fundamental solutionsexterior Helmholtz problemcirculant matrixfast Fourier transformaxisymmetric domain
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 5 , Special Issue 4 , August 2013 , pp. 477 - 493
Copyright
Copyright © Global-Science Press 2013

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References

[1]Alves, Carlos J. S., Vitor, M. A. and Leitão, , Crack analysis using an enriched MFS domain decomposition technique, Eng. Anal. Bound. Elem., 30 (2006), pp. 160–166.Google Scholar
[2]Alves, Carlos J.S. and Valtchev, Svilen S., Numerical comparsion of two meshfree methods for acoustic wave scattering, Eng. Anal. Bound. Elem., 29 (2005), pp. 371–382.Google Scholar
[3]Brebbia, C. A. and Wrobel, L. C., The boundary element method, Comput. Methods Fluid, London, Pentech Press, Ltd.,1980, pp. 26–48.Google Scholar
[4]Bin-Mohsin, B. and Lesnic, D., Determination of inner boundaries in modified Helmholtz inverse geometric problems using the method of fundamentalsolutions, Math. Comput. Simulation, 82 (2012), pp. 1445–1458.Google Scholar
[5]Bayliss, A. and Turke, E., Radiation boundary conditions for wave-like equations, Commun. Pure Appl. Math., 33 (2006), pp. 707–725.Google Scholar
[6]Barnett, A. H. and Betcke, T., Satbility and convergence of the method of fundamental solutions for Helmholtz problems on analytic domains, J. Comput. Phys., 227 (2008), pp. 7003–7026.Google Scholar
[7]Chen, C. S., Cho, H. A. and Golberg, M. A., Somments on the ill-conditioning of the method of fundamental solutions, Eng. Anal. Bound. Elem., 30 (2009), pp. 405–410.Google Scholar
[8]Drombosky, T. W., Meyer, A. L. and Ling, L., Applicability of the method of fundamental solutions, Eng. Anal. Bound. Elem., 33 (2009), pp. 637–643.CrossRefGoogle Scholar
[9]Davis, P. J., Circulant Matrices, John Wiley & Sons, New York, Chichester, Brisbane, 1979.Google Scholar
[10]Fairweather, G., Karageorghis, A. and Smyrlis, Y. S., A matrix decomposition MFS algorithm for axisymmetric biharmonic problems, Adv. Comput. Math., 23 (2005), pp. 55–71.Google Scholar
[11]Fairweather, G. and Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math., 9 (1998), pp. 69–95.Google Scholar
[12]Fairweather, G., Karageorghis, A. and Martin, P. A., The method of fundamental solutions for scattering and radiation problems, Eng. Anal. Bound. Elem., 27 (2003), pp. 759–769.Google Scholar
[13]Golberg, M. A., The method of fundamental solutions for Poisson’s equation, Eng. Anal. Bound. Elem., 16 (1995), pp. 205–213.CrossRefGoogle Scholar
[14]Golberg, M. A. and Chen, C. S., The method of fundamental solutions for potential, Helmholtz and diffusion problems, in: Golberg, M.A. (Eds.), Boundary intergral methods-numerical and mathematical aspects, Computational Mechanics Publications, 1998, pp. 103–176.Google Scholar
[15]Heryudono, A. R. H. and Driscoll, T. A., Radial basis function interpolation on irregular domain through conformal transplanation, J. Sci. Comput., 44 (2010), pp. 286–300.Google Scholar
[16]Hon, Y. C. and Wei, T., A fundamental solution method for inverse heat conduction problem, Eng. Anal. Bound. Elem., 28 (2004), pp. 489–495.Google Scholar
[17]Karageorghis, A., Chen, C. S. and Smyrlis, Y. S., Matrix decomposition RBF algorithm for solving 3D elliptic problems, Eng. Anal. Bound. Elem., 33 (2009), pp. 1368–1373.Google Scholar
[18]Karageorghis, A. and Smyrlis, Y. S., Conformal mapping for the efficient MFS solution of Dirichlet boundary value problems, Computing, 83 (2008), pp. 1–24.Google Scholar
[19]Karageorghis, A. and Fairweather, G., The method of fundamental solutions for axisym-metric potential problems, Int. J. Numer. Methods Eng., 44 (1999), pp. 1653–1669.3.0.CO;2-1>CrossRefGoogle Scholar
[20]Karageorghis, A. and Fairweather, G., The method of fundamental solutions for axisym-metric elasticity problems, Comput. Mech., 25 (2000), pp. 524–532.Google Scholar
[21]Karageorghis, A. and Fairweather, G., The method of fundamental solutions for axisymmetric acoustic scattering and radiation problems, J. Acoustic Soc. Am., 104 (1998), pp. 3212–3218.Google Scholar
[22]Kitagawa, T., On the numerical stability of the method of fundamental solution applied to the Dirichlet problem, Japan J. Industrial Appl. Math., 5 (1988), pp. 123–133.Google Scholar
[23]Li, X. L. and Zhu, J. L., The method of fundamental solutions for nonlinear elliptic problems, Eng. Anal. Bound. Elem., 33 (2009), pp. 322–329.Google Scholar
[24]Li, J. C. and Hon, Y. C., Domain decomposition for radial basis meshless methods, Numer. Methods Partial Differential Equations, 20 (2004), pp. 450–462.Google Scholar
[25]Mera, N. S., The method of fundamental solutions for the backward heat conduction problem, Inverse Prob. Sci. Eng., 13 (2005), pp. 65–78.CrossRefGoogle Scholar
[26]Neumaier, A., Solving ill-conditioned and singular linear systems: a tutorial on regularization, SIAM Rev., 40 (1998), pp. 636–666.Google Scholar
[27]Ramachandran, P. A., Method of fundamental solutions: singular value deconposition analysis, Commun. Numer. Methods Eng., 18 (2002), pp. 789–801.Google Scholar
[28]Smylis, Y. S. and Karageorghis, A., Some aspects of the method of fundamental solutions for certain harmonic problems, J. Sci. Comput., 16 (2001), pp. 341–371.CrossRefGoogle Scholar
[29]Smylis, Y. S. and Karageorghis, A., Some aspects of the method of fundamental solutions for certain biharmonic problems, Comput. Model. Eng. Sci., 4 (2003), pp. 535–550.Google Scholar
[30]Seybert, A. F., Soenarko, B., Rizzo, F. J. and Shippy, D. J., An advanced computational method for radiation and scattering of acoustic waves in three dimensions, The Journal of Acoustical Society of America, 77 (1985), pp. 362–368.Google Scholar
[31]Tsangaris, TH., Smyrlis, Y. S. and Karageorghis, A., A matrix decomposition mfs algorithm for problems in hollow axisymmetric domains, J. Sci. Comput., 28 (2006), pp. 31–50.Google Scholar
[32]Wei, T., Hon, Y. C. and Ling, L., Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators, Eng. Anal. Bound. Elem., 31 (2007), pp. 373–385.Google Scholar
[33]Young, D. L., Tsai, C. C., Chen, C. W. and Fan, C. M., The method of fundamental solutions and condition number analysis for inverse problems of Laplace equations, Comput. Math. Appl., 55 (2008), pp. 1189–1200.Google Scholar
[34]Young, D. L., Fan, C. M., Tsai, C. C. and Chen, C. W., The method of fundamental solutions and domain decomposition method for degenerate seepage flownet problems, Journal of the Chinese Institute of Engineering, 20 (2006), pp. 63–73.Google Scholar
[35]Zienkiewicz, O. C., Kelly, D. W. and Bettess, P., The sommerfeld (radiation) condition on infinite domains and its modelling in numerical procedures, Comput. Methods Appl. Sci. Eng., 704 (1979), pp. 169–203.Google Scholar