Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T08:39:26.810Z Has data issue: false hasContentIssue false
  • English
  • Français

Fast Solution of Three-Dimensional Modified Helmholtz Equations by the Method of Fundamental Solutions

Part of: Partial differential equations, boundary value problems

Published online by Cambridge University Press:  21 July 2016

Ji Lin ,
C. S. Chen  and
Chein-Shan Liu
Ji Lin*
Affiliation:
State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, International Center for Simulation Software in Engineering and Sciences, College of Mechanics and Materials, Hohai University, Nanjing 211100, China
C. S. Chen*
Affiliation:
Department of Mathematics, University of Southern Mississippi, Hattiesburg, MS 39406, USA
Chein-Shan Liu*
Affiliation:
State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, International Center for Simulation Software in Engineering and Sciences, College of Mechanics and Materials, Hohai University, Nanjing 211100, China Department of Civil Engineering, National Taiwan University, Taipei 106-17, Taiwan
*
*Corresponding author. Email addresses:[email protected] (J. Lin), [email protected] (C. S. Chen), [email protected] (C.-S. Liu)
*Corresponding author. Email addresses:[email protected] (J. Lin), [email protected] (C. S. Chen), [email protected] (C.-S. Liu)
*Corresponding author. Email addresses:[email protected] (J. Lin), [email protected] (C. S. Chen), [email protected] (C.-S. Liu)
Get access

Abstract

This paper describes an application of the recently developed sparse scheme of the method of fundamental solutions (MFS) for the simulation of three-dimensional modified Helmholtz problems. The solution to the given problems is approximated by a two-step strategy which consists of evaluating the particular solution and the homogeneous solution. The homogeneous solution is approximated by the traditional MFS. The original dense system of the MFS formulation is condensed into a sparse system based on the exponential decay of the fundamental solutions. Hence, the homogeneous solution can be efficiently obtained. The method of particular solutions with polyharmonic spline radial basis functions and the localized method of approximate particular solutions in combination with the Gaussian radial basis function are employed to approximate the particular solution. Three numerical examples including a near singular problem are presented to show the simplicity and effectiveness of this approach.

Keywords

Method of fundamental solutionsexponential decaymodified Helmholtz problemsparse schemelocalized method of approximate particular solutionsmethod of particular solutions

MSC classification

Secondary: 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Research Article
Information
Communications in Computational Physics , Volume 20 , Issue 2 , August 2016 , pp. 512 - 533
Copyright
Copyright © Global-Science Press 2016 

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] Ihlenburg, F., and Babuska, I., Finite element solution of the Helmholtz equation with high wave number Part I: The h-version of the FEM, Computers & Mathematics with Applications, 30(1995), 9–37.Google Scholar
[2] Chen, J.T., Lee, J.W., and Leu, S.Y, Analytical and numerical investigation for true and spurious eigensolutions of an elliptical membrane using the real-part dual BIEM/BEM, Meccanica, 47(2012), 1103–1117.CrossRefGoogle Scholar
[3] Yang, K., Feng, W.Z., and Gao, X.W., A new approach for computing hyper-singular interface stresses in IIBEM for solving multi-medium elasticity problems, Comput. Methods Appl. Mech. Eng., 287(2015), 54–68.Google Scholar
[4] Liu, Y.J., Zhang, D., and Rizzo, F.J., Nearly singular and hypersingular integrals in the boundary element method, Boundary Elements XV, 1(1993), 453–468.Google Scholar
[5] Atluri, S.N., and Shen, S., The meshless local petrov-galerkin (MLPG) method: a simple & less-costly alternative to the finite element and boundary element methods, CMES-Comp. Model. Eng., 3(2002), 11–52.Google Scholar
[6] Wang, J.G., and Liu, G.R., A point interpolation meshless method based on radial basis functions, Int. J. Numer. Meth. Eng., 54(2002), 1623–1648.CrossRefGoogle Scholar
[7] Mirzaei, D., and Dehghan, M., A meshless based method for solution of integral equations, Appl. Numer. Math., 60(2010), 245–262.Google Scholar
[8] Karageorghis, A., and Fairweather, G., The method of fundamental solutions for axisymmetric acoustic scattering and radiation problems, J. Acoust. Soc. Am., 104(1998), 3212–3218.CrossRefGoogle Scholar
[9] Tadeu, A., Simoes, N., and Simoes, I., Coupling BEM/TBEM and MFS for the simulation of transient conduction heat transfer, Int. J. Numer. Meth. Eng., 84(2010), 179–213.Google Scholar
[10] Lin, J., Chen, W., and Chen, C.S., A new scheme for the solution of reaction diffusion and wave propagation problems, Appl. Math. Model., 38(2014), 5651–5664.Google Scholar
[11] Kupradze, V.D., and Aleksidze, M.A., The method of functional equations for the approximate solution of certain boundary value problems, USSR Computational Mathematics and Mathematical Physics, 4(1964), 82–126.Google Scholar
[12] Fairweather, G., and Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math., 9(1998), 69–95.Google Scholar
[13] Marin, L., and Karageorghis, A., The MFS-MPS for two-dimensional steady-state thermoelasticity problems, Eng. Anal. Bound. Elem., 37(2013), 1004–1020.Google Scholar
[14] Wen, P.H., Chen, C.S., The method of particular solutions for solving scalar wave equations, Int. J. Numer. Meth. Bio., 26(2010), 1878–1889.Google Scholar
[15] Cao, L., Qin, Q.H., and Zhao, N., Application of DRM-Trefftz and DRM-MFS to transient heat conduction analysis, Recent Patents on Space Technology, 2(2010), 41–50.Google Scholar
[16] Partridge, P.W., and Brebbia, C.A., (Eds.), Dual reciprocity boundary element method, Springer Science & Business Media, (2012).Google Scholar
[17] Chen, W., Fu, Z.J., and Jin, B.T., A truly boundary-only meshfree method for inhomogeneous problems based on recursive composite multiple reciprocity technique, Eng. Anal. Bound. Elem., 34(2010), 196–205.Google Scholar
[18] Fu, Z.J., Chen, W., and Yang, H.T., Boundary particle method for Laplace transformed time fractional diffusion equations, J. Comput. Phys., 235(2013), 52–66.Google Scholar
[19] Lin, J., Chen, W., Wang, F., A new investigation into regularization techniques for the method of fundamental solutions, Math. Comput. Simulat., 81(2011), 1144–1152.CrossRefGoogle Scholar
[20] Liu, C.S., Improving the ill-conditioning of the method of fundamental solutions for 2D laplace equation, CMES-Comp. Model. Eng., 851(2009), 1–17.Google Scholar
[21] Liu, C.S., A two-side equilibration method to reduce the condition number of an ill-posed linear system, CMES-Comp. Model. Eng., 91(2013), 17–42.Google Scholar
[22] Lin, J., Chen, W., and Sun, L.L., Simulation of elastic wave propagation in layered materials by the method of fundamental solutions, Eng. Anal. Bound. Elem., 57(2015), 88–95.Google Scholar
[23] Gu, Y., Gao, H., Chen, W., Liu, C., Zhang, C., and He, X., Fast-multipole accelerated singular boundary method for large-scale three-dimensional potential problems, Int. J. Heat Mass Tran., 90(2015), 291–301.Google Scholar
[24] Wei, X., Chen, W., and Chen, B., An ACA acceleratedMFS for potential problems, Eng. Anal. Bound. Elem., 41(2014), 90–97.Google Scholar
[25] Smyrlis, Y.S., and Karageorghis, A., A matrix decomposition MFS algorithm for axisymmetric potential problems, Eng. Anal. Bound. Elem., 28(2004), 463–474.Google Scholar
[26] Chen, C.S., Jiang, X.R., Chen, W., and Yao, G.M., Fast solution for solving the modified Helmholtz equation with the method of fundamental solutions, Commun. Comput. Phys., 17(2015), 867–886.Google Scholar
[27] Yao, G., Kolibal, J., and Chen, C.S., A localized approach for the method of approximate particular solutions, Computers & Mathematics with Applications, 61(2011), 2376–2387.Google Scholar
[28] Lamichhane, A.R., and Chen, C.S., The closed-form particular solutions for Laplace and biharmonic operators using a Gaussian function, Appl. Math. Lett., 46(2015), 50–56.Google Scholar
[29] Rippa, S., An algorithm for selecting a good value for the parameter c in radial basis function interpolation, Adv. Comput. Math., 11(1999), 193–210.Google Scholar
[30] Fasshauer, G.E., and Zhang, J.G., On choosing optimal shape parameters for RBF approximation, Numer. Algorithms, 45(2007), 345–368.CrossRefGoogle Scholar
[31] Johnston, R.L., and Fairweather, G., The method of fundamental solutions for problems in potential flow, Appl. Math. Model., 8(1984), 265–270.Google Scholar
[32] Mathon, R., and Johnston, R.L., The approximate solution of elliptic boundary-value problems by fundamental solutions, SIAM J. Numer. Anal., 14(1977), 638–650.CrossRefGoogle Scholar
[33] Gorzelańczyk, P., and Kolodziej, J.A., Some remarks concerning the shape of the source contour with application of the method of fundamental solutions to elastic torsion of prismatic rods, Eng. Anal. Bound. Elem., 32(2008), 64–75.Google Scholar
[34] Kolodziej, J.A., and Zielinski, A.P., Boundary collocation techniques and their application in engineering, WIT Press, (2009).Google Scholar
[35] Alves, C.J., On the choice of source points in the method of fundamental solutions, Eng. Anal. Bound. Elem., 33(2009), 1348–1361.Google Scholar
[36] Cisilino, A.P., and Sensale, B., Application of a simulated annealing algorithm in the optimal placement of the source points in the method of the fundamental solutions, Comput. Mech., 28(2002), 129–136.Google Scholar
[37] Nishimura, R., Nishimori, K., and Ishihara, N., Determining the arrangement of fictitious charges in charge simulation method using genetic algorithms, J. Electrostat., 49(2000), 95–105.Google Scholar
[38] Nishimura, R., Nishimori, K., and Ishihara, N., Automatic arrangement of fictitious charges and contour points in charge simulation method for polar coordinate system, J. Electrostat., 51(2001), 618–624.Google Scholar
[39] Nishimura, R., Nishihara, M., Nishimori, K., and Ishihara, N., Automatic arrangement of fictitious charges and contour points in charge simulation method for two spherical electrodes, J. Electrostat., 57(2003), 337–346.Google Scholar
[40] Liu, C.S., An equilibrated method of fundamental solutions to choose the best source points for the Laplace equation, Eng. Anal. Bound. Elem., 36(2012), 1235–1245.Google Scholar
[41] Chen, C.S., Karageorghis, A., and Li, Y., On choosing the location of the sources in the MFS, Numer. Algorithms, (2015), 1–24.Google Scholar
[42] Golberg, M.A., Muleshkov, A.S., Chen, C.S., and Cheng, A.H.-D., Polynomial particular solutions for certain kind of partial differential operators, Numer. Meth. Part. D. E., 19(2003), 112–133.CrossRefGoogle Scholar
[43] Chen, C.S., Fan, C.M., and Wen, P.H., The method of approximate particular solutions for solving certain partial differential equations, Numer. Meth. Part. D. E., 28(2012), 506–522.Google Scholar
[44] Jiang, T., Li, M., and Chen, C.S., The method of particular solutions for solving inverse problems of a nonhomogeneous convection-diffusion equation with variable coefficients, Numer. Heat Tr. A-Appl., 61(2012), 338–352.Google Scholar