Skip to main content Accessibility help

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

  • Ji Lin (a1), C. S. Chen (a2) and Chein-Shan Liu (a1) (a3)

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.

Corresponding author
*Corresponding author. Email addresses: (J. Lin), (C. S. Chen), (C.-S. Liu)
Hide All
[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), 937.
[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), 11031117.
[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), 5468.
[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), 453468.
[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), 1152.
[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), 16231648.
[7] Mirzaei, D., and Dehghan, M., A meshless based method for solution of integral equations, Appl. Numer. Math., 60(2010), 245262.
[8] Karageorghis, A., and Fairweather, G., The method of fundamental solutions for axisymmetric acoustic scattering and radiation problems, J. Acoust. Soc. Am., 104(1998), 32123218.
[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), 179213.
[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), 56515664.
[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), 82126.
[12] Fairweather, G., and Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math., 9(1998), 6995.
[13] Marin, L., and Karageorghis, A., The MFS-MPS for two-dimensional steady-state thermoelasticity problems, Eng. Anal. Bound. Elem., 37(2013), 10041020.
[14] Wen, P.H., Chen, C.S., The method of particular solutions for solving scalar wave equations, Int. J. Numer. Meth. Bio., 26(2010), 18781889.
[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), 4150.
[16] Partridge, P.W., and Brebbia, C.A., (Eds.), Dual reciprocity boundary element method, Springer Science & Business Media, (2012).
[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), 196205.
[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), 5266.
[19] Lin, J., Chen, W., Wang, F., A new investigation into regularization techniques for the method of fundamental solutions, Math. Comput. Simulat., 81(2011), 11441152.
[20] Liu, C.S., Improving the ill-conditioning of the method of fundamental solutions for 2D laplace equation, CMES-Comp. Model. Eng., 851(2009), 117.
[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), 1742.
[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), 8895.
[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), 291301.
[24] Wei, X., Chen, W., and Chen, B., An ACA acceleratedMFS for potential problems, Eng. Anal. Bound. Elem., 41(2014), 9097.
[25] Smyrlis, Y.S., and Karageorghis, A., A matrix decomposition MFS algorithm for axisymmetric potential problems, Eng. Anal. Bound. Elem., 28(2004), 463474.
[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), 867886.
[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), 23762387.
[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), 5056.
[29] Rippa, S., An algorithm for selecting a good value for the parameter c in radial basis function interpolation, Adv. Comput. Math., 11(1999), 193210.
[30] Fasshauer, G.E., and Zhang, J.G., On choosing optimal shape parameters for RBF approximation, Numer. Algorithms, 45(2007), 345368.
[31] Johnston, R.L., and Fairweather, G., The method of fundamental solutions for problems in potential flow, Appl. Math. Model., 8(1984), 265270.
[32] Mathon, R., and Johnston, R.L., The approximate solution of elliptic boundary-value problems by fundamental solutions, SIAM J. Numer. Anal., 14(1977), 638650.
[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), 6475.
[34] Kolodziej, J.A., and Zielinski, A.P., Boundary collocation techniques and their application in engineering, WIT Press, (2009).
[35] Alves, C.J., On the choice of source points in the method of fundamental solutions, Eng. Anal. Bound. Elem., 33(2009), 13481361.
[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), 129136.
[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), 95105.
[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), 618624.
[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), 337346.
[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), 12351245.
[41] Chen, C.S., Karageorghis, A., and Li, Y., On choosing the location of the sources in the MFS, Numer. Algorithms, (2015), 124.
[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), 112133.
[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), 506522.
[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), 338352.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Communications in Computational Physics
  • ISSN: 1815-2406
  • EISSN: 1991-7120
  • URL: /core/journals/communications-in-computational-physics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed