Acta Mechanica Solida Sinica

, Volume 20, Issue 1, pp 21–29 | Cite as

Some problems with the method of fundamental solution using radial basis functions

  • Hui Wang
  • Qinghua Qin


The present work describes the application of the method of fundamental solutions (MFS) along with the analog equation method (AEM) and radial basis function (RBF) approximation for solving the 2D isotropic and anisotropic Helmholtz problems with different wave numbers. The AEM is used to convert the original governing equation into the classical Poisson’s equation, and the MFS and RBF approximations are used to derive the homogeneous and particular solutions, respectively. Finally, the satisfaction of the solution consisting of the homogeneous and particular parts to the related governing equation and boundary conditions can produce a system of linear equations, which can be solved with the singular value decomposition (SVD) technique. In the computation, such crucial factors related to the MFS-RBF as the location of the virtual boundary, the differential and integrating strategies, and the variation of shape parameters in multi-quadric (MQ) are fully analyzed to provide useful reference.

Key words

meshless method analog equation method method of fundamental solution radial basis function singular value decomposition Helmholtz equation 


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  1. [1]
    Kupradze V.D. and Aleksidze M.A., The method of functional equations for the approximate solution of certain boundary value problems. USSR Comput. Math. Math. Phys, 1964, 4: 82–126.MathSciNetCrossRefGoogle Scholar
  2. [2]
    Golberg M.A. and Chen C.S., The method of fundamental solutions for potential, Helmholtz and diffusion problems//Boundary Integral Methods-Numerical and Mathematical Aspects, ed. Golberg M.A., Boston/Southampton: Computational Mechanics Publications, 1999: 105–176.Google Scholar
  3. [3]
    Fairweather G. and Karageorghis A., The method of fundamental solutions for elliptic boundary value problems. Adv. Comput. Math., 1998, 9: 69–95.MathSciNetCrossRefGoogle Scholar
  4. [4]
    Poullikkas A., Karageorghis A. and Georgiou G., The method of fundamental solutions for three-dimensional elastostatics problems. Computers and Structures, 2002, 80: 365–370.MathSciNetCrossRefGoogle Scholar
  5. [5]
    Karageorghis A., The Method of fundamental solutions for the calculation of the eigenvalues of the Helmholtz equation. Applied Mathematics Letters, 2001, 14: 837–842.MathSciNetCrossRefGoogle Scholar
  6. [6]
    Li Xin, On convergence of the method of fundamental solutions for solving the Dirichlet problem of Poisson’s equation. Advances in Computational Mathematics, 2005, 23: 265–277.MathSciNetCrossRefGoogle Scholar
  7. [7]
    Peter Mitic, Youssef F. Rashed, Convergence and stability of the method of meshless fundamental solutions using an array of randomly distributed sources. Eng. Anal. Bound. Elem., 2004, 28: 143–153.CrossRefGoogle Scholar
  8. [8]
    Golberg M.A., Chen C.S. and Bowman H., Some recent results and proposals for the use of radial basis functions in the BEM. Eng. Anal. Bound. Elem., 1999, 23: 285–296.CrossRefGoogle Scholar
  9. [9]
    Golberg M.A., Chen C.S. and Karur S.R., Improved multiquadric approximation for partial differential equations. Eng. Anal. Bound. Elem., 1996, 18: 9–17.CrossRefGoogle Scholar
  10. [10]
    Li J.C., Mathematical justification for RBF-MFS. Eng. Anal. Bound. Elem., 2001, 25: 897–901.CrossRefGoogle Scholar
  11. [11]
    Katsikadelis J. T., The analog equation method—a powerful BEM—based solution technique for solving linear and nonlinear engineering problems//Brebbia CA, ed., Boundary Element Method XVI, Southampton: CLM Publications, 1994: 167–182.zbMATHGoogle Scholar
  12. [12]
    Kansa E.J., Multiquadrics: A scattered data approximation scheme with applications to computational fluid dynamics. Comput. Math. Appl., 1990, 19: 147–161.MathSciNetCrossRefGoogle Scholar
  13. [13]
    Patridge P.W., Brebbia C.A. and Wrobel L.W., The Dual Reciprocity Boundary Element Method. Southampton: Computational Mechanics Publication, 1992: 69–75.CrossRefGoogle Scholar
  14. [14]
    Zou J, Li Z L et al., Boundary element method for model analysis of 2-D composite structure. Acta Mechanica Solida Sinica, 1998, 11: 63–71.Google Scholar
  15. [15]
    Chen W., Tanaka M., A meshless, integration-free and boundary-only RBF technique. Computers & Mathematics with Applications, 2002, 43: 379–391.MathSciNetCrossRefGoogle Scholar
  16. [16]
    Sun H C, Zhang L Z, et al., Nonsingularity Boundary Element Methods. Dalian: Dalian University of Technology Press, 1999: 185–189.Google Scholar
  17. [17]
    William H. Press, Saul A. Teukolsky, et al. Numerical recipes in C (2nd ed.). Cambridge University Press, 2001: 59–70.Google Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics and Technology 2007

Authors and Affiliations

  • Hui Wang
    • 1
    • 2
  • Qinghua Qin
    • 3
  1. 1.College of Civil Engineering and ArchitectureHenan University of TechnologyZhengzhouChina
  2. 2.Department of MechanicsTianjin UniversityTianjinChina
  3. 3.Department of EngineeringAustralian National UniversityCanberraAustralia

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