ANZIAM J. 49 (2007), no. 1, pp. 131–150.

A comparative study of the direct boundary element method and the dual reciprocity boundary element method in solving the Helmholtz equation

Song-Ping Zhu Yinglong Zhang
School of Mathematics and Applied Statistics
University of Wollongong
NSW 2522
Department of Environmental & Biomolecular Systems
OGI School of Science & Engineering
Oregon Health & Science University
OR 97006, USA.
Received 11 June 2006


In this paper, we compare the direct boundary element method (BEM) and the dual reciprocity boundary element method (DRBEM) for solving the direct interior Helmholtz problem, in terms of their numerical accuracy and efficiency, as well as their applicability and reliability in the frequency domain. For BEM formulation, there are two possible choices for fundamental solutions, which can lead to quite different conclusions in terms of their reliability in the frequency domain. For DRBEM formulation, it is shown that although the DBREM can correctly predict eigenfrequencies even for higher modes, it fails to yield a reasonably accurate numerical solution for the problem when the frequency is higher than the first eigenfrequency.

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2000 Mathematics Subject Classification: primary 65N38; secondary 35Q35
(Metadata: XML, RSS, BibTeX) MathSciNet: MR2378154 Z'blatt-MATH: pre05243896
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  1. J. O. Adeyeye, M. J. M. Bernal and K. E. Pitman, “An improved boundary integral equation method for Helmholtz problems”, Int. J. Num. Meth. Engng. 21 (1985) 779–787. MR792583
  2. S. Amini, P. J. Harris and D. T. Wilton, Coupled Boundary and Finite Elements Methods for the Solution of the Dynamic Fluid-Structure Interaction Problem, Volume 77 of Lecture Notes in Engineering (Springer-Verlag, Berlin, Heidelberg, New York, 1992).
  3. C. A. Brebbia, J. C. F. Telles and L. C. Wrobel, Boundary Element Techniques (Springer-Verlag, Berlin, 1984). MR0934922
  4. A. J. Burton and G. F. Miller, “The application for integral equation methods to the numerical solution of some exterior boundary-value problems”, Proc. Roy. Soc. London Ser. A. 323 (1971) 201–210. MR495032
  5. D. Colton., “The inverse scattering problem for time-harmonic acoustic waves”, SIAM Rev. 26 (1984) 323–350. MR750453
  6. M. S. Ingber and A. K. Mitra, “Grid redistribution based on measurable error indicators for direct boundary element method”, Engng. Ana. Boundary Elements 9 (1991) 13–19.
  7. N. Kamiya and E. Andoh., “Boundary element eigenvalue analysis by standard routine”, Boundary Elements XV 1 (1993) 375–383.
  8. O. D. Kellog, Foundations of Potential Theory (Springer-Verlag, Berlin, 1929). MR222317
  9. S. M. Kirkup and S. Amini, “Solution of the Helmholtz eigenvalue problem via the boundary element method”, Int. J. Numer. Methods Eng. 36 (1993) 321–330.
  10. R. E. Klainman and G. F. Roach, “Boundary integral equations for three dimensional Helmholtz equation”, SIAM Rev. 16 (1974) 214–236. MR380087
  11. G. De Mey, “Calculation of eigenvalues of the Helmholtz equation by an integral equation”, Int. J. Num. Meth. Engng. 10 (1976) 59–66.
  12. G. De Mey., “A simplified integral equation method for the calculation of the eigenvalues of Helmholtz equation”, Int. J. Num. Meth. Engng. 10 (1976) 1340–1342.
  13. A. J. Nowak and C. A. Brebbia, “Solving Helmholtz equation by multiple reciprocity method”, in Computer and Experiments in Fluid Flow (eds. G. M. Carlomangno and C. A. Brebbia), (Computational Mechanics Publication, Southampton, 1989) 265–270. MR1072142
  14. A. J. Nowak and P. W. Partridge, “Comparison of the dual reciprocity and the multiple reciprocity methods”, Engng. Ana. Boundary Elements 10 (1992) 155–160.
  15. P. W. Partridge and C. A. Brebbia, “The dual reciprocity boundary element method for the Helmholtz equation”, in Mechanical and Electrical Engineering (eds. C. A. Brebbia and A. Chaudouet-Miranda), (Computational Mechanics Publication, Southampton, Boston, 1990). MR1160792
  16. M. Rezayat, D. J. Shippy and F. J. Rizzo, “On time-harmonic elastic wave analysis by the boundary element method for moderate to high frequencies”, Comp. Meth. Appl. Mech. Eng. 55 (1986) 349–367.
  17. M. G. Salvadori and M. L. Baron, Numerical Methods in Engineering (Prentice-Hall, Englewood Cliffs, N.J., 1961). MR0070253
  18. H. A. Schenck, “Improved integral formulation for acoustic radiation problems”, J. Acoust. Soc. Am. 44 (1968) 41–58.
  19. R. P. Shaw, “Boundary integral equation method applied to wave problems”, in Developments in Boundary Element Methods 1 (eds. P. K. Banerjee and R. Buterfield), (Applied Science Publisher Ltd., London, 1979) Ch. 6, 121–154.
  20. G. R. C. Tai and R. P. Shaw, “Helmholtz equation eigenvalues and eigenmodes for arbitrary domains”, J. Acoust. Soc. Am. 56 (1974) 796–804, (also Rep. No. 90, SUNY, Buffalo).
  21. R. Wait and A. R. Mitchell, Finite element analysis and applications (John Wiley and Sons, Chichester, 1985). MR817440
  22. S.-P. Zhu and G. Moule, “Numerical calculation of forces induced by short-crested waves on a vertical cylinder of arbitrary cross-sections”, Ocean Eng. 21 (1994) 645–662.
  23. O. C. Zienkiewicz and R. L. Taylor, The finite element method, Vols. 1 and 2 (McGraw-Hill, London, 1988). MR1897985, MR1897986
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