Skip to main content
Log in

Diagonal form fast multipole boundary element method for 2D acoustic problems based on Burton-Miller boundary integral equation formulation and its applications

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

This paper describes formulation and implementation of the fast multipole boundary element method (FMBEM) for 2D acoustic problems. The kernel function expansion theory is summarized, and four building blocks of the FMBEM are described in details. They are moment calculation, moment to moment translation, moment to local translation, and local to local translation. A data structure for the quad-tree construction is proposed which can facilitate implementation. An analytical moment expression is derived, which is more accurate, stable, and efficient than direct numerical computation. Numerical examples are presented to demonstrate the accuracy and efficiency of the FMBEM, and radiation of a 2D vibration rail mode is simulated using the FMBEM.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Copley, L. G. Integral equation method for radiation from vibrating bodies. Journal of the Acoustical Society of America, 41(4A), 807–816 (1967)

    Article  Google Scholar 

  2. Schenck, H. A. Improved integral formulation for acoustic radiation problems. Journal of the Acoustical Society of America, 44(1), 41–48 (1968)

    Article  Google Scholar 

  3. Meyer, W. L., Bell, W. A., Zinn, B. T., and Stallybrass, M. P. Boundary integral solutions of three dimensional acoustic radiation problems. Journal of Sound and Vibration, 59(2), 245–262 (1978)

    Article  MATH  Google Scholar 

  4. Terai, T. On calculation of sound fields around three dimensional objects by integral equation methods. Journal of Sound and Vibration, 69(1), 71–100 (1980)

    Article  MATH  Google Scholar 

  5. Rokhlin, V. Rapid solution of integral equations of classical potential theory. Journal of Computational Physics, 60(2), 187–207 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  6. Greengard, L. and Rokhlin, V. A fast algorithm for particle simulations. Journal of Computational Physics, 73(2), 325–348 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  7. Greengard, L. The Rapid Evaluation of Potential Fields in Particle Systems, MIT Press, Cambridge (1988)

    MATH  Google Scholar 

  8. Rokhlin, V. Rapid solution of integral equations of scattering theory in two dimensions. Journal of Computational Physics, 86(2), 414–439 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  9. Amini, S. and Profit, A. T. J. Analysis of a diagonal form of the fast multipole algorithm for scattering theory. BIT Numerical Mathematics, 39(4), 585–602 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Amini, S. and Profit, A. T. J. Multi-level fast multipole solution of the scattering problem. Engineering Analysis with Boundary Elements, 27(5), 547–564 (2003)

    Article  MATH  Google Scholar 

  11. Chen, J. T. and Chen, K. H. Applications of the dual integral formulation in conjunction with fast multipole method in large-scale problems for 2D exterior acoustics. Engineering Analysis with Boundary Elements, 28(6), 685–709 (2004)

    Article  MATH  Google Scholar 

  12. Shen, L. and Liu, Y. J. An adaptive fast multipole boundary element method for three-dimensional acoustic wave problems based on the Burton-Miller formulation. Computational Mechanics, 40(3), 461–472 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bapat, M. S., Shen, L., and Liu, Y. J. Adaptive fast multipole boundary element method for three-dimensional half-space acoustic wave problems. Engineering Analysis with Boundary Elements, 33(8–9), 1113–1123 (2009)

    Article  MathSciNet  Google Scholar 

  14. Liu, Y. J. Fast Multipole Boundary Element Method: Theory and Applications in Engineering, Cambridge University Press, Cambridge (2009)

    Book  Google Scholar 

  15. Crutchfield, W., Gimbutas, Z., Greengard, L., Huang, J., Rokhlin, V., Yarvin, N., and Zhao, J. Remarks on the implementation of wideband FMM for the Helmholtz equation in two dimensions. Contemporary Mathematics, 408, 99–110 (2006)

    MathSciNet  Google Scholar 

  16. Cheng, H., Crutchfield, W. Y., Gimbutas, Z., Greengard, L. F., Ethridge, J. F., Huang, J., Rokhlin, V., Yarvin, N., and Zhao, J. A wideband fast multipole method for the Helmholtz equation in three dimensions. Journal of Computational Physics, 216(1), 300–325 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gumerov, N. A. and Duraiswami, R. A broadband fast multipole accelerated boundary element method for the three dimensional Helmholtz equation. Journal of the Acoustical Society of America, 125(1), 191–205 (2009)

    Article  Google Scholar 

  18. Liu, Y. J. and Nishimura, N. The fast multipole boundary element method for potential problems: a tutorial. Engineering Analysis with Boundary Elements, 30(5), 371–381 (2006)

    Article  MATH  Google Scholar 

  19. Ciskowski, R. D. and Brebbia, C. A. Boundary Element Methods in Acoustics, 1st ed., Springer, New York (1991)

    MATH  Google Scholar 

  20. Burton, A. J. and Miller, G. F. The application of integral equation methods to the numerical solution of some exterior boundary-value problems. Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences, 323(1553), 201–210 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kress, R. Minimizing the condition number of boundary integral operators in acoustic and electromagnetic scattering. Quarterly Journal of Mechanics and Applied Mathematics, 38(2), 323–341 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  22. Colton, D. and Kress, R. Integral Equation Methods in Scattering Theory, Wiley, New York (1983)

    MATH  Google Scholar 

  23. Abramowitz, M. and Stegun, I. A. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, U.S. Govt. Print. Off., Washington (1964)

    MATH  Google Scholar 

  24. Saad, Y. and Schultz, M. H. GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM Journal on Scientific and Statistical Computing, 7(3), 856–869 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sonneveld, P. GGS: a fast Lanczos-type solver for nonsymmetric linear systems. SIAM Journal on Scientific and Statistical Computing, 10, 36–52 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  26. Labreuche, C. A convergence theorem for the fast multipole method for 2 dimensional scattering problems. Mathematics of Computation, 67(222), 553–591 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  27. Amini, S. and Profit, A. Analysis of the truncation errors in the fast multipole method for scattering problems. Journal of Computational and Applied Mathematics, 115(1–2), 23–33 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  28. Wu, H. J., Jiang, W. K., and Liu, Y. J. Analysis of numerical integration error for Bessel integral identity in fast multipole method for 2D Helmholtz equation. Journal of Shanghai Jiaotong University (Science), 15(6), 690–693 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Coifman, R., Rokhlin, V., and Wandzura, S. The fast multipole method for the wave equation: a pedestrian prescription. Antennas and Propagation Magazine, IEEE, 35(3), 7–12 (1993)

    Article  Google Scholar 

  30. Jakob-Chien, R. and Alpert, B. K. A fast spherical filter with uniform resolution. Journal of Computational Physics, 136(2), 580–584 (1997)

    Article  MATH  Google Scholar 

  31. Morse, P. M. and Ingard, K. U. Theoretical Acoustics, Princeton University Press, Princeton, New Jersey (1987)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Wei-kang Jiang  (蒋伟康).

Additional information

Project supported by the National Natural Science Foundation of China (No. 11074170) and the State Key Laboratory Foundation of Shanghai Jiao Tong University (No.MSVMS201105)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wu, Hj., Jiang, Wk. & Liu, Y.J. Diagonal form fast multipole boundary element method for 2D acoustic problems based on Burton-Miller boundary integral equation formulation and its applications. Appl. Math. Mech.-Engl. Ed. 32, 981–996 (2011). https://doi.org/10.1007/s10483-011-1474-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-011-1474-7

Key words

Chinese Library Classification

2010 Mathematics Subject Classification

Navigation