Skip to main content
SpringerLink
Log in

Boundary Element Method

  • Chapter
  • First Online:
Computational Simulation in Architectural and Environmental Acoustics

Abstract

This chapter gives an outline of acoustic analysis using the boundary element method (BEM). In the first section, the fundamentals of the BEM and its application to sound field analysis are explained. The second section presents two advanced techniques, the indirect approach with degenerate boundary and the domain decomposition method. The third section introduces a new application of the fast multipole method, as a fast solution technique for large-scale problems.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. C.A. Brebbia, The Boundary Element Method for Engineers (Pentech Press, London, 1978)

    Google Scholar 

  2. R.D. Ciskowski, C.A. Brebbia (eds.), Boundary Elements in Acoustics (Computational Mechanics Publications and Elsevier Applied Science, London, 1991)

    Google Scholar 

  3. O. von Estorff (ed.), Boundary Elements in Acoustics: Advances and Applications (WIT Press, Southampton, 2000)

    Google Scholar 

  4. T.W. Wu (ed.), Boundary Element Acoustics: Fundamentals and Computer Codes (WIT Press, Southampton, 2000)

    Google Scholar 

  5. S. Marburg, B. Nolte (eds.), Computational Acoustic of Noise Propagation in Fluids—Finite and Boundary Element Methods (Springer, Berlin, 2008)

    Google Scholar 

  6. D.E. Freund, R. Farrell, A variational principle for the scattered wave. J. Acoust. Soc. Am. 87, 1847–1860 (1990)

    Article  MathSciNet  Google Scholar 

  7. Z.S. Chen, G. Hofstetter, H.A. Mang, A symmetric Galerkin formulation of the boundary element method for acoustic radiation and scattering. J. Comput. Acoust. 5, 219–241 (1997)

    Article  Google Scholar 

  8. M. Abramowitz, I.A. Stegun (eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover Publications, New York, 1965)

    Google Scholar 

  9. P.C. Hammer, O.P. Marlowe, A.H. Stroud, Numerical integration over simplexes and cones. Math. Tables Aids Comput. 10, 130–137 (1956)

    Article  MATH  MathSciNet  Google Scholar 

  10. T. Terai, On calculation of sound fields around three dimensional objects by integral equation methods. J. Sound Vib. 69, 71–100 (1980)

    Article  MATH  Google Scholar 

  11. K. Hayami, H. Matsumoto, A numerical quadrature for nearly singular boundary element integrals. Eng. Anal. Bound. Elem. 13, 143–154 (1994)

    Article  Google Scholar 

  12. A.J. Burton, G.F. Miller, The application of integral equation methods to the numerical solution of some exterior boundary value problems. Proc. R. Soc. Lond. Ser. A 323, 201–210 (1971)

    Google Scholar 

  13. H.A. Schenck, Improved integral formulation for acoustic radiation problems. J. Acoust. Soc. Am. 44, 41–58 (1968)

    Article  Google Scholar 

  14. K.A. Cuneface, G. Koopmann, A boundary element method for acoustic radiation valid for all wavenumbers. J. Acoust. Soc. Am. 85, 39–48 (1989)

    Article  Google Scholar 

  15. T. Terai, Y. Kawai, The application of Kirchhoff’s formula to the numerical calculation of transient response in an enclosures. J. Acoust. Soc. Jpn. (E) 11, 1–10 (1990)

    Article  Google Scholar 

  16. M.J. Bluck, S.P. Walker, Analysis of three-dimensional transient acoustic wave propagation using the boundary integral equation method. Int. J. Numer. Meth. Eng. 39, 1419–1431 (1996)

    Article  MATH  Google Scholar 

  17. A.A. Ergin, B. Shanker, E. Michielssen, Analysis of transient wave scattering from rigid bodies using a burton-miller approach. J. Acoust. Soc. Am. 106, 2396–2404 (1999)

    Article  Google Scholar 

  18. J.A. Hargreaves, T. Cox, A transient boundary element method for acoustic scattering from mixed regular and thin rigid bodies. Acta Acustica United Acustica 95, 678–689 (2009)

    Article  Google Scholar 

  19. Y. Kawai, T. Terai, A numerical method for the calculation of transient acoustic wave scattering from thin rigid plates. J. Sound Vib. 141, 83–96 (1990)

    Article  Google Scholar 

  20. P.J.T. Filippi, Layer potentials and acoustic diffraction. J. Sound Vib. 54, 473–500 (1977)

    Article  MATH  Google Scholar 

  21. H. Utsuno, T.W. Wu, A.F. Seybert, T. Tanaka, Prediction of sound fields in cavities with sound absorbing materials. AIAA J 28, 1870–1876 (1990)

    Google Scholar 

  22. A.F. Seybert, C.Y.R. Cheng, T.W. Wu, The solution of coupled interior/exterior acoustic problems using the boundary element method. J. Acoust. Soc. Am. 88, 1612–1618 (1990)

    Article  Google Scholar 

  23. C.Y.R. Cheng, A.F. Seybert, T.W. Wu, A multidomain boundary element solution for silencer and muffler performance prediction. J. Sound Vib. 151, 119–129 (1991)

    Article  Google Scholar 

  24. A.F. Seybert, Z.H. Jia, T.W. Wu, Solving knife-edge scattering problems using singular boundary elements. J. Acoust. Soc. Am. 91, 1278–1283 (1992)

    Article  Google Scholar 

  25. E. Sarradj, Multi-domain boundary element method for sound fields in and around porous absorbers. Acta Acustica United Acustica 89, 21–27 (2003)

    Google Scholar 

  26. Y. Yasuda, S. Sakamoto, T. Sakuma, Application of the fast multipole boundary element method to sound field analysis using a domain decomposition approach, in Proceedings of Inter-Noise 2006, no. 624 (Honolulu, 2006)

    Google Scholar 

  27. Y. Kosaka, T. Sakuma, Boundary element analysis using thin plate modeling—on calculation of a thin plate with different boundary conditions for both surfaces. in Proceedings of ASJ Autumn Meeting, 979–980 (2005) (in Japanese)

    Google Scholar 

  28. T. Masumoto, T. Oshima, Y. Yasuda, T. Sakuma, M. Kabuto, M. Akiyama, HRTF caulculation in the full audible frequency range using FMBEM, in Proceedings of 19th International Congress on Acoustics, COM-06-012 (Madrid, 2007)

    Google Scholar 

  29. N. Kamiya, H. Iwase, E. Kita, Parallel implementation of boundary element method with domain decomposition. Eng. Anal. Bound. Elem. 18, 209–216 (1996)

    Article  Google Scholar 

  30. M.E. Delany, E.N. Bazley, Acoustical properties of fibrous absorbent materials. Appl. Acoust. 3, 105–116 (1970)

    Article  Google Scholar 

  31. Y. Miki, Acoustical properties of porous materials—modifications of Delany-Bazley models. J. Acoust. Soc. Jpn. (E) 11, 19–24 (1990)

    Article  Google Scholar 

  32. P.M. Morse, K.U. Ingard, Theoretical Acoustics, Chap. 6 (McGraw-Hill Inc, New York, 1968)

    Google Scholar 

  33. V. Rokhlin, Rapid solution of integral equations of classical potential theory. J. Comput. Phys. 60, 187–207 (1983)

    Article  MathSciNet  Google Scholar 

  34. L. Greengard, The Rapid Evaluation of Potential Fields in Paticle Systems (The MIT Press, Cambridge, 1987)

    Google Scholar 

  35. J.K. Salmon, M.S. Warren, G.S. Winckelmans, Fast parallel tree codes for gravitational and fluid dynamical N-body problems. Int. J. Supercomput. Appl. 8, 124–142 (1994)

    Article  Google Scholar 

  36. W.C. Chew, J.M. Jin, C.C. Lu, E. Michelssen, J.M. Song, Fast solution methods in electromagnetics. IEEE Trans. Antennas Propag. 45, 533–543 (1997)

    Article  Google Scholar 

  37. H. Schwichtenberg, G. Winter, H. Wallmeier, Acceleration of molecular mechanic simulation by parallelization and fast multipole techniques. Parallel Comput. 25, 535–546 (1999)

    Article  MATH  Google Scholar 

  38. K. Hayami, S.A. Sauter, A formulation of the panel clustering method for the three-dimensional elastostatic problem. JASCOME 13, 125–130 (1996)

    Google Scholar 

  39. Y. Fu, K.J. Klimkowski, G.J. Robin, E. Berger, J.C. Browne, J.K. Singer, R.A. Van De Geijn, K.S. Vemaganti, A fast solution method for three-dimensional many-particle preblems of linear elasticity. Int. J. Numer. Meth. Eng. 42, 1215–1229 (1998)

    Article  MATH  Google Scholar 

  40. N. Nishimura, K. Yoshida, S. Kobayashi, A fast multipole boundary integral equation method for crack problems in 3D. Eng. Anal. Bound. Elem. 23, 97–105 (1999)

    Article  MATH  Google Scholar 

  41. A. Buchau, W. Rieger, W.M. Rucker, BEM computations using the fast multipole method in combination with higher order elements and the Galerkin method. IEEE Trans. Magn. 37(5), 3181–3185 (2001)

    Article  Google Scholar 

  42. T. Sakuma, Y. Yasuda, Fast multipole boundary element method for large-scale steady-state sound field analysis. Part I: Setup and validation. Acta Acustica United Acustica 88(4), 513–525 (2002)

    Google Scholar 

  43. Y. Yasuda, T. Sakuma, Fast multipole boundary element method for large-scale steady-state sound field analysis. Part II: Examination of numerical items. Acta Acustica United Acustica 89(1), 28–38 (2003)

    Google Scholar 

  44. S. Schneider, Application of fast methods for acoustic scattering and radiation problems. J. Comput. Acoust. 11, 387–401 (2003)

    Article  Google Scholar 

  45. M. Fischer, U. Gauger, L. Gaul, A multipole Galerkin boundary element method for acoustics. Eng. Anal. Bound. Elem. 28, 155–162 (2004)

    Article  MATH  Google Scholar 

  46. Y. Yasuda, T. Sakuma, An effective setting of hierarchical cell structure for the fast multipole boundary element method. J. Comput. Acoust. 13(1), 47–70 (2005)

    Article  MATH  Google Scholar 

  47. Y. Yasuda, T. Sakuma, A technique for plane-symmetric sound field analysis in the fast multipole boundary element method. J. Comput. Acoust. 13(1), 71–85 (2005)

    Article  MATH  Google Scholar 

  48. M. Fischer, L. Gaul, Application of the fast multipole BEM for structural-acoustic simulations. J. Comput. Acoust. 13(1), 87–98 (2005)

    Article  MATH  Google Scholar 

  49. H. Cheng, W.Y. Crutchfield, Z. Gimbutas, L.F. Greengard, J.F. Ethridge, J. Huang, V. Rokhlin, N. Yarvin, J. Zhao, A wideband fast multipole method for the Helmholtz equation in three dimensions. J. Comput. Phys. 216, 300–325 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  51. M.S. Tong, W.C. Chew, Multilevel fast multipole algorithm for acoustic wave scattering by truncated ground with trenches. J. Acoust. Soc. Am. 123, 2513–2521 (2008)

    Article  Google Scholar 

  52. N.A. Gumerov, R. Duraiswami, A broadband fast multipole accelerated boundary element method for the three dimensional Helmholtz equation. J. Acoust. Soc. Am. 125, 191–205 (2009)

    Article  Google Scholar 

  53. M.S. Bapat, L. Shen, Y.J. Liu, Adaptive fast multipole boundary element method for three-dimensional half-space acoustic wave problems. Eng. Anal. Bound. Elem. 33, 1113–1123 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  54. Y. Yasuda, T. Oshima, T. Sakuma, A. Gunawan, T. Masumoto, Fast multipole boundary element method for low-frequency acoustic problems based on a variety of formulations. J. Comput. Acoust. 18(4), 363–395 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  55. Y. Yasuda, K. Higuchi, T. Oshima, T. Sakuma, Efficient technique in low-frequency fast multipole boundary element method for plane-symmetric acoustic problems. Eng. Anal. Bound. Elem. 36(10), 1493–1501 (2012)

    Article  MathSciNet  Google Scholar 

  56. N.A. Gumerov, R. Duraiswami, Fast Multipole Methods for the Helmholtz Equation in Three Dimensions (Elsevier, Amsterdam, 2004)

    Google Scholar 

  57. Y. Saad, H.A. van der Vorst, Iterative solution of linear systems in the 20th century. J. Comput. Appl. Math. 123, 1–33 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  58. C.C. Lu, W.C. Chew, A multilevel algorithm for a boundary integral equation of wave scattering. Microw. Opt. Technol. Lett. 7, 466–470 (1994)

    Article  Google Scholar 

  59. J.M. Song, W.C. Chew, Multilevel fast-multipole algolithm for solving combined field integral equations of electromagnetic scattering. Microw. Opt. Technol. Lett. 10, 14–19 (1995)

    Article  Google Scholar 

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

    Article  Google Scholar 

  61. N.A. Gumerov, R. Duraiswami, Recursions for the computation of multipole translation and rotation coefficients for the 3-D Helmholtz equation. SIAM J. Sci. Comput. 25, 1344–1381 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  62. V. Rokhlin, Diagonal forms of translation operators for the Helmholtz equation in three dimensions. Appl. Comput. Harmon. Anal. 1, 82–93 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  63. L. Greengard, J. Huang, V. Rokhlin, S. Wandzura, Accelerating fast multipole methods for the Helmholtz equation at low frequencies. IEEE Comput. Sci. Eng. 5, 32–38 (1998)

    Article  Google Scholar 

  64. E. Darve, P. Havé, Efficient fast multipole method for low-frequency scattering. J. Comput. Phys. 197, 341–363 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  65. M.A. Epton, B. Dembart, Multipole translation theory for the three-dimensional Laplace and Helmholtz equations. SIAM J. Sci. Comput. 16(4), 865–897 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  66. E. Darve, The fast multipole method I: error analysis and asymptotic complexity. SIAM J. Numer. Anal. 38(1), 98–128 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  67. S. Koc, W.C. Chew, Calculation of acoustical scattering from a cluster of scatterers. J. Acoust. Soc. Am. 103(2), 721–734 (1997)

    Article  Google Scholar 

  68. A. Brandt, Multilevel computations of integral transforms and particle interactions with oscillatory kernels. Comput. Phys. Commun. 65, 24–38 (1991)

    Article  MATH  Google Scholar 

  69. S. Schneider, S. Marburg, Performance of iterative solvers for acoustic problems. Part II. Acceleration by ILU-type preconditioner. Eng. Anal. Bound. Elem. 27, 751–757 (2003)

    Article  MATH  Google Scholar 

  70. Y. Yasuda, S. Sakamoto, Y. Kosaka, T. Sakuma, N. Okamoto, T. Oshima, Numerical analysis of large-scale sound fields using iterative methods. Part I: Application of Krylov subspace methods to boundary element analysis. J. Comput. Acoust. 15(4), 449–471 (2007)

    Article  Google Scholar 

  71. S. Marburg, B. Nolte (eds.), Computational Acoustics of Noise Propagation in Fluids—Finite and Boundary Element Methods, Chap. 12 (Springer, Berlin, 2008)

    Google Scholar 

  72. T. Oshima, Y. Yasuda, T. Sakuma, A high efficiency implementation of fast multipole boundary element method, in Proceedings of 18th International Congress on Acoustics, vol. I, pp. 487–490 (Kyoto, 2004)

    Google Scholar 

  73. T. Masumoto, A. Gunawan, T. Oshima, Y. Yasuda, T. Sakuma, Coupling analysis between FMBEM-based acoustic and modal-based structural models -convergence behavior of iterative solutions, in Proceedings of Inter-Noise 2008, no. 0308 (Shanghai, 2008)

    Google Scholar 

  74. Y. Yasuda, T. Sakuma, Analysis of sound fields in porous materials using the fast multipole BEM, in Proceedings of Inter-Noise 2008, no. 0531 (Shanghai, 2008)

    Google Scholar 

  75. Y. Yasuda, K. Eda, T. Sakuma, T. Oshima, A fast multipole BE analysis of a small room with sound absorbers using domain decomposition approach, in Proceedings of Inter-Noise 2010, no. 0503 (Lisbon, 2010)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Engineering, Kanagawa University, 3-27-1 Rokkakubashi, Kanagawa-ku, Yokohama, 221-8686, Japan

    Yosuke Yasuda

  2. Graduate School of Frontier Sciences, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa-shi, Chiba, 277-8563, Japan

    Tetsuya Sakuma

Authors
  1. Yosuke Yasuda
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tetsuya Sakuma
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yosuke Yasuda .

Editor information

Editors and Affiliations

  1. The University of Tokyo, Tokyo, Japan

    Tetsuya Sakuma

  2. The University of Tokyo, Tokyo, Japan

    Shinichi Sakamoto

  3. Oita University, Oita, Japan

    Toru Otsuru

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Japan

About this chapter

Cite this chapter

Yasuda, Y., Sakuma, T. (2014). Boundary Element Method. In: Sakuma, T., Sakamoto, S., Otsuru, T. (eds) Computational Simulation in Architectural and Environmental Acoustics. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54454-8_4

Download citation

  • DOI: https://doi.org/10.1007/978-4-431-54454-8_4

  • Published:

  • Publisher Name: Springer, Tokyo

  • Print ISBN: 978-4-431-54453-1

  • Online ISBN: 978-4-431-54454-8

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation