Development of the Fast Multipole Boundary Element Method for Acoustic Wave Problems

  • Yijun LiuEmail author
  • Liang Shen
  • Milind Bapat


In this chapter, we review some recent development of the fast multipole boundary element method (BEM) for solving large-scale acoustic wave problems in both 2-D and 3-D domains. First, we review the boundary integral equation (BIE) formulations for acoustic wave problems. The Burton-Miller BIE formulation is emphasized, which uses a linear combination of the conventional BIE and hypersingular BIE. Next, the fast multipole formulations for solving the BEM equations are provided for both 2-D and 3-D problems. Several numerical examples are presented to demonstrate the effectiveness and efficiency of the developed fast multipole BEM for solving large-scale acoustic wave problems, including scattering and radiation problems, and half-space problems.


Boundary Element Method Boundary Integral Equation Helmholtz Equation Fast Multipole Method Boundary Element Method Model 
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  1. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 10th ed (United States Department of Commerce: U.S. Government Printing Office, Washington, D.C., 1972).zbMATHGoogle Scholar
  2. S. Amini, “On the choice of the coupling parameter in boundary integral formulations of the exterior acoustic problem,” Appl. Anal., 35, No. 75–92 (1990).zbMATHCrossRefMathSciNetGoogle Scholar
  3. A. J. Burton and G. F. Miller, “The application of integral equation methods to the numerical solution of some exterior boundary-value problems,” Proc. R. Soc. Lond. A, 323, No. 201–210 (1971).zbMATHCrossRefMathSciNetGoogle Scholar
  4. J. T. Chen and K. H. Chen, “Applications of the dual integral formulation in conjunction with fast multipole method in large-scale problems for 2D exterior acoustics,” Eng Anal Bound Elem, 28, No. 6, 685–709 (2004).zbMATHCrossRefGoogle Scholar
  5. S. H. Chen and Y. J. Liu, “A unified boundary element method for the analysis of sound and shell-like structure interactions. I. Formulation and verification,” J. Acoust. Soc. Am., 103, No. 3, 1247–1254 (1999).CrossRefGoogle Scholar
  6. S. H. Chen, Y. J. Liu, and X. Y. Dou, “A unified boundary element method for the analysis of sound and shell-like structure interactions. II. Efficient solution techniques,” J. Acoust. Soc. Am., 108, No. 6, 2738–2745 (2000).CrossRefGoogle Scholar
  7. R. D. Ciskowski and C. A. Brebbia, Boundary Element Methods in Acoustics (Kluwer Academic Publishers, New York, 1991).zbMATHGoogle Scholar
  8. R. Coifman, V. Rokhlin, and S. Wandzura, “The fast multipole method for the wave equation: A pedestrian prescription,” IEEE Antennas Propagat. Mag., 35, No. 3, 7–12 (1993).CrossRefGoogle Scholar
  9. K. A. Cunefare and G. Koopmann, “A boundary element method for acoustic radiation valid for all wavenumbers,” J. Acoust. Soc. Am., 85, No. 1, 39–48 (1989).CrossRefGoogle Scholar
  10. K. A. Cunefare and G. H. Koopmann, “A boundary element approach to optimization of active noise control sources on three-dimensional structures,” J. Vib. Acoust., 113, July, 387–394 (1991).CrossRefGoogle Scholar
  11. E. Darve and P. Havé, “Efficient fast multipole method for low-frequency scattering,” J. Comput. Phys., 197, No. 1, 341–363 (2004).zbMATHCrossRefMathSciNetGoogle Scholar
  12. N. Engheta, W. D. Murphy, V. Rokhlin, and M. S. Vassiliou, “The fast multipole method (FMM) for electromagnetic scattering problems,” IEEE Trans. Ant. Propag., 40, No. 6, 634–641 (1992).zbMATHCrossRefMathSciNetGoogle Scholar
  13. M. Epton and B. Dembart, “Multipole translation theory for the three dimensional Laplace and Helmholtz equations,” SIAM J Sci Comput, 16, No. 865–897 (1995).zbMATHCrossRefMathSciNetGoogle Scholar
  14. G. C. Everstine and F. M. Henderson, “Coupled finite element/boundary element approach for fluid structure interaction,” J. Acoust. Soc. Am., 87, No. 5, 1938–1947 (1990).CrossRefGoogle Scholar
  15. M. Fischer, U. Gauger, and L. Gaul, “A multipole Galerkin boundary element method for acoustics,” Eng Anal Bound Elem, 28, No. 155–162 (2004).zbMATHCrossRefGoogle Scholar
  16. L. Greengard, J. Huang, V. Rokhlin, and S. Wandzura, “Accelerating fast multipole methods for the helmholtz equation at low frequencies,” IEEE Comput. Sci. Eng., 5, No. 3, 32–38 (1998).CrossRefGoogle Scholar
  17. L. F. Greengard, The Rapid Evaluation of Potential Fields in Particle Systems (The MIT Press, Cambridge, 1988).zbMATHGoogle Scholar
  18. L. F. Greengard and V. Rokhlin, “A fast algorithm for particle simulations,” J. Comput. Phys., 73, No. 2, 325–348 (1987).zbMATHCrossRefMathSciNetGoogle Scholar
  19. N. A. Gumerov and R. Duraiswami, “Recursions for the computation of multipole translation and rotation coefficients for the 3-D Helmholtz equation,” SIAM J. Sci. Comput., 25, No. 4, 1344–1381 (2003).zbMATHCrossRefMathSciNetGoogle Scholar
  20. M. F. Gyure and M. A. Stalzer, “A prescription for the multilevel Helmholtz FMM,” IEEE Comput. Sci. Eng., 5, No. 3, 39–47 (1998).CrossRefGoogle Scholar
  21. D. S. Jones, “Integral equations for the exterior acoustic problem,” Q. J. Mech. Appl. Math., 27, No. 129–142 (1974).zbMATHCrossRefGoogle Scholar
  22. R. Kress, “Minimizing the condition number of boundary integral operators in acoustic and electromagnetic scattering,” Quart. J. Mech. Appl. Math., 38, No. 2, 323–341 (1985).zbMATHCrossRefMathSciNetGoogle Scholar
  23. G. Krishnasamy, T. J. Rudolphi, L. W. Schmerr, and F. J. Rizzo, “Hypersingular boundary integral equations: some applications in acoustic and elastic wave scattering,” J. App. Mech., 57, No. 404–414 (1990).zbMATHCrossRefMathSciNetGoogle Scholar
  24. R. E. Kleinman and G. F. Roach, “Boundary integral equations for the three-dimensional Helmholtz equation,” SIAM Rev., 16, No. 214–236 (1974).zbMATHCrossRefMathSciNetGoogle Scholar
  25. S. Koc and W. C. Chew, “Calculation of acoustical scattering from a cluster of scatterers,” J. Acoust. Soc. Am., 103, No. 2, 721–734 (1998).CrossRefGoogle Scholar
  26. C. Lu and W. Chew, “Fast algorithm for solving hybrid integral equations,” IEE Proc. H, 140, No. 455–460 (1993).Google Scholar
  27. Y. J. Liu, “Development and applications of hypersingular boundary integral equations for 3-D acoustics and elastodynamics”, Ph.D., Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign (1992).Google Scholar
  28. Y. J. Liu and F. J. Rizzo, “A weakly-singular form of the hypersingular boundary integral equation applied to 3-D acoustic wave problems,” Comput. Methods Appl.Mech. Eng., 96, No. 271–287 (1992).zbMATHCrossRefMathSciNetGoogle Scholar
  29. Y. J. Liu and S. H. Chen, “A new form of the hypersingular boundary integral equation for 3-D acoustics and its implementation with C$0$ boundary elements,” Comput. Methods Appl.Mech. Eng., 173, No. 3–4, 375–386 (1999).zbMATHCrossRefGoogle Scholar
  30. R. Martinez, “The thin-shape breakdown (TSB) of the Helmholtz integral equation,” J. Acoust. Soc. Am., 90, No. 5, 2728–2738 (1991).CrossRefGoogle Scholar
  31. W. L. Meyer, W. A. Bell, B. T. Zinn, and M. P. Stallybrass, “Boundary integral solutions of three dimensional acoustic radiation problems,” J. Sound Vib., 59, No. 245–262 (1978).zbMATHCrossRefGoogle Scholar
  32. N. Nishimura, “Fast multipole accelerated boundary integral equation methods,” Appl. Mech. Rev., 55, No. 4 (July), 299–324 (2002).CrossRefGoogle Scholar
  33. V. Rokhlin, “Rapid solution of integral equations of classical potential theory,” J. Comp. Phys., 60, No. 187–207 (1985).zbMATHCrossRefMathSciNetGoogle Scholar
  34. V. Rokhlin, “Rapid solution of integral equations of scattering theory in two dimensions,” J. Comput. Phys., 86, No. 2, 414–439 (1990).zbMATHCrossRefMathSciNetGoogle Scholar
  35. V. Rokhlin, “Diagonal forms of translation operators for the Helmholtz equation in three dimensions,” Appl. Comput. Harmon. Anal., 1, No. 1, 82–93 (1993).zbMATHCrossRefMathSciNetGoogle Scholar
  36. H. A. Schenck, “Improved integral formulation for acoustic radiation problems,” J. Acoust. Soc. Am., 44, No. 41–58 (1968).CrossRefGoogle Scholar
  37. A. F. Seybert, B. Soenarko, F. J. Rizzo, and D. J. Shippy, “An advanced computational method for radiation and scattering of acoustic waves in three dimensions,” J. Acoust. Soc. Am., 77, No. 2, 362–368 (1985).zbMATHCrossRefGoogle Scholar
  38. A. F. Seybert and T. K. Rengarajan, “The use of CHIEF to obtain unique solutions for acoustic radiation using boundary integral equations,” J. Acoust. Soc. Am., 81, No. 1299–1306 (1987).CrossRefGoogle Scholar
  39. L. Shen and 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, No. 3, 461–472 (2007).zbMATHCrossRefGoogle Scholar
  40. L. Shen and Y. J. Liu, “An adaptive fast multipole boundary element method for 3-D half-space acoustic wave problems,” in review, (2009).Google Scholar
  41. M. A. Tournour and N. Atalla, “Efficient evaluation of the acoustic radiation using multipole expansion,” Int. J. Numer. Methods Eng. 46, No. 6, 825–837 (1999).zbMATHCrossRefGoogle Scholar
  42. F. Ursell, “On the exterior problems of acoustics,” Proc. Cambridge Philos. Soc., 74, No. 117–125 (1973).zbMATHCrossRefMathSciNetGoogle Scholar
  43. R. Wagner and W. Chew, “A ray-propagation fast multipole algorithm,” Microwave Opt. Technol. Lett., 7, No. 435–438 (1994).CrossRefGoogle Scholar
  44. T. W. Wu, A. F. Seybert, and G. C. Wan, “On the numerical implementation of a Cauchy principal value integral to insure a unique solution for acoustic radiation and scattering,” J. Acoust. Soc. Am., 90, No. 1, 554–560 (1991).CrossRefGoogle Scholar
  45. S.-A. Yang, “Acoustic scattering by a hard and soft body across a wide frequency range by the Helmholtz integral equation method,” J. Acoust. Soc. Am., 102, No. 5, Pt. 1, November, 2511–2520 (1997).CrossRefGoogle Scholar
  46. K. Yoshida, “Applications of fast multipole method to boundary integral equation method”, Ph.D. Dissertation, Department of Global Environment Engineering, Kyoto University (2001).Google Scholar

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© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of CincinnatiCincinnatiUSA

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