Recent Advances in Boundary Element Methods pp 287-303 | Cite as

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

## Abstract

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.

## Keywords

Boundary Element Method Boundary Integral Equation Helmholtz Equation Fast Multipole Method Boundary Element Method Model## Preview

Unable to display preview. Download preview PDF.

## References

- 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 - 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 - 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 - 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 - 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 - 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 - R. D. Ciskowski and C. A. Brebbia,
*Boundary Element Methods in Acoustics*(Kluwer Academic Publishers, New York, 1991).zbMATHGoogle Scholar - 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 - 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 - 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 - E. Darve and P. Havé, “Efficient fast multipole method for low-frequency scattering,”
*J. Comput. Phys.*,**197**, No. 1, 341–363 (2004).zbMATHCrossRefMathSciNetGoogle Scholar - 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 - 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 - 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 - 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 - 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 - L. F. Greengard, The Rapid Evaluation of Potential Fields in Particle Systems (The MIT Press, Cambridge, 1988).zbMATHGoogle Scholar
- L. F. Greengard and V. Rokhlin, “A fast algorithm for particle simulations,”
*J. Comput. Phys.*,**73**, No. 2, 325–348 (1987).zbMATHCrossRefMathSciNetGoogle Scholar - 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 - 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 - D. S. Jones, “Integral equations for the exterior acoustic problem,”
*Q. J. Mech. Appl. Math.*,**27**, No. 129–142 (1974).zbMATHCrossRefGoogle Scholar - 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 - 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 - 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 - 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 - C. Lu and W. Chew, “Fast algorithm for solving hybrid integral equations,”
*IEE Proc. H*,**140**, No. 455–460 (1993).Google Scholar - 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
- 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 - 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 - R. Martinez, “The thin-shape breakdown (TSB) of the Helmholtz integral equation,”
*J. Acoust. Soc. Am.*,**90**, No. 5, 2728–2738 (1991).CrossRefGoogle Scholar - 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 - N. Nishimura, “Fast multipole accelerated boundary integral equation methods,”
*Appl. Mech. Rev.*,**55**, No. 4 (July), 299–324 (2002).CrossRefGoogle Scholar - V. Rokhlin, “Rapid solution of integral equations of classical potential theory,”
*J. Comp. Phys.*,**60**, No. 187–207 (1985).zbMATHCrossRefMathSciNetGoogle Scholar - V. Rokhlin, “Rapid solution of integral equations of scattering theory in two dimensions,”
*J. Comput. Phys.*,**86**, No. 2, 414–439 (1990).zbMATHCrossRefMathSciNetGoogle Scholar - 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 - H. A. Schenck, “Improved integral formulation for acoustic radiation problems,”
*J. Acoust. Soc. Am.*,**44**, No. 41–58 (1968).CrossRefGoogle Scholar - 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 - 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 - 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 - 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 - 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 - F. Ursell, “On the exterior problems of acoustics,”
*Proc. Cambridge Philos. Soc.*,**74**, No. 117–125 (1973).zbMATHCrossRefMathSciNetGoogle Scholar - R. Wagner and W. Chew, “A ray-propagation fast multipole algorithm,”
*Microwave Opt. Technol. Lett.*,**7**, No. 435–438 (1994).CrossRefGoogle Scholar - 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 - 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 - 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