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.
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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).
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).
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).
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).
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).
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).
R. D. Ciskowski and C. A. Brebbia, Boundary Element Methods in Acoustics (Kluwer Academic Publishers, New York, 1991).
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).
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).
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).
E. Darve and P. Havé, “Efficient fast multipole method for low-frequency scattering,” J. Comput. Phys., 197, No. 1, 341–363 (2004).
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).
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).
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).
M. Fischer, U. Gauger, and L. Gaul, “A multipole Galerkin boundary element method for acoustics,” Eng Anal Bound Elem, 28, No. 155–162 (2004).
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).
L. F. Greengard, The Rapid Evaluation of Potential Fields in Particle Systems (The MIT Press, Cambridge, 1988).
L. F. Greengard and V. Rokhlin, “A fast algorithm for particle simulations,” J. Comput. Phys., 73, No. 2, 325–348 (1987).
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).
M. F. Gyure and M. A. Stalzer, “A prescription for the multilevel Helmholtz FMM,” IEEE Comput. Sci. Eng., 5, No. 3, 39–47 (1998).
D. S. Jones, “Integral equations for the exterior acoustic problem,” Q. J. Mech. Appl. Math., 27, No. 129–142 (1974).
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).
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).
R. E. Kleinman and G. F. Roach, “Boundary integral equations for the three-dimensional Helmholtz equation,” SIAM Rev., 16, No. 214–236 (1974).
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).
C. Lu and W. Chew, “Fast algorithm for solving hybrid integral equations,” IEE Proc. H, 140, No. 455–460 (1993).
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).
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).
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).
R. Martinez, “The thin-shape breakdown (TSB) of the Helmholtz integral equation,” J. Acoust. Soc. Am., 90, No. 5, 2728–2738 (1991).
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).
N. Nishimura, “Fast multipole accelerated boundary integral equation methods,” Appl. Mech. Rev., 55, No. 4 (July), 299–324 (2002).
V. Rokhlin, “Rapid solution of integral equations of classical potential theory,” J. Comp. Phys., 60, No. 187–207 (1985).
V. Rokhlin, “Rapid solution of integral equations of scattering theory in two dimensions,” J. Comput. Phys., 86, No. 2, 414–439 (1990).
V. Rokhlin, “Diagonal forms of translation operators for the Helmholtz equation in three dimensions,” Appl. Comput. Harmon. Anal., 1, No. 1, 82–93 (1993).
H. A. Schenck, “Improved integral formulation for acoustic radiation problems,” J. Acoust. Soc. Am., 44, No. 41–58 (1968).
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).
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).
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).
L. Shen and Y. J. Liu, “An adaptive fast multipole boundary element method for 3-D half-space acoustic wave problems,” in review, (2009).
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).
F. Ursell, “On the exterior problems of acoustics,” Proc. Cambridge Philos. Soc., 74, No. 117–125 (1973).
R. Wagner and W. Chew, “A ray-propagation fast multipole algorithm,” Microwave Opt. Technol. Lett., 7, No. 435–438 (1994).
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).
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).
K. Yoshida, “Applications of fast multipole method to boundary integral equation method”, Ph.D. Dissertation, Department of Global Environment Engineering, Kyoto University (2001).
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Liu, Y., Shen, L., Bapat, M. (2009). Development of the Fast Multipole Boundary Element Method for Acoustic Wave Problems. In: Manolis, G.D., Polyzos, D. (eds) Recent Advances in Boundary Element Methods. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-9710-2_19
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DOI: https://doi.org/10.1007/978-1-4020-9710-2_19
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