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An adaptive fast multipole boundary element method for three-dimensional acoustic wave problems based on the Burton–Miller formulation

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Abstract

The high solution costs and non-uniqueness difficulties in the boundary element method (BEM) based on the conventional boundary integral equation (CBIE) formulation are two main weaknesses in the BEM for solving exterior acoustic wave problems. To tackle these two weaknesses, an adaptive fast multipole boundary element method (FMBEM) based on the Burton–Miller formulation for 3-D acoustics is presented in this paper. In this adaptive FMBEM, the Burton–Miller formulation using a linear combination of the CBIE and hypersingular BIE (HBIE) is applied to overcome the non-uniqueness difficulties. The iterative solver generalized minimal residual (GMRES) and fast multipole method (FMM) are adopted to improve the overall computational efficiency. This adaptive FMBEM for acoustics is an extension of the adaptive FMBEM for 3-D potential problems developed by the authors recently. Several examples on large-scale acoustic radiation and scattering problems are presented in this paper which show that the developed adaptive FMBEM can be several times faster than the non-adaptive FMBEM while maintaining the accuracies of the BEM.

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References

  1. Brebbia CA, Dominguez J (1989) Boundary elements—an introductory course. McGraw Hill, New York

    MATH  Google Scholar 

  2. Banerjee PK (1994) The boundary element methods in engineering. McGraw-Hill, New York

    Google Scholar 

  3. Copley LG (1967) Integral equation method for radiation from vibrating bodies. J Acoust Soc Am 41:807–810

    Article  Google Scholar 

  4. Schenck HA (1968) Improved integral formulation for acoustic radiation problems. J Acoust Soc Am 44:41–58

    Article  Google Scholar 

  5. Burton AJ, Miller GF (1971) The application of the integral equation methods to the numerical solution of some exterior boundary-value problems. In: Proceedings of the Royal Society of London, Series A, Math Phys Sci 323(1553):201–210

  6. Meyer WL, Bell WA, Zinn BT (1978) Boundary integral solutions of three dimensional acoustic radiation problems. J Sound Vib 59(2):245–262

    Article  MATH  Google Scholar 

  7. Terai T (1980) On calculation of sound fields around three dimensional objects by integral equation methods. J Sound Vib 69(1):71–100

    Article  MATH  Google Scholar 

  8. Gentle JE (1998) Gaussian elimination. In Numerical linear algebra for applications in statistics. Springer, Berlin Heidelberg New York, pp 87–91

  9. Saad Y, Schultz MH (1986) GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J Sci Statist Comput 7:856–869

    Article  MATH  MathSciNet  Google Scholar 

  10. Sonneveld P (1989) CGS: a fast Lanczos-type solver for nonsymmetric linear systems. SIAM J Sci Statist Comput 10:36–52

    Article  MATH  MathSciNet  Google Scholar 

  11. Beylkin G, Coifman A, Rokhlin V (1991) Fast wavelet transforms and numerical algorithms I. Comm Pure Appl Math XLIV:141–183

    Article  MathSciNet  Google Scholar 

  12. Hackbusch W (1999) A sparse matrix arithmetic based on H-matrices I. Introduction to H-matrices. Computing 62(2):89–108

    Article  MATH  MathSciNet  Google Scholar 

  13. Golub G, Loan CV (1996) Matrix computations, 3rd edn. The Johns Hopkins University Press, Baltimore

    MATH  Google Scholar 

  14. Greengard L, Rokhlin V (1987) A fast algorithm for particle simulations. J Comput Phys 73:325–348

    Article  MATH  MathSciNet  Google Scholar 

  15. Rokhlin V (1988) A fast algorithm for the discrete Laplace transformation. J Complex 4(1):12–32

    Article  MATH  MathSciNet  Google Scholar 

  16. White CA, Head-Gordon M (1994) Derivation and efficient implementation of the fast multipole method. J Chem Phys 101(8):6593–6605

    Article  Google Scholar 

  17. White CA, et al. (1994) The continues fast multipole method. Chem Phys Lett 230(1–2):8–16

    Article  Google Scholar 

  18. White CA, Head-Gordon M (1996) Rotating around the quartic angular momentum barrier in fast multipole method calculations. J Chem Phys 105(12):5061–5067

    Article  Google Scholar 

  19. Beatson R, Greengard L (1996) A short course on fast multipole methods. In: Ainsworth M, et al. (eds) Wavelets, multilevel methods and elliptic PDEs. Oxford University Press, pp 1–37

  20. Cheng H, Greengard L, Rokhlin V (1999) A fast adaptive multipole algorithm in three dimensions. J Comput Phys 155(1):468–498

    Article  MATH  MathSciNet  Google Scholar 

  21. Rokhlin V (1990) Rapid solution of integral equations of scattering theory in two dimensions. J Comput Phys 86(2):414–439

    Article  MATH  MathSciNet  Google Scholar 

  22. Engheta N, et al. (1992) The fast multipole method (FMM) for electromagnetic scattering problems. IEEE Trans Ant Propag 40(6):634–641

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  25. Lu C, Chew W (1993) Fast algorithm for solving hybrid integral equations. IEE Proceedings-H 140(6):455–460

    Google Scholar 

  26. Wagner R, Chew W (1994) A ray-propagation fast multipole algorithm. Microwave Opt Technol Lett 7:435–438

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  28. Rahola J (1996) Diagonal forms of the translation operators in the fast multipole algorithm for scattering problems. BIT 36:333–358

    Article  MATH  MathSciNet  Google Scholar 

  29. Chen YH, Chew WC, Zeroug S (1997) Fast multipole method as an efficient solver for 2D elastic wave surface integral equations. Comput Mech 20(6):495–506

    Article  MATH  Google Scholar 

  30. Song J, Lu C-C, Chew WC (1997) Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects. IEEE Trans Ant Propag 45(10):1488–1493

    Article  Google Scholar 

  31. Koc S, Chew WC (1998) Calculation of acoustical scattering from a cluster of scatterers. J Acoust Soc Am 103(2):721–734

    Article  Google Scholar 

  32. Gyure MF, Stalzer MA (1998) A prescription for the multilevel Helmholtz FMM. IEEE Comput Sci Eng 5(3): 39–47

    Article  Google Scholar 

  33. Greengard L, et al (1998) Accelerating fast multipole methods for the Helmholtz equation at low frequencies. IEEE Comput Sci Eng 5(3):32–38

    Article  Google Scholar 

  34. Tournour MA, Atalla N (1999) Efficient evaluation of the acoustic radiation using multipole expansion. Int J Numer Methods Eng 46(6):825–837

    Article  MATH  Google Scholar 

  35. Darve E (2000) The fast multipole method: Numerical implementation. J Comput Phys 160(1):195–240

    Article  MATH  MathSciNet  Google Scholar 

  36. Gumerov NA, Duraiswami R (2003) Recursions for the computation of multipole translation and rotation coefficients for the 3-D Helmholtz equation. SIAM J Sci Comput 25(4):1344–1381

    Article  MATH  MathSciNet  Google Scholar 

  37. Darve E, Havé P (2004) Efficient fast multipole method for low-frequency scattering. J Comput Phys 197(1):341–363

    Article  MATH  MathSciNet  Google Scholar 

  38. Fischer M, Gauger U, Gaul L (2004) A multipole Galerkin boundary element method for acoustics. Eng Anal Bound Elem 28(2):155–162

    Article  MATH  Google Scholar 

  39. Chen JT, Chen KH (2004) 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(6):685–709

    Article  MATH  Google Scholar 

  40. Chew WC (1992) Recurrence relations for three-dimensional scalar and addition theorem. J Electromagn Waves Appl 6:133–142

    Article  Google Scholar 

  41. Nishimura N (2002) Fast multipole accelerated boundary integral equation methods. Appl Mech Rev 55(4):299–324

    Article  Google Scholar 

  42. Chen J, Hong H-K (1999) Review of dual boundary element methods with emphasis on hypersingular integrals and divergent series. Appl Mech Rev 52(1):17–33

    Article  MathSciNet  Google Scholar 

  43. Guiggiani M (1998) Formulation and numerical treatment of boundary integral equations with hypersingular kernels. In: Sladek V, Sladek J (eds) Singular integrals in boundary element methods. Computational Mechanics Publications, Boston

    Google Scholar 

  44. Krishnasamy GL, et al. (1990) Hypersingular boundary integral equations: some applications in acoustic and elastic wave scattering. J Appl Mech 57:404–414

    MATH  MathSciNet  Google Scholar 

  45. Liu YJ, Rizzo FJ (1992) A weakly singular form of the hypersingular boundary integral equation applied to 3D acoustic wave problems. Comput Methods Appl Mech Eng 96: 271–287

    Article  MATH  MathSciNet  Google Scholar 

  46. Yang SA (2002) Evaluation of 2D Green’s boundary formula and its normal derivative using Legendre polynomials, with an application to acoustic scattering problems. Int J Numer Methods Eng 53:905–927

    Article  Google Scholar 

  47. Shen L, Liu YJ (2006) An adaptive fast multipole boundary element method for three-dimensional potential problems. Comput Mech (in press)

  48. Kress R (1985) Minimizing the condition number of boundary integral operators in acoustic and electromagnetic scattering. Q J Mech Appl Math 38:323–341

    Article  MATH  MathSciNet  Google Scholar 

  49. Yoshida K-i (2001) Applications of fast multipole method to boundary integral equation method. Department of Global Environment Engineering, Kyoto University

  50. Nishida T, Hayami K (1997) Application of the fast multipole method to the 3D BEM analysis of electron guns. In: Marchettia M, Brebbia CA, Aliabadi MH (eds) Boundary elements XIX. Computational Mechanics Publications, Boston, pp 613–622

    Google Scholar 

  51. Chen K, Harris PJ (2001) Efficient preconditioners for iterative solution of the boundary element equations for the three-dimensional Helmholtz equation. Appl Numer Math 36(4):475–489

    Article  MATH  MathSciNet  Google Scholar 

  52. Seybert AF, Wu TW (1989) Modified Helmholtz integral equation for bodies sitting on an infinite plane. J Acoust Soc Am 85(1):19–23

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Computer-Aided Engineering Research Laboratory, Department of Mechanical Engineering, University of Cincinnati, Cincinnati, OH, 45221-0072, USA

    L. Shen & Y. J. Liu

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  1. L. Shen
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  2. Y. J. Liu
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Correspondence to Y. J. Liu.

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Shen, L., Liu, Y.J. 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). https://doi.org/10.1007/s00466-006-0121-2

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  • DOI: https://doi.org/10.1007/s00466-006-0121-2

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