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Convolution Quadrature Galerkin Method for the Exterior Neumann Problem of the Wave Equation

  • D. J. Chappell
Chapter

Abstract

The wave equation is important for many real-world applications in timedomain linear acoustics, including scattering from aircraft components and submarines and radiation from loudspeakers. The latter example forms the underlying motivation for the present study. Here the problem is to be solved on an unbounded exterior domain, and so the boundary integral method is a powerful tool for reducing this to an integral equation on the boundary of the radiating or scattering object.

Time-domain boundary integral methods have been employed to solve wave propagation problems since the 1960s [Fr62]. Since then, increasing computer power has made numerical solutions possible over longer run times, and so long-time instabilities in the time marching numerical solutions have become evident [Bi99, Ry85]. A number of methods have been suggested to resolve this such as time averaging [Ry90] and modified time stepping [Bi99]. Using an implicit formulation with high order interpolation and quadrature was also found to give stable results for all practical purposes [Bl96, Do98]. Terrasse et al. [HaD03] obtained stable results using a Galerkin approach and used an energy identity to prove stability of the Galerkin approximation. A stable Burton–Miller type integral equation formulation has also been developed in the time domain [ChHa06, Er99]. In addition, the convolution quadrature method of Lubich [Lu88a, Lu88b] has been applied to a number of problems [Ab06, Ban08, Sc01] and has been shown to give stable numerical results. However, computations for the wave equation tend to be for either two-dimensional or very simple three-dimensional cases such as spheres.

Keywords

Boundary Element Method Boundary Integral Equation Boundary Integral Equation Method Boundary Integral Method Boundary Integral Formulation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [Ab06]
    Abreu, A.I., Mansur, W.J., Carrer, J.A.: Initial conditions contribution in a BEM formulation based on the convolution quadrature method. Internat. J. Numer. Methods Engng., 67, 417-434 (2006).MATHCrossRefMathSciNetGoogle Scholar
  2. [Bam86]
    Bamberger, A., Ha-Duong, T.: Formulation variationnelle pour le calcul de la diffraction d’une onde acoustique par une surface rigide. Math. Methods Appl. Sci., 8, 598-608 (1986).MATHCrossRefMathSciNetGoogle Scholar
  3. [Ban08]
    Banjai, L., Sauter, S.: Rapid solution of the wave equation on unbounded domains. SIAM J. Numer. Anal., 47, 227-249 (2008).CrossRefMathSciNetGoogle Scholar
  4. [Bi99]
    Birgisson, B., Siebrits, E., Pierce, A.P.: Elastodynamic direct boundary element methods with enhanced numerical stability properties. Internat. J. Numer. Methods Engng., 46, 871-888 (1999).MATHCrossRefGoogle Scholar
  5. [Bl96]
    Bluck, M.J., Walker, S.P.: Analysis of three-dimensional transient acoustic wave propagation using the boundary integral equation method. Internat. J. Numer. Methods Engng., 39, 1419-1431 (1996).MATHCrossRefGoogle Scholar
  6. [Ch08]
    Chappell, D.J.: A convolution quadrature Galerkin boundary element method for the exterior Neumann problem of the wave equation. Math. Meth. Appl. Sci., 32:1585-1608 (2009).MATHCrossRefMathSciNetGoogle Scholar
  7. [ChHa06]
    Chappell, D.J., Harris, P.J., Henwood, D., Chakrabarti, R.: A stable boundary element method for modeling transient acoustic radiation. J. Acoust. Soc. Amer., 120, 74-80 (2006).CrossRefGoogle Scholar
  8. [Ci87]
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam (1987).Google Scholar
  9. [Co04]
    Costabel, M.: Time-dependent problems with the boundary integral equation method, in Encyclopedia of Computational Mechanics, Stein, E., De Borst, R., Hughes, T.J.R., eds., Wiley, New York (2004).Google Scholar
  10. [Do98]
    Dodson, S.J., Walker, S.P., Bluck, M.J.: Implicitness and stability of time domain integral equation scattering analysis. Appl. Computat. Electromagnetics Soc. J., 13, 291-301 (1998).Google Scholar
  11. [Er99]
    Ergin, A.A., Shanker, B., Michielssen, E.: Analysis of transient wave scattering from rigid bodies using a Burton-Miller approach. J. Acoust. Soc. Amer., 106, 2396-2404 (1999).CrossRefGoogle Scholar
  12. [Fr62]
    Friedman, M.B., Shaw, R.: Diffraction of pulses by cylindrical objects of arbitrary cross section. J. Appl. Mech., 29, 40-46 (1962).MATHMathSciNetGoogle Scholar
  13. [Ha08]
    Hackbusch, W., Kress, W., Sauter, S.: Sparse convolution quadrature for time domain boundary integral formulations of the wave equation. IMA J. Numer. Anal., 29:158-179 (2009).MATHCrossRefMathSciNetGoogle Scholar
  14. [HaD03]
    Ha-Duong, T., Ludwig, B., Terrasse, I.: A Galerkin BEM for transient acoustic scattering by an absorbing obstacle. Internat. J. Numer. Methods Engng., 57, 1845-1882 (2003).MATHCrossRefMathSciNetGoogle Scholar
  15. [Lu88a]
    Lubich, C.: Convolution quadrature and discretized operational calculus. I. Numer. Math., 52, 129-145 (1988).MATHCrossRefMathSciNetGoogle Scholar
  16. [Lu88b]
    Lubich, C.: Convolution quadrature and discretized operational calculus. II. Numer. Math., 52, 413-425 (1988).MATHCrossRefMathSciNetGoogle Scholar
  17. [Lu94]
    Lubich, C.: On the multistep time discretization of linear initial-boundary value problems and their boundary integral equations. Numer. Math., 67, 365-369 (1994).MATHCrossRefMathSciNetGoogle Scholar
  18. [Mc00]
    Mclean, W.: Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, London (2000).MATHGoogle Scholar
  19. [Ry85]
    Rynne, B.P.: Stability and convergence of time marching methods in scattering problems. IMA J. Appl. Math., 35, 297-310 (1985).MATHCrossRefMathSciNetGoogle Scholar
  20. [Ry90]
    Rynne, B.P., Smith, P.D.: Stability of time marching algorithms for the electric field integral equation. J. Electromagnetic Waves Appl., 4, 1181-1205 (1990).CrossRefGoogle Scholar
  21. [Sc01]
    Schantz, M.: Application of 3D time domain boundary element formulation to wave propagation in poroelastic solids. Engng. Anal. Boundary Elements, 25, 363-376 (2001).CrossRefGoogle Scholar

Copyright information

© Birkhäuser Boston 2010

Authors and Affiliations

  1. 1.University of NottinghamNottinghamUK

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