Skip to main content

Sparse Convolution Quadrature for Time Domain Boundary Integral Formulations of the Wave Equation by Cutoff and Panel-Clustering

  • Chapter

Part of the Lecture Notes in Applied and Computational Mechanics book series (LNACM,volume 29)

Abstract

We consider the wave equation in a time domain boundary integral formulation. To obtain a stable time discretization, we employ the convolution quadrature method in time, developed by Lubich. In space, a Galerkin boundary element method is considered. The resulting Galerkin matrices are fully populated and the computational complexity is proportional to N log2 NM 2, where M is the number of spatial unknowns and N is the number of time steps.

We present two ways of reducing these costs. The first is an a priori cutoff strategy, which allows to replace a substantial part of the matrices by 0. The second is a panel clustering approximation, which further reduces the storage and computational cost by approximating subblocks by low rank matrices.

Keywords

  • Boundary Element Method
  • Convolution Quadrature
  • Versus B95L
  • Galerkin Boundary Element Method
  • Convolution Quadrature Method

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.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-540-47533-0_5
  • Chapter length: 22 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   189.00
Price excludes VAT (USA)
  • ISBN: 978-3-540-47533-0
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   249.99
Price excludes VAT (USA)
Hardcover Book
USD   249.99
Price excludes VAT (USA)

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. Bamberger, T. Ha-Duong: Formulation variationelle espace-temps pour le calcul par potentiel retardé d’une onde acoustique. Math. Meth. Appl. Sci. 8 (1986) 405–435, 598–608.

    MATH  CrossRef  MathSciNet  Google Scholar 

  2. B. Birgisson, E. Siebrits, A. Pierce: Elastodynamic direct boundary element methods with enhanced numerical stability properties. Internat. J. Numer. Methods Engrg. 46 (1999) 871–888.

    MATH  CrossRef  MathSciNet  Google Scholar 

  3. M. Bluck, S. Walker: Analysis of three-dimensional transient acoustic wave propagation using the boundary integral equation method. Internat. J. Numer. Methods Engrg. 39 (1996) 1419–1431.

    MATH  CrossRef  Google Scholar 

  4. P. Ciarlet: The finite element method for elliptic problems. North-Holland, 1987.

    Google Scholar 

  5. M. Costabel: Developments in boundary element methods for time-dependent problems. In: Problems and Methods in Mathematical Physics (L. Jentsch, F. Tröltzsch eds.), B.G. Teubner, Leipzig, pp. 17–32, 1994.

    Google Scholar 

  6. P. Davies: Numerical stability and convergence of approximations of retarded potential integral equations. SIAM J. Numer. Anal. 31 (1994) 856–875.

    MATH  CrossRef  MathSciNet  Google Scholar 

  7. P. Davies: Averaging techniques for time marching schemes for retarded potential integral equations. Appl. Numer. Math. 23 (1997) 291–310.

    MATH  CrossRef  MathSciNet  Google Scholar 

  8. P. Davies, D. Duncan: Stability and convergence of collocation schemes for retarded potential integral equations. SIAM J. Numer. Anal. 42 (2004) 1167–1188.

    MATH  CrossRef  MathSciNet  Google Scholar 

  9. Y. Ding, A. Forestier, T. Ha-Duong: A Galerkin scheme for the time domain integral equation of acoustic scattering from a hard surface. J. Acoust. Soc. Am. 86 (1989) 1566–1572.

    CrossRef  Google Scholar 

  10. A. Ergin, B. Shanker, E. Michielssen: Fast analysis of transient acoustic wave scattering from rigid bodies using the multilevel plane wave time domain algorithm. J. Acoust. Soc. Am. 117 (2000) 1168–1178.

    CrossRef  Google Scholar 

  11. M. Friedman, R. Shaw: Diffraction of pulses by cylindrical obstacles of arbitrary cross section. J. Appl. Mech. 29 (1962) 40–46.

    MATH  MathSciNet  Google Scholar 

  12. T. Ha-Duong: On retarded potential boundary integral equations and their discretization. In: Computational Methods in Wave Propagation, Vol. 31 (M. Ainsworth, P. Davies, D. Duncan, P. Martin, B. Rynne eds.), Heidelberg, Springer, pp. 301–336, 2003.

    Google Scholar 

  13. T. Ha-Duong, B. Ludwig, I. Terrasse: A Galerkin BEM for transient acoustic scattering by an absorbing obstacle. Internat. J. Numer. Methods Engrg. 57 (2003) 1845–1882.

    MATH  CrossRef  MathSciNet  Google Scholar 

  14. W. Hackbusch, W. Kress, S. Sauter: Sparse convolution quadrature for time domain boundary integral formulations of the wave equation. Technical Report 116, Max-Planck-Institut, Leipzig, Germany, 2005.

    Google Scholar 

  15. W. Hackbusch, Z. Nowak: On the fast matrix multiplication in the boundary element method by panel-clustering. Numer. Math, 54 (1989) 463–491.

    MATH  CrossRef  MathSciNet  Google Scholar 

  16. E. Hairer, C. Lubich, M. Schlichte: Fast numerical solution of nonlinear Volterra convolution equations. SIAM J. Sci. Stat. Comput. 6 (1985) 532–541.

    MATH  CrossRef  MathSciNet  Google Scholar 

  17. P. Henrici: Fast Fourier methods in computational complex analysis. SIAM Review 21 (1979) 481–527.

    MATH  CrossRef  MathSciNet  Google Scholar 

  18. C. Lubich: Convolution quadrature and discretized operational calculus I. Numer. Math. 52 (1988) 129–145.

    MATH  CrossRef  MathSciNet  Google Scholar 

  19. C. Lubich: Convolution quadrature and discretized operational calculus II. Numer. Math. 52 (1988) 413–425.

    MATH  CrossRef  MathSciNet  Google Scholar 

  20. C. Lubich: On the multistep time discretization of linear initial-boundary value problems and their boundary integral equations. Numer. Math. 67 (1994) 365–389.

    MATH  CrossRef  MathSciNet  Google Scholar 

  21. C. Lubich, R. Schneider: Time discretization of parabolic boundary integral equations. Numer. Math. 63 (1992) 455–481.

    MATH  CrossRef  MathSciNet  Google Scholar 

  22. E. Miller: An overview of time-domain integral equations models in electromagnetics. J. of Electromagnetic Waves and Appl. 1 (1987) 269–293.

    CrossRef  Google Scholar 

  23. B. Rynne, P. Smith. Stability of time marching algorithms for the electric field integral equation. J. of Electromagnetic Waves and Appl. 4 (1990) 1181–1205.

    CrossRef  Google Scholar 

  24. S. Sauter, C. Schwab: Randelementmethoden. Analyse, Numerik und Implementierung schneller Algorithmen. B. G. Teubner, Stuttgart, Leipzig, Wiesbaden, 2004.

    Google Scholar 

  25. M. Schanz: Wave Propagation in Viscoelastic and Poroelastic Continua. A Boundary Element Approach. Lecture Notes in Applied and Computational Mechanics, Vol. 2, Springer, Heidelberg, 2001.

    MATH  Google Scholar 

  26. M. Schanz, H. Antes: Application of operational quadrature methods in time domain boundary element methods. Meccanica 32 (1997) 179–186.

    MATH  CrossRef  Google Scholar 

  27. M. Schanz, H. Antes, T. Rüberg: Convolution quadrature boundary element method for quasi-static visco-and poroelastic continua. Computers & Structures 83 (2005) 673–684.

    CrossRef  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2007 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Hackbusch, W., Kress, W., Sauter, S.A. (2007). Sparse Convolution Quadrature for Time Domain Boundary Integral Formulations of the Wave Equation by Cutoff and Panel-Clustering. In: Schanz, M., Steinbach, O. (eds) Boundary Element Analysis. Lecture Notes in Applied and Computational Mechanics, vol 29. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-47533-0_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-47533-0_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-47465-4

  • Online ISBN: 978-3-540-47533-0

  • eBook Packages: EngineeringEngineering (R0)