Advertisement

Time-Domain BEM

  • Joachim Gwinner
  • Ernst Peter Stephan
Chapter
Part of the Springer Series in Computational Mathematics book series (SSCM, volume 52)

Abstract

Time-domain Galerkin boundary elements provide an efficient tool for numerical solution of boundary value problems for the homogeneous wave equation. In Sect. 13.1 we present from [193] a time-domain Galerkin BEM for the wave equation outside a Lipschitz obstacle in an absorbing half-space.A priori error estimates from [193] and a posteriori error estimates from [194] are given in Sect. 13.2

References

  1. 19.
    I. Babuška, B.Q. Guo, Regularity of the solution of elliptic problems with piecewise analytic data. I. Boundary value problems for linear elliptic equation of second order. SIAM J. Math. Anal. 19, 172–203 (1988)MathSciNetCrossRefGoogle Scholar
  2. 29.
    A. Bamberger, T. Ha-Duong, Formulation variationnelle espace-temps pour le calcul par potentiel retardé de la diffraction d’une onde acoustique. I. Math. Methods Appl. Sci. 8, 405–435 (1986)Google Scholar
  3. 30.
    L. Banjai, S. Sauter, Rapid solution of the wave equation in unbounded domains. SIAM J. Numer. Anal. 47, 227–249 (2008/09)MathSciNetCrossRefGoogle Scholar
  4. 31.
    L. Banjai, M. Schanz, Wave Propagation Problems Treated with Convolution Quadrature and BEM. Fast Boundary Element Methods in Engineering and Industrial Applications. Lect. Notes Appl. Comput. Mech., vol. 63 (Springer, Heidelberg, 2012), pp. 145–184Google Scholar
  5. 34.
    L. Banz, H. Gimperlein, Z. Nezhi, E.P. Stephan, Time domain BEM for sound radiation of tires. Comput. Mech. 58, 45–57 (2016)MathSciNetCrossRefGoogle Scholar
  6. 74.
    C. Carstensen, Efficiency of a posteriori BEM-error estimates for first-kind integral equations on quasi-uniform meshes. Math. Comput. 65, 69–84 (1996)Google Scholar
  7. 92.
    C. Carstensen, E.P. Stephan, A posteriori error estimates for boundary element methods. Math. Comput. 64, 483–500 (1995)MathSciNetCrossRefGoogle Scholar
  8. 94.
    J. Chabassier, A. Chaigne, P. Joly, Time domain simulation of a piano. Part 1: Model description. ESAIM Math. Model. Numer. Anal. 48, 1241–1278 (2014)MathSciNetCrossRefGoogle Scholar
  9. 117.
    M. Costabel, Time-Dependent Problems with a Boundary Integral Equation Method. Encyclopedia of Computational Mechanics, 2004Google Scholar
  10. 190.
    H. Gimperlein, M. Maischak, E.P. Stephan, Adaptive time domain boundary element methods with engineering applications. J. Integr. Equ. Appl. 29, 75–105 (2017)MathSciNetCrossRefGoogle Scholar
  11. 191.
    H. Gimperlein, F. Meyer, C. Özdemir, D. Stark, E.P. Stephan, Boundary elements with mesh refinements for the wave equation. Numer. Math. (2018)Google Scholar
  12. 192.
    H. Gimperlein, F. Meyer, C. Özdemir, D. Stark, E.P. Stephan, Time domain boundary elements for dynamic contact problems. Comput. Methods Appl. Mech. Eng. 333, 147–175 (2018)MathSciNetCrossRefGoogle Scholar
  13. 193.
    H. Gimperlein, Z. Nezhi, E.P. Stephan, A priori error estimates for a time-dependent boundary element method for the acoustic wave equation in a half-space. Math. Methods Appl. Sci. 40, 448–462 (2017)MathSciNetCrossRefGoogle Scholar
  14. 194.
    H. Gimperlein, C. Özdemir, D. Stark, E.P. Stephan, A residual a posteriori error estimate for the time-domain boundary element method. Preprint 2017Google Scholar
  15. 195.
    H. Gimperlein, C. Özdemir, E.P. Stephan, Time domain boundary element methods for the Neumann problem: error estimates and acoustics problems. J. Comput. Math. 36, 70–89 (2018)MathSciNetCrossRefGoogle Scholar
  16. 197.
    M. Gläfke, Adaptive methods for time domain boundary integral equations, Ph.D. thesis, Brunel University, 2012Google Scholar
  17. 223.
    T. Ha-Duong, On Retarded Potential Boundary Integral Equations and Their Discretisation. Topics in Computational Wave Propagation. Lect. Notes Comput. Sci. Eng., vol. 31 (Springer, Berlin, 2003), pp. 301–336Google Scholar
  18. 224.
    T. Ha-Duong, B. Ludwig, I. Terrasse, A Galerkin BEM for transient acoustic scattering by an absorbing obstacle. Int. J. Numer. Methods Eng. 57, 1845–1882 (2003)MathSciNetCrossRefGoogle Scholar
  19. 268.
    A.Yu. Kokotov, P. Neittaanmäki, B.A. Plamenevskii, The Neumann problem for the wave equation in a cone. J. Math. Sci. (New York) 102, 4400–4428 (2000)MathSciNetCrossRefGoogle Scholar
  20. 269.
    A.Yu. Kokotov, P. Neittaanmäki, B.A. Plamenevskii, Diffraction on a cone: the asymptotic of solutions near the vertex.. J. Math. Sci. (New York) 109, 1894–1910 (2002)Google Scholar
  21. 329.
    M. Ochmann, Closed form solutions for the acoustical impulse response over a masslike or an absorbing plane. J. Acoust. Soc. Am. 129, 3502–3512 (2011)CrossRefGoogle Scholar
  22. 331.
    E. Ostermann, Numerical methods for space-time variational formulations of retarded potential boundary integral equations, Ph.D. thesis, Leibniz Universität Hannover, 2009Google Scholar
  23. 358.
    S. Sauter, A. Veit, Retarded boundary integral equations on the sphere: exact and numerical solution. IMA J. Numer. Anal. 34, 675–699 (2014)MathSciNetCrossRefGoogle Scholar
  24. 361.
    F.-J. Sayas, Retarded Potentials and Time Domain Boundary Integral Equations. A Road Map. Springer Series in Computational Mathematics, vol. 50 (Springer, Cham, 2016)CrossRefGoogle Scholar
  25. 402.
    E.P. Stephan, M. Maischak, E. Ostermann, Transient boundary element method and numerical evaluation of retarded potentials. In: Computational Science - ICCS 2008, 8th International Conference, Kraków, Poland, June 23–25, 2008, Proceedings, Part II, 2008, pp. 321–330CrossRefGoogle Scholar
  26. 403.
    E.P. Stephan, M. Maischak, E. Ostermann, TD-BEM for sound radiation in three dimensions and the numerical evaluation of retarded potentials. In: International Conference on Acoustics, NAG/DAGA, 2009Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Joachim Gwinner
    • 1
  • Ernst Peter Stephan
    • 2
  1. 1.Fakultät für Luft- und RaumfahrttechnikUniversität der Bundeswehr MünchenNeubiberg/MünchenGermany
  2. 2.Institut für Angewandte MathematikLeibniz Universität HannoverHannoverGermany

Personalised recommendations