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Energetic BEM for the Numerical Solution of 2D Hard Scattering Problems of Damped Waves by Open Arcs

  • Alessandra AimiEmail author
  • Mauro Diligenti
  • Chiara Guardasoni
Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 30)

Abstract

The energetic boundary element method (BEM) is a discretization technique for the numerical solution of wave propagation problems, introduced and applied in the last decade to scalar wave propagation inside bounded domains or outside bounded obstacles, in 1D, 2D, and 3D space dimension.

The differential initial-boundary value problem at hand is converted into a space–time boundary integral equations (BIEs), then written in a weak form through considerations on energy and discretized by a Galerkin approach.

The paper will focus on the extension of 2D wave problems of hard scattering by open arcs to the more involved case of damped waves propagation, taking into account both viscous and material damping.

Details will be given on the algebraic reformulation of Energetic BEM, i.e., on the so-called time-marching procedure that gives rise to a linear system whose matrix has a Toeplitz lower triangular block structure.

Numerical results confirm accuracy and stability of the proposed technique, already proved for the numerical treatment of undamped wave propagation problems in several space dimensions and for the 1D damped case.

Keywords

Damped waves Energetic boundary element method FFT 

Notes

Acknowledgements

The authors are grateful to INdAM-GNCS for its financial support through Research Projects funding.

References

  1. 1.
    Aimi, A., Diligenti, M.: A new space-time energetic formulation for wave propagation analysis in layered media by BEMs. Int. J. Numer. Methods Eng. 75, 1102–1132 (2008)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Aimi, A., Diligenti, M., Guardasoni, C.: Numerical integration schemes for space-time hypersingular integrals in energetic Galerkin BEM. Num. Alg. 55(2–3), 145–170 (2010)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Aimi, A., Diligenti, M., Panizzi, S.: Energetic Galerkin BEM for wave propagation Neumann exterior problems. CMES 58(2), 185–219 (2010)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Aimi, A., Diligenti, M., Frangi, A., Guardasoni, C.: Neumann exterior wave propagation problems: computational aspects of 3D energetic Galerkin BEM. Comp. Mech. 51(4), 475–493 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Aimi, A., Diligenti, M., Guardasoni, C.: Energetic BEM-FEM coupling for the numerical solution of the damped wave equation. Adv. Comput. Math. 43, 627–651 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Aimi, A., Diligenti, M., Guardasoni, C.: Energetic BEM for the numerical analysis of 2D Dirichlet damped wave propagation exterior problems. Commun. Appl. Ind. Math. 8(1), 103–127 (2017)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bamberger, A., Ha Duong, T.: Formulation variationelle espace-temps pour le calcul par potential retardé de la diffraction d’une onde acoustique (I). Math. Meth. Appl. Sci. 8, 405–435 (1986)CrossRefGoogle Scholar
  8. 8.
    Bamberger, A., Ha Duong, T.: Formulation variationelle pour le calcul de la diffraction d’une onde acoustique par une surface rigide. Math. Meth. Appl. Sci. 8, 598–608 (1986)CrossRefGoogle Scholar
  9. 9.
    Banerjee, P., Butterfield, P.: Boundary Element Methods in Engineering. McGraw-Hill, London (1981)zbMATHGoogle Scholar
  10. 10.
    Becache, E.: A variational Boundary Integral Equation method for an elastodynamic antiplane crack. Int. J. Numer. Methods Eng. 36, 969–984 (1993)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bini, D.: Matrix Structure and Applications. Les cours du CIRM 4(1), 1–45 (2014)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Borodin, A., Munro, I.: The Computational Complexity of Algebraic and Numeric Problems. American Elsevier, New York (1975)zbMATHGoogle Scholar
  13. 13.
    Chaillat, S., Desiderio, L., Ciarlet, P.: Theory and implementation of H-matrix based iterative and direct solvers for Helmholtz and elastodynamic oscillatory kernels. J. Comput. Phys. 341, 429–446 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Costabel, M.: Time-dependent problems with the boundary integral equation method. In: Stein, E., et al. (eds.) Encyclopedia of Computational Mechanics, pp. 1–28. Wiley, New York (2004)Google Scholar
  15. 15.
    Ha Duong, T.: On retarded potential boundary integral equations and their discretization. In: Ainsworth, M., et al. (eds.) Topics in Computational Wave Propagation. Direct and Inverse Problems, pp. 301–336. Springer, Berlin (2003)CrossRefGoogle Scholar
  16. 16.
    Hairer, E., Lubich, C., Schlichte, M.: Fast numerical solution of nonlinear Volterra convolution equations. SIAM J. Sci. Stat. Comput. 6, 532–541 (1985)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Hartmann, F.: Introduction to Boundary Element Theory Method in Engineering. McGraw-Hill, London (1981)Google Scholar
  18. 18.
    Mayer, S.: Plasmonics: Fundamentals and Applications. Springer, Berlin (2007)CrossRefGoogle Scholar
  19. 19.
    Oran Brigham, E.: The Fast Fourier Transform and Its Applications. Prentice Hall, Englewood Cliffs (1988)Google Scholar
  20. 20.
    Rao, S.: Mechanical Vibrations. Addison-Wesley Publishing, Reading (2010)Google Scholar
  21. 21.
    Stephan, E., Suri, M.: On the convergence of the p-version of the Boundary Element Galerkin Method. Math. Comput. 52(185), 31–48 (1989)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Alessandra Aimi
    • 1
    Email author
  • Mauro Diligenti
    • 1
  • Chiara Guardasoni
    • 1
  1. 1.Dept. of Mathematical, Physical and Computer SciencesUniversity of ParmaParmaItaly

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