Advertisement

On Multiple Modes of Propagation of High-Order Finite Element Methods for the Acoustic Wave Equation

  • S. P. OliveiraEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 119)

Abstract

Earlier analyses of numerical dispersion of high-order finite element methods (HO-FEM) for acoustic and elastic wave propagation pointed out the presence of multiple modes of propagation. The number of modes increases with the polynomial degree of the finite element space, and since they were regarded as numerical artifacts, the use of HO-FEM was discouraged on wave propagation problems. Later on, alternative techniques showed that numerical dispersion decreases with the polynomial degree, and were supported by the success of spectral element methods on seismic wave propagation. This work concerns the interpretation of multiple propagation modes, which are solutions of an eigenvalue problem arising from the HO-FEM discretization of the wave equation as approximations to an eigenvalue problem associated with the continuous wave equation. By considering a continuous version of the standard periodic plane wave whose amplitude depends on the element grid, there are multiple combinations of the amplitude coefficients that yield exact solutions to the acoustic wave equation. Hence, modes regarded as non-physical can be associated with feasible propagation modes. Under this point of view, one can separately analyze each propagation mode or focus on the acoustical (constant amplitude) mode.

Notes

Acknowledgements

The author is supported by CNPq under grant 306083/2014-0 and is a collaborator of INCT-GP.

References

  1. 1.
    N. Abboud, P. Pinsky, Finite element dispersion analysis for the three-dimensional second-order scalar wave equation. Int. J. Numer. Methods Eng. 35(6), 1183–1218 (1992)CrossRefzbMATHGoogle Scholar
  2. 2.
    M. Ainsworth, Discrete dispersion relation for hp-version finite element approximation at high wave number. SIAM J. Numer. Anal. 42(2), 553–575 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    M. Ainsworth, H.A. Wajid, Dispersive and dissipative behavior of the spectral element method. SIAM J. Numer. Anal. 47(5), 3910–3937 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    T. Belytschko, R. Mullen, On dispersive properties of finite element solutions, in Modern Problems in Elastic Wave Propagation, ed. by J. Miklowitz, J. Achenbach (Wiley, New York, NY, 1978), pp. 67–82Google Scholar
  5. 5.
    L. Brillouin, Wave Propagation in Periodic Structures. Electric Filters and Crystal Lattices, 2nd edn. (Dover publications, New York, NY, 1953)Google Scholar
  6. 6.
    B. Cathers, B. O’Connor, The group velocity of some numerical schemes. Int. J. Numer. Methods Fluids 5(3), 201–224 (1985)CrossRefzbMATHGoogle Scholar
  7. 7.
    G. Cohen, P. Joly, N. Tordjman, Eléments finis d’ordre élevé avec condensation de masse pour l’équation des ondes en dimension 1. Rapport de recherche RR-2323 (1994)Google Scholar
  8. 8.
    M. Cullen, The use of quadratic finite element methods and irregular grids in the solution of hyperbolic problems. J. Comput. Phys. 45(2), 221–245 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    J. De Basabe, M. Sen, Grid dispersion and stability criteria of some common finite-element methods for acoustic and elastic wave equations. Geophysics 72(6), T81–T95 (2007)CrossRefGoogle Scholar
  10. 10.
    D. Durran, Wave propagation in quadratic-finite-element approximations to hyperbolic equations. J. Comput. Phys. 159(2), 448–455 (2000)CrossRefzbMATHGoogle Scholar
  11. 11.
    G. Gabard, R. Astley, M. Ben Tahar, Stability and accuracy of finite element methods for flow acoustics: I. General theory and application to one-dimensional propagation. Int. J. Numer. Methods Eng. 63(7), 947–973 (2005)zbMATHGoogle Scholar
  12. 12.
    P.M. Gresho, R.L. Lee, Comments on ‘the group velocity of some numerical schemes’. Int. J. Numer. Methods Fluids 7(12), 1357–1362 (1987)CrossRefzbMATHGoogle Scholar
  13. 13.
    D. Komatitsch, J. Tromp, Introduction to the spectral-element method for 3-D seismic wave propagation. Geophys. J. Int. 139(3), 806–822 (1999)CrossRefGoogle Scholar
  14. 14.
    K. Marfurt, Appendix - analysis of higher order finite-element methods, in Numerical Modeling of Seismic Wave Propagation, ed. by K. Kelly, K. Marfurt. Geophysics Reprint Series, vol. 13 (Society of Exploration Geophysicists, Tulsa, OK, 1990), pp. 516–520Google Scholar
  15. 15.
    R.C. Moura, S.J. Sherwin, J. Peiró, Linear dispersion–diffusion analysis and its application to under-resolved turbulence simulations using discontinuous Galerkin spectral/hp methods. J. Comput. Phys. 298, 695–710 (2015)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    W. Mulder, Spurious modes in finite-element discretizations of the wave equation may not be all that bad. Appl. Numer. Math. 30(4), 425–445 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    S.P. Oliveira, G. Seriani, DFT modal analysis of spectral element methods for the 2D elastic wave equation. J. Comput. Appl. Math. 234(6), 1717–1724 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    W. Scott, Errors due to spatial discretization and numerical precision in the finite-element method. IEEE Trans. Antennas Propagat. 42(11), 1565–1570 (1994)CrossRefGoogle Scholar
  19. 19.
    G. Seriani, S.P. Oliveira, Dispersion analysis of spectral element methods for acoustic wave propagation. J. Comput. Acoust. 16(4), 531–561 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    G. Seriani, S.P. Oliveira, Dispersion analysis of spectral element methods for elastic wave propagation. Wave Motion 45(6), 729–744 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    G. Seriani, E. Priolo, Spectral element method for acoustic wave simulation in heterogeneous media. Finite Elem. Anal. Des. 16(3–4), 337–348 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    L. Thompson, A review of finite-element methods for time-harmonic acoustics. J. Acoust. Soc. Am. 119(3), 1315–1330 (2005)CrossRefGoogle Scholar
  23. 23.
    L. Thompson, P. Pinsky, Complex wavenumber Fourier analysis of the p-version finite element method. Comput. Mech. 13, 255–275 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    L.N. Trefethen, D. Bau, Numerical Linear Algebra (SIAM, Philadelphia, 1997)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade Federal do ParanáCuritiba-PRBrazil

Personalised recommendations