1887
Volume 68, Issue 8
  • E-ISSN: 1365-2478

Abstract

ABSTRACT

The representation of a force or moment point source in a spectral finite‐element code for modelling elastic wave propagation becomes fundamentally different in degenerate cases where the source is located on the boundary of an element. This difference is related to the fact that the finite‐element basis functions are continuous across element boundaries, but their derivatives are not. A method is presented that effectively deals with this problem. Tests on one‐dimensional elements show that the numerical errors for a force source follow the expected convergence rate in terms of the element size, apart from isolated cases where superconvergence occurs. For a moment source, the method also converges but one order of accuracy is lost, probably because of the reduced regularity of the problem. Numerical tests in three dimensions on continuous mass‐lumped tetrahedral elements show a similar error behaviour as in the one‐dimensional case, although in three dimensions the loss of accuracy for the moment source is not a severe as a full order.

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2020-07-30
2024-04-23
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References

  1. Anderson, J.E., Brytik, V. and Ayeni, G. (2015) Numerical temporal dispersion corrections for broadband temporal simulation, RTM and FWI. In: SEG Technical Program Expanded Abstracts, pp. 1096–1100.
    [Google Scholar]
  2. Chin‐Joe‐Kong, M.J.S., Mulder, W.A. and van Veldhuizen, M. (1999) Higher‐order triangular and tetrahedral finite elements with mass lumping for solving the wave equation. Journal of Engineering Mathematics, 35, 405–426.
    [Google Scholar]
  3. Cohen, G., Joly, P. and Tordjman, N. (1995) Higher order triangular finite elements with mass lumping for the wave equation. In: Cohen, G., Bécache, E., Joly, P. and Roberts, J.E. (Eds.) Proceedings of the Third International Conference on Mathematical and Numerical Aspects of Wave Propagation. Philadelphia: SIAM, pp. 270–279.
    [Google Scholar]
  4. Cohen, G., Joly, P., Roberts, J.E. and Tordjman, N. (2001) Higher order triangular finite elements with mass lumping for the wave equation. SIAM Journal on Numerical Analysis, 38(6), 2047–2078.
    [Google Scholar]
  5. Cui, T., Leng, W., Lin, D., Ma, S. and Zhang, L. (2017) High order mass‐lumping finite elements on simplexes. Numerical Mathematics: Theory, Methods and Applications, 10(2), 331–350.
    [Google Scholar]
  6. Dablain, M.A. (1986) The application of high‐order differencing to the scalar wave equation. Geophysics, 51(1), 54–66.
    [Google Scholar]
  7. Fichtner, A. (2011) Full Seismic Waveform Modelling and Inversion, Berlin: Springer‐Verlag.
    [Google Scholar]
  8. Geevers, S., Mulder, W.A. and van der Vegt, J.J.W. (2018) New higher‐order mass‐lumped tetrahedral elements for wave propagation modelling. SIAM Journal on Scientific Computing, 40(5), A2830–A2857.
    [Google Scholar]
  9. Geevers, S., Mulder, W.A. and van der Vegt, J.J.W. (2019) Efficient quadrature rules for computing the stiffness matrices of mass‐lumped tetrahedral elements for linear wave problems. SIAM Journal on Scientific Computing, 41(2), A1041–A1065.
    [Google Scholar]
  10. Koene, E.F.M., Robertsson, J.O.A., Broggini, F. and Andersson, F. (2017) Eliminating time dispersion from seismic wave modeling. Geophysical Journal International, 213(1), 169–180.
    [Google Scholar]
  11. Komatitsch, D. and Vilotte, J.P. (1998) The spectral‐element method: an efficient tool to simulate the seismic response of 2‐D and 3‐D geological structures. Bulletin of the Seismological Society of America, 88(2), 368–392.
    [Google Scholar]
  12. Lax, P. and Wendroff, B. (1960) Systems of conservation laws. Communications on Pure and Applied Mathematics, 31(2), 217–237.
    [Google Scholar]
  13. Lesage, A.C., Aubry, R., Houzeaux, G., Polo, M. Araya and Cela, J.M. (2010) 3D spectral element method combined with h‐refinement. In: 72nd EAGE Conference & Exhibition, Barcelona, Spain, Extended Abstracts, C047.
    [Google Scholar]
  14. Liu, Y., Teng, J., Xu, T. and Badal, J. (2017) Higher‐order triangular spectral element method with optimized cubature points for seismic wavefield modeling. Journal of Computational Physics, 336, 458–480.
    [Google Scholar]
  15. Mulder, W.A. (1996) A comparison between higher‐order finite elements and finite differences for solving the wave equation. In: Désidéri, J.‐A., LeTallec, P., Oñate, E., Périaux, J. and Stein, E. (Eds.) Proceedings of the Second ECCOMAS Conference on Numerical Methods in Engineering, Chichester: John Wiley & Sons, pp. 344–350.
    [Google Scholar]
  16. Mulder, W.A. (2013) New triangular mass‐lumped finite elements of degree six for wave propagation. Progress in Electromagnetics Research, 141, 671–692.
    [Google Scholar]
  17. Mulder, W.A. and Shamasundar, R. (2016) Performance of continuous mass‐lumped tetrahedral elements for elastic wave propagation with and without global assembly. Geophysical Journal International, 207(1), 414–421.
    [Google Scholar]
  18. Mulder, W.A., Zhebel, E. and Minisini, S. (2014) Time‐stepping stability of continuous and discontinuous finite‐element methods for 3‐D wave propagation. Geophysical Journal International, 196(2), 1123–1133.
    [Google Scholar]
  19. Qin, Y., Quiring, S. and Nauta, M. (2017) Temporal dispersion correction and prediction by using spectral mapping. In: 79th EAGE Conference & Exhibition, Paris, France Extended Abstracts, Th P1 10. https://doi.org/10.3997/2214-4609.201700677.
    [Google Scholar]
  20. Shubin, G.R. and Bell, J.B. (1987) A modified equation approach to constructing fourth order methods for acoustic wave propagation. SIAM Journal on Scientific and Statistical Computing, 8(2), 135–151.
    [Google Scholar]
  21. Stork, C. (2013) Eliminating nearly all dispersion error from FD modeling and RTM with minimal cost increase. In: 75th EAGE Conference & Exhibition incorporating SPE EUROPEC, Extended Abstract. https://doi.org/10.3997/2214-4609.20130478.
    [Google Scholar]
  22. Strang, G. and Fix, G. (1973) An Analysis of the Finite Element Method, Englewood Cliffs, NJ: Prentice‐Hall.
    [Google Scholar]
  23. Tordjman, N. (1995) Élements finis d'order élevé avec condensation de masse pour l'equation des ondes. Ph.D. thesis, L'Université Paris IX Dauphine.
    [Google Scholar]
  24. von Kowalevsky, S. (1875) Zur Theorie der partiellen Differentialgleichung. Journal für die reine und angewandte Mathematik, 80, 1–32.
    [Google Scholar]
  25. Wang, M. and Xu, S. (2015) Time dispersion transforms in finite difference of wave propagation. In: 77th EAGE Conference & Exhibition, Extended Abstract.
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  • Article Type: Research Article
Keyword(s): Computing aspects; Elastics; Mathematical formulation; Modelling; Seismics

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