1887
  1. Home
  2. A-Z Publications
  3. Geophysical Prospecting
  4. Volume 72, Issue 3
  5. Article
Volume 72, Issue 3
  • E-ISSN: 1365-2478
PDF

Abstract

Abstract

Finite elements with mass lumping allow for explicit time stepping when modelling wave propagation and can be more efficient than finite differences in complex geological settings. In two dimensions on quadrilaterals, spectral elements are the obvious choice. Triangles offer more flexibility for meshing, but the construction of polynomial elements is less straightforward. The elements have to be augmented with higher‐degree polynomials in the interior to preserve accuracy after lumping of the mass matrix. With the classic accuracy criterion, triangular elements suitable for mass lumping up to a polynomial degree 9 were found. With a newer, less restrictive criterion, new elements were constructed of degree 5–7. Some of these are more efficient than the older ones. To assess which of all these elements performs best, the acoustic wave equation is solved for a homogeneous model on a square and on a domain with corners, as well as on a heterogeneous example with topography. The accuracy and runtimes are measured using either higher‐order time stepping or second‐order time stepping with dispersion correction. For elements of polynomial degree 2 and higher, the latter is more efficient. Among the various finite elements, the degree‐4 element appears to be a good choice.

© 2024 European Association of Geoscientists & Engineers. © 2023 Shell Global Solutions International B.V. Geophysical Prospecting published by John Wiley & Sons Ltd on behalf of European Association of Geoscientists & Engineers.
Loading

Article metrics loading...

/content/journals/10.1111/1365-2478.13383
2024-02-21
2025-04-19

  • From This Site

    /content/journals/10.1111/1365-2478.13383
    dcterms_title,dcterms_subject,pub_keyword
    -contentType:Journal -contentType:Contributor -contentType:Concept -contentType:Institution
    10
    5
Loading full text...

Full text loading...

/deliver/fulltext/gpr/72/3/gpr13383.html?itemId=/content/journals/10.1111/1365-2478.13383&mimeType=html&fmt=ahah

References

  1. Anderson, J.E., Brytik, V. & Ayeni, G. (2015) Numerical temporal dispersion corrections for broadband temporal simulation, RTM and FWI. In SEG Technical Program Expanded Abstracts. Houston, TX: Society of Exploration Geophysicists, pp. 1096–1100.
     [Google Scholar]
  2. Chin‐Joe‐Kong, M.J.S., Mulder, W.A. & 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. Ciarlet, P.G. (1978) The finite element method for elliptic problems, Studies in Mathematics and Its Applications, volume 4. Amsterdam, The Netherlands: North‐Holland.
     [Google Scholar]
  4. Cohen, G., Joly, P., Roberts, J.E. & 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. Cohen, G.C. (2002) Higher‐order numerical methods for transient wave equations. Berlin, Germany: Springer.
     [Google Scholar]
  6. Courant, R., Friedrichs, K. & Lewy, H. (1928) Über die partiellen Differenzengleichungen der mathematischen Physik. Mathematische Annalen, 100(1), 32–74.
     [Google Scholar]
  7. Crouzeix, M. & Raviart, P.‐A. (1973) Conforming and non conforming finite element methods for solving the stationary Stokes equations. Revue française d'automatique informatique recherche opérationnelle. Analyse numérique, 7(R3), 33–75.
     [Google Scholar]
  8. Cui, T., Leng, W., Lin, D., Ma, S. & Zhang, L. (2017) High order mass‐lumping finite elements on simplexes. Numerical Mathematics: Theory, Methods and Applications, 10(2), 331–350.
     [Google Scholar]
  9. Dablain, M.A. (1986) The application of high‐order differencing to the scalar wave equation. Geophysics, 51(1), 54–66.
     [Google Scholar]
  10. Dai, N., Wu, W. & Liu, H. (2014) Solutions to numerical dispersion error of time FD in RTM. In SEG Technical Program Expanded Abstracts 2014. Houston, TX: Society of Exploration Geophysicists, pp. 4027–4031.
     [Google Scholar]
  11. Geevers, S., Mulder, W.A. & van der Vegt, J.J.W. (2018a) Dispersion properties of explicit finite element methods for wave propagation modelling on tetrahedral meshes. Journal of Scientific Computing, 77(1), 372–396.
     [Google Scholar]
  12. Geevers, S., Mulder, W.A. & van der Vegt, J.J.W. (2018b) New higher‐order mass‐lumped tetrahedral elements for wave propagation modelling. SIAM Journal on Scientific Computing, 40(5), A2830–A2857.
     [Google Scholar]
  13. Geevers, S., Mulder, W.A. & 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]
  14. Koene, E.F.M., Robertsson, J.O.A., Broggini, F. & Andersson, F. (2018) Eliminating time dispersion from seismic wave modeling. Geophysical Journal International, 213(1), 169–180.
     [Google Scholar]
  15. Komatitsch, D. & 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]
  16. Lax, P. & Wendroff, B. (1960) Systems of conservation laws. Communications on Pure and Applied Mathematics, 31(2), 217–237.
     [Google Scholar]
  17. Li, T.Y. (2003) Numerical solution of polynomial systems by homotopy continuation methods. In Handbook of Numerical Analysis, volume 11. Amsterdam: Elsevier, pp. 209–304.
     [Google Scholar]
  18. Li, Y.E., Wong, M. & Clapp, R. (2016) Equivalent accuracy at a fraction of the cost: overcoming temporal dispersion. Geophysics, 81(5), T189–T196.
     [Google Scholar]
  19. Liu, Y., Teng, J., Xu, T. & 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]
  20. Marchildon, A.L. & Zingg, D.W. (2022) Unisolvency for polynomial interpolation in simplices with symmetrical nodal distributions. Journal of Scientific Computing, 92(2), 1–24.
     [Google Scholar]
  21. Mittet, R. (2017) On the internal interfaces in finite‐difference schemes. Geophysics, 82(4), T159–T182.
     [Google Scholar]
  22. 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. & Stein, E. (Eds.), Proceedings of the Second ECCOMAS Conference on Numerical Methods in Engineering, Chichester, U.K.: John Wiley & Sons, pp. 344–350.
     [Google Scholar]
  23. Mulder, W.A. (1999) Spurious modes in finite‐element discretisations of the wave equation may not be all that bad. Applied Numerical Mathematics, 30(4), 425–445.
     [Google Scholar]
  24. Mulder, W.A. (2001) Higher‐order mass‐lumped finite elements for the wave equation. Journal of Computational Acoustics, 9(2), 671–680.
     [Google Scholar]
  25. 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]
  26. Mulder, W.A. (2020) On the error behaviour of force and moment sources in simplicial spectral finite elements. Geophysical Prospecting, 68, 2598–2603.
     [Google Scholar]
  27. Mulder, W.A. (2022a) Efficiency of old and new triangular finite elements for wavefield modelling in time. In 83rd EAGE Conference and Exhibition 2022, Extended Abstracts. Houten, the Netherlands: European Association of Geoscientists & Engineers, Volume 2022, pp. 1–5.
     [Google Scholar]
  28. Mulder, W.A. (2022b) More continuous mass‐lumped triangular finite elements. Journal of Scientific Computing, 92(2), 38.
     [Google Scholar]
  29. Mulder, W.A. (2023a) Temporal dispersion correction for wave‐propagation modelling with a series approach. Geophysical Prospecting. Submitted.
  30. Mulder, W.A. (2023b) Unisolvence of symmetric node patterns for polynomial spaces on the simplex. Journal of Scientific Computing, 95(2), 1–25. https://doi.org/10.1007/s10915-023-02161-1
     [Google Scholar]
  31. Mulder, W.A. & 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]
  32. Orszag, S.A. (1980) Spectral methods for problems in complex geometries. Journal of Computational Physics, 37(1), 70–92.
     [Google Scholar]
  33. Qin, Y., Quiring, S. & Nauta, M. (2017) Temporal dispersion correction and prediction by using spectral mapping. In 79th EAGE Conference & Exhibition, Paris, France, Extended Abstracts, Th P1 10. Houten, the Netherlands: European Association of Geoscientists & Engineers, Volume 2017, pp. 1–5.
     [Google Scholar]
  34. Shubin, G.R. & 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]
  35. 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. Houten, the Netherlands: European Association of Geoscientists & Engineers, cp‐348‐01021.
  36. Tordjman, N. (1995) Élements finis d'order élevé avec condensation de masse pour l'equation des ondes. PhD thesis, L'Université Paris IX Dauphine.
     [Google Scholar]
  37. Tschöke, K. & Gravenkamp, H. (2018) On the numerical convergence and performance of different spatial discretization techniques for transient elastodynamic wave propagation problems. Wave Motion, 82, 62–85.
     [Google Scholar]
  38. von Kowalevsky, S. (1875) Zur Theorie der partiellen Differentialgleichung. Journal für die reine und angewandte Mathematik, 80, 1–32.
     [Google Scholar]
  39. von Mises, R. & Pollaczek‐Geiringer, H. (1929) Praktische Verfahren der Gleichungsauflösung. Zeitschrift für Angewandte Mathematik und Mechanik, 9(2), 152–164.
     [Google Scholar]
  40. Wang, M. & Xu, S. (2015) Time dispersion transforms in finite difference of wave propagation. In 77th EAGE Conference & Exhibition, Extended Abstract. Houten, the Netherlands: European Association of Geoscientists & Engineers, Volume 2015, pp. 1–5.
     [Google Scholar]
  41. Wolfram Research, Inc. (2016) Mathematica, Version 10.4. Champaign, IL.
     [Google Scholar]
  42. Xu, Z., Jiao, K., Cheng, X., Sun, D., King, R., Nichols, D. & Vigh, D. (2017) Time‐dispersion filter for finite‐difference modeling and reverse time migration. In SEG Technical Program Expanded Abstracts 2017. Houston, TX: Society of Exploration Geophysicists, pp. 4448–4452.
     [Google Scholar]
/content/journals/10.1111/1365-2478.13383
Loading
Performance of old and new mass‐lumped triangular finite elements for wavefield modelling
Geophysical Prospecting 72, 885 (2024); https://doi.org/10.1111/1365-2478.13383
/content/journals/10.1111/1365-2478.13383
/content/journals/10.1111/1365-2478.13383
Loading

Data & Media loading...

  • Article Type: Research Article
Keyword(s): acoustics; computing aspects; modelling; numerical study; seismics; wave

Most Cited This Month Most Cited RSS feed

This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error