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Computational Geosciences

, Volume 23, Issue 5, pp 997–1010 | Cite as

An efficient high-order multiscale finite element method for frequency-domain elastic wave modeling

  • Shubin Fu
  • Kai GaoEmail author
  • Richard L. GibsonJr.
  • Eric T. Chung
Original Paper
  • 68 Downloads

Abstract

Solving the frequency-domain elastic wave equation in highly heterogeneous and complex media is computationally challenging. Conventional methods for solving the elastic wave Helmholtz equation usually lead to a large-dimensional linear system that is difficult to solve without specialized and sophisticated techniques. Based on the multiscale finite element theory, we develop a novel method to solve the frequency-domain elastic wave equation in complex media. The key feature of our method is employing high-order multiscale basis functions defined by solving local linear problems to achieve model reduction, which eventually leads to a linear system with significantly reduced dimensions. Solving this reduced linear system therefore results in obvious computational time reduction. We use three 2D examples to verify the accuracy and efficiency of our high-order multiscale finite element method for solving the Helmholtz equation in complex isotropic and anisotropic elastic media. The results show that our new method can approximate the fine-scale reference solution on the coarse mesh with high accuracy and significantly reduced computational time at the linear system solving stage.

Keywords

Elastic wave modeling Multiscale method High-order accuracy Strongly heterogeneous media Helmholtz equation Computational efficiency 

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References

  1. 1.
    Amestoy, P.R., Duff, I.S., L’Excellent, J.-Y., Rouet, F. -H.: Parallel computation of entries of A − 1. SIAM J. Sci. Comput. 37(2), C268–C284 (2015)Google Scholar
  2. 2.
    Bao, H., Bielak, J., Ghattas, O., Kallivokas, L.F., O’Hallaron, D.R., Shewchuk, J.R., Xu, J.: Earthquake ground motion modeling on parallel computers. In: Proceedings of the 1996 ACM/IEEE Conference on Supercomputing (1996)Google Scholar
  3. 3.
    Barucq, H., Calandra, H., Chaumont-Frelet, T., Gout, C., Valentin, F.: The multiscale hybrid mixed method for the Helmholtz equation. HAL-Inria,, no. hal-00930139 (2018)Google Scholar
  4. 4.
    Berenger, J.-P.: A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114 (2), 185–200 (1994)Google Scholar
  5. 5.
    Brossier, R., Operto, S., Virieux, J.: Seismic imaging of complex onshore structures by 2D elastic frequency-domain full-waveform inversion. Geophysics 74(6), WCC105–WCC118 (2009)Google Scholar
  6. 6.
    Capdeville, Y., Marigo, J.-J.: Second order homogenization of the elastic wave equation for non-periodic layered media. Geophys. J. Int. 170(2), 823–838 (2007)Google Scholar
  7. 7.
    Capdeville, Y., Guillot, L., Marigo, J.-J.: 2-D non-periodic homogenization to upscale elastic media for P-SV waves. Geophys. J. Int. 182(2), 903–922 (2010)Google Scholar
  8. 8.
    Carcione, J.M.: Wave Fields in Real Media. Wave Propagation in Anisotropic, Anelastic Porous and Electromagnetic Media, 3rd edn. Elsevier, Amsterdam, Netherlands (2015)Google Scholar
  9. 9.
    Cessenat, O., Després, B.: Using plane waves as base functions for solving time harmonic equations with the ultra weak variational formulation. J. Comput. Acoust. 11(02), 227–238 (2003)Google Scholar
  10. 10.
    Chaumont-Frelet, T., Nicaise, S.: Wavenumber explicit convergence analysis for finite element discretizations of general wave propagation problems. HAL-Inria,, no. hal-01685388 (2018)Google Scholar
  11. 11.
    Chaumont-Frelet, T., Valentin, F.: A multiscale hybrid-mixed method for the Helmholtz equation. HAL-Inria,, no. hal-01698914 (2018)Google Scholar
  12. 12.
    Chaumont-Frelet, T.: Finite element approximation of Helmholtz problems with application to seismic wave propagation. Ph.D. thesis, INSA Rouen and Inria project-team Magique3D (2015)Google Scholar
  13. 13.
    Chaumont-Frelet, T.: On high order methods for the heterogeneous Helmholtz equation. Comput. Math. Appl. 72(9), 2203–2225 (2016)Google Scholar
  14. 14.
    Chen, Z., Cheng, D., Wu, T.: A dispersion minimizing finite difference scheme and preconditioned solver for the 3D helmholtz equation. J. Comput. Phys. 231(24), 8152–8175 (2012)Google Scholar
  15. 15.
    Chung, E.T., Efendiev, Y., Leung, W.T.: Generalized multiscale finite element methods for wave propagation in heterogeneous media. Multiscale Model. Simul. 12(4), 1691–1721 (2014)Google Scholar
  16. 16.
    Chung, E.T., Lam, C.Y., Qian, J.: A staggered discontinuous Galerkin method for the simulation of seismic waves with surface topography. Geophysics 80(4), T119–T135 (2015)Google Scholar
  17. 17.
    Collino, F., Tsogka, C.: Application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media. Geophysics 66(1), 294–307 (2001)Google Scholar
  18. 18.
    Efendiev, Y., Hou, T.Y., Ginting, V.: 12 Multiscale finite element methods for nonlinear problems and their applications. Commun. Math. Sci. 2(4), 553–589 (2004)Google Scholar
  19. 19.
    Efendiev, Y., Galvis, J., Wu, X.-H.: Multiscale finite element methods for high-contrast problems using local spectral basis functions. J. Comput. Phys. 230(4), 937–955 (2011)Google Scholar
  20. 20.
    Efendiev, Y., Galvis, J., Hou, T.Y.: Generalized multiscale finite element methods (GMsFEM). J. Comput. Phys. 251, 116–135 (2013)Google Scholar
  21. 21.
    Engquist, B., Ying, L.: Sweeping preconditioner for the Helmholtz equation. Hierarchical matrix representation. Commun. Pure Appl. Math. 64(5), 697–735 (2011)Google Scholar
  22. 22.
    Engquist, B.: Sweeping preconditioner for the Helmholtz equation. Moving perfectly matched layers. Multiscale Model. Simul. 9(2), 686–710 (2011)Google Scholar
  23. 23.
    Erlangga, Y.A., Oosterlee, C.W., Vuik, C.: A novel multigrid based preconditioner for heterogeneous Helmholtz problems. SIAM J. Sci. Comput. 27(4), 1471–1492 (2006)Google Scholar
  24. 24.
    Fang, J., Qian, J., Zepeda-Núñez, L., Zhao, H.: Learning dominant wave directions for plane wave methods for high-frequency Helmholtz equations. Res. Math. Sci. 4(1), 9 (2017)Google Scholar
  25. 25.
    Farhat, C., Harari, I., Hetmaniuk, U.: A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime. Comput. Methods Appl. Mech. Eng. 192(11), 1389–1419 (2003)Google Scholar
  26. 26.
    Fichtner, A.: Full Seismic Waveform Modelling and Inversion. Springer, Berlin (2011)Google Scholar
  27. 27.
    Fornberg, B.: The pseudospectral method. Accurate representation of interfaces in elastic wave calculations. Geophyscis 53(5), 625–637 (1988)Google Scholar
  28. 28.
    Fu, S., Gao, K.: A fast solver for the Helmholtz equation based on the generalized multiscale finite-element method. Geophys. J. Int. 211(2), 819–835 (2017)Google Scholar
  29. 29.
    Fu, S., Efendiev, Y., Gao, K., Gibson, R.L. Jr: Multiscale modeling of acoustic wave propagation in 2D heterogeneous media using local spectral basis functions. SEG Technical Program Expanded Abstracts 2013, 3553–3558 (2013)Google Scholar
  30. 30.
    Gao, K., Chung, E.T., Gibson, R.L., Fu, S., Efendiev, Y.: A numerical homogenization method for heterogeneous, anisotropic elastic media based on multiscale theory. Geophysics 80(4), D385–D401 (2015)Google Scholar
  31. 31.
    Gao, K., Fu, S., Chung, E.T.: An efficient multiscale finite-element method for frequency-domain seismic wave propagation. Bull. Seismol. Soc. Am. 108(2), 966 (2018)Google Scholar
  32. 32.
    Gao, K.: A high-order multiscale finite-element method for time-domain acoustic-wave modeling. J. Comput. Phys. 360, 120–136 (2018)Google Scholar
  33. 33.
    Gibson, R.L. Jr., Gao, K., Chung, E., Efendiev, Y.: Multiscale modeling of acoustic wave propagation in 2D media. Geophysics 79(2), T61 (2014)Google Scholar
  34. 34.
    Gozani, J., Nachshon, A., Turkel, E.: Conjugate gradient coupled with multigrid for an indefinite problem. Advances in Computer Methods for Partial Differential Equations V, 425–427 (1984)Google Scholar
  35. 35.
    Hiptmair, R., Moiola, A., Perugia, I.: Plane wave discontinuous Galerkin methods for the 2d Helmholtz equation. Analysis of the p-version. SIAM J. Numer. Anal. 49(1), 264–284 (2011)Google Scholar
  36. 36.
    Hudson, J.A., Liu, E., Crampin, S.: The mean transmission properties of a fault with imperfect facial contact. Geophys. J. Int. 129(3), 720–726 (1997)Google Scholar
  37. 37.
    Imbert-Gérard, L.-M., Després, B.: A generalized plane-wave numerical method for smooth nonconstant coefficients. IMA J. Numer. Anal. 34(3), 1072–1103 (2014)Google Scholar
  38. 38.
    Komatitsch, D., Tromp, J.: Introduction to the spectral element method for three-dimensional seismic wave propagation. Geophys. J. Int. 139(3), 806–822 (1999)Google Scholar
  39. 39.
    Komatitsch, D., Tromp, J.: Spectral-element simulations of global seismic wave propagation – I. Validation: Geophys. J. Int. 149(2), 390–412 (2002)Google Scholar
  40. 40.
    Levander, A.R.: Fourth-order finite-difference P-SV seismograms. Geophysics 53(11), 1425–1436 (1988)Google Scholar
  41. 41.
    Li, Y., Métivier, L., Brossier, R., Han, B., Virieux, J.: 2D and 3D frequency-domain elastic wave modeling in complex media with a parallel iterative solver. Geophysics 80(3), T101–T118 (2015)Google Scholar
  42. 42.
    Li, X.S.: An overview of SuperLU. Algorithms, implementation, and user interface. ACM Trans. Math. Softw. 31(3), 302–325 (2005)Google Scholar
  43. 43.
    Lisitsa, V., Vishnevskiy, D.: Lebedev scheme for the numerical simulation of wave propagation in 3D anisotropic elasticity. Geophys. Prospect. 58(4), 619–635 (2010)Google Scholar
  44. 44.
    Martin, R., Komatitsch, D.: An unsplit convolutional perfectly matched layer technique improved at grazing incidence for the viscoelastic wave equation. Geophys. J. Int. 179(1), 333–344 (2009)Google Scholar
  45. 45.
    Melenk, J., Sauter, S.: Wavenumber explicit convergence analysis for Galerkin discretizations of the Helmholtz equation. SIAM J. Numer. Anal. 49(3), 1210–1243 (2011)Google Scholar
  46. 46.
    Moczo, P., Kristek, J., Galis, M., Chaljub, E., Etienne, V.: 3-D finite-difference, finite-element, discontinuous-Galerkin and spectral-element schemes analysed for their accuracy with respect to P-wave to S-wave speed ratio. Geophys. J. Int. 187(3), 1645–1667 (2011)Google Scholar
  47. 47.
    MUMPS: MUMPS – multifrontal massively parallel solver users’ guide, version 5.1.1. http://mumps.enseeiht.fr/(2017)
  48. 48.
    Olson, L.N., Schroder, J.B.: Smoothed aggregation for Helmholtz problems. Numer. Linear Algebra Appl. 17(2-3), 361–386 (2010)Google Scholar
  49. 49.
    Operto, S., Virieux, J., Amestoy, P., L’Excellent, J.-Y., Giraud, L., Ali, H.B.H.: 3D finite-difference frequency-domain modeling of visco-acoustic wave propagation using a massively parallel direct solver. A feasibility study. Geophysics 72(5), SM195–SM211 (2007)Google Scholar
  50. 50.
    Poulson, J., Engquist, B., Li, S., Ying, L.: A parallel sweeping preconditioner for heterogeneous 3D Helmholtz equations. SIAM J. Sci. Comput. 35(3), C194–C212 (2013)Google Scholar
  51. 51.
    Pratt, R.G.: Seismic waveform inversion in the frequency domain, part 1. Theory and verification in a physical scale model. Geophysics 64(3), 888–901 (1999)Google Scholar
  52. 52.
    Saenger, E.H., Gold, N., Shapiro, S.A.: Modeling the propagation of elastic waves using a modified finite-difference grid. Wave Motion 31(1), 77–92 (2000)Google Scholar
  53. 53.
    Shi, L., Zhou, Y., Wang, J. -M., Zhuang, M., Liu, N., Liu, Q.H.: Spectral element method for elastic and acoustic waves in frequency domain. J. Comput. Phys. 327, 19–38 (2016)Google Scholar
  54. 54.
    Singer, I., Turkel, E.: High-order finite difference methods for the Helmholtz equation. Comput. Methods Appl. Mech. Eng. 163(1), 343–358 (1998)Google Scholar
  55. 55.
    Stolk, C.C.: A rapidly converging domain decomposition method for the Helmholtz equation. J. Comput. Phys. 241, 240–252 (2013)Google Scholar
  56. 56.
    Sutmann, G.: Compact finite difference schemes of sixth order for the Helmholtz equation. J. Comput. Appl. Math. 203(1), 15–31 (2007)Google Scholar
  57. 57.
    Thompson, L.L., Pinsky, P.M.: A Galerkin least-squares finite element method for the two-dimensional Helmholtz equation. Int. J. Numer. Methods Eng. 38(3), 371–397 (1995)Google Scholar
  58. 58.
    Tsuji, P., Poulson, J., Engquist, B., Ying, L.: Sweeping preconditioners for elastic wave propagation with spectral element methods. ESAIM Math. Modell. Numer. Anal. 48(2), 433–447 (2014)Google Scholar
  59. 59.
    Turkel, E., Farhat, C., Hetmaniuk, U.: Improved accuracy for the Helmholtz equation in unbounded domains. Int. J. Numer. Methods Eng. 59(15), 1963–1988 (2004)Google Scholar
  60. 60.
    Turkel, E., Gordon, D., Gordon, R., Tsynkov, S.: Compact 2D and 3D sixth order schemes for the Helmholtz equation with variable wave number. J. Comput. Phys. 232(1), 272–287 (2013)Google Scholar
  61. 61.
    Vigh, D., Jiao, K., Watts, D., Sun, D.: Elastic full-waveform inversion application using multicomponent measurements of seismic data collection. Geophysics 79(2), R63–R77 (2014)Google Scholar
  62. 62.
    Virieux, J., Operto, S.: An overview of full-waveform inversion in exploration geophysics. Geophysics 74 (6), WCC1–WCC26 (2009)Google Scholar
  63. 63.
    Virieux, J.: P-SV wave propagation in heterogeneous media. Velocity-stress finite-difference method. Geophysics 51(4), 889–901 (1986)Google Scholar
  64. 64.
    Wang, S., de Hoop, M.V., Xia, J.: On 3D modeling of seismic wave propagation via a structured parallel multifrontal direct Helmholtz solver. Geophys. Prospect. 59(5), 857–873 (2011)Google Scholar
  65. 65.
    Wang, S., Xia, J., de Hoop, M.V., Li, X.S.: Massively parallel structured direct solver for equations describing time-harmonic qP-polarized waves in TTI media. Geophysics 77(3), T69–T82 (2012)Google Scholar
  66. 66.
    Wilcox, L.C., Stadler, G., Burstedde, C., Ghattas, O.: A high-order discontinuous Galerkin method for wave propagation through coupled elastic-acoustic media. J. Comput. Phys. 229(24), 9373–9396 (2010)Google Scholar
  67. 67.
    Yang, D., Teng, J., Zhang, Z., Liu, E.: A nearly analytic discrete method for acoustic and elastic wave equations in anisotropic media. Bull. Seismol. Soc. Am. 93(2), 882–890 (2003)Google Scholar
  68. 68.
    Zhan, Q., Ren, Q., Zhuang, M., Sun, Q., Liu, Q.H.: An exact Riemann solver for wave propagation in arbitrary anisotropic elastic media with fluid coupling. Comput. Methods Appl. Mech. Eng. 329, 24–39 (2018)Google Scholar
  69. 69.
    Zhu, L., Wu, H.: Preasymptotic error analysis of CIP-FEM and FEM for Helmholtz equation with high wave number. Part II. hp version. SIAM J. Numer. Anal. 51(3), 1828–1852 (2013)Google Scholar

Copyright information

© This is a U.S. Government work and not under copyright protection in the US; foreign copyright protection may apply 2019

Authors and Affiliations

  • Shubin Fu
    • 1
  • Kai Gao
    • 2
    Email author
  • Richard L. GibsonJr.
    • 3
  • Eric T. Chung
    • 1
  1. 1.Department of MathematicsThe Chinese University of Hong KongSha TinHong Kong
  2. 2.Geophysics GroupLos Alamos National LaboratoryLos AlamosUSA
  3. 3.Department of Geology and GeophysicsTexas A&M UniversityCollege StationUSA

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