Skip to main content
SpringerLink
Log in

Boundary Conforming Chimera Meshes to Account for Surface Topography and Curved Interfaces in Geological Media

  • Published:
Lobachevskii Journal of Mathematics Aims and scope Submit manuscript

Abstract

Nowadays, solving seismic exploration problems requires taking into account the scattering of seismic waves at boundaries and contact boundaries of complex shape. For the subsequent solution of inverse problems, it is important to save computing resources as much as possible. A way to do this while saving the accuracy of calculations is to use Chimera or overlapping computational meshes. In this way, the calculation is carried out in a background Cartesian grid, and curvilinear structured boundary conforming Chimera meshes are constructed along the boundaries and contact boundaries. Interpolation is carried out between the Cartesian and Chimera curvilinear grids. The issue arises of the optimal way to generate this type of computational grids when solving this class of problems. In this work, we consider 5 types of generation the Chimera computational grids for various boundary geometries and present the results of a comparative analysis in dependence of various parameters, such as the maximal applicable Courant number, calculation speed, numerical error, numerical order of convergence, etc.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22

REFERENCES

  1. V. Lisitsa, V. Tcheverda, and C. Botter, ‘‘Combination of the discontinuous Galerkin method with finite differences for simulation of seismic wave propagation,’’ J. Comput. Phys. 311, 142–157 (2016). https://doi.org/10.1016/j.jcp.2016.02.005

    Article  MathSciNet  Google Scholar 

  2. I. B. Petrov and A. S. Kholodov, ‘‘Numerical investigation of certain dynamical problems of the mechanics of a deformable solid body by the grid-characteristic method,’’ Zh. Vychisl. Mat. Mat. Fiz. 24, 722–739 (1984). https://doi.org/10.1016/0041-5553(84)90044-2

    Article  MathSciNet  Google Scholar 

  3. N. Khokhlov, A. Favorskaya, V. Stetsyuk, and I. Mitskovets, ‘‘Grid-characteristic method using Chimera meshes for simulation of elastic waves scattering on geological fractured zones,’’ J. Comput. Phys. 446, 110637 (2021). https://doi.org/10.1016/j.jcp.2021.110637

  4. A. V. Favorskaya, N. I. Khokhlov, and I. B. Petrov, ‘‘Grid-characteristic method on joint structured regular and curved grids for modeling coupled elastic and acoustic wave phenomena in objects of complex shape,’’ Lobachevskii J. Math. 41, 512–525 (2020). https://doi.org/10.1134/S1995080220040083

    Article  MathSciNet  Google Scholar 

  5. A. V. Favorskaya, M. S. Zhdanov, N. I. Khokhlov, and I. B. Petrov, ‘‘Modeling the wave phenomena in acoustic and elastic media with sharp variations of physical properties using the grid-characteristic method,’’ Geophys. Prospect. 66, 1485–1502 (2018). https://doi.org/10.1111/1365-2478.12639

    Article  Google Scholar 

  6. B. Poursartip, A. Fathi, and J. L. Tassoulas, ‘‘Large-scale simulation of seismic wave motion: A review,’’ Soil Dyn. Earthquake Eng. 129, 105909 (2020). https://doi.org/10.1016/j.soildyn.2019.105909

  7. Q. Li, G. Wu, J. Wu, and P. Duan, ‘‘Finite difference seismic forward modeling method for fluid-solid coupled media with irregular seabed interface,’’ J. Geophys. Eng. 16, 198–214 (2019). https://doi.org/10.1093/jge/gxy017

    Article  Google Scholar 

  8. W. Zhang, Z. Zhang, and X. Chen, ‘‘Three-dimensional elastic wave numerical modelling in the presence of surface topography by a collocated-grid finite-difference method on curvilinear grids,’’ Geophys. J. Int. 190, 358–378 (2012). https://doi.org/10.1111/j.1365-246X.2012.05472.x

    Article  Google Scholar 

  9. Y. C. Sun, H. Ren, X. Z. Zheng, N. Li, W. Zhang, Q. Huang, and X. Chen, ‘‘2-D poroelastic wave modelling with a topographic free surface by the curvilinear grid finite-difference method,’’ Geophys. J. Int. 218, 1961–1982 (2019). https://doi.org/10.1093/gji/ggz263

    Article  Google Scholar 

  10. L. Zhang, S. Wang, and N. A. Petersson, ‘‘Elastic wave propagation in curvilinear coordinates with mesh refinement interfaces by a fourth order finite difference method,’’ SIAM J. Sci. Comput. 43, A1472–A1496 (2021). https://doi.org/10.1137/20M1339702

    Article  MathSciNet  Google Scholar 

  11. Y. C. Sun, W. Zhang, and X. Chen, ‘‘Seismic-wave modeling in the presence of surface topography in 2D general anisotropic media by a curvilinear grid finite-difference method,’’ Bull. Seismol. Soc. Am. 106, 1036–1054 (2016). https://doi.org/10.1785/0120150285

    Article  Google Scholar 

  12. O. O’Reilly, T. Y. Yeh, K. B. Olsen, Z. Hu, A. Breuer, D. Roten, and C. A. Goulet, ‘‘A high-order finite-difference method on staggered curvilinear grids for seismic wave propagation applications with topography,’’ Bull. Seismol. Soc. Am. 112, 3–22 (2022). https://doi.org/10.1785/0120210096

    Article  Google Scholar 

  13. K. Wang, S. Peng, Y. Lu, and X. Cui, ‘‘The velocity-stress finite-difference method with a rotated staggered grid applied to seismic wave propagation in a fractured medium,’’ Geophys. 85, T89–T100 (2020). https://doi.org/10.1190/geo2019-0186.1

    Article  Google Scholar 

  14. C. P. Solano, D. Donno, and H. Chauris, ‘‘Finite-difference strategy for elastic wave modelling on curved staggered grids,’’ Comput. Geosci. 20, 245–264 (2016). https://doi.org/10.1007/s10596-016-9561-8

    Article  MathSciNet  Google Scholar 

  15. W. Boscheri, M. Dumbser, M. Ioriatti, I. Peshkov, and E. Romenski, ‘‘A structure-preserving staggered semi-implicit finite volume scheme for continuum mechanics,’’ J. Comput. Phys. 424, 109866 (2021). https://doi.org/10.1016/j.jcp.2020.109866

  16. Y. Qu, J. Zhang, S. Eisentrager, and C. Song, ‘‘A time-domain approach for the simulation of three-dimensional seismic wave propagation using the scaled boundary finite element method,’’ Soil Dyn. Earthquake Eng. 152, 107011 (2022). https://doi.org/10.1016/j.soildyn.2021.107011

  17. C. Hong, X. Wang, G. Zhao, Z. Xue, F. Deng, Q. Gu, Z. Song, L. Yuan, X. Meng, S. Liang, and S. Yang, ‘‘Discontinuous finite element method for efficient three-dimensional elastic wave simulation,’’ J. Geophys. Eng. 18, 98–112 (2021). https://doi.org/10.1093/jge/gxaa070

    Article  Google Scholar 

  18. P. F. Antonietti, I. Mazzieri, and F. Migliorini, ‘‘A space-time discontinuous Galerkin method for the elastic wave equation,’’ J. Comput. Phys. 419, 109685 (2020). https://doi.org/10.1016/j.jcp.2020.109685

  19. Y. Xu, X. Chen, W. Zhang, and X. Pan, ‘‘An adaptive modal discontinuous Galerkin finite element parallel method using unsplit multi-axial perfectly matched layer for seismic wave modeling,’’ Commun. Comput. Phys. 31, 1083–1113 (2022). https://doi.org/10.4208/cicp.OA-2021-0118

    Article  MathSciNet  Google Scholar 

  20. X. He, D. Yang, C. Qiu, Y. Zhou, and X. Ma, ‘‘An efficient discontinuous Galerkin method using a tetrahedral mesh for 3D seismic wave modeling,’’ Bull. Seismol. Soc. Am. 112, 1197–1223 (2022). https://doi.org/10.1785/0120210229

    Article  Google Scholar 

  21. S. Liu, D. Yang, X. Xu, X. Li, W. Shen, and Y. Liu, ‘‘Three-dimensional element-by-element parallel spectral-element method for seismic wave modeling,’’ Chin. J. Geophys. 64, 993–1005 (2021). https://doi.org/10.6038/cjg2021O0405

    Article  Google Scholar 

  22. E. D. Mercerat, J. P. Vilotte, and F. J. Sanchez-Sesma, ‘‘Triangular spectral element simulation of two-dimensional elastic wave propagation using unstructured triangular grids,’’ Geophys. J. Int. 166, 679–698 (2006). https://doi.org/10.1111/j.1365-246X.2006.03006.x

    Article  Google Scholar 

  23. J. Liu, E. Kausel, and X. W. Liu, ‘‘Using pseudo-spectral method on curved grids for SH-wave modeling of irregular free-surface,’’ J. Appl. Geophys. 140, 42–51 (2017). https://doi.org/10.1016/j.jappgeo.2017.03.004

    Article  Google Scholar 

  24. I. E. Kvasov and I. B. Petrov, ‘‘High-performance computer simulation of wave processes in geological media in seismic exploration,’’ Comput. Math. Math. Phys. 52, 302–313 (2012). https://doi.org/10.1134/S096554251202011X

    Article  Google Scholar 

  25. A. Gupta, R. Sharma, A. Thakur, and P. Gulia, ‘‘Metamaterial foundation for seismic wave attenuation for low and wide frequency band,’’ Sci. Rep. 13, 2293 (2023). https://doi.org/10.1038/s41598-023-27678-1

    Article  Google Scholar 

  26. N. I. Khokhlov, A. Favorskaya, and V. Furgailo ‘‘Grid-characteristic method on overlapping curvilinear meshes for modeling elastic waves scattering on geological fractures,’’ Minerals 12, 1597 (2022). https://doi.org/10.3390/min12121597

    Article  Google Scholar 

  27. A. Favorskaya and N. Khokhlov, ‘‘Accounting for curved boundaries in rocks by using curvilinear and Chimera grids,’’ Proc. Comput. Sci. 192, 3787–3794 (2021). https://doi.org/10.1016/j.procs.2021.09.153

    Article  Google Scholar 

  28. I. B. Petrov and A. V. Favorskaya, ‘‘Joint modeling of wave phenomena by applying the grid-characteristic method and the discontinuous Galerkin method,’’ Dokl. Math. 106, 356–360 (2022). https://doi.org/10.1134/S1064562422050179

    Article  MathSciNet  Google Scholar 

  29. A. V. Favorskaya and I. B. Petrov, ‘‘Combination of grid-characteristic method on regular computational meshes with discontinuous Galerkin method for simulation of elastic wave propagation,’’ Lobachevskii J. Math. 42, 1652–1660 (2021). https://doi.org/10.1134/S1995080221070076

    Article  MathSciNet  Google Scholar 

  30. W. Chan, ‘‘Overset grid technology development at NASA Ames Research Center,’’ Comput. Fluids 38, 496–503 (2009). https://doi.org/10.1016/j.compfluid.2008.06.009

    Article  Google Scholar 

  31. U. M. Mayer, A. Popp, A. Gerstenberger, and W. A. Wall, ‘‘3D fluid-structure-contact interaction based on a combined XFEM FSI and dual mortar contact approach,’’ Comput. Mech. 46, 53–67 (2010). https://doi.org/10.1007/s00466-010-0486-0

    Article  MathSciNet  Google Scholar 

  32. Y. Zhang, S. C. Yim, and F. Del Pin, ‘‘A nonoverlapping heterogeneous domain decomposition method for three-dimensional gravity wave impact problems,’’ Comput. Fluids 106, 154–170 (2015). https://doi.org/10.1016/j.compfluid.2014.09.005

    Article  MathSciNet  Google Scholar 

  33. V. T. Nguyen, D. T. Vu, W. G. Park, and C. M. Jung, ‘‘Navier–Stokes solver for water entry bodies with moving Chimera grid method in 6DOF motions,’’ Comput. Fluids 140, 19–38 (2016). https://doi.org/10.1016/j.compfluid.2016.09.005

    Article  MathSciNet  Google Scholar 

  34. M. Discacciati, B. J. Evans, and M. Giacomini, ‘‘An overlapping domain decomposition method for the solution of parametric elliptic problems via proper generalized decomposition,’’ Comput. Methods Appl. Mech. Eng. 418, 116484 (2024). https://doi.org/10.1016/j.cma.2023.116484

  35. A. Sharma, S. Ananthan, J. Sitaraman, S. Thomas, and M. A. Sprague, ‘‘Overset meshes for incompressible flows: On preserving accuracy of underlying discretizations,’’ J. Comput. Phys. 428, 109987 (2021). https://doi.org/10.1016/j.jcp.2020.109987

  36. J. Huang, T. Hu, J. Song, Y. Li, Z. Yu, and L. Liu, ‘‘A novel hybrid method based on discontinuous Galerkin method and staggered-grid method for scalar wavefield modelling with rough topography,’’ Geophys. Prospect. 70, 441–458 (2022). https://doi.org/10.1111/1365-2478.13171

    Article  Google Scholar 

  37. J. Vamaraju, M. K. Sen, J. de Basabe, and M. Wheeler, ‘‘A hybrid Galerkin finite element method for seismic wave propagation in fractured media,’’ Geophys. J. Int. 221, 857–878 (2020). https://doi.org/10.1093/gji/ggaa037

    Article  Google Scholar 

  38. E. A. Pesnya, A. V. Favorskaya, and A. A. Kozhemyachenko, ‘‘Implicit hybrid grid-characteristic method for modeling dynamic processes in acoustic medium,’’ Lobachevskii J. Math. 43, 1032–1042 (2022). https://doi.org/10.1134/S1995080222070204

    Article  MathSciNet  Google Scholar 

  39. A. Kozhemyachenko, A. Favorskaya, E. Pesnya, and V. Stetsyuk, ‘‘Modification of the grid-characteristic method on Chimera meshes for 3D problems of railway non-destructive testing,’’ Lobachevskii J. Math. 44, 376–386 (2023). https://doi.org/10.1134/S1995080223010262

    Article  MathSciNet  Google Scholar 

  40. A. V. Favorskaya and N. Khokhlov, ‘‘Using Chimera grids to describe boundaries of complex shape,’’ Smart Innov. Syst. Technol. 309, 249–258 (2022). https://doi.org/10.1007/978-981-19-3444-5_22

    Article  Google Scholar 

  41. V. Golubev, A. V. Shevchenko, and I. B. Petrov, ‘‘Raising convergence order of grid-characteristic schemes for 2D linear elasticity problems using operator splitting,’’ Comput. Res. Model. 14, 899–910 (2022). https://doi.org/10.20537/2076-7633-2022-14-4-899-910

    Article  Google Scholar 

  42. A. V. Favorskaya, ‘‘Simulation of the human head ultrasound study by grid-characteristic method on analytically generated curved meshes,’’ Smart Innov. Syst. Technol. 214, 249–263 (2021). https://doi.org/10.1007/978-981-33-4709-0_21

    Article  Google Scholar 

Download references

ACKNOWLEDGMENTS

This work has been carried out using computing resources of the federal collective usage center Complex for Simulation and Data Processing for Mega-science Facilities at NRC ‘‘Kurchatov Institute,’’ https://ckp.nrcki.ru/.

Funding

The research was supported by the Russian Science Foundation grant no. 20-71-10028, https://rscf.ru/project/20-71-10028/.

Author information

Authors and Affiliations

  1. Moscow Institute of Physics and Technology, 141701, Dolgoprudny, Moscow oblast, Russia

    A. V. Favorskaya, N. I. Khokhlov, V. I. Golubev & A. V. Shevchenko

Authors
  1. A. V. Favorskaya
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. I. Khokhlov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. V. I. Golubev
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. A. V. Shevchenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to A. V. Favorskaya, N. I. Khokhlov, V. I. Golubev or A. V. Shevchenko.

Ethics declarations

The authors of this work declare that they have no conflicts of interest.

Additional information

Publisher’s Note.

Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

(Submitted by I. B. Petrov)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Favorskaya, A.V., Khokhlov, N.I., Golubev, V.I. et al. Boundary Conforming Chimera Meshes to Account for Surface Topography and Curved Interfaces in Geological Media. Lobachevskii J Math 45, 191–212 (2024). https://doi.org/10.1134/S1995080224010141

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1995080224010141

Keywords:

Search

Navigation