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Methods for Calculating the Dynamics of Layered and Block Media with Nonlinear Contact Conditions

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Advances in Theory and Practice of Computational Mechanics

Part of the book series: Smart Innovation, Systems and Technologies ((SIST,volume 173))

Abstract

Continual models of solid media with a discrete set of slip planes (layered, block media) and with nonlinear type slip conditions at the contact boundaries of structural elements are constructed. The constitutive equations of the resulting system of equations contain a small viscosity parameter in the denominator of nonlinear free terms. For a stable numerical solution of a system of differential equations, an explicit–implicit method is proposed with the explicit approximation of the equations of motion and implicit approximation of the constitutive relations containing a small parameter. From implicit nonlinear difference approximations analytically using the perturbation method, various effective formulas for the correction of stress components after an “elastic” time step are obtained. To calculate the “elastic” step, we used a grid-characteristic method on hexahedral grids, which allowed us to increase significantly the speed of calculations and simulate a non-stationary three-dimensional problem of generating a response from an oriented layered or block cracked cluster located in a homogeneous medium.

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References

  1. Nikitin, I.S.: Dynamic models of layered and block media with slip, friction and separation. Mech. Solids 43(4), 652–661 (2008)

    Article  Google Scholar 

  2. Burago, N.G., Zhuravlev, A.B., Nikitin, I.S.: Continuum model and method of calculating for dynamics of inelastic layered medium. Math. Model. Comput. Simul. 11(3), 59–74 (2019)

    Article  MathSciNet  Google Scholar 

  3. Burago, N.G., Nikitin, I.S.: Improved model of a layered medium with slip on the contact boundaries. J. Appl. Math. Mech. 80(2), 164–172 (2016)

    Article  MathSciNet  Google Scholar 

  4. Nikitin, I.S., Burago, N.G., Nikitin, A.D.: Continuum model of the layered medium with slippage and nonlinear conditions at the interlayer boundaries. Solid State Phenom. 258, 137–140 (2017)

    Article  Google Scholar 

  5. Virieux, J., Calandra, H., Plessix, R.E: A review of the spectral, pseudo-spectral, finite-difference and finite-element modelling techniques for geophysical imaging. Geophys. Prospect. 59(5), 794–813 (2011)

    Google Scholar 

  6. Carcione, J.M., Herman, C.G., Kroode, A.P.E: Seismic modeling. Geophysics 67(4), 1304–1325 (2002)

    Google Scholar 

  7. Burago, N.G., Nikitin, I.S., Yakushev, V.L.: Hybrid numerical method with adaptive overlapping meshes for solving nonstationary problems in continuum mechanics. Comput. Math. Math. Phys. 56(6), 1065–1074 (2016)

    Article  MathSciNet  Google Scholar 

  8. Lisitsa, V., Tcheverda, V., Botter, C.: Combination of the discontinuous Galerkin method with finite differences for simulation of seismic wave propagation. J. Comput. Phys. 311, 142–157 (2016)

    Article  MathSciNet  Google Scholar 

  9. Massau, J.: Memoire sur L’integration Graphique des Equations aux Derivess Partielles. F. Meyer-van Loo, Ghent (1899)

    MATH  Google Scholar 

  10. Zhukov, A.I.: Using the Method of Characteristics for the numerical solution of one-dimensional problems of gas dynamics. Tr. Mat. Inst. Akad. Nauk SSSR 58, 4–150 (1960)

    Google Scholar 

  11. Butler, D.S.: The numerical solution of hyperbolic systems of partial differential equations of three independent variables. Proc. Roy. Soc. London. Ser. A 255(1281), 232–241 (1960)

    Article  MathSciNet  Google Scholar 

  12. Beklemysheva, K.A., Vasyukov, A.V., Golubev, V.I., Zhuravlev, Y.I.: On the estimation of seismic resistance of modern composite oil pipeline elements. Dokl. Math. 97(2), 184–187 (2018)

    Article  MathSciNet  Google Scholar 

  13. Golubev, V.I., Golubeva, YuA: Full-wave simulation of the earthquake initiation process. CEUR Work. Proc. 2267, 346–350 (2018)

    Google Scholar 

  14. Golubev, V., Khokhlov, N., Grigorievyh, D., Favorskaya, A.: Numerical simulation of destruction processes by the grid-characteristic method. Procedia Comput. Sci. 126, 1281–1288 (2018)

    Article  Google Scholar 

  15. Muratov, M.V., Petrov, I.B.: Application of fractures mathematical models in exploration seismology problems modeling. In: Petrov, I.B., Favorskaya, A.V., Favorskaya, M.N., Simakov, S.S., Jain, L.C. (eds.) Proceedings of the Conference on 50 years of the Development of Grid-characteristic Method, SIST, vol. 133, pp. 120–131. Springer, Berlin (2019)

    Google Scholar 

  16. Favorskaya, A.V., Zhdanov, M.S., Khokhlov, N.I., Petrov, I.B.: Modeling the wave phenomena in acoustic and elastic media with sharp variations of physical properties using the grid-characteristic method. Geophys. Prospect. 66(8), 1485–1502 (2018)

    Article  Google Scholar 

  17. Favorskaya, A.V., Kabisov, S.V., Petrov, I.B.: Modeling of ultrasonic waves in fractured rails with an explicit approach. Dokl. Math. 98(1), 401–404 (2018)

    Article  Google Scholar 

  18. Golubev, V.I.: The usage of grid-characteristic method in seismic migration problems. In: Petrov, I.B., Favorskaya, A.V., Favorskaya, M.N., Simakov, S.S., Jain, L.C. (eds.) Proceedings of the Conference on 50 years of the Development of Grid-characteristic Method, SIST, vol. 133, pp. 143–155. Springer, Berlin (2019)

    Google Scholar 

  19. Golubev, V.I., Voinov, O.Y., Petrov, I.B.: Migration of seismic data for multi-layered fractured geological media using elastic approach. In: 19th Science and Applied Research Conference Oil and Gas Geological Exploration and Development, pp. 43756.1–43756.5 (in Russian) (2017)

    Google Scholar 

  20. Khokhlov, N.I., Golubev, V.I.: On the class of compact grid-characteristic schemes. In: Petrov, I.B., Favorskaya, A.V., Favorskaya, M.N., Simakov, S.S., Jain, L.C. (eds.) Proceedings of the Conference on 50 years of the Development of Grid-characteristic Method, SIST, vol. 133, pp. 64–77. Springer, Berlin (2019)

    Google Scholar 

  21. Ivanov, A.M., Khokhlov, N.I.: Efficient inter-process communication in parallel implementation of grid-characteristic method. In: Petrov, I.B., Favorskaya, A.V., Favorskaya, M.N., Simakov, S.S., Jain, L.C. (eds.) Proceedings of the Conference on 50 years of the Development of Grid-characteristic Method, SIST, vol. 133, pp. 91–102. Springer, Berlin (2019)

    Google Scholar 

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Acknowledgements

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”, http://ckp.nrcki.ru/. This work was carried out with the financial support of the Russian Science Foundation, project no. 19-71-10060.

Author information

Authors and Affiliations

  1. Institute of Computer Aided Design of the RAS, 19/18, Vtoraya Brestskaya ul., Moscow, 123056, Russian Federation

    Ilia S. Nikitin & Alexander D. Nikitin

  2. Ishlinsky Institute for Problems in Mechanics of the RAS, 101, b1, pr. Vernadskogo, Moscow, 119526, Russian Federation

    Nikolay G. Burago

  3. Moscow Institute of Physics and Technology (National Research University), 9, Institutsriy per., Dolgoprudny, Moscow Region, 141700, Russian Federation

    Vasily I. Golubev

Authors
  1. Ilia S. Nikitin
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  2. Nikolay G. Burago
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  3. Vasily I. Golubev
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  4. Alexander D. Nikitin
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Corresponding author

Correspondence to Ilia S. Nikitin .

Editor information

Editors and Affiliations

  1. University of Technology Sydney, Sydney, Australia

    Lakhmi C. Jain

  2. Department of Informatics and Computer Techniques, Institute of Informatics and Telecommunications, Reshetnev Siberian State University of Science and Technology, Krasnoyarsk, Russia

    Margarita N. Favorskaya

  3. Institute of Computer Aided Design of the Russian Academy of Sciences (ICAD RAS), Moscow, Russia

    Ilia S. Nikitin

  4. Department of Numerical Mathematics and Computer Programming, Moscow Aviation Institute (National Research University), Moscow, Russia

    Dmitry L. Reviznikov

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Nikitin, I.S., Burago, N.G., Golubev, V.I., Nikitin, A.D. (2020). Methods for Calculating the Dynamics of Layered and Block Media with Nonlinear Contact Conditions. In: Jain, L., Favorskaya, M., Nikitin, I., Reviznikov, D. (eds) Advances in Theory and Practice of Computational Mechanics. Smart Innovation, Systems and Technologies, vol 173. Springer, Singapore. https://doi.org/10.1007/978-981-15-2600-8_13

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