Abstract
This chapter is devoted to the investigation of the dynamic loading of fractured media occurred in many applications like the seismic survey process, non-invasive material quality control, and fatigue failure of samples. Two different approaches were described. The first one is based on the continual model of solid media with a discrete set of slip planes and with nonlinear-type slip conditions at them. 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 used with an explicit approximation of motion equations and an implicit approximation of constitutive relations containing a small parameter. The second one is based on the explicit crack positioning inside the computational grid. With the help of correct contact conditions, unfilled and fluid-filled fractures are described. To solve numerically the elastic system of equations, the grid-characteristic method on structured grids is used. It allows to set precisely all necessary boundary conditions. Both approaches were successfully applied for the simulation of the seismic survey problem in fractured media. Full-wave numerical solutions were obtained and compared. The striker–target interaction problem was simulated using Lagrange grids. Based on the analysis of the dynamic stress state, the process of the new crack initiation was described. The influence of the material strength on it was investigated. Positions of damages were compared with the delamination region.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Golubev, V.I., Muratov, M.V., Petrov, I.B.: Different approaches for solving inverse seismic problems in fractured media. In: Jain, L.C., Favorskaya, M.N., Nikitin, I.S., Reviznikov, D.L. (eds.) Advances in Theory and Practice of Computational Mechanics: Proceedings of the 21st International Conference on Computational Mechanics and Modern Applied Software Systems, SIST, vol. 173, pp. 199–212 (2020)
Beklemysheva, K.A., Kazakov, A.O., Petrov, I.B.: Numerical modeling of ultrasound phased array for non-destructive testing of composites. Lobachevskii J. Math. 40(4), 415–424 (2019)
Beklemysheva, K.A., Golubev, V.I., Vasyukov, A.V., Petrov, I.B.: Numerical modeling of the seismic influence on an underwater composite oil pipeline. Math. Models Comput. Simul. 11(5), 715–721 (2019)
Sadovskii, V.M., Sadovskaya, O.V.: Numerical algorithm based on implicit finite-difference schemes for analysis of dynamic processes in blocky media. Russian J. Numer. Anal. Math. Model. 33(2), 111–121 (2018)
Chentsov, E.P., Sadovskii, V.M., Sadovskaya, O.V.: Modeling of wave processes in a blocky medium with fluid-saturated porous interlayers. AIP Conf. Proc. 1895, 080002 (2017)
Sadovskii, V.M., Sadovskaya, O.V., Lukyanov, A.A.: Modeling of wave processes in blocky media with porous and fluid-saturated interlayers. J. Comput. Phys. 345, 834–855 (2017)
Nikitin, I.S.: Dynamic models of layered and block media with slip, friction and separation. Mech. Solids 43(4), 652–661 (2008)
Burago, N.G., Zhuravlev, A.B., Nikitin, I.S.: Continuum model and method of calculating for dynamics of inelastic layered medium. Math. Models Comput. Simul. 11(3), 59–74 (2019)
Nikitin, I.S., Burago, N.G., Golubev, V.I., Nikitin, A.D.: Mathematical modeling of the dynamics of layered and block media with nonlinear contact conditions on supercomputers. J. Phys.: Conf. Ser. 1392(1), 012057 (2019)
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)
Carcione, J.M., Herman, C.G., Kroode, P.E.: Y2K Review article: seismic modeling. Rev. Lit. Arts Am. 67(4), 1304–1325 (2002)
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)
Massau, J.: Memoire sur L’integration Graphique des E ́quations aux Deriv ́ess Partielles. F. Meyer-van Loo, Ghent (1899)
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)
Butler, D.S.: The numerical solution of hyperbolic systems of partial differential equations of three independent variables. Proc. R. Soc. London Ser. A 255(1281), 232–241 (1960)
Golubev, V.I., Shevchenko, A.V., Petrov, I.B.: Taking into account fluid saturation of bottom sediments in marine seismic survey. Doklady Math. 100(2), 488–490 (2019)
Yabe, T., Aoki, T., Sakaguchi, G., Wang, P.Y., Ishikawa, T.: The compact CIP (Cubic-Interpolated Pseudo-Particle) method as a general hyperbolic solver. Comput. Fluids 19, 421–431 (1991)
Azarova, O.A.: Complex conservative difference schemes for computing supersonic flows past simple aerodynamic forms. Comput. Math. Math. Phys. 55, 2025–2049 (2015)
Sakhalkar, S.V., Bogy, D.B.: A model for lubricant transfer from media to head during Heat-Assisted Magnetic Recording (HAMR) writing. Tribol. Lett. 65, art. no. 166 (2017)
Mittal, H.V.R., Ray, R.K.: Solving immersed interface problems using a new interfacial points-based finite difference approach. SIAM J. Sci. Comput. 40(3), A1860–A1883 (2018)
Mittal, H.V.R., Kalita, J.C., Ray, R.K.: A class of finite-difference schemes for interface problems with an HOC approach. Int. J. Numer. Methods Fluids 82(9), 567–606 (2016)
Golubev, V.I., Khokhlov, N.I., Nikitin, I.S., Churyakov, M.A.: Application of compact grid-characteristic schemes for acoustic problems. J. Phys.: Conf. Ser. 1479, 012058.1–012058.11 (2020)
Nikitin, I.S., Burago, N.G., Golubev, V.I., Nikitin, A.D.: Continual models of layered and block media with slippage and delamination. Procedia Struct. Integr. 23, 125–130 (2019)
Nikitin, I.S., Burago, N.G., Golubev, V.I., Nikitin, A.D.: Methods for calculating the dynamics of layered and block media with nonlinear contact conditions. In: Jain, L.C., Favorskaya, M.N., Nikitin, I.S., Reviznikov, D.L. (eds.) Advances in Theory and Practice of Computational Mechanics: Proceedings of the 21st International Conference on Computational Mechanics and Modern Applied Software Systems, SIST, vol. 173, pp. 171–183 (2020)
Khokhlov, N., Stognii, P.: Novel approach to modeling the seismic waves in the areas with complex fractured geological structures. Minerals 10, 122–139 (2020)
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
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Nikitin, I.S., Golubev, V.I., Golubeva, Y.A., Miryakha, V.A. (2021). Numerical Comparison of Different Approaches for the Fractured Medium Simulation. In: Favorskaya, M.N., Favorskaya, A.V., Petrov, I.B., Jain, L.C. (eds) Smart Modelling For Engineering Systems. Smart Innovation, Systems and Technologies, vol 214. Springer, Singapore. https://doi.org/10.1007/978-981-33-4709-0_8
Download citation
DOI: https://doi.org/10.1007/978-981-33-4709-0_8
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-33-4708-3
Online ISBN: 978-981-33-4709-0
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)