DOI: https://doi.org/10.26089/NumMet.v22r104

One-dimensional finite difference schemes for splitting method realization in axisymmetric equations of the dynamics of elastic medium

Authors

  • V.M. Sadovskii
  • O.V. Sadovskaya
  • E.A. Efimov

Keywords:

elastic medium
direct seismic problem
cylindrical waves
finite difference scheme
splitting method
monotonicity
dissipativity
parallel computing

Abstract

We construct efficient finite difference shock-capturing schemes for the solution of direct seismic problems in axisymmetric formulation. When parallelizing the algorithms implementing the schemes on multiprocessor computing systems, the two-cyclic splitting method with respect to the spatial variables is used. One-dimensional systems of equations are solved at the stages of splitting on the basis of explicit gridcharacteristic schemes and an implicit finite difference scheme of the “predictor–corrector” type with controllable artificial energy dissipation. The verification of algorithms and programs is fulfilled on the exact solutions of one-dimensional problems describing traveling monochromatic waves. The comparison of the results showed the advantages of the scheme with controllable energy dissipation in terms of the accuracy of computing smooth solutions and the advisability of application of explicit monotone schemes when calculating discontinuities.

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Published

2021-03-11

Issue

Vol. 22 (2021): Issue 1.

Section

Parallel software tools and technologies

Author Biographies

V.M. Sadovskii

Institute of Computational Modeling of SB RAS (ICM SB RAS)
• Director

O.V. Sadovskaya

Institute of Computational Modeling of SB RAS (ICM SB RAS)
• Senior Researcher

E.A. Efimov

Siberian Federal University,
School of Mathematics and Computer Science
• PhD Student

References

  1. V. M. Sadovskii, O. V. Sadovskaya, and E. A. Efimov, “Analysis of Seismic Waves Excited in Near-Surface Soils by Means of the Electromagnetic Pulse Source ’Yenisei’,” Mater. Phys. Mech. 42 (5), 544-557 (2019).
  2. O. V. Sadovskaya and V. M. Sadovskii, “Supercomputing Analysis of Seismic Efficiency of the Electromagnetic Pulse Source ’Yenisei’,” in AIP Conf. Proc. (American Institute of Physics, College Park, 2019), Vol. 2164, Iss. 1, 110011-1-110011-9.
  3. E. A. Efimov, V. M. Sadovskii, and O. V. Sadovskaya, “Mathematical Modeling of the Impact of a Pulse Seismic Source on Geological Media,” in AIP Conf. Proc. (American Institute of Physics, College Park, 2020), Vol. 2302, Iss. 1, 120002-1-120002-8.
  4. S. K. Godunov, A. V. Zabrodin, M. Ya. Ivanov, et al., Numerical Solution of Multidimensional Problems of Gas Dynamics (Nauka, Moscow, 1976) [in Russian].
  5. S. K. Godunov and V. S. Ryabenkii, Difference Schemes (Nauka, Moscow, 1976; North-Holland, Amsterdam, 1987).
  6. O. V. Sadovskaya and V. M. Sadovskii, “Parallel Implementation of an Algorithm for the Computation of Elasto-Plastic Waves in a Granular Medium,” Vychisl. Metody Programm. 6, 209-216 (2005).
  7. O. Sadovskaya and V. Sadovskii, Mathematical Modeling in Mechanics of Granular Materials (Fizmatlit, Moscow, 2008; Springer, Heidelberg, 2012).
  8. O. V. Sadovskaya and V. M. Sadovskii, “Numerical Implementation of Mathematical Model of the Dynamics of a Porous Medium on Supercomputers of Cluster Architecture,” in AIP Conf. Proc. (American Institute of Physics, College Park, 2015), Vol. 1684, Iss. 1, 070005-1-070005-9.
  9. V. M. Sadovskii and O. V. Sadovskaya, “Analyzing the Deformation of a Porous Medium with Account for the Collapse of Pores,” Zh. Prikl. Mekh. Tekh. Fiz. 57 (5), 53-65 (2016) [J. Appl. Mech. Tech. Phys. 57 (5), 808-818 (2016)].
  10. M. P. Varygina, M. A. Pokhabova, O. V. Sadovskaya, and V. M. Sadovskii, “Numerical Algorithms for the Analysis of Elastic Waves in Block Media with Thin Interlayers,” Vychisl. Metody Programm. 12 (4), 435-442 (2011).
  11. V. M. Sadovskii and O. V. Sadovskaya, “Modeling of Elastic Waves in a Blocky Medium Based on Equations of the Cosserat Continuum,” Wave Motion 52, 138-150 (2015).
  12. V. M. Sadovskii, O. V. Sadovskaya, and A. A. Lukyanov, “Modeling of Wave Processes in Blocky Media with Porous and Fluid-Saturated Interlayers,” J. Comput. Phys. 345, 834-855 (2017).
  13. V. M. Sadovskii and O. V. Sadovskaya, “Supercomputer Modeling of Wave Propagation in Blocky Media Accounting Fractures of Interlayers,” in Nonlinear Wave Dynamics of Materials and Structures (Springer, Cham, 2020), Vol. 122, pp. 379-398.
  14. P. F. Sabodash and R. A. Cherednichenko, “Application of the Method of Spatial Characteristics to the Solution of Axially Symmetric Problems Relating to the Propagation of Elastic Waves,” Zh. Prikl. Mekh. Tekh. Fiz. 12 (4), 101-109 (1971) [J. Appl. Mech. Tech. Phys. 12 (4), 571-577 (1971)].
  15. K. M. Magomedov and A. S. Kholodov, Grid-Characteristic Numerical Methods (Nauka, Moscow, 1988) [in Russian].
  16. I. B. Petrov and A. S. Kholodov, “Numerical Study of Some Dynamic Problems of the Mechanics of a Deformable Rigid Body by the Mesh-Characteristic Method,” Zh. Vychisl. Mat. Mat. Fiz. 24 (5), 722-739 (1984) [USSR Comput. Math. Math. Phys. 24 (3), 61-73 (1984)].
  17. I. B. Petrov, A. G. Tormasov, and A. S. Kholodov, “On the Use of Hybrid Grid-Characteristic Schemes for the Numerical Solution of Three-Dimensional Problems in the Dynamics of a Deformable Solid,” Zh. Vychisl. Mat. Mat. Fiz. 30 (8), 1237-1244 (1990) [USSR Comput. Math. Math. Phys. 30 (4), 191-196 (1990)].
  18. V. N. Kukudzhanov, “On Numerical Solution of the Problems of Propagation of Elasto-Viscoplastic Waves,” in Propagation of Elastic and Elastoplastic Waves (Nauka, Alma-Ata, 1973), pp. 223-230.
  19. V. N. Kukudzhanov, Difference Methods for the Solution of Problems of Mechanics of Deformable Media (MFTI, Moscow, 1992) [in Russian].
  20. V. N. Kukudzhanov, Numerical Continuum Mechanics (Fizmatlit, Moscow, 2008; De Gruyter, Berlin, 2013).
  21. N. G. Burago, I. S. Nikitin, and A. B. Zhuravlev, “Continuum Model and Method of Calculating for Dynamics of Inelastic Layered Medium,” Mat. Model. 30 (11), 59-74 (2018) [Math. Models Comput. Simul. 11 (13), 488-498 (2019)].
  22. I. S. Nikitin, N. G. Burago, and A. D. Nikitin, “Explicit-Implicit Schemes for Solving the problems of the Dynamics of Isotropic and Anisotropic Elastoviscoplastic Media,” J. Phys.: Conf. Ser. 1158, 032039-1-032039-8 (2019).
  23. I. S. Nikitin, Theory of Inelastic Layered and Blocky Media (Fizmatlit, Moscow, 2019) [in Russian].
  24. N. G. Burago and V. N. Kukudzhanov, “A Numerical Method for Solving Axisymmetric Problems for Geometrically Nonlinear Elastic-Plastic Shells of Revolution,” Stroit. Mekh. Raschet Sooruzh., N 5, 44-49 (1976).
  25. N. G. Burago and V. N. Kukudzhanov, “Buckling and Supercritical Deformations of Elastic-Plastic Shells under Axial Symmetry,” in Collection of Numerical Methods in the Mechanics of Deformable Solids (Computing Center of the USSR Academy of Sciences, Moscow, 1978), pp. 47-66.
  26. V. G. Bazhenov and E. V. Igonicheva, Nonlinear Processes of Shock Buckling of Elastic Structural Elements in the Form of Orthotropic Shells of Rotation (Publishing House of UNN, N. Novgorod, 1991) [in Russian].
  27. V. G. Bazhenov and D. T. Chekmarev, Solving Problems of Unsteady Dynamics of Plates and Shells by the Variational-Difference Method (Publishing House of UNN, N. Novgorod, 2000) [in Russian].
  28. V. G. Bazhenov and V. L. Kotov, Mathematical Modeling of Non-Stationary Processes of Impact and Penetration of Axisymmetric Bodies and Identification of Soil Media Properties (Fizmatlit, Moscow, 2011) [in Russian].
  29. G. I. Marchuk, “Splitting and Alternating Direction Methods,” in Handbook of Numerical Analysis (Elsevier, North-Holland, 1990), Vol. 1, pp. 197-462.
  30. A. G. Kulikovskii, N. V. Pogorelov, and A. Yu. Semenov, Mathematical Aspects of Numerical Solution of Hyperbolic Systems (Fizmatlit, Moscow, 2001; CRC Press, Boca Raton, 2001).
  31. Yu. P. Popov and A. A. Samarskii, “Completely Conservative Difference Schemes,” Zh. Vychisl. Mat. Mat. Fiz. 9 (4), 953-958 (1969) [USSR Comput. Math. Math. Phys. 9 (4), 296-305 (1969)].
  32. A. A. Samarskii, The Theory of Difference Schemes (Nauka, Moscow, 1977; CRC Press, New York, 2001).
  33. B. L. Rozhdestvenskii and N. N. Janenko, Systems of Quasilinear Equations and Their Applications to Gas Dynamics (Nauka, Moscow, 1978; American Mathematical Society, Providence, 1983).
  34. G. V. Ivanov and V. D. Kurguzov, “Schemes for Solving One-Dimensional Problems of the Dynamics of Inhomogeneous Elastic Bodies on the Basis of Approximation by Linear Polynomials,” in Dynamics of Continuum Medium (Lavrentyev Inst. of Hydrodynamics, Novosibirsk, 1981), Issue 49, pp. 27-44.
  35. G. V. Ivanov, Yu. M. Volchkov, I. O. Bogulskii, et al., Numerical Solution of Dynamic Elastic-Plastic Problems of Deformable Solids (Sib. Univ. Izd., Novosibirsk, 2002) [in Russian].
  36. V. M. Sadovskii, O. V. Sadovskaya, and E. A. Efimov, “Finite Difference Schemes for Modelling the Propagation of Axisymmetric Elastic Longitudinal Waves,” J. Sib. Feder. Univ.: Math. Phys. 13 (5), 644-654 (2020).

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