Finite difference schemes for modelling the propagation of axisymmetric elastic longitudinal waves

Vladimir M. Sadovskii ; Oxana V. Sadovskaya ; Evgenii A. Efimov

Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 13 (2020) no. 5, pp. 644-654.

Résumé

An efficient finite difference shock-capturing scheme for the solution of direct seismic problems is constructed. Problem formulation is based on equations of the dynamics of elastic medium with axial symmetry. When implementating the scheme on multiprocessor computing systems, the two-cyclic splitting method with respect to spatial variables is used. One-dimensional systems of equations that arise in the context of splitting procedure are represented as subsystems for longitudinal, transverse and torsional waves. The case of longitudinal waves is considered in this paper. The results of simulations with the use of explicit grid-characteristic schemes and implicit schemes of the "predictor–corrector" type with controllable dissipation of energy are compared with exact solutions that describe propagation of monochromatic waves.

Détail

Keywords: elastic medium, cylindrical waves, splitting method, finite difference scheme, monotonicity, dissipativity, parallel computing.

Bibliographie

[1] S.K. Godunov, A.V. Zabrodin, M. Ya.Ivanov, A.N. Kraiko, G.P. Prokopov, Numerical Solution of Multidimensional Problems of Gas Dynamics, Nauka, M., 1976 (in Russian) | MR

[2] O. Sadovskaya, V. Sadovskii, Mathematical Modeling in Mechanics of Granular Materials, Advanced Structured Materials, 21, Springer, Heidelberg–New York–Dordrecht–London, 2012 | DOI | MR | Zbl

[3] O.V. Sadovskaya, V.M. Sadovskii, “Numerical implementation of mathematical model of the dynamics of a porous medium on supercomputers of cluster architecture”, AIP Conf. Proc., 1684, 2015, 070005-1–070005-9 | DOI

[4] V.M. Sadovskii, O.V. Sadovskaya, “Analyzing the deformation of a porous medium with account for the collapse of pores”, J. Appl. Mech. Techn. Phys., 57:5 (2016), 808–818 | DOI | MR

[5] V.M. Sadovskii, O.V. Sadovskaya, “Modeling of elastic waves in a blocky medium based on equations of the Cosserat continuum”, Wave Motion, 52 (2015), 138–150 | DOI | MR | Zbl

[6] V.M. Sadovskii, O.V. Sadovskaya, A.A. Lukyanov, “Modeling of wave processes in blocky media with porous and fluid-saturated interlayers”, J. Comput. Phys., 345 (2017), 834–855 | DOI | MR | Zbl

[7] V.M. Sadovskii, O.V. Sadovskaya, “Supercomputer Modeling of Wave Propagation in Blocky Media Accounting Fractures of Interlayers”, Nonlinear Wave Dynamics of Materials and Structures, chapt. 22, Advanced Structured Materials, 122, eds. H. Altenbach, V.A. Eremeyev, I.S. Pavlov, A.V. Porubov, Springer, Cham, 2020, 379–398 | DOI | MR

[8] P.F. Sabodash, R.A. Cherednichenko, “Application of the method of spatial characteristics to the solution of axisymmetric problems on the propagation of elastic waves”, Prikl. Mekh. Tehn. Fiz., 12:4 (1971), 101–109 (in Russian) | MR

[9] K.M. Magomedov, A.S. Kholodov, Grid-Characteristic Numerical Methods, Nauka, M., 1988 (in Russian) | MR

[10] V.N. Kukudzhanov, Difference Methods for the Solution of Problems of Mechanics of Deformable Media, MFTI, M., 1992 (in Russian)

[11] V.N. Kukudzhanov, Numerical Continuum Mechanics, De Gruyter Studies in Mathematical Physics, 15, De Gruyter, Berlin–Boston, 2013 | DOI | MR | Zbl

[12] N.G. Burago, A.B. Zhuravlev, I.S. Nikitin, “Continuum model and method of calculating for dynamics of inelastic layered medium”, Math. Models Comput. Simul., 11:3 (2019), 488–498 | DOI | MR

[13] I.S. Nikitin, Theory of Inelastic Layered and Blocky Media, Fizmatlit, M., 2019 (in Russian)

[14] N.G. Burago, V.N. Kukudzhanov, “Buckling and Supercritical Deformations of Elastic-Plastic Shells under Axial Symmetry”, Collection of Numerical Methods in the Mechanics of Deformable Solids, Computing Center of the USSR Academy of Sciences, M., 1978, 47–66 (in Russian)

[15] V.G. Bazhenov, E.V. Igonicheva, Nonlinear Processes of Shock Buckling of Elastic Structural Elements in the Form of Orthotropic Shells of Rotation, UNN, N. Novgorod, 1991 (in Russian)

[16] V.G. Bazhenov, D.T. Chekmarev, Solution of the Problems of Non-Stationary Dynamics of Plates and Shells by the Variational-Difference Method, UNN, N. Novgorod, 2000 (in Russian) | MR

[17] G.I. Marchuk, “Splitting and Alternating Direction Methods”, Handbook of Numerical Analysis, v. 1, eds. P. G. Ciarlet, J.-L. Lions, North-Holland, Elsevier, 1990, 197–462 | DOI | MR | Zbl

[18] A.G. Kulikovskii, N.V. Pogorelov, A. Yu.Semenov, Mathematical Aspects of Numerical Solution of Hyperbolic Systems, Monographs and Surveys in Pure and Applied Mathematics, 118, Chapman Hall/CRC, Boca Raton–London–New York–Wasington, 2001 | DOI | MR | Zbl

[19] Yu.P. Popov, A.A. Samarskii, “Completely conservative difference schemes”, USSR Comput. Math. Math. Phys., 9:4 (1969), 296–305 | DOI | MR

[20] A.A. Samarskii, The Theory of Difference Schemes, Pure and Applied Mathematics, CRC Press, New York–Basel, 2001 | DOI | MR

[21] B.L. Roždestvenskii, N.N. Janenko, Systems of Quasilinear Equations and Their Applications to Gas Dynamics, Translations of Mathematical Monographs, 55, American Mathematical Society, Providence, 1983 | DOI | MR

[22] G.V. Ivanov, V.D. Kurguzov, “Schemes for solving one-dimensional problems of the dynamics of inhomogeneous elastic bodies on the basis of approximation by linear polynomials”, Dynam. Contin. Medium, 49 (1981), 27–44 (in Russian) | MR | Zbl

[23] G.V. Ivanov, Yu.M. Volchkov, I.O. Bogulskii, S.A. Anisimov, V.D. Kurguzov, Numerical Solution of Dynamic Elastic-Plastic Problems of Deformable Solids, Sib. Univ. Izd., Novosibirsk, 2002 (in Russian) | MR