Homogeneous difference schemes for the coupled problems of hydrodynamics and elasticity

I.P. Tsygvintsev^a*, A.Yu. Krukovskiy^a**, Yu.A. Poveshchenko^a,b*** ,
V.A. Gasilov^a,b**** , D.S. Boykov^a***** , S.B. Popov^a******
^a Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, 125047 Russia
^b National Research Nuclear University MEPhI (Moscow Engineering Physics Institute ), Moscow, 115409 Russia
E-mail: *IliaTsygvintsev@gmail.com, **Alexander-krukovskiy@yandex.ru, ***hecon@mail.ru, ****vgasilov@gmail.com, *****boykovds@gmail.com, ******popovsb@yandex.ru
Received June 12, 2019

Full text pdf

DOI: 10.26907/2541-7746.2019.3.377-392

For citation: Tsygvintsev I.P., Krukovskiy A.Yu., Poveshchenko Yu.A., Gasilov V.A., Boykov D.S., Popov S.B. Homogeneous difference schemes for the coupled problems of hydrodynamics and elasticity. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2019, vol. 161, no. 3, pp. 377–392. doi: 10.26907/2541-
7746.2019.3.377-392. (In Russian)

Abstract

Finite-difference approximations of elastic forces on the staggered moving grid were constructed. For the displacement vectors at the irregular grids in which topological and geometrical structures are subjected to minimal reasonable restrictions, with regard to the finite-difference schemes of the elasticity theory problems, approximations of the vector analysis operators in plane and cylindrical geometries were constructed. Taking into account the energy balance of the medium, the families of integral consistent approximations of the vector analysis operators, which are sufficient for the discrete modeling of these processes considering the space curvature caused by the cylindrical geometry of the system, were built. The schemes, both using a stress tensor in the full form and dividing it into volumetric and deviator components, were studied. This separation is used to construct homogeneous equations that are applicable for solid body and vaporized phase. The linear theory of elasticity was used. The resulting expressions for the elastic forces were presented in the explicit form for two-dimensional flat and axisymmetric geometries for a mesh consisting of triangular and quadrangular cells. Generalization of the method for other cases (non-linear strain tensor, non-Hookean relation between strain and stress, full 3D geometry, etc.) can be performed by analogy, but this was not a subject of the current paper. Using the model problem, comparison between different temporal discretizations for the obtained ordinary differential equations system was carried out. In particular, we considered fully implicit approximation, conservative implicit approximation (Crank–Nicolson method), and explicit approximation, which is similar to the “leap-frog” method. The analysis of full energy imbalance and calculation costs showed that the latter is more advantageous. The analysis of the effectiveness of various temporal approximations was performed via numerical experiments.

Keywords: finite-difference scheme, strain tensor, support-operator method, Lagrange staggered grid

References

1. Samarskii A.A., Tishkin V.F., Favorskii A.P., Shashkov M.Yu. Operator difference schemes. Differ. Uravn., 1981, vol. 17, no. 7, pp. 1317–1327. (In Russian)

2. Samarskii A.A., Tishkin V.F., Favorskii A.P., Shashkov M.Yu. Representation of difference schemes of mathematical physics in operator form. Dokl. Akad. Nauk SSSR, 1981, vol. 258, no. 5, pp. 1092–1096. (In Russian)

3. Korshiya T.K., Tishkin V.F., Samarskii A.A., Favorskii A.P., Shashkov M.Yu. Variatsionno-operatornye raznostnye skhemy dlya uravnenii matematicheskoi fiziki [Variational Operator Difference Schemes for Equations of Mathematical Physics]. Tbilisi, Izd. Tbilis. Univ., 1983. 143 p. (In Russian)

4. Denisov A.A., Koldoba A.V., Poveshchenko Yu.A. The convergence to generalized solutions of difference schemes of the reference-operator method for Poisson's equation. USSR Comput. Math. Math. Phys., 1989, vol. 29, no. 2, pp. 32–38. doi: 10.1016/0041-5553(89)90005-0.

5. Denisov A.A., Koldoba A.V., Poveshchenko Yu.A. Convergence of difference schemes of the reference-operator method to generalized solutions of the axisymmetric Poisson equation. USSR Comput. Math. Math. Phys., 1990, vol. 30, no. 5, pp. 140–147. doi: 10.1016/0041-5553(90)90172-O.

6. Samarskii A.A., Koldoba A.V., Poveshchenko Yu.A., Tishkin V.F., Favorskii A.P. Raznostnye skhemy na neregulyarnykh setkakh [Difference Schemes on Irregular Grids]. Minsk, Kriterii, 1996. 273 p. (In Russian)

7. Koldoba A.V., Poveshchenko Yu.A., Gasilova I.V., Dorofeeva E.Yu. Numerical schemes of the support operators method for elasticity theory equations. Mat. Model., 2012, vol. 24, no. 12, pp. 86–96. (In Russian)

8. Samarskii A.A., Popov Yu.P. Raznostnye metody resheniya zadach gazovoi dinamiki [Difference Methods for Solving Gas Dynamics Problems]. Moscow, Nauka, 1992. 424 p. (In Russian)

9. Nikiforov A.F., Novikov V.G., Uvarov V.B. Kvantovo-statisticheskie modeli vysokotemperaturnoi plazmy [Quantum-Statistical Models of High-Temperature Plasma]. Moscow, Fizmatlit, 2000. 400 p. (In Russian)

10. Landau L.D., Lifshits E.M. Theory of Elasticity. Course of Theoretical Physics. Vol. 7. Oxford, Pergamon Press, 1970. 165 p.

11. Goloviznin V.M., Samarskii A.A., Favorskii A.P. A variational approach to the construction of finite-difference mathematical models in hydrodynamics. Dokl. Akad. Nauk SSSR, 1977, vol. 235, no. 6, pp. 1285–1288. (In Russian)

12. Gasilov V.A., Krukovskii A.Yu., Novikov V.G., Romanov I.V., Tsygvintsev I.P. Numerical simulation of current flow in a vacuum diode with laser ignition. Keldysh Inst. Prepr., 2013, no. 78. 20 p. (In Russian)

13. Gasilov V.A., Krukovskii A.Yu., Poveshchenko Yu.A., Tsygvintsev I.P. Homogeneous difference schemes for solving related problems of hydrodynamics and elasticity. Keldysh Inst. Prepr., 2018, no. 13. 17 p. (In Russian)

14. Poveshchenko Yu.A., Gasilov V.A., Ladonkina M.E., Podryga V.O., Nasekin I.S. Difference schemes of the support operator method for equations of the theory of elasticity in cylindrical geometry. Keldysh Inst. Prepr., 2018, no. 142. 22 p. (In Russian)

15. Samarskii A.A., Nikolaev E.S. Numerical Methods for Grid Equations. Vol. 2: Iterative methods. Basel, Boston, Berlin, Birkhauser Verlag, 1989. 502 p.

The content is available under the license Creative Commons Attribution 4.0 License.