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Splitting Scheme for Poroelasticity and Thermoelasticity Problems

  • Alexandr E. KolesovEmail author
  • Petr N. Vabishchevich
  • Maria V. Vasilyeva
  • Victor F. Gornov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9045)

Abstract

We consider an unconditionally stable splitting scheme for solving coupled systems of equations arising in poroelasticity and thermoelasticity problems. The scheme is based on splitting the systems of equation into physical processes, which means the transition to the new time level is associated with solving separate sub-problems for displacement and pressure/temperature. The stability of the scheme is achieved by switching to three-level finite-difference scheme with weight. We present stability estimates of the scheme based on Samarskii’s theory of stability for operator-difference schemes. We provide numerical experiments supporting the stability estimates of the splitting scheme.

Keywords

Time Level Implicit Scheme Finite Element Discretization Split Scheme Thermoelasticity Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

This work is supported by CJSC OptoGan (contract N02.G25.31.0090); RFBR (project N13-01-00719A); The Ministry of Education and Science of Russian Federation (contract RFMEFI5791X0026).

References

  1. 1.
    Armero, F.: Formulation and finite element implementation of a multiplicative model of coupled poro-plasticity at finite strains under fully saturated conditions. Comput. Methods Appl. Mech. Eng. 171(3), 205–241 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Barry, S., Mercer, G.: Exact solutions for two-dimensional time-dependent flow and deformation within a poroelastic medium. J. Appl. Mech. 66, 1–5 (1999)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Gaspar, F., Grigoriev, A., Vabishchevich, P.: Explicit-implicit splitting schemes for some systems of evolutionary equations. Int. J. Numer. Anal. Model. 11(2), 346–357 (2014)MathSciNetGoogle Scholar
  4. 4.
    Gaspar, F., Lisbona, F., Vabishchevich, P.: A finite difference analysis of biot’s consolidation model. Appl. Numer. Math. 44(4), 487–506 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Haga, J.B., Osnes, H., Langtangen, H.P.: On the causes of pressure oscillations in low-permeable and low-compressible porous media. Int. J. Numer. Anal. Meth. Geomech. 36(12), 1507–1522 (2012)CrossRefGoogle Scholar
  6. 6.
    Jha, B., Juanes, R.: A locally conservative finite element framework for the simulation of coupled flow and reservoir geomechanics. Acta Geotech. 2(3), 139–153 (2007)CrossRefGoogle Scholar
  7. 7.
    Lisbona, F., Vabishchevich, P., Gaspar, F., Oosterlee, C.: An efficient multigrid solver for a reformulated version of the poroelasticity system. Comput. Methods Appl. Mech. Eng. 196(8), 1447–1457 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Marchuk, G.I.: Splitting and alternating direction methods. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. I, pp. 197–462. North-Holland, Amsterdam (1990)Google Scholar
  9. 9.
    Samarskii, A.A.: The Theory of Difference Schemes. Marcel Dekker, New York (2001)zbMATHCrossRefGoogle Scholar
  10. 10.
    Samarskii, A.A., Matus, P.P., Vabishchevich, P.N.: Difference Schemes with Operator Factors. Kluwer Academic Publisher, The Netherlands (2002)zbMATHCrossRefGoogle Scholar
  11. 11.
    Vabishchevich, P.N.: Additive Operator-Difference Schemes: Splitting schemes. de Gruyter, Berlin (2013)CrossRefGoogle Scholar
  12. 12.
    Wang, H.: Theory of Linear Poroelasticity with Applications to Geomechanics and Hydrogeology. Princeton University Press, Princeton (2000)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Alexandr E. Kolesov
    • 1
    Email author
  • Petr N. Vabishchevich
    • 1
    • 2
  • Maria V. Vasilyeva
    • 1
  • Victor F. Gornov
    • 3
  1. 1.North-Eastern Federal UniversityYakutskRussia
  2. 2.Nuclear Safety Institute, RASMoscowRussia
  3. 3.JSC Insolar-InvestMoscowRussia

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