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Convergence of \(\text{ dG(1) }\) in elastoplastic evolution

  • Carsten Carstensen
  • Dongjie Liu
  • Jochen Alberty
Article
  • 24 Downloads

Abstract

The discontinuous Galerkin (dG) methodology provides a hierarchy of time discretization schemes for evolutionary problems such as elastoplasticity with the Prandtl-Reuß  flow rule. A dG time discretization has been proposed for a variational inequality in the context of rate-independent inelastic material behaviour in Alberty and Carstensen in (CMME 191:4949–4968, 2002) with the help of duality in convex analysis to justify certain jump terms. This paper establishes the first a priori error analysis for the dG(1) scheme with discontinuous piecewise linear polynomials in the temporal and lowest-order finite elements for the spatial discretization. Compared to a generalized mid-point rule, the dG(1) formulation distributes the action of the material law in the form of the variational inequality in time and so it introduces an error in the material law. This may result in a suboptimal convergence rate for the dG(1) scheme and this paper shows that the stress error in the \(L^\infty (L^2)\) norm is merely \(O(h+k^{3/2})\) based on a seemingly sharp error analysis. The numerical investigation for a benchmark problem with known analytic solution provides empirical evidence of a higher convergence rate of the dG(1) scheme compared to dG(0).

Mathematics Subject Classification

65N30 65R20 73C50 

Notes

Acknowledgements

The work has been written, while the authors enjoyed the hospitality of the Hausdorff Research Institute of Mathematics in Bonn, Germany, during the Hausdorff Trimester Program Multiscale Problems: Algorithms, Numerical Analysis and Computation. The research of the first author (CC) has been supported by the Deutsche Forschungsgemeinschaft in the Priority Program 1748 ‘Reliable simulation techniques in solid mechanics. Development of non-standard discretization methods, mechanical and mathematical analysis’ under the project ‘Foundation and application of generalized mixed FEM towards nonlinear problems in solid mechanics’ (CA 151/22-1). The corresponding author’s research was supported by National Natural Science Foundation of China (Nos. 11571226, 11671313). The authors thank Simone Hock for a collaboration at the early stage of this research and the anonymous referees for valuable remarks that improved the presentation.

References

  1. 1.
    Alberty, J.: Zeitdiskretisierungsverfahren für elastoplastische Probleme der Kontinuumsmechanik. Ph.D. thesis, University of Kiel, FRG, Tenea Verlag (2001)Google Scholar
  2. 2.
    Alberty, J., Carstensen, C.: Numerical analysis of time-depending primal elastoplastic with hardening. SIAM J. Numer. Anal. 37, 1271–1294 (2000)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Alberty, J., Carstensen, C.: Discontinuous Galerkin time discretization in elastoplasticity: motivation, numerical algorithms and applications. Comp. Methods Appl. Mech. Eng. 191, 4949–4968 (2002)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Alberty, J., Carstensen, C., Zarrabi, D.: Adaptive numerical analysis in primal elastoplasticity with hardening. Comput. Methods Appl. Mech. Eng. 171, 175–204 (1999)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bartels, Sören, Mielke, Alexander, Roubíček, Tomáš: Quasi-static small-strain plasticity in the limit of vanishing hardening and its numerical approximation. SIAM J. Numer. Anal. 50, 951–976 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 15, Texts in Applied Mathematics. Springer, New York (1994)CrossRefGoogle Scholar
  7. 7.
    Carstensen, C.: Domain decomposition for a non-smooth convex minimisation problem and its application to plasticity. NLAA 4, 1–13 (1997)Google Scholar
  8. 8.
    Carstensen, C.: Numerical analysis of the primal problem of elastoplasticity with hardening. Numer. Math. 82, 577–597 (1999)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)zbMATHGoogle Scholar
  10. 10.
    Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976)zbMATHGoogle Scholar
  11. 11.
    Evans, L.C.: Partial Differential Equations, 19, Graduate Studies in Mathematics. American Mathematical Society, Providence (1998)Google Scholar
  12. 12.
    Han, W., Reddy, B.D.: Plasticity, 2nd edn. IAM, Springer, New York (2013)CrossRefGoogle Scholar
  13. 13.
    Johnson, C.: Existence theorems for plasticity problems. J. Math. pures et appl. 55, 431–444 (1976)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Johnson, C.: On plasticity with hardening. J. Math. Anal. Appl. 62, 325–336 (1978)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Rindler, Filip, Schwarzacher, Sebastian, Süli, Endre: Regularity and approximation of strong solutions to rate-independent systems. Math. Models Methods Appl. Sci. 27, 2511–2556 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Seregin, G.A.: On the regularity of the minimizers of some variational problems of plasticity theory. St. Petersb. Math. J. 4, 989–1020 (1993)zbMATHGoogle Scholar
  17. 17.
    Simo, J.C., Hughes, T.J.R.: Computational Inelasticity. Springer, New York (1998)zbMATHGoogle Scholar
  18. 18.
    Suquet, P.M.: Discontinuities and plasticity, Nonsmooth Mechanics and Applications, CISM Courses, 302, pp. 279–341. Springer, New York (1988)Google Scholar
  19. 19.
    Temam, R.: Mathematical Problems in Plasticity. Gauthier-Villars, Paris (1985)zbMATHGoogle Scholar
  20. 20.
    Thomee, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer, Berlin (2006)zbMATHGoogle Scholar
  21. 21.
    Zeidler E.: Nonlinear Functional Analysis and Its Applications iii and iv. Springer, New York (1985,1988)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Carsten Carstensen
    • 1
  • Dongjie Liu
    • 2
  • Jochen Alberty
    • 3
  1. 1.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany
  2. 2.Department of Mathematics, College of SciencesShanghai UniversityShanghaiPeople’s Republic of China
  3. 3.FrankfurtGermany

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