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, 56:6 | Cite as

Finite element discretization of local minimization schemes for rate-independent evolutions

  • Christian MeyerEmail author
  • Michael Sievers
Article
  • 31 Downloads

Abstract

This paper is concerned with a space-time discretization of a rate-independent evolution governed by a non-smooth dissipation and a non-convex energy functional. For the time discretization, we apply the local minimization scheme introduced in Efendiev and Mielke (J Convex Anal 13(1):151–167, 2006), which is known to resolve time discontinuities, which may show up due to the non-convex energy. The spatial discretization is performed by classical linear finite elements. We show that accumulation points of the sequence of discrete solutions for mesh size tending to zero exist and are so-called parametrized solutions of the continuous problem. The discrete problems are solved by means of a mass lumping scheme for the non-smooth dissipation functional in combination with a semi-smooth Newton method. A numerical test indicates the efficiency of this approach. In addition, we compared the local minimization scheme with a time stepping scheme for global energetic solutions, which shows that both schemes yield different solutions with differing time discontinuities.

Keywords

Rate independent evolutions Parametrized solutions Finite elements Semi-smooth Newton methods 

Mathematics Subject Classification

65J08 65J15 65M60 

Notes

Acknowledgements

The authors are very grateful to Dorothee Knees (University of Kassel) for various helpful discussions.

References

  1. 1.
    Attouch, H.: Variational Convergence for Functions and Operators. Pitman Advanced Publishing Program, Boston (1984)zbMATHGoogle Scholar
  2. 2.
    Efendiev, M.A., Mielke, A.: On the rate-independent limit of systems with dry friction and small viscosity. J. Convex Anal. 13(1), 151–167 (2006)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Gaspoz, F.D., Heine, C.-J., Siebert, K.G.: Optimal grading of the newest vertex bisection and \(H^1\)-stability of the \(L^2\)-projection. IMA J. Numer. Anal 36(3), 1217–1241 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints, vol. 23. Springer, Berlin (2008)zbMATHGoogle Scholar
  5. 5.
    Ito, K., Kunisch, K.: Lagrange multiplier approach to variational problems and applications. SIAM, Philadelphia (2008)CrossRefGoogle Scholar
  6. 6.
    Knees, D.: Convergence Analysis in Time-Discretization Schemes for Rate-Independent Systems. Version: (2017)—submitted to ESAIM:COCV. https://arxiv.org/pdf/1712.06851.pdf
  7. 7.
    Knees, D., Rossi, R., Zanini, C.: A vanishing viscosity approach to a rate-independent damage model. Math. Models Methods Appl. Sci. 23(04), 565–616 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Mielke, A.: Energetic formulation of multiplicative elasto-plasticity using dissipation distances. Contin. Mech. Thermodyn. 15(4), 351–382 (2003)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Mielke, A.: Differential, energetic, and metric formulations for rate-independent processes. Nonlinear PDE’s Appl. 2028, 87–167 (2011)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Mielke, A., Paoli, L., Petrov, A., Stefanelli, U.: Error estimates for space-time discretizations of a rate-independent variational inequality. SIAM J. Numer. Anal. 48(5), 1625–1646 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Mielke, A., Roubíc̆ek, T.: Numerical approaches to rate-independent processes and applications in inelasticity. ESAIM: M2AN 43(3), 399–428 (2009)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Mielke, A., Roubíc̆ek, T.: Rate-Independent Systems: Theory and Application. Springer, New York (2015)CrossRefGoogle Scholar
  13. 13.
    Mielke, A., Rossi, R., Savaré, G.: BV solutions and viscosity approximations of rate-independent systems. ESAIM: Control Optim. Calc. Var. 18(1), 36–80 (2012)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Mielke, A., Rossi, R., Savaré, G.: Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems. J. Eur. Math. Soc. 18, 2107–2165 (2016)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Mielke, A., Theil, F.: On rate-independent hysteresis models. NoDEA: Nonlinear Differ. Equ. Appl. 11(2), 151–189 (2004)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Mielke, A., Zelik, S.: On the vanishing-viscosity limit in parabolic systems with rate-independent dissipation terms. Ann. Sc. Norm. Super. Pisa Cl. Sci. 13(1), 67–135 (2014)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Negri, M.: Quasi-static rate-independent evolutions: characterization, existence, approximation and application to fracture mechanics. ESAIM Control Optim. Calc. Var. 20(4), 983–1008 (2014).  https://doi.org/10.1051/cocv/2014004 MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Negri, M., Scala, R.: A quasi-static evolution generated by local energy minimizers for an elastic material with a cohesive interface. Nonlinear Anal. Real World Appl. 38, 271–305 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Istituto di Informatica e Telematica (IIT) 2019

Authors and Affiliations

  1. 1.Fakultät für MathematikTechnische Universität DortmundDortmundGermany

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