Numerical Algorithms for the Simulation of Finite Plasticity with Microstructures

  • Carsten Carstensen
  • Dietmar Gallistl
  • Boris Krämer
Part of the Lecture Notes in Applied and Computational Mechanics book series (LNACM, volume 78)


This article reports on recent developments in the analysis of finite element methods for nonlinear PDEs with enforced microstructures. The first part studies the convergence of an adaptive finite element scheme for the two-well problem in elasticity. The analysis is based on the relaxation of the classical model energy by its quasiconvex envelope. The second part aims at the computation of guaranteed lower energy bounds for the two-well problem with nonconforming finite element methods that involve a stabilization for the discrete linear Green strain tensor. The third part of the paper investigates an adaptive discontinuous Galerkin method for a degenerate convex problem from topology optimization and establishes some equivalence to nonconforming finite element schemes.


Topology Optimization Posteriori Error Discontinuous Galerkin Method Energy Error Posteriori Error Estimator 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [BC08]
    Bartels, S., Carstensen, C.: A convergent adaptive finite element method for an optimal design problem. Numer. Math. 108(3), 359–385 (2008)CrossRefMATHMathSciNetGoogle Scholar
  2. [BC10]
    Boiger, W., Carstensen, C.: On the strong convergence of gradients in stabilised degenerate convex minimisation problems. SIAM J. Numer. Anal. 47(6), 4569–4580 (2010)CrossRefMATHMathSciNetGoogle Scholar
  3. [BC14]
    Boiger, W., Carstensen, C.: A posteriori error analysis of stabilised FEM for degenerate convex minimisation problems under weak regularity assumptions. Advanced Modeling and Simulation in Engineering Sciences 1(5) (2014)Google Scholar
  4. [BJ87]
    Ball, J.M., James, R.D.: Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal. 100(1), 13–52 (1987)CrossRefMATHMathSciNetGoogle Scholar
  5. [BJ92]
    Ball, J.M., James, R.D.: Proposed experimental tests of a theory of fine microstructure and the two-well problem. Phil. Trans. R. Soc. Lond. A 338, 389–450 (1992)CrossRefMATHGoogle Scholar
  6. [BKK00]
    Ball, J.M., Kirchheim, B., Kristensen, J.: Regularity of quasiconvex envelopes. Calc. Var. Partial Differential Equations 11(4), 333–359 (2000)CrossRefMATHMathSciNetGoogle Scholar
  7. [Car01]
    Carstensen, C.: Numerical analysis of microstructure. In: Theory and Numerics of Differential Equations (Durham 2000). Universitext, pp. 59–126. Springer, Berlin (2001)CrossRefGoogle Scholar
  8. [Car08]
    Carstensen, C.: Convergence of an adaptive FEM for a class of degenerate convex minimization problems. IMA J. Numer. Anal. 28(3), 423–439 (2008)CrossRefMATHMathSciNetGoogle Scholar
  9. [CD04]
    Carstensen, C., Dolzmann, G.: An a priori error estimate for finite element discretizations in nonlinear elasticity for polyconvex materials under small loads. Numer. Math. 97(1), 67–80 (2004)CrossRefMATHMathSciNetGoogle Scholar
  10. [CD14]
    Carstensen, C., Dolzmann, G.: Convergence of adaptive finite element methods for a nonconvex double-well minimisation problem. Math. Comp. (2014)Google Scholar
  11. [CDK11]
    Conti, S., Dolzmann, G., Kreisbeck, C.: Asymptotic behavior of crystal plasticity with one slip system in the limit of rigid elasticity. SIAM J. Math. Anal. 43, 2337–2353 (2011)CrossRefMATHMathSciNetGoogle Scholar
  12. [CDK13]
    Conti, S., Dolzmann, G., Kreisbeck, C.: Relaxation of a model in finite plasticity with two slip systems. Math. Models Methods Appl. Sci. 23(11), 2111–2128 (2013)CrossRefMATHMathSciNetGoogle Scholar
  13. [CDK15]
    Conti, S., Dolzmann, G., Kreisbeck, C.: Variational modeling of slip: From crystal plasticity to geological strata. In: Hackl, K., Conti, S. (eds.) Analysis and Computation of Microstructure in Finite Plasticity. LNACM, vol. 78, pp. 31–62. Springer, Heidelberg (2015)Google Scholar
  14. [CG14]
    Carstensen, C., Gallistl, D.: Guaranteed lower eigenvalue bounds for the biharmonic equation. Numer. Math. 126(1), 33–51 (2014)CrossRefMATHMathSciNetGoogle Scholar
  15. [CGR12]
    Carstensen, C., Günther, D., Rabus, H.: Mixed finite element method for a degenerate convex variational problem from topology optimization. SIAM J. Numer. Anal. 50(2), 522–543 (2012)CrossRefMATHMathSciNetGoogle Scholar
  16. [Chi00]
    Chipot, M.: Elements of nonlinear analysis. Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Verlag, Basel (2000)CrossRefMATHGoogle Scholar
  17. [CHM02]
    Carstensen, C., Hackl, K., Mielke, A.: Non-convex potentials and microstructures in finite-strain plasticity. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 458(2018), 299–317 (2002)CrossRefMATHMathSciNetGoogle Scholar
  18. [CJ03]
    Carstensen, C., Jochimsen, K.: Adaptive finite element methods for microstructures? Numerical experiments for a 2-well benchmark. Computing 71(2), 175–204 (2003)CrossRefMATHMathSciNetGoogle Scholar
  19. [CK88]
    Chipot, M., Kinderlehrer, D.: Equilibrium configurations of crystals. Arch. Rational Mech. Anal. 103(3), 237–277 (1988)CrossRefMATHMathSciNetGoogle Scholar
  20. [CL15]
    Carstensen, C., Liu, D.J.: Nonconforming FEMs for an optimal design problem. SIAM J. Numer. Anal. (2015) (in press)Google Scholar
  21. [CM02]
    Carstensen, C., Müller, S.: Local stress regularity in scalar nonconvex variational problems. SIAM J. Math. Anal. 34(2), 495–509 (2002)CrossRefMATHMathSciNetGoogle Scholar
  22. [CP97]
    Carstensen, C., Plecháč, P.: Numerical solution of the scalar double-well problem allowing microstructure. Math. Comp. 66(219), 997–1026 (1997)CrossRefMATHMathSciNetGoogle Scholar
  23. [CP00]
    Carstensen, C., Plecháč, P.: Numerical analysis of compatible phase transitions in elastic solids. SIAM J. Numer. Anal. 37(6), 2061–2081 (2000)CrossRefMATHMathSciNetGoogle Scholar
  24. [CP01]
    Carstensen, C., Plecháč, P.: Numerical analysis of a relaxed variational model of hysteresis in two-phase solids. M2AN Math. Model. Numer. Anal. 35(5), 865–878 (2001)CrossRefMATHMathSciNetGoogle Scholar
  25. [CS15]
    Carstensen, C., Schedensack, M.: Medius analysis and comparison results for first-order finite element methods in linear elasticity. IMA J. Numer. Anal. (published online, 2015), doi:10.1093/imanum/dru048Google Scholar
  26. [Dac08]
    Dacorogna, B.: Direct methods in the calculus of variations, 2nd edn. Applied Mathematical Sciences, vol. 78. Springer, New York (2008)MATHGoogle Scholar
  27. [DE12]
    Di Pietro, D.A., Ern, A.: Mathematical aspects of discontinuous Galerkin methods. Mathématiques & Applications (Berlin), vol. 69. Springer, Heidelberg (2012)CrossRefMATHGoogle Scholar
  28. [Dör96]
    Dörfler, W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33(3), 1106–1124 (1996)CrossRefMATHMathSciNetGoogle Scholar
  29. [Fri94]
    Friesecke, G.: A necessary and sufficient condition for nonattainment and formation of microstructure almost everywhere in scalar variational problems. Proc. Roy. Soc. Edinburgh Sect. A 124(3), 437–471 (1994)CrossRefMATHMathSciNetGoogle Scholar
  30. [GKH15]
    Günther, C., Kochmann, D., Hackl, K.: Rate-independent versus viscous evolution of laminate microstructures in finite crystal plasticity. In: Hackl, K., Conti, S. (eds.) Analysis and Computation of Microstructure in Finite Plasticity. LNACM, vol. 78, pp. 63–88. Springer, Heidelberg (2015)Google Scholar
  31. [HHM12]
    Hackl, K., Heinz, S., Mielke, A.: A model for the evolution of laminates in finite-strain elastoplasticity. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 92(11-12), 888–909 (2012)CrossRefMATHMathSciNetGoogle Scholar
  32. [HL03]
    Hansbo, P., Larson, M.G.: Discontinuous Galerkin and the Crouzeix-Raviart element: application to elasticity. M2AN Math. Model. Numer. Anal. 37(1), 63–72 (2003)CrossRefMATHMathSciNetGoogle Scholar
  33. [HM12]
    Hildebrand, F., Miehe, C.: Variational phase field modeling of laminate deformation microstructure in finite gradient crystal plasticity. Proc. Appl. Math. Mech. 12(1), 37–40 (2012)CrossRefGoogle Scholar
  34. [KK11]
    Kochmann, D., Hackl, K.: The evolution of laminates in finite crystal plasticity: a variational approach. Continuum Mechanics and Thermodynamics 23, 63–85 (2011)CrossRefMATHMathSciNetGoogle Scholar
  35. [Koh91]
    Kohn, R.V.: The relaxation of a double-well energy. Contin. Mech. Thermodyn. 3(3), 193–236 (1991)CrossRefMATHMathSciNetGoogle Scholar
  36. [KP91]
    Kinderlehrer, D., Pedregal, P.: Characterizations of Young measures generated by gradients. Arch. Rational Mech. Anal. 115(4), 329–365 (1991)CrossRefMATHMathSciNetGoogle Scholar
  37. [KS86a]
    Kohn, R.V., Strang, G.: Optimal design and relaxation of variational problems. I. Comm. Pure Appl. Math. 39(1), 113–137 (1986)CrossRefMATHMathSciNetGoogle Scholar
  38. [KS86b]
    Kohn, R.V., Strang, G.: Optimal design and relaxation of variational problems. II. Comm. Pure Appl. Math. 39(1), 139–182 (1986)CrossRefMATHGoogle Scholar
  39. [KS86c]
    Kohn, R.V., Strang, G.: Optimal design and relaxation of variational problems. III. Comm. Pure Appl. Math. 39(3), 353–377 (1986)CrossRefMATHMathSciNetGoogle Scholar
  40. [LC88]
    Lurie, K.A., Cherkaev, A.V.: On a certain variational problem of phase equilibrium. In: Material instabilities in Continuum Mechanics (Edinburgh, 1985–1986), pp. 257–268. Oxford Univ. Press, New York (1988)Google Scholar
  41. [Lus96]
    Luskin, M.: On the computation of crystalline microstructure. Acta Numerica 5, 191–257 (1996)CrossRefMathSciNetGoogle Scholar
  42. [Mie15]
    Mielke, A.: Variational approaches and methods for dissipative material models with multiple scales. In: Hackl, K., Conti, S. (eds.) Analysis and Computation of Microstructure in Finite Plasticity. LNACM, vol. 78, pp. 125–156. Springer, Heidelberg (2015)Google Scholar
  43. [MNS02]
    Morin, P., Nochetto, R.H., Siebert, K.G.: Convergence of adaptive finite element methods. SIAM Rev. 44(4), 631–658 (2002)CrossRefMATHMathSciNetGoogle Scholar
  44. [MŠ99]
    Müller, S., Šverák, V.: Convex integration with constraints and applications to phase transitions and partial differential equations. J. Eur. Math. Soc. (JEMS) 1(4), 393–422 (1999)CrossRefMATHMathSciNetGoogle Scholar
  45. [Pip91]
    Pipkin, A.C.: Elastic materials with two preferred states. Quart. J. Mech. Appl. Math. 44(1), 1–15 (1991)CrossRefMATHMathSciNetGoogle Scholar
  46. [Ser96]
    Seregin, G.A.: The uniqueness of solutions of some variational problems of the theory of phase equilibrium in solid bodies. J. Math. Sci. 80(6), 2333–2348 (1996); Nonlinear boundary-value problems and some questions of function theoryCrossRefMathSciNetGoogle Scholar
  47. [Ser98]
    Seregin, G.A.: A variational problem on the phase equilibrium of an elastic body. St. Petersbg. Math. J. 10, 477–506 (1998)MathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Carsten Carstensen
    • 1
  • Dietmar Gallistl
    • 2
  • Boris Krämer
    • 1
  1. 1.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany
  2. 2.Institut für Numerische SimulationUniversität BonnBonnGermany

Personalised recommendations