Numerische Mathematik

, Volume 96, Issue 4, pp 641–660 | Cite as

Multiscale resolution in the computation of crystalline microstructure

  • Sören Bartels
  • Andreas ProhlEmail author


This paper addresses the numerical approximation of microstructures in crystalline phase transitions without surface energy. It is shown that branching of different variants near interfaces of twinned martensite and austenite phases leads to reduced energies in finite element approximations. Such behavior of minimizing deformations is understood for an extended model that involves surface energies. Moreover, the closely related question of the role of different growth conditions of the employed bulk energy is discussed. By explicit construction of discrete deformations in lowest order finite element spaces we prove upper bounds for the energy and thereby clarify the question of the dependence of the convergence rate upon growth conditions and lamination orders. For first order laminates the estimates are optimal.


Phase Transition Austenite Martensite Surface Energy Lamination 
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  1. 1.
    Ball, J.M., James, R.D.: Fine phase mixtures as minimizers of energy. Arch. Rat. Mech. Anal. 100, 13–52 (1987)MathSciNetGoogle Scholar
  2. 2.
    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)Google Scholar
  3. 3.
    Bartels, S.: Numerical analysis of some non-convex variational problems. PhD- Thesis, Universität Kiel, 2001. Available at Scholar
  4. 4.
    Belgacem, H.B., Conti, S., DeSimone, A., Müller, S.: Rigourous bounds for the Föppl-von Kármán theory of isotropically compressed plates. J. Nonlinear Sci. 10, 661–683 (2000)CrossRefGoogle Scholar
  5. 5.
    Carstensen, C., Plecháč, P.: Numerical solution of the scalar double-well problem allowing microstructure. Math. Comp. 66, 997–1026 (1997)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Carstensen, C., Roubíček, T.: Numerical approximation of young measures in non-convex variational problems. Numer. Math. 84, 395–415 (2000)MathSciNetGoogle Scholar
  7. 7.
    Chipot, M., Collins, C.: Numerical approximations in variational problems with potential wells, SIAM J. Numer. Anal. 29, 1002–1019 (1992)MathSciNetGoogle Scholar
  8. 8.
    Chipot, M.: The appearance of microstructures in problems with incompatible wells and their numerical approach. Numer. Math. 83, 325–352 (1999)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Chipot, M., Collins, C., Kinderlehrer, D.: Numerical analysis of oscillations in multiple well problems. Numer. Math. 70, 259–282 (1995)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Chipot, M., Müller, S.: Sharp energy estimates for finite element approximations of non-convex problems in: Variations of domain and free-boundary problems in solid mechanics (Paris),~pp. 317–325 (1997)Google Scholar
  11. 11.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Amsterdam: North-Holland, 1978Google Scholar
  12. 12.
    Matthias, G.K., Prohl, A.: A discontinuous finite element method for solving a multi-well problem. SIAM J. Numer. Anal. 37, 246–268 (1999)CrossRefGoogle Scholar
  13. 13.
    Kohn, R.V., Müller, S.: Branching of twins near an austenite/twinned martensite interface. Phil. Mag. 66A, 697–715 (1994)Google Scholar
  14. 14.
    Kohn, R.V., Müller, S.: Surface energy and microstructure in coherent phase transitions. Comm. Pure Appl. Math. 47, 405–435 (1994)MathSciNetGoogle Scholar
  15. 15.
    Kružík, M.: Numerical approach to double well problems. SIAM J. Numer. Anal. 35, 1833–1849 (1998)CrossRefGoogle Scholar
  16. 16.
    Li, B.: Finite element analysis of a class of stress-free martensitic microstructures. Math. Comp., 2002. In pressGoogle Scholar
  17. 17.
    Li, B., Luskin, M.: Nonconforming Finite element approximation of crystalline microstructure. Math. Comp. 67, 917–946 (1998)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Li, B., Luskin, M.: Theory and computation for the microstructure near the interface between twinned layers and a pure variant of martensite. Mat. Sci. Eng. A, 273–275, 237–240 (1999)Google Scholar
  19. 19.
    Luskin, M.: Approximation of a laminated microstructure for a rotationally invariant, double well energy density. Numer. Math. 75, 205–221 (1997)CrossRefGoogle Scholar
  20. 20.
    Luskin, M.: On the computation of crystalline microstructure. Acta Numerica, 1996Google Scholar
  21. 21.
    Müller, S.: Variational models for microstructure and phase transitions. Springer. Lect. Notes Math. 1713, 85–210 (1999)Google Scholar
  22. 22.
    Pedregal, P.: On the numerical analysis of nonconvex variational problems. Num. Math. 74, 325–336 (1996)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Prohl, A.: An adaptive finite element method for solving a double well problem describing crystalline microstructure. M2AN 33, 781–796 (1999)Google Scholar
  24. 24.
    Shield, T.W.: Needles in Martensites. needlesGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Mathematisches Seminar, Christian-Albrechts-Universität zu KielKielGermany
  2. 2.Department of MathematicsETHZZurichSwitzerland

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