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A subgrid projection method for relaxation of non-attainable differential inclusions

  • D. Cox
  • P. Klouček
  • D. R. Reynolds
Conference paper

Summary

We propose a subgrid projection method which is suitable for computing microstructures describing non-attainable infima of variational integrals. We document that a descent method in combination with the proposed method yields relaxing sequences which converge in weak-* topology as well as in the sense of approximate Young measures. We show that a sufficient condition for our method to work is to asymptotically form white noise. We present an example which shows that this requires symmetry in the target function.

Keywords

White Noise Shape Memory Alloy Steep Descent Target Function Free Energy Density 
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References

  1. [1]
    Abeyaratne, R., Chu, C., James, R.D. (1996): Kinetics of materials with wiggly energies: theory and application to the evolution of twinning microstructures in Cu-Al-Ni shape memory alloy. Philos. Mag. A 73, 457–498CrossRefGoogle Scholar
  2. [2]
    Bekker, A., Brinson, L.C., Issen, K. (1998): Localized and diffuse thermoinduced phase transformation in 1D shape memory alloys. J. Intelligent Material Syst. Struct. 9, 355–365CrossRefGoogle Scholar
  3. [3]
    Cox, D., Klouček, P., Reynolds, D.R. (2001): The non-local relaxation of non-attainable differential inclusions using subgrid projection method: one dimensional theory and computations. SIAM J. Sci. Comput., submittedGoogle Scholar
  4. [4]
    Cox, D., Klouček, P., Reynolds, D.R. (2002): The computational modeling of crystalline materials using a stochastic variational principle. In: Sloot, P.M.A. et al. (eds.): Computational science — ICCS 2002. (Lecture Notes in Computer Science, vol. 2330). Springer, Berlin, pp. 461–469CrossRefGoogle Scholar
  5. [5]
    Jordan, R., Kinderlehrer, D., Otto, F. (1999): Dynamics of the Fokker-Planck equation. Phase Transitions 69, 271–288CrossRefGoogle Scholar
  6. [6]
    Klouček, P. (1999): The relaxation of non-quasiconvex variational integrals. Numer. Math. 82, 281–311MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Klouček, P. (2000): The steepest descent minimization of double-well stored energies does not yield vectorial microstructures. M2AN Math. Model. Numer. Anal., to appearGoogle Scholar
  8. [8]
    Oberaigner, E.R., Tanaka, K., Fischer, F.D. (1996): Kinetics on the micro-and macrolevels in polycrystalline alloy materials during martensitic transformation. Acta Mech. 116, 171–186MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Italia 2003

Authors and Affiliations

  • D. Cox
    • 1
  • P. Klouček
    • 2
  • D. R. Reynolds
    • 1
  1. 1.Rice UniversityHoustonUSA
  2. 2.Department of Computational and Applied MathematicsRice UniversityHoustonUSA

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