Advertisement

Advancements in the Computational Calculus of Variations

  • Carsten Carstensen
  • Cataldo ManigrassoEmail author
Conference paper
  • 669 Downloads
Part of the IUTAM Bookseries book series (IUTAMBOOK, volume 21)

Abstract

The aim of this paper is a survey on some state-of-the-art techniques in the numerical analysis of minimisation problems in nonlinear elasticity and on the concepts of quasi and polyconvexity in the calculus of variations. The issue of the approximation of singular minimisers is addressed via a penalty finite element method (PFEM) and macroscopic deformations are computed via the relaxation finite element method (RFEM). New numerical results are presented for the scalar doublewell problem as a benchmark in computational microstructures with adaptive meshrefining, in order to recover optimal convergence in the presence of singularities at interfaces.

Keywords

Posteriori Error Nonlinear Elasticity Posteriori Error Estimate Young Measure Relax Problem 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal. 63, 1977, 337–403.zbMATHCrossRefGoogle Scholar
  2. 2.
    J.M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Philos. Trans. Roy. Soc. London Ser. A 306(1496), 1982, 557–611.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    J.M. Ball, A version of the fundamental theorem for Young measures, in 1Partial Differential Equations and Continuum Models of Phase Transitions, Proceedings, M. Rascle, D. Serre, and M. Slemrod (Eds.), Springer Lecture Notes in Physics, Vol. 359, Springer, 1989, pp. 207–215.Google Scholar
  4. 4.
    J.M. Ball and G. Knowles, A numerical method for detecting singular minimisers, Numer. Math. 51, 1987, 181–197.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    J.M. Ball and V.J. Mizel, Singular minimisers for regular one-dimensional problems in the calculus of variations, Bull. Am. Math. Soc., New Ser. 11, 1984, 143–146.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods, Acta Numerica, Cambridge University Press,2001, pp. 1102.Google Scholar
  7. 7.
    A. Braides, Gamma-Convergence for Beginners, Oxford University Press, USA, 2002.zbMATHCrossRefGoogle Scholar
  8. 8.
    C. Carstensen, Convergence of adaptive FEM for a class of degenerate convex minimisation problems, IMA J. Num. Math., 2008.Google Scholar
  9. 9.
    C. Carstensen and G. Dolzmann, An a priori error estimate for finite element discretisations in nonlinear elasticity for polyconvex materials under small loads, Numer. Math. 97, 2004, 67–80.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    C. Carstensen and K. Jochimsen, Adaptive finite element error control for non-convex minimization problems: Numerical two-well model example allowing microstructures, Computing 71, 2003, 175–204.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    C. Carstensen and Y. Kondratyuk, Adaptive finite element methods for uniformly convex minimisation problems of optimal complexity, in preparation, 2008.Google Scholar
  12. 12.
    C. Carstensen and C. Ortner, Computation of the Lavrentiev phenomenon, in preparation, 2008.Google Scholar
  13. 13.
    C. Carstensen and P. Plecháč, Numerical solution of the scalar double-well problem allowing microstructure, Math. Comp. 66, 1997, 997–1026.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    B. Dacorogna, Direct Methods in the Calculus of Variations, Springer, 2007.Google Scholar
  15. 15.
    G. Dolzmann, Variational Methods for Crystalline Microstructure – Analysis and Computation, Lecture Notes in Mathematics, Vol. 1803, Springer, 2003.Google Scholar
  16. 16.
    L.C. Evans, Partial Differential Equations, American Mathematical Society, 2002.Google Scholar
  17. 17.
    M. Foss, W.J. Hrusa and V.J. Mizel, The Lavrentiev gap phenomenon in nonlinear elasticity, Arch. Ration. Mech. Anal. 167(4), 2003, 337–365.zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    J. Kristensen, On the non-locality of quasiconvexity. Ann. Inst. H. Poincaré Anal. Non Linéaire 16(1), 1999, 1–13.zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    C.B. Morrey, Quasiconvexity and the lower semicontinuity of multiple integrals, Pacific J. Math. 2, 1952, 25–53.zbMATHMathSciNetGoogle Scholar
  20. 20.
    S. Müller, private communication.Google Scholar
  21. 21.
    P.V. Negrón-Marrero, A numerical method for detecting singular minimisers of multidimensional problems in nonlinear elasticity, Numer. Math. 58(2), 1990, 135–144.zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    V. Šverák, Rank-one convexity does not imply quasiconvexity, Proc. Royal Soc. Edinburgh, Sec. A. Math. 120(1–2), 1992, 185–189.zbMATHGoogle Scholar
  23. 23.
    R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh Refinement Techniques, Wiley-Teubner, 1996.zbMATHGoogle Scholar
  24. 24.
    K. Zhang, Energy minimisers in nonlinear elastostatics and the implicit function theorem, Arch. Rat. Mech. Anal. 114, 1991, 95–117.zbMATHCrossRefGoogle Scholar
  25. 25.
    O.C. Zienkiewicz and J.Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique, Int. J. Numer. Meth. Engrg. 33, 1992, 1331–1364.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Humboldt-Universität zu BerlinBerlinGermany
  2. 2.Department of Computational Science and EngineeringYonsei UniversitySeoulKorea

Personalised recommendations