Applications of Mathematics

, Volume 61, Issue 5, pp 527–564

Guaranteed and computable bounds of the limit load for variational problems with linear growth energy functionals

  • Jaroslav Haslinger
  • Sergey Repin
  • Stanislav Sysala
Article

Abstract

The paper is concerned with guaranteed and computable bounds of the limit (or safety) load, which is one of the most important quantitative characteristics of mathematical models associated with linear growth functionals. We suggest a new method for getting such bounds and illustrate its performance. First, the main ideas are demonstrated with the paradigm of a simple variational problem with a linear growth functional defined on a set of scalar valued functions. Then, the method is extended to classical plasticity models governed by vonMises and Drucker-Prager yield laws. The efficiency of the proposed approach is confirmed by several numerical experiments.

Keywords

functionals with linear growth limit load truncation method perfect plasticity 

MSC 2010

49M15 74C05 74S05 90C25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    A. Caboussat, R. Glowinski: Numerical solution of a variational problem arising in stress analysis: the vector case. Discrete Contin. Dyn. Syst. 27 (2010), 1447–1472.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    M. Cermak, J. Haslinger, T. Kozubek, S. Sysala: Discretization and numerical realization of contact problems for elastic-perfectly plastic bodies: Part II—numerical realization, limit analysis. ZAMM, Z. Angew. Math. Mech. 95 (2015), 1348–1371.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    W. F. Chen, X. L. Liu: Limit Analysis in Soil Mechanics. Elsevier, 1990.Google Scholar
  4. [4]
    E. Christiansen: Limit analysis of collapse states. Handbook of Numerical Analysis, Volume IV: Finite Element Methods (part 2), Numerical Methods for Solids (part 2) (P. G. Ciarlet, eds.). North-Holland, Amsterdam, 1996, pp. 193–312.CrossRefGoogle Scholar
  5. [5]
    E. A. de Souza Neto, D. Perić, D. R. J. Owen: Computational Methods for Plasticity: Theory and Applications. Wiley, 2008.CrossRefGoogle Scholar
  6. [6]
    U. Dierkes, S. Hildebrandt, F. Sauvigny: Minimal Surfaces. Grundlehren der Mathematischen Wissenschaften 339, Springer, Dordrecht, 2010.MATHGoogle Scholar
  7. [7]
    G. Duvaut, J. L. Lions: Inequalities in Mechanics and Physics. Grundlehren der Mathematischen Wissenschaften 219, Springer, Berlin, 1976.Google Scholar
  8. [8]
    I. Ekeland, R. Temam: Convex Analysis and Variational Problems. Études Mathématiques, Dunod; Gauthier-Villars, Paris, 1974. (In French.)MATHGoogle Scholar
  9. [9]
    R. Finn: Equilibrium Capillary Surfaces. Grundlehren der Mathematischen Wissenschaften 284, Springer, New York, 1986.Google Scholar
  10. [10]
    S. Fučík, A. Kufner: Nonlinear Differential Equations. Studies in Applied Mechanics 2, Elsevier Scientific Publishing Company, Amsterdam, 1980.MATHGoogle Scholar
  11. [11]
    E. Giusti: Minimal Surfaces and Functions of Bounded Variations. Monographs in Mathematics 80, Birkhäuser, Basel, 1984.Google Scholar
  12. [12]
    P. Hansbo: A discontinuous finite element method for elasto-plasticity. Int. J. Numer. Methods Biomed. Eng. 26 (2010), 780–789.MathSciNetMATHGoogle Scholar
  13. [13]
    J. Haslinger, S. Repin, S. Sysala: A reliable incremental method of computing the limit load in deformation plasticity based on compliance: Continuous and discrete setting. J. Comput. Appl. Math. 303 (2016), 156–170.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    C. Johnson, R. Scott: A finite element method for problems in perfect plasticity using discontinuous trial functions. Nonlinear Finite Element Analysis in StructuralMechanics (W. Wunderlich, et al., eds.). Proc. Europe-U. S. Workshop, Bochum, 1980, Springer, Berlin, 1981, pp. 307–324.CrossRefGoogle Scholar
  15. [15]
    M. A. Krasnosel’skii: Topological Methods in the Theory of Nonlinear Integral Equations. Pergamon Press, Oxford, 1964.Google Scholar
  16. [16]
    D. W. Langbein: Capillary Surfaces: Shape, Stability, Dynamics, in Particular under Weightlessness. Springer Tracts in Modern Physics 178, Springer, Berlin, 2002.Google Scholar
  17. [17]
    F. Liu, J. Zhao: Limit analysis of slope stability by rigid finite-element method and linear programming considering rotational failure. Int. J. Geomech. 13 (2013), 827–839.CrossRefGoogle Scholar
  18. [18]
    J. C. C. Nitsche: Lectures on Minimal Surfaces: Volume 1: Introduction, Fundamentals, Geometry and Basic Boundary Value Problems. Revised, extended and updated by the author. Cambridge University Press, Cambridge, 2011.Google Scholar
  19. [19]
    E. Ramm: Strategies for tracing nonlinear response near limit points. Nonlinear Finite Element Analysis in Structural Mechanics (W. Wunderlich, eds.). Proc. Europe-U. S. Workshop, Bochum, 1980, Springer, Berlin, 1981, pp. 63–89.CrossRefGoogle Scholar
  20. [20]
    S. Repin, G. Seregin: Existence of a weak solution of the minimax problem arising in Coulomb-Mohr plasticity. Nonlinear Evolution Equations (N. N. Uraltseva, ed.). Am. Math. Soc. Ser. 2, 164, American Mathematical Society, Providence, 1995, pp. 189–220.CrossRefGoogle Scholar
  21. [21]
    R. T. Rockafellar: Convex Analysis. Princeton University Press, Princeton, 1970.CrossRefMATHGoogle Scholar
  22. [22]
    S. W. Sloan: Lower bound limit analysis using finite elements and linear programming. Int. J. Numer. Anal. Methods Geomech. 12 (1988), 61–77.CrossRefMATHGoogle Scholar
  23. [23]
    P.-M. Suquet: Existence et régularité des solutions des équations de la plasticité parfaite. C. R. Acad. Sci., Paris, Sér. A 286 (1978), 1201–1204. (In French.)MathSciNetMATHGoogle Scholar
  24. [24]
    S. Sysala: Properties and simplifications of constitutive time-discretized elastoplastic operators. ZAMM, Z. Angew. Math. Mech. 94 (2014), 233–255.MathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    S. Sysala, M. Cermak, T. Koudelka, J. Kruis, J. Zeman, R. Blaheta: Subdifferentialbased implicit return-mapping operators in computational plasticity. ZAMM, Z. Angew. Math. Mech. 96 (2016), 1–21, DOI 10.1002/zamm.201500305.CrossRefGoogle Scholar
  26. [26]
    S. Sysala, J. Haslinger, I. Hlaváček, M. Cermak: Discretization and numerical realization of contact problems for elastic-perfectly plastic bodies: PART I–discretization, limit analysis. ZAMM, Z. Angew. Math. Mech. 95 (2015), 333–353.MathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    R. Temam: Mathematical Problems in Plasticity. Gauthier-Villars, Montrouge, 1983.MATHGoogle Scholar
  28. [28]
    X. Yu, F. Tin-Loi: A simple mixed finite element for static limit analysis. Comput. Struct. 84 (2006), 1906–1917.CrossRefGoogle Scholar
  29. [29]
    O. C. Zienkiewicz, R. L. Taylor: The Finite Element Method. Vol. 2. Solid Mechanics. Butterworth-Heinemann, Oxford, 2000.MATHGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2016

Authors and Affiliations

  • Jaroslav Haslinger
    • 1
    • 2
  • Sergey Repin
    • 3
    • 4
    • 5
  • Stanislav Sysala
    • 2
  1. 1.Department of Numerical MathematicsCharles UniversityPraha 8Czech Republic
  2. 2.Institute of Geonics of the Czech Academy of SciencesDepartment of Applied Mathematics and Computer Science & Department IT4InnovationsOstrava-PorubaCzech Republic
  3. 3.St. Petersburg Department of V.A. Steklov Institute of Mathematics of the Russian Academy of SciencesSt. PetersburgRussia
  4. 4.St. Petersburg Polytechnic University of Peter The GreateSt. PetersburgRussia
  5. 5.University of JyväskyläJyväskyläFinland

Personalised recommendations