Pełnotekstowe zasoby PLDML oraz innych baz dziedzinowych są już dostępne w nowej Bibliotece Nauki.
Zapraszamy na


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników


2009 | 7 | 3 | 506-519

Tytuł artykułu

Functional a posteriori error estimates for incremental models in elasto-plasticity

Treść / Zawartość

Warianty tytułu

Języki publikacji



We consider incremental problem arising in elasto-plastic models with isotropic hardening. Our goal is to derive computable and guaranteed bounds of the difference between the exact solution and any function in the admissible (energy) class of the problem considered. Such estimates are obtained by an advanced version of the variational approach earlier used for linear boundary-value problems and nonlinear variational problems with convex functionals [24, 30]. They do no contain mesh-dependent constants and are valid for any conforming approximations regardless of the method used for their derivation. It is shown that the structure of error majorant reflects properties of the exact solution so that the majorant vanishes only if an approximate solution coincides with the exact one. Moreover, it possesses necessary continuity properties, so that any sequence of approximations converging to the exact solution in the energy space generates a sequence of positive numbers (explicitly computable by the majorant functional) that tends to zero.


  • V.A. Steklov Institute of Mathematics in St.-Petersburg
  • University of Bergen


  • [1] Ainsworth M., Oden J.T., A posteriori error estimation in finite element analysis, Pure and Applied Mathematics, A Wiley-Interscience Series of Texts, Monographs, and Tracts, Wiley and Sons, New York, 2000
  • [2] Alberty J., Carstensen C., Numerical analysis of time-depending primal elastoplasticity with hardening, SIAM J. Numer. Anal., 2000, 37, 1271–1294
  • [3] Alberty J., Carstensen C., Zarrabi D., Adaptive numerical analysis in primal elastoplasticity with hardening, Comput. Methods Appl. Mech. Engrg., 1999, 171, 175–204
  • [4] Babuška I., Strouboulis T., The finite element method and its reliability, Oxford University Press, New York, 2001
  • [5] Bangerth W., Rannacher R., Adaptive finite element methods for differential equations, Birkhäuser, Berlin, 2003
  • [6] Bensoussan A., Frehse J., Asymptotic behaviour of Norton-Hoff’s law in plasticity theory and H1 regularity, In: Lions J.L. (Ed.) et al., Boundary value problems for partial differential equations and applications, Dedicated to Enrico Magenes on the occasion of his 70th birthday, Paris: Masson. Res. Notes Appl. Math., 1993, 29, 3–25
  • [7] Bildhauer M., Fuchs M., Repin S., A posteriori error estimates for stationary slow flows of power-law fluids, Journal of Non-Newtonian Fluid Mechanics, 2007, 142, 112–122
  • [8] Bildhauer M., Fuchs M., Repin S., A functional type a posteriori error analysis for Ramberg-Osgood Model, ZAMM Z. Angew. Math. Mech., 2007, 87(11–12), 860–876
  • [9] Blaheta R., Numerical methods in elasto-plasticity, Comput. Methods Appl. Mech. Engrg., 1997, 147, 167–185
  • [10] Brokate M., Carstensen C., Valdman J., A quasi-static boundary value problem in multi-surface elastoplasticity, I, Analysis, Math. Methods Appl. Sci., 2004, 27, 1697–1710
  • [11] Brokate M., Carstensen C., Valdman J., A quasi-static boundary value problem in multi-surface elastoplasticity, II, Numerical solution, Math. Methods Appl. Sci., 2005, 28, 881–901
  • [12] Brokate M., Sprekels J., Hysteresis and phase transitions, Springer, New York, 1996
  • [13] Carstensen C., Numerical analysis of the primal problem of elastoplasticity with hardening, Numer. Math., 1999, 82(4), 577–597
  • [14] Carstensen C., Orlando A., Valdman J., A convergent adaptive finite element method for the primal problem of elastoplasticity, Internat. J. Numer. Methods Engrg., 2006, 67(13), 1851–1887
  • [15] Ekeland I., Teman R., Convex analysis and variational problems, North-Holland, Oxford, 1976
  • [16] Fuchs M., Repin S., Estimates for the deviation from the exact solutions of variational problems modeling certain classes of generalized Newtonian fluids, Math. Methods Appl. Sci., 2006, 29, 2225–2244
  • [17] Glowinski R., Lions J.L., Tremolieres R., Analyse numerique des inequations variationnelles, Dunod, Paris, 1976 (in French)
  • [18] Gruber P., Valdman J., Implementation of an elastoplastic solver based on the Moreau, Yosida theorem, Math. Comput. Simulation, 2007, 76(1–3), 73–81
  • [19] Gruber P., Valdman J., Solution of one-time-step problems in elastoplasticity by a slant Newton method, SIAM J. Sci. Comput., 2009, 31(2), 1558–1580
  • [20] Han W., Reddy B.D., Computational plasticity: the variational basis and numerical analysis, Comput. Methods Appl. Mech. Engrg., 1995, 283–400
  • [21] Hofinger A., Valdman J., Numerical solution of the two-yield elastoplastic minimization problem, Computing, 2007, 81, 35–52
  • [22] Krejčí P., Hysteresis, convexity and dissipation in hyperbolic equations, GAKUTO Internat. Ser. Math. Sci. Appl., Vol. 8, Gakkotosho, Tokyo, 1996
  • [23] Lions J.L., Stampacchia G., Variational inequalities, Comm. Pure Appl. Math., 1967, XX(3), 493–519
  • [24] Neittaanmäki P., Repin S., Reliable methods for computer simulation, Error control and a posteriori estimates, Elsevier, New York, 2004
  • [25] Rannacher R., Suttmeier F.T., A posteriori error estimation and mesh adaptation for finite element models in elastoplasticity, Comput. Methods Appl. Mech. Engrg., 1999, 176, 333–361
  • [26] Repin S., A priori error estimates of variational-difference methods for Hencky plasticity problems, Zap. Nauchn. Semin. POMI, 1995, 221, 226–234 (in Russian), English translation: J. Math. Sci., New York, 1997, 87(2), 3421-3427
  • [27] Repin S.I., Errors of finite element methods for perfectly elasto-plastic problems, Math. Models Meth. Appl. Sci., 1996, 6(5), 587–604
  • [28] Repin S., A posteriori estimates for approximate solutions of variational problems with strongly convex functionals, Problems of Mathematical Analysis, 1997, 17, 199–226 (in Russian), English translation: J. Math. Sci., 1999, 97(4), 4311–4328
  • [29] Repin S., A posteriori error estimation for variational problems with uniformly convex functionals, Math. Comp., 2000, 69(230), 481–500
  • [30] Repin S., A posteriori estimates for partial differential equations, Walter de Gruyter Verlag, Berlin, 2008
  • [31] Repin S.I., Seregin G.A., Error estimates for stresses in the finite element analysis of the two-dimensional elastoplastic problems, Internat. J. Engrg. Sci., 1995, 33(2), 255–268
  • [32] Repin S., Valdman J., Functional a posteriori error estimates for problems with nonlinear boundary conditions, J. Numer. Math., 2008, 16(1), 51–81
  • [33] Repin S.I., Xanthis L.S., A posteriori error estimation for elasto-plastic problems based on duality theory, Comput. Methods Appl. Mech. Engrg., 1996, 138, 317–339
  • [34] Seregin G., On the regularity of weak solutions of variational problems of plasticity theory, Algebra i Analiz, 1990, 2(2), 121–140 (in Russian), English translation: Leningrad Mathematical Journal, 1991, 2(2), 321-338
  • [35] Simo J.C., Hughes T.J.R., Computational inelasticity, Springer-Verlag New York, 1998
  • [36] Valdman J., Minimization of functional majorant in a posteriori error analysis based on H(div) multigrid-preconditioned CG method, Advances in Numerical Analysis, to appear
  • [37] Wieners Ch., Nonlinear solution methods for infinitesimal perfect plasticity, ZAMM Z. Angew. Math. Mech., 2007, 87(8–9), 643–660

Typ dokumentu



Identyfikator YADDA

JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.