Functional a posteriori error estimates for incremental models in elasto-plasticity. (English) Zbl 1269.74202

Summary: 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. 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.


74S05 Finite element methods applied to problems in solid mechanics
65K15 Numerical methods for variational inequalities and related problems
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)
Full Text: DOI


[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 http://dx.doi.org/10.1137/S0036142998341301 · Zbl 1049.74010
[3] Alberty J., Carstensen C., Zarrabi D., Adaptive numerical analysis in primal elastoplasticity with hardening, Comput. Methods Appl. Mech. Engrg., 1999, 171, 175-204 http://dx.doi.org/10.1016/S0045-7825(98)00210-2 · Zbl 0956.74049
[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 · Zbl 1020.65058
[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 · Zbl 0831.35047
[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 http://dx.doi.org/10.1016/j.jnnfm.2006.06.001 · Zbl 1109.76007
[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 http://dx.doi.org/10.1002/zamm.200710350 · Zbl 1128.74006
[9] Blaheta R., Numerical methods in elasto-plasticity, Comput. Methods Appl. Mech. Engrg., 1997, 147, 167-185 http://dx.doi.org/10.1016/S0045-7825(97)00012-1 · Zbl 0887.73017
[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 http://dx.doi.org/10.1002/mma.524 · Zbl 1074.74013
[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 http://dx.doi.org/10.1002/mma.593 · Zbl 1112.74007
[12] Brokate M., Sprekels J., Hysteresis and phase transitions, Springer, New York, 1996 · Zbl 0951.74002
[13] Carstensen C., Numerical analysis of the primal problem of elastoplasticity with hardening, Numer. Math., 1999, 82(4), 577-597 http://dx.doi.org/10.1007/s002110050431 · Zbl 0947.74061
[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 http://dx.doi.org/10.1002/nme.1686 · Zbl 1127.74040
[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 http://dx.doi.org/10.1002/mma.773 · Zbl 1105.76049
[17] Glowinski R., Lions J.L., Tremolieres R., Analyse numerique des inequations variationnelles, Dunod, Paris, 1976 (in French) · Zbl 0358.65091
[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 http://dx.doi.org/10.1016/j.matcom.2007.01.036 · Zbl 1132.74045
[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 http://dx.doi.org/10.1137/070690079 · Zbl 1186.74025
[20] Han W., Reddy B.D., Computational plasticity: the variational basis and numerical analysis, Comput. Methods Appl. Mech. Engrg., 1995, 283-400 · Zbl 0847.73078
[21] Hofinger A., Valdman J., Numerical solution of the two-yield elastoplastic minimization problem, Computing, 2007, 81, 35-52 http://dx.doi.org/10.1007/s00607-007-0242-2 · Zbl 1177.74167
[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 http://dx.doi.org/10.1002/cpa.3160200302 · Zbl 0152.34601
[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 http://dx.doi.org/10.1016/S0045-7825(98)00344-2 · Zbl 0954.74070
[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 · Zbl 0894.73228
[27] Repin S.I., Errors of finite element methods for perfectly elasto-plastic problems, Math. Models Meth. Appl. Sci., 1996, 6(5), 587-604 http://dx.doi.org/10.1142/S0218202596000237 · Zbl 0856.73071
[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 · Zbl 0941.65059
[29] Repin S., A posteriori error estimation for variational problems with uniformly convex functionals, Math. Comp., 2000, 69(230), 481-500 http://dx.doi.org/10.1090/S0025-5718-99-01190-4 · Zbl 0949.65070
[30] Repin S., A posteriori estimates for partial differential equations, Walter de Gruyter Verlag, Berlin, 2008 http://dx.doi.org/10.1515/9783110203042 · Zbl 1162.65001
[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 http://dx.doi.org/10.1016/0020-7225(94)00057-Q · Zbl 0899.73532
[32] Repin S., Valdman J., Functional a posteriori error estimates for problems with nonlinear boundary conditions, J. Numer. Math., 2008, 16(1), 51-81 http://dx.doi.org/10.1515/JNUM.2008.003 · Zbl 1146.65054
[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 http://dx.doi.org/10.1016/S0045-7825(96)01136-X · Zbl 0886.73082
[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 · Zbl 0704.73104
[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 · Zbl 1200.65095
[37] Wieners Ch., Nonlinear solution methods for infinitesimal perfect plasticity, ZAMM Z. Angew. Math. Mech., 2007, 87(8-9), 643-660 http://dx.doi.org/10.1002/zamm.200610339 · Zbl 1128.74008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.