A posteriori error estimation for elasto-plastic problems based on duality theory. (English) Zbl 0886.73082

Repin, Sergey I.; Xanthis, Leonidas S.

Comput. Methods Appl. Mech. Eng. 138, No. 1-4, 317-339 (1996).

We introduce a new approach to a posteriori error estimation for elasto-plastic problems based on the duality theory of the calculus of variations. We show that, in spite of the prevailing view, duality methods provide a viable way for obtaining computable a posteriori error estimates for nonlinear boundary value problems without directly solving the dual problem. Rigorous mathematical analysis leads to what we call duality error estimators consisting of two parts: the error in the constitutive law and the error in the equilibrium equations. The duality error estimators hold for any conforming approximation of the exact solution regardless of whether or not they satisfy the Galerkin orthogonality condition. In particular, they encompass the familiar smoothening or gradient averaging techniques commonly used in practice.

MSC:

74S30	Other numerical methods in solid mechanics (MSC2010)
74P10	Optimization of other properties in solid mechanics
74C99	Plastic materials, materials of stress-rate and internal-variable type

Keywords:

calculus of variations; nonlinear boundary value problems; duality error estimators; conforming approximation

References:

[1]	Ainsworth, M.; Craig, A., A posteriori error estimators in the finite element method, Numer. Math., 60, 429-463 (1992) · Zbl 0757.65109
[2]	Ainsworth, M.; Oden, J. T., A unified approach to a posteriori error estimation using element residual methods, Numer. Math., 65, 23-50 (1993) · Zbl 0797.65080
[3]	Ainsworth, M.; Oden, J. T., A posteriori error estimation in finite element analysis, (Technical report (1995), University of Leicester) · Zbl 0789.65083
[4]	Anzellotti, O.; Giaquinta, M., Existence of the displacements field for an elastic-plastic body subjected to Hencky’s law and von Mises yield condition, Manuscripta Math., 32, 101-136 (1980) · Zbl 0465.73022
[5]	Argyris, J. H., Elasto-plastic matrix displacement analysis of three-dimensional media, J. Roy. Aeronaut. Soc., 69, 633-635 (1965)
[6]	Argyris, J. H.; Doltsinis, I. St., A primer on superplasticity in natural formulation, Comput. Methods Appl. Mech. Engrg., 14, 259-294 (1978) · Zbl 0389.73076
[7]	Argyris, J. H.; Mlejnek, H.-P., (Die Methode der Finite Elemente, III (1988)) · Zbl 0373.01008
[8]	Arthurs, A. M., Complementary Variational Principles (1980), Clarendon Press: Clarendon Press Oxford · Zbl 0437.35001
[9]	Ascher, U. M.; Mattheij, R. M.M.; Russell, R. D., Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (1988), Prentice Hall: Prentice Hall Englewood Cliffs, NJ · Zbl 0666.65056
[10]	Aubin, J. P.; Burchard, H. G., Some aspects of the method of the hypercicle applied to elliptic variational problems, (Hubbard, B., Numerical Solutions of Partial Differential Equations — II, SYNSPADE 1970 (1971), Academic Press: Academic Press New York, London)
[11]	Babuška, I.; Duran, R.; Rodriguez, R., Analysis of the efficiency of an a posteriori error estimator for linear triangular finite elements, SIAM J. Numer. Anal., 29, 4, 947-964 (1992) · Zbl 0759.65069
[12]	Babuška, I.; Planck, L.; Rodriguez, R., Basic problems of a posteriori error estimation, Comput Methods Appl. Mech. Engrg., 101, 97-112 (1992) · Zbl 0787.65079
[13]	Babuǎka, I.; Rheinboldt, W. C., A-posteriori error estimates for the finite element method, Int. J. Numer. Methods Engrg., 12, 1597-1615 (1978) · Zbl 0396.65068
[14]	Babuška, I.; Rheinboldt, W. C., Error estimates for adaptive finite element computations, SIAM J. Numer. Anal., 15, 4, 736-754 (1978) · Zbl 0398.65069
[15]	Babuška, I.; Rheinboldt, W. C., A posteriori error analysis of finite element solutions for one-dimensional problems, SIAM J. Numer. Anal., 18, 3, 565-589 (1981) · Zbl 0487.65060
[16]	Babuška, I.; Rodriguez, R., The problem of the selection of an a posteriori error indicator based on smoothening techniques, Int. J. Numer. Methods Engrg., 36, 539-567 (1993) · Zbl 0770.65065
[17]	Babuška, I.; Strouboulis, T.; Upadhyay, C. S., A model study of the quality of a posteriori error estimators for linear elliptic problems. Error estimation in the interior of patchwise uniform grids of triangles, Comput. Methods Appl. Mech. Engrg., 114, 4, 307-378 (1994)
[18]	(Babuǎka, I.; Zienkiewicz, O. C.; Gago, J.; de A. Oliveira, E. R., Accuracy Estimates and Adaptive Refinements in Finite Element Computations (1986), Wiley: Wiley New York)
[19]	Bank, R. E.; Weiser, A., Some a posteriori error estimators for elliptic partial differential equations, Math. Comput., 44, 283-301 (1985) · Zbl 0569.65079
[20]	Carathéodory, C., (Calculus of Variations and Partial Differential Equations of the First Order, Volumes 1 and 2 (1967), Holden Day: Holden Day San-Francisco, CA), (First edition in German: B.G. Teubner, Berlin, 1935)
[21]	Ciarlet, P. G., The Finite Element Method for Elliptic Problems (1978), North-Holland: North-Holland Amsterdam, New York, Oxford · Zbl 0445.73043
[22]	Ciarlet, P. G.; Schultz, M. H.; Varga, R. S., Numerical methods of high-order accuracy for nonlinear boundary value problems, Numer. Math., 9, 394-430 (1967) · Zbl 0155.20403
[23]	(Cottle, R. W.; Giannessi, F.; Lions, J-L., Variational Inequalities and Complementary Problems, Theory and Applications (1980), Wiley: Wiley New York)
[24]	de Veubeke, B. Fraeijs, Displacement and equilibrium models in the finite element method, (Zienkiewicz, O. C.; Holister, G. S., Stress Analysis (1965), Wiley: Wiley London, New York), 145-197 · Zbl 0245.73031
[25]	Duvaut, G.; Lions, J. L., Inequalities in Mechanics and Physics (1976), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York · Zbl 0331.35002
[26]	Ekeland, I.; Temam, R., Convex Analysis and Variational Problems (1976), North-Holland: North-Holland Amsterdam, Oxford, New-York
[27]	Glowinski, R., Numerical Methods for Nonlinear Variational Problems (1984), Springer-Verlag: Springer-Verlag New-York, Berlin, Heidelberg, Tokyo · Zbl 0575.65123
[28]	Han, W., A posteriori error analysis for linearization of nonlinear elliptic problems and their discretizations, Math. Methods Appl. Sci., 17, 487-508 (1994) · Zbl 0799.35082
[29]	Iljushin, A. A., Some problems of plastic deformations, Prikl. Mat. Mekh., 7, 245-272 (1943), (in Russian) · Zbl 0063.02959
[30]	Johnson, C.; Hansbo, P., Adaptive finite element methods in computational mechanics, Comput. Methods Appl. Mech. Engrg., 101, 143-181 (1992) · Zbl 0778.73071
[31]	Johnson, C.; Mercier, B., Some equilibrium finite element methods for two-dimensional elasticity problems, Numer. Math., 30, 101-116 (1978) · Zbl 0427.73072
[32]	Kelly, D. W., The self-equilibration of residuals and complementary a posteriori error estimates in the finite element method, Int. J. Numer. Methods Engrg., 20, 1491-1506 (1984) · Zbl 0575.65100
[33]	Kohn, R.; Temam, R., Dual spaces of stresses and strains with applications to Hencky plasticity, Appl. Math. Optim., 10, 1-35 (1983) · Zbl 0532.73039
[34]	Ladevèze, P.; Leguillon, D., Error estimate procedure in the finite element method and applications, SIAM J. Numer. Anal., 20, 3, 485-509 (1983) · Zbl 0582.65078
[35]	Ladevèze, P.; Pelle, J.-P.; Rougeot, Ph., Error estimation and mesh optimization for classical finite elements, Engrg. Comput., 8, 69-80 (1991)
[36]	Mikhlin, S. G., Variational Methods in Mathematical Physics (1964), Pergamon: Pergamon Oxford · Zbl 0119.19002
[37]	Mikhlin, S. G., Error Analysis in Numerical Processes (1991), Wiley: Wiley New York
[38]	Mosolov, P. P.; Myasnikov, V. P., Mechanics of Rigid Plastic Bodies (1981), Nauka: Nauka Moskow, (in Russian) · Zbl 0551.73002
[39]	Nadai, A., Plastic behaviour of metals in the strain hardening range, J. Appl. Phys., 8, 205-213 (1937) · JFM 64.1425.01
[40]	Oden, J. T.; Ainsworth, M.; Wu, W., An a posteriori error estimate for finite element approximations of the Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 111, 185-202 (1993) · Zbl 0844.76056
[41]	Oden, J. T.; Demkowicz, L.; Rachowicz, W.; Westermann, T. A., Towards a universal \(h-p\) adaptive finite element strategy. Part 2. A posteriori error estimation, Comput. Methods Appl. Mech. Engrg., 77, 113-180 (1989) · Zbl 0723.73075
[42]	Pousin, J.; Rappaz, J., Consistency, stability, a priori and a posteriori errors for Petrov-Galerkin methods applied to nonlinear problems, Numer. Math., 69, 2, 213-232 (1994) · Zbl 0822.65034
[43]	Repin, S. I., Numerical analysis of nonsmooth variational problems of perfect plasticity, Russ. J. Numer. Anal. Math. Modell., 9, 33-46 (1994) · Zbl 0820.73079
[44]	Repin, S. I., A priori error estimates of variational-difference methods for Hencky plasticity problems, Zapiski Nauchnych Seminarov POMI, 221, 226-234 (1995), (in Russian, English translation in the Journal of Mathematical Sciences 1995) · Zbl 0894.73228
[45]	Repin, S. I., Errors of finite element methods for perfectly elasto-plastic problems, \(M^3\) AS — Math. Models Methods Appl. Sci. (1996), to appear · Zbl 0856.73071
[46]	Repin, S. I.; Seregin, G. A., Error estimates for stresses in the finite element analysis of the two-dimensional elasto-plastic problems, Int. J. Engrg. Sci., 33, 2, 255-268 (1995) · Zbl 0899.73532
[47]	Repin, S. I.; Xanthis, L. S., A posteriori error estimation for nonlinear variational problems, C.R. Acad. Sci. Paris, Série I (1996), to appear · Zbl 0886.73082
[49]	Schwab, C.; Xanthis, L. S., The method of arbitrary lines — an \(h-p\) error analysis for singular problems, C.R. Acad. Sci. Paris, Série I, 315, 1421-1426 (1992) · Zbl 0782.65137
[50]	Strang, G.; Fix, G., An Analysis of the Finite Element Method (1973), Prentice Hall: Prentice Hall Englewood Cliffs, NJ · Zbl 0278.65116
[51]	Suquet, P., Existence et régularité des solutions des équations de la plasticité parfaite, C.R. Acad. Sci., Paris, Série A, 1201-1204 (1978) · Zbl 0378.35057
[52]	(R. Temam, Problèmes Mathématique en Plasticité (1983), Bordas: Bordas Paris) · Zbl 0547.73026
[53]	Temam, R.; Strang, G., Functions of bounded deformation, Arch. Rational Mech. Anal., 75, 7-21 (1980) · Zbl 0472.73031
[54]	Verfürth, R., A posteriori error estimates for nonlinear problems. Finite element discretisations of elliptic equations, Math. Comput., 62, 445-475 (1994) · Zbl 0799.65112
[55]	Verfürth, R., A posteriori error estimation and adaptive mesh-refinement techniques, J. Comput. Appl. Math., 50, 67-83 (1994) · Zbl 0811.65089
[56]	Weil, H., The method of orthogonal projections in potential theory, Duke Math. J., 7 (1940)
[57]	Xanthis, L. S.; Schwab, C., The method of arbitrary lines, C.R. Acad. Sci. Paris, Série I, 312, 181-187 (1991) · Zbl 0722.65058
[59]	Zienkiewicz, O. C.; Zhu, J. Z., A simple error estimator and adaptive procedure for practical ingineering analysis, Int. J. Numer. Methods Engrg., 24, 337-357 (1987) · Zbl 0602.73063
[60]	Zienkiewicz, O. C.; Zhu, J. Z., The superconvergent patch recovery (SPR) and adaptive finite element refinement, Comput. Methods Appl. Mech. Engrg., 101, 207-224 (1992) · Zbl 0779.73078