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Anisotropic discretization- and model-error estimation in solid mechanics by local Neumann problems. (English) Zbl 0954.74072
Summary: First, we give a survey of existing residuum-based error estimators and error indicators. Generally, residual error estimators (which have at least upper bound in contrast to indicators) can be locally computed from residua of equilibrium and stress-jumps at element interfaces using Dirichlet or Neumann conditions for element patches or individual elements (REM). Another equivalent method for error estimation can be derived from a posteriori computed improved boundary tractions which provide exact equilibrium of elements. They are computed from local Neumann problems and yield improved local solution by testing with higher test functions. This method is called posterior equilibrium method and yields better efficiency indices than REM for the investigated problem, even in case of locking. Also, the important feature of model adaptivity via hierarchical error estimation can only be realized by this strategy. Global Neumann-type estimators and examples are presented for elastic and elastoplastic deformations, controlling equilibrium in tangential space.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
65N15 Error bounds for boundary value problems involving PDEs
74B05 Classical linear elasticity
74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)
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[1] Babuška, I.; Rheinboldt, W.C., Error estimates for adaptive finite element computations, SIAM J. numer. anal., 15, 736-754, (1978) · Zbl 0398.65069
[2] Babuška, I.; Strouboulis, T.; Gangaraj, S.K.; Upadhyay, C.S., A-posteriori estimation and adaptive control of the pollution-error in the h-version of the finite element method, Int. J. numer. methods engrg., 38, 4207-4235, (1995) · Zbl 0844.65078
[3] Bank, R.E.; Weiser, A., Some a posteriori error estimators for elliptic partial differential equations, Math. comput., 44, 283-301, (1985) · Zbl 0569.65079
[4] Barthold, F.-J.; Schmidt, M.; Stein, E., Error estimation and mesh adaptivity for elasto-plastic deformations, (), 597-602, Part 1
[5] Brink, U., Adaptive gemischte finite elemente in der linearen finiten elastostatik und deren kopplung mit randelementen, ()
[6] Brink, U.; Stein, E., A posteriori error estimation in large-strain elasticity using equilibrated local Neumann problems, Comput. methods appl. mech. engrg., 161, 77-101, (1998) · Zbl 0943.74062
[7] Bufler, H.; Stein, E., Zur plattenberechnung mittels finiter elements, Ing. archiv., 39, 248-260, (1970) · Zbl 0197.21901
[8] Gruttmann, F.; Stein, E.; Wagner, W., A generalized FE-method for non-linear composite shell with 2d- and 3d-modeling, (), 2533-2538
[9] Johnson, C.; Hansbo, P., Adaptive finite element methods in computational mechanics, Comput. methods appl. mech. engrg., 101, 143-181, (1992) · Zbl 0778.73071
[10] Ladevèze, P.; Leguillon, D., Error estimate procedure in the finite element method and applications, SIAM J. numer. anal., 20, 485-509, (1983) · Zbl 0582.65078
[11] Ladevèze, P.; Maunder, E.A.W., A general methodology for recovering equilibration finite element tractions and stress fields for plate and solid elements, (), 34-35 · Zbl 0886.73065
[12] Ohnimus, S., Theorie und numerik dimensions- und modelladaptiver finite-elemente-methoden von flächentragwerken, (1996), Berichte des Instituts für Baumechanik und Numerische Mechanik der Universität Hannover, F96/6
[13] S. Ohnimus, E. Stein and E. Walhorn, Local error estimates in FEM for displacements and stresses in linear elasticity by solving local Neumann problems, Int. J. Numer. Meth. Engrg., submitted. · Zbl 1057.74042
[14] Ramm, E.; Cirak, F., Adaptivity for nonlinear thin-walled structures, (), 145-163
[15] Rannacher, R.; Suttmeier, F.-T., A feed-back approach to error control in finite element method: application to linear elasticity, (), 1-24 · Zbl 0894.73170
[16] Stein, E.; Ahmad, R., An equilibrium method for stress calculation using finite element displacements models, Comput. methods appl. mech. engrg., 10, 175-198, (1977) · Zbl 0347.73058
[17] Stein, E.; Barthold, F.-J.; Ohnimus, S.; Schmidt, M., Adaptive finite elements in elastoplasticity with mechanical error indicators and Neumann-type estimators, ()
[18] Stein, E.; Ohnimus, S., Dimensional adaptivity in linear elasticity with hierarchical test-spaces for h-and p-refinement processes, Engrg. comput., 12, 107-119, (1996)
[19] Stein, E.; Ohnimus, S., Coupled model- and solution-adaptivity in the finite-element method, Comput. methods appl. mech. engrg., 150, 327-350, (1997) · Zbl 0926.74127
[20] Stein, E.; Ohnimus, S., Equilibrium method for postprocessing and error estimation in the finite element method, First Int. Workshop on Trefftz Method, 30 May-1 June, 1996, Cracov, Poland, Comput. assist. mech. engrg. sci., 4, 645-666, (1997) · Zbl 0974.74066
[21] Verführth, R., A review of a posteriori error estimation and adaptive mesh-refinement techniques, (1996), Wiley & Teubner Verlag
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