Keywords: stress local error, finite element analysis, linear elasticity.

A major concern in the mechanical field has always been to control the quality of finite element computations. For linear analysis, several methods have been developped. Three main approaches have been introduced: error in the constitutive relation (see [1]), residuals methods [2] and estimators based on stress recovery [3].

Although these error estimators have proved to be efficient in the estimation of the global error, they do not involve local errors, for instance on the stresses. Such global information is totally insufficient for mechanical design. The accuracy of the solution is assessed in terms of local quantities that are of practical interest for engineers in many design criteria.

The knowledge of the local error is both an industrial priority and a key issue in research. The concept of pollution was introduced by Babuska and Strouboulis [4,5]. In [6,7,8,9], other approaches leading to local error estimation have been proposed. All these methods involve calculation of an influence function. This influence function, obtained through an approximate computation leads to quantities of interest such as stress, displacement, intensity factor, ...

A different approach for stress local error evaluation exists. This approach was introduced and implemented in [8] for 2D problems. It is based on the concept of error on the constitutive relation and associated techniques to recover enhanced statically admissible stress field from the finite element solution and the data.

The principle of error on the constitutive relation relies on splitting the equations of the problem to be solved in two groups:

• the admissibility conditions (kinematic constraints and equilibrium equations),
• the constitutive relation

The error on the constitutive relation characterizes the quality of an approximate solution that satisfies the first group of equations (admissibility conditions). In elasticity the error is measured by:

and is an upper-bound of the exact global error on the displacement field or on the stress field.

The finite element displacement field satisfies the kinematic constraints and generaly we simply use . The key point of the application of the error on the constitutive relation for the F.E.M is the construction of a statically admissible stress field from the finite element solution and the data. The quality of the error estimator depends strongly on the quality of the recovered admissible stress field.

A technique of construction of an enhanced statically admissible stress field was developped in [10] and extented to 2D plasticity problems in [11]. This technique leads to a new generation of error on the constitutive relation that presents very good global effectivity indexes as well as very good local effectivity indexes (i.e. effectivity indexe computed on a mesh element).

This enhanced technique is used to compute an upper bound of the local errors on stresses [8]. A first extension to 3-D problems considering linear tetrahedric elements was performed in [12]. In this paper a further step is done studying other classical 3-D finite elements. It is shown that the local constitutive error estimator gives a satisfactory practical upper bound on significants examples.

1

P. Ladevèze. Constitutive relation error estimators for time-dependent non linear fe analysis. Comp. Meth. in Applied Mech. and Engrg., 188:775-788, 2000. doi:10.1016/S0045-7825(99)00361-8

2

I. Babuska and W.C. Rheinboldt. A posteriori estimates for the finite element method. Int. J. for Num. Meth. in Engrg., 12:1597-1615, 1978. doi:10.1002/nme.1620121010

3

O.C. Zienkiewicz and J.Z. Zhu. A simple error estimator and adaptive procedure for practical enineering analysis. Int. J. for Num. Meth. in Engrg., 24:337-357, 1987. doi:10.1002/nme.1620240206

4

I. Babuska, T. Strouboulis, C.S. Upadhyay, and S.K. Gangaraj. A posteriori estimation and adapative control of the pollution error in the h-version of the finite element method. Int. J. for Num. Meth. in Engrg., 38:4207-4235, 1995. doi:10.1002/nme.1620382408

5

I. Babuska, T. Strouboulis, S.K. Gangaraj, K. Copps, and D.K. Datta. A posteriori estimation of the error estimate. In P. Ladevèze and J.T. Oden, editors, Advances in adaptive computational methods in mechanics, pages 155-197. Studies in Applied Mechanics 47, Elsevier, 1998.

6

R. Rannacher and F.T. Stuttmeier. A feedback approach to error control in finite element methods: Application to linear elasticity. Computational Mechanic, 19:434-446, 1997. doi:10.1007/s004660050191

7

J. Peraire and A. Patera. Bounds for linear-functional outputs of coercive partial differential equations: local indicators and adaptive refinement. In P. Ladevèze and J.T. Oden, editors, Advances in Adaptive Computational Methods, pages 199-216. Elsevier, 1998. doi:10.1016/S0922-5382(98)80011-1

8

P. Ladevèze, P. Rougeot, P. Blanchard, and J.P. Moreau. Local error estimators for finite element analysis. Comp. Meth. in Applied Mech. and Engrg., 176:231-246, 1999. doi:10.1016/S0045-7825(98)00339-9

9

S. Prudhomme and J.T. Oden. On goal-oriented error estimation for elliptic problems : application to the control of pointwise errors. Comp. Meth. in Applied Mech. and Engrg., 176:313-331, 1999. doi:10.1016/S0045-7825(98)00343-0

10

P. Ladevèze and P. Rougeot. New advances on a posteriori error on constitutive relation in f.e. analysis. Comp. Meth. in Applied Mech. and Engrg., 150:239-249, 1997. doi:10.1016/S0045-7825(97)00089-3

11

L. Gallimard, P. Ladevèze, and J.P. Pelle. An enhanced error estimator on constitutive relation for plasticity problems. Computers and Structures, 78:801-810, 2000. doi:10.1016/S0045-7949(00)00056-0

12

E. Florentin, L. Gallimard, P. Ladevèze, and J.P. Pelle. Stress local upper bound for 2d and 3d elasticity problems. In Proc. European Congress on Computational Methods in Applied Science and Engineering, Barcelona, septembre 2000, 2000.

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