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By Pierre LadevA?ze, Jean P. Pelle

Modeling and simulation are significant to a mechanical engineer's activity. Increasingly advanced versions are getting used in many instances on an everyday basis. This revolution is the results of the extreme growth in computing device know-how by way of either and software program. This paintings bargains with the keep an eye on of the hypotheses top from a mechanical version, often coming from continuum mechanics, to a numerical version, i.e. the mastery of the mechanical computation technique itself.  specific realization is given to structural research which, during this context, is the main complex area. an important a part of this paintings is devoted to the applying of mistakes estimators to the keep an eye on of many of the parameters concerned with a calculation, starting with the parameters relating to the mesh.

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42) 3. 1 Introduction In this chapter, we give an overview of the two other main methods which have been proposed for estimating discretization errors for linear problems. [LADEVEZE-ODEN, 1998] gave a precise idea of the state of the art. 2 is devoted to estimators based on the residuals of the equilibrium equations first proposed by B A B U S K A and R H E I N B O L D T [BABUSKA-RHEINBOLDT , 1978 ]. Several variants of these (of which one of the most recent can be found in [DIEZ-EGOZENZ -HUERTA, 1998]) have since been developed by many authors: explicit estimators and implicit estimators by element or by groups of elements.

These equilibrium deficiencies, also called residuals, are explained below along with their connection with the error in the solution. 1) The residuals equation. 2) with: if where if is not a side of , : is the element adjacent to element E with respect to side is a side of : ; ; The residuals measure the discontinuity of the stress vector in the at the interface between two elements, and the nonsatisfaction of direction the equilibrium conditions . 3) Thus, the error is the solution of an elasticity problem whose applied loads (in the interior and on the interfaces of the mesh being used) are the residuals of the equilibrium equations.

The denomination “energy theorems” is not really appropriate; these theorems are related to the global formulation of the constitutive relation, not to the notion of energy. MOREAU [MOREAU , 1966, 1974] and to the further developments by NAYROLES [NAYROLES, 1973]. 1 Construction of an admissible pair The calculation of the error in constitutive relation cannot be made directly on the pair because this pair is not admissible. 1. Principle of utilization. 1): the data of the reference problem; The constitutive relation error method for linear problems 39 the finite element solution .

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