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We present several applications for 2D or axisymmetric elasticity problems of a method to control the quality of a finite element computation, and to optimize the choice of meshes. The method used, which is very general, is based (i) on the concept of error in constitutive relation and (ii) on explicit techniques to construct admissible fields. Illustrative examples are shown for several 2D or axisymmetric elements (3 or 6 node triangles, 4 or 8 node quadrilaterals). They have been achieved with our code ESTEREF, a post‐processor of error computation and mesh optimization which can be interfaced with any finite element code.

In this paper we present a new method of mesh optimization which automatically accounts for steep gradients. With this method, the user needs no previous knowledge of the problem. The method is based on the concept of error in the constitutive relation, coupled with an h‐version remeshing procedure. The steep gradient regions are detected by using the local errors, which are taken into account using the finite energy element. Consequently the procedure can be extended to all estimators of discretization errors. It is implemented in our code ESTEREF, a post‐processor of error computation and mesh optimization that can be used with any finite element code. Numerous examples show the capabilities of the proposed method.

A method for controlling the quality of finite element analyses for plate structures is proposed herein. It is based on the concept of error in the constitutive relation as well as on associated techniques for constructing admissible displacement‐stress fields with respect to a reference model. In this study, the chosen model is either Reissner‐Mindlin’s or Kirchhoff‐Love’s model. The finite element used is the DKT element; these error estimators allow us to determine that this element converges for Kirchhoff‐Love’s model. Once these error estimators have been identified, techniques of adaptive meshing developed in 2D are applied and several examples are presented.

The purpose of this paper is to acquire strict upper and lower bounds on quantities of slender beams on Winkler foundation in finite element analysis.

It leans on the dual analysis wherein the constitutive relation error (CRE) is used to perform goal-oriented error estimation. Due to the coupling of the displacement field and the stress field in the equilibrium equations of the beam, the prolongation condition for the stress field which is the key ingredient of CRE estimation is not directly applicable. To circumvent this difficulty, an approximate problem and the solution thereof are introduced, enabling the CRE estimation to proceed. It is shown that the strict bounding property for CRE estimation is preserved and strict bounds of quantities of the beam are obtainable thereafter.

Numerical examples are presented to validate the strict upper and lower bounds for quantities of beams on elastic foundation by dual analysis.

This paper deals with one-dimensional (1D) beams on Winkler foundation. Nevertheless, the present work can be naturally extended to analysis of shells and 2D and 3D reaction-diffusion problems for future research.

CRE estimation is extended to analysis of beams on elastic foundation by a decoupling strategy; strict upper bounds of global energy norm error for beams on elastic foundation are obtained; strict bounds of quantities for beams on elastic foundation are also obtained; unified representation and corresponding dual analysis of various quantities of the beam are presented; rigorous derivation of admissible stresses for beams is given.

Many industrial analyses require the resolution of complex nonlinear problems. For such calculations, error‐controlled adaptive strategies must be used to improve the quality of the results. In this paper, adaptive strategies for nonlinear calculations in plasticity based on an enhanced error on the constitutive relation are presented. We focus on the adaptivity of the mesh and of the time discretization.

In this paper, we focus on the quality of a 2D elastic finite element analysis.

Our objective is to control the discretization parameters in order to achieve a prescribed local quality level over a dimensioning zone. The method is based on the concept of constitutive relation error.

The method is illustrated through 2D test examples and shows clearly that in terms of cost, this technique provides an additional benefit compared to previous methods.

The saving would be even more significant if this mesh adaptation technique were applied in three dimensions. Indeed, in 3D problems, the computing cost is vital and, in general, it is this cost that sets the limits.

This tool is directly usable in the design stage.

The new tool developed guarantees a local quality level prescribed by the user.

This paper aims to deal with the verification of local quantities of interest obtained through linear elastic finite element analysis. A technique is presented for determining the most accurate error estimation. This technique enables one to address industrial‐size problems while keeping computing costs reasonable.

The concept of error in constitutive relation is used to assess the quality of the finite element solution. The key issue is the construction of admissible fields. The objective is to show that it is possible to build admissible fields using a new method. These fields are obtained by using a high‐quality construction over a limited zone while the construction is less refined and less expensive elsewhere.

Numerical tests are presented in order to illustrate a very satisfying presented methodology. It shows clearly how to take advantage of the method to treat large examples. They clearly show the interest of this new method to treat large examples.

The paper demonstrates clearly that verification of large finite element problem must have dedicated methods in order to be applicable.

This paper aims to focus on the local quality of outputs of interest computed by a finite element analysis in linear elasticity.

In particular outputs of interest are studied which do not depend linearly on the solution of the problem considered such as the L2‐norm of the stress and the von Mises' stress. The method is based on the concept of error in the constitutive relation.

The method is illustrated through 2D test examples and shows that the proposed error estimator leads in practice to upper bounds of the output of interest being studied.

This tool is directly usable in the design stage. It can be used to develop efficient adaptive techniques.

The interest of this paper is to provide an estimation of the local quality of L2‐norm of the stress and the Von Mises' stress as well as practical upper bounds for these quantities.

The analysis of error estimation is addressed in the framework of viscoplasticity problems, this is to say, of incompressible and non‐linear materials. Firstly, Zienkiewicz—Zhu (Z²) type error estimators are studied. They are based on the comparison between the finite element solution and a continuous solution which is computed by smoothing technique. From numerical examples, it is shown that the choice of a finite difference smoothing method (Orkisz’ method) improves the precision and the efficiency of this type of estimator. Then a Δ estimator is introduced. It makes it possible to take into account the fact that the smoothed solution does not verify the balance equations. On the other hand, it leads us to introduce estimators for the velocity error according to the L² and L^∞norms, since in metal forming this error is as important as the energy error. These estimators are applied to an industrial problem of extrusion, demonstrating all the potential of the adaptive remeshing method for forming processes.

Simplified methods are often employed for the analysis of reinforced concrete beams (R‐C beams). A three‐dimensional problem (3D) is often transformed into a two‐dimensional problem (2D) with some assumptions which are usually established in static. The essential reason for this simplification lies in the fact that the 3D finite element analysis is so expensive that it is impossible to study directly the non‐linear behaviour of R‐C beams in many cases. Our purpose is to present a specific method which allows the direct 3D analysis of R‐C beams with a suitable numerical cost. First, the 3D linear heterogeneous beam theory is briefly recalled as well as the continuum damage model used for concrete. Second, the non‐linear behaviour of concrete is introduced in the 3D beam theory. Several numerical examples illustrate the effectiveness of the method.