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An extension of the concept of error on constitutive relation is proposed to the case of Mindlin plate finite element computations. The error of the performed analysis is estimated from the incompatibility in relation with the constitutive equation of admissible fields calculated from the finite element results. In a first stage, loads and moments densities leading to the equilibrium of each element are computed on the element edges as the sums of densities derived from the finite element solution and of densities with a resultant equal to zero on each element edge. Then strictly statically admissible stress resultants are calculated within each element. Both of the two stages allow an optimization for the statically admissible field in order to get a more accurate error. The calculations are local which is very interesting especially in case of complex structure analyses with a large number of degrees of freedom for which adaptivity is an important feature. A set of examples shows the efficiency of the proposed estimator and the good adaptation of the error on constitutive law method to Mindlin plate analysis.

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.

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.

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.

In this paper, we discuss the application of the constitutive relation error (CRE) to model updating and validation in the context of uncertain measurements. First, a parallel is drawn between the CRE method and a general theory for inverse problems proposed by Tarantola. Then, an extension of the classical CRE method considering uncertain measurements is proposed. It is shown that the proposed mechanics‐based approach for model validation is very effective in filtering noise in the experimental data. The method is applied to an industrial structure, the SYLDA5, which is a satellite support for Ariane5. The results demonstrate the robustness of the method in actual industrial situations.

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.

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 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 present an approach for calibrating the numerical models of dynamical systems that have spatially localized nonlinear components. The approach implements the extended constitutive relation error (ECRE) method using multi-harmonic coefficients and is conceived to separate the errors in the representation of the global, linear and local, nonlinear components of the dynamical system through a two-step process.

The first step focuses on the system’s predominantly linear dynamic response under a low magnitude periodic excitation. In this step, the discrepancy between measured and predicted multi-harmonic coefficients is calculated in terms of residual energy. This residual energy is in turn used to spatially locate errors in the model, through which one can identify the erroneous model inputs which govern the linear behavior that need to be calibrated. The second step involves measuring the system’s nonlinear dynamic response under a high magnitude periodic excitation. In this step, the response measurements under both low and high magnitude excitation are used to iteratively calibrate the identified linear and nonlinear input parameters.

When model error is present in both linear and nonlinear components, the proposed iterative combined multi-harmonic balance method (MHB)-ECRE calibration approach has shown superiority to the conventional MHB-ECRE method, while providing more reliable calibration results of the nonlinear parameter with less dependency on a priori knowledge of the associated linear system.

This two-step process is advantageous as it reduces the confounding effects of the uncertain model parameters associated with the linear and locally nonlinear components of the system.

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.