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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 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.

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.

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.

Traditional adaptive analysis based on a coarse mesh, using finite element method (FEM) analysis, produces the original solution. Then post-processing the result and figuring out the regions should be refined and these regions refined once. Finally, this new mesh is used to get the solution of first refinement. After several iterations of above procedures, we can achieve the last result that is closer to the true solution, which takes time, making adaptive scheme inpractical to engineering application. The paper aims to discuss these issues.

This paper based on FEM proposes a multi-level refinement strategy with a refinement strategy and an indicator. The proposed indicator uses value of the maximum difference of strain energy density among the elements that associated with one node, and divides all nodes into several categories based on the value. A multi-level refinement strategy is proposed according to which category the node belongs to refine different elements to different times rather than whether refine or not.

Multi-level refinement strategy takes full use of the numerical calculation, resulting in the whole adaptive analysis that only need to iterate twice while other schemes must iterate more times. Using much less times of numerical calculation and approaches, more accurate solution, making adaptive analysis more practical to engineering.

Multi-level refinement strategy takes full use of the numerical calculation, resulting in the whole adaptive analysis only need iterate twice while other schemes must iterate more times. using much less times of numerical calculation and approaches more accurate solution, making adaptive analysis more practical to engineering.

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 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.

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 present a variational mesh h-adaption approach for strongly coupled thermomechanical problems.

The mesh is adapted by local subdivision controlled by an energy criterion. Thermal and thermomechanical problems are of interest here. In particular, steady and transient purely thermal problems, transient strongly coupled thermoelasticity and thermoplasticity problems are investigated.

Different test cases are performed to test the robustness of the algorithm for the problems listed above. It is found that a better cost-effectiveness can be obtained with that approach compared to a uniform refining procedure. Because the algorithm is based on a set of tolerance parameters, parametric analyses and a study of their respective influence on the mesh adaption are carried out. This detailed analysis is performed on unidimensional problems, and a final example is provided in two dimensions.

This work presents an original approach for independent h-adaption of a mechanical and a thermal mesh in strongly coupled problems, based on an incremental variational formulation. The approach does not rely on (or attempt to provide) error estimation in the classical sense. It could merely be considered to provide an error indicator. Instead, it provides a practical methodology to adapt the mesh on the basis of the variational structure of the underlying mathematical problem.

A complete thermo‐mechanical model for the simulation of the inertia welding process of two similar parts is described. The material behaviour is represented by an incompressible viscoplastic Norton—Hoff law in which the rheological parameters are dependent on temperature. The friction law was determined experimentally and depends on the prescribed pressure and the relative rotating velocity between the two parts. The mechanical problem is solved considering the virtual work principle including inertia terms. The computation of the three components of the velocity field such as radial, longitudinal and rotational velocity, in an axisymmetric approximation allows to take into account the torsional effects. The domain is updated based on a Lagrangian formulation. The non‐linear heat transfer equation with boundary conditions (convection, radiation and friction flux) is solved separately for each time step. Error estimators on mechanical and thermal computation are devised to adapt the mesh in an automatic way. Finally, numerical results concerning evolution of parts shape, strain, temperature, rotating velocity, upsetting are compared with actual industrial welds.