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How do consumers choose to throw away food from a refrigerator? An observational study in controlled conditions. 7. European Conference on Sensory and Consumer Research (Eurosense).

In this Note, we consider an inflated orthotropic linearly elastic generalized membrane shell submitted to an outer surface perturbation. We obtain the strong convergence towards the solution of a well-posed “2D” problem of the mean value in the membrane thickness 2 e of the “3D” scaled displacements, as e approaches zero. To cite this article: R. Luce et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).

We consider a family of linearly elastic shells of the first kind (as defined in [2]), also known as non inhibited pure bending shells [7]. This family is indexed by the half-thickness ε. When ε approaches zero, the averages across the thickness of the shell of the covariant components of the displacement of the points of the shell converge strongly towards the solution of a "2D generalized membrane shell problem" provided the applied forces satisfy admissibility conditions [1,3]. The identification of the admissible applied forces usually requires delicate analysis. In the first part of this paper, we simplify the general admissibility conditions when applied forces h are surface forces only, and obtain conditions that no longer depend on ε [5]: find hαβ = hαβ in L2(ω) such that for all η = (ηi) in V(ω), ∫ω hi ηi dω = ∫ω hαβγαβ(η)dω where ω is a domain of ℝ2, θ is in [Formula: see text] and [Formula: see text] is the middle surface of the shells, where (γαβ (η)) is the linearized strain tensor of S and V(ω) = {η ∈ H1(ω), η = 0 on γ0}, the shells being clamped along Γ0 = θ(γ0). In the second part, since the simplified admissibility formulation does not allow to conclude directly to the existence of hαβ, we seek sufficient conditions on h for hαβ to exist in L2(ω). In order to get them, we impose more regularity to hαβ and boundary conditions. Under these assumptions, we can obtain from the weak formulation a system of PDE's with hαβ as unknowns. The existence of solutions depends both on the geometry of the shell and on the choice of h. We carry through the study of four representative geometries of shells and identify in each case a special admissibility functional space for h.

In this first part we propose and analyse a model for the study of two-dimensional incompressible Navier–Stokes equations with a temperature-dependent viscosity. The flow is supposed in a mixed convection regime and considers an outflow region, leading to a strongly coupled problem between the Navier–Stokes and energy equations, which will be justified theoretically. The coupling in the continuous problem is treated by an outer temperature fixed point strategy. Existence results for a particular variational formulation follows from this study. Further, a particular uniqueness result for small data is also obtained. Copyright © 2007 John Wiley & Sons, Ltd.

In this second part, we analyse the associated discrete problem arising from a conforming finite element method formulation of the mathematical model presented in the first part. Thus, existence and uniqueness of the discrete solution when using small enough data are stated following the same strategy used in the continuous case, with a Cea's type error estimate established as the main result. Some numerical experiments, steady and unsteady, are performed, which allow us to validate the previous mathematical model and its discrete approximation. Copyright © 2007 John Wiley & Sons, Ltd.

This paper deals with the solution of problems involving partial differential equations in $\mathbb{R}^3$ . For three dimensional case, methods are useful if they require neither domain boundary regularity nor regularity for the exact solution of the problem. A new domain decomposition method is therefore presented which uses low degree finite elements. The numerical approximation of the solution is easy, and optimal error bounds are obtained according to suitable norms.

We apply the boundary integral equation method and a primal mixed finite element approach to study the weak solvability and Galerkin approximations of linear interior transmission problems arising in potential theory and elastostatics. The existence and uniqueness of solution of the resulting weak formulations and of the associated discrete schemes are derived by using the classical theory for variational problems with constraints. Suitable finite element subspaces of Lagrange type satisfying the compatibility conditions are utilized for defining the Galerkin scheme. The error analysis and corresponding rates of convergence are also provided.

Resume On s'interesse a l'approximation par elements finis non conformes d'une classe de problemes elliptiques dependant d'un petit parametre ɛ. On etablit la convergence uniforme par rapport a ɛ et h, ce qui generalise un resultat connu dans le cas des elements finis conformes (voir [3]).

On propose et on analyse une nouvelle methode d'elements finis hybrides en decomposition de domaines pour l'approximation de la solution d'un probleme elliptique. Dans cette methode, le domaine est divise en plusieurs sous-domaines et on utilise sur chaque sous-domaine une discretisation de type elements finis. Les raccords sont obtenus a l'aide de deux multiplicateurs de Lagrange et un espace joint. Des resultats d'ordre optimal sont obtenus pour un probleme modele du second ordre.

International audience; The aim of this paper is to present a new class of finite elements on quadrilaterals where the approximation is polynomial on each element K. In the case of Lagrange finite elements, the degrees of freedom are the values at the vertices and in the case of mixed finite elements the degrees of freedom are the mean values of the fluxes on each side. The degres of freedom are the same as those of classical finite elements. However, in general, with this kind of finite elements,the resolution of second order elliptic problems leads to non conforming approximations. In the particular case when the finite elements are parallelograms, we can notice that our method is conform and coincides with the classical finite elements on structured meshes. First, a motivation for the study of the Pseudo-conforming polynomial finite elements method is given, and the convergence of the method established. Then, numerical results that confirm the error estimates, predicted by the theory, are presented.

The aim of this paper is to develop a new class of finite elements on quadrilaterals and hexahedra. The degrees of freedom are the values at the vertices and the approximation is polynomial on each element K. In general, with this kind of finite elements, the resolution of second order elliptic prob- lems leads to non-conforming approximations.Degrees of freedom are the same than those of isoparametric finite elements. The convergence of the method is analyzed and the theory is confirmed by some numerical results. Note that in the particular case when the finite elements are parallelotopes, the method is conforming and coincides with the classical finite elements on structured meshes.

We prove a discrete extension theorem for the equilibrium finite elements spaces. The most important part of the work consists of establishing the basic properties used for proving the extension theorem.

How are the stresses and the deformations of an inflated structure when external forces are applied to it. This thesis answers that question when the structure is a long and thin shell composed of two congruent cylindrical orthotropical membranes which are glued together along rigid straight or curved axes and inflated by a perfect gas. From a mechanical modelling in linearized elasticity, we established a 3D variatinnal problem well posed in classical Sobolev spaces. Then we let the thickness parameter tend to zero, in order to obtain a 2D asymptotic model, well posed as long as the external forces check special properties -they are then called admissible forces- in spaces obtained by completion. We proved that the mean value in the thickness of the 3D solution strongly tends to the solution of the 2D asymptotic problem. We also identified sufficient conditions of admissibility. The numerical analysis showed that the error is a priori estimated in the energy norm. This, as confirmed by numerical simulations, creates problems as far as displacements are concerned, but is satisfying from the stresses point of view.; Quelles sont les déformations et les contraintes que subissent une structure gonflée précontrainte lorsqu'elle est soumise à des perturbations de son environnement extérieur? Cette thèse répond à cette interrogation dans le cas d'un lobe formé de deux membranes orthotropes, fixées sur leurs longueurs à deux axes rigides, successivement rectilignes puis légèrement arrondis, et sur leurs largeurs à des arcs de cercle également rigides. L'intérieur est gonflé par un gaz supposé parfait. A partir d'une modélisation mécanique en élasticité linéaire, nous avons établi un modèle mathématique rigoureux en 3D. Puis en faisant tendre l'épaisseur des membranes vers zéro, nous avons obtenu un modèle asymptotique 2D, bien posé pour certaines forces, dites admissibles, dans des espaces obtenus par complétion. Nous avons alors démontré que la suite des valeurs moyennes dans l'épaisseur des solutions des problèmes 3D converge fortement vers l'unique solution du problème asymptotique. Nous avons de plus mis en évidence des conditions suffisantes d'admissibilité des forces extérieures. L'analyse numérique du modèle asymptotique a montré que les estimations d'erreur a priori se font dans la norme de l'énergie. Ce qui, comme l'ont confirmé quelques essais numériques, ne va pas sans poser de problèmes quand on s'intéresse aux déplacements, mais est tout à fait satisfaisant du point de vue des contraintes.

How are the stresses and the deformations of an inflated structure when external forces are applied to it. This thesis answers that question when the structure is a long and thin shell composed of two congruent cylindrical orthotropical membranes which are glued together along rigid straight or curved axes and inflated by a perfect gas. From a mechanical modelling in linearized elasticity, we established a 3D variatinnal problem well posed in classical Sobolev spaces. Then we let the thickness parameter tend to zero, in order to obtain a 2D asymptotic model, well posed as long as the external forces check special properties -they are then called admissible forces- in spaces obtained by completion. We proved that the mean value in the thickness of the 3D solution strongly tends to the solution of the 2D asymptotic problem. We also identified sufficient conditions of admissibility. The numerical analysis showed that the error is a priori estimated in the energy norm. This, as confirmed by numerical simulations, creates problems as far as displacements are concerned, but is satisfying from the stresses point of view.; Quelles sont les déformations et les contraintes que subissent une structure gonflée précontrainte lorsqu'elle est soumise à des perturbations de son environnement extérieur? Cette thèse répond à cette interrogation dans le cas d'un lobe formé de deux membranes orthotropes, fixées sur leurs longueurs à deux axes rigides, successivement rectilignes puis légèrement arrondis, et sur leurs largeurs à des arcs de cercle également rigides. L'intérieur est gonflé par un gaz supposé parfait. A partir d'une modélisation mécanique en élasticité linéaire, nous avons établi un modèle mathématique rigoureux en 3D. Puis en faisant tendre l'épaisseur des membranes vers zéro, nous avons obtenu un modèle asymptotique 2D, bien posé pour certaines forces, dites admissibles, dans des espaces obtenus par complétion. Nous avons alors démontré que la suite des valeurs moyennes dans l'épaisseur des solutions des problèmes 3D converge fortement vers l'unique solution du problème asymptotique. Nous avons de plus mis en évidence des conditions suffisantes d'admissibilité des forces extérieures. L'analyse numérique du modèle asymptotique a montré que les estimations d'erreur a priori se font dans la norme de l'énergie. Ce qui, comme l'ont confirmé quelques essais numériques, ne va pas sans poser de problèmes quand on s'intéresse aux déplacements, mais est tout à fait satisfaisant du point de vue des contraintes.