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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.

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 analyze the numerical approximation of a class of elliptic problems which depend on a small parameter $\varepsilon$ . We give a generalization to the nonconforming case of a recent result established by Chenais and Paumier for a conforming discretization. For both the situations where numerical integration is used or not, a uniform convergence in $\varepsilon$ and h is proved, numerical locking being thus avoided. Important tools in the proof of such a result are compactness properties for nonconforming spaces as well as the passage to the limit problem.