Evariste Sanchez-Palencia

Anisotropic finite element estimates and local locking for shells : parabolic case

Abstract We consider a thin shell with thickness 2e with developable middle surface. A boundary is free along a generator. The loading is normal and non-zero along this generator. The energy concentrates inside a thin layer with thickness O (η) , η=e1/4. The structure of the layer is described by a penalty problem so that a locking phenomenon may appear. We give estimates of the error for the approximation with anisotropic finite elements of step O (H) and O (ηH) along and accross the layer respectively. For the same error, the corresponding triangulation includes a number of elements smaller, in the ratio η, to that of an isotropic mesh. The form and intensity of the terms of the local locking are made evident.

New cases of propagation of singularities along characteristic boundaries for model problems of shell theory

Abstract We continue a previous work [1] on propagation of singularities for model problems of thin shell theory in the parabolic and hyperbolic cases. The singularities along the characteristic boundaries are considered using extensions of the solutions out of the domain, adapted to either free or fixed boundaries. The corresponding transport equations are given except for the case of a characteristic fixed boundary for a hyperbolic shell, where the phenomenon is non local, but depends on the whole domain.

Propagation of singularities along a characteristic boundary for a model problem of shell theory and relation with the boundary layer

Abstract We consider the propagation of singularities for a differential system which constitutes a simplified model of thin shells with developable middle surface (parabolic case). Extensions of the solutions out of the domain allow us to consider either boundary or internal singularities. The properties of propagation of singularities and their relation with the structure of the boundary layers are given. We remove a mistake in [2], Sectionxa06.1, concerning the analyticity of solutions (in fact they are in the Gevrey class of order 3).

Remarks on ellipticity and regularity for non-inhibited shells and other constrained systems

Abstract We examine ellipticity properties for three examples of constrained systems with a Lagrange multiplier. The system of incompressible elasticity is in the restricted context of the Grubb-Geymonat elliptic systems. The Reissner-Mindlin system of plates is in the Agmon-Douglis-Nirenberg context. The system of thin shells which are not geometrically rigid is elliptic or hyperbolic at elliptic or hyperbolic points of the surface, respectively; moreover, this property is only concerned with equations, without boundary conditions. Some properties of the wave fronts and propagation of singularities are given.