John Cagnol

Uniform stability in structural acoustic models with flexible curved walls

Abstract The aim of this paper is twofold. First, we develop an explicit extension of the Kirchhoff model for thin shells, based on the model developed by Michel Delfour and Jean-Paul Zolesio. This model relies heavily on the oriented distance function which describes the geometry. Once this model is established, we investigate the uniform stability of a structural acoustic model with structural damping. The result no longer requires that the active wall be a plate. It can be virtually any shell, provided that the shell is thin enough to accommodate the curvatures.

On the Free Boundary Conditions for a Dynamic Shell Model Based on Intrinsic Differential Geometry

The mathematical theory behind the modeling of shells is a crucial issue in many engineering problems. Here, the authors derive the free boundary conditions and associated strong form of a dynamic shallow Kirchhoff shell model based on the intrinsic geometry methods of Michael Delfour and Jean-Paul Zolésio. This model relies on the oriented distance function which describes the geometry. This is an extension of the work done in [J. Cagnol, I. Lasiecka, C. Lebiedzik and J.-P. Zolésio (2002). Uniform stability in structural acoustic models with flexible curved walls. J. Differential Equation, 186(1), 88–121.], where the model was derived for clamped boundary conditions only. In the current article, manipulations with the model result in a cleaner form where the displacement of the shell and shell boundary is written explicitly in terms of standard tangential operators.

Shape Control in Hyperbolic Problems

The vibration of a shell constrained to be in a specific configuration and the vibration of a shell with that shape at its natural reference position are known to be different by physicists. We consider a shell which is under a large displacement and a small deformation, that gives a constrained shell Ω in a static equilibrium. We consider a small vibration of Ω and we are interested in the equation of that vibration. We first investigate a new exact model for constrained shell that is p(d, ∞) which is the counterpart of the model developed by Delfour and Zolesio for shells in the reference configuration. Then we study the regularity of the solution in the interior domain as well as the the boundary regularity. Both regularities were proven to be interesting for shape differentiability (and thus shape control) in hyperbolic equations, we will recall the shape differentiability results for the wave equation.

Hidden shape derivative in the wave equation with Dirichlet boundary condition

Abstract We establish the shape differentiability for the solution to the wave equation with Dirichlet boundary condition. This is well known for elliptic and parabolic problems. In the hyperbolic situation, the implicit functions theorem does not work, but the hidden regularity (see [5]) gives part of the result.

Static equilibrium of hyperlestic thin shell: symbolic and numerical computation

We here examine the natural shapes of an hyperelastic thin shell called a Carpentiers joint, when the terminal position is known. More specifically we study a rectangular strip that is a flexible thin shell with a constant curvature in its width and a null curvature in its length, at its unconstrained state. We use the theory of large displacement and small strain for hyperelastic material. We first consider an appropriate parameterization of the joint. Then we compute the Green-St Venant strain tensor with a symbolic computation system and we generate the numerical code to compute the elastic energy. In particular, we make strong use of symbolic elements to resolve some problems with zero division. Numerical minimization of this energy is used to find the shape and a couple of simulation are presented.

A Uniqueness Theorem for a Classical Nonlinear Shallow Shell Model

The main goal of this paper is to establish the uniqueness of solutions of finite energy for a classical dynamic nonlinear thin shallow shell model with clamped boundary conditions. The static representation of the model is an extension of a Koiter shallow shell model. Until now, this has been an open problem in the literature. The primary difficulty is due to a lack of regularity in the nonlinear terms. Indeed the nonlinear terms are not locally Lipshitz with respect to the energy norm. The proof of the theorem relies on sharp PDE estimates that are used to prove uniqueness in a lower topology than the space of finite energy.

Free Boundary Conditions for Intrinsic Shell Models

We derive the free boundary conditions and associated strong form of a shallow Kirchhoff shell model based on the intrinsic geometry methods of Michael Delfour and Jean-Paul Zolesio. Manipulations with the model result in a cleaner form where the displacement of the shell and shell boundary is written explicitly in terms of standard tangential operators.

Vibration of a pre-constrained elastic thin shell II: Intrinsic exact model

Abstract We study the vibration of an elastic thin shell which is pre-constrained by a large displacement with a small deformation. In this second Note we come up with an exact model p ( d ,∞) in intrinsic geometry. We take advantage of the exactness of the model for the existence and regularity of its the solutions. To cite this article: J. Cagnol, J.-P. Zolesio, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 251–256.