Folding of Thin Walled Tubes as a Free Gradient Discontinuity Problem

The aim of this paper is to describe buckling deformations of hollow cylinders whose buckled configurations consist of inextensional deformations of their original middle surface, through minimization of two competing energy forms: a bulk elastic energy and an interface bending energy. These minimal energy configurations are obtained through descent minimization within the class of folding deformations. The non-standard problem of minimization over variable triangulations is considered.     

Exit cycling for the Van der Pol oscillator and quasipotential calculations

We discuss the phenomenon of cycling for noise induced escape to a unstable periodic orbit. The presence of cycling is shown to follow from qualitative properties of two quasipotential functions. A method of numerically evaluating these quasipotential functions is described, and applied to the Van der Pol oscillator as an example. Figures resulting from these calculations reveal that nonconvergent cycling of exit measures does occur for the Van der Pol example.     

A Quasi-linear Viscoelastic Constitutive Equation for the Brain: Application to Hydrocephalus

Hydrocephalus is a condition which occurs when an excessive accumulation of cerebrospinal fluid in the brain causes enlargement of the ventricular cavities. Modern treatments of shunt implantation are effective, but have an unacceptably high rate of failure in most reported series. One of the common factors causing shunt failure is the misplacement of the proximal catheter's tip, which can be remedied if the healed configuration of the ventricular space can be predicted. In a recent study we have shown that this is accomplished by a mathematical model which requires as input the knowledge of the speed at which the ventricular walls move inwardly. In this paper we report on a theoretical method of calculating this speed and show that it will become of great practical usefulness as soon as more experimental results become available.     

A reformulation of the strong ellipticity conditions for unconstrained hyperelastic media

The conditions for the strong ellipticity of the equilibrium equations of compressible, isotropic, nonlinearly elastic solids (established by Simpson and Spector [1]) are expressed in terms of the stored-energy function regarded as a function of the principal stretches. The applicability of this reformulation is illustrated with the help of two specific examples.     

A Numerical Verification of Nontrivial Solutions for the Heat Convection Problem

A computer assisted proof of the existence of nontrivial steady-state solutions for the two-dimensional Rayleigh-Bénard convection is described. The method is based on an infinite dimensional fixed-point theorem using a Newton-like operator. This paper also proposes a numerical verification algorithm which generates automatically on a computer a set including the exact nontrivial solution. All discussed numerical examples take into account of the effects of rounding errors in the floating point computations.     

A frequency–time domain finite element model for tidal circulation based on the least-squares harmonic analysis method

An iterative type harmonic finite element model is developed for solving the full non-linear form of the shallow water equations. The scheme iteratively updates time histories of the non-linear terms which are then harmonically decomposed and used as forcing terms for the linear sets of equations which result from the harmonic separation of the shallow water equations. A least-squares harmonic analysis procedure is used to decompose the non-linear forcing terms. This procedure allows for the very efficient separation of extremely closely spaced harmonics, since it is highly selective with respect to the frequencies it considers. In addition tailoring the procedure and using very specific time steps and sampling periods significantly reduces the number of time samplings points required. In conjunction with the iterative nature of our scheme, the least-squares procedure makes the scheme entirely general, allows for the direct assessment of all tidal constituents, including compound tides, and permits the clear cut and complete investigation of their mutual interaction through the non-linearities. In addition this procedure readily computes very-low-frequency or residual type circulations. The FE formulation used shows a very low degree of spurious oscillations while remaining quite simple to implement. This control on nodal oscillations is especially important due to the energy transfer mechanisms involved in this type of iterative scheme. In an example application the effects of the various non-linear overtide and compound tide type interactions are examined. It is demonstrated that not only are compound tides significant relative to the overtides, but they also influence the overtides.     

A harmonic balance method for the non-linear vibration of viscoelastic shellsVibrations non linéaires des structures viscoélastiques par une méthode de la balance harmonique

In this Note, we deal with the non-linear vibration of viscoelastic shell structures. Coupling a harmonic balance method with a one mode Galerkin's procedure, one obtains an amplitude equation depending on two complex coefficients. These are determined by solving a classical eigenvalue problem and two linear problems. This permits us to characterize the evolution of the loss factor with the vibration amplitude. To validate our approach, the amplitude-frequency and the amplitude-loss factor relationships are illustrated in the case of a circular ring. To cite this article: E.H. Boutyour et al., C. R. Mecanique 334 (2006).