Bruno Cochelin

A critical review of asymptotic numerical methods

SummaryVarious sorts of asymptotic-numerical methods have been propsed in the literature: the reduced basis technique, direct computation of series or the use of Padé approximants. The efficiency of the method may also depend on the chosen path parameter, on the order of truncature and on alternative parameters. In this paper, we compare the three classes of asymptotic-numerical method, with a view to define the “best” numerical strategy.

Résolution des équations de Navier-Stokes et Détection des bifurcations stationnaires par une Méthode Asymptotique-Numérique

ABSTRACTPerturbation methods (asymptotic expansions) are usually considered as powerful methods for solving many kinds of non-linear problems. However, these methods are very often applied in a purely analytic framework, and the calculation is limited to the first few terms of the series. Since a few years, we have shown that the combination of perturbation techniques and finite element method can lead to a robust numerical method for some categories of non-linear problems. In this paper, we apply these techniques to compute branches of stationary solutions of Navier-Stokes equations and to detect stationary bifurcations.

Model reduction method: an application to the buckling analysis of laminated rubber bearings

In this paper, we apply a model reduction method to find the equilibrium state at finite strain of geometrically complex structures which have periodic properties in one direction and exhibit a non-linear material behavior. This method, based on a finite-element approach, consists in projecting the unknowns fields onto a polynomial basis in order to reduce the size of the problem. This method was combined with a continuation resolution scheme to find the instabilities of a laminated rubber bearing subjected to compression loading. Comparisons with standard finite-element models show the reliability of the present method.

Stability of thin-shell structures and imperfection sensitivity analysis with the Asymptotic Numerical Method

This paper is concerned with stability behaviour and imperfection sensitivity of thin elastic shells. The aim is to determine the reduction of the critical buckling load as a function of the imperfection amplitude. For this purpose, the direct calculation of the so-called fold line connecting all the limit points of the equilibrium branches when the imperfection varies is performed. This fold line is the solution of an extended system demanding the criticality of the equilibrium. The Asymptotic Numerical Method is used as an alternative to Newton-like incremental-iterative procedures for solving this extended system. It results in a very robust and efficient path-following algorithm that takes the singularity of the tangent stiffness matrix into account. Two specific types of imperfections are detailed and several numerical examples are discussed.

An Asymptotic Numerical Method to Compute Stationary Navier-Stokes Solutions and Bifurcation Points

We proposed in this paper a numerical method to compute Hopf bifurcation points on stationary Navier-Stokes solution branches. This bifurcation corresponds to a transition between a stationary solution and a time-periodic one (J.E. Marsden and M.McCracken, 1976). The solutions of the stationary Navier-Stokes are determined with an Asymptotic-Numerical Method (N. Damil and M. Potier-Ferry, 1990) which is a combination between a perturbation technique and the Finite Element Method. We introduce a bifurcation’s indicator which has the peculiarity to be null at the Hopf bifurcation points.