Jean-Marc Cadou

Automatic detection and branch switching methods for steady bifurcation in fluid mechanics

This paper deals with the computation of steady bifurcations in the framework of 2D incompressible Navier-Stokes flow. We first propose a numerical method to accurately detect the critical Reynolds number where this kind of bifurcation appears. From this singular value, we introduce a numerical tool to compute all the steady bifurcated branches. All these algorithms are based on the Asymptotic Numerical Method [1,2]. The critical values are determined by using a bifurcation indicator [3-5] and the bifurcated branches are computed by using an augmented system which was first introduced in solid mechanics [4,6]. Several numerical examples from 2D Navier-Stokes show the reliability and the efficiency of the proposed methods.

Forced harmonic response of viscoelastic sandwich beams by a reduction method

ABSTRACT The aim of this article is to develop a reduction method to determine the forced harmonic response of viscoelastic sandwich structures at a reasonable computational cost. The numerical resolution is based on the asymptotic numerical method. This type of problem is complex, and its number of degrees of freedom is double the number of the undamped structure, leading to a high computational time. To address the problem, three reduction methods are evaluated, which differ from the projection operator. Numerical tests have been performed in the case of cantilever sandwich beams. The comparison of the results obtained by the reduction order resolution with those given by the full order resolution shows both a good agreement and a significant reduction in computational cost.

Numerical comparisons of high-order nonlinear solvers for the transient Navier-Stokes equations based on homotopy and perturbation techniques

Efficient solvers for the unsteady Navier-Stokes equations are presented. A classic time-stepping scheme is combined with high-order nonlinear solvers coupling homotopy and a perturbation technique. Polynomial and rational representations are used to approximate the unknowns of the problem. A pseudo-residual criterion is proposed to improve the efficiency of the solvers. The numerical example considered in this paper is the time-periodic two-dimensional flow around a circular cylinder. Comparisons with the classical first order Newton-Raphson solver are performed. Numerical results reveal that a lower number of matrix factorization is needed with the proposed methods, decreasing the computational effort.

Stability Analysis of 2D Flows by Numerical Tools Based on the Asymptotic Numerical Method: Application to the One and Two Sides Lid Driven Cavity

This work deals with the complete stability analysis in the case of two dimensional fluid flows searching for steady and Hopf bifurcations.The stability analysis starts with the computation of steady bifurcations. It is realized by first considering the monitoring of an indicator which is a scalar function. The indicator is computed via a perturbation method: the Asymptotic Numerical Method. Steady bifurcation point corresponds to the zero of this indicator. From this singular point, all the steady bifurcated branches are computed by using the perturbation method.Then the stability analysis is pursued with the computation of Hopf bifurcations by a hybrid method. This latter consists in coupling a continuation method with a direct Newton solver.The continuation method allows the bifurcation indicator to be determined using alternating reduced order and full size steps resolution. The quantities coming from the bifurcation indicator are used as initial approximations for the direct method iterations.Then an augmented system whose solutions are Hopf bifurcation is solved by a direct method of Newton kind.The examples of the one side and two sides lid driven cavities show the reliability and the efficiency of the proposed numerical tools.Copyright