An algorithm for the computation of multiple Hopf bifurcation points based on Pade approximants

International audienceRecently, a numerical method was proposed to compute a Hopf bifurcation point in fluid mechanics. This numerical method associates a bifurcation indicator and a Newton method. The former gives initial guesses to the iterative method. These initial values are the minima of the bifurcation indicator. However, sometimes, these minima do not lead to the convergence of the Newton method. Moreover, as only a single initial guess is obtained for each computation of the indicator, the computational time to obtain a Hopf bifurcation point can be quite long. The present algorithm is an enhancement of the previous one. It consists in automatically computing several initial guesses for each indicator curve. The majority of these initial values leads to the convergence of the Newton method. This method is evaluated through the problem of the lid-driven cavity with several aspect ratios in the framework of the finite element analysis of the 2D Navier-Stokes equations. The results prove the efficiency and the robustness of the proposed algorithm

A numerical algorithm coupling a bifurcating indicator and a direct method for the computation of Hopf bifurcation points in fluid mechanics

International audienceThis paper deals with the computation of Hopf bifurcation points in fluid mechanics. This computation is done by coupling a bifurcation indicator proposed recently and a direct method which consists in solving an augmented system whose solutions are Hopf bifurcation points. The bifurcation indicator gives initial critical values (Reynolds number, Strouhal frequency) for the direct method iterations. Some classical numerical examples from fluid mechanics, in two dimensions, are studied to demonstrate the efficiency and the reliability of such an algorithm

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

International audienceThis 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. The critical values are determined by using a bifurcation indicator and the bifurcated branches are computed by using an augmented system which was first introduced in solid mechanics. Several numerical examples from 2D Navier-Stokes show the reliability and the efficiency of the proposed methods

Techniques de réduction de modèles pour les vibrations non linéaires de plaques minces amorties

National audienceSee http://hal.archives-ouvertes.fr/docs/00/59/28/65/ANNEX/r_QO25E817.pd

Méthode de Relaxation Dynamique incrémentale

National audienceSee http://hal.archives-ouvertes.fr/docs/00/59/29/46/ANNEX/r_ZL662YL9.pd

Numerical study of dynamic relaxation with kinetic damping applied to inflatable fabric structures with extensions for 3D solid element and non-linear behavior

International audienceThis work mainly deals with the numerical study of inflatable fabric structures. As implicit integration schemes can lead to numerical difficulties such as singular stiffness matrices, explicit schemes are preferred. Since the final objective of this study is to obtain the final shape of a structure, a dynamic relaxation (DR) method is used. These methods allow us to obtain the final and stable shape of the inflatable fabric structures without doing so many time increments, which is the case when using a classical explicit integration method. Han and Lee [5] proposed an extension of the DR method stated by Barnes [13] suitable for triangular elements and elastic behavior. There are two main contributions in this paper. Firstly, we propose a modification of Han and Lee's method, allowing it to be used with any kind of membrane or solid finite elements and any reversible behavior. Secondly, we propose to rewrite the expression initially introduced by Barnes. Furthermore, these proposals are adapted for incremental loadings, allowing this way to obtain the pseudo-equilibriums of the intermediate phases. Numerical examples from academic problems (rectangular and circular membranes) show the efficiency and the reliability of proposed methods, with linear elasticity behavior, and also with a non-linear incremental behavior and finite deformation states

Simulation 2D de la fissuration dans un matériau ductile endommageable avec X-FEM

Ce travail est consacré à la simulation numérique de la propagation d’une fissure dans un matériau dont la rupture résulte de la création, la croissance et la coalescence de vides. Nous avons ainsi considéré une plaque (cas des déformations planes) soumise à un chargement de type Mode I. Le comportement du matériau est décrit via un modèle de type Gurson [1] (en réalité GTN, voir Ref. [3]) prenant en compte les effets combinés de l’écrouissage, de l’adoucissement thermique, de la viscoplasticité et de l’endommagement par croissance de vides. Le matériau étudié est un acier. La méthode des éléments finis étendu (X-FEM) [2] a été retenue pour décrire les discontinuités fortes induites par la présence d’une fissure dans le maillage. Une méthode de propagation originale est proposée afin de coupler la X-FEM et le comportement fortement non linéaire du matériau. La simulation numérique de ce problème est réalisée à l’aide du code de calculs industriel par éléments finis, Abaqus. Cela à impliqué le développement i) d’un élément fini utilisateur (UEL dans Abaqus) afin de décrire les conséquences cinématiques de la présence d’une fissure et ii) d’un matériau utilisateur (UMAT dans Abaqus) pour décrire le comportement du matériau. Un schéma d’intégration implicite est utilisé dans le code de calculs. Considérant quelques simplifications, le travail présenté reproduit la propagation d’une fissure en 2D résultant de la croissance de vides induits par l’endommagement. [1] A.L. Gurson. Continuum theory of ductile rupture by void nucleation and growth : Part I - Yield criteria and flow rules for porous ductile media, J. Eng. Mat. Tech., 2-15, 1977. [2] N. Moës, J. Dolbow, T. Belytschko. A finite element method for crack growth without remeshing. Int. J. Num. Methods Eng., 131-150,1999. [3] V. Tvergaard, A. Needleman. Analysis of the cup-cone fracture in a round tensile bar, Acta Metall., page157- page169, 1984

Nonlinear forced vibration of damped plates by an asymptotic numerical method

International audienceThis work deals with damped nonlinear forced vibrations of thin elastic rectangular plates subjected to harmonic excitation by an asymptotic numerical method. Using the harmonic balance method and Hamilton’s principle, the governing equation is converted into a static formulation. A mixed formulation is used to transform the problem from cubic nonlinearity to quadratic one sequence. Displacement, stress and frequency are represented by power series with respect to a path parameter. Equating the like powers of this parameter, the nonlinear governing equation is transformed into a sequence of linear problems with the same stiffness matrix. Through a single matrix inversion, a considerable number of terms of the perturbation series can easily be computed with a limited computation time. The starting point, corresponding to a regular solution, is obtained by the Newton–Raphson method. In order to increase the step length, Padé approximants are used. Numerical tests are presented and compared with numerical and analytical results in the literature, for different boundary conditions, excitations and damping coefficients

Calculs de points de bifurcation et de branches bifurquées pour les écoulements stationnaires tridimensionnels de fluide Newtonien incompressible

La Méthode Asymptotique Numérique est utilisée dans cette étude pour détecter d'éventuels points de bifurcation stationnaire et calculer les branches solutions émanant de ces derniers. L'étude est réalisée dans le cadre des équations de Navier-Stokes tridimensionnelles régissant l'écoulement de fluide Newtonien incompressible. L'implémentation est réalisée au sein d'ELMER, un logiciel open-source de simulation multi-physiques par éléments finis

Simulation numérique de recherche de forme : application aux gilets de sauvetage gonflables

Cet article présente une application de la recherche de forme aux gilets de sauvetage gonflables. La recherche de forme est le nom générique désignant le processus de conception de la forme globale des structures légères instables telles que les structures gonflables. L'état d'équilibre statique des gilets gonflables est recherché pour analyser la forme générale, le volume contenu et la localisation des zones de plis. La méthode de relaxation dynamique avec amortissement cinétique permet d'éviter le problème de singularité de la matrice de raideur et le problème d'instabilité locale dans les zones de plis. Sa rapidité de convergence est très dépendante de la formulation de la matrice des masses. Dans cet article, plusieurs expressions des masses sont testées et leurs performances comparées entre elles