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In this research work, numerical time perturbation methods are applied on nonlinear ODE. Solutions are sought in the form of power series using time as the perturbation parameter. This time integration approach with continuation procedures allows to obtain analytical continuous approximated solutions. Asymptotic Numerical Method and new resummations techniques of divergent series namely Borel–Pade–Laplace and Inverse Factorial series are studied. A comparison with classic integration scheme is presented in order to evaluate the robustness and the effectiveness of these algorithms. Full details are given regarding first- and second-order derivative of resummation techniques.

International audience; In this work, perturbation method and Padé approximants are used to compute the eigenvalues of linear problems. This algorithm is based on the introduction of perturbation loads in the eigenvalue problem. This modified problem is solved by using the perturbation method and the Padé approximants. The computation of the eigenvalues consists then on finding the roots of the numerator of a rational fraction. To demonstrate the efficiency of the proposed method, examples of linear vibrations of plates and shells are considered. The obtained results showed that the proposed method can deal with problems with close or multiple eigenvalues. Furthermore, the results also showed that the proposed algorithm is less sensitive to the ill-conditioning of the matrices.

Summary This paper deals with bifurcation analysis methods based on the Asymptotic Numerical Method [35, 19]. It is used to investigate three-dimensional instabilities in a sudden expansion. To do so, high performance computing is implemented in ELMER an open-source multi-physical software. In this work, velocity-pressure mixed vectors are used with ANM based methods, remarks are made for the branch-switching method in the case of symmetry breaking bifurcation and new three-dimensional instabilities results are presented for the sudden expansion ratio E=3. Critical Reynolds numbers for primary bifurcations are studied with the evolution of a geometric parameter. New values are computed which reveal new trends that complete the work of [46]. Several kinds of bifurcation are depicted and tracked with the evolution of the span-wise aspect ratio. One of them relies on a fully three-dimensional effect [15, 46] as it breaks both span-wise and top-bottom symmetries. This bifurcation is found for smaller aspect ratios than expected. Furthermore, a critical Reynolds number is found for the aspect ratio Ai=1, that was not previously reported. Finally, primary and secondary bifurcations are efficiently detected and all post-bifurcated branches are followed. It makes it possible to plot a complete bifurcation diagram for this three-dimensional case. This article is protected by copyright. All rights reserved.

The phenomenon of void coalescence induced strain localization plays an important role in the ductile failure process as the intermediate stage between diffuse damage and crack formation. The aim of the present work is double: (i) giving a framework for numerically treating the failure of ductile structures in the context of the X-FEM and (ii) assessing three different enrichment methods allowing for reproducing the intermediate stage of void coalescence induced localization within X-FEM.

In this paper, the vibration behaviour and damping performances of Flax Fiber Reinforced Epoxy (FFRE) are investigated. Static tests using FFRE composite beams are carried out leading to the identification of elastic properties of each layer. Then, three FFRE composite plates are elaborated and used in experimental vibration tests to identify their eigenfrequencies and their modal damping. In numerical part, a constant complex representation of the stiffness is assumed and the loss factors are supposed constants and identified from the first experimental vibration mode. The homogenization technique and the finite element method are applied to describe their damped dynamic behaviour. The resolution of the resulting non-linear problem is carried out using the Asymptotic Numerical Method (ANM). Experimental results show that the modal damping is greater when the flax fibers are oriented at 90 ° . Comparing numerical and experimental results, the proposed finite element modelling enables to estimate the damped eigenfrequencies with high accuracy. However, the predicted modal loss factors are over-estimated compared to experimental ones. It is concluded that the present modelling should be improved considering the frequency dependence of damping.

International audience; There are many recent studies concerning reduced-order computational methods, especially reductions by projection on a small-sized basis. But it is difficult to control the quality of the solutions if the basis is fixed once and for all. This is why we attempt to define efficient and low-cost strategies for correction and updating of the basis. These correction steps re-use previously computed quantities such as: vectors and triangulated matrices. The proposed algorithms use alternately full and reduced-size steps, allowing a strong reduction in the number of full-size tangent matrices. Two classes of applications are discussed. First, we consider an algorithm for determining Hopf bifurcation points in 2D Navier-Stokes equations, but which requires time-consuming preliminary frequency-dependent calculations. New reduction-correction procedures are applied to reduce these preliminary computations. The second application concerns the response curves of viscoelastic structures. A key point is the definition of the reduced basis. Vector Taylor series are computed within the asymptotic numerical method and the relevance of this set of vectors is analysed.

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.

The aim of this study is to compute damped eigenfrequencies and modes of a vibroacoustic interior problem with fluid-structure coupling. Damping is introduced using a sandwich structure with viscoelastic core. In this paper, the coupled problem is solved by a High Order Newton (HON) solver, based on homotopy and perturbation techniques. Thus, the initial non-linear problem is turned into a set of linear algebraic systems, easier to solve. Comparison of the results obtained by the HON solver with those obtained by the Newton classical method highlights the efficiency of the proposed method. Finally, the mechanical behavior of the coupled damped problem is studied.

International audience; The present work aims at numerically predicting the current residual strength of large engineering structures made of ductile metals against accidental failure. With this aim in view, the challenge consists in reproducing within a unified finite element-based methodology the successive steps of micro-voiding-induced damage, strain localization and crack propagation, if any. A key ingredient for a predictive ductile fracture model is the proper numerical treatment of the critical transition phase of damage-induced strain localization inside a narrow band. For this purpose, the strong discontinuity cohesive model and the eXtended Finite Element Method are combined. A propagation algorithm is proposed and studied in the context of ductile materials. Physics-motivated criteria to pass from the phase of more or less diffuse damage to strain localization and from strain localization to crack propoagation are proposed. Finally, a 2D numerical example is shown to study the performance of the failure analysis model when implemented into an engineering finite element computation code, namely Abaqus.

International audience; This paper concerns the study of vibroacoustic problems. By considering a displacement-pressure formulation, a non-symmetric eigenvalue problem is obtained. In order to solve it, three numerical schemes are compared: the classical ARPACK solver, an indicator method (initially proposed in B. Claude et al. Comptes Rendus Mécaniques, 2017, 345(2)) which has the property to be null at the eigenvalues, and an original method based on the analysis of Taylor series expansions near a singularity. Numerical results show all the evaluated numerical methods give accurate results but the indicator method requires the lowest computational times. Nevertheless, the original method based on the behavior of the perturbation method close to eigenvalues seems to be a very promising technique.

This paper deals with the computation of backbone curves bifurcated from a Hopf bifurcation point in the framework of nonlinear free vibrations of a rotating flexible beams. The intrinsic and geometrical equations of motion for anisotropic beams subjected to large displacements are used and transformed with Galerkin and harmonic balance methods to one quadratic algebraic equation involving one parameter, the pulsation. The latter is treated with the asymptotic numerical method using Pade approximants. An algorithm, equivalent to the Lyapunov–Schmidt reduction is proposed, to compute the bifurcated branches accurately from a Hopf bifurcation point, with singularity of co-rank 2, related to a conservative and gyroscopic dynamical system steady state, toward a nonlinear periodic state. Numerical tests dealing with clamped, isotropic and composite, rotating beams show the reliability of the proposed method reinforced by accurate results.

International audience; Dans ce travail nous nous intéressons aux problèmes de vibroacoustique interne avec couplage fluide-structure. Le solide est un sandwich composé d’une couche visco-élastique comprise entre deux couches élastiques. Ce problème sera résolu par une méthode basée sur les techniques d’homotopie et de perturbation. Ceci nécessite dans un premier temps la résolution du problème non amorti, que l’on propose de résoudre avec la méthode asymptotique numérique

International audience; In this paper, an efficient and robust numerical method is proposed to solve non-symmetric eigenvalue problems resulting from the spatial discretization with the finite element method of a vibroacoustic interior problem. The proposed method relies on a perturbation method. Finding the eigenvalues consists in determining zero values of a scalar that depends on angular frequency. Numerical tests show that the proposed method is not sensitive to poorly conditioned matrices resulting from the displacement-pressure formulation. Moreover, the computational times required with this method are lower than those needed with a classical technique such as, for example, the Arnoldi method. (C) 2016 Academie des sciences. Published by Elsevier Masson SAS. This is an open access article under the CC BY-NC-ND license.

This work deals with the computation of steady bifurcation points in 2D incompressible Newtonian fluid flows. The problem is modeled with the Navier–Stokes equations with an evolving geometric parameter. The aim of the present study is to propose a reliable and efficient numerical method for parametric steady bifurcation calculations. The numerical algorithm is based on the coupling of a continuation method with a homotopy technique. The continuation method lies on the asymptotic numerical method with Pade approximants for an initial linear stability analysis with an initial geometric configuration. The homotopy technique completes the calculation with the computation of critical Reynolds numbers for different discrete values of the geometric parameter. Two classical numerical problems are approached. The first one is the flow in sudden expansion. The geometric parameter is the height of the expansion inlet. The second problem is the flow in a divergent/convergent channel. In this case, the geometric parameter is the length of the channel. Comparisons of results with those obtained from the literature are performed, showing the efficiency of the proposed algorithm. The aim of this study is to determine the critical Reynolds numbers of the flow using few computations for each geometric parameter.

This work deals with forced vibration of nonlinear rotating anisotropic beams with uniform cross sections. Coupling the Galerkin method with the balance harmonic method, the nonlinear intrinsic and geometrically exact equations of motion for anisotropic beams subjected to large displacements, are converted into a static formulation. This latter is treated with two continuation methods. The first one is the asymptotic-numerical method, where power series expansions and Pade approximants are used to represent the generalized vector of displacement and the frequency. The second one is the pseudo-arclength continuation method. Numerical tests dealing with isotropic and anisotropic beams are considered. The natural frequencies obtained for prismatic beams are compared with the literature. Response curves are obtained and the nonlinearity is investigated for various geometrical conditions, excitation amplitudes and kinematical conditions. The nonlinearity related to the angular speed for prismatic isotropic beam is thus identified. The stability of the solution branch is examined, in the frequency domain using the Floquet theory.

This work deals with the computation of Hopf bifurcation points in the framework of two-dimensional fluid flows. These bifurcation points are determined by using a Hybrid method [1] which associates an indicator curve and a Newton method. The indicator provides initial values for the Newton method. As the calculus of this indicator is time consuming, we suggest using an algorithm to save computational time. This algorithm alternates reduced order and full size step resolution which are all carried out by using a pertubation method. Hence, the computed vectors on the full size problem are used to define the reduced order model. As the low-dimensional model has a finite validity range, we propose a simple criterion which makes it possible to know when the basis has to be updated. The latter phase is carried out by going through a new full step which permits to build a new basis and, thus, compute a supplementary part of the indicator curve. Some numerical tests, such as the classical lid-driven cavity or the flow in a channel, permit to fix the optimum values of the parameters for the proposed method. The objective of this study is to save computational time without modifying the performance of the Hybrid method initially introduced in Ref. [1] . These numerical methods are applied to 2D fluid flows (flow in a channel and the 2D lid-driven cavity). Our conclusions, therefore, hold only for these kinds of problem.

International audience; In this paper, an efficient asymptotic basis is proposed to reduce the forced dynamic problem of viscoelastic sandwich plates. The numerical resolution is based on the asymptotic numerical method (ANM) and finite element method (FEM). Numerical tests have been performed in the case of sandwich plates with Young's modulus variable with respect to the frequency. The comparison of the results obtained in the reduced order model with those given in the full order model shows both a good agreement and a significant reduction in computational cost.

In this paper, we study the bending nonlinear free vibrations of a centrifugally stiffened beam with uniform cross-section and constant angular velocity. The nonlinear intrinsic equations of motion used here are geometrically exact and specific to beams exhibiting large amplitude displacements and rotations associated with small strains. Based on the Timoshenko beam model, these equations are derived from Hamilton's principle, in which the warping is considered. All coupling terms are considered including Coriolis terms. The studied beams are isotropic with clamped-free boundary conditions. By combining the Galerkin method with the harmonic balance method, the equations of motion are converted into a quadratic function treated with a continuation method the Asymptotic Numerical Method, where the generalized displacement vector is presented as a series expansion. While analysing the effect of the angular velocity, we determine the amplitude versus frequency variations which are plotted as backbone curves. Considering the first lagging and flapping modes, the changes in beam behaviour from hardening to softening are investigated and identified as a function of the angular velocity and the effect of shear. Particular attention is paid to high angular velocities for both Euler-Bernoulli and Timoshenko beams and the natural frequencies so obtained are compared with the results available in the literature. (C) 2016 Elsevier Ltd. All rights reserved.

In this paper, free vibrations of a rotating clamped Euler-Bernoulli beams with uniform cross section are studied using continuation method, namely asymptotic numerical method. The governing equations of motion are derived using Lagrange's method. The kinetic and strain energy expression are derived from Rayleigh-Ritz method using a set of hybrid variables and based on a linear deflection assumption. The derived equations are transformed in two eigenvalue problems, where the first is a linear gyroscopic eigenvalue problem and presents the coupled lagging and stretch motions through gyroscopic terms. While the second is standard eigenvalue problem and corresponds to the flapping motion. Those two eigenvalue problems are transformed into two functionals treated by continuation method, the Asymptotic Numerical Method. New method proposed for the solution of the linear gyroscopic system based on an augmented system, which transforms the original problem to a standard form with real symmetric matrices. By using some techniques to resolve these singular problems by the continuation method, evolution curves of the natural frequencies against dimensionless angular velocity are determined. At high angular velocity, some singular points, due to the linear elastic assumption, are computed. Numerical tests of convergence are conducted and the obtained results are compared to the exact values. Results obtained by continuation are compared to those computed with discrete eigenvalue problem. (C) 2016 Elsevier Ltd. All rights reserved.

International audience; 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.

SUMMARY Recently, 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. Copyright © 2011 John Wiley & Sons, Ltd.

This work concerns the computation of the nonlinear solutions of forced vibration of damped plates. In a recent work (Boumediene et al. in Comput Struct 87:1508---1515, 2009), a numerical method coupling an asymptotic numerical method (ANM), harmonic balance method and Finite Element method was proposed to resolve this type of problem. The harmonic balance method transforms the dynamic equations to equivalent static ones which are solved by using a perturbation method (ANM) and the finite element method. The numerical results presented in reference (Boumediene et al. in Comput Struct 87:1508---1515, 2009) show that the ANM is very efficient and permits one to obtain the nonlinear solutions with few matrix triangulation numbers compared to a classical incremental iterative method. However, putting a great number of harmonics (6 or greater) into the load vector leads to tangent matrices with a great size. The computational time necessary for the triangulation of such matrices can then be large. In this paper, reduced order models are proposed to decrease the size of these matrices and consequently the computational time. We consider two reduced bases. In the first one, the reduced basis is obtained by the resolution of a classical eigenvalue problem. The second one is obtained by using the nonlinear solutions computed during the first step of the calculus which is realized with the ANM. Several classical benchmarks of nonlinear damped plates are presented to show the efficiency of the proposed numerical methods.

An iterative solver is proposed to solve the family of linear equations arising from the numerical computation of non-linear problems. This solver relies on two quantities coming from previous steps of the computations: the preconditioning matrix is a matrix that has been factorized at an earlier step and previously computed vectors yield a reduced basis. The principle is to define an increment in two sub-steps. In the first sub-step, only the projection of the unknown on a reduced subspace is incremented and the projection of the equation on the reduced subspace is satisfied exactly. In the second sub-step, the full equation is solved approximately with the help of the preconditioner. Last, the convergence of the sequences is accelerated by a well-known method, the modified minimal polynomial extrapolation. This algorithm assessed by classical benchmarks coming from shell buckling analysis. Finally, its insertion in path following techniques is discussed. This leads to non-linear solvers with few matrix factorizations and few iterations. Copyright © 2009 John Wiley & Sons, Ltd.

L’objectif de ce travail est la mise au point de techniques numeriques pour la resolution de problemes non lineaires a tres grand nombre de degres de liberte. Nous proposons de coupler des techniques multigrilles a la methode asymptotique numerique (MAN) qui associe une methode de perturbation a la methode des elements finis.

Two original algorithms are proposed for the computation of bifurcation points in fluid mechanics. These algorithms consist of finding the zero values of a specific indicator. To compute this indicator a perturbation method is used which leads to an analytical expression of this indicator. Two kinds of instability are considered: stationary and Hopf bifurcations. To prove the efficiency and advantages of such numerical methods several numerical tests are discussed.

High-order iterative algorithms have been proposed recently (Commun. Numer. Methods Eng. 15 (1999) 701; Comput. Methods Appl. Mech. Eng. 190 (2000) 1845-1858). This technique relies on a homotopy transformation and on calculations of series and of Pade approximants. So one can define a high-order Newton algorithm that is efficient, robust and that often converges after a single iteration. Other algorithms denoted as L algorithms do not require any triangulation of a large matrix, but they are not as efficient as the previous one. In this paper, a reduced basis technique is used to improve the latter method, that is a sort of Newton method on the reduced subspace and a sort of L algorithm on the whole space. This new method is evaluated in the finite element analysis of thin shell buckling. So new efficient predictor-correctors are introduced that do not require any matrix triangulation in the correction phase.

This paper deals with the convergence acceleration of a sequence of vectors. This acceleration is achieved by using Minimal Polynomial Extrapolation Method (J Comput Appl Math 2000; 122:149–165) or by the vectorial Pade approximants, that have been considered in the framework of high-order iterative algorithms (Commun Numer Methods Eng 1999; 15:701–708; Comput Methods Appl Mech Eng 2000; 190:1845–1858). It is established that, in some cases, these two procedures are mathematically equivalent. In the practice, such techniques are affected by numerical instabilities. So the numerical robustness of the two methods is discussed, on the basis of a test emanating from non-linear elastic thin shell analysis. Copyright © 2004 John Wiley & Sons, Ltd.

The Asymptotic Numerical Method (ANM) is a family of algorithms based on the com- putation of series to solve non-linear problems. How to use at best the informations included in the series ? How to define in an optimal way prediction-correction algorithms within ANM ? In this paper, the knowledges about these questions are reviewed. A complete bibliography is also presented, that shows a wide application field and the many variants offered by ANM algorithms. RESUME. La Methode Asymptotique-Numerique (MAN) est une famille d'algorithmes bases sur le calcul de series entieres pour la resolution de problemes non-lineaires. Comment utiliser au mieux les informations contenues dans une serie ? Comment definir de facon optimale des algorithmes de prediction-correction dans le cadre de la MAN ? Dans cet article, nous discutons l'etat de l'art sur ces deux questions. Nous presentons aussi une etude bibliographique complete montrant un vaste champ d'application et les nombreuses possibilites offertes par la MAN.

On se propose dans cet article de coupler la Methode Asymptotique Numerique (MAN) avec une technique de sous-structuration et un calcul explicite de la matrice d'interface. De cette maniere, on obtient une decomposition exacte de la matrice tangente, qui est souhaitable puisque la MAN implique une resolution de nombreux problemes lineaires avec une meme matrice. Le cout de calcul de cette condensation est analyse et compare au cout de traitement des seconds membres de la MAN. Des exemples sont analyses, issus de modeles de post-flambage de plaques longues.

In this paper, we propose a new class of bi-grid algorithm to solve large scale linear algebraic equations. This method is based on homotopy, perturbation technique and Pade approximants. To cite this article: R. El Mokhtari et al., C. R. Mecanique 330 (2002) 825–830.

New predictor-corrector algorithms are presented for the computation of solution paths of non-linear partial differential equations. The predictors and the correctors are based on perturbation techniques and Pade approximants. This extends the Asymptotic Numerical Method (ANM), which is an efficient high-order continuation technique without corrector. The efficiency and the reliability of the new technique are assessed by several examples within thin shell theory and Navier-Stokes equations. Many variants have been tested to establish an optimal algorithm.

International audience; The present work is devoted to the numerical simulation of crack propagation in engineering materials whose failure results from void initiation, growth and coalescence. The behaviour of the plate material is described via a Gurson type model accounting for the combined effects of strain hardening, thermal softening, viscoplasticity and void growth induced damage. The eXtended Finite Element Method has been retained to describe the kinematic consequences of the crack propagation across the mesh. The crack is assumed to propagate as soon as the stored energy around the crack tip reaches a critical value. The related crack length is estimated using an exhaustion method. The constitutive model and the extended finite elements were both implemented in the engineering FE computation code Abaqus as user subroutines. The numerical simulation of a notched plate and an asymmetrically notched plate under tension loading has been conducted. While making some simplifications, the present work reproduces numerically the 2D propagation of a crack resulting from void growth induced damage.

This work deals with the vibrations of sandwich visco-elastic structures. The central visco-elastic layer included between two metallic layers leads to solve a problem of complex and non-linear eigenvalues. Indeed the core material depends on the frequency. Several methods exist to solve this type of problem. But the dimension of matrices to be manipulated can become very large and generate substantial computational times. A reduction technique is proposed herein. It is applied and compared to the high order Newton method.

International audience; The aim of this paper is to develop a reduction method to determine the modal characteristics of viscoelastic sandwich structures. The method is based on the high order Newton algorithm and reduction techniques. Numerical tests have been performed in the case of sandwich beams and cylindrical shells. The comparison of the results obtained by the reduction method with those given by direct simulation shows both a good agreement and a significant reduction in computational cost.

In this paper, we propose a new class of algorithms to solve large scale linear algebraic equations. This method is based on homotopy, perturbation technique and Pade approximants.

Une nouvelle technique de base reduite est proposee afin de resoudre une large classe de problemes non lineaires. L'idee principale est de reduire les problemes lineaires obtenus par une technique ...

Resume Nous proposons quelques techniques de regularisation permettant d'adapter les methodes asymptotiques numeriques (developpement en series couple avec une discretisation par elements finis) a des problemes fortement non lineaires. Des applications seront presentees pour un probleme d'elasticite avec contact, en dynamique des fluides visqueux et pour le gonflement hydraulique d'une plaque viscoplastique.

This work concerns numerical simulations of Power-law fluids. This non-linear problem is solved by using the Asymptotic-Numerical Method (ANM). As this problem is strongly non-linear, we show how the ANM can be used (introduction of new variables, regularization parameter). A numerical method to compute critical Reynolds numbers, bifurcation points, is also proposed. This method makes it possible to determine accurate critical Reynolds without increasing the computational times. Several numerical examples help to demonstrate the efficiency and the reliability of the proposed methods.

This work deals with forced vibration of nonlinear rotating composite beams with uniform cross-section. Coupling the Galerkin method with the balance harmonic method, the nonlinear intrinsic and geometrical exact equations of motion for anisotropic beams are converted into a static formulation, which is treated with the continuation method; the asymptotic numerical method, where power series expansions and Pade approximants are used to represent the generalized vector of displacement and the frequency. Response curves are obtained and the nonlinearity is studied for various angular speed. Internal resonance flexion-flexion is found and the angular speed effect on the coupling between modes is investigated.

This work deals with the computation of the non-linear solutions of the vibration of damped plates by coupling a harmonic balance method and the asymptotic numerical method. These computations can lead to lengthy central processing unit (CPU) times if the solution sought contains an important number of harmonics. In this study, we propose two reduced order models which can be applied to solve this type of problem. Both reduced methods are based on a first computation carried out with a small number of harmonics (here two). Numerical examples of plate vibration show that these algorithms help save a great deal of computational time and can be applied to problems involving numerous harmonics. Ce travail porte sur le calcul des solutions non lineaires de vibration des plaques minces amorties par le couplage de la methode d’Equilibrage Harmonique et la Methode Asymptotique Numerique (MAN). Ces calculs peuvent conduire a un temps significatif si la solution recherchee contient un nombre important d’harmoniques...

This paper deals with the numerical study of bifurcations in the two-dimensional (2D) lid-driven cavity (LDC). Two specific geometries are considered. The first geometry is the two-sided non-facing (2SNF) cavity: the velocity is imposed on the upper and the left side of the cavity. The second geometry is the four-sided (4S) cavity where all the sides have a prescribed motion. For the first time, the linear stability analysis is performed by coupling two specific algorithms. The first one is dedicated to the computation of the stationary bifurcations and the bifurcated branches. Then, a second algorithm is dedicated to the computation of Hopf bifurcations. In this study, for both problems, it is shown that the flow becomes asymmetric via a stationary bifurcation. The critical Reynolds numbers are close to 1070 and 130, respectively, for the 2SNF and the 4S cavity. Following the stationary bifurcated branches, supplementary results concerning the stability are found. Firstly, for both examples, a second stationary bifurcation appears on the unstable solution, for a Reynolds number equal to 1890 and 360, respectively, for the 2NSF and the 4S cavity. Secondly, a second stationary bifurcation is found on the stable solutions of the 4S LDC for a critical Reynolds number close to 860. Nevertheless, no Hopf bifurcation has been found on this stable bifurcated branch for Reynolds numbers between 130 and 1000. Concerning the 2SNF LDC, Hopf bifurcation points have been determined on these stable bifurcated solutions. The first bifurcation occurs for a Reynolds number close to 3000 and a Strouhal number equal to 0.47.

International audience; 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] and [2]. The critical values are determined by using a bifurcation indicator [3], [4] and [5] and the bifurcated branches are computed by using an augmented system which was first introduced in solid mechanics [4] and [6]. Several numerical examples from 2D Navier-Stokes show the reliability and the efficiency of the proposed methods.

International audience; This 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.

International audience; 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 leur forme, 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 sur un cas de gilet de sauvetage gonflable.

This paper deals with the computation of Hopf bifurcation points in fluid mechanics. This computation is done by coupling a bifurcation indicator proposed recently (Cadou et al., 2006) [1] and a direct method (Jackson, 1987; Jepson, 1981) [2] , [3] 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.

International audience; This 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.

International audience; This work deals with the convergence acceleration of iterative nonlinear methods. Two convergence accelerating techniques are evaluated: the Modified Mininal Polynomial Extrapolation Method (MMPE) and the Padé approximants. The algorithms studied in this work are iterative correctors: Newton’s modified method, a high-order iterative corrector presented in Damil et al. (Commun Numer Methods Eng 15:701–708, 1999) and an original algorithm for vibration of viscoelastic structures. We first describe the iterative algorithms for the considered nonlinear problems. Secondly, the two accelerating techniques are presented. Finally, through several numerical tests from the thin shell theory, Navier–Stokes equations and vibration of viscoelastic shells, the advantages and drawbacks of each accelerating technique is discussed.