Crushing Modes of Aluminium Tubes under Axial Compression

A numerical study of the crushing of circular aluminium tubes with and without aluminium foam fillers has been carried out to investigate their buckling behaviours under axial compression. A crushing mode classification chart has been established for empty tubes. The influence of boundary conditions on crushing mode has also been investigated. The effect of foam filler on the crushing mode of tubes filled with foam was then examined. The predicted results would assist the design of crashworthy tube components with the preferred crushing mode with the maximum energy absorption

Robust Multiscale Identification of Apparent Elastic Properties at Mesoscale for Random Heterogeneous Materials with Multiscale Field Measurements

The aim of this work is to efficiently and robustly solve the statistical inverse problem related to the identification of the elastic properties at both macroscopic and mesoscopic scales of heterogeneous anisotropic materials with a complex microstructure that usually cannot be properly described in terms of their mechanical constituents at microscale. Within the context of linear elasticity theory, the apparent elasticity tensor field at a given mesoscale is modeled by a prior non-Gaussian tensor-valued random field. A general methodology using multiscale displacement field measurements simultaneously made at both macroscale and mesoscale has been recently proposed for the identification the hyperparameters of such a prior stochastic model by solving a multiscale statistical inverse problem using a stochastic computational model and some information from displacement fields at both macroscale and mesoscale. This paper contributes to the improvement of the computational efficiency, accuracy and robustness of such a method by introducing (i) a mesoscopic numerical indicator related to the spatial correlation length(s) of kinematic fields, allowing the time-consuming global optimization algorithm (genetic algorithm) used in a previous work to be replaced with a more efficient algorithm and (ii) an ad hoc stochastic representation of the hyperparameters involved in the prior stochastic model in order to enhance both the robustness and the precision of the statistical inverse identification method. Finally, the proposed improved method is first validated on in silico materials within the framework of 2D plane stress and 3D linear elasticity (using multiscale simulated data obtained through numerical computations) and then exemplified on a real heterogeneous biological material (beef cortical bone) within the framework of 2D plane stress linear elasticity (using multiscale experimental data obtained through mechanical testing monitored by digital image correlation)

Méthode multi-échelle avec patchs pour la propagation d'incertitudes localisées dans les modèles stochastiques

National audienceNous présentons une stratégie basée sur une méthode multiéchelle avec patch afin de traiter des problèmes stochastiques où les sources d'incertitudes sont nombreuses et dont les modèles associés sont des modèles multiéchelles complexes de grande dimension stochastique. La méthode exploite l'aspect localisé des incertitudes en séparant les échelles ce qui permet d'améliorer à la fois le conditionnement du problème et la convergence des méthodes d'approximation de tenseur utilisées pour résoudre les problèmes stochastiques de grande dimension aux niveaux local et global

A posteriori error estimation and adaptive strategy for PGD model reduction applied to parametrized linear parabolic problems

We define an a posteriori verification procedure that enables to control and certify PGD-based model reduction techniques applied to parametrized linear elliptic or parabolic problems. Using the concept of constitutive relation error, it provides guaranteed and fully computable global/goal-oriented error estimates taking both discretization and PGD truncation errors into account. Splitting the error sources, it also leads to a natural greedy adaptive strategy which can be driven in order to optimize the accuracy of PGD approximations. The focus of the paper is on two technical points: (i) construction of equilibrated fields required to compute guaranteed error bounds; (ii) error splitting and adaptive process when performing PGD-based model reduction. Performances of the proposed verification and adaptation tools are shown on several multi-parameter mechanical problems

Identification of the Mechanical Properties of Particle Boards and Stochastic Simulation of the Behavior of Furniture

International audienceIn furniture industry, the numerical simulation allows the design and optimization of wood-based structures, thus avoiding expensive experimental campaigns. Most of wood-based furniture present some particular features in terms of material properties and geometries. On the one hand, the properties of timber materials (such as particule boards) are strongly heterogeneous and anisotropic. On the other hand, the furniture are often made of simply-shaped elements and then can be represented by an assembly of plates and/or beams. The present work deals with those specific features and presents the identification of the elastic properties of particle boards from digital image correlation (DIC) [1] as well as the simulation of the mechanical behavior of furniture.First, three-point bending tests based on Timoshenko's beam theory are performed on different samples cut from a prototype desk for the identification of the material properties using DIC techniques. Secondly, a probabilistic model for the uncertain material parameters is constructed by using the Maximum Entropy (MaxEnt) principle [2] combined with a Markov Chain Monte-Carlo (MCMC) method based on Metropolis-Hastings algorithm for generating realizations of the underlying random variables. Thirdly, numerical virtual tests are performed to propagate the uncertainties in the material properties through the model and assess the impact of such variabilities on the response of the structure. Lastly, several real tests were previously conducted on the desk in order to validate the proposed numerical approach. Quite good agreement is observed between the numerical computations and the experimental measurements.[1] L. Chevalier, S. Calloch, F. Hild and Y. Marco, Digital image correlation used to ana- lyze the multiaxial behavior of rubber-like materials. European Journal of Mechanics- A/Solids. Vol. 20(2), pp. 169–187, 2001.[2] C. Soize, Uncertainty Quantication: An Accelerated Course with Advanced Applica- tions in Computational Engineering. Springer, Vol. 47, 2017

Méthode de décomposition de domaine multiéchelle pour la résolution de problèmes stochastiques non-linéaires en grande dimension avec sources d'incertitudes et non-linéarités localisées

National audienceDuring the last few decades, functional approaches for uncertainty quantification and propagation (henceforth known as spectral stochastic methods) have been a subject of growing interest and are currently employed for solving complex multiscale stochastic problems. Such multiscale stochastic models may exhibit some variabilities in the material properties (affecting the operator), the boundary conditions (affecting the loading and the source term) or the geometry at different scales. Classical monoscale approaches based on adaptive remeshing (e.g. mesh refinement) or enrichement techniques may lead to high computational costs and memory storage. Conversely, multiscale approaches based on patches allow to take the high solution complexity into account by operating a separation of scales. A multiscale coupling statregy devoted to stochastic problems featuring localized uncertainties hase been recently proposed in [1]. It relies on an overlapping domain decomposition method and leads to a global-local (two-scale) formulation of the stochastic problem. The associated global-local iterative algorithm requires the successive solution of a series of simplified global problems (with deterministic operators) defined on a deterministic domain and complex local problems (with uncertain operators and/or geometry) defined on subdomains of interest called patches. Convergence and robustness properties of the algorithm have been analyzed in [1] for linear elliptic stochastic problems.In the present work, the method is extended to non-linear elliptic stochastic problems with locaized variabilities and non-linearities. Convergence and robustness results of the global-local iterative algorithm are shown for a class of non-linear elliptic stochastic problems. The multiscale coupling approach appears to be flexible and non-intrusive allowing to handle different models, approximation spaces and dedicated solvers at both local and global levels. The stochastic local problems are solved in parallel using a least-squares minimization method (non-intrusive sampling approach) which asks for the evaluation of samples of local solutions thanks to non-linear deterministic codes. Greedy algorithms proposed in [2] and dedicated to the solution of high-dimensional stochastic problems allow the adaptive construction of sparse polynomial approximations. The approximation error of local solutions is controlled by adapting the stochastic approximation basis and the number of samples. The performances of the multiscale domain decomposition method are illustrated through numerical examples carried out on a non-linear diffusion-reaction stochastic problem with localized random material heterogeneities.References[1] M. Chevreuil, A. Nouy, and E. Safatly. A multiscale method with patch for the solution of stochastic partial differential equations with localized uncertainties. Computer Methods in Applied Mechanics and Engineering, 255(0):255–274, 2013.[2] A. Chkifa, A. Cohen, R. DeVore, and C. Schwab. Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs. ESAIM: Mathematical Modelling and Numerical Analysis, 47(1):253–280, 2013.Durant les deux dernières décennies, les approches fonctionnelles pour la quantification et la propagation d'incertitudes (désormais connues sous le nom de méthodes spectrales stochastiques) ont suscité un intérêt grandissant et sont actuellement employées pour résoudre des problèmes stochastiques multiéchelles complexes. De tels modèles stochastiques multiéchelles peuvent présenter des variabilités au niveau des propriétés matériaux (affectant l'opérateur), des conditions aux limites (affectant le chargement et le terme source) ou de la géométrie à différentes échelles. Les approches monoéchelles classiques basées sur des techniques de raffinement ou d'enrichissement peuvent s'avérer coûteuses en temps de calcul et capacité mémoire. A l'inverse, les approches multiéchelles avec patchs permettent de prendre en compte la grande complexité des solutions en introduisant une séparation d'échelles. Une stratégie de couplage multiéchelle dédiée aux problèmes stochastiques présentant des sources d'incertitudes localisées a été récemment proposée dans [1]. Celle-ci s'appuie sur une méthode de décomposition de domaine avec recouvrement conduisant à une formulation globale-locale (à deux échelles) du problème stochastique. L'algorithme itératif global-local associé requiert la résolution successive d'une série de problèmes globaux simplifiés (avec opérateurs déterministes) définis sur un domaine déterministe et de problèmes locaux complexes (avec opérateurs et/ou géométrie incertains) définis sur des sous-domaines d'intérêt appelés patchs. Les propriétés de convergence et de robustesse de l'algorithme ont été analysées dans [1] dans le cadre des problèmes stochastiques elliptiques linéaires.Dans ce travail, la méthode est étendue au cas des problèmes stochastiques elliptiques non-linéaires en présence d'incertitudes et de non-linéarités localisées. La convergence et la robustesse de l'algorithme itératif global-local sont démontrées pour une classe de problèmes stochastiques elliptiques non-linéaires. L'approche de couplage multi-échelle est flexible et non-intrusive dans la mesure où elle permet la prise en compte de modèles différents, d'espaces d'approximation et de solveurs dédiés aux niveaux global et local. Les problèmes locaux stochastiques sont résolus en parallèle en utilisant une méthode de minimisation au sens des moindres carrés (approche non-intrusive basée sur de l'échantillonnage) qui nécessite l'évaluation d'échantillons des solutions locales à l'aide d'un code déterministe non-linéaire. Des algorithmes gloutons proposés dans [2] et dédiés à la résolution de problèmes stochastiques en grande dimension permettent la construction adaptative d'approximations polynomiales creuses. L'erreur d'approximation des solutions locales est contrôlée en adaptant la base d'approximation stochastique et le nombre d'échantillons. Les performances de la méthode de décomposition de domaine multiéchelle sont illustrées à partir d'exemples numériques sur un problème stochastique de diffusion-réaction non-linéaire présentant des hétérogénéités matériaux aléatoires localisées.Références[1] M. Chevreuil, A. Nouy et E. Safatly. A multiscale method with patch for the solution of stochastic partial differential equations with localized uncertainties. Computer Methods in Applied Mechanics and Engineering, 255(0):255–274, 2013.[2] A. Chkifa, A. Cohen, R. DeVore et C. Schwab. Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs. ESAIM: Mathematical Modelling and Numerical Analysis, 47(1):253–280, 2013