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

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

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

Multiscale method with patches for the propagation of localized uncertainties in (semi-)linear elliptic and parabolic equations

Séminaire du Laboratoire de Mécanique des Structures et des Systèmes Couplés (LMSSC), Conservatoire National des Arts et Métiers (CNAM)Uncertainty quantification in structural computation has gained an ever increasing interest in the scientific community during the last decades. The propagation of localized uncertainties through stochastic computational models is critical for the design and analysis of mechanical structures and components. The input uncertainties in the computational model may stem from intrinsic variabilities in the material properties or epistemic variabilities resulting from partial or limited information about the geometry, the boundary or initial conditions. Classical monoscale approaches require either local refinement or enrichment techniques to cope with the high complexity of multiscale stochastic problems. On the contrary, multiscale coupling approaches based on substructuring, domain decomposition or multigrid methods have been designed to solve such intractable high-dimensional problems. A multiscale method based on patches has been recently proposed in [1] for the solution of linear stochastic multiscale models (with localized uncertainties) and extended to a broader class of non-linear stochastic multiscale models (with localized uncertainties and non-linearities) in [2]. It relies on a partition of the domain into several subdomains of interest (called patches) containing the different sources of uncertainties and possible non-linearities, and a complementary subdomain. The multiscale solution is then computed using a global-local iterative algorithm that involves the solution of a sequence of linear global problems (with possibly deterministic operators and uncertain right-hand sides) over a deterministic domain and (non-)linear local problems (with uncertain operators and right-hand sides) over patches.In the present work, the method is extended to time evolution problems featuring localized uncertainties. The convergence of the iterative algorithm is analyzed in the stationary case. The proposed multiscale approach offers the possibility to use different global and local computational models with suitable space and time discretizations (finite element spaces, numerical time integration schemes) as well as stand-alone global and local solvers. At the local level, the stochastic problems are solved using sampling-based approaches combined with adaptive sparse approximation methods [3] in order to efficiently derive sparse representations of high-dimensional stochastic local solutions with controlled accuracy. The performances of the multiscale domain decomposition method are illustrated through numerical experiments carried out on a stationary non-linear diffusion-reaction stochastic problem and a transient linear advection-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. Nouy and F. Pled. A multiscale method for semi-linear elliptic equations with localized uncertainties and non-linearities. ESAIM: M2AN (submitted), 2017.[3] A. Chkifa, A. Cohen, and C. Schwab. Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs. Journal de Mathématiques Pures et Appliquées, 103(2):400–428, 2015