Model-Free Data-Driven Methods in Mechanics: Material Data Identification and Solvers

This paper presents an integrated model-free data-driven approach to solid mechanics, allowing to perform numerical simulations on structures on the basis of measures of displacement fields on representative samples, without postulating a specific constitutive model. A material data identification procedure, allowing to infer strain-stress pairs from displacement fields and boundary conditions, is used to build a material database from a set of mutiaxial tests on a non-conventional sample. This database is in turn used by a data-driven solver, based on an algorithm minimizing the distance between manifolds of compatible and balanced mechanical states and the given database, to predict the response of structures of the same material, with arbitrary geometry and boundary conditions. Examples illustrate this modelling cycle and demonstrate how the data-driven identification method allows importance sampling of the material state space, yielding faster convergence of simulation results with increasing database size, when compared to synthetic material databases with regular sampling patterns.Comment: Revised versio

An overview of the proper generalized decomposition with applications in computational rheology

We review the foundations and applications of the proper generalized decomposition (PGD), a powerful model reduction technique that computes a priori by means of successive enrichment a separated representation of the unknown field. The computational complexity of the PGD scales linearly with the dimension of the space wherein the model is defined, which is in marked contrast with the exponential scaling of standard grid-based methods. First introduced in the context of computational rheology by Ammar et al. [3] and [4], the PGD has since been further developed and applied in a variety of applications ranging from the solution of the Schrödinger equation of quantum mechanics to the analysis of laminate composites. In this paper, we illustrate the use of the PGD in four problem categories related to computational rheology: (i) the direct solution of the Fokker-Planck equation for complex fluids in configuration spaces of high dimension, (ii) the development of very efficient non-incremental algorithms for transient problems, (iii) the fully three-dimensional solution of problems defined in degenerate plate or shell-like domains often encountered in polymer processing or composites manufacturing, and finally (iv) the solution of multidimensional parametric models obtained by introducing various sources of problem variability as additional coordinates

Deim-based pgd for multi-parametric nonlinear model reduction

A new technique for efficiently solving parametric nonlinear reduced order models in the Proper Generalized Decomposition (PGD) framework is presented here. This technique is based on the Discrete Empirical Interpolation Method (DEIM)[1], and thus the nonlinear term is interpolated using the reduced basis instead of being fully evaluated. The DEIM has already been demonstrated to provide satisfactory results in terms of computational complexity decrease when combined with the Proper Orthogonal Decomposition (POD). However, in the POD case the reduced basis is a posteriori known as it comes from several pre-computed snapshots. On the contrary, the PGD is an a priori model reduction method. This makes the DEIM-PGD coupling rather delicate, because different choices are possible as it is analyzed in this work

Molecular imaging for stem cell therapy in the brain

In this work, we suggest a method, based on the data driven computational mechanics framework introduced by Kirchdoerfer and Ortiz, to extract representative strain-stress couples from a collection of non-homogenous full field measurements corresponding to different loading conditions

Data-Driven Identification for Linear-Viscoelastic Materials

Data-driven Identification (DDI) is a technique that allows to estimate the stresses of a sample and the behavior of the material solely by the use of strain information, avoiding the bias imposed by an empirical constitutive model. In this work, we extend the applicability of DDI from elasticity to linear-viscoelastic materials by extending the dimensionality of the problem. Rather than estimating the state of the elements considering an instantaneous value of strain-stress, we include the strain history of the sample in order to account for the viscosity effect. We also combine the method with data analysis techniques such as Kernel Principal Component Analysis to improve the estimation of stresses. Preliminary results in modeled samples show a clear improvement on the estimation of stresses when compared against the original formulation of the algorithm, allowing us to obtain results in cases where the original DDI fails to do so

Numerical Tools for the Control of the Unsteady Heating of an Airfoil

International audienceThis paper concerns the real time control of the boundary layer on an aircraft wing. This new approach consists in heating the surface in an unsteady regime using electrically resistant strips embedded in the wing skin. The control of the boundary layer's separation and transition point will provide a reduction in friction drag, and hence a reduction in fuel consumption. This new method consists in applying the required thermal power in the different strips in order to ensure the desired temperatures on the aircraft wing. We also have to determine the optimum size of these strips (length, width and distance between two strips). This implies finding the best mathematical model corresponding to the physics enabling us to facilitate the calculation for any type of material used for the wings. Secondly, the heating being unsteady, and, as during a flight the flow conditions or the ambient temperatures vary, the thermal power needed changes and must be chosen as fast as possible in order to ensure optimal operating conditions

Mesurer des déplacements ET des contraintes par correlation mécanique d'images

Depuis ses premiers développements, la corrélation d'images a suscité un engouement très fort dans la communauté de la mécanique expérimentale. La possibilité d'accéder à la mesure d'un champ de déformation hétérogène offre en effet des perspectives nombreuses. Néanmoins, l'établissement d'une relation avec les contraintes ne peut être réalisé que dans des configurations où une solution analytique est connue. Ceci limite donc a priori la portée de l'approche aux cas où le champ de contrainte est homogène et aux cas où l'on peut supposer le comportement élastique et disposer d'une solution analytique ( fissure, essai brésilien par exemple). Si l'on sort de ce cadre, il est nécessaire de recourir à la simulation numérique. En couplant mesure de champs par corrélation d'images et simulation numérique, on peut accéder au champ de contraintes. Cela suppose de connaître la loi de comportement du matériau. Mais on peut alors ajuster les paramètres de cette loi pour minimiser l'écart entre mesure et simulation. Cette approche est prometteuse mais elle suppose l'écriture d'une loi de comportement pour le matériau étudié ce qui peut parfois être limitant. On propose ici un nouvelle approche qui permet à partir d'images, d'accéder au champ de contrainte sans a priori sur le comportement. Cela repose sur une paramétrisation du déplacement adéquate en termes de contraintes. En plus, du champ de contrainte, on accède à un champ de déformation anélastique mesurant l'écart à la compatibilité vis-à-vis d'une déformation élastique dérivée du champ de contrainte. Des résultats illustreront les possibilités de la méthode par des exemples

The Proper Generalized Decomposition (PGD) as a numerical procedure to solve 3D cracked plates in linear elastic fracture mechanics

In this work, we present a new approach to solve linear elastic crack problems in plates using the so-called Proper Generalized Decomposition (PGD). In contrast to the standard FE method, the method enables to solve the crack problem in an efficient way by obtaining a single solution in which the Poisson's ratio v and the plate thickness B are non-fixed parameters. This permits to analyze the influence of v and B in the 3D solutions at roughly the cost of a series expansion of 2D analyses. Computationally, the PGD solution is less expensive than a full 3D standard FE analysis for typical discretizations used in practice to capture singularities in 3D crack problems. In order to verify the effectiveness of the proposed approach, the method is applied to cracked plates in mode I with a straight-through crack and a quarter-elliptical corner crack, validating J-integral results with different reference solutions.The authors thank the Ministerio de Ciencia y Tecnologia for the support received in the framework of the projects DPI2010-20990, DPI2010-20542 and to the Generalitat Valenciana, Programme PROMETEO 2012/023.Giner Maravilla, E.; Bognet, B.; Ródenas, J.; Leygue, A.; Fuenmayor Fernández, FJ.; Chinesta Soria, FJ. (2013). The Proper Generalized Decomposition (PGD) as a numerical procedure to solve 3D cracked plates in linear elastic fracture mechanics. International Journal of Solids and Structures. 50(10):1710-1720. https://doi.org/10.1016/j.ijsolstr.2013.01.039S17101720501

Vademecum-based GFEM (V-GFEM): optimal enrichment for transient problems

This paper proposes a generalized finite element method based on the use of parametric solutions as enrichment functions. These parametric solutions are precomputed off-line and stored in memory in the form of a computational vademecum so that they can be used on-line with negligible cost. This renders a more efficient computational method than traditional finite element methods at performing simulations of processes. One key issue of the proposed method is the efficient computation of the parametric enrichments. These are computed and efficiently stored in memory by employing proper generalized decompositions. Although the presented method can be broadly applied, it is particularly well suited in manufacturing processes involving localized physics that depend on many parameters, such as welding. After introducing the vademecum-generalized finite element method formulation, we present some numerical examples related to the simulation of thermal models encountered in welding processes