Regularized regressions for parametric models based on separated representations

Regressions created from experimental or simulated data enable the construction of metamodels, widely used in a variety of engineering applications. Many engineering problems involve multi-parametric physics whose corresponding multi-parametric solutions can be viewed as a sort of computational vademecum that, once computed offline, can be then used in a variety of real-time engineering applications including optimization, inverse analysis, uncertainty propagation or simulation based control. Sometimes, these multi-parametric problems can be solved by using advanced model order reduction—MOR-techniques. However, solving these multi-parametric problems can be very costly. In that case, one possibility consists in solving the problem for a sample of the parametric values and creating a regression from all the computed solutions. The solution for any choice of the parameters is then inferred from the prediction of the regression model. However, addressing high-dimensionality at the low data limit, ensuring accuracy and avoiding overfitting constitutes a difficult challenge. The present paper aims at proposing and discussing different advanced regressions based on the proper generalized decomposition (PGD) enabling the just referred features. In particular, new PGD strategies are developed adding different regularizations to the s-PGD method. In addition, the ANOVA-based PGD is proposed to ally them

Sur la prise en compte des variations de chargement et de géométrie dans les assemblages

L'objet de ce travail est de développer une méthode spécialisée pour l'étude paramétrique d'assemblages incluant de multiples zones de contact frottant. Cette méthode tire parti du fait que les non-linéarités et/ou les incertitudes sont localisées dans les liaisons. On utilise une représentation unifiée de ces zones de liaison sous forme d'interfaces en introduisant une décomposition de l'assemblage. L'algorithme de résolution itératif utilisé (méthode LATIN) permet un découplage du traitement des non-linéarités et/ou incertitudes locales du traitement des problèmes linéaires globaux. Chaque configuration d'un assemblage correspond à un choix de valeurs des paramètres décrivant les chargements extérieurs et les géométries des zones de contact. Plutôt que de réaliser un calcul complet pour chaque jeu de paramètres, nous utilisons la capacité de la méthode LATIN à réutiliser simplement la solution d'un problème donné (associé à un jeu de paramètres) pour en résoudre d'autres. Les exemples proposés concernent des assemblages de structures 3D avec des liaisons par contact frottant

Méthode des éléments finis : Développements récents adaptés au calcul de structures complexes

National audienceLe développement et la commercialisation de nombreux codes de calcul par la méthode des éléments finis rendent la simulation de systèmes mécaniques accessible à tous, même aux plus novices d'entre nous. Les systèmes de modélisation et de CAO permettent aisément d'obtenir un modèle éléments finis, même pour des géométries complexes. De plus, les systèmes de post-traitement graphiques tracent des cartes d'isovaleurs (de contraintes, de déformations, …) avec une qualité et un nombre de couleurs impressionnants. Néanmoins il convient de rester lucide et d'être conscients de la qualité des calculs que nous menons. Même si elle se banalise, la méthode des éléments finis reste un outil complexe et savoir bien la maîtriser n'est pas chose aisée. Nous présentons ici quelques notions sur le contrôle des calculs et sur certains outils en cours de développement pour donner à l'utilisateur l'assurance d'une qualité de son modèle. Un exemple de maîtrise de modèle élastique 3D est détaillé. De plus, nous essayons de donner quelques notions sur les difficultés rencontrées lors de l'étude de structures complexes qui n'est pas encore, de nos jours, à la portée de tous. Par "complexe", nous entendons complexité du système mécanique (géométrie, liaisons, …), complexité des sollicitations appliquées (chargements divers, dynamiques, cycliques, …, grands déplacements, …) et complexité du modèle de matériau utilisé (plasticité, viscoplasticité, endommagement, …).Nous présentons les problèmes numériques rencontrés lors de l'étude, par la méthode des éléments finis, de structures complexes et nous donnons quelques pistes pour faciliter le traitement de certains types de problèmes. Un exemple de méthode adaptée au calcul d'assemblages complexes de structures élastiques est proposé. Après avoir donné quelques idées sur les problèmes de taille et de coût de calcul, nous présentons les perspectives offertes par les calculateurs à architecture parallèle modernes. Un exemple illustre ces possibilités

Port-metriplectic neural networks: thermodynamics-informed machine learning of complex physical systems

We develop inductive biases for the machine learning of complex physical systems based on the port-Hamiltonian formalism. To satisfy by construction the principles of thermodynamics in the learned physics (conservation of energy, non-negative entropy production), we modify accordingly the port-Hamiltonian formalism so as to achieve a port-metriplectic one. We show that the constructed networks are able to learn the physics of complex systems by parts, thus alleviating the burden associated to the experimental characterization and posterior learning process of this kind of systems. Predictions can be done, however, at the scale of the complete system. Examples are shown on the performance of the proposed technique.Comment: 9 pages, 5 figure

On a multiscale strategy and its optimization for the simulation of combined delamination and buckling

This paper investigates a computational strategy for studying the interactions between multiple through-the-width delaminations and global or local buckling in composite laminates taking into account possible contact between the delaminated surfaces. In order to achieve an accurate prediction of the quasi-static response, a very refined discretization of the structure is required, leading to the resolution of very large and highly nonlinear numerical problems. In this paper, a nonlinear finite element formulation along with a parallel iterative scheme based on a multiscale domain decomposition are used for the computation of 3D mesoscale models. Previous works by the authors already dealt with the simulation of multiscale delamination assuming small perturbations. This paper presents the formulation used to include geometric nonlinearities into this existing multiscale framework and discusses the adaptations that need to be made to the iterative process in order to ensure the rapid convergence and the scalability of the method in the presence of buckling and delamination. These various adaptations are illustrated by simulations involving large numbers of DOFs

Recalage de structures légères par une approximation polynomiale en vue de leur contrôle actif

Le recalage de structures légères en vue du contrôle de leurs vibrations est une problématique importante. Un des outils mécaniques performant dans ce domaine est le recalage basé sur l'erreur en relation de comportement. Cet outil est adapté ici au contrôle actif en contexte incertain de telles structures grâce une description polynomiale de l'algorithme de calcul : les inconnues des polynômes sont la variabilité des paramètres (matériau ou conditions aux limites) qu'on souhaite recaler

Modular parametric PGD enabling online solution of partial differential equations

In the present work, a new methodology is proposed for building surrogate parametric models of engineering systems based on modular assembly of pre-solved modules. Each module is a generic parametric solution considering parametric geometry, material and boundary conditions. By assembling these modules and satisfying continuity constraints at the interfaces, a parametric surrogate model of the full problem can be obtained. In the present paper, the PGD technique in connection with NURBS geometry representation is used to create a parametric model for each module. In this technique, the NURBS objects allow to map the governing boundary value problem from a parametric non-regular domain into a regular reference domain and the PGD is used to create a reduced model in the reference domain. In the assembly stage, an optimization problem is solved to satisfy the continuity constraints at the interfaces. The proposed procedure is based on the offline--online paradigm: the offline stage consists of creating multiple pre-solved modules which can be afterwards assembled in almost real-time during the online stage, enabling quick evaluations of the full system response. To show the potential of the proposed approach some numerical examples in heat conduction and structural plates under bending are presented

Describing and Modeling Rough Composites Surfaces by Using Topological Data Analysis and Fractional Brownian Motion

Many composite manufacturing processes employ the consolidation of pre-impregnated preforms. However, in order to obtain adequate performance of the formed part, intimate contact and molecular diffusion across the different composites’ preform layers must be ensured. The latter takes place as soon as the intimate contact occurs and the temperature remains high enough during the molecular reptation characteristic time. The former, in turn, depends on the applied compression force, the temperature and the composite rheology, which, during the processing, induce the flow of asperities, promoting the intimate contact. Thus, the initial roughness and its evolution during the process, become critical factors in the composite consolidation. Processing optimization and control are needed for an adequate model, enabling it to infer the consolidation degree from the material and process features. The parameters associated with the process are easily identifiable and measurable (e.g., temperature, compression force, process time, ⋯). The ones concerning the materials are also accessible; however, describing the surface roughness remains an issue. Usual statistical descriptors are too poor and, moreover, they are too far from the involved physics. The present paper focuses on the use of advanced descriptors out-performing usual statistical descriptors, in particular those based on the use of homology persistence (at the heart of the so-called topological data analysis—TDA), and their connection with fractional Brownian surfaces. The latter constitutes a performance surface generator able to represent the surface evolution all along the consolidation process, as the present paper emphasizes

Modeling systems from partial observations

Modeling systems from collected data faces two main difficulties: the first one concerns the choice of measurable variables that will define the learnt model features, which should be the ones concerned by the addressed physics, optimally neither more nor less than the essential ones. The second one is linked to accessibility to data since, generally, only limited parts of the system are accessible to perform measurements. This work revisits some aspects related to the observation, description, and modeling of systems that are only partially accessible and shows that a model can be defined when the loading in unresolved degrees of freedom remains unaltered in the different experiments

Application of PGD separation of space to create a reduced-order model of a lithium-ion cell structure

Lithium-ion cells can be considered a laminate of thin plies comprising the anode, separator, and cathode. Lithium-ion cells are vulnerable toward out-of-plane loading. When simulating such structures under out-of-plane mechanical loads, subordinate approaches such as shells or plates are sub-optimal because they are blind toward out-of-plane strains and stresses. On the other hand, the use of solid elements leads to limitations in terms of computational efficiency independent of the time integration method. In this paper, the bottlenecks of both (implicit and explicit) methods are discussed, and an alternative approach is shown. Proper generalized decomposition (PGD) is used for this purpose. This computational method makes it possible to divide the problem into the characteristic in-plane and out-of-plane behaviors. The separation of space achieved with this method is demonstrated on a static linearized problem of a lithium-ion cell structure. The results are compared with conventional solution approaches. Moreover, an in-plane/out-of-plane separated representation is also built using proper orthogonal decomposition (POD). This simply serves to compare the in-plane and out-of-plane behaviors estimated by the PGD and does not allow computational advantages relative to conventional techniques. Finally, the time savings and the resulting deviations are discussed