Nous présentons dans cette thèse nos travaux sur la réduction d'ordre appliquée à des simulations d'aérothermie. Nous considérons le couplage entre les équations de Navier-Stokes et une équations d'énergie de type advection-diffusion. Les paramètres physiques considérés nous obligent à considéré l'introduction d'opérateurs de stabilisation de type SUPG ou GLS. Le but étant d'ajouter une diffusion numérique dans la direction du champs de convection, afin de supprimer les oscillations non-phyisques. Nous présentons également notre stratégie de résolution basée sur la méthode des bases réduite (RBM). Afin de retrouver une décomposition affine, essentielle pour l'application de la RBM, nous avons implémenté une version discrète de la méthode d'interpolation empirique (EIM). Cette variante permet de la construction d'approximation affine pour des opérateurs complexes. Nous utilisons notamment cette méthode pour la réduction des opérateurs de stabilisations. Cependant, la construction des bases EIM pour des problèmes non-linéaires implique un grand nombre de résolution éléments finis. Pour pallier à ce problème, nous mettons en oeuvre les récents développement de l'algorithme de coconstruction entre EIM et RBM (SER). / We present in this thesis our work on model order reduction for aerothermal simulations. We consider the coupling between the incompressible Navier-Stokes equations and an advection-diffusion equation for the temperature. Since the physical parameters induce high Reynolds and Peclet numbers, we have to introduce stabilization operators in the formulation to deal with the well known numerical stability issue. The chosen stabilization, applied to both fluid and heat equations, is the usual Streamline-Upwind/Petrov-Galerkin (SUPG) which add artificial diffusivity in the direction of the convection field. We also introduce our order reduction strategy for this model, based on the Reduced Basis Method (RBM). To recover an affine decomposition for this complex model, we implemented a discrete variation of the Empirical Interpolation Method (EIM) which is a discrete version of the original EIM. This variant allows building an approximated affine decomposition for complex operators such as in the case of SUPG. We also use this method for the non-linear operators induced by the shock capturing method. The construction of an EIM basis for non-linear operators involves a potentially huge number of non-linear FEM resolutions - depending on the size of the sampling. Even if this basis is built during an offline phase, we usually can not afford such expensive computational cost. We took advantage of the recent development of the Simultaneous EIM Reduced basis algorithm (SER) to tackle this issue.

O objetivo deste estudo é apresentar uma revisão sobre técnicas de redução da ordem de modelos dinâmicos lineares e invariantes no tempo. Com a implementação de tais técnicas, o autor mostra que é possível reproduzir as principais características da resposta de um modelo de alta ordem através de um modelo de ordem reduzida com menor número de graus de liberdade. Uma metodologia para redução da ordem de modelos de sistemas dinâmicos foi apresentada. Os processos envolvidos nesta metodologia foram descritos, os quais são: técnicas para realizar projeções do sistema em diferentes bases, selecionar os graus de liberdade que são bons candidatos a eliminação, eliminar graus de liberdade do modelo completo e implementar correções na resposta do modelo reduzido. Foram apresentadas maneiras de quantificar as similitudes entre as respostas dos modelos completo e reduzido através de métricas de representatividade. Para implementar e estudar as técnicas de redução apresentadas, o autor elaborou dois modelos para estudo de caso: um modelo para estudo da dinâmica vertical de um veículo de passeio e outro modelo para estudo da dinâmica longitudinal de um trem. Diferentes técnicas de redução foram implementadas a partir dos dois modelos para estudo de caso e os resultados foram comparados através das métricas de representatividade. O critério proposto pelo autor para quantificar desempenho de um modelo reduzido foi utilizado para determinar de maneira objetiva o modelo reduzido mais adequado para cada aplicação. Como contribuição neste trabalho, o autor propôs uma definição de desempenho de um modelo reduzido e um método para quantificar o mesmo, além de duas novas métricas para mensurar a capacidade do modelo reduzido em reproduzir os máximos sobre-sinais e tempos de acomodação do modelo completo. / The objective of this work is to present a revision about model order reduction techniques applied to linear, time invariant dynamic systems. With the implementation of these techniques, the author shows that it is possible to reproduce the main characteristics of the response of a high order dynamic system using a reduced order model with fewer degrees of freedom. A model order reduction methodology was presented. The processes which are involved in this methodology were described, which are: techniques for projection onto different basis, selection of the most suitable degrees of freedom to be reduced, elimination of degrees of freedom from the high order model, implementation of corrections at the reduced model response. The author showed ways to quantify the similarities between the responses of the complete and reduced models using representativeness metrics. In order to implement and study the presented model order reduction techniques, the author developed two case study models: one model to study the vertical dynamics of a passenger car and another model to study the longitudinal dynamics of a train. Different model order reduction techniques were implemented and its results were compared using representativeness metrics and the performance of the reduced models. The criteria proposed by the author to quantify the performance of a reduced order model was used to objectively determine the most suitable reduced order model for each application. The author proposed, as contribution at this work, a definition of the reduced order model performance, a method to quantify its performance and two new metrics to measure the capacity of the reduced model to reproduce the overshoots and settling times of the complete model.

La finesse de la résolution spectrale et spatiale des images hyperspectrales en font des données de très grande dimension. C'est également le cas d'autres types de données, où leur taille tend à augmenter pour de plus en plus d'applications. La complexité des données provenant de l'hétérogénéité spectrale et spatiale, de la non gaussianité du bruit et des processus physiques sous-jacents, renforcent la richesse des informations présentes sur une image hyperspectrale. Exploiter ces informations demande alors des outils statistiques adaptés aux grandes données mais aussi à leur nature non gaussienne. Des méthodes reposant sur la théorie des matrices aléatoires, théorie adaptée aux données de grande dimension, et reposant sur la robustesse, adaptée aux données non gaussiennes, sont ainsi proposées dans cette thèse, pour des applications à l'imagerie hyperspectrale. Cette thèse propose d'améliorer deux aspects du traitement des images hyperspectrales : l'estimation du nombre d'endmembers ou de l'ordre du modèle et le problème du démélange spectral. En ce qui concerne l'estimation du nombre d'endmembers, trois nouveaux algorithmes adaptés au modèle choisi sont proposés, le dernier présentant de meilleures performances que les deux autres, en raison de sa plus grande robustesse.Une application au domaine de la finance est également proposée. Pour le démélange spectral, trois méthodes sont proposées, qui tiennent comptent des diff érentes particularités possibles des images hyperspectrales. Cette thèse a permis de montrer que la théorie des matrices aléatoires présente un grand intérêt pour le traitement des images hyperspectrales. Les méthodes développées peuvent également s'appliquer à d'autres domaines nécessitant le traitement de données de grandes dimensions. / Hyperspectral imaging generates large data due to the spectral and spatial high resolution, as it is the case for more and more other kinds of applications. For hyperspectral imaging, the data complexity comes from the spectral and spatial heterogeneity, the non-gaussianity of the noise and other physical processes. Nevertheless, this complexity enhances the wealth of collected informations, that need to be processed with adapted methods. Random matrix theory and robust processes are here suggested for hyperspectral imaging application: the random matrix theory is adapted to large data and the robustness enables to better take into account the non-gaussianity of the data. This thesis aims to enhance the model order selection on a hyperspectral image and the unmixing problem. As the model order selection is concerned, three new algorithms are developped, and the last one, more robust, gives better performances. One financial application is also presented. As for the unmixing problem, three methods that take into account the peculierities of hyperspectral imaging are suggested. The random matrix theory is of great interest for hyperspectral image processing, as demonstrated in this thesis. Differents methods developped here can be applied to other field of signal processing requiring the processing of large data.

The continuous miniaturization of electronic systems using the three-dimensional (3D) integration technique has brought in new challenges for the computer-aided design and modeling of 3D integrated circuits (ICs) and systems. The major challenges for the modeling and analysis of 3D integrated systems mainly stem from four aspects: (a) the interaction between the electrical and thermal domains in an integrated system, (b) the increasing modeling complexity arising from 3D systems requires the development of multiscale techniques for the modeling and analysis of DC voltage drop, thermal gradients, and electromagnetic behaviors, (c) efficient modeling of microfluidic cooling, and (d) the demand of performing fast thermal simulation with varying design parameters. Addressing these challenges for the electrical/thermal modeling and analysis of 3D systems necessitates the development of novel numerical modeling methods. This dissertation mainly focuses on developing efficient electrical and thermal numerical modeling and co-simulation methods for 3D integrated systems. The developed numerical methods can be classified into three categories. The first category aims to investigate the interaction between electrical and thermal characteristics for power delivery networks (PDNs) in steady state and the thermal effect on characteristics of through-silicon via (TSV) arrays at high frequencies. The steady-state electrical-thermal interaction for PDNs is addressed by developing a voltage drop-thermal co-simulation method while the thermal effect on TSV characteristics is studied by proposing a thermal-electrical analysis approach for TSV arrays. The second category of numerical methods focuses on developing multiscale modeling approaches for the voltage drop and thermal analysis. A multiscale modeling method based on the finite-element non-conformal domain decomposition technique has been developed for the voltage drop and thermal analysis of 3D systems. The proposed method allows the modeling of a 3D multiscale system using independent mesh grids in sub-domains. As a result, the system unknowns can be greatly reduced. In addition, to improve the simulation efficiency, the cascadic multigrid solving approach has been adopted for the voltage drop-thermal co-simulation with a large number of unknowns. The focus of the last category is to develop fast thermal simulation methods using compact models and model order reduction (MOR). To overcome the computational cost using the computational fluid dynamics simulation, a finite-volume compact thermal model has been developed for the microchannel-based fluidic cooling. This compact thermal model enables the fast thermal simulation of 3D ICs with a large number of microchannels for early-stage design. In addition, a system-level thermal modeling method using domain decomposition and model order reduction is developed for both the steady-state and transient thermal analysis. The proposed approach can efficiently support thermal modeling with varying design parameters without using parameterized MOR techniques.

O objetivo deste estudo é apresentar uma revisão sobre técnicas de redução da ordem de modelos dinâmicos lineares e invariantes no tempo. Com a implementação de tais técnicas, o autor mostra que é possível reproduzir as principais características da resposta de um modelo de alta ordem através de um modelo de ordem reduzida com menor número de graus de liberdade. Uma metodologia para redução da ordem de modelos de sistemas dinâmicos foi apresentada. Os processos envolvidos nesta metodologia foram descritos, os quais são: técnicas para realizar projeções do sistema em diferentes bases, selecionar os graus de liberdade que são bons candidatos a eliminação, eliminar graus de liberdade do modelo completo e implementar correções na resposta do modelo reduzido. Foram apresentadas maneiras de quantificar as similitudes entre as respostas dos modelos completo e reduzido através de métricas de representatividade. Para implementar e estudar as técnicas de redução apresentadas, o autor elaborou dois modelos para estudo de caso: um modelo para estudo da dinâmica vertical de um veículo de passeio e outro modelo para estudo da dinâmica longitudinal de um trem. Diferentes técnicas de redução foram implementadas a partir dos dois modelos para estudo de caso e os resultados foram comparados através das métricas de representatividade. O critério proposto pelo autor para quantificar desempenho de um modelo reduzido foi utilizado para determinar de maneira objetiva o modelo reduzido mais adequado para cada aplicação. Como contribuição neste trabalho, o autor propôs uma definição de desempenho de um modelo reduzido e um método para quantificar o mesmo, além de duas novas métricas para mensurar a capacidade do modelo reduzido em reproduzir os máximos sobre-sinais e tempos de acomodação do modelo completo. / The objective of this work is to present a revision about model order reduction techniques applied to linear, time invariant dynamic systems. With the implementation of these techniques, the author shows that it is possible to reproduce the main characteristics of the response of a high order dynamic system using a reduced order model with fewer degrees of freedom. A model order reduction methodology was presented. The processes which are involved in this methodology were described, which are: techniques for projection onto different basis, selection of the most suitable degrees of freedom to be reduced, elimination of degrees of freedom from the high order model, implementation of corrections at the reduced model response. The author showed ways to quantify the similarities between the responses of the complete and reduced models using representativeness metrics. In order to implement and study the presented model order reduction techniques, the author developed two case study models: one model to study the vertical dynamics of a passenger car and another model to study the longitudinal dynamics of a train. Different model order reduction techniques were implemented and its results were compared using representativeness metrics and the performance of the reduced models. The criteria proposed by the author to quantify the performance of a reduced order model was used to objectively determine the most suitable reduced order model for each application. The author proposed, as contribution at this work, a definition of the reduced order model performance, a method to quantify its performance and two new metrics to measure the capacity of the reduced model to reproduce the overshoots and settling times of the complete model.

Abstract The concept of resting state functional magnetic resonance imaging (fMRI) is built onto an original finding in 1995 that brain hemispheres present synchronous signal fluctuations with distinct patterns. fMRI measurements rely on blood oxygenation changes that indirectly mirror neural activity. Therefore, the origin of functional connectivity patterns, resting state networks (RSNs), has been a widely debated research question and numerous contributing factors have been identified. According to current understanding the fluctuations reflect maintenance of the system integrity in addition to spontaneous thought and action processes in the resting state. A popular method to study the functional connectivity in resting state fMRI is spatial independent component analysis (ICA) that decomposes signal sources into statistically independent components. The dichotomy of functional specialization versus functional integration has a correspondence in fMRI studies where RSNs play the integrative viewpoint of brain function. Although canonical large-scale RSNs are broadly distributed they also express modularity that can be accomplished by ICA with a high number of estimated components. The characteristics of high ICA dimensionality are broadly investigated in the thesis. An enduring issue in resting state research has been the confounding noise sources like motion and cardiorespiratory processes which may hamper the analysis. In this thesis the ability of ICA to separate these noise sources from the default mode network, a major RSN, is studied. Additionally, the suitability of ICA for full frequency spectrum analysis, a relatively rare setting in biosignal analysis, is investigated. The results of the thesis support the viewpoint of ICA as a robust analysis method for functional connectivity analysis. Cardiorespiratory and motion induced noise did not confound the functional connectivity analyses with ICA. High dimensional ICA provided better signal source separation, revealed the modular structure of the RSNs and pinpointed the specific aberrations in the autism spectrum disorder population. ICA was also found applicable for fully explorative analysis in both the spatial and temporal domains and indicated functional connectivity changes induced by transcranial bright light stimulation. / Tiivistelmä Konsepti lepotilan tutkimisesta toiminnallisella magneettikuvauksella (engl. functional magnetic resonance imaging, fMRI) on rakentunut vuonna 1995 tehdylle löydökselle aivopuoliskojen välillä synkronisesta signaalivaihtelusta. Mittaukset perustuvat veren hapetuksen muutoksiin, jotka epäsuorasti heijastelevat hermostollista toimintaa. Tämän takia toiminnallisen kytkennällisyyden muodot, lepotilaverkostot, ovat olleet laajasti väitelty tutkimusaihe ja monia verkostoihin vaikuttavia tekijöitä onkin tunnistettu. Nykykäsityksen mukaan signaalivaihtelut lepotilassa heijastelevat järjestelmän yhtenäisyyden ylläpitoa spontaanin ajattelun ja toiminnan lisäksi. Suosittu menetelmä toiminnallisen kytkennällisyyden tutkimiseen lepotilan fMRI:ssä on spatiaalinen itsenäisten komponenttien analyysi (engl. independent component analysis, ICA), joka hajottaa signaalilähteet tilastollisesti itsenäisiin komponentteihin. Aivotoiminnan mallintamisessa kahtiajaolla toiminnalliseen erikoistumiseen ja toiminnalliseen integraatioon on vastaavuus fMRI-tutkimukseen, jossa lepotilaverkostot vastaavat toiminnallisen integraation näkökulmasta. Vaikka kanoniset lepotilaverkostot ovat laaja-alaisia, ne ovat toisaalta modulaarisia, jota voidaan tutkia tutkimalla korkean komponenttimäärän ICA-hajotelmaa. Korkea- dimensioisen ICA-hajotelman ominaisuuksia tutkitaan laajasti tässä väitöskirjassa. Kestoaihe lepotilatutkimuksessa on ollut analyysiä hankaloittavien kohinalähteiden kuten liikkeen ja kardiorespiratoristen prosessien vaikutus. Väitöskirjassa tutkitaan ICA:n kykyä erotella kohinalähteitä ’default mode’ -verkostosta, joka on merkittävin lepotilaverkosto. Lisäksi tutkitaan ICA:n soveltuvuutta täyden taajuuskaistan analysointiin, joka on verrattain harvinaista biosignaalien analyysissä. Väitöskirjan tulokset tukevat näkemystä ICA:n suorituskyvystä toiminnallisen kytkennällisyyden analyysissä. Kardiorespiratorinen ja liikkeestä lähtöisin oleva kohina ei häirinnyt merkittävästi ICA-tuloksia. Korkeadimensioinen ICA tarjosi paremman erottelun signaalilähteille, paljasti lepotilaverkostojen modulaarisen rakenteen ja määritti erityisen poikkeaman autismin kirjon oireyhtymän populaatiossa. ICA:n havaittiin olevan soveltuva täyseksploratiiviselle analyysille ajassa ja avaruudessa; tulos viittaa toiminnallisen kytkennällisyyden muutoksiin kallon läpäisevän kirkasvalostimulaation aikaansaamana.

Cette thèse a pour but l'étude de la mise en œuvre de commandes par asservissement visuel pour le contrôle actif d'un écoulement de Poiseuille. D'un point de vue général, le contrôle d'écoulements vise à modifier ou à maintenir l'état de l'écoulement, malgré une éventuelle perturbation extérieure. Une des situations d'intérêt concerne par exemple la transition vers la turbulence où l'écoulement peut devenir turbulent avec la croissance de sa densité d'énergie cinétique. La réduction de la traînée est également une application potentielle dans des problèmes d'ingénierie. Un des buts applicatifs de cette thèse cherchera ainsi à minimiser à la fois la densité d'énergie cinétique et la traînée. Des modèles numériques peuvent être utilisés pour générer un modèle d'état des équations aux dérivées partielles d'un écoulement de Poiseuille. Le modèle d'état considéré dans cette thèse s'appuie sur une représentation spectrale afin de transformer les équations aux dérivées partielles originelles en un système d'équations différentielles ordinaires. Le vecteur d'état rassemble dans notre cas la vitesse et la vorticité. Les signaux de commande dépendent eux de conditions aux limites de type Dirichlet non homogènes qui correspondent à des actions de soufflage/aspiration. Le nombre de degrés de liberté commandé du problème correspond à la dimension du signal de commande. La densité d'énergie cinétique et la traînée sont modélisées en fonction du vecteur d'état et du signal de commande. Dans cette thèse nous avons plus particulièrement considéré un asservissement visuel partitionné. Celui-ci est appliqué au modèle d'état de l'écoulement avec deux degrés de liberté afin de minimiser simultanément la densité d'énergie cinétique et la traînée. La traînée, contrairement à l'énergie cinétique, diminue de façon monotone en fonction du temps. Une augmentation du nombre de degrés de liberté permet d'améliorer la décroissance de la densité d'énergie cinétique. Lorsque le nombre de degré de liberté correspond à la dimension du vecteur d'état, et en s'appuyant sur une commande par asservissement visuel, nous montrons que la densité d'énergie cinétique décroit de façon monotone au cours du temps. Le modèle d'état de l'écoulement de Poiseuille vit dans un espace de très grande dimension. Par conséquent, il est nécessaire d'un point de vue pratique de réduire la dimension du contrôleur. Nous démontrons que la loi de commande s'appuyant sur un modèle réduit peut être appliquée au système complet. Dans ce cas la densité d'énergie cinétique décroit presque de façon monotone au cours du temps en utilisant une commande par asservissement visuel à deux degrés de liberté. / The visual servoing control approach is formulated for the flow control of the plane Poiseuille flow. Generally, the flow control can lead the flow from its current state to a desired state. In transition to turbulence, the growth of kinetic energy density can lead the flow to turbulence. Moreover, the drag reduction is a potential application in the engineering applications. Therefore, this thesis aims to minimize the kinetic energy density and the skin friction drag. The governing equations of the plane Poiseuille flow are modeled to a standard form in the automatic control. More precisely, the partial differential equations of the plane Poiseuille flow are transformed to a state space representation by using the spectral method. The streamwise and spanwise directions are discretized based on the Fourier series while the wall-normal direction is discretized based on the Chebyshev polynomials. The state vector involves the wall-normal velocity and vorticity. The control signals depend on the inhomogeneous Dirichlet boundary conditions which correspond to blowing/suction boundary control. The number of independent control signals is called the number of the degree of freedom. Moreover, the skin-friction drag and the kinetic energy density are modeled as a function of the state vector. The goal is to minimize both the skin-friction drag and the kinetic energy density by appropriate methods. The partitioned visual servoing control is used to minimize, simultaneously, the skin-friction drag and the kinetic energy density with two degrees of freedom. As a result, the behavior of the skin-friction drag monotonically decreases in time. However, the behavior of the kinetic energy density does not monotonically decrease in time, the similar results from the other methods such as: PID and LQR controls. Therefore, the number of the degree of freedom increases, which leads to the improvement of the kinetic energy density. In addition, when the number of the degree of freedom equals the number of state vector, the kinetic energy density monotonically decreases in time by using the visual servoing control. The dimension of linearized plane Poiseuille flow is large, therefore, we need to reduce the order of controller. We demonstrate that the control law based on a mode reduction can be applied for the full system. Moreover, the kinetic energy density almost will monotonically decreases in time even using two degrees of freedom when the visual servoing control is designed based on the model order reduction.

La réduction d'ordre de modèles d'un problème de contact demeure un sujet de recherche important en mécanique numérique des solides.Nous proposons une extension de l'hyper-réduction avec domaine d'intégration réduit à la mécanique du contact sans frottement s'écrivant à l'aide d'une formulation mixte.Comme la zone de contact potentiel se limite au domaine réduit, nous faisons le choix de prendre comme base réduite pour la variable duale (représentative des forces de contact) la base du modèle d'ordre plein restreinte.Nous obtenons ainsi un modèle hyper-réduit hybride avec une approximation de la variable primale par des modes empiriques et de la variable duale par les fonctions de base des éléments finis. Si nécessaire, la condition inf-sup de ce modèle peut être forcée par une approximation hybride la variable primale. Cela mène à une stratégie hybride combinant un modèle d'ordre hyper-réduit et un modèle d'ordre plein permettant l'obtention d'une meilleure approximation de la solution sur la zone de contact.Un post-traitement permettant la reconstruction des multiplicateurs de Lagrange sur l'ensemble de la zone de contact est également introduit.De manière à optimiser la sélection des snapshots, un indicateur d'erreur simple et efficace est avancé pour être couplé à un algorithme glouton. / The model order reduction of mechanical problems involving contact remains an important issue in computational solid mechanics.An extension of the hyper-reduction method based on a reduced integration domain to frictionless contact problems written by a mixed formulation is proposed.As the potential contact zone is naturally reduced through the reduced domain, the dual reduced basis is chosen as the restriction of the dual full-order model basis.A hybrid hyper-reduced model combining empirical modes for primal variables with finite element approximation for dual variables is then obtained.If necessary, the inf-sup condition of this hybrid saddle point problem can be enforced by extending the hybrid approximation to the primal variables. This leads to a hybrid hyper-reduced/full-order model strategy. By this way, a better approximation on the potential contact zone is furthermore obtained.A post-treatment dedicated to the reconstruction of the contact forces on the whole domain is introduced.In order to optimize the snapshots selection, an efficient error indicator is coupled to a greedy sampling algorithm leading to a robust reduced-order model.

Virtualization of products means the representation of some of their properties by models. In a stronger digitalized world, these models will gain a much broader use than models had in engineering so far. Even for one modelling aspect different models of the same product will be used, depending on the specific need of the model user. That need may change in the course of product life, between first product concepts till over the different phases of development, to product use, maintenance, or even recycling. Since a digitalized world use of these diverse models will not be limited to experts model consistency will play a much stronger role. Model hierarchies will play a stronger role and can serve also as means for teaching product users a deeper understanding of product properties. A consistent model hierarchy leading from a simple to a more advanced property representation can support this learning process. In this paper perturbation methods are analyzed as a means for setting up model hierarchies in a consistent manner. This is studied by models for the behavior of a electrohydraulic drive, which consists of a variable speed motor, a pump, a double stroke cylinder and a counterbalance valve. Model hierarchy is achieved by model reduction in the sense of perturbation theory. The use of these different models for different questions in a system design context and their interrelations are exemplified.

L'objet de ce projet de recherche est de prédire la durée de vie d'éléments mécaniques qui sont soumis à des phénomènes de fatigue cyclique. L'idée est de développer un schéma numérique novateur pour prédire la rupture de structures sous de tels chargements. Le modèle est basé sur la mécanique des milieux continus qui introduit des variables internes pour décrire l'évolution de l'endommagement. Le défi repose dans le traitement des cycles de chargement pour la prédiction de la durée de vie, particulièrement pour la prédiction de la durée de vie résiduelle de structures existantes. Les approches traditionnelles de l'analyse de la fatigue sont basées sur des méthodes phénoménologiques utilisant des relations empiriques. De telles méthodes considèrent des approximations simplificatrices et sont incapables de prendre en compte aisément des géométries ou des charges complexes associées à des problèmes d'ingénierie réels. Une approche basée sur la description de l'évolution thermodynamique d'un milieu continu est donc utilisée pour modéliser le comportement en fatigue. Cela permet de considérer efficacement des problèmes d'ingénierie complexe et la détérioration des propriétés du matériau due à la fatigue peut être quantifiée à l'aide de variables internes. Cependant, cette approche peut être numériquement coûteuse et, par conséquent, des approches numériques sophistiquées doivent être utilisées.La stratégie numérique sur laquelle ce projet est basé est singulière par rapport aux schémas incrémentaux en temps usuellement utilisés pour résoudre des problèmes élasto-(visco)plastique avec endommagement dans le cadre de la mécanique des milieux continus. Cette stratégie numérique appelée méthode LATIN (Large Time Increment method) est une méthode non-incrémentale qui recherche la solution de manière itérative sur l'ensemble du domaine spacio-temporel. Une importante innovation de la méthode LATIN est d'incorporer une stratégie de réduction de modèle adaptative pour réduire de manière très importante le coût numérique. La Décomposition Propre Généralisée (PGD) est une stratégie de réduction de modèle a priori qui sépare les quantités d'intérêt spacio-temporelles en deux composantes indépendantes, l'une dépendant du temps, l'autre de l'espace, et estime itérativement les approximations de ces deux composantes. L'utilisation de l'approche LATIN-PGD a montré son efficacité depuis des années pour résoudre des problèmes élasto-(visco)plastiques. La première partie de ce projet vise à étendre cette approche aux modèles incorporant de l'endommagement.Bien que l'utilisation de la PGD réduise les coûts numériques, le gain n'est pas suffisant pour permettre de résoudre des problèmes considérant un grand nombre de cycles de chargement, le temps de calcul peut être très conséquent, rendant les simulations de problèmes de fatigue intraitables même en utilisant les techniques LATIN-PGD. Cette limite peut être dépassée en introduisant une approche multi-échelle en temps, qui prend en compte l'évolution rapide des quantités d'intérêt lors d'un cycle et leur évolution lente au cours de l'ensemble des cycles. Une description type « éléments finis » en temps est proposée, où l'ensemble du domaine temporel est discrétisé en éléments temporels, et seulement les cycles nodaux, qui forment les limites des éléments, sont calculés en utilisant la technique LATIN-PGD. Puis, des fonctions de forme classiques sont utilisées pour interpoler les quantités d'intérêt à l'intérieur des éléments temporels. Cette stratégie LATIN-PGD à deux échelles permet de réduire le coût numérique de manière significative, et peut être utilisée pour simuler l'évolution de l'endommagement dans une structure soumise à un chargement de fatigue comportant un très grand nombre de cycles. / The motivation of the research project is to predict the life time of mechanical components that are subjected to cyclic fatigue phenomena. The idea herein is to develop an innovative numerical scheme to predict failure of structures under such loading. The model is based on classical continuum damage mechanics introducing internal variables which describe the damage evolution. The challenge lies in the treatment of large number of load cycles for the life time prediction, particularly the residual life time for existing structures.Traditional approaches for fatigue analysis are based on phenomenological methods and deal with the usage of empirical relations. Such methods consider simplistic approximations and are unable to take into account complex geometries, and complicated loadings which occur in real-life engineering problems. A thermodynamically consistent continuum-based approach is therefore used for modelling the fatigue behaviour. This allows to consider complicated geometries and loads quite efficiently and the deterioration of the material properties due to fatigue can be quantified using internal variables. However, this approach can be computationally expensive and hence sophisticated numerical frameworks should be used.The numerical strategy used in this project is different when compared to regular time incremental schemes used for solving elasto-(visco)plastic-damage problems in continuum framework. This numerical strategy is called Large Time Increment (LATIN) method, which is a non-incremental method and builds the solution iteratively for the complete space-time domain. An important feature of the LATIN method is to incorporate an on-the-fly model reduction strategy to reduce drastically the numerical cost. Proper generalised decomposition (PGD), being a priori a model reduction strategy, separates the quantities of interest with respect to space and time, and computes iteratively the spatial and temporal approximations. LATIN-PGD framework has been effectively used over the years to solve elasto-(visco)plastic problems. Herein, the first effort is to solve continuum damage problems using LATIN-PGD techniques. Although, usage of PGD reduces the numerical cost, the benefit is not enough to solve problems involving large number of load cycles and computational time can be severely high, making simulations of fatigue problems infeasible. This can be overcome by using a multi-time scale approach, that takes into account the rapid evolution of the quantities of interest within a load cycle and their slow evolution along the load cycles. A finite element like description with respect to time is proposed, where the whole time domain is discretised into time elements, and only the nodal cycles, which form the boundary of the time elements, are calculated using LATIN-PGD technique. Thereby, classical shape functions are used to interpolate within the time element. This two-scale LATIN-PGD strategy enables the reduction of the computational cost remarkably, and can be used to simulate damage evolution in a structure under fatigue loading for a very large number of cycles.