Alors que la simulation numérique prend aujourd'hui une place essentielle dans de nombreuses branches de l'ingénierie, les évolutions incroyables des moyens de calculs peinent à compenser la complexité croissante des modèles que les ingénieurs sont amenés à traiter. Dans ce contexte, les modèles réduits sont de véritables outils d'aide à la décision car ils permettent, une fois construits, d'évaluer un très grand nombre de scénarios en temps quasi réel. En particulier, la méthode PGD (Proper Generalized Decomposition) initiée au LMT a connu de très nombreux développements (problèmes non linéaires, multiéchelles, multiphysiques...) et conduit à des gains en temps CPU pouvant atteindre plusieurs ordres de grandeur.Malheureusement, l'essor de ces modèles réduits est actuellement freiné par la difficulté à les calculer lorsque le nombre de paramètres à prendre en compte augmente. Toutes les techniques de réduction de modèles actuelles (PGD comprise) peinent à traiter des problèmes à très grand nombre de paramètres (la limite actuelle tourne autour de la vingtaine de paramètres), ce qui constitue un verrou scientifique majeur pour l'essor de ces techniques. Cette thèse présente une adaptation de la méthode PGD qui permet le traitement de tels problèmes.Trois contributions principales ont permis d'atteindre de telles performances. D'une part, une nouvelle structure de données plus proche de la physique du problème a été développée. Elle introduit deux échelles de représentation des fonctions paramétriques et donne son nom à la méthode : la Parameter-Multiscale PGD. Par ailleurs, une discrétisation spatiale discontinue particulièrement adaptée à nos méthodes de résolution a été implémentée, la WTDG (Weak Trefftz Discontinuous Galerkin). Enfin de nouveaux algorithmes ont été développés pour construire des modèles réduits qui permettent des gains de temps conséquents pour des problèmes ayant jusqu'à mille paramètres. / Numerical simulation is nowadays a major tool in a large number of engineering fields. Nevertheless, even the recent incredible improvements of the computational power can hardly compensate the increasing complexity of the models used by engineers. In this context, Reduced Order Models (ROM) can be major decision-maker tools because, once they have been computed, they can be used to evaluate a very large number of test cases in a duration close to real time. The PGD (Proper Generalized Decomposition) in particular, is a method introduced at the LMT which has been adapted to many cases (non-linear problems, multiscale, multiphysics) and leads to savings of CPU time reaching several orders of magnitude.Unfortunately, it is currently difficult to build ROM with an increasing number of parameters. All the actual model reduction technics (including the PGD) can hardly solve problems with a high number of parameters (the current limit is about twenty parameters). It is a major barrier to a larger development of these methods. This PhD thesis presents a new methodology based on the PGD able to take into account high numbers of parameters.This goal has been achived thanks to three major contributions. First, a new data structure faithfull to mecanical properties of the problem has been developed. To that end, two different scales are introduced in the parametric space, giving its name to our method : Parameter-Multiscale PGD. Furthermore, the WTDG (Weak Treffz Discontinuous Galerkin) method has been inpemented. It is a discontinuous spatial discretisation adapted to our resolution techniques. Finally, new algorithms have been developed to built reduced order models of problems taking into account up to one thousand parameters.

Embryonic biopsy routinely involves the removal of one or two blastomeres in the preimplantation genetic diagnosis (PGD) procedure to determine the presence of a specific disease. The rapid development of the PGD technique and stem cell research has led to great demand for highly automated high precision equipment for cellular component micro-extraction. This thesis presents the development of an automated microsurgery system for embryo biopsy. While the ultimate objective of this research is to improve the so called “take-home-baby rate”, the primary focus of this research, however, is devoted to demonstrate the automation of the first two steps in the embryo biopsy procedure: embryo immobilization and embryo perforation with a piezoelectric actuated micro-cutter mounted on a five DOF micromanipulator. A biological embryo holding device incorporating a unique configuration of fluidic channels is designed to increase embryo mobility in order to overcome friction force while maintaining a low suction flow rate and pressure. The validity of this design is demonstrated by good qualitative agreement between the experimental and simulation results. 3D nonlinear equations of motion of a micro-needle driven longitudinally by a piezoelectric actuator are developed based on Kane’s method . The longitudinal vibration of a micro-needle results in excitation of its out-of-plane, lateral eigenmodes at low damping coefficients. The dynamic model is in good agreement with experimental observations. This model is exploited further to describe the response of an immersed glass micropipette with imbedded mercury in piezo-assisted intracytoplasmic sperm injection (ICSI). Furthermore, piezoelectric actuator dynamic nonlinearity introduced by hysteresis is addressed in this research. A new model is proposed to characterize the rate-dependent hysteresis based on Duffing's equation. A nonlinear capacitor element is incorporated into a linear second-order system to predict the relationship between an input state and a hysteretic output. The proposed hysteresis model is verified experimentally. Based on this approach, a new electromechanical piezoelectric actuator model is proposed. A vision-assisted controller for embryo perforation is proposed by implementing a vision tracking and robust autofocusing algorithm using the particle swarm optimization (PSO) method. The performance of the proposed visual-based controller demonstrated experimentally to be effective in providing accurate embryo and micro-needle 3D positioning. Finally, an automated embryo perforation with the proposed mechanical approach was conducted successfully.

Embryonic biopsy routinely involves the removal of one or two blastomeres in the preimplantation genetic diagnosis (PGD) procedure to determine the presence of a specific disease. The rapid development of the PGD technique and stem cell research has led to great demand for highly automated high precision equipment for cellular component micro-extraction. This thesis presents the development of an automated microsurgery system for embryo biopsy. While the ultimate objective of this research is to improve the so called “take-home-baby rate”, the primary focus of this research, however, is devoted to demonstrate the automation of the first two steps in the embryo biopsy procedure: embryo immobilization and embryo perforation with a piezoelectric actuated micro-cutter mounted on a five DOF micromanipulator. A biological embryo holding device incorporating a unique configuration of fluidic channels is designed to increase embryo mobility in order to overcome friction force while maintaining a low suction flow rate and pressure. The validity of this design is demonstrated by good qualitative agreement between the experimental and simulation results. 3D nonlinear equations of motion of a micro-needle driven longitudinally by a piezoelectric actuator are developed based on Kane’s method . The longitudinal vibration of a micro-needle results in excitation of its out-of-plane, lateral eigenmodes at low damping coefficients. The dynamic model is in good agreement with experimental observations. This model is exploited further to describe the response of an immersed glass micropipette with imbedded mercury in piezo-assisted intracytoplasmic sperm injection (ICSI). Furthermore, piezoelectric actuator dynamic nonlinearity introduced by hysteresis is addressed in this research. A new model is proposed to characterize the rate-dependent hysteresis based on Duffing's equation. A nonlinear capacitor element is incorporated into a linear second-order system to predict the relationship between an input state and a hysteretic output. The proposed hysteresis model is verified experimentally. Based on this approach, a new electromechanical piezoelectric actuator model is proposed. A vision-assisted controller for embryo perforation is proposed by implementing a vision tracking and robust autofocusing algorithm using the particle swarm optimization (PSO) method. The performance of the proposed visual-based controller demonstrated experimentally to be effective in providing accurate embryo and micro-needle 3D positioning. Finally, an automated embryo perforation with the proposed mechanical approach was conducted successfully.

The following work presents the ongoing philosophical debate regarding the use of biotechnology to improve human capabilities and attempts to apply the insights drawn from these debates to the regulation of non-medical uses of reproductive technology. After presenting the criticisms towards a hardline approach that would ban all attempts to enhance humans, we evaluate various alternative frameworks and adopt a framework called Genetic Justice, which can be understood as an application of Rawlsian distributive justice to biological assets. After improving on this moral framework by incorporating various institutional considerations into it, we apply it to the evaluation of non-medical uses of a screening technique called preimplantation genetic diagnosis and propose recommendations for its regulation.

Le travail présenté dans cette thèse s’inscrit dans le contexte du développement de nouveaux procédés pour la mise en forme de matériaux composites pour répondre aux contraintes industrielles de gain de temps et d’énergie. Le procédé de chauffage par micro-onde reposant sur un principe de chauffage volumique permet de garantir ces avantages. L’inconvénient majeur est que le comportement du champ électrique lors de l’interaction entre matériaux composites et micro-ondes est peu connu. L’objectif de cette thèse est donc de proposer des solutions numériques pour approfondir la connaissance du procédé et la physique associée afin de mettre en avant ses capacités quant à un développement industriel. Pour ce faire, le travail réalisé est axé autour de trois verrous. En premier lieu, une nouvelle méthode numérique est proposée afin de résoudre les équations de Maxwell en 3D dans un stratifié composite. Prendre en compte le comportement 3D est essentiel afin de décrire avec précision l’impact des différentes couches du matériau et de leur épaisseur sur la propagation du champ électrique. Dans un second temps, la simulation électromagnétique est couplée à une résolution thermique afin de simuler le procédé de chauffage d’une pièce composite. Différents paramètres du procédé sont étudiés afin de mettre en avant les plus sensibles pour l’utilisation du chauffage micro-onde pour les matériaux composites. Enfin, l’attention du lecteur est portée sur le développement d’une méthode de simulation pour un contrôle en temps réel du procédé en comparaison avec des essais expérimentaux. / The context of the present work is the development of new processes for the heating and forming of composite materials in order to provide an answer to the industrials needs for less energy and less time-consuming processes. In that sense, microwave heating is perfect match as it relies on volumetric heating. The major drawback is that the behaviour of the electric field while interacting with composite material is poorly known. Therefore, the main objective of this thesis is to provide numerical solutions to go more deeply in the understanding of such process and put forward its capabilities for an industrial development.To fulfil this objective, the work is oriented over three main axes. First, an innovative simulation tool is presented in order to solve the Maxwell’s equations in a thin multi layered domain. Taking into account the 3D behaviour of the electric field is a major issue in order to describe precisely the impact of the different plies of the laminate on the propagation of the electric field.Then, the electromagnetic simulation is coupled with a thermal simulation in order to simulate the full heating process of a composite part. Parameters of the process are investigated to bring forward the most crucialones. Finally, real-time control of the process is tackled using a model order reduction simulation technique. These results are compared to experimental work on two sets of samples.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2016-10-13T11:36:18Z No. of bitstreams: 1 julianapolisseni.pdf: 645848 bytes, checksum: 5362a586a32461dda31a0e7f5d2e9fef (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-10-22T12:55:07Z (GMT) No. of bitstreams: 1 julianapolisseni.pdf: 645848 bytes, checksum: 5362a586a32461dda31a0e7f5d2e9fef (MD5) / Made available in DSpace on 2016-10-22T12:55:07Z (GMT). No. of bitstreams: 1 julianapolisseni.pdf: 645848 bytes, checksum: 5362a586a32461dda31a0e7f5d2e9fef (MD5) Previous issue date: 2008-08-29 / O impacto da técnica de biópsia sobre o desenvolvimento embrionário posterior ainda foi pouco estudado e também não foi estabelecido um protocolo único e universal para o PGD. Outro ponto a considerar é a pequena quantidade de DNA genômico disponível para se realizar os estudos genéticos, pelo número reduzido de células obtidas pela técnica de biópsia. Metodologias que utilizam whole genome amplification (WGA) vêm sendo desenvolvidas. Utilizando este método é possível gerar microgramas de DNA partindo de pequenas quantidades de DNA. Então, o objetivo deste estudo foi avaliar o desenvolvimento de embriões bovinos submetidos à biópsia nos estádios de 8-16 células e o uso da técnica de amplificação de todo o genoma nos blastômeros retirados, para posterior identificação do sexo através do PCR. Um grupo de 706 complexos cumulus-ovócitos (CCOs) de bovinos foram maturados e fertilizados in vitro, em estufa incubadora a 38,8 °C, 95% de umidade e 5% de CO2. Os prováveis zigotos foram semi-desnudados e cultivados no meio CR2aa sob as mesmas condições da fertilização. No quarto dia após a fertilização, embriões bovinos com 8-16 células foram aleatoriamnte distribuídos entre dois grupos: controle (n=103) e biópsia (n=92). O número de células removidas correspondeu à quarta parte do embrião. Os blastômeros retirados foram submetidos à amplificação de todo o genoma seguido por PCR. Os embriões retornaram ao meio de cultivo para avaliação do desenvolvimento embrionário. Avaliou-se pelo teste do qui-quadrado a produção de blastocisto no oitavo dia de cultivo, a taxa de eclosão no décimo dia de cultivo, a eficiência da amplificação de todo o genoma e da identificação do sexo nos blastômeros removidos. O número de células embrionárias foi analisado por análise de variância ANOVA (Instituto SAS, Inc., Cary, NC, USA). A proporção de blastocisto no oitavo dia de cultivo foi avaliada pelo teste de Wilcoxon. Diferenças foram consideradas significantes se P<0,05. Um total de 92 embriões bovinos foi utilizado para realização da técnica de biópsia e apresentaram produção de blastocisto de 53,3%, com 44,9% de taxa de eclosão. Essas taxas foram semelhantes (P>0,05) às dos 103 embriões do grupo controle (66,0% e 42,6%, respectivamente). Não houve variação no número de células embrionárias entre os grupos e também não ocorreu diferença na proporção de blastocisto entre os grupos controle e biópsia no oitavo dia de cultivo (P>0,05). Os blastômeros retirados foram submetidos à amplificação de todo o genoma, com 98,2% de eficiência. Entretanto, só foi possível identificar o sexo em 59% das amostras. Conclui-se que a técnica de biópsia realizada em embriões bovinos de 816 células não afetou o desenvolvimento embrionário subseqüente nem a qualidade embrionária dos embriões. A utilização do kit de amplificação de todo o genoma mostrou-se eficiente nos blastômeros retirados, sendo possível identificar o sexo em mais da metade dos embriões. . / The impact of biopsy technique on posterior embryo development is still poorly studied and there is no single universally practiced protocol for implementing PGD. Another thing to consider is the small amount of genomic DNA available to perform the genetic study, due to the reduced number of cells obtained from the biopsy. Alternatively, methodologies using whole genome amplification (WGA) have been developed. Using these methods, it is possible to generate microgram quantities of DNA starting with a little genomic DNA. The aim of this study was to evaluate the effect of biopsy at 8-to 16-cell bovine embryos on subsequent development and WGA on removed blastomeres. A group of 706 complexes cumulus oocytes (CCOs) obtained from local slaughterhouse ovaries were matured and fertilized in vitro, in incubator at 38.8°C with 95% humidified air and 5% of CO2. The zygotes were semidenuded and cultured in CR2aa medium under the same conditions as with in vitro fertilization. On the fourth day after fertilization the 8-to 16-cell embryos were distributed randomly across two groups: control (n=103) and biopsy (n=92). The amount of cells removed cells was one-fourth the embryo of blastomeres. The removed blastomeres were submitted to whole genome amplification (WGA) followed by PCR. The embryos were returned to culture for development evaluation. The chisquare test was used for statistic evaluation of the results of blastocyst rate on the eight day after fertilization and hatching rate on tenth day between biopsy and control group, to test the WGA efficiency and to determine the sex proportion of biopsied samples submitted to PCR. The number of cell per embryo was analyzed by ANOVA with SAS (SAS Institute, Inc., Cary, NC, USA). The blastocyst proportion was evaluated with Wilcoxon test. Differences were considered significant if P<0.05. A total of 92 embryos were submitted to biopsy technique. The blastocyst production was 53.3%, with 44.9% of hatching rate. This rate was similar to control group (66.0% and 42.6%) found on 103 embryos. Overall, no impact was detected on embryo quality in blastocyst cell number between the two groups. The proportion of blastocyst in different stages of development on the 8th day did not differ among groups (P>0.05). Removed blastomeres were submitted to WGA, resulting in 98.2% of efficiency. However, only 59% of samples were sexed by PCR. In conclusion biopsy of 8-to 16-cell bovine embryos did not affect subsequent development. WGA was successful in removed blastomeres and was possible to determinate the sex in the samples.

Afin de tenir compte de l'amorçage et de la propagation des fissures en fatigue, il est nécessaire de connaître l'historique des différentes quantités mécaniques dans la zone d'amorçage. Cela nécessite une connaissance des états mécaniques cycliques stabilisés. D'un point de vue numérique, les simulations numériques d'agrégats polycristallin avec les méthodes de résolution classiques ne sont réalisées que pour quelques cycles. Ce travail présente le développement des méthodes accélérées de calcul pour réduire le temps de calcul de la méthode des Éléments Finis dans le cas des simulations d'agrégats polycristallins soumis à un chargement cyclique. La première idée est de maintenir la matrice de rigidité constante afin d'effectuer une seule factorisation. Un premier algorithme a été écrit dans ce sens avec une résolution incrémentale et non incrémentale. La deuxième proposition est d'utiliser une méthode de réduction dimensionnelle en espace/temps couplé à la méthode des éléments finis. La PGD a été choisie. Cette méthode permet de découpler les variables spatiales et temporelles et les champs de déplacement ne sont calculés que pour un certain nombre de modes. Deux possibilités d'utilisation de la méthode PGD dans le cadre de la plasticité ont été proposées. La troisième proposition consiste à utiliser la stratégie de réduction adaptative APR comme solveur afin de résoudre un modèle d'ordre réduit en termes de nombre de ddl. Une stratégie incrémentale d'amélioration de la qualité de la base pour un certain intervalle de temps choisi a été mis en place dans ce cadre. Quatre possibilités d'utilisation de la méthode APR ont été proposées. L'analyse des performances des différentes méthodes est effectuée tout d'abord sur un problème élasto-plastique classique tridimensionnel présentant un défaut sphérique, ensuite à l'échelle de la microstructure avec un calcul visco-élasto-plastique d'agrégats polycristallins tridimensionnels. Les analyses ont montré que les réponses mécaniques macroscopique et mésoscopique des méthodes de réduction de modèle sont très proches de la méthode incrémentale. Le temps des simulations a été réduit par les différentes méthodes. Les gains sont plus importants quand on augmente la taille des maillages et la non linéarité du problème. Toutefois, l'idée de garder la matrice de rigidité constante avec une résolution incrémentale a permis un gain de temps encore plus conséquent à l'échelle de la microstructure. / In order to take account of fatigue cracks initiation and growth, it is necessary to know the history of the various mechanical quantities in fatigue initiation site. This requires knowledge of the stabilized cyclic mechanical states. From a numerical approach, numerical simulation of polycrystalline aggregates with conventional resolution methods are only carried out for a few cycles. This work presents the development of accelerated numerical methods to reduce the computation time of the Finite Element method in the case of numerical simulation of polycrystalline aggregates under cyclic loading. The first idea is to keep a constant stiffness matrix during overall simulation in order to get just one single factorization to carry out. An algorithm has been proposed in this sense with an incremental and non incremental resolution. The second proposal is based on the use of a model reduction method coupled with the finite element method to solve space/time problem. The PGD has been selected. This method allows to decouple spatial and time coordinates and the displacement fields are computed for a certain number of modes. Two possibilities have been proposed to implement the PGD method in the context of plasticity. The third proposal is to use an a priori adaptative approach based on the use of APR strategy to solve a reduced order model in terms of number of degrees of freedom. An incremental adaptive strategie has been proposed in order to master the quality of the reduced-basis for a certain time steps. Four possibilities of using the APR method have been proposed. The applicability and the performance of the different methods have been first analyzed on a conventional three-dimensional elastoplastic problem with a spherical defect, then on the scale of the microstructure with numerical simulation of polycrystalline aggregates under cyclic elasto-visco-plastic loading. The analyzes have shown that the macroscopic and mesoscopic mechanical responses of the model reduction methods are very close to the incremental method. The simulation time has been reduced by the different methods. The time gains are more significant by increasing the size of the meshes and the non-linearity of the problem. Nevertheless, the idea of keeping a constant stiffness matrix with the incremental method has enabled more CPU time saving at the microstructural scale.

Ces travaux de thèse sont consacrés au développement d'un algorithme de résolution de problèmes non-linéaires pour lesquels il existe une variabilité sur certains paramètres du modèle ou du chargement définis par leur intervalle de définition. Le cadre d'étude est le projet SINAPS@, qui a pour but d'évaluer les incertitudes dans les structures de génie civil, et de quantifier leur influence sur la réponse mécanique globale d’une structure sujette à un aléa sismique. Contrairement aux approches statistiques ou probabilistes classiques, une résolution déterministique est privilégiée dans notre étude. Cependant, afin de réduire le coût de calcul de cette famille de problèmes, une approche de type réduction de modèle PGD est mise en place, pour laquelle les paramètres incertains sont considérés comme des variables supplémentaires du problème. Cette méthode est mise en place au sein de l'algorithme LATIN, qui utilise une approche itérative pour résoudre le caractère non-linéaire des équations rencontrées lors de la résolution du problème mécanique. Ces travaux présentent donc l'extension de l'algorithme classique temps-espace LATIN-PGD à des problèmes paramétriques, pour lesquels les paramètres sont considérés comme des variables additionnelles dans la définition des quantités d’intérêt, ainsi que l'application de cette méthode à un modèle endommageant avec refermeture de fissure, présentant une variabilité à la fois sur des paramètres matériaux et sur l'amplitude du chargement. La faisabilité de ce couplage est illustrée par des exemples numériques sur des structures en béton armé pour divers types de chargement cycliques (traction—compression, flexion). / This thesis is dedicated to the development of an algorithm for the resolution of nonlinear problems for which there is a variability on some of the model parameters or on the loading conditions, which are only described by their intervals of variation. This study is part of the SINAPS@ project, which aims at evaluating the uncertainties in civil engineering structures and to quantify their influence on the global mechanical response of a structure to a seismic hazard. Unlike statistical or probabilistic approaches, we rely here on a deterministic approach. However, in order to reduce the computation cost of such problems, a PGD-based reduced-order modeling approach is implemented, for which the uncertain parameters are considered as additional variables of the problem. This method was implemented into the LATIN algorithm, which uses an iterative approach to solve the nonlinear aspect of the equations of the mechanical problem. This work present the extension of the classical time-space LATIN—PGD algorithm to parametric problems for which the parameters are considered as additional variables in the definition of the quantities of interest, as well as the application of such method to a damage model with unilateral effect, highlighting a variability on both material parameters and the loading amplitude. The feasibility of such coupling is illustrated on numerical examples for reinforced concrete structures subjected to different types of cyclic loading conditions (tension—compression, bending).

Aujourd'hui, les outils de simulations numériques sont omniprésents dans l'industrie, que ce soit dans l'industrie aéronautique, aérospatiale, ferroviaire ou automobile.En effet leur utilisation limite la fabrication (souvent très couteuse) de prototype, et permettent ainsi de réduire les couts et d'accélérer la phase de conception d'un nouveau produit.Dans le cadre des applications acoustiques, il est souvent recommandé de calculer la réponse sur l'ensemble de la bande de fréquences d'intérêt.Les outils de simulation numérique sur une bande de fréquences impliquent généralement plusieurs calculs à fréquences fixes.Même si c'est un moyen simple et naturel pour répondre au problème posé, une telle stratégie peut facilement conduire à des calculs énormes. Cela est particulièrement vrai dans la gamme des moyennes fréquences, où la réponse est particulièrement sensible à la fréquence.Dans cette thèse, la PGD (Proper Generalized Decomposition), est appliqué pour trouver une représentation fonctionnelle distincte de la fréquence et de l'amplitude des inconnues de la TVRC (théorie variationnelle des rayons complexes) sur une bande de fréquences donnée.Ceci permet d'obtenir une réponse de haute qualité sur une large bande en moyennes fréquences, sans nécessité de discrétisation fine de la plage de fréquences considérées. En outre, la représentation PGD de la solution permet d'économiser une grande quantité d'espace en terme de stockage de données. Dans un second temps, la technique PGD a été étendue à la stochastique. / In many industrial contexts, such as aerospace applications or cars design, numerical prediction techniquesbecome more and more useful. They restrict the use of real prototypes to a minimum and make easier thedesign phase. In such industries and in the specific for acoustic, engineers are interested in computing theresponses of systems on frequency bands. In order to predict the vibration behavior of systems overfrequency bands, standard numerical techniques usually involve many frequency-fixed computations, atmany different frequencies. Although it is a straightforward and natural mean to answer to the posed problem,such a strategy can easily lead to huge computations, and the amount of data to store often increasessignificantly. This is particularly true in the context of medium frequency bands, where these responses havea strong sensitivity to the frequency. In this work PGD (Proper Generalized Decomposition), in a first time, isapplied to found a separate functional representation over frequency and space of the unknown amplitude ofVTCR (Variational Theory of Complex Rays) formulation on a reduced frequency space. This allows tocalculate an high quality mid-frequency response over a wide band without a fine frequency discretization,saving computational resources. Moreover the PGD representation of the solution allows to save a hugeamount of space in term of stored data. In a second time, PGD technique as been applied to extend itspeculiarity to mid-frequency wide band with uncertainty.

Dans ces travaux , nous avons deux principaux objectifs physique et numérique. Le problème physique consiste à trouver la solution de Rayleigh-Bénard pour des fluides newtoniens et non-newtoniens. Dans la présente étude, une présentation générale des résultats de la convection de Rayleigh-Bénard (RBC) est donnée dans le cas des fluides newtoniens et non-newtoniens tels que des fluides rhéofluidifiants modélisés par la loi puissance et des fluides viscoplastiques (fluides de Bingham, Herschel-Bulkley et Casson), en régime permanent et transitoire. Dans le cas des fluides viscoplastiques, les modèles macroscopiques ne prenant pas bien en compte la réalité physique de la contrainte seuil ont fait l'objet d'une modélisation. Un modèle mesoscopique proposé par Hébraud et Lequeux a été utilisé. Le problème numérique consiste à développer la méthode de résolution PGD (Proper Generalized Decomposition) pour résoudre les modèles non linéaires couplés transitoires, dans le cas du problème de Rayleigh-Bénard. Cette méthode est également utilisée pour résoudre le problème RBC paramétrique en y ajoutant quelques variables physiques comme coordonnées supplémentaires. Par ailleurs, dans le cas des fluides non-newtoniens, nous avons utilisé la PGD pour résoudre les équations mesoscopiques et macroscopiques couplées. / In this research work we are looking for two main physical and numerical purposes. The physical problem is to find the solution of Rayleigh Bénard convection for several conditions dependent on fluid thermo-physical properties such as temperature, viscosity and initial and boundary conditions. Continuing previous research works in this study we have provided the results of Rayleigh Bénard convection for Newtonian, Power-law and viscoplastic fluids (Bingham, Herschel-Bulkley and Casson) and for steady state and transient conditions. We also solve this problem for Nano and soft glassy materials. In some cases the results are interesting not only as a part of the Rayleigh Bénard convection analysis but also on a larger scale as a part of the heat transfer and mechanical fluid analysis such as viscoplastic and soft glassy material studies. Numerically, it was interesting to develop Proper Generalized Decomposition (PGD) method for solving transient coupled non-linear models, in particular the one related to the Rayleigh–Bénard flow. This model also was used to solve RBC problem parametrically by adding some physical properties as extra coordinates. For soft glassy material we used PGD to connect micro and macro equations together.