abstract: The main objective of this study is to develop an innovative system in the form of a sandwich panel type composite with textile reinforced skins and aerated concrete core. Existing theoretical concepts along with extensive experimental investigations were utilized to characterize the behavior of cement based systems in the presence of individual fibers and textile yarns. Part of this thesis is based on a material model developed here in Arizona State University to simulate experimental flexural response and back calculate tensile response. This concept is based on a constitutive law consisting of a tri-linear tension model with residual strength and a bilinear elastic perfectly plastic compression stress strain model. This parametric model was used to characterize Textile Reinforced Concrete (TRC) with aramid, carbon, alkali resistant glass, polypropylene TRC and hybrid systems of aramid and polypropylene. The same material model was also used to characterize long term durability issues with glass fiber reinforced concrete (GFRC). Historical data associated with effect of temperature dependency in aging of GFRC composites were used. An experimental study was conducted to understand the behavior of aerated concrete systems under high stain rate impact loading. Test setup was modeled on a free fall drop of an instrumented hammer using three point bending configuration. Two types of aerated concrete: autoclaved aerated concrete (AAC) and polymeric fiber-reinforced aerated concrete (FRAC) were tested and compared in terms of their impact behavior. The effect of impact energy on the mechanical properties was investigated for various drop heights and different specimen sizes. Both materials showed similar flexural load carrying capacity under impact, however, flexural toughness of fiber-reinforced aerated concrete was proved to be several degrees higher in magnitude than that provided by plain autoclaved aerated concrete. Effect of specimen size and drop height on the impact response of AAC and FRAC was studied and discussed. Results obtained were compared to the performance of sandwich beams with AR glass textile skins with aerated concrete core under similar impact conditions. After this extensive study it was concluded that this type of sandwich composite could be effectively used in low cost sustainable infrastructure projects. / Dissertation/Thesis / M.S. Civil and Environmental Engineering 2012

The experimental identification of mechanical properties is crucial in mechanics for understanding material behavior and for the development of numerical models. Classical identification procedures employ standard shaped specimens, assume that the mechanical fields in the object are homogeneous, and recover global properties. Thus, multiple tests are required for full characterization of a heterogeneous object, leading to a time consuming and costly process. The development of non-contact, full-field measurement techniques from which complex kinematic fields can be recorded has opened the door to a new way of thinking. From the identification point of view, suitable methods can be used to process these complex kinematic fields in order to recover multiple spatially varying parameters through one test or a few tests. The requirement is the development of identification techniques that can process these complex experimental data. This thesis introduces a novel identification technique called the constitutive compatibility method. The key idea is to define stresses as compatible with the observed kinematic field through the chosen class of constitutive equation, making possible the uncoupling of the identification of stress from the identification of the material parameters. This uncoupling leads to parametrized solutions in cases where 5 the solution is non-unique (due to unknown traction boundary conditions) as demonstrated on 2D numerical examples. First the theory is outlined and the method is demonstrated in 2D applications. Second, the method is implemented within a domain decomposition framework in order to reduce the cost for processing very large problems. Finally, it is extended to 3D numerical examples. Promising results are shown for 2D and 3D problems

Nesta tese é proposto um modelo fenomenológico eletromecânico com relação constitutiva variacional para representar numericamente a resposta ativa/passiva do tecido muscular esquelético, a fadiga muscular de diferentes tipos de fibras e a combinação de contrações. Na literatura não são encontrados modelos que representem esses comportamentos combinados. Para atingir este objetivo são definidos conjuntos de variáveis auxiliares e variáveis internas para desencadear e mapear a evolução das principais características do tecido muscular esquelético. É considerado que o tecido se contrai localmente, através da propagação de um potencial de ação elétrico advindo do sistema nervoso através de uma ou mais unidades motoras. A proposta de propagação do potencial de ação é construída com a criação de critérios de malha e discretização temporal, eliminando o erro dentro desses limites. Com a propagação do potencial de ação é informado ao modelo constitutivo se localmente o tecido está ativo, através de uma variável de acoplamento binária. A relação constitutiva mecânica utiliza este parâmetro para iniciar a contração muscular, que depende da evolução de uma variável interna. A evolução dessa variável substitui a função de ativação encontrada em outros modelos. Também é utilizado um segundo potencial dissipativo para representar a eficiência metabólica do tecido resultando na representação da fadiga muscular. Um conjunto de potenciais elásticos e dissipativos são utilizados para representar diferentes níveis de forças após combinações de contração do tipo isométrica-concêntrica-isométrica e isométrica-excêntrica-isométrica para diferentes velocidades e alterações de comprimento. A proposta é verificada frente ao ajuste de parâmetros de experimentos reais, obtendo boa representação. Através da implementação no método de elementos finitos é possível observar o comportamento do modelo proposto em geometrias tridimensionais. / In this thesis an electromechanical phenomenological model with variational constitutive relation is proposed to numerically represent the active/passive response of skeletal muscle tissue, muscle fatigue of different types of fibers and contraction combination. In the literature there are no models that represent these combined behaviors. To achieve this goal, sets of auxiliary variables and internal variables are defined to trigger and map the evolution of the main characteristics of skeletal muscle tissue. The tissue is considered to contract locally by propagating an electrical action potential from the nervous system through one or more motor units. The proposed propagation of the action potential is defined with the creation of mesh criteria and temporal discretization, eliminating the error within these limits. With the propagation of the action potential it is informed to the constitutive model if the tissue is locally active, through a binary coupling variable. The mechanical constitutive relation uses this parameter to initiate muscle contraction, which depends on the evolution of an internal variable. The evolution of this variable replaces the activation function found in other models. A second dissipative potential is also used to represent the metabolic efficiency of the tissue resulting in the representation of muscle fatigue. A set of elastic and dissipative potentials are used to represent different levels of forces after isometric-concentric-isometric and isometric-eccentric-isometric contraction combinations for different velocities and length changes. The proposed model capabilities are verified with the parameters adjustment to reproduce real experiments, obtaining good representation. The implementation in the finite element method allows to observe the behavior of the proposed model in three-dimensional geometries.

Les matériaux composites sont de plus en plus importants dans la communauté de l’industrie. Pour l’identification du comportement mécanique anisotrope et hétérogène de certains composite, les mesures de champs cinématiques sont répandues dans la communauté mécanique, parce qu’elles offrent de riches informations en déplacement ou déformation permettant d’exploiter des essais hétérogènes. Dans les dernières années, certaines identifications inverses spécifiques ont été proposées, mais certains de ces méthodes ne considèrent le problème d’identification que sur la zone de mesure ou ont généralement besoin de conditions limites pour être mises en œuvre. Cependant, mener l’identification sur l’ensemble de l’éprouvette (au-delà de la zone de mesure) offre la possibilité de tenir compte d’information de conditions limites supplémentaire, même si dans le même temps, cela amène à traiter d’autres bords sous conditions limites. L’Erreur en Relation de Comportement Modifiée (ERC-M) permet de traiter ce type de cas en prenant en compte l’ensemble des informations disponibles d’un point de vue théorique et expérimental sans hypothèse supplémentaire. Pour l’identification des propriétés élastiques, une première stratégie mono-échelle a été proposée et permet effectivement de traiter les différents types de conditions limites ainsi que leur absence. Pour un matériau hétérogène, les mesures ne sont pas assez riches pour permettre l’identification des propriétés élastiques hétérogènes en dehors de la zone de mesure. Une approche multi-échelle est donc proposée permettant de tenir compte de toute l’éprouvette, dans laquelle des propriétés micros hétérogènes sont recherchées sur la zone de mesure et des propriétés macros homogènes sur l’ensemble du domaine. / Composite materials are more and more important in the industry community. For the identification of the heterogeneous and anisotropic mechanical behavior of some composite, the full-field measurements are widespread in the mechanics community, because they can offer a very rich information of displacement or strain to exploit heterogeneous tests. During last thirty years, some specific inverse identification strategies have been proposed but most of these methods consider the identification problem only on the measured area and would usually need boundary conditions to be performed. However, the boundary conditions are not always completely known and might not be on the measurement zone. There is hence a need of identification methods allowing both the identification over the whole specimen and the dealing of missing boundary conditions. The Modified Constitutive Relation Error (M-CRE) allows dealing with such situations through the taking into account of the whole available information from a theoretical and experimental point of view without additional assumption. For the identification of elastic properties, a first mono-scale strategy has been proposed and actually used to process different types of boundary conditions as well as their absence. Considering the identification of heterogeneous elastic properties, the lack of information outside the measurement zone prevents from identifying heterogeneous properties in this area. It leads us to propose a multi-scale approach where micro heterogeneous properties are sought at the measurement level and macro homogeneous ones at the specimen level.

Des nombreuses industries composites cherchent à étendre le champ d'application des composites structuraux à fibres continues pour d'autre marché que l'aéronautique. Être compétitif sur ces marchés potentiels, tel l'automobile, implique de se conformer à des contraintes différentes; de coût, de temps de mise en œuvre, de trace écologique. Dans ce cadre les résines thermodurcissables sont remplacées par des résines thermoplastiques permettant des temps de cycle cours. Ces temps de cycle cours favorisent l'apparition de défauts potentiellement important et une question récurrente dans ce cadre est: Comment traiter des défauts? Un des aspects de cette question est celui de la détection et de la caractérisation des défauts. L'approche actuelle est celle des techniques non destructives. Celles ci sont longues, coûteuses et complexes et apparaissent donc mal adaptées aux contraintes de coûts et de temps caractéristique de la plupart des produits grands diffusions. Ce problème a motivé ce travail de thèse: est-il possible de détecter et d'estimer l'effet des défauts sans avoir recours à des techniques d'essais non destructifs complexes et coûteuses? Une réponse acceptable pourrait conduire à moins de précision, mais devrait fournir des informations quantitatives suffisantes pour les applications. La voie étudiée dans cette thèse est celle de l'exploitation par approches inverse d'essais mécaniques statiques suivis par mesure de champs par corrélation d'images numériques. Seuls la localisation et la caractérisation de grands défauts est visée. Des essais de flambement ont été choisis en raison de leur sensibilité supposée aux défauts. Parmi les approches inverses, nous nous sommes concentrés sur la méthode de l'erreur en relation de comportement modifiée (MCRE). En effet cette méthode a, par le passé, montrée de très bonnes propriétés de localisation dans le cas d'essais de vibrations couvrant plusieurs fréquences propres. En vue d'obtenir une formulation simple et rapide nous avons privilégié une exploitation approchée des essais par l'intermédiaire d'une approche linéarisée du flambage. Une extension de la MCRE dans ce cadre a été proposée. Cette formulation pose, notamment la difficulté de traiter des résultats expérimentaux qui se présentent non pas sous la forme d'une réponse de flambage pure mais, en raison des défauts géométriques, d'une réponse non-linéaire dés le début des essais. Pour essayer de contourner cette difficulté une exploitation des essais au moyen du diagramme de Southwell a été utilisée. Elle permet, au moins pour des éprouvettes élancées, de déterminer la charge critique théorique d'une éprouvette sans défauts géométriques mais avec défauts matériels. La stéréo corrélation d'images numériques (StereoDIC) est exploitée pour reconstruire la déformée la forme de l'échantillon pendant le test. Les intérêts et les limites de la méthodologie sont discutés sur la base d'essais numériques et l'essai flambage est comparé aux essais de traction, flexion ou de vibration. Il est montré que l'exploitation d'essais de flambage par l'approche proposée est efficace au moins dans le cas d'imperfections géométriques modérées. Finalement la méthode est exploitée dans le cas d'essais expérimentaux à la fois pour un spécimen presque parfait et pour une pièce défectueuse, où une zone d'ondulation de la fibre est introduite. Les résultats obtenus sont encourageants. / Composite materials are nowadays extending their operational field to industrial applications other than aeronautics. New potential markets, such as automotive, imply the need to comply with different constraints; reduced cost and production time become more binding, taking the lead over the complete absence of defects. The drawback to fast automatized procedure is the higher defectiveness of the components produced, a deeper control of the part is therefore needed. Non-destructive techniques are expensive both in terms of cost and time and therefore the main question we tried to answer in this thesis is: is it possible to detect and estimate the effect of defects without resorting to the complex and time-consuming NDT techniques? An acceptable answer may potentially lead to a lower precision but should guarantee sufficient quantitative information for these applications. The thesis aims at exploring possibilities to use classical mechanical test combined with Digital Image correlation and inverse procedure to localize and characterized possible (large) defects. Buckling tests have been chosen at first due their supposed sensitivity to defects. Among the possible inverse technique, we have chosen to extend the so-called Modified Error in Constitutive Relation to the case of buckling because, in the case of vibration tests performed with several frequencies, the MCRE proved to have very good localization properties. The dedicated formulation of the MCRE for linearized buckling requires a post-processing of the non-linear experimental results. The Southwell plot is here employed to reconstruct the eigenvalue, the critical load, of the equivalent eigenvalue problem (i.e. the solution of the problem with material defect and no geometrical ones) and the Stereo Digital Image Correlation (StereoDIC) is exploited to reconstruct the deformed shape of the specimen during the test, used as mode. The interests and limits of the methodology are discussed notably through the comparison of numerical results using the MCRE in case of traction, flexion or vibration tests. It is shown that the linearized buckling based MCRE technique proves well for pseudo-experimental measurements at least for moderate geometrical imperfections. In addition first experiments have been performed; the defects are characterized from real experimental specimens, both for a nominally perfect specimen and for a defective one, where a zone of fibre waviness is induced. Stereo Digital Image Correlation (StereoDIC) is exploited to reconstruct the deformed shape of the specimen during the test, this shape being used as an approximation of the buckling mode. While on the first one no defects are detected, on the flawed specimen the localized area is in reasonable agreement with the area affected by fibre undulations.

[pt] A modelagem de materiais biológicos para o entendimento de diversas práticas cirúrgicas na medicina, tem se mostrado a cada dia mais importante. Na cirurgia plástica, reparadora ou estética, o conhecimento do comportamento da pele humana, in vivo, é de fundamental importância nos processos cirúrgicos. Nos últimos anos, mais precisamente desde 1957, uma técnica realizada pela cirurgia plástica, conhecida como expansão de pele, vem sendo cada vez mais utilizada. Esta técnica tem como objetivo a obtenção de uma quantidade adicional de pele, para a utilização na recuperação de queimados, reconstrução mamária, retirada de pele com aparência indesejada e outras aplicações. Este trabalho tem como objetivo fornecer aos médicos o conhecimento do comportamento da pele, através de uma análise numérica e experimental, quando submetida ao processo de expansão. Durante o desenvolvimento deste trabalho, estaremos mostrando a metodologia adotada para estudar a pele humana e os resultados alcançados com o acompanhamento de pacientes submetidos a este tipo de processo cirúrgico. Para atingir o objetivo, foram estudadas diversas equações constitutivas, procurando as melhores constantes viscoelásticas, que descrevam o comportamento da pele sob expansão. No estudo realizado, avaliamos ainda as mudanças das constantes viscoelásticas durante a evolução da expansão e o comportamento da pele em diferentes regiões do corpo humano. Nos resultados apresentados, encontramos erros máximos para as constantes da equação constitutiva adotada, na ordem de 28 por cento. Estudos recentes envolvendo a pele humana apresentam erros na ordem de 75 por cento, para as constantes elásticas do material. Com a análise dos resultados, mostramos o comportamento da pele humana, em seres vivos, em diferentes regiões do corpo. O conhecimento de uma equação constitutiva, ou seja, da relação entre as tensões e deformações que ocorrem na pele durante o processo de expansão, proporcionará aos médicos a possibilidade de avaliar as diversas técnicas utilizadas na cirurgia plástica em geral : a expansão de pele, a reconstrução mamária, análise de suturas, análise de cicatrizes, comportamento da pele durante uma incisão, cirurgias estéticas da face (face lifting), e outras aplicações. / [en] The modeling of biological materials is really important today, for the agreement of diverse practical surgical in the medicine. In the area of plastic surgery, the knowledge of the behavior of the human skin has a basic importance in the surgical processes. Since 1957, one technique in plastic surgery, known as skin expansion, has been much used. This technique has as objective the attainment of an additional amount of skin, for the use in the burned patients, breast reconstruction, removal of skin with not desired appearance is and other applications. This work has as objective supply to doctors the knowledge of the skin behavior, through a numerical and experimental analysis, when submitted to the expansion process. During the development of this work, we will show the methodology used to study the human skin and the results reached with the accompaniment of patients submitted to this type of surgical process. To reach the objective, different constitutive equations had been studied, looking for the best viscoelastic constants, to describe the behavior of the skin under expansion. In the study, we also evaluate the changes of the constants during the evolution of the expansion and the behavior of the skin in different regions of the human body. In the presented results, we find maximum errors for the constants of the adopted constitutive equation, to be in 28 percent order. Recent studies involving the human skin present errors in the order of 75 percent, for the elastic constants of the material. With the analysis of the results, we will be show to the behavior of the human skin, in vivo, in different regions of the body. The knowledge of a constitutive equation, that is, of the behavior of the stresses and strains that occur in the skin during the expansion process, will provide to the doctors the possibility to evaluate the diverse techniques used in the plastic surgery: The expansion of skin, breast reconstruction, analysis of sutures, analysis of scars, behavior of the skin during an incision, face lifting, and other applications.

La caractérisation des propriétés matérielles basée sur le dialogue essai-calcul requiert des précautions quant à la prise en compte des grandeurs physiques mises en jeu, aussi bien mesurées que calculées. Les mesures de champs cinématiques délivrent des informations qui permettent de rendre compte du comportement local du matériau et constituent une entrée riche pour ce type de dialogue axé sur la confrontation modèle - mesure. L’objet de ce travail est de discuter l’identification de propriétés élastiques homogènes ou hétérogènes, basée sur l’exploitation de mesures de champs, qui soulève la problématique de la prise en compte de mesures perturbées et d’hypothèses fortes de modélisation. Le but de cette étude est de discuter deux manières différentes pour aborder le problème. La première consiste à traiter les mesures en amont de l’étape d’identification, afin de filtrer les perturbations de mesures, et est basée sur l’outil d’approximation diffuse. La seconde consiste à développer une méthode d’identification inverse à partir d’essais quasi-statiques cherchant à tenir compte du caractère incertain des mesures, dès sa formulation, en utilisant l’ensemble des informations théoriques et expérimentales disponibles, en particulier en l’absence d’informations sur les conditions aux limites. Cette méthode est basée sur les principes de l’erreur en relation de comportement modifiée, amenant à résoudre des problèmes éléments finis non standards, et a conduit à mettre en œuvre des méthodes de résolution adaptées. L’étude a permis de souligner la robustesse de la méthode face aux incertitudes de mesure, notamment en comparaison avec le recalage par éléments finis. / The identification of material properties based on the confrontation between the experimental data and their numerical counterpart needs to pay special attention to their reliability and then to take some precautions. Especially, when dealing with full-field measurements which provide rich kinematic information enabling new possibilities for the characterization of material constitutive parameters. The aim of this work is to discuss the identification of the elastic parameters, for either homogeneous or heterogeneous materials, leading to the more general problem of the taking into account of noisy experimental data and strong model hypotheses. Two different approaches are addressed in this study. The first one consists in the processing of the experimental data upstream of the identification stage, by filtering them through a diffuse approximation algorithm. The second approach focuses on the development of an inverse identification strategy from static tests. This method takes into account the uncertainty of the measurements from the beginning of the formulation and is based on the modified constitutive relation error principles. It involves the resolution of non standard finite element problems using specific resolution methods. The study pointed out the robustness of the proposed method towards measurement errors, in particular when compared to the finite element model updating method.

Depuis plusieurs années, les évolutions considérables survenues au niveau des moyens de calcul à disposition ont entraîné de nouvelles pratiques de simulation des structures mécaniques. Parmi ces nouvelles pratiques celle qui motive ces travaux de thèse est le paradigme Dynamic Data Driven Application Systems (DDDAS). L’idée fondatrice de cette approche est de mettre en place un dialogue entre un système physique et son modèle numérique. L’objectif est alors de (i) permettre un recalage du modèle numérique à l’aide de mesures faites sur le système physique ; (ii) contrôler l’évolution du système physique à l’aide de la prédiction faite par la simulation numérique. La difficulté majeure est de réaliser ce dialogue en temps réel. Ces travaux de thèse se focalisent sur l’étape de recalage de modèle du paradigme DDDAS. La problématique est alors de développer des méthodes et outils de résolution de problèmes inverses prenant en compte diverses contraintes à savoir : (i) une robustesse vis-à-vis des données corrompues ; (ii) une généricité permettant de considérer une grande variété de problèmes et de modèles mécaniques ; (iii) un temps de calcul réduit afin de tendre vers un recalage de modèle en temps réel. Le point de départ de ces travaux est l’Erreur en Relation de Comportement modifiée, approche énergétique dédiée à la résolution des problèmes inverses en mécanique, s’étant notamment illustrée par sa grande robustesse vis-à-vis des bruits de mesure. Dans un premier temps, afin de garantir un processus d’identification rapide, nous avons couplé l’Erreur en Relation de Comportement modifiée avec la réduction de modèle PGD dans le cadre de modèle linéaire, permettant ainsi de mettre en place un processus d’identification rapide et automatique. Ensuite, dans le but d’être appliquée au paradigme DDDAS, nous avons développé une démarche d’identification reposant sur un processus d’assimilation de données (le filtre de Kalman) et utilisant l’Erreur en Relation de Comportement modifiée comme opérateur d’observation toujours dans le cadre de problèmes linéaires. Nous avons ensuite étendu cette méthode d’assimilation de données à la problématique de l’identification de champs de paramètres en introduisant une séparation des discrétisations spatiales et des outils provenant de l’adaptation de maillage. Nous avons ensuite abordé la problématique des modèles mécaniques non-linéaires, au travers de modèles d’endommagement et de visco-plasticité. Pour cela nous avons dans un premier temps reformulé et étendu le concept d’Erreur en Relation de Comportement modifiée à ce cadre non-linéaire matériau et nous avons mis en place un processus de résolution dédié, s’inspirant de la méthode LaTIn. Pour finir nous avons introduit cette reformulation de l’Erreur en Relation de Comportement modifiée au sein de la méthode d’assimilation de données développée précédemment afin de traiter le recalage dynamique de modèles non-linéaires. / For several years, the considerable changes that have occurredin computing tools have led to new practices in the simulation of mechanical structures. Among them, the motivation for this work is the Dynamic Data Driven Application Systems paradigm (DDDAS). The founding idea of this approach is to establish a dialogue between a physical system and its numericalmodel. The objective is then to (i) allow a calibration of the numerical model by means of measurements performed on the physical system; (ii) control the evolution of the physical system using theprediction given by numerical simulation. The major difficulty is to realize this dialogue in real time. This work focuses on the model updating step of the DDDAS paradigm. The problem is then to develop methods and tools to solve inverse problems taking into account various constraints, namely: (i) robustness with respect to corrupted data; (ii) genericity for considering a wide variety of problems and mechanical models; (iii) a reduced computation time in order to tend towards a real-time model updating.The starting point of this work is the modified Constitutive Relation Error, an energetic approach dedicated to the solution of inverse problems in mechanics, notably illustrated by its robustness with respect to measurement noises. First, in order to guarantee a fast identification process, we have coupled the modified Constitutive Relation Error with the PGD model reduction in the linear model framework, thus enabling a fast and automatic identification process. Then, in order to be applied to the DDDAS paradigm, we have developed an identification method based on a data assimilation process (the Kalman filter) and using the modified Constitutive Relation Error as an observer alwayswithin the framework of linear problems. We have then extended this data assimilation approach to the problem of the identification of parameter fields by introducing a separation of the spatial discretizations and by introducing tools resulting from the mesh adaptation framework. We have then addressed the problem of non-linear mechanical models, through damage and visco-plasticitymodels. To this end, we have first recast and extended the concept of the modified Constitutive Relation Error to this nonlinear material framework and we have implemented a dedicated resolution process, based on the LaTIn method. Finally, we have introduced this reformulation of the modified Constitutive Relation Error in the previously data assimilation method in order to process the model updating of nonlinear models.

De nombreux problèmes de mécanique des structures nécessitent la résolution de plusieurs problèmes numériques semblables. Une approche itérative de type réduction de modèle, la Proper Generalized Decomposition (PGD), permet de déterminer l’ensemble des solutions en une fois, par l’introduction de paramètres supplémentaires. Cependant, un frein majeur à son utilisation dans le monde industriel est l’absence d’estimateur d’erreur robuste permettant de mesurer la qualité des solutions obtenues. L’approche retenue s’appuie sur le concept d’erreur en relation de comportement. Cette méthode consiste à construire des champs admissibles, assurant ainsi l’aspect conservatif et garanti de l’estimation de l’erreur en réutilisant le maximum d’outils employés dans le cadre éléments finis. La possibilité de quantifier l’importance des différentes sources d’erreur (réduction et discrétisation) permet de plus de piloter les principales stratégies de résolution PGD. Deux stratégies ont été proposées dans ces travaux. La première s’est principalement limitée à post-traiter une solution PGD pour construire une estimation de l’erreur commise, de façon non intrusive pour les codes PGD existants. La seconde consiste en une nouvelle stratégie PGD fournissant une approximation améliorée couplée à une estimation de l’erreur commise. Les diverses études comparatives sont menées dans le cadre des problèmes linéaires thermiques et en élasticité. Ces travaux ont également permis d’optimiser les méthodes de construction de champs admissibles en substituant la résolution de nombreux problèmes semblables par une solution PGD, exploitée comme un abaque. / Many structural mechanics problems require the resolution of several similar numerical problems. An iterative model reduction approach, the Proper Generalized Decomposition (PGD), enables the control of the main solutions at once, by the introduction of additional parameters. However, a major drawback to its use in the industrial world is the absence of a robust error estimator to measure the quality of the solutions obtained.The approach used is based on the concept of constitutive relation error. This method consists in constructing admissible fields, thus ensuring the conservative and guaranteed aspect of the estimation of the error by reusing the maximum number of tools used in the finite elements framework. The ability to quantify the importance of the different sources of error (reduction and discretization) allows to control the main strategies of PGD resolution.Two strategies have been proposed in this work. The first was limited to post-processing a PGD solution to construct an estimate of the error committed, in a non-intrusively way for existing PGD codes. The second consists of a new PGD strategy providing an improved approximation associated with an estimate of the error committed. The various comparative studies are carried out in the context of linear thermal and elasticity problems.This work also allowed us to optimize the admissible fields construction methods by substituting the resolution of many similar problems by a PGD solution, exploited as a virtual chart.

Le contrôle de la qualité des calculs de structure suscite un intérêt croissant dans les processus de conception et de certification. Il repose sur l'utilisation d'estimateurs d'erreur, dont la mise en pratique entraîne un sur-coût numérique souvent prohibitif sur des calculs de grande taille. Le présent travail propose une nouvelle procédure permettant l'obtention d'une estimation garantie de l'erreur de discrétisation dans le cadre de problèmes linéaires élastiques résolus au moyen d'approches par décomposition de domaine. La méthode repose sur l'extension du concept d'erreur en relation de comportement au cadre des décompositions de domaine sans recouvrement, en s'appuyant sur la construction de champs admissibles aux interfaces. Son développement dans le cadre des approches FETI et BDD permet d'accéder à une mesure pertinente de l'erreur de discrétisation bien avant convergence du solveur lié à la décomposition de domaine. Une extension de la procédure d'estimation aux problèmes hétérogènes est également proposée. Le comportement de la méthode est illustré et discuté sur plusieurs exemples numériques en dimension 2. / The control of the quality of mechanical computations arouses a growing interest in both design and certification processes. It relies on error estimators the use of which leads to often prohibitive additional numerical costs on large computations. The present work puts forward a new procedure enabling to obtain a guaranteed estimation of discretization error in the setting of linear elastic problems solved by domain decomposition approaches. The method relies on the extension of the constitutive relation error concept to the framework of non-overlapping domain decomposition through the recovery of admissible interface fields. Its development within the framework of the FETI and BDD approaches allows to obtain a relevant estimation of discretization error well before the convergence of the solver linked to the domain decomposition. An extension of the estimation procedure to heterogeneous problems is also proposed. The behaviour of the method is illustrated and assessed on several numerical examples in 2 dimension.