Dans les réacteurs nucléaires à eau sous-pression, le combustible est constitué de pastilles d’oxyde d’uranium (UO2), dont le comportement ne peut être simulé qu'à travers une modélisation multi-échelles et multi-physiques, tenant compte à la fois de la thermo-mécanique et de la physico-chimie relative aux produits de fission. L’évolution récente des modèles et des moyens de calcul a permis de développer les simulations à l’échelle de la microstructure et d’accroitre les possibilités de couplage. Ce travail concerne le développement d'un modèle de comportement thermo-mécanique de l’UO2 à l’échelle du polycristal. Le comportement du VER est analysé en termes de réponse effective et de phénomènes de localisation. Nous nous intéressons notamment aux valeurs de pression hydrostatique, qui pilotent le transport des produits de fission. La robustesse des résultats obtenus en fonction du choix du maillage éléments finis est étudiée. Une série de calculs est présentée afin de trouver un compromis satisfaisant en termes de discrétisation pour une estimation correcte des contraintes locales. Une première étude propose de retrouver des mesures expérimentales de dé cohésion intergranulaire sur le combustible en introduisant des modèles de zones cohésives dans le VER. Afin d'analyser le comportement micromécanique de l’UO2 en irradiation, un chargement de type rampe de puissance est appliqué au polycristal. L’analyse des distributions locales de contraintes donne lieu à une discussion sur l’effet de l’incompatibilité intergranulaire sur le transport des produits de fission. / In Pressurized Water Reactors (PWR), fuel is generally composed of uranium dioxide pellets piled up within a zirconium tubular cladding. Modeling of the fuel behavior in nominal and accidental conditions requires multi-scale models in order to take into account both the thermo-mechanical behavior of the pellets and the effects of fission gases. Recent development of micromecanical tools of simulation has improved coupling possibilities. Our study proposes to set a full micromechanical model for uranium dioxide dealing with both mechanics and fission products transport at the scale of a polycristalline aggregate. Both the effective behavior of the RVE and stress localization effects are studied. Hydrostatic pressure, which directly controls the diffusion of fission gases, is given a particular focus. The numerical robustness of results is also debated in terms of mesh refinement. A series of simulations leads to a satisfying compromise between accuracy and calculation time. A study compares experimental measurement of intergranular crack opening and simulation results obtained using cohesive models. The micromecanical behavior of uranium dioxide during irradiation is analysed by submitting the polycristalline RVE to transient irradiation. The local stress distribution leads to a debate on the consequences of intergranular strain incompatibility on fission gases diffusion.

Cette thèse s'intéresse aux mécanismes de déformations et d’endommagement dans les alliages de coulée pour l’industrie automobile. Les études ont porté sur un alliage modèle contenant 12% de silicium et des éléments d’addition. Il a été fabriqué par le procédé squeeze-casting (coulée sous haute pression) au LKR de l’Université Technologique de Vienne avec laquelle cette thèse s’est déroulée en collaboration. Des essais de traction in situ dans le MEB ainsi que sous tomographie aux rayons X ont permis de suivre l’évolution des microstructures pendant le chargement. Une attention particulière a été portée sur les mécanismes d’initiation et de propagation de l’endommagement. Par ailleurs la technique de la corrélation d’images numérique 2D spécialement développée pour être appliquée à des images MEB a été utilisée pour mesurer les mécanismes d’endommagement à l’échelle des inclusions. Une attention particulièreest portée sur les mécanismes de localisation de la déformation. La corrélation d’images numériques 3D appliquée aux images tomographiques a permis une mesure précise de l’évolution de l’endommagement au cours du chargement. Les données expérimentales obtenues ont permis d’adapter un modèle d’endommagement de type GTN. Une dernière étape s’est consacrée à une modélisation micro mécaniques des microstructures. Des méthodes de caractérisations basées sur des hypothèses statistiques ont permis de mesurer et d’identifier des paramètres microstructuraux. Au regard des mécanismes de déformations et de la morphologie des particules de seconde phases, des microstructures modèles sont proposées pour simuler le comportement du matériau réel. Cette démarche servant essentiellement à simplifier les modèles éléments finis afin de gagner en temps de calcul, elle sert aussi à insérer des mécanismes complexes afin de serapprocher le plus possible du comportement réel du matériau. / This thesis focuses on the mechanisms of deformation and damage in cast aluminum alloysfor automotive industry. Studies were carried on a model alloy containing 12% silicon and added element like iron and nickel. The material was prepared by squeeze-casting process at the Vienna University of Technology in Austria with which this thesis was held in collaboration. In situ tensile test with X-rays tomographic observations as well as with Scanning Electron Microscope observations were performed. They allowed following microstructure evolution under mechanical loading. Particular attention was paid to the mechanisms of initiation and propagation of damage. Furthermore the technic of Digital Image Correlation (DIC) developed to be applied to SEM images was used to visualize 2D strain localization mechanisms at second phases particles scale. DIC were also applied on tomographic images and allowed to measure accurately in 3D damage evolution under loading. Experimental data obtained from DIC were used to identify parameters of a Gurson Tvergaard Needleman model and allow to simulate by Finite Element Modeling the damage evolution. At last stages micro mechanical studies were performed by FEM using microstructure obtained from tomographic images. Algorithms were developed to measure and identify statistics parameters of the microstructure. In consideration of the mechanism of deformations and second phase particles morphology, generatedmicrostructures are proposed to simulate the behavior of the material. This approach aims to simplify the FEM models by reducing the number of degree of Freedom and, then reduce computation time. It also enables to insert complex mechanisms in the model to be as close as possible the actual behavior of the material.

La communauté des métamatériaux est entrée en effervescence depuis la publication de deux articles de Science en 2006 par Pendry et Leonhardt dans lesquels il est proposé de réaliser des systèmes exotiques tels que les capes d’invisibilité par transformation d’espace. En effet, l’invariance de forme des équations de Maxwell permet une équivalence entre géométrie déformée et présence d’un matériau aux propriétés particulières. Depuis, de nombreux exemples expérimentaux ont montré la faisabilité de tels systèmes transformés. L’invariance de forme se retrouve également dans d’autres phénomènes physiques et les transformations d’espace ont par ailleurs été appliquées à plusieurs disciplines telles que l’acoustique, l’élasto-dynamique ou la propagation d’ondes de surface. Nous présentons ici les transformations d’espace appliquées à l’équation de la chaleur. Au cours de notre étude, nous nous intéressons aux transformations menant aux capes d’invisibilité thermiques et aux concentrateurs thermiques. Ces systèmes sont constitués de matériaux anisotropes et hétérogènes ce qui les rend difficile à réaliser. Nous utilisons donc la théorie de l’homogénéisation à deux échelles qui permet d’approcher le comportement de ces systèmes par une alternance de couches de matériaux isotropes. Nous suivons une démarche systématique d’évaluation quantitative des performances de nos systèmes approchés dans une optique d’une ingénierie à haut niveau de métamatériaux thermiques. Un modèle de tapis thermique à 50 couches est proposé dont les résultats expérimentaux sont attendus. / The metamaterials community has been heavily excited since the publication of two articles by Pendry and Leonhardt in 2006 in which exotic devices such as invisibility cloaks are proposed to be implemented by space transformation. Indeed, the form invariance of the Maxwell equations allow for an equivalence between a deformed geometry and a material with specific properties. Since then, several experimental studies have shown the feasibility of such transformed devices. The form invariance is also found in other physical domains such and the space transformations were applied to mulitphyscial phenomena such acoustic wave propagation, elasto-dynamic wave and surface wave propagation. We present in this work the space transformation applied to the heat equation. Throughout our study, we focus on the transformations leading to thermal invisibility cloaks and thermal concentrators. Those transformed devices are made of anisotropic heterogeneous materials which make them difficult to practically design. Therefore, we make use of the two-scale homogenization theory allowing to approach the behavior of those devices with an alternate set of isotropic materials. We systematically try to evaluate quantitatively the performance of our approximate devices by defining an effectiveness criterion to achieve high level of mthermal metamaterials engineering. We present a the end a model of a 50-layer carpet cloak whose first results are to be expected.

Ce travail est dédié au calcul des propriétés de transfert thermique et de transport dans les milieux hétérogènes périodiques. Les résultats sont établis dans le cadre d'homogénéisation périodique pour lequel les propriétés macroscopiques sont obtenues par la résolution de problèmes élémentaires pour la cellule irréductible. Plusieurs contributions sont ainsi apportées, visant à établir de nouvelles estimations par des approches analytiques ou en développant des méthodes numériques adaptées. Ainsi dans une première partie, on s'intéresse à la modélisation des propriétés non linéaires de filtration dans les milieux poreux. A l'échelle microscopique l'écoulement est régi par l'équation de Navier-Stokes. En développant la solution en série, on obtient par homogénéisation, une loi de filtration polynomiale. Tous les coefficients constitutifs de cette loi sont alors obtenus en résolvant en cascade des problèmes élémentaires sur la cellule à l'aide de schémas itératifs utilisant sur la transformée de Fourier rapide. On propose ensuite de nouvelles expressions analytiques pour les propriétés de conductivité thermique de composites périodiques renforcés par des inclusions sphériques. On résout l'équation intégrale de Lippmann-Schwinger par des développements en série de Neumann et en choisissant une polarisation constante dans les inclusions. Des expressions analytiques sont alors obtenues pour diverses configurations spatiales : réseaux cubiques et répartitions aléatoires isotropes. Dans la dernière partie de ce travail, on détermine les propriétés de transfert thermique par conduction et convection dans les milieux poreux saturés par un fluide. A nouveau, on propose des schémas de résolution basés sur la transformée de Fourier rapide pour le calcul du tenseur de diffusivité de milieux poreux / In this work, we determine the macroscopic properties of thermal transfer and mass transport in periodic heterogeneous materials. All the results are established in the framework of periodic homogenization, for which, the macroscopic properties are deduced by solving elementary problems for the irreducible cell. Various contributions are provided, leading to the derivation of new closed-form expressions for the effective properties or by developing numerical tools. In the first part, we determine the nonlinear filtration properties of porous media. At the microscopic scale, the fluid flow obeys to the Navier-Stokes equation. By expanding the solution into power series, we obtain, after homogenization, a polynomial type macroscopic filtration law. All the constitutive coefficients of are determined by solving a hierarchy of cell problems by means of a numerical approach based on the Fast Fourier Transform algorithm. The problem of conductivity of periodic composites reinforced by spherical inclusions is thereafter considered by an analytic approach. We solve the Lippmann-Schwinger integral equation using Neumann series and a constant polarization in the inclusion. Closed-form estimate of the macroscopic conductivity are then obtain for different spatial configurations: cubic lattice and isotropic distribution of inclusions. In the last part, we determine the thermal transfer properties by conduction and convection of porous media fulfilled by a viscous fluid. Again, numerical tools based on FFT are considered to solve the unit cell problems and to compute the diffusivity tensor

Orientador: Marcelo Cristianini / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia de Alimentos / Made available in DSpace on 2018-08-22T10:26:08Z (GMT). No. of bitstreams: 1 Oliveira_MiguelMeirellesde_M.pdf: 1688064 bytes, checksum: 7da6b05069a8467a053c06082ae009f5 (MD5) Previous issue date: 2013 / Resumo: Leites fermentados probióticos são considerados alimentos funcionais, devendo conferir proteção às culturas. A alta pressão dinâmica (APD) vem sendo investigada como alternativa na produção de leite fermentado, pois altera a funcionalidade de alguns constituintes. Este trabalho objetivou avaliar o uso de leite submetido à APD na cinética de fermentação de S. thermophilus e L. acidophilus, bem como a viabilidade das culturas, as características físico-químicas e reológicas durante a estocagem refrigerada do leite fermentado probiótico.Leite (3% de gordura) foi submetido ao processo de APD (100MPa e 200MPa) e comparados com controle (15/5MPa), e avaliada a cinética de fermentação (pH), atividade proteolítica e a viabilidade durante a fermentação do L. acidophilus (Cap.2). Outro experimento foi realizado com leite (2% de gordura) submetido à APD (50/5MPa, 100/5MPa, 150/5MPa e 180/5MPa) e comparado com o controle (15/5MPa), no qual foi avaliada a cinética de fermentação de S. thermophilus em co-cultura com L. acidophilus (Cap.3), bem como o comportamento reológico (Cap.3) e as características físico-químicas (Cap.4) do leite fermentado probiótico durante a estocagem refrigerada. A cinética de fermentação de Lactobacillus acidophilus foi alterada pela APD (100MPa e 200MPa) em comparação com o controle (15/5MPa), aumentando consideravelmente o tempo de fermentação com o aumento da pressão, sendo que após 24 horas de fermentação as amostras apresentaram valores de pH 4,49 (15/5MPa), pH 4,72 (100MPa) e 4,93 (200MPa). A atividade proteolítica foi alterada, sendo que pressões elevadas reduziram a atividade, atingindo no final da fermentação uma redução de 21,39% (100MPa) e 35,89% (200MPa), quando comparadas com o controle. Além disso, as amostras submetidas à APD apresentaram menor número de células viáveis de L. acidophilus no fim da fermentação em relação ao controle (0,28 ciclos log) (p<0,05). A cinética de fermentação do Streptococcus thermophilus e Lactobacillus acidophilus não foi alterada pela APD. O comportamento reológico durante a fermentação de leite processado à 100/5MPa e 180/5MPa apresentou correlação positiva entre a pressão usada com a consistência do leite fermentado, gerando um produto com maior consistência. Além disso, o processo de APD aumentou a capacidade de retenção de água (CRA) em 9,15% e reduziu a sinérese espontânea do produto em 31%, entre o controle e a amostras processada a 180/5MPa. A viabilidade do S. thermophilus não foi (p<0,05) influenciada pela APD (50/5MPa, 100/5MPa, 50/5MPa e 180/5MPa) em comparação com a homogeneização convencional (15/5MPa) durante a estocagem refrigerada, reduzindo a contagem em aproximadamente 0,2 ciclos log em todos os tratamentos estudados. A viabilidade do L. acidophilus foi maior na amostra processada a 180/5MPa a partir do 14° dia, em comparação ao controle (15/5MPa). O pH apresentou uma pequena redução (p<0,05) somente na amostra processada a 180/5MPa no ultimo dia. A atividade proteolítica das culturas não foi influenciada pela ADP. Os resultados demonstraram que a APD pode melhorar a textura de leite fermentado, aumentar a capacidade de retenção de água, reduzir a sinérese e alterar a cinética de fermentação, dependendo da cultura utilizada. No entanto, a viabilidade das culturas, a redução do pH e a atividade proteolítica não sofreram alterações expressivas pela APD, apesar de apresentar diferenças significativas / Abstract: Probiotics fermented milk are considered functional food, should provide protection to cultures. Dynamic high pressure (DHP) is being investigated as an alternative in the production of fermented milk, because changing the functionality of some constituents. This study evaluated the use of milk submitted to DHP in fermentation kinetics of S. thermophilus and L. acidophilus, as well as the viability of the cultures, the physico-chemical and rheological during refrigerated storage of fermented milk probiotic. Milk (3% fat) was subjected to the process of DHP (100MPa and 200MPa) and compared with control (15/5MPa), and evaluated the kinetics of fermentation (pH), proteolytic activity and viability during fermentation of L. acidophilus (Cap.2). Another experiment was performed with milk (2% fat) subject to DHP (50/5MPa, 100/5MPa, and 150/5MPa 180/5MPa) and compared with the control (15/5MPa), in which was evaluated the kinetics fermentation of S. thermophilus in co-cultured with L. acidophilus (Cap.3), and rheological behavior (Cap.3) and the physico-chemical (Cap.4) of probiotic fermented milk during refrigerated storage. The fermentation kinetics of Lactobacillus acidophilus was modified by DHP (100MPa and 200MPa), comparated to control (15/5 MPa) considerably increasing the fermentation time the higher the pressure used, and that after 24 hours of fermentation samples showed pH 4.49 (15/5MPa), pH 4.72 (100MPa) 4.93 (200MPa). The proteolytic activity was changed, with elevated pressures reducing proteolytic activity at the end of the fermentation, reaching a reduction of 21.39% (100MPa) and 35.89% (200MPa) compared with the control. In addition, the samples submitted to APD had lower number of viable probiotics in the end of fermentation compared to control (0,28 log cycles) (p<0.05). The fermentation kinetics of Streptococcus thermophilus and Lactobacillus acidophilus was not altered by DHP. The rheological behavior during fermentation in milk processed by DHP (100/5 MPa and 180/5 MPa) demonstrated a positive correlation between the pressure and consistency of the fermented milk. Moreover, the process of DHP favored water-holding capacity (WHC) in 10.13% and reduced syneresis spontaneous in 31% between the control and 180/5 MPa. The viability of S. thermophilus was not influenced by DHP (50/5MPa, 100/5MPa, 150/5MPa and 180/5MPa), comparated to conventional homogenization (15/5 MPa) during refrigerated storage, reducing the count in approximately 0.2 log cycles in all treatments. The viability of L. acidophilus was higher in the processed sample 180/5MPa from day 14, compared to the control (15/5 MPa). The pH showed a small reduction in sample subject to 180/5 MPa at the final day of storage. The proteolytic activity of the culture was not influenced by DHP. The results showed that the DHP may improve the texture of fermented milk, increase water-holding capacity, reduce syneresis and alter the kinetics of the fermentation. However, viability of cultures, the decrease in pH and proteolytic activity did not change by DHP, despite showing significant differences / Mestrado / Tecnologia de Alimentos / Mestre em Tecnologia de Alimentos

Orientador: Renato Pavanello / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecânica / Made available in DSpace on 2018-08-28T11:58:24Z (GMT). No. of bitstreams: 1 Calixto_TainanKhalilLeite_M.pdf: 15775150 bytes, checksum: e5e4d45becc32f922a1fd31b8f9bd3ab (MD5) Previous issue date: 2015 / Resumo: A utilização de materiais de alta performance se tornou uma realidade em diversos campos da engenharia, como na indústria automotiva e aeroespacial, devido aos avanços nas técnicas de manufatura aditiva. Por outro lado, sabe-se que a otimização topológica estrutural é uma ferramenta de desenvolvimento de estruturas com ampla aplicação industrial. Dentre os vários métodos de otimização topológica existentes, a otimização estrutural evolucionária tem se destacado pela sua versatilidade, podendo ser utilizada em diversos tipos de problemas de engenharia. Na tentativa de combinar esses campos, este trabalho consiste no estudo do método de otimização evolucionária BESO (Bi-directional Evolutionary Structural Optimization) aplicado a sistemas bidimensionais multiescala a fim de se projetar as topologias ótimas, em ambas as escalas, de uma estrutura. A análise do modelo multiescala é feita através do método da homogeneização, onde o padrão do material microestrutural é considerado periódico. O algoritmo implementado pode buscar dois objetivos distintos: a minimização da flexibilidade média, que resulta na maximização da rigidez global; ou a maximização da frequência fundamental. Resultados numéricos do algoritmo são apresentados para o projeto de materiais, onde apenas a microestrutura é otimizada, e de estruturas, na qual otimiza-se as topologias nas duas escalas. Para a análise do desempenho do método de otimização multiescala, são propostos um índice de eficiência estrutural e uma metodologia de fabricação de estruturas periódicas / Abstract: High-performance materials utilization became a reality in many fields of actual engineering, such as in automotive and aerospace industries, due to advances in additive manufacturing techniques. In the other hand, structural topology optimization is a powerful tool for the structure development with wide industrial application. Among the various optimization methods, evolutionary structural optimization stands out for its versatility and it can be used in many engineering problems. As an attempt to combine these fields, this work intends to study the Bi-directional evolutionary Structural Optimization method applied to two-dimensional multi-scale systems in order to design the optimal topologies of structures in both scales. The analysis of multi-scale model is made using the homogenization method, where the pattern of the micro-structural material is considered periodic. The implemented algorithm can use two different objective function: mean compliance minimization, which results in maximizing the global stiffness; or fundamental frequency maximization. Numerical results are presented for material design, where only the micro structure is optimized, and for structural design, in which the topologies in both scales are optimized. Deepening the study in multiscale optimization, it is proposed an index to analyse the structural efficiency and also a manufacturing methodology of periodic structures / Mestrado / Mecanica dos Sólidos e Projeto Mecanico / Mestre em Engenharia Mecânica

Ossos são sólidos não homogêneos com estruturas altamente complexas que requerem uma modelagem multiescala para entender seu comportamento eletromecânico e seus mecanismos de remodelamento. O objetivo deste trabalho é encontrar expressões analíticas para as propriedades elástica, piezoelétrica e dielétrica efetivas de osso cortical modelando-o em duas escalas: microscópica e macroscópica. Utiliza-se o Método de Homogeneização Assintótica (MHA) para calcular as constantes eletromecânicas efetivas deste material. O MHA produz um procedimento em duas escalas que permite obter as propriedades efetivas de um material compósito contendo uma distribuição periódica de furos cilíndricos circulares unidirecionais em uma matriz piezoelétrica linear e transversalmente isotrópica. O material da matriz pertence à classe de simetria cristalina 622. Os furos estão centrados em células de uma matriz periódica de secções transversais quadradas e a periodicidade é a mesma em duas direções perpendiculares. O compósito piezoelétrico está sob cisalhamento antiplano acoplado a um campo elétrico plano. Os problemas locais que surgem da análise em duas escalas usando o MHA são resolvidos por meio de um método da teoria de variáveis complexas, o qual permite expandir as soluções correspondentes em séries de potências de funções elípticas de Weierstrass. Os coeficientes das séries são determinados das soluções de sistemas lineares infinitos de equações algébricas. Truncando estes sistemas infinitos até uma ordem finita de aproximação, obtêm-se fórmulas analíticas para as constantes efetivas elástica, piezoelétrica e dielétrica, que dependem da fração de volume dos furos e de um fator de acoplamento eletromecânico da matriz. Os resultados numéricos obtidos a partir destas fórmulas são comparados com resultados obtidos pelas fórmulas calculadas via método de Mori-Tanaka e apresentam boa concordância. A boa concordância entre todas as curvas obtidas via MHA sugere que a expressão correspondente da primeira aproximação fornece uma fórmula muito simples para calcular o fator de acoplamento efetivo do compósito. Os resultados são úteis na mecânica de osso. / Bones are inhomogeneous solids with highly complex structures that require multiscale modeling to understand its electromechanical behavior and its remodeling mechanisms. The objective of this work is to find analytical expressions for the effective elastic, piezoelectric, and dielectric properties of cortical bone by modeling it on two scales: microscopic and macroscopic. We use Asymptotic Homogenization Method (AHM) to calculate the effective electromechanical constants of this material. The AHM yields a two-scale procedure to obtain the effective properties of a composite material containing a periodic distribution of unidirectional circular cylindrical holes in a linear transversely isotropic piezoelectric matrix. The matrix material belongs to the symmetry crystal class 622. The holes are centered in a periodic array of cells of square cross sections and the periodicity is the same in two perpendicular directions. The piezoelectric composite is under antiplane shear deformation together with in-plane electric field. Local problems that arise from the two-scale analysis using the AHM are solved by means of a complex variable method, which allows us to expand the corresponding solutions in power series of Weierstrass elliptic functions. The coefficients of these series are determined from the solutions of infinite systems of linear algebraic equations. Truncating the infinite systems up to a finite, but otherwise arbitrary, order of approximation, we obtain analytical formulas for effective elastic, piezoelectric, and dielectric properties, which depend on both the volume fraction of the holes and an electromechanical coupling factor of the matrix. Numerical results obtained from these formulas are compared with results obtained by the Mori-Tanaka approach and show good agreement. The good agreement between all curves obtained via AHM suggests that the corresponding expression of first approximation provides a very simple formula to calculate the effective coupling factor of the composite. The results are useful in bone mechanics.

L'objet de cette thèse est l'étude et la mise en œuvre d'une méthode d’éléments finis multi-échelles pour la simulation d'écoulements miscibles en milieux poreux. La définition des fonctions de base multi-échelles suit l'idée introduite par F. Ouaki. La nouveauté de ce travail consiste à combiner cette approche multi-échelle avec des éléments finis de type Galerkine Discontinus (DG) de façon à pouvoir utiliser ces nouveaux éléments sur des maillages non-conformes composés de mailles de formes diverses. Nous rappelons, dans un premier temps, le principe des méthodes DG et montrons comment ces méthodes peuvent être utilisées pour discrétiser une équation de convection-diffusion instationnaire identique à celle rencontrée dans le problème d'écoulement considéré dans ce travail. Après avoir vérifié l'existence et l'unicité d'une solution à ce problème, nous redémontrons la convergence des méthodes DG vers cette solution en établissant une estimation d'erreur a priori. Nous introduisons, ensuite, les éléments finis multi-échelles non conformes et détaillons leur mise en œuvre sur ce problème de convection-diffusion. En supposant les conditions aux limites et les paramètres du problème périodiques, nous montrons une nouvelle estimation d'erreur a priori pour cette méthode. Dans une seconde partie, nous considérons le problème d'écoulement complet où l'équation considérée dans la première partie est résolue de manière couplée avec l'équation de Darcy. Nous introduisons différents cas tests inspirés de modèles d'écoulements rencontrés en géosciences et comparons les solutions obtenues avec les deux méthodes DG, à savoir la méthode classique utilisant un seul maillage et la méthode étudiée ici. Nous proposons de nouvelles conditions aux limites pour la résolution des problèmes de cellule qui permettent, par rapport à des conditions aux limites linéaires plus classiquement utilisées, de mieux reproduire les variations des solutions le long des interfaces du maillage grossier. Les résultats de ces tests montrent que la méthode multi-échelle proposée permet de calculer des solutions proches de celles obtenues avec la méthode DG sur un seul maillage et de réduire, de façon significative, la taille du système linéaire à résoudre à chaque pas de temps. / This work deals with the study and the implementation of a multiscale finite element method for the simulation of miscible flows in porous media. The definition of the multiscale basis functions is based on the idea introduced by F. Ouaki. The novelty of this work lies in the combination of this multiscale approach with Discontinuous Galerkin methods (DG) so that these new finite elements can be used on nonconforming meshes composed of cells with various shapes. We first recall the basics of DG methods and their application to the discretisation of a convection-diffusion equation that arises in the flow problem considered in this work. After establishing the existence and uniqueness of a solution to the continuous problem, we prove again the convergence of DG methods towards this solution by establishing an a priori error estimate. We then introduce the nonconforming multiscale finite element method and explain how it can be implemented for this convection-diffusion problem. Assuming that the boundary conditions and the parameters of the problem are periodic, we prove a new a priori error estimate for this method. In a second part, we consider the whole flow problem where the equation, studied in the first part of that work, is coupled and simultaneously solved with Darcy equation. We introduce various synthetic test cases which are close to flow problems encountered in geosciences and compare the solutions obtained with both DG methods, namely the classical method based on the use of a single mesh and the one studied here. For the resolution of the cell problems, we propose new boundary conditions which, compared to classical linear conditions, allow us to better reproduce the variations of the solutions on the interfaces of the coarse mesh. The results of these tests show that the multiscale method enables us to calculate solutions which are close to the ones obtained withDG methods on a single mesh and also enables us to reduce significantly the size of the linear system that has to be solved at each time step.

This diploma thesis focuses on realization of a computational model of fibre composite with elastomer matrix and on homogenization of properties of this composite. The work deals with computational modelling of strain-stress states which arise in mechanical tests of composites. The composites investigated by mechanical tests comprise of hyperelastic rubber matrix and steel reinforcing fibres. Computational modelling is carried out at two levels of the model. First, with three-dimensional modelling of fibres and matrix as two different materials and, second, using a homogenized model of composite; this constitutive model describes the composite as a homogeneous anisotropic material. It means that properties of fibres are encompassed into strain energy density by the mathematical formulation of the constitutive model. Further, the work deals with computational modelling of mechanical tests of hyperelastic isotropic materials used for identification of their material parameters and for verification of the selected constitutive model of material. For particular hyperelastic material, simulations of tests were carried out, namely of uniaxial tension, biaxial tension, uniaxial compression, biaxial compression, pure shear and uniaxial tension with constrained transversal strain (planar tension). Parameters of the constitutive model were determined of experimental input data. Verification of the constitutive model was carried out by comparison of the data acquired by experiments with the results of simulations of mechanical tests in FE program system Ansys. Then the authentic constitutive model of material was used for description of matrix behaviour in models of mechanical tests of composite material and results were compared with experimental data. Principal objectives which I want to attain are following: • to acquaint with the constitutive models of hyperelastic isotropic and anisotropic materials and identification of their perameters on base of mechanical tests. • to create computational models of testing specimens of composite “ rubber – steel fibre“ for different fibre arrangements and to use the created computational models in simulations of chosen tests. • to test the possibilities of computational modelling of composites with application of homogenized properties and to compare the results of both approaches. Results which were attained: • the computational models were created with the fibres modelled; the strain – stress characteristics are qualitatively corresponding to experiments, and quantitative difference is 20% - 40% (see (4.3)). • the computational models based on homogenization of properties were tested and gave results corresponding to the models with modelled fibres (see (4.4)) with a good accuracy.

Cette thèse est dédiée à l'analyse mathématique et la simulation numérique des équations intervenant dans la modélisation de l’électrophysiologie cardiaque. D'abord, nous donnons une justification mathématique rigoureuse du processus d’homogénéisation périodique à l’aide de la méthode d'éclatement périodique. Nous considérons des conductivités électriques tensorielles qui dépendent de l’espace et des modèles ioniques non linéaires physiologiques et phénoménologiques. Nous montrons l'existence et l'unicité d’une solution du modèle microscopique en utilisant une approche constructive de Faedo- Galerkin suivie par un argument de compacité dans L2. Ensuite, nous montrons la convergence de la suite de solutions du problème microscopique vers la solution du problème macroscopique. À cause des termes non linéaires sur la variété oscillante, nous utilisons l’opérateur d’éclatement sur la surface et un argument de compacité de type Kolmogorov pour les modèles phénoménologiques et de type Minty pour les modèles physiologiques. En outre, nous considérons le modèle monodomaine couplé au modèle physiologique de Beeler-Reuter. Nous proposons un schéma volumes finis et nous analysons sa convergence. D'abord, nous dérivons la formulation variationnelle discrète correspondante et nous montrons l'existence et l'unicité de sa solution. Par compacité, nous obtenons la convergence de la solution discrète. Comme le schéma TPFA (two point flux approximation) est inefficace pour approcher les flux diffusifs avec des tenseurs anisotropes, nous proposons et analysons, ensuite, un schéma combiné non-linéaire qui préserve le principe de maximum. Ce schéma est basé sur l’utilisation d’un flux numérique de Godunov pour le terme de diffusion assurant que les solutions discrètes soient bornées sans restriction sur le maillage du domaine spatial ni sur les coefficients de transmissibilité. Enfin, dans la perspective d'étudier la solvabilité des modèles électromécaniques couplés avec des modèles ioniques physiologiques, nous considérons un modèle avec une description linéarisée de la réponse élastique passive du tissu cardiaque, une linéarisation de la contrainte d'incompressibilité et une approximation tronquée des diffusivités non linéaires intervenant dans les équations du modèle bidomaine. La preuve utilise des approximations par des systèmes non-dégénérés et la méthode Faedo-Galerkin suivie par un argument de compacité. / This thesis is concerned with the mathematical analysis and numerical simulation of cardiac electrophysiology models. We use the unfolding method of homogenization to rigorously derive the macroscopic bidomain equations. We consider tensorial and space dependent conductivities and physiological and simplified ionic models. Using the Faedo-Galerkin approach followed by compactness, we prove the existence and uniqueness of solution to the microscopic bidomain model. The convergence of a sequence of solutions of the microscopic model to the solution of the macroscopic model is then obtained. Due to the nonlinear terms on the oscillating manifold, the boundary unfolding operator is used as well as a Kolmogorov compactness argument for the simplified models and a Minty type argument for the physiological models. Furthermore, we consider the monodomain model coupled to Beeler- Reuter's ionic model. We propose a finite volume scheme and analyze its convergence. First, we show existence and uniqueness of its solution. By compactness, the convergence of the discrete solution is obtained. Since the two-point flux approximation (TPFA) scheme is inefficient in approximating anisotropic diffusion fluxes, we propose and analyze a nonlinear combined scheme that preserves the maximum principle. In this scheme, a Godunov approximation to the diffusion term ensures that the solutions are bounded without any restriction on the transmissibilities or on the mesh. Finally, in view of adressing the solvability of cardiac electromechanics coupled to physiological ionic models, we considered a model with a linearized description of the passive elastic response of cardiac tissue, a linearized incompressibility constraint, and a truncated approximation of the nonlinear diffusivities appearing in the bidomain equations. The existence proof is done using nondegenerate approximation systems and the Faedo-Galerkin method followed by a compactness argument.