Dans cette thèse on étudie quelques modèles macroscopiques pour des milieux conducteurs ou élastiques fortement hétérogènes et anisotropes obtenus par homogénéisation. Nous considérons le cas de l’homogénéisation périodique. En particulier pour le système de l’élasticité linéarisée modélisant les petites déformations d’un matériau fibré, nous étudions l’effet de l’anisotropie du matériau sur le modèle macroscopique et nous montrons que sous l’effet conjugué des conditions aux limites et de l’anisotropie des fibres, le système modélisant les déplacements à l’échelle macroscopique fait intervenir des termes non standard. Nous considérons plusieurs scalings et deux situations géométriques : dans la première le rayon des fibres cylindriques est du même ordre de grandeur que la taille de la période du milieu et dans la seconde la rayon est petit devant la période. Les résultats obtenus dans les deux cas, indépendants d’hypothèses de symétrie sur le matériau, permettent de retrouver les résultats déjà connus dans le cas de matériaux isotropes. / In this thesis we study some macroscopic models for drivers or elastic media highly heterogeneous and anisotropic obtained by homogenization. We consider the case of periodic homogenization. In particular the system of linearized elasticity modeling small deformations of a fiber material, we study the effect of material anisotropy on the macroscopic model and show that the combined effect of the boundary conditions and the anisotropy of the fiber system modeling movement at the macroscopic scale involves non-standard terms. We consider several scalings and two geometric situations: in the first radius of cylindrical fibers is of the same order of magnitude as the size of the middle period and in the second the radius is small compared to the period. The results obtained in both cases, independent of symmetry assumptions on the material used to find the results already known in the case of isotropic materials.

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In this dissertation we develop and analyze numerical method to solve general elliptic boundary value problems with many scales. The numerical method presented is intended to capture the small scales effect on the large scale solution without resolving the small scale details, which is done through the construction of a multiscale map. The multiscale method is more effective when the coarse element size is larger than the small scale length. To guarantee a numerical conservation, a finite volume element method is used to construct the global problem. Analysis of the multiscale method is separately done for cases of linear and nonlinear coefficients. For linear coefficients, the multiscale finite volume element method is viewed as a perturbation of multiscale finite element method. The analysis uses substantially the existing finite element results and techniques. The multiscale method for nonlinear coefficients will be analyzed in the finite element sense. A class of correctors corresponding to the multiscale method will be discussed. In turn, the analysis will rely on approximation properties of this correctors. Several numerical experiments verifying the theoretical results will be given. Finally we will present several applications of the multiscale method in the flow in porous media. Problems that we will consider are multiphase immiscible flow, multicomponent miscible flow, and soil infiltration in saturated/unsaturated flow.

The objective of the following research is to investigate the problem of information management and conveyed experience on the World Wide Web (WWW) when multi-modal sensors and media are available. After studying related areas of work about the web and heterogeneous media, it became apparent that one of the main challenges of the area is the semantic unification of heterogeneous media. This thesis will introduce an event-based model to semantically unify media. An event is defined as something of significance that takes place at a given time and location. Using this definition and the corresponding model, a system will be designed to illustrate practical use cases for events.

In this dissertation we develop and analyze numerical method to solve general elliptic boundary value problems with many scales. The numerical method presented is intended to capture the small scales effect on the large scale solution without resolving the small scale details, which is done through the construction of a multiscale map. The multiscale method is more effective when the coarse element size is larger than the small scale length. To guarantee a numerical conservation, a finite volume element method is used to construct the global problem. Analysis of the multiscale method is separately done for cases of linear and nonlinear coefficients. For linear coefficients, the multiscale finite volume element method is viewed as a perturbation of multiscale finite element method. The analysis uses substantially the existing finite element results and techniques. The multiscale method for nonlinear coefficients will be analyzed in the finite element sense. A class of correctors corresponding to the multiscale method will be discussed. In turn, the analysis will rely on approximation properties of this correctors. Several numerical experiments verifying the theoretical results will be given. Finally we will present several applications of the multiscale method in the flow in porous media. Problems that we will consider are multiphase immiscible flow, multicomponent miscible flow, and soil infiltration in saturated/unsaturated flow.

The use of Digital Image Correlation (DIC) technique has become increasingly popular for displacement measurements and for characterizing localized material deformation. In this study, a three-dimensional digital image correlation analysis (3D-DIC) was performed to investigate the displacements on the surface of isotropically consolidated and drained sand specimens during triaxial compression tests. The deformation of a representative volume of the material captured by 3D-DIC is used for the estimation of the kinematic and volumetric conditions of the specimen at different stages of deformation, combined with the readings of the global axial compression of the specimen. This allowed for the characterization of a Mohr-Coulomb plasticity model with hardening and softening laws. In addition, a two-dimensional axisymmetric finite element model was built to simulate the actual experimental conditions, including both the global and local kinematics effects captured by 3D digital image correlation analysis on the boundary of the specimen. A comparison between the axisymmetic model predictions and the experimental observations showed good agreement, for both the global and local behavior, in the case of different sand specimen configuration, including loose, dense and half-loose half-dense specimens.

In this dissertation, we develop multiscale finite element methods and uncertainty quantification technique for Richards' equation, a mathematical model to describe fluid flow in unsaturated porous media. Both coarse-level and fine-level numerical computation techniques are presented. To develop an accurate coarse-scale numerical method, we need to construct an effective multiscale map that is able to capture the multiscale features of the large-scale solution without resolving the small scale details. With a careful choice of the coarse spaces for multiscale finite element methods, we can significantly reduce errors. We introduce several methods to construct coarse spaces for multiscale finite element methods. A coarse space based on local spectral problems is also presented. The construction of coarse spaces begins with an initial choice of multiscale basis functions supported in coarse regions. These basis functions are complemented using weighted local spectral eigenfunctions. These newly constructed basis functions can capture the small scale features of the solution within a coarse-grid block and give us an accurate coarse-scale solution. However, it is expensive to compute the local basis functions for each parameter value for a nonlinear equation. To overcome this difficulty, local reduced basis method is discussed, which provides smaller dimension spaces with which to compute the basis functions. Robust solution techniques for Richards' equation at a fine scale are discussed. We construct iterative solvers for Richards' equation, whose number of iterations is independent of the contrast. We employ two-level domain decomposition pre-conditioners to solve linear systems arising in approximation of problems with high contrast. We show that, by using the local spectral coarse space for the preconditioners, the number of iterations for these solvers is independent of the physical properties of the media. Several numerical experiments are given to support the theoretical results. Last, we present numerical methods for uncertainty quantification applications for Richards' equation. Numerical methods combined with stochastic solution techniques are proposed to sample conductivities of porous media given in integrated data. Our proposed algorithm is based on upscaling techniques and the Markov chain Monte Carlo method. Sampling results are presented to prove the efficiency and accuracy of our algorithm.

Ce travail propose de nouvelles contributions aux méthodes d’homogénéisation avec des applications aux composite, aux polycristaux et aux milieux poreux. Les propriétés effectives sont déterminées en résolvant un problème élémentaire sur la cellule unitaire que l’on peut reformuler avec l’équation de Lippmann-Schwinger (LS). Celle-ci est résolue en utilisant des développements en série de Neumann. Plusieurs approches sont alors proposées pour calculer les différents termes de la série, en utilisant des approches analytiques ou numériques. Ainsi, dans les deux premiers chapitres, on établit une famille d’équation LS pour la polarisation dans le contexte de la conductivité thermique et de l’élasticité. L’opérateur de cette équation est optimisé afin d’obtenir la meilleur convergence de la série de Neumann et par conséquent la meilleur estimation des propriétés effectives du composite. L’estimation proposée est basée à la fois sur une série tronquée et une estimation du résidu de la série de Neumann. Le travail présenté au chapitre 3 concerne le calcul des propriétés de transport de masse en milieu poreux. De manière classique, la loi de filtration est donnée par la loi de Darcy à l’échelle macroscopique. Dans ce travail, on calcule les termes correctifs à l’équation de Darcy lorsque la condition de stricte séparation des échelles n’est pas vérifiée. Ces termes correctifs sont calculés numériquement en résolvant une équation LS et en utilisant un schéma itératif basé sur la transformée de Fourier Rapide (TFR). Finalement, au chapitre 4, on détermine numériquement des bornes pour les propriétés élastiques des polycristaux en utilisant toujours les approches basées sur la TFR. L’approche proposée permet de tenir compte de la géométrie exacte de la cellule de Voronoi en utilisant les expressions exactes des fonctions formes pour des polygones et des polyèdres. La méthode est appliquée à des polycristaux constitués de monocristaux cubiques / This work proposed some contributions to the homogenization methods with applications to composites materials, polycristals and porous media. The effective properties are determined by solving the unit cell problem and the corresponding Lippmann-Schwinger (LS) equation. The latter is solved by means of Neumann series. Different approaches are considered to evaluate each terms of the series using analytic or numerical approaches. In the first two chapters, we formulate a general class of LS equations for the polarization in the case of conductivity and then elasticity. The operator of the latter is optimized to obtain the best convergence of the associated Neumann series and then of the better estimate of the effective of the composite. The estimate is based on both a truncated Neumann series and an approximation of its residual. In chapter 3, we deal with the mass transport properties of porous media. Classically, the filtration law is given by the Darcy equation at the macroscopic scale. In the present work we compute the corrective terms of the Darcy equation in the situation of no strict scale separation. These corrective terms are determined numerically by solving a LS equation with a fast Fourier Transform (FFT) based iterative scheme. Finally, in chapter 4, we derivative numerically some bounds for the elastic properties of polycristals still by means of an FFT iterative scheme. The approach uses an exact description of the voronoi-unit cell geometry by using the shape functions of polygons and polyhedrons. The method is applied to polycristals constituted of cubic single crystals

In this thesis, the propagation of transient waves in heterogeneous media and the tuning of periodic elastic materials are studied. The behaviour of time harmonic waves in complex media is a well understood phenomenon. The primary aim of this text is to gain a deeper understanding into the propagation of transient waves in periodic media. The secondary aim is to explore the time harmonic behaviour of two dimensional pre-stressed elastic media and investigate the plausibility of band gap tuning. We begin this text by investigating the reflection of pulses from a semi-infinite set of point masses (we call 'beads') on a string. The reflected pulse is formulated using Fourier transforms which involve the harmonic reflection coefficient. We find that the reflected amplitude of a harmonic wave depends on its frequency. We then ask whether it is possible to find an effective reflection coefficient by assuming the beaded portion of the string is given by some effective homogeneous medium. An effective reflection coefficient is found by assuming the homogeneous medium has the wavenumber given by the infinite beaded string. This effective reflection coefficient is compared to the exact reflection coefficient found using the Wiener-Hopf technique. The results from studying the reflection problem gave inspiration to chapter 4, which focuses on the time dependent forcing of an infinite beaded string that is initially at rest. We again use the Fourier transform to find a time dependent solution. The z-transform is then used, after sampling the solution at the bead positions. We impose a sinusoidal loading which is switched on at a specified time. In doing this we are able to explore how the system behaves differently when excited in a stop band, a pass band and at a frequency on the edge between the two. An exact solution for the infinite beaded string is found at any point in time by expanding the branch points of the solution as a series of poles. We compare this exact solution to the long time asymptotics. The energy input into the system is studied with the results from the exact solution and long time approximation showing agreement. Interesting behaviour is discovered on the two edges between stop and pass bands. In chapter 5 the effect of a nonlinear elastic pre-stress on the wave band structure of a two dimensional phononic crystal is investigated. In this chapter we restrict ourselves to incompressible materials with the strain energy functions used being the neo-Hookean, Mooney-Rivlin and Fung. The method of small-on-large is used to derive the equation for incremental elastic waves and then the plane wave expansion method is used to find the band structure. Finally, chapter 6 focuses on the same geometry with a compressible elastic material. The strain energy function used is the one suggested by Levinson and Burgess. We use the theory of small-on-large to derive the incremental equations for coupled small amplitude pressure and shear waves in this material. In both compressible and incompressible materials we show how it is possible to control the stop bands in a material by applying a large elastic pre-stress.

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