<p>The present thesis is devoted to the homogenization of certain elliptic and parabolic partial differential equations by means of appropriate generalizations of the notion of two-scale convergence. Since homogenization is defined in terms of H-convergence, we desire to find the H-limits of sequences of periodic monotone parabolic operators with two spatial scales and an arbitrary number of temporal scales and the H-limits of sequences of two-dimensional possibly non-periodic linear elliptic operators by utilizing the theories for evolution-multiscale convergence and λ-scale convergence, respectively, which are generalizations of the classical two-scale convergence mode and custom-made to treat homogenization problems of the prescribed kinds. Concerning the multiscaled parabolic problems, we find that the result of the homogenization depends on the behavior of the temporal scale functions. The temporal scale functions considered in the thesis may, in the sense explained in the text, be slow or rapid and in resonance or not in resonance with respect to the spatial scale function. The homogenization for the possibly non-periodic elliptic problems gives the same result as for the corresponding periodic problems but with the exception that the local gradient operator is everywhere substituted by a differential operator consisting of a product of the local gradient operator and matrix describing the geometry and which depends, effectively, parametrically on the global variable.</p>

The present thesis is devoted to the homogenization of certain elliptic and parabolic partial differential equations by means of appropriate generalizations of the notion of two-scale convergence. Since homogenization is defined in terms of H-convergence, we desire to find the H-limits of sequences of periodic monotone parabolic operators with two spatial scales and an arbitrary number of temporal scales and the H-limits of sequences of two-dimensional possibly non-periodic linear elliptic operators by utilizing the theories for evolution-multiscale convergence and λ-scale convergence, respectively, which are generalizations of the classical two-scale convergence mode and custom-made to treat homogenization problems of the prescribed kinds. Concerning the multiscaled parabolic problems, we find that the result of the homogenization depends on the behavior of the temporal scale functions. The temporal scale functions considered in the thesis may, in the sense explained in the text, be slow or rapid and in resonance or not in resonance with respect to the spatial scale function. The homogenization for the possibly non-periodic elliptic problems gives the same result as for the corresponding periodic problems but with the exception that the local gradient operator is everywhere substituted by a differential operator consisting of a product of the local gradient operator and matrix describing the geometry and which depends, effectively, parametrically on the global variable.

Este trabalho propõe uma técnica de modelagem multiescala concorrente do concreto considerando duas escalas distintas: a mesoescala, onde o concreto é modelado como um material heterogêneo, e a macroescala, na qual o concreto é tratado como um material homogêneo. A heterogeneidade da estrutura mesoscópica do concreto é idealizada considerando três fases distintas, compostas pelos agregados graúdos e argamassa (matriz), estes considerados materiais homogêneos, e zona de transição interfacial (ZTI), tratada como a parte mais fraca entre as três fases. O agregado graúdo é gerado a partir de uma curva granulométrica e posicionado na matriz de forma aleatória. Seu comportamento mecânico é descrito por um modelo constitutivo elástico-linear, devido a sua maior resistência quando comparado com as outras duas fases do concreto. Elementos finitos contínuos com alta relação de aspecto em conjunto com um modelo constitutivo de dano são usados para representar o comportamento não linear do concreto, decorrente da iniciação de fissuras na ZTI e posterior propagação para a matriz, dando lugar à formação de macrofissuras. Os elementos finitos de interface com alta relação de aspecto são inseridos entre todos os elementos regulares da matriz e entre os da matriz e agregados, representando a ZTI, tornando-se potenciais caminhos de propagação de fissuras. No estado limite, quando a espessura do elemento de interface tende a zero (h ?0) e, consequentemente, a relação de aspecto tende a infinito, estes elementos apresentam a mesma cinemática da aproximação contínua de descontinuidades fortes (ACDF), sendo apropriados para representar a formação de descontinuidades associados a fissuras, similar aos modelos coesivos. Um modelo de dano à tração é proposto para representar o comportamento mecânico não linear das interfaces, associado à formação de fissuras, ou até mesmo ao eventual fechamento destas. A fim de contornar os problemas causados pela malha de elementos finitos de transição entre as malhas da macro e da mesoescala, que, em geral, apresentam diferenças expressivas 5 de refinamento, utiliza-se uma técnica recente de acoplamento de malhas não conformes. Esta técnica é baseada na definição de elementos finitos de acoplamento (EFAs), os quais são capazes de estabelecer a continuidade de deslocamento entre malhas geradas de forma completamente independentes, sem aumentar a quantidade total de graus de liberdade do problema, podendo ser utilizados tanto para acoplar malhas não sobrepostas quanto sobrepostas. Para tornar possível a análise em multiescala em casos nos quais a região de localização de deformações não pode ser definida a priori, propõe-se uma técnica multiescala adaptativa. Nesta abordagem, usa-se a distribuição de tensões da escala macroscópica como um indicador para alterar a modelagem das regiões críticas, substituindo-se a macroescala pela mesoescala durante a análise. Consequentemente, a malha macroscópica é automaticamente substituída por uma malha mesoscópica, onde o comportamento não linear está na iminência de ocorrer. Testes numéricos são desenvolvidos para mostrar a capacidade do modelo proposto de representar o processo de iniciação e propagação de fissuras na região tracionada do concreto. Os resultados numéricos são comparados com os resultados experimentais ou com aqueles obtidos através da simulação direta em mesoescala (SDM). / A concurrent multiscale analysis of concrete is presented, in which two distinct scales are considered: the mesoscale, where the concrete is modeled as a heterogeneous material and the macroscale that treats the concrete as a homogeneous material. The mesostructure heterogeneities are idealized as three phase materials composed of the coarse aggregates, mortar matrix and the interfacial transition zone (ITZ). The coarse aggregates are generated from a grading curve and placed into the mortar matrix randomly. Their behavior is described using an elastic-linear constitutive model due to their significant higher strength when compared with the other two phases of the concrete. Special continuum finite elements with a high aspect ratio and a damage constitutive model are used to describe the nonlinear behavior associated to the propagation of cracks, which initiates in the ITZ and then propagates to the mortar matrix given place to a macro-crack formation. These interface elements with a high aspect ratio are inserted in between all regular finite elements of the mortar matrix and in between the mortar matrix and aggregate elements, representing the ITZ. In the limit case, when the thickness of interface elements tends to zero (h ?0) and consequently the aspect ratio tends to infinite, these elements present the same kinematics as the continuous strong discontinuity approach (CSDA), so that they are suitable to represent the formation of discontinuities associated to cracks, similar to cohesive models. A tensile damage model is proposed to model the nonlinear mechanical behavior of the interfaces, associated to the crack formation and also to the possible crack closure. To avoid transition meshes between the macro and the mesoscale meshes, a new technique for coupling non-matching meshes is used. This technique is based on the definition of coupling finite elements (CFEs), which can ensure the continuity of displacement between independent meshes, without increasing the total number of degrees of freedom of the problem. This technique can be used to couple non-overlapping and overlapping meshes.To make possible the concurrent multiscale analysis, where the strain localization region cannot be defined a priori, an adaptive multiscale model is proposed. In this approach the macroscale stress distribution is used as an indicator to properly change from the macroscale to the mesoscale modeling in the critical regions during the analysis. Consequently, the macroscopic mesh is automatically replaced by a mesoscopic mesh where the nonlinear behavior is imminent. A variety of tests are performed to show the ability of the proposed methodology in predicting the behavior of initiation and propagation of cracks in the tensile region of the concrete. The numerical results are compared with the experimental ones or with those obtained by the direct simulation in mesoscale (DSM).

Dans cette thèse nous introduisons une approche nouvelle pour la reconstruction d’un front d’ondes en Optique Adaptative (OA), à partir des données de gradients à basse résolution en provenance de l’analyseur de front d’ondes, et en utilisant une approche non-linéaire issue du Formalisme Multiéchelles Mi-crocanonique (FMM). Le FMM est issu de concepts établis en physique statistique, il est naturellement approprié à l’étude des propriétés multiéchelles des signaux naturels complexes, principalement grâce à l’estimation numérique précise des exposants critiques localisés géométriquement, appelés exposants de singularité. Ces exposants quantifient le degré de prédictabilité localement en chaque point du domaine du signal, et ils renseignent sur la dynamique du système associé. Nous montrons qu’une analyse multirésolution opérée sur les exposants de singularité d’une phase turbulente haute résolution (obtenus par modèle ou à partir des données) permet de propager, le long des échelles, les gradients en basse résolution issus de l’analyseur du front d’ondes jusqu’à une résolution plus élevée. Nous comparons nos résultats à ceux obtenus par les approches linéaires, ce qui nous permet de proposer une approche novatrice à la reconstruction de fronts d’onde en Optique Adaptative. / In this thesis, we introduce a new approach to wavefront phase reconstruction in Adaptive Optics (AO) from the low-resolution gradient measurements provided by a wavefront sensor, using a non-linear approach derived from the Microcanonical Multiscale Formalism (MMF). MMF comes from established concepts in statistical physics, it is naturally suited to the study of multiscale properties of complex natural signals, mainly due to the precise numerical estimate of geometrically localized critical exponents, called the singularity exponents. These exponents quantify the degree of predictability, locally, at each point of the signal domain, and they provide information on the dynamics of the associated system. We show that multiresolution analysis carried out on the singularity exponents of a high-resolution turbulent phase (obtained by model or from data) allows a propagation along the scales of the gradients in low-resolution (obtained from the wavefront sensor), to a higher resolution. We compare our results with those obtained by linear approaches, which allows us to offer an innovative approach to wavefront phase reconstruction in Adaptive Optics.

Atmospheric boundary layer flows are characterized by the coexistence of a broad range of scales. These scales cover from synoptic- (100-5000 km) and meso-scales (1-100 km) up to three-dimensional micro-scale turbulence (less than a few kilometers). This multiscale nature inherent to atmospheric flows clearly determines the behaviour of the atmospheric boundary layer, whose structure and evolution are of major importance for the wind energy community. This PhD thesis is focused on the development of a numerical methodology that allows to include contribution from all the above mentioned scales, with the purpose of improving the representation of boundary layer processes. The multiscale numerical methodology is developed based on a numerical weather prediction (NWP) model, the Weather Research and Forecasting (WRF) model.<p><p>Prior to the development of the multiscale numerical methodology, one-year of sonic anemometer and wind LiDAR measurements from the FINO1 offshore platform are analyzed. A comprehensive database of offshore measurements in the lowest 250 m of the boundary layer is developed after quality data check and correction for flow distortion effects by the measurement mast, allowing the characterization of the offshore conditions at FINO1. Spectral analysis of high frequency sonic anemometer measurements is used to estimate a robust averaing time for the turbulent fluxes that minimizes non-universal contributions from mesoscale structures but captures the contribution from boundary layer turbulence, employing the Ogive function concept. A stability classification of the measurements is carried out based on the Obukhov length. Results compare well to other surface layer observational studies while vertical wind speed profiles exhibit the expected stability-dependency.<p><p>Although NWP models have been extensively used for weather forecasting purposes, a comprehensive analysis of its suitability to meet the wind energy requirements needs to be carried out. The applicability of the WRF mesoscale model to reproduce offshore boundary layer characteristics is evaluated and validated against field measurements from FINO1. The ability of six planetary boundary layer (PBL) parameterizations to account for stability effects is analyzed. Overall, PBL parameterizations are rather accurate in reproducing the vertical structure of the boundary layer for convective and neutral stabilities. However, difficulties are found under stable stratifications, due to the general tendency of PBL formulations to be overdiffusive and therefore, not capable to develope the strong vertical gradients found in the observations. A low-level jet and a very shallow boundary layer cases are simulated to provide further insights into the limits of the parameterizations.<p><p>Large-eddy simulations (LES) based on averaged conditions from a convective episode at FINO1 are conducted to understand the mechanisms of transition and equilibration that occur in turbulent one-way nested simulations. The nonlinear backscatter and anisotropy subgrid scale model with a prognostic turbulent kinetic energy equation is found to be capable of providing similar results when performing one-way nested large-eddy simulations to a reference stand-alone domain using periodic lateral boundary conditions. A good agreement is obtained in terms of velocity shear and turbulent fluxes of heat and momentum, while velocity variances are overestimated. A considerable streamwise fetch is needed following each domain transition for appropriate energy levels to be reached at high wavelengths and for the solution to reach quasi-stationary results. A pile-up of energy is observed at low wavelengths on the first nested domain, mitigated by the inclusion of a second nested domain with higher resolution that allows the development of an appropriate turbulent energy cascade.<p><p>As the final step towards developing the multiscale capabilities of WRF, the specific problem of the transition from meso- to micro-scales in atmospheric models is addressed. The challenge is to generate turbulence on inner LES domain from smooth mesoscale inflow. Several new methods are proposed to trigger the development of turbulent features. The inclusion of adequate potential temperature perturbations near the inflow boundaries of the LES domain results in a very good agreement of mean velocity profiles, variances and turbulent fluxes, as well as velocity spectra, when compared to periodic stand-alone simulations. This perturbation method allows an efficient generation of fully developed turbulence and is tested under a broad range of atmospheric stabilities: convective, neutral and stable conditions, showing successful results in all the regimes. / Doctorat en Sciences de l'ingénieur / info:eu-repo/semantics/nonPublished

A l'heure actuelle, les nouvelles contraintes de la réglementation thermique en vigueur ne cessent de s'adapter au contexte économique global pour lequel la recherche d'une efficacité énergétique dans le bâtiment est devenue incontournable. Pour répondre à ces défis, des Matériaux intelligents à Changement de Phase (MCP) ont fait leur apparition sur le marché de la construction. Grâce à leur capacité de stockage de l'énergie, les MCP sont de plus en plus associés aux matériaux de construction classiques (béton, plâtre, etc.) afin d'améliorer leur inertie thermique et apporter un meilleur confort aux usagers. Pour ce faire, les propriétés thermophysiques intrinsèques aux MCP doivent être suffisamment maitrisées afin de pouvoir contrôler les propriétés du produit composite final. Dans ce contexte, cette thèse est une contribution ayant pour objectif de développer des méthodologies spécifiques pour une meilleure caractérisation des MCP et des béton-MCP. Une panoplie d'approches expérimentales a été présentée pour l'identification des propriétés thermophysiques des MCP et pour identifier l'effet d'incorporation et de l'endommagement de ces matériaux sur les propriétés thermiques et mécaniques de béton. Plusieurs modèles d'homogénéisation ont été utilisés afin de prédire le comportement thermique des bétons-MCP en utilisant les propriétés thermiques moyennées obtenues expérimentalement. Une étude probabiliste paramétrique a été menée afin de prendre en compte les incertitudes liées à la dispersion aléatoire des mesures expérimentales de propriétés thermiques du béton-MCP. Les résultats issus des essais expérimentaux ont été intégrés dans le cadre d'une étude numérique par la Méthode des Volumes finis (MVF) afin d'étudier le mécanisme de transfert de chaleur à travers une paroi en béton-MCP / The thermal policies have been kept to fit the new economic in a global context particularly in terms of buildings energy efficiency. To meet these challenges, different technologies have been used such as the Phase Change Materials (PCMs) which have the ability to store and release energy. PCMs are generally used with conventional building materials in order to improve their thermal inertia and provide better comfort to users. To enhance the properties of the final composite, the PCMs thermo-physical properties must be sufficiently controlled. In this context, this thesis is a contribution aimed to develop specific methodologies for better characterization of PCM and PCM-concrete. Different experimental approaches will be presented for the identification of PCMs thermophysical properties and to identify the effect of the incorporation and the damage of these materials on the thermal and mechanical properties of concrete. A multiscale modelling considering the average of experimental thermal properties was applied to predict the thermal behaviour of PCMs-concrete. A probabilistic study of experimental uncertainties will be also conducted to assess the level of confidence of the impact of PCM on the thermodynamic properties of PCM-concrete. A numerical study was conducted using experimental data to study the heat transfer through a PCM-concrete wall

In this thesis we consider two great challenges in computer simulations of partial differential equations: multiscale data, varying over multiple scales in space and time, and data uncertainty, due to lack of or inexact measurements. We develop a multiscale method based on a coarse scale correction, using localized fine scale computations. We prove that the error in the solution produced by the multiscale method decays independently of the fine scale variation in the data or the computational domain. We consider the following aspects of multiscale methods: continuous and discontinuous underlying numerical methods, adaptivity, convection-diffusion problems, Petrov-Galerkin formulation, and complex geometries. For uncertainty quantification problems we consider the estimation of p-quantiles and failure probability. We use spatial a posteriori error estimates to develop and improve variance reduction techniques for Monte Carlo methods. We improve standard Monte Carlo methods for computing p-quantiles and multilevel Monte Carlo methods for computing failure probability.

The field of modelling and processing of 'images' has fairly recently become important, even crucial, to areas of science, medicine, and engineering. The inevitable explosion of imaging modalities and approaches stemming from this fact has become a rich source of mathematical applications. <br /><br /> 'Imaging' is quite broad, and suffers somewhat from this broadness. The general question of 'what is an image?' or perhaps 'what is a natural image?' turns out to be difficult to address. To make real headway one may need to strongly constrain the class of images being considered, as will be done in part of this thesis. On the other hand there are general principles that can guide research in many areas. One such principle considered is the assertion that (classes of) images have multiscale relationships, whether at a pixel level, between features, or other variants. There are both practical (in terms of computational complexity) and more philosophical reasons (mimicking the human visual system, for example) that suggest looking at such methods. Looking at scaling relationships may also have the advantage of opening a problem up to many mathematical tools. <br /><br /> This thesis will detail two investigations into multiscale relationships, in quite different areas. One will involve Iterated Function Systems (IFS), and the other a stochastic approach to reconstruction of binary images (binary phase descriptions of porous media). The use of IFS in this context, which has often been called 'fractal image coding', has been primarily viewed as an image compression technique. We will re-visit this approach, proposing it as a more general tool. Some study of the implications of that idea will be presented, along with applications inferred by the results. In the area of reconstruction of binary porous media, a novel, multiscale, hierarchical annealing approach is proposed and investigated.

This thesis mainly deals with questions concerning the convergence of some sequences of elliptic and parabolic linear and non-linear operators by means of G-convergence and homogenization. In particular, we study operators with oscillations in several spatial and temporal scales. Our main tools are multiscale techniques, developed from the method of two-scale convergence and adapted to the problems studied. For certain classes of parabolic equations we distinguish different cases of homogenization for different relations between the frequencies of oscillations in space and time by means of different sets of local problems. The features and fundamental character of two-scale convergence are discussed and some of its key properties are investigated. Moreover, results are presented concerning cases when the G-limit can be identified for some linear elliptic and parabolic problems where no periodicity assumptions are made.

Des matériaux poreux à architecture complexe et de couleur blanche ont été synthétisés, en combinant la physico-chimie des fluides complexes (émulsions, mésophase lyotropes) avec la chimie sol-gel. Ce procédé est connu sous le nom de chimie intégrative. En contrôlant la taille des objets diffusants (diamètres des pores) et en augmentant l’indice de réfraction, nous souhaitons augmenter le caractère diffusant de ces matériaux, générant ainsi diffusion et localisation de la lumière. Toutes les caractérisations structurales et optiques ont été réalisées. En utilisant des modèles physiques, nous avons analysé les résultats et obtenu les paramètres critiques de transport (transport moyen, longueur d’onde d’adsorption et constante du diffusion). Ces matériaux présentent un fort comportement multidiffusif et éventuellement de localisation de la lumière. Ces matériaux très diffusants sont des candidats pour la génération de lasers aléatoires. Dans cette optique, nous les avons infiltrés avec de la rhodamine-6G (chromophores) et quantifié leurs propriétés comme lasers aléatoires. / Disordered, porous, white, hierarchical materials have been synthesized using a sol-gel process combined with the physical chemistry of complex fluids (emulsion, lyotrope mesophase). The whole process is known as integrative chemistry. By tuning the size of the scatters (pore diameters) and increasing the refractive index contrast, we want to increase the scattering strength of our materials, thus promoting light scattering/localization. The structural and optical characterizations have been performed. By using well established theories, we have analyzed our results and obtain the transport parameters (transport mean free path, absorption length and diffusion constant). The materials exhibit a strong multiple-diffusive behavior and an eventual localization of light. These strongly scattering materials would be of potential interest for random lasing applications. Therefore, we infiltrated them with Rhodamine 6G laser dyes and quantified their random lasing performances.