This thesis is divided into two parts: In the first one (Chapters 1 and 2), we deal with problems arising from quantitative homogenization of the random elliptic operator in divergence form $-\\nabla \\cdot a \\nabla$. In Chapter 1 we study existence and stochastic bounds for the Green function $G$ associated to $-\\nabla \\cdot a \\nabla$ in the case of systems. Without assuming any regularity on the coefficient field $a= a(x)$, we prove that for every (measurable) uniformly elliptic tensor field $a$ and for almost every point $y \\in \\mathbb{R}^d$, there exists a unique Green\''s function centred in $y$ associated to the vectorial operator $-\\nabla \\cdot a\\nabla $ in $\\mathbb^d$, $d> 2$. In addition, we prove that if we introduce a shift-invariant ensemble $\\langle\\cdot \\rangle$ over the set of uniformly elliptic tensor fields, then $\\nabla G$ and its mixed derivatives $\\nabla \\nabla G$ satisfy optimal pointwise $L^1$-bounds in probability. Chapter 2 deals with the homogenization of $-\\nabla \\cdot a \\nabla$ to $-\\nabla \\ah \\nabla$ in the sense that we study the large-scale behaviour of $a$-harmonic functions in exterior domains $\\$ by comparing them with functions which are $\\ah$-harmonic. More precisely, we make use of the first and second-order correctors to compare an $a$-harmonic function $u$ to the two-scale expansion of suitable $\\ah$-harmonic function $u_h$. We show that there is a direct correspondence between the rate of the sublinear growth of the correctors and the smallness of the relative homogenization error $u- u_h$. The theory of stochastic homogenization of elliptic operators admits an equivalent probabilistic counterpart, which follows from the link between parabolic equations with elliptic operators in divergence form and random walks. This allows to reformulate the problem of homogenization in terms of invariance principle for random walks. The second part of thesis (Chapters 3 and 4) focusses on this interplay between probabilistic and analytic approaches and aims at exploiting it to study invariance principles in the case of degenerate random conductance models and systems of interacting particles. In Chapter 3 we study a random conductance model where we assume that the conductances are independent, stationary and bounded from above but not uniformly away from $0$. We give a simple necessary and sufficient condition for the relaxation of the environment seen by the particle to be diffusive in the sense of every polynomial moment. As a consequence, we derive polynomial moment estimates on the corrector which imply that the discrete elliptic operator homogenises or, equivalently, that the random conductance model satisfies a quenched invariance principle. In Chapter 4 we turn to a more complicated model, namely the symmetric exclusion process. We show a diffusive upper bound on the transition probability of a tagged particle in this process. The proof relies on optimal spectral gap estimates for the dynamics in finite volume, which are of independent interest. We also show off-diagonal estimates of Carne-Varopoulos type.

This thesis covers the basics in the stochastic homogenization of elliptic partial differential equations, from underlying theory up to numerical ap- proaches. In particular, we introduce and analyze a combination of the Fourier-Galerkin method in the spatial domain with a collocation method in the stochastic domain. The material coefficients are assumed to depend on a finite number of random variables. We present a comparison of the Monte Carlo method with the full tensor grid and sparse grid collocation method for two applications. The first one is the checkerboard problem with continuous random variables, the other considers the material coefficients to be described in terms of an autocorrelation function.

In a Hilbert space setting homogenization of evolutionary equations is discussed. In order to do so, a suitable topology on material laws is introduced and several properties of that topology are shown. With those properties homogenization theorems of a large class of linear evolutionary problems of classical mathematical physics can be obtained. The results are exemplified by the equations of piezo-electro-magnetism.

Diese Arbeit liefert ein dreidimensionales numerisches Homogenisierungskonzept, welches beim Übergang von der Mikro- zur Makroskala einen Wechsel in der Kontinuumsbeschreibung beinhaltet. Während für die Beschreibung der Makroskala das verallgemeinerte Cosserat-Kontinuum verwendet wird, basiert die Mikroskala auf der klassischen Cauchy-Theorie. Um das homogene Cosserat-Ersatzmaterial im Rahmen numerischer Simulationen nutzen zu können, erfolgt die Implementierung geeigneter Finiter Elemente in das Programmsystem Abaqus und deren Verifikation. Neben der Diskussion der bei der Homogenisierung beobachteten Effekte werden anhand eines idealisierten Modells eines biaxialverstärkten Mehrlagengestrickes die Vorteile gegenüber der klassischen Herangehensweise aufgezeigt. / This contribution provides a threedimensional homogenization approach which includes the switch of the continuum theory during the scale transition. Whereas the microscopic scale is described in the framework of the classical Cauchy theory, the macroscopic scale is based on the generalized Cosserat continuum. In order to use the obtained homogeneous Cosserat material, suitable finite elements are implemented in the commercial program system Abaqus followed by an appropriate verification. Beside the discussion of the arising effects the advantages of this approach compared to the classical procedure are shown by means of an idealized model of a biaxial woven fabric.

Ce mémoire aborde les questions relatives à la mise en place de procédures de conception rapide, fiable et automatisée des volumes élémentaires représentatifs (VER) d’un matériau composite à microstructure complexe (matrice/inclusions), et de la détermination de leurs propriétés homogénéisées ou effectives. Nous avons conçu et développé des algorithmes conduisant à des outils efficaces permettant la génération aléatoire de tels matériaux à inclusions sphériques, cylindriques, elliptiques ou toute combinaison de celles-ci. Ces outils sont également capables d’altérer les inclusions : inflation, déflation, arrachements aléatoires, ondulation et de les pelliculer permettant ainsi de générer des VER s’approchant des matériaux composites fabriqués. Un soin particulier a été porté sur la génération de VER périodiques. Les caractéristiques homogénéisées ou propriétés effectives de matériaux constitués de tels VER périodiques peuvent alors être déterminées selon le principe d’homogénéisation périodique, soit par une méthode basée sur un schéma itératif utilisant la FFT (Transformation de Fourier Rapide) via l’équation de Lippmann-Schwinger, soit par une méthode d’éléments finis. Le caractère aléatoire de la génération nous amène à réaliser des études en moyenne à partir d’un ensemble de paramètres morphologiques déterminé : nombre d’inclusions, type et forme, fraction volumique, orientation des inclusions, prise en compte d’une éventuelle altération. Deux études particulières sur la conductivité thermique apparente ont été menées, la première sur les composites à inclusions sphériques pelliculées de façon à déterminer l’influence de l’épaisseur de la pellicule et la seconde sur les composites de type stratifié en polymère et fibre de carbone, cousu par un fil de cuivre pour évaluer l'apport de la couture en cuivre selon la fibre de carbone utilisée. / This thesis focuses on setting up of fast, reliable and automated approaches to design representative volume elements (RVE) of composite materials with complex microstructures (matrix/inclusions) and the evaluation of their effective properties via a homogenization process. We developed algorithms and efficient tools for the random generation of such materials. Inclusions shapes may be spherical, cylindrical, elliptical or any combinations of them. Inflation, deflation, dislocation, undulation and coating are also available to generate RVE. The aim is to approach realistic materials subjected to be damaged during production. Particular attention has been focused on the periodic RVE generation.The homogenized characteristics or effective properties of materials formed from such periodic RVE may then be determined according to the principle of periodic homogenization, by an iterative scheme using FFT (Fast Fourier Transform) via the integral Lippmann-Schwinger or by a finite elements method.The stochastic generation of RVE and the set of morphological parameters studied: number of inclusions, type and shape, volume fraction, orientation of the inclusions lead to achieve an average process. Moreover, a special study has been led to take into account the behavior of altered inclusions. Furthermore, we studied two particular cases on the apparent thermal conductivity of the composite, the first for coated spherical inclusions in order to determine the influence of the layer thickness and the second for laminated polymer and carbon fiber composite sewn by a copper wire, in order to determine the influence of the sewing contribution according to the carbon fiber used.

Many biological populations reside in increasingly fragmented landscapes, which arise from human activities and natural causes. Landscape characteristics may change abruptly in space and create sharp transitions (interfaces) in landscape quality. How patchy landscape affects ecosystem diversity and stability depends, among other things, on how individuals move through the landscape. Individuals adjust their movement behaviour to local habitat quality and show preferences for some habitat types over others. In this dissertation, we focus on how landscape composition and the movement behaviour at an interface between habitat patches of different quality affects the steady states of a single species and a predator-prey system. First, we consider a model for population dynamics in a habitat consisting of two homogeneous one-dimensional patches in a coupled ecological reaction-diffusion equation. Several recent publications by other authors explored how individual movement behaviour affects population-level dynamics in a framework of reaction-diffusion systems that are coupled through discontinuous boundary conditions. The movement between patches is incorporated into the interface conditions. While most of those works are based on linear analysis, we study positive steady states of the nonlinear equations. We establish the existence, uniqueness and global asymptotic stability of the steady state, and we classify their qualitative shape depending on movement behaviour. We clarify the role of nonrandom movement in this context, and we apply our analysis to a previous result where it was shown that a randomly diffusing population in a continuously varying habitat can exceed the carrying capacity at steady state. In particular, we apply our results to study the question of why and under which conditions the total population abundance at steady state may exceed the total carrying capacity of the landscape. Secondly, we model population dynamics with a predator-prey system in a coupled ecological reaction-diffusion equation in a heterogeneous landscape to study Turing patterns that emerge from diffusion-driven instability (DDI). We derive the DDI conditions, which consist of necessary and sufficient conditions for initiation of spatial patterns in a one-dimensional homogeneous landscape. We use a finite difference scheme method to numerically explore the general conditions using the May model, and we present numerical simulations to illustrate our results. Then we extend our studies on Turing-pattern formation by considering a predator-prey system on an infinite patchy periodic landscape. The movement between patches is incorporated into the interface conditions that link the reaction-diffusion equations between patches. We use a homogenization technique to obtain an analytically tractable approximate model and determine Turing-pattern formation conditions. We use numerical simulations to present our results from this approximation method for this model. With this tool, we then explore how differential movement and habitat preference of both species in this model (prey and predator) affect DDI.

In numerical multiscale methods, one relies on a coupling between macroscopic model and a microscopic model. The macroscopic model does not include the microscopic properties that the microscopic model offers and that are vital for the desired solution. Such microscopic properties include parameters like material coefficients and fluxes which may variate microscopically in the material. The effective values of this data can be computed by running local microscale simulations while averaging the microscopic data. One desires the effect of the microscopic coefficients on a macroscopic scale, and this can be done using classical homogenisation theory. One method in the homogenization theory is to use local elliptic cell problems in order to compute the homogenized constants and this results in <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Clambda%20/R" data-classname="equation_inline" data-title="" /> error where <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Clambda" data-classname="equation" /> is the wavelength of the microscopic variations and <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?R" data-classname="mimetex" data-title="" /> is the size of the simulation domain. However, one could greatly improve the accuracy by a slight modification in the homogenisation elliptic PDE and use a filter in the averaging process to get much better orders of error. The modification relates the elliptic PDE to a parabolic one, that could be solved and integrated in time to get the elliptic PDE's solution.   In this thesis I apply the modified elliptic cell homogenization method with a qth order filter to compute the homogenized diffusion constant in a 2d Poisson equation on a rectangular domain. Two cases were simulated. The diffusion coefficients used in the first case was a deterministic 2d matrix function and in the second case I used stochastic 2d matrix function, which results in a 2d stochastic differential equation (SDE). In the second case two methods were used to determine the expected value of the homogenized constants, firstly the multi-level Monte Carlo (MLMC) and secondly its generalization multi-index Monte Carlo (MIMC). The performance of MLMC and MIMC is then compared when used in the process of the homogenization.   In the homogenization process the finite element notations in 2d were used to estimate a solution of the Poisson equation. The grid spatial steps were varied in a first order differences in MLMC (square mesh) and first order mixed differences in MIMC (which allows for rectangular mesh).

No description available.

The packaging does not serve as a mere containment but also can be designed to play a key role in preserving the product from quality-deteriorating factors, including oxygen, light irradiation, and foodborne pathogenic microorganisms (e.g., Escherichia coli). There has been a growing interest in employing ultra-porous metal-organic frameworks (MOF) with visible light-responsive antibacterial mechanisms to generate reactive oxygen species (ROS) that can eliminate bacteria via an oxidative burst. MOF is made of inorganic metal ions/nodes/clusters/secondary building units linked by organic bridge ligands, where titanium dioxide (TiO2) and tetrakis(4-carboxyphenyl)porphyrin) (TCPP) were selected for these components, respectively. TiO2 is an exceptional UV-A/B/C-blocker; meanwhile, TCPP dye performs a remarkable photocatalytic ability even under visible light, on top of its macro-heterocyclic structure that is ideal as a MOF linker. Both have good compatibility but suffer from the notorious tendency to self-quench/aggregate. The incorporation of MOF-based conjugates into a polymeric matrix, like cellulose, is among the proven-successful solutions. Cellulose is the Earth's most abundant and naturally biodegradable, and cellulose nanofibril (CNF) was particularly chosen for its high specific surface area and surface activity. However, a straightforward, cheap, and environmentally friendly approach of multicycle homogenization (0-25 passes) was conducted to solve neat cellulose's challenge of natural hydrophilicity, where low pressure (<10 MPa) was applied to prevent the common over-shearing effect. The antibacterial efficacy of CNF films functionalized with TiO2-TCPP conjugate on inhibiting E. coli growth was analyzed with and without light of different intensities (3000 and 6000 lux). The positive impacts of CNFs' promoted fibrillation and subsequent inter/intra-molecular hydrogen bonding post-homogenization were evidenced in an array of functional properties, i.e., crystallinity, TiO2-TCPP conjugate dispersion, surface smoothness, mechanical properties, thermal stability, hydrophobicity, oxygen barrier (comparable to ethylene-vinyl alcohol (EVOH), a commercial high-barrier polymer), and 100%-antibacterial rate (under 6000 lux after 72 hours). Varying optimum cycles of homogenization demonstrated the prospect of the proposed homogenization approach in preparing CNF with diverse processability and applicability. These findings also exhibited a promising potential for a myriad of high-barrier, UV-blocking, and/or visible light-responsive antibacterial film applications, including food packaging and biomedical. / Doctor of Philosophy / Packaging is useful not only as a container but can also be designed to help prevent products from being spoiled due to various reasons such as oxidation, light, and bacterial contamination. Researchers have discovered the promising antibacterial feature of the metal-organic framework (MOF). Packaging made with MOF technology can harness light and oxygen in the environment to produce a special form of oxygen called reactive oxygen species (ROS) that can kill unwanted bacteria. MOF is an extremely porous sponge-like material made of two ingredients: an inorganic metal cluster and an organic linker; in this study, titanium dioxide (TiO2) and a porphyrin called TCPP were selected, respectively. TiO2 is an excellent ultraviolet blocker, while TCPP has a unique, ring-like geometry that is ideal for use as a linker and an antimicrobial feature that works well under the visible light spectrum. The pair are compatible but still suffer from MOF's notorious challenge, where it tends to clump together because of its tiny size. To resolve this problem, TiO2-TCPP MOF can be deposited evenly in a cast made of polymer. Cellulose has been proven to work effectively as a polymeric cast; moreover, it is natural, biodegradable, and in abundant supply. A type of nanosized cellulose—cellulose nanofibril (CNF)—was specifically chosen because its high surface area and activity are useful when blended with other materials. However, cellulose is naturally a poor water-repellant that is not ideal for packaging applications. As a solution, cellulose can be treated with a homogenization technique by passing the material through a very narrow hole under high pressure. Homogenization can be problematic as it possibly damages the cellulose's structure, and its high pressure can also be expensive and energy consuming. Therefore, low pressure with multiple cycles was applied in this work. CNF-TiO2-TCPP films were tested for their ability to slow down E. coli bacteria growth with and without light of varying brightness to compare its light-sensitive antimicrobial feature. Homogenization was found helpful in producing higher-quality CNF, which improved several of the film's final characteristics, including an even material dispersion, structural order, smoothness, strength, heat resistance, and water repellency. Most importantly, it produced films with oxygen barrier ability comparable to commercial high-barrier plastics and completely eliminated bacteria after 72 hours. The optimum number of homogenization cycles was found to be dependent on the desired characteristics and application. Overall, these findings carry a promising potential for a variety of applications, including food packaging and the biomedical field.

Many geologic processes occur in association with hydrothermal fluids and some of these fluids are eventually trapped as fluid inclusions in minerals formed during the process. Fluid inclusions provide valuable information on the pressure, temperature and fluid composition (PTX) of the environment of formation, hence understanding PTX properties of the fluid inclusions is required. The most important step of a fluid inclusion study is the identification of Fluid Inclusion Assemblages (FIA) that represent the finest (shortest time duration) geologic event that can be constrained using fluid inclusions. Homogenization temperature data obtained from fluid inclusions is often used to reconstruct temperature history of a geologic event. The precision with which fluid inclusions constrain the temperatures of geologic events depends on the precision with which the temperature of a fluid inclusion assemblage can be determined. Synthetic fluid inclusions trapped in the one-fluid-phase field are formed at a known and relatively constant temperature. However, microthermometry of synthetic fluid inclusions often reveals Th variations of about ± 1- 4 degrees Centigrade, or one order of magnitude larger than the precision of the measurement for an individual inclusion. The same range in Th was observed in well-constrained natural FIAs where the inclusions are assumed to have been trapped at the same time. The observed small variations are the result of the effect of the fluid inclusion size on the bubble collapsing temperature. As inclusions are heated the vapor bubble is getting smaller until the pressure difference between the pressure of the vapor and the confining pressure reaches a critical value and the bubble collapses. It was observed that smaller inclusions reach critical bubble radius and critical pressure differences at lower temperatures than larger inclusions within the same FIA. Homogenization temperature (Th) variations depend on many factors that vary within different geological environments. In order to determine minimum and acceptable Th ranges fro FIAs formed in different environments we investigated several geologic environments including sedimentary, metamorphic, and magmatic hydrothermal environments. The observed minimum Th ranges range from 1-4 degrees Centigrade and acceptable Th range from 5-25 degrees Centigrade. The variations are mostly caused by the fluid inclusion size, natural temperature and pressure fluctuations during the formation of an FIA and reequilibration after trapping. Fluid inclusions containing H₂O-CO₂-NaCl are common in many geologic environments and knowing the salinity of these inclusions is important to interpret PVTX properties of the fluids. A technique that combines Raman spectroscopy and microthermometry of individual inclusions was developed to determine the salinity of these inclusions. In order to determine the salinity, the pressure and temperature within the inclusion must be known. The pressure within the inclusions is determined using the splitting in the Fermi diad of the Raman spectra of the CO₂ at the clathrate melting temperature. Applying the technique with to synthetic fluid inclusions with known salinity suggests that the technique is valid and useable to determine salinity of H₂O-CO₂-NaCl fluid inclusions with unknown salinity. / Ph. D.