In this thesis we investigate some partial differential equations with respect to G-convergence and homogenization. We study a few monotone parabolic equations that contain periodic oscillations on several scales, and also some linear elliptic and parabolic problems where there are no periodicity assumptions. To begin with, we examine parabolic equations with multiple scales regarding the existence and uniqueness of the solution, in view of the properties of some monotone operators. We then consider G-convergence for elliptic and parabolic operators and recall some results that guarantee the existence of a well-posed limit problem. Then we proceed with some classical homogenization techniques that allow an explicit characterization of the limit operator in periodic cases. In this context, we prove G-convergence and homogenization results for a monotone parabolic problem with oscillations on two scales in the space variable. Then we consider two-scale convergence and the homogenization method based on this notion, and also its generalization to multiple scales. This is further extended to the case that allows oscillations in space as well as in time. We prove homogenization results for a monotone parabolic problem with oscillations on two spatial scales and one temporal scale, and for a linear parabolic problem where oscillations occur on one scale in space and two scales in time. Finally, we study some linear elliptic and parabolic problems where no periodicity assumptions are made and where the coefficients are created by certain integral operators. Here we prove results concerning when the G-limit may be obtained immediately and is equal to a certain weak limit of the sequence of coefficients.

Triply Periodic Co-continuous piezocomposites, which consist of a ferroelectric-ceramic phase and an elastic-polymer phase continuously interconnected in three dimensions (3D), are emerging flexible piezoelectric materials with high efficiency in absorbing and converting multi-directional mechanical stimuli into electrical signals. Current co-continuous piezocomposites cannot be achieved with controlled piezoelectric properties due to the limited capability of traditional fabrication methods in carefully controlling the morphology of each phase, additive manufacturing such as Suspension-Enclosing Projection-Stereolithography process thus was selected. Porous ceramic skeleton with randomly distributed grain size is commonly observed in sintered ceramic skeleton fabricated by additive manufacturing. The effective piezoelectric properties of the piezocomposites were thus studied utilizing a two-scale method. Through analyzing the simulated results of different process parameters, optimal parameters of 3D printing processes including post-processes was subsequently suggested.

We study a two-scale coupled system consisting of a macroscopic elliptic equation and a microscopic parabolic equation. This system models the interplay between a gas and liquid close to equilibrium within a porous medium with distributed microstructures. We use formal homogenization arguments to derive the target system. We start by proving well-posedness and inverse estimates for the two-scale system. We follow up by proposing a Galerkin scheme which is continuous in time and discrete in space, for which we obtain well-posedness, a priori error estimates and convergence rates. Finally, we propose a numerical error reduction strategy by refining the grid based on residual error estimators.

Context. In computer graphics, realistic looking fluid is often desired. Simulating realistic fluids is a time consuming and computationally expensive task, therefore, much research has been devoted to reducing the simulation time while maintaining the realism. Two of the more recent optimization algorithms within particle based simulations are two-scale simulation and regional time stepping (RTS). Both of them are based on the predictive-corrective incompressible smoothed particle hydrodynamics (PCISPH) algorithm. Objectives. These algorithms improve on two separate aspects of PCISPH, two-scale simulation reduces the number of particles and RTS focuses computational power on regions of the fluid where it is most needed. In this paper we have developed and investigated the performance of an algorithm combining them, utilizing both optimizations. Methods. We implemented both of the base algorithms, as well as PCISPH, before combining them. Therefore we had equal conditions for all algorithms when we performed our experiments, which consisted of measuring the time it took to run each algorithm in three different scene configurations. Results. Results showed that our combined algorithm on average was faster than the other three algorithms. However, our implementation of two-scale simulation gave results inconsistent with the original paper, showing a slower time than even PCISPH. This invalidates the results for our combined algorithm since it utilizes the same implementation. Conclusions. We see that our combined algorithm has potential to speed up fluid simulations, but since the two-scale implementation was incorrect, our results are inconclusive.

By Banach algebra we mean Banach space enriched with a multiplication operation. It is a mathematical structure that is used in the non-periodic homogenization of composite materials. The theory of classical homogenization studies materials assuming the periodicity of the structure. For real materials, the assumption of a periodicity is not enough and is replaced by the so-called an abstract hypothesis based on a concept composed mainly of the spectrum of Banach algebra and Sigma convergence. This theory was introduced in 2004.

Numerické řešení matematických modelů popisujících chování materiálů s jemnou strukturou (kompozitní materiály, jemně perforované materiály, atp.) obvykle vyžaduje velký výpočetní výkon. Proto se při numerickém modelování původní materiál nahrazuje ekvivalentním materiálem homogenním. V této práci je k nalezení homogenizovaného materiálu použita dvojškálová konvergence založena na tzv. rozvinovacím operátoru (anglicky unfolding operator). Tento operátor poprvé použil J. Casado-Díaz. V disertační práci je operátor definován jiným způsobem, než jak uvádí původní autor. To dovoluje pro něj dokázat některé nové vlastnosti. Analogicky je definován operátor pro funkce definované na perforovaných oblastech a jsou dokázány jeho vlastnosti. Na závěr je rozvinovací operátor použit k nalezení homogenizovaného řešení speciální skupiny diferenciálních problémů s integrální okrajovou podmínkou. Odvozené homogenizované řešení je ilustrováno na numerických experimentech.

Mean-field games (MFGs) study the behavior of rational and indistinguishable agents in a large population. Agents seek to minimize their cost based upon statis- tical information on the population’s distribution. In this dissertation, we study the homogenization of a stationary first-order MFG and seek to find a numerical method to solve the homogenized problem. More precisely, we characterize the asymptotic behavior of a first-order stationary MFG with a periodically oscillating potential. Our main tool is the two-scale convergence. Using this convergence, we rigorously derive the two-scale homogenized and the homogenized MFG problems. Moreover, we prove existence and uniqueness of the solution to these limit problems. Next, we notice that the homogenized problem resembles the problem involving effective Hamiltoni- ans and Mather measures, which arise in several problems, including homogenization of Hamilton–Jacobi equations, nonlinear control systems, and Aubry–Mather theory. Thus, we develop algorithms to solve the homogenized problem, the effective Hamil- tonians, and Mather measures. To do that, we construct the Hessian Riemannian flow. We prove the convergence of the Hessian Riemannian flow in the continuous setting. For the discrete case, we give both the existence and the convergence of the Hessian Riemannian flow. In addition, we explore a variant of Newton’s method that greatly improves the performance of the Hessian Riemannian flow. In our numerical experiments, we see that our algorithms preserve the non-negativity of Mather mea- sures and are more stable than related methods in problems that are close to singular. Furthermore, our method also provides a way to approximate stationary MFGs.

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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.

Les travaux de cette thèse concernent l'homogénéisation d'équations de Maxwell harmoniques tridimensionnelles, modélisant la propagation d'une onde électromagnétique issue de la foudre, de l'air dans le matériau composite. La problématique des composites étant, par exemple en aéronautique, l'évacuation de la foudre et la protection contre les agressions électromagnétiques. Nous considérons une structure constituée de fibres de carbone incluses dans une résine époxy qui sera elle-même nano chargée. Rendant ainsi le composite électriquement conducteur. Afin d'obtenir le problème homogénéisé nous utilisons l'analyse asymptotique à deux échelles. Puis nous justifions mathématiquement le résultat par la convergence à deux échelles. La solution du champ électrique est approchée par l'addition du champ électrique moyen et le champ correcteur, dépendant de la microstructure, et solution des problèmes de cellule. Dans la deuxième partie, nous proposons une validation numérique du modèle simplifié en 2D via des simulations avec le logiciel libre d'éléments finis Freefem ++. Trois cas tests seront présentés avant de valider la méthode d'homogénéisation. Enfin, en guise d'illustration du modèle, deux exemples d'agressions électromagnétiques : l'arc en retour de foudre de type A et une impulsion électromagnétique nucléaire seront testées dans le domaine fréquentiel. / The work of this thesis concerns the homogenization of three-dimensional harmonic Maxwell equations, modeling the propagation of an electromagnetic wave originating from lightning, from air in the composite material. The problem of composites being, for example in aeronautics, the evacuation of the lightning and the protection against the electromagnetic aggressions. We consider a structure made of carbon fibers included in an epoxy resin which will itself be nano- charged. Thus rendering the composite electrically conductive. In order to obtain the homogenized problem, we use asymptotic analysis at two scales. Then we mathematically justify the result by two-scale convergence. The solution of the electric field is approximated by the addition of the average electric field and the correct field, depending on the microstructure, and solution of the cell problems. In the second part, we propose a numerical validation of the simplified model in 2D via simulations with the free finite element software Freefem ++. Three test cases will be presented before validating the homogenization method. Finally, as an illustration of the model, two examples of electromagnetic aggression: the Type A lightning bolt and a nuclear electromagnetic pulse will be tested in the frequency domain.