Many Hitachi ABB Power Grids (HAPG) products vibrate and emit noise, but legal limitations of these parameters fuel the development of energy absorption methods used on products. Methods such as Constrained Layer Damping (CLD). Modern CLD research focuses on the application of micro-architectured structures in the damping layer and improving numerical models for streamlined product optimization. Focus was on developing a numerical computational homogenization, Finite Element Analysis (FEA), method for investigating the mechanical properties of a linear viscoelastic material model applied to a micro-architectured octet lattice. Computational homogenization of a Representative Volume Element (RVE) is performed in COMSOL Multiphysics 5.6. Initially, a linear elastic and then a linear viscoelastic material model applied to an octet lattice structure is homogenized for several lattice nominal densities. Linear isotropic viscoelastic bulk material properties were extracted from stress relaxation tests of SLA 3D-printed Formlabs Standard Black Methacrylate Resin measured in a Shimadzu AGS-X series universal test frame 20kN Table Top Model. Extracted properties were applied to a Prony Series code with one term, imitating a viscoelastic material of Standard Linear Solid (SLS) type. Recreated numerical relaxation tests validated the method of applying the viscoelastic material model in the computational model before applying the material model to an octet lattice structure. An eigenfrequency analysis was performed on SLA 3D-printed octetlattice blocks of the same Standard Resin. The computationally homogenized linear elastic octet lattice results were validated using analytical Euler-Bernoulli beam theory for lower lattice densities. Engineering constants E, G, and v analyzed from the homogenized viscoelastic octet lattices displayed a dependency on the nominal density agreeing with literature. The numerical viscoelastic relaxation time was also found to be independent of the nominal density. Experimental eigenfrequency data was also collected from three viscoelastic octet lattice blocks and is suggested to be compared with numerical results in a future study.

Brassica napus (AACC, 2n = 38) is an allotetraploid species derived from the parentel diploid species Brassica rapa (AA, 2n = 20) and Brassica oleracea (CC, 2n = 18). The aim of thesis was to carry out the genetic and epigenetic analysis of high-copy rRNA genes (or rDNA) in several varieties of hybrid species B. napus. The experiments involved determining the ratio of parental genes in hybrids, sequencing and methylation analysis of the promoter region of rDNA. Using Southern hybridization, it was revealed significant variability in the number of parental rDNA units between each variety. Data from sequence analysis were in good agreement with the results of Southern blot. Genetic recombination between parental rDNA units was revealed in one variety by DNA sequencing of promotor region. To study methylation, bisulfite sequencing was performed. It was found out that rDNA units of B. rapa origin have a higher value of methylation than units originated from B. oleracea.

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.

Cette thèse est consacrée principalement à l’étude de quelques problèmes elliptiques singuliers dans un domaine Ωɛ*, périodiquement perforé par des trous de taille ɛ. On montre l’existence et l’unicité d’une solution, pour tout ɛ fixé, ainsi que des résultats d’homogénéisation et correcteurs pour le problème singulier suivant :{█(-div (A (x/ɛ,uɛ)∇uɛ)=fζ(uɛ) dans Ωɛ*@uɛ=0 sur Γɛ0@@(A (x/ɛ,uɛ)∇uɛ)υ+ɛγρ (x/ɛ) h(uɛ)= ɛg (x/ɛ) sur Γɛ1@)┤Où l’on prescrit des conditions de Dirichlet homogènes sur la frontière extérieure Γɛ0 et des conditions de Robin non linéaires sur la frontière des trous Γɛ1. Le champ matriciel quasi linéaire A est elliptique, borné, périodique dans la primière variable et de Carathéodory. Le terme singulier non linéaire est le produit d’une fonction continue ζ (singulier en zéro) et de f, dont la sommabilité dépend de la croissance de ζ près de sa singularité. Le terme de bord non linéaire h est une fonction croissante de classe C1, ρ et g sont des fonctions périodiques non négatives avec sommabilité convenables. Pour étudier le comportement asymptotique du problème quand ɛ -> 0, on applique la méthode de l’éclatement périodique due à D. Cioranescu-A. Damlamian-G. Griso (cf. D. Cioranescu-A. Damlamian-P. Donato-G. Griso-R. Zaki pour les domaines perforés). Enfin, on montre l’existence et l’unicité de la solution faible pour la même équation, dans un domaine à deux composants Ω = Ω1 υ Ω2 υ Γ, étant Γ l’interface entre le composant connecté Ω1 et les inclusions Ω2. Plus précisément on considère{█(-div (A(x, u)∇u)+ λu=fζ(u) dans Ω\Γ,@u=0 sur δΩ@(A(x, u1)∇u1)υ1= (A(x, u2)∇u2)υ1 sur Γ,@(A(x, u1)∇u1)υ1= -h(u1-u2) sur Γ@)┤Où λ est un réel non négatif et h représente le coefficient de proportionnalité entre le flux de chaleur et le saut de la solution, et il est supposé être borné et non négatif sur Γ. / This thesis is mainly devoted to the study of some singular elliptic problems posed in perforated domains. Denoting by Ωɛ* e domain perforated by ɛ-periodic holes of ɛ-size, we prove existence and uniqueness of the solution , for fixed ɛ, as well as homogenization and correctors results for the following singular problem :{█(-div (A (x/ɛ,uɛ)∇uɛ)=fζ(uɛ) dans Ωɛ*@uɛ=0 sur Γɛ0@@(A (x/ɛ,uɛ)∇uɛ)υ+ɛγρ (x/ɛ) h(uɛ)= ɛg (x/ɛ) sur Γɛ1@)┤Where homogeneous Dirichlet and nonlinear Robin conditions are prescribed on the exterior boundary Γɛ0 and on the boundary of the holles Γɛ1, respectively. The quasilinear matrix field A is elliptic, bounded, periodic in the first variable and Carathéodory. The nonlinear singular lower order ter mis the product of a continuous function ζ (singular in zero) and f whose summability depends on the growth of ζ near its singularity. The nonlinear boundary term h is a C1 increasing function, ρ and g are periodic nonnegative functions with prescribed summabilities. To investigate the asymptotic behaviour of the problem, as ɛ -> 0, we apply the Periodic Unfolding Method by D. Cioranescu-A. Damlamian-G. Griso, adapted to perforated domains by D. Cioranescu-A. Damlamian-P. Donato-G. Griso-R. Zaki. Finally, we show existence and uniqueness of a weak solution of the same equation in a two-component domain Ω = Ω1 υ Ω2 υ Γ, being Γ the interface between the connected component Ω1 and the inclusions Ω2. More precisely we consider{█(-div (A(x, u)∇u)+ λu=fζ(u) dans Ω\Γ,@u=0 sur δΩ@(A(x, u1)∇u1)υ1= (A(x, u2)∇u2)υ1 sur Γ,@(A(x, u1)∇u1)υ1= -h(u1-u2) sur Γ@)┤Where ν1 is the unit external vector to Ω1 and λ a nonnegative real number. Here h represents the proportionality coefficient between the continuous heat flux and the jump of the solution and it is assumed to be bounded and nonnegative on Γ.

In this thesis we study a parameter identification problem for a stationary diffusion equation posed in heterogeneous media. This problem is closely related to the Calderón problem with anisotropic conductivities. The anisotropic case is particularly difficult and is ill-posed both in regards to uniqueness of solution and stability on the data. Since the present problem is posed in heterogeneous media, we can take advantage of multiscale modelling and the tools of homogenization theory in the study of the inverse problem, unlike the original Calderón problem. We investigate the possibilities of combining the theory of the Calderón problem with homogenization theory in order to obtain a well-posed parameter identification. We find that homogenization theory indeed can be used to make progress towards a well-posed identification of the diffusion coefficient. The success of the method is, however, dependent both on the precise structure of the heterogeneous media and on the modelling of the measurements in the invese problem framework. We have in mind a particular problem formulation which is motivated by an experiment to determine effective coefficients of materials used in food packaging. This experiment comes with a set of requirements on both the heterogeneous media and on the method for making measurements that, unfortunately, are in conflict with the currently available results for well-posedness. We study also an optimization approach to solving the inverse problem under these application specific requirements. Some progress towards well-posedness of the optimization problem is made by proving existence of minimizer, again with homogenization theory playing a key role in obtaining the result. In a proof-of-concept computational study this optimization approach is implemented and compared to two other optimization problems. For the two tested heterogeneous media, the only optimization method that manages to identify reasonably well the diffusion coefficient is the one which makes use of homogenization theory.

In dieser Arbeit wird für Strukturen, die im makroskopischen aufgrund unterschiedlicher Materialeigenschaften oder komplexer Geometrien eine hohe Netzfeinheit für Finite-Elemente-Berechnungen benötigen, eine neue Möglichkeit zur Berechnung effektiver Materialkennwerte vorgestellt. Durch einen modalen Ansatz, bei dem, je nach Struktur analytisch oder numerisch, mit Hilfe der modalen Kennwerte die Formänderungsenergie eines repräsentativen Volumens der Originalstruktur mit der Formänderungsenergie eines äquivalenten homogen Vergleichsvolumens verglichen wird, können effektive Materialkennwerte ermittelt und daran anschließend eine Finite-Elemente-Berechnung mit einem im Vergleich zum Originalmodell sehr viel gröberen Netz durchgeführt werden, was eine enorme Zeiteinsparung mit sich bringt. Weiterhin enthält die vorgestellte Methode die Möglichkeit, unsichere Eingabeparameter wie Geometrieabmessungen oder Materialkennwerte mit Hilfe der polynomialen Chaos Expansion zu approximieren, um Möglichkeiten zur Aussage bzgl. der daraus resultierenden Verteilungen modaler Kenngrößen auf eine schnelle und effektive Weise zu gewinnen.

This thesis is split into two parts, the first one is concerned with some problems in stochastic homogenization and the second addresses a problem in singular SPDEs. In the part on stochastic homogenization we are interested in developing large-scale regularity theories for random linear elliptic operators by using estimates for the homogenization error to transfer regularity from the homogenized operator to the heterogeneous one at large scales. In the whole-space case this has been done by Gloria, Neukamm, and Otto through means of a homogenization-inspired Campanato iteration. Here we are specifically interested in boundary regularity and as a model setting we consider random linear elliptic operators on the half-space with either homogeneous Dirichlet or Neumann boundary data. In each case we obtain a large-scale regularity theory and the main technical difficulty turns out to be the construction of a sublinear homogenization corrector that is adapted to the boundary data. The case of Dirichlet boundary data is taken from a joint work with Julian Fischer. In an attempt to head towards a percolation setting, we have also included a chapter concerned with the large-scale behaviour of harmonic functions on a domain with random holes assuming that these are 'well-spaced'. In the second part of this thesis we would like to provide a pathwise solution theory for a singular quasilinear parabolic initial value problem with a periodic forcing. The difficulty here is that the roughness of the data limits the regularity the solution such that it is not possible to define the nonlinear terms in the equation. A well-posedness result, therefore, comes with two steps: 1) Giving meaning to the nonlinear terms and 2) Showing that with this meaning the equation has a solution operator with some continuity properties. The solution theory that we develop in this contribution is a perturbative result in the sense that we think of the solution of the initial value problem as a perturbation of the solution of an associated periodic problem, which has already been handled in a work by Otto and Weber. The analysis in this part relies entirely on estimates for the heat semigroup. The results in the second part of this thesis will be in an upcoming joint work with Felix Otto and Jonas Sauer.

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.

The mathematical modelling of the reaction and diffusion mechanism of lipophilic toxic compounds in the mammalian cell is a challenging task because of its considerable complexity and variation in the architecture of the cell. The heterogeneity of the cell regarding the enzyme distribution participating in the bio-transformation, makes the modelling even more difficult. In order to reduce the complexity of the model, and to make it less computationally expensive and numerically treatable, Homogenization techniques have been used. The resulting complex system of Partial Differential Equations (PDEs), generated from the model in 2-dimensional axi-symmetric setting is implemented in Comsol Multiphysics. The numerical results obtained from the model show a nice agreement with the in vitro cell experimental results. The model can be extended to more complex reaction systems and also to 3-dimensional space. For the reduction of complexity and computational cost, we have implemented a model of mixed PDEs and Ordinary Differential Equations (ODEs). We call this model as Non-Standard Compartment Model. Then the model is further reduced to a system of ODEs only, which is a Standard Compartment Model. The numerical results of the PDE Model have been qualitatively verified by using the Compartment Modeling approach. The quantitative analysis of the results of the Compartment Model shows that it cannot fully capture the features of metabolic system considered in general. Hence we need a more sophisticated model using PDEs for our homogenized cell model. / Computational Modelling of the Mammalian Cell and Membrane Protein Enzymology

The issue of accurately determining the effective properties of composite materials has received the attention of numerous researchers in the last few decades and continues to be in the forefront of material research. Micromechanics models have been proven to be very useful tools for design and analysis of composite materials. In the present work, a versatile micromechanics modeling framework, namely, the Variational Asymptotic Method for Unit Cell Homogenization (VAMUCH), has been invented and various micromechancis models have been constructed in light of this novel framework. Considering the periodicity as a small parameter, we can formulate the variational statements of the unit cell through an asymptotic expansion of the energy functional. It is shown that the governing differential equations and periodic boundary conditions of mathematical homogenization theories (MHT) can be reproduced from this variational statement. Finally, we employed the finite element method to solve the numerical solution of the constrained minimization problem. If the local fields within the unit cell are of interest, the proposed models can also accurately recover those fields based on the global behavior. In comparison to other existing models, the advantages of VAMUCH are: (1) it invokes only two essential assumptions within the concept of micromechanics for heterogeneous material with identifiable unit cells; (2) it has an inherent variational nature and its numerical implementation is shown to be straightforward; (3) it calculates the different material properties in different directions simultaneously, which is more efficient than those approaches requiring multiple runs under different loading conditions; and (4) it calculates the effective properties and the local fields directly with the same accuracy as the fluctuation functions. No postprocessing calculations such as stress averaging and strain averaging are needed. The present theory is implemented in the computer program VAMUCH, a versatile engineering code for the homogenization of heterogeneous materials. This new micromechanics modeling approach has been successfully applied to predict the effective properties of composite materials including elastic properties, coefficients of thermal expansion, and specific heat and the effective properties of piezoelectric and electro-magneto-elastic composites. This approach has also been extended to the prediction of the nonlinear response of multiphase composites. Numerous examples have been utilized to clearly demonstrate its application and accuracy as a general-purpose micromechanical analysis tool.