Böhm, Michael
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Böhm, Michael
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Böhm, Michael
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Item-typ:Veröffentlichung, Two-scale models for reactive transport and evolving microstructure(2008-06-26); ; ; Reactive transport in materials with a complex microstructure is a phenomenon that frequently occurs in nature and in technical applications. An example are chemical reactions taking place in the pores of concrete structures. Since these processes are highly influenced by the geometry of the microstructure, it is a difficult task to understand and predict their behaviour on a macroscopic scale. It has been shown by periodic homogenisation that for certain situations two-scale models are appropriate. Periodic homogenisation is an averaging method that is limited to a uniform and constant microstructure. However, in general a porous medium contains multiple phases that evolve in time. For instance, the air-water mixture in an unsaturated porous medium varies due to chemical reactions or non-constant boundary conditions. In certain applications, such effects need to be captured in a mathematical model. In this thesis, two-scale models are formulated for reactive transport in a porous medium with an evolving phase configuration. The reaction-diffusion problem is motivated by a chemical degradation process in concrete structures. The model equations are formally derived from a pore-scale model by homogenisation, but are defined on a time-dependent geometry. Two different settings are considered: In the first one, the evolution of the pore geometry is independent of the reaction-diffusion process and can be considered as a-priori given. This leads to a semilinear two-scale system of parabolic PDEs. In the second setting, the pore water evolves due to the reaction itself, which results in a quasilinear model. Using a transformation to a reference configuration, it is proven that both two-scale models are well-posed. A simple numerical implementation that works with both settings is suggested and validated. The qualitative behaviour of the solutions of the two-scale models are illustrated for an academic example. In summary, the results give a solid theoretical fundament for further important tasks such as the identification of model parameters or the design of efficient numerical algorithms suitable for real-world applications.Dissertation436 529 - Some of the metrics are blocked by yourconsent settings
Item-typ:Veröffentlichung, Derivation of boundary conditions at a curved contact interface between a free fluid and a porous medium via homogenisation theory(2009-09-14); In soil chemistry or marine microbiology (for example when dealing with marine aggregates), one often encounters situations where porous bodies are suspended in a fluid. In this context, the question of boundary conditions for the fluid velocity and pressure at the porous-liquid interface arises. Up to the present, only results for straight interfaces are known. In this work, the behaviour of a free fluid above a porous medium is investigated, where the interface between the two flow regions is assumed to be curved. By carrying out a coordinate transformation, we obtain the description of the flow in a domain with a straight boundary. We assume the geometry in this domain to be epsilon-periodic. Using periodic homogenisation, the effective behaviour of the solution of the transformed partial differential equations in the porous part is obtained, yielding a Darcy law with a non-constant permeability matrix. The boundary layer approach of Jäger and Mikelic is then generalized to construct corrections at the interface. Finally, this allows us to obtain the fluid behaviour at the porous-liquid interface: Whereas the velocity in normal direction is continuous over the interface, a jump appears in tangential direction. The magnitude of this jump can explicitely be calculated and seems to be related to the slope of the interface. Therefore the results indicate a generalized law of Beavers and Joseph.Diplomarbeit393 158 - Some of the metrics are blocked by yourconsent settings
Item-typ:Veröffentlichung, Extension Operators for Sobolev Spaces on Periodic Domains, Their Applications, and Homogenization of a Phase Field Model for Phase Transitions in Porous Media(2016-07-12); ; ; The first part of this thesis is concerned with extension operators for Sobolev spaces on periodic domains and their applications. When homogenizing nonlinear partial differential equations in periodic domains by two-scale convergence, the need for uniformly bounded families of extension operators often arises. In this thesis, new extension operators that allow for estimates in the whole domain, even if the complement of the periodic domain is connected, are constructed. These extension operators exist if the domain is generalized rectangular. They are useful for homogenization problems with flux boundary conditions. Additionally, the existence of extension operators that respect zero and nonnegative traces on the exterior boundary is shown. These can be applied to problems with mixed or Dirichlet boundary conditions. Making use of this type of extension operators, uniform Poincare and Korn inequalities for functions with mixed boundary values in periodic domains are proven. Furthermore, a generalization of a compactness theorem of Meirmanov and Zimin, which is similar to the well-known Lions-Aubin compactness theorem and which is applicable in periodic domains, is presented. The above results are then applied to the homogenization by two-scale convergence of some quasilinear partial differential equations and variational inequalities with operators of Leray-Lions type in periodic domains with mixed boundary conditions. In the second part, a phase field model for phase transitions on the pore scale of a porous medium is introduced. The existence and uniqueness of weak solutions is shown. Uniform a-priori estimates are established by making use of extension operators of the type of those that have previously been constructed in the first part of this thesis. By using two-scale convergence and again applying the extension operators, a homogenized phase field model is obtained. Finally, by applying the method of formal asymptotic expansion, a macroscopic sharp interface model for phase transitions in porous media is derived. Possible applications include the melting of permafrost soil and the frost attack on concrete.Dissertation805 1125 - Some of the metrics are blocked by yourconsent settings
Item-typ:Veröffentlichung, Homogenization of Thermoelasticity Systems Describing Phase Transformations(2018-04-13); ; ; This thesis is concerned with the mathematical homogenization of thermoelasticity models with moving boundary describing solid-solid phase transformations occurring in highly heterogeneous, two-phase media. In the first part of this thesis, existence and uniqueness of weak solutions are established under the assumption that the changes in the geometry, which are due to the moving boundary, are given a priori. This is achieved after a transformation of coordinates to a fixed referential geometry. In addition, uniform a priori estimates are provided. Via an argument utilizing the concept of two-scale convergence, a corresponding homogenized model with distributed time and space dependent microstructures is derived. Quantitative error estimates measuring the accuracy and efficacy of the homogenized model are investigated. While such estimates seem not to be obtainable in the fully coupled setting, optimal convergence rates are proven for some special scenarios where the coupling mechanisms between the mechanical part and the heat part are simplified. In the second part, a more general scenario, in which the geometric changes are not assumed to be prescribed at the outset, is considered. Starting with the normal velocity of the interface separating the competing phases, a specific transformation of coordinates, the so-called Hanzawa transformation, is constructed. This is achieved by (i) solving a non-linear system of ODEs characterizing the motion of the interface and (ii) using the Implicit Function Theorem to arrive at the height function parametrizing this motion. Based on uniform estimates for the functions related to the transformation of coordinates, the strong two-scale convergence of these functions is shown. Finally, these results are used to establish the corresponding homogenized model.Dissertation528 306 - Some of the metrics are blocked by yourconsent settings
Item-typ:Veröffentlichung, Eine konsistente Plattentheorie zweiter Ordnung für monotropes Material(2010-01-05); In der Arbeit wird eine Flächentragwerkstheorie für eine homogen monotrope, linear elastische Platte konstanter Dicke modelliert. Die Theorie berücksichtigt dabei Schubspannungseinflüsse. Eine solche Theorie wird als Theorie zweiter Ordnung, oder als Theorie vom Reissnerschen Typ bezeichnet. Die Modellierung basiert auf dem Ansatz der konsistenten Approximation. Ein besonderer Schwerpunkt wird in der Arbeit auf eine mathematisch rigorose Modellierung gelegt. Zunächst leitet man aus den Grundgleichungen der dreidimensionalen linearen Elastizitätstheorie äquivalente zweidimensionale Formulierungen mittels Fourier-Reihenentwicklungen bezüglich einer Basis aus skalierten Legendre-Polynomen in Dickenrichtung der Platte ab. Durch den konsistenten Abbruch der entstehenden Potenzreihen des Quadrates des Plattenparameters gelangt man zu einer Hierarchie von partiellen Differentialgleichungssystemen steigender Approximationsgenauigkeit. Die Anzahl der Unbekannten Verschiebungskoeffizientenfunktionen steigt mit dem Grad der Approximationsgenauigkeit. Durch eine Pseudoreduktion wird die Anzahl der Unbekannten anschließend wieder reduziert, was jedoch die Ordnung der partiellen Differtialgleichungen erhöht. Die finale Theorie ist ein gekoppeltes System zweier Differtialgleichungen in zwei Unbekannten. Für den Spezialfall der Isotropie ist die Theorie als Approximation erster Ordnung der Kichhoff-Theorie und als Approximation zweier Ordnung der Reissner-Mindlin-Theorie äquivalent.Diplomarbeit535 345 - Some of the metrics are blocked by yourconsent settings
Item-typ:Veröffentlichung, Homogenization of a System of Nonlinear Multi-Species Diffusion-Reaction Equations in an H^{1,p} Setting(2013-05-17); ; ; The processes of chemical transport in porous media are extensively studied in the fields of applied mathematics, material science, chemical engineering etc. A porous medium (e.g. concrete, soil, rocks, reservoir etc.) is a multiscale material/medium where the heterogeneities present in the medium are characterized by the micro scale and the global behaviors of the medium are observed by the macro scale. The upscaling from the micro scale to the macro scale can be done via averaging methods. The transport process in a porous medium is a complex phenomena. In this thesis, the heterogeneities inside a porous medium are assumed to be periodically distributed and diffusion-reaction of a finite number of chemical species are investigated. Two different models are proposed in this work. In model M1, diffusion-reaction of mobile chemical species are considered. The chemical processes are modeled via mass action kinetics and the modeling leads to a system of multi-species diffusion-reaction equations (nonlinear partial differential equations) at the micro scale. For this system of equations, existence of a unique positive global weak solution is proved by the help of a Lyapunov functional and Schaefer's fixed point theorem. The upscaled model of this system is obtained using periodic homogenization which is an averaging method. In model M2, we consider diffusion-advection-reaction of two different types of mobile species (type I and type II). The type II species are supplied via dissolution process due to the presence of immobile species on the surface of the solid parts. The presence of mobile and the immobile species make the model complex and the modeling yields a coupled system of nonlinear partial differential equations. The existence of a unique positive global weak solution of this complex system is shown. Finally, with the help of periodic homogenization, model M2 is upscaled from the micro scale to the macro scale. Numerical simulations are conducted for both models separately. For the purpose of illustration, we restrict ourselves to relatively simple 2-dimensional situations. For models M1 and M2, simulation results at the micro scale and at the macro scale are compared.Dissertation328 185 - Some of the metrics are blocked by yourconsent settings
Item-typ:Veröffentlichung, Homogenization Techniques for Lower Dimensional Structures(2012-08-31); ; ; This thesis is concerned with extensions and applications of the theory of periodic unfolding in the field of (mathematical) homogenization. The first part extends the applicability of homogenization in domains with evolving microstructure to the case of evolving hypersurfaces: We consider a diffusion-reaction equation inside a perforated domain, where also surface diffusion and reaction takes place. Upon a transformation to a referential geometry, we (formally) obtain a transformed set of equations. We show that homogenization techniques can be applied to this transformed formulation. Special emphasis is placed on possible nonlinear reaction rates on the surface, a fact which requires special results for estimation and convergence results. In the limit, we obtain a macroscopic system, where each point of the domain is coupled to a system posed in the reference (micro-)geometry. Additionally, this reference geometry is evolving. In a second part, we are concerned with an extension of the notion of periodic unfolding to some Riemannian manifolds: We develop a notion of periodicity on nonflat structures in a local fashion with the help of a special atlas. If this atlas satisfies a compatibility condition, unfolding operators can be defined which operate on the manifold. We show that continuity and compactness theorems hold, generalizing the well-known results from the established theory. As an application of this newly developed results, we apply the unfolding operators to a strongly elliptic model problem. Again, we obtain a generalization of results well-known in homogenization. Moreover, we are also able to show some additional smoothness-properties of the solution of the cell problem, and we construct an equivalence relation for different atlases. With respect to this relation, the limit problem is independent of the parametrization of the manifold.Dissertation712 828