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A Consistent Large Eddy Approach for Lattice Boltzmann Methods and its Application to Complex Flows
(2015)
Lattice Boltzmann Methods have shown to be promising tools for solving fluid flow problems. This is related to the advantages of these methods, which are among others, the simplicity in handling complex geometries and the high efficiency in calculating transient flows. Lattice Boltzmann Methods are mesoscopic methods, based on discrete particle dynamics. This is in contrast to conventional Computational Fluid Dynamics methods, which are based on the solution of the continuum equations. Calculations of turbulent flows in engineering depend in general on modeling, since resolving of all turbulent scales is and will be in near future far beyond the computational possibilities. One of the most auspicious modeling approaches is the large eddy simulation, in which the large, inhomogeneous turbulence structures are directly computed and the smaller, more homogeneous structures are modeled.
In this thesis, a consistent large eddy approach for the Lattice Boltzmann Method is introduced. This large eddy model includes, besides a subgrid scale model, appropriate boundary conditions for wall resolved and wall modeled calculations. It also provides conditions for turbulent domain inlets. For the case of wall modeled simulations, a two layer wall model is derived in the Lattice Boltzmann context. Turbulent inlet conditions are achieved by means of a synthetic turbulence technique within the Lattice Boltzmann Method.
The proposed approach is implemented in the Lattice Boltzmann based CFD package SAM-Lattice, which has been created in the course of this work. SAM-Lattice is feasible of the calculation of incompressible or weakly compressible, isothermal flows of engineering interest in complex three dimensional domains. Special design targets of SAM-Lattice are high automatization and high performance.
Validation of the suggested large eddy Lattice Boltzmann scheme is performed for pump intake flows, which have not yet been treated by LBM. Even though, this numerical method is very suitable for this kind of vortical flows in complicated domains. In general, applications of LBM to hydrodynamic engineering problems are rare. The results of the pump intake validation cases reveal that the proposed numerical approach is able to represent qualitatively and quantitatively the very complex flows in the intakes. The findings provided in this thesis can serve as the basis for a broader application of LBM in hydrodynamic engineering problems.
The present thesis describes the development and validation of a viscosity adaption method for the numerical simulation of non-Newtonian fluids on the basis of the Lattice Boltzmann Method (LBM), as well as the development and verification of the related software bundle SAM-Lattice.
By now, Lattice Boltzmann Methods are established as an alternative approach to classical computational fluid dynamics
methods. The LBM has been shown to be an accurate and efficient tool for the numerical simulation of weakly compressible or incompressible fluids. Fields of application reach from turbulent simulations through thermal problems to acoustic calculations among others. The transient nature of the method and the need for a regular grid based, non body conformal discretization makes the LBM ideally suitable for simulations involving complex solids. Such geometries are common, for instance, in the food processing industry, where fluids are mixed by static mixers or agitators. Those fluid flows are often laminar and non-Newtonian.
This work is motivated by the immense practical use of the Lattice Boltzmann Method, which is limited due to stability issues. The stability of the method is mainly influenced by the discretization and the viscosity of the fluid. Thus, simulations of non-Newtonian fluids, whose kinematic viscosity depend on the shear rate, are problematic. Several authors have shown that the LBM is capable of simulating those fluids. However, the vast majority of the simulations in the literature are carried out for simple geometries and/or moderate shear rates, where the LBM is still stable. Special care has to be taken for practical non-Newtonian Lattice Boltzmann simulations in order to keep them stable. A straightforward way is to truncate the modeled viscosity range by numerical stability criteria. This is an effective approach, but from the physical point of view the viscosity bounds are chosen arbitrarily. Moreover, these bounds depend on and vary with the grid and time step size and, therefore, with the simulation Mach number, which is freely chosen at the start of the simulation. Consequently, the modeled viscosity range may not fit to the actual range of the physical problem, because the correct simulation Mach number is unknown a priori. A way around is, to perform precursor simulations on a fixed grid to determine a possible time step size and simulation Mach number, respectively. These precursor simulations can be time consuming and expensive, especially for complex cases and a number of operating points. This makes the LBM unattractive for use in practical simulations of non-Newtonian fluids.
The essential novelty of the method, developed in the course of this thesis, is that the numerically modeled viscosity range is consistently adapted to the actual physically exhibited viscosity range through change of the simulation time step and the simulation Mach number, respectively, while the simulation is running. The algorithm is robust, independent of the Mach number the simulation was started with, and applicable for stationary flows as well as transient flows. The method for the viscosity adaption will be referred to as the "viscosity adaption method (VAM)" and the combination with LBM leads to the "viscosity adaptive LBM (VALBM)".
Besides the introduction of the VALBM, a goal of this thesis is to offer assistance in the spirit of a theory guide to students and assistant researchers concerning the theory of the Lattice Boltzmann Method and its implementation in SAM-Lattice. In Chapter 2, the mathematical foundation of the LBM is given and the route from the BGK approximation of the Boltzmann equation to the Lattice Boltzmann (BGK) equation is delineated in detail.
The derivation is restricted to isothermal flows only. Restrictions of the method, such as low Mach number flows are highlighted and the accuracy of the method is discussed.
SAM-Lattice is a C++ software bundle developed by the author and his colleague Dipl.-Ing. Andreas Schneider. It is a highly automated package for the simulation of isothermal flows of incompressible or weakly compressible fluids in 3D on the basis of the Lattice Boltzmann Method. By the time of writing of this thesis, SAM-Lattice comprises 5 components. The main components are the highly automated lattice generator SamGenerator and the Lattice Boltzmann solver SamSolver. Postprocessing is done with ParaSam, which is our extension of the
open source visualization software ParaView. Additionally, domain decomposition for MPI
parallelism is done by SamDecomposer, which makes use of the graph partitioning library MeTiS. Finally, all mentioned components can be controlled through a user friendly GUI (SamLattice) implemented by the author using QT, including features to visually track output data.
In Chapter 3, some fundamental aspects on the implementation of the main components, including the corresponding flow charts will be discussed. Actual details on the implementation are given in the comprehensive programmers guides to SamGenerator and SamSolver.
In order to ensure the functionality of the implementation of SamSolver, the solver is verified in Chapter 4 for Stokes's First Problem, the suddenly accelerated plate, and for Stokes's Second Problem, the oscillating plate, both for Newtonian fluids. Non-Newtonian fluids are modeled in SamSolver with the power-law model according to Ostwald de Waele. The implementation for non-Newtonian fluids is verified for the Hagen-Poiseuille channel flow in conjunction with a convergence analysis of the method. At the same time, the local grid refinement as it is implemented in SamSolver, is verified. Finally, the verification of higher order boundary conditions is done for the 3D Hagen-Poiseuille pipe flow for both Newtonian and non-Newtonian fluids.
In Chapter 5, the theory of the viscosity adaption method is introduced. For the adaption process, a target collision frequency or target simulation Mach number must be chosen and the distributions must be rescaled according to the modified time step size. A convenient choice is one of the stability bounds. The time step size for the adaption step is deduced from the target collision frequency \(\Omega_t\) and the currently minimal or maximal shear rate in the system, while obeying auxiliary conditions for the simulation Mach number. The adaption is done in the collision step of the Lattice Boltzmann algorithm. We use the transformation matrices of the MRT model to map from distribution space to moment space and vice versa. The actual scaling of the distributions is conducted on the back mapping, because we use the transformation matrix on the basis of the new adaption time step size. It follows an additional rescaling of the non-equilibrium part of the distributions, because of the form of the definition for the discrete stress tensor in the LBM context. For that reason it is clear, that the VAM is applicable for the SRT model as well as the MRT model, where there is virtually no extra cost in the latter case. Also, in Chapter 5, the multi level treatment will be discussed.
Depending on the target collision frequency and the target Mach number, the VAM can be used to optimally use the viscosity range that can be modeled within the stability bounds or it can be used to drastically accelerate the simulation. This is shown in Chapter 6. The viscosity adaptive LBM is verified in the stationary case for the Hagen-Poiseuille channel flow and in the transient case for the Wormersley flow, i.e., the pulsatile 3D Hagen-Poiseuille pipe flow. Although, the VAM is used here for fluids that can be modeled with the power-law approach, the implementation of the VALBM is straightforward for other non-Newtonian models, e.g., the Carreau-Yasuda or Cross model. In the same chapter, the VALBM is validated for the case of a propeller viscosimeter developed at the chair SAM. To this end, the experimental data of the torque on the impeller of three shear thinning non-Newtonian liquids serve for the validation. The VALBM shows excellent agreement with experimental data for all of the investigated fluids and in every operating point. For reasons of comparison, a series of standard LBM simulations is carried out with different simulation Mach numbers, which partly show errors of several hundred percent. Moreover, in Chapter 7, a sensitivity analysis on the parameters used within the VAM is conducted for the simulation of the propeller viscosimeter.
Finally, the accuracy of non-Newtonian Lattice Boltzmann simulations with the SRT and the MRT model is analyzed in detail. Previous work for Newtonian fluids indicate that depending on the numerical value of the collision frequency \(\Omega\), additional artificial viscosity is introduced due to the finite difference scheme, which negatively influences the accuracy. For the non-Newtonian case, an error estimate in the form of a functional is derived on the basis of a series expansion of the Lattice Boltzmann equation. This functional can be solved analytically for the case of the Hagen-Poiseuille channel flow of non-Newtonian fluids. The estimation of the error minimum is excellent in regions where the \(\Omega\) error is the dominant source of error as opposed to the compressibility error.
Result of this dissertation is a verified and validated software bundle on the basis of the viscosity adaptive Lattice Boltzmann Method. The work restricts itself on the simulation of isothermal, laminar flows with small Mach numbers. As further research goals, the testing of the VALBM with minimal error estimate and the investigation of the VALBM in the case of turbulent flows is suggested.
Real-time systems are systems that have to react correctly to stimuli from the environment within given timing constraints.
Today, real-time systems are employed everywhere in industry, not only in safety-critical systems but also in, e.g., communication, entertainment, and multimedia systems.
With the advent of multicore platforms, new challenges on the efficient exploitation of real-time systems have arisen:
First, there is the need for effective scheduling algorithms that feature low overheads to improve the use of the computational resources of real-time systems.
The goal of these algorithms is to ensure timely execution of tasks, i.e., to provide runtime guarantees.
Additionally, many systems require their scheduling algorithm to flexibly react to unforeseen events.
Second, the inherent parallelism of multicore systems leads to contention for shared hardware resources and complicates system analysis.
At any time, multiple applications run with varying resource requirements and compete for the scarce resources of the system.
As a result, there is a need for an adaptive resource management.
Achieving and implementing an effective and efficient resource management is a challenging task.
The main goal of resource management is to guarantee a minimum resource availability to real-time applications.
A further goal is to fulfill global optimization objectives, e.g., maximization of the global system performance, or the user perceived quality of service.
In this thesis, we derive methods based on the slot shifting algorithm.
Slot shifting provides flexible scheduling of time-constrained applications and can react to unforeseen events in time-triggered systems.
For this reason, we aim at designing slot shifting based algorithms targeted for multicore systems to tackle the aforementioned challenges.
The main contribution of this thesis is to present two global slot shifting algorithms targeted for multicore systems.
Additionally, we extend slot shifting algorithms to improve their runtime behavior, or to handle non-preemptive firm aperiodic tasks.
In a variety of experiments, the effectiveness and efficiency of the algorithms are evaluated and confirmed.
Finally, the thesis presents an implementation of a slot-shifting-based logic into a resource management framework for multicore systems.
Thus, the thesis closes the circle and successfully bridges the gap between real-time scheduling theory and real-world implementations.
We prove applicability of the slot shifting algorithm to effectively and efficiently perform adaptive resource management on multicore systems.
In this dissertation, we discuss how to price American-style options. Our aim is to study and improve the regression-based Monte Carlo methods. In order to have good benchmarks to compare with them, we also study the tree methods.
In the second chapter, we investigate the tree methods specifically. We do research firstly within the Black-Scholes model and then within the Heston model. In the Black-Scholes model, based on Müller's work, we illustrate how to price one dimensional and multidimensional American options, American Asian options, American lookback options, American barrier options and so on. In the Heston model, based on Sayer's research, we implement his algorithm to price one dimensional American options. In this way, we have good benchmarks of various American-style options and put them all in the appendix.
In the third chapter, we focus on the regression-based Monte Carlo methods theoretically and numerically. Firstly, we introduce two variations, the so called "Tsitsiklis-Roy method" and the "Longstaff-Schwartz method". Secondly, we illustrate the approximation of American option by its Bermudan counterpart. Thirdly we explain the source of low bias and high bias. Fourthly we compare these two methods using in-the-money paths and all paths. Fifthly, we examine the effect using different number and form of basis functions. Finally, we study the Andersen-Broadie method and present the lower and upper bounds.
In the fourth chapter, we study two machine learning techniques to improve the regression part of the Monte Carlo methods: Gaussian kernel method and kernel-based support vector machine. In order to choose a proper smooth parameter, we compare fixed bandwidth, global optimum and suboptimum from a finite set. We also point out that scaling the training data to [0,1] can avoid numerical difficulty. When out-of-sample paths of stock prices are simulated, the kernel method is robust and even performs better in several cases than the Tsitsiklis-Roy method and the Longstaff-Schwartz method. The support vector machine can keep on improving the kernel method and needs less representations of old stock prices during prediction of option continuation value for a new stock price.
In the fifth chapter, we switch to the hardware (FGPA) implementation of the Longstaff-Schwartz method and propose novel reversion formulas for the stock price and volatility within the Black-Scholes and Heston models. The test for this formula within the Black-Scholes model shows that the storage of data is reduced and also the corresponding energy consumption.
Schon jetzt durchlaufen mehr als vier von fünf chemischen Produkten bei der Herstellung einen Katalysezyklus. In zunehmendem Maße findet man katalytische Anwendungen neben dem Einsatz in der Synthesechemie auch in den Biowissenschaften, beim Klima- und Umweltschutz sowie zur Energieversorgung. Durch gezieltes Ligandendesign werden dabei kontinuierlich bekannte Katalysatorsysteme optimiert und die Anwendungsbreite erweitert. Für zweizähnige, pyrimidinhaltige Ligandensysteme ist aus anderen Forschungsarbeiten der AG Thiel eine intramolekularen C-H-Aktivierung im Pyrimidinring bekannt, die zu einer carbanionkoordination am Übergangsmetallzentrum führt. Diese Reaktivität wurde im Rahmen dieser Arbeit mit der stabilisierenden Wirkung eines N-heterocyclischen Carbenliganden (NHC) zu einem neuen Ligandensystem kombiniert. Verschiedene Imidazoliumvorstufen neuer NHC-Liganden mit einem in der 2-Position aminosubstituierten Pyrimidinring als N-Substituent wurden über zwei neu erarbeitete Syntheserouten dargestellt und mit verschiedenen Übergangsmetallvorstufen umgesetzt. In Palladium(II)-Komplexen von pyrimidinyl- und mesitylsubstituierten NHC-Liganden wurden verschiedene Koordinationsmodi in Abhängigkeit von der verwendeten Synthesemethode beobachtet. Über Silber-Carben-Komplexe als Carben-Transferreagenzien konnten für verschieden tertiär amino- und mesitylsubstituierten Liganden die nicht C-H-aktivierten, d.h. C,N-koordinierten Palladiumkomplexe dargestellt werden. Eine direkte Umsetzung der ionischen Imidazoliumverbindungen mit Palladiumvorstufen wie PdCl2 führte in Pyridin und Pyridinderivaten als Lösungsmittel bei Reaktionstemperaturen direkt zu einer C-H-Aktivierung im Pyrimidinring des Liganden. Der leicht basische Pyridinligand stabilisiert während der Komplexbildung die hochreaktive, C-H-aktivierte Spezies und verhindert so Neben- und Zersetzungsreaktionen. Über die Abspaltung des labilen Pyridinliganden durch Erhitzen in schwach koordinierenden Lösungsmitteln wurden die zweikernigen, unlöslichen, pyridinfreien Palladiumkomplexe erhalten und mittels Festkörper-NMR-Spektroskopie charakterisiert. Diese Reaktion ist vollständig reversibel und wurde zum Einführen verschiedener Pyridinderivate als labile Liganden genutzt. In schwach koordinierenden Lösungsmitteln mit einem Siedepunkt < 80 °C, wie THF, wurde durch eine direkte Umsetzung der ionischen Vorstufen der Liganden mit PdCl2 eine weitere Art von Pd(II)-Komplexen erhalten, für welche die Strukturformel eines N-koordinierten Palladates postuliert wurde. In NMR-spektroskopischen Experimenten wurde die Reversibilität der C-H-Aktivierung im Pyrimidinring der Pd(II)-Komplexe in Abhängigkeit von pH-Wert und Temperatur nachgewiesen. Auch hier erwies sich der stabilisierende Pyridinligand für die C-H-Aktivierung und HCl-Eliminierung als notwendig. Die Rückreaktion wurde unter schwach sauren Reaktionsbedingungen bei Raumtemperatur über eine NHC-gebundene, pyridinhaltige Spezies, strukturanalog der literaturbekannten PEPPSI-Komplexe, beobachtet.
Für die stark Lewis-aciden Übergangsmetallzentren Iridium (III) und Ruthenium (II) wurden mit den entsprechenden ionischen Ligandenvorstufen über in situ dargestellte Silber-Carben-Komplexe ausschließlich die C-H-aktivierten, C,N-koordinierten Halbsandwichkomplexe der neuen 2-Amino-4-(imidazolylidenyl)pyrimidinliganden erhalten, trotz variierter Reaktionsbedingungen. Die C-H-Aktivierung mit anschließender HCl-Eliminierung erfolgte bei diesen Übergangsmetallzentren bereits bei Raumtemperatur irreversibel.
In Rahmen dieser Arbeit wurde außerdem die Notwendigkeit eines sterisch anspruchsvollen, stabilisierenden Mesitylrestes am NHC-Liganden für stabile und isolierbare C-H-aktivierte Komplexe beobachtet. Mit anderen, sterisch weniger anspruchsvollen Resten an dieser Position des Liganden wurden unter den Reaktionsbedingungen für potentielle C-H-Aktivierungen nur Zersetzungsprodukte erhalten. Von jedem Komplextyp der neuen C-H-aktivierten Übergangsmetallkomplexe wurden messbare Kristalle für eine Kristallstrukturanalyse erhalten, welche tiefere Einblicke in die Bindungssituation der neuen Liganden ermöglichte.
Die C-H-aktivierten Übergangsmetallkomplexe der neuen Liganden zeigen sehr gute Aktivitäten in verschiedenen katalytischen Anwendungen. Neben der stabilisierenden Wirkung des NHC mit starkem σ-Donorcharakter wird die hohe Elektronendichte am Übergangsmetallzentrum durch die Koordination des Carbanions weiter erhöht. Unter optimierten Bedingungen wurden in der Suzuki-Miyaura-Kupplung mit geringeren Katalysatorkonzentrationen der C-H-aktivierten Pd(II)-Komplexe eine große Bandbreite von sterisch und elektronisch gehinderten Chlorarylen mit verschiedenen Boronsäurederivaten erfolgreich zu Biarylen umgesetzt. Mit den C-H-aktivierten Ru(II)- und Ir(III)-Halbsandwichkomplexen der neuen Liganden wurden in der katalytischen Transferhydrierung von Acetophenon bereits bei geringen Katalysatorkonzentrationen von 0.15 mol% sehr hohe Ausbeuten erhalten. Die katalytisch hochaktiven Komplexe zeichneten sich außerdem durch eine hohe Stabilität unter den optimierten Reaktionsbedingungen aus. Die C-H-Aktivierung weist zwar keine Abhängigkeit vom sterischen Anspruch der variierten tertiären Aminosubstituenten auf, wurde aber für die anderen Reste in der 2-Position des Pyrimidinrings nicht beobachtet.
Open distributed systems are a class of distributed systems where (i) only partial information about the environment, in which they are running, is present, (ii) new resources may become available at runtime, and (iii) a subsystem may become aware of other subsystems after some interaction. Modeling and implementing such systems correctly is a complex task due to the openness and the dynamicity aspects. One way to ensure that the resulting systems behave correctly is to utilize formal verification.
Formal verification requires an adequate semantic model of the implementation, a specification of the desired behavior, and a reasoning technique. The actor model is a semantic model that captures the challenging aspects of open distributed systems by utilizing actors as universal primitives to represent system entities and allowing them to create new actors and to communicate by sending directed messages as reply to received messages. To enable compositional reasoning, where the reasoning task is reduced to independent verification of the system parts, semantic entities at a higher level of abstraction than actors are needed.
This thesis proposes an automaton model and combines sound reasoning techniques to compositionally verify implementations of open actor systems. Based on I/O automata, the model allows automata to be created dynamically and captures dynamic changes in communication patterns. Each automaton represents either an actor or a group of actors. The specification of the desired behavior is given constructively as an automaton. As the basis for compositionality, we formalize a component notion based on the static structure of the implementation instead of the dynamic entities (the actors) occurring in the system execution. The reasoning proceeds in two stages. The first stage establishes the connection between the automata representing single actors and their implementation description by means of weakest liberal preconditions. The second stage employs this result as the basis for verifying whether a component specification is satisfied. The verification is done by building a simulation relation from the automaton representing the implementation to the component's automaton. Finally, we validate the compositional verification approach through a number of examples by proving correctness of their actor implementations with respect to system specifications.
An efficient multiscale approach is established in order to compute the macroscopic response of nonlinear composites. The micro problem is rewritten in an integral form of the Lippmann-Schwinger type and solved efficiently by Fast Fourier Transforms. Using realistic microstructure models complex nonlinear effects are reproduced and validated with measured data of fiber reinforced plastics. The micro problem is integrated in a Finite Element framework which is used to solve the macroscale. The scale coupling technique and a consistent numerical algorithm is established. The method provides an efficient way to determine the macroscopic response considering arbitrary microstructures, constitutive behaviors and loading conditions.
The overall goal of the work is to simulate rarefied flows inside geometries with moving boundaries. The behavior of a rarefied flow is characterized through the Knudsen number \(Kn\), which can be very small (\(Kn < 0.01\) continuum flow) or larger (\(Kn > 1\) molecular flow). The transition region (\(0.01 < Kn < 1\)) is referred to as the transition flow regime.
Continuum flows are mainly simulated by using commercial CFD methods, which are used to solve the Euler equations. In the case of molecular flows one uses statistical methods, such as the Direct Simulation Monte Carlo (DSMC) method. In the transition region Euler equations are not adequate to model gas flows. Because of the rapid increase of particle collisions the DSMC method tends to fail, as well
Therefore, we develop a deterministic method, which is suitable to simulate problems of rarefied gases for any Knudsen number and is appropriate to simulate flows inside geometries with moving boundaries. Thus, the method we use is the Finite Pointset Method (FPM), which is a mesh-free numerical method developed at the ITWM Kaiserslautern and is mainly used to solve fluid dynamical problems.
More precisely, we develop a method in the FPM framework to solve the BGK model equation, which is a simplification of the Boltzmann equation. This equation is mainly used to describe rarefied flows.
The FPM based method is implemented for one and two dimensional physical and velocity space and different ranges of the Knudsen number. Numerical examples are shown for problems with moving boundaries. It is seen, that our method is superior to regular grid methods with respect to the implementation of boundary conditions. Furthermore, our results are comparable to reference solutions gained through CFD- and DSMC methods, respectevly.
The Wilkie model is a stochastic asset model, developed by A.D. Wilkie in 1984 with a purpose to explore the behaviour of investment factors of insurers within the United Kingdom. Even so, there is still no analysis that studies the Wilkie model in a portfolio optimization framework thus far. Originally, the Wilkie model is considering a discrete-time horizon and we apply the concept of Wilkie model to develop a suitable ARIMA model for Malaysian data by using Box-Jenkins methodology. We obtained the estimated parameters for each sub model within the Wilkie model that suits the case of Malaysia, and permits us to analyse the result based on statistics and economics view. We then tend to review the continuous time case which was initially introduced by Terence Chan in 1998. The continuous-time Wilkie model inspired is then being employed to develop the wealth equation of a portfolio that consists of a bond and a stock. We are interested in building portfolios based on three well-known trading strategies, a self-financing strategy, a constant growth optimal strategy as well as a buy-and-hold strategy. In dealing with the portfolio optimization problems, we use the stochastic control technique consisting of the maximization problem itself, the Hamilton-Jacobi-equation, the solution to the Hamilton-Jacobi-equation and finally the verification theorem. In finding the optimal portfolio, we obtained the specific solution of the Hamilton-Jacobi-equation and proved the solution via the verification theorem. For a simple buy-and-hold strategy, we use the mean-variance analysis to solve the portfolio optimization problem.
Die Gewichtsreduktion im Strukturbereich stellt einen zentralen Optimierungsansatz in der Luftfahrtindustrie dar, der vor allem durch adäquate Fügetechnologien genutzt werden kann. Ausgehend vom aktuellen Stand der Technik im Helikopterbau, dem Nieten, gilt es durch die Verwendung einer innovativen Fügetechnologie das volle technologische Leistungsvermögen hinsichtlich Performance, Qualität und Kosten zu nutzen. Dazu wurde das Induktionsschweißen als die potentialreichste Fügetechno-logie für den Helikopterbau bewertet. Um dieses identifizierte Potential für eine Luftfahrtfertigungstechnologie nutzbar zu machen, ist es unerlässlich, das Indukti-onsschweißen an die Luftfahrtanforderungen anzupassen. Vor allem in den Berei-chen Nachweisbarkeit, Leistungsfähigkeit und Kosten wurden daher Fragestellungen identifiziert deren Beantwortung den Kern dieser Arbeit darstellt.
Beim Induktionsschweißen werden faserverstärkte Thermoplaste durch ein Auf-schmelzen der Matrix und ein anschließendes Abkühlen unter Druck gefügt. Die Erwärmung des sich in einem alternierenden elektromagnetischen Feld (EMF) befindlichen Laminats erfolgt dabei durch die Einkopplung eines elektrischen Stroms in die Fasern.
Das zentrale Element zur Erreichung der geforderten Leistungsfähigkeit stellt die interlaminare Temperaturverteilung dar, welche es nachzuweisen gilt. Dieser Nach-weis wurde durch ein umfassendes analytisches Modell realisiert, das eine höchst-präzise interlaminare Temperaturberechnung ermöglicht. Die Kernaussagen, welche aus dem Modell abgeleitet werden, sind die dickenabhängige Erwärmung des Laminats, die EMF-Semipermeabilität der Laminatoberflächen und der nicht expo-nentielle Abfall sowie der nicht lineare Verlauf der Temperatur in Dickenrichtung. Die Validierung der analytischen Modellierung gelang nur durch die Identifikation einer EMF-toleranten, hochdynamischen Temperaturmesstechnologie, welche mit hoher Auflösung interlaminar eingesetzt werden kann.
Auf Grundlage der Modellergebnisse wurden die optimalen Schweißparameter definiert, auf deren Basis die Leistungsfähigkeit der Induktionsschweißtechnologie mit circa 36 MPa Scherfestigkeit bestätigt wurde. Durch eine Sensibilitätsanalyse konnte weiterhin der Einfluss der Parameter Generatorleistung, Kühlvolumen, Anpressdruck, Induktorabstand, Fehlereinschlüsse und Geschwindigkeit bestimmt werden. Aufgrund der im aktuellen Anlagenaufbau nicht vorhanden Parameterüber-wachung und aufgrund des hohen Prozesseinflusses erwies sich dabei der Induktor-abstand als der kritischste Faktor.
Etwaige dadurch auftretende qualitative Mängel können durch den schlanken, maßgeschneiderten Einsatz einer Kombination aus der Ultraschalluntersuchung, einer in der Luftfahrt standardmäßig eingesetzten Qualitätssicherungsmethode und eines progressiven Inline-Prozesskontrollansatzes detektiert werden.
Parallel zur mechanischen Leistungsfähigkeit der Technologie stand der Einfluss der Temperaturverteilung auf die Oberflächenqualität im Fokus. Durch die umfangreiche theoretische und experimentelle Analyse bereits bekannter und neu entwickelter Temperaturoptimierungsmethoden konnte mit der Kühlung der Oberfläche mit temperatur- und volumenvariablen Druckluftströmen eine effiziente Methode zur zielführenden Lösung der bestehenden Problemstellung ermittelt werden.
Die Anwendbarkeit der Induktionsschweißtechnologie konnte auch durch eine Kostenrechnung am Beispiel eines helikopterspezifischen Musterbauteils bestätigt werden.
The main goal of this thesis is twofold. First, the thesis aims at bridging the gap between existing Pattern Recognition (PR) methods of automatic signature verification and the requirements for their application in forensic science. This gap, attributed by various factors ranging from system definition to evaluation, prevents automatic methods from being used by Forensic Handwriting Examiners (FHEs). Second, the thesis presents novel signature verification methods developed particularly considering the implications of forensic casework, and outperforming the state-of-the-art PR methods.
The first goal of the thesis is attributed by four important factors, i.e., data, terminology, output reporting, and how evaluation of automatic systems is carried out today. It is argued that traditionally the signature data used in PR are not actual/close representative of the real world data (especially that available in forensic cases). The systems trained on such data are, therefore, not suitable for forensic environments. This situation can be tackled by providing more realistic data to PR researchers. To this end, various signature and handwriting datasets are gathered in collaboration with FHEs and are made publicly available through the course of this thesis. A special attention is given to disguised signatures--where authentic authors purposefully make their signatures look like a forgery. This genre was at large neglected in PR research previously.
The terminology used, in the two communities - PR and FHEs, differ greatly. In fact, even in PR, there is no standard terminology and people often differ in the usage of various terms particularly related to various types of forged signatures/handwriting. The thesis presents a new terminology that is equally useful for both forensic scientists and PR researchers. The proposed terminology is hoped to increase the general acceptability of automatic signature analysis systems in forensic science.
The outputs reported by general signature verification systems are not acceptable for FHEs and courts as they are either binary (yes/no) or score (raw evidence) based on similarity/difference. The thesis describes that automatic systems should rather report the probability of observing the evidence (e.g., a certain similarity/difference score) given the signature belongs to the acclaimed identity, and the probability of observing the same evidence given the signature does not belong to the acclaimed identity. This will take automatic systems from hard decisions to soft decisions, thereby enabling them to report likelihood ratios that actually represent the evidential value of the score rather than the raw score (evidence).
When automatic systems report soft decisions (as in the form of likelihood ratios), the thesis argues that there must be some methods to evaluate such systems. This thesis presents one such adaptation. The thesis argues that the state-of-the-art evaluation methods, like equal error rate and area under curve, do not address the needs of forensic science. These needs require an assessment of the evidential value of signature verification, rather than a hard/pure classification (accept/reject binary decision). The thesis demonstrates and validates a relatively simple adaptation of the current verification methods based on the Bayesian inference dependent calibration of continuous scores rather than hard classifications (binary and/or score based classification).
The second goal of this thesis is to introduce various local features based techniques which are capable of performing signature verification in forensic cases and reporting results as anticipated by FHEs and courts. This is an important contribution of the thesis because of the following two reasons. First, to the best of author's knowledge, local feature descriptors are for the first time used for development of signature verification systems for forensic environments (particularly considering disguised signatures). Previously, such methods have been heavily used for recognition tasks, rather than verification of writing behaviors, such as character and digit recognition. Second, the proposed methods not only report the more traditional decisions (like scores-usually reported in PR) but also the Bayesian inference based likelihood ratios (suitable for courts and forensic cases).
Furthermore, the thesis also provides a detailed man vs. machine comparison for signature verification tasks. The men, in this comparison, are forensic scientists serving as forensic handwriting examiners and having experience of varying number of years. The machines are the local features based methods proposed in this thesis, along with various other state-of-the-art signature verification systems. The proposed methods clearly outperform the state-of-the-art systems, and sometimes the human experts.
Finally, the thesis details various tasks that have been performed in the areas closely related to signature verification and its application in forensic casework. These include, developing novel local feature based methods for extraction of signatures/handwritten text from document images, hyper-spectral image analysis for extraction of signatures from forensic documents, and analysis of on-line signatures acquired through specialized pens equipped with Accelerometer and Gyroscope. These tasks are important as they enable the thesis to take PR systems one step further close to direct application in forensic cases.
Im Rahmen der vorliegenden Arbeit konnten sowohl konzeptionelle Beiträge zur Berechnung
der magnetischen Anisotropie von mehrkernigen Übergangsmetallkomplexen mittels
DFT-Methoden geleistet, sowie zwei Molekülsysteme eingehend untersucht werden.
Auf der Ebene der Methodenweiterentwicklung konnten die Matrixelemente des Spin-Bahn
mean-field-Operators als effektivem Einteilchenoperator implementiert werden. Diese wurden
sowohl mit SMP als auch mit MPI parallelisiert, so dass eine sehr effektive Berechnung
auch bei großen Systemen möglich ist. Bei Verwendung von Hybridfunktionalen erhöht sich
die Rechenzeit einer sopt-Rechnung durch die Berechnung dieser Integrale um etwa
einen Faktor drei. Die restliche Rechenzeit wird durch das Lösen des CHF-Systems
dominiert.
Bei Verwendung reiner Dichtefunktionale dauern die Rechnungen sehr viel kürzer, da hier
kein CHF-System zu lösen ist. Dadurch verteuert die Berechnung der SOMF-Matrixelemente
diese Rechnungen prozentual deutlich mehr, aber dennoch sind Rechnungen mit reinen
Dichtefunktionalen sehr viel schneller fertig als mit Hybridfunktionalen.
Weiterhin konnte eine neue Methode zur Berechnung von Einzelionentensoren entworfen und
implementiert werden, die mit lokalisierten Orbitalen arbeitet. Diese Methode konnte mit
der bereits bestehenden Methode, bei der die SOC an einzelnen Zentren "ausgeschaltet" wird,
verglichen werden. Die Ergebnisse besitzen in etwa dieselbe
Qualität.
Für die Zuordnung einzelner, lokalisierter Orbitale gibt es keine physikalisch eindeutige
Lösung. Es konnte jedoch ein Zuordnungsschema auf der Basis von Abständen und Winkeln
entwickelt werden, das in den meisten Fällen der chemischen Intuition entspricht, so dass
einem oktaedrisch koordinierten Metallzentrum sechs Bindungselektronenpaare zugeordnet
werden.
Die so entwickelte Methode liefert eine bequeme und einfache Möglichkeit zur Berechnung
aller benötigter Einzelionentensoren im Zuge einer einzigen Rechnung ohne nennenswerten
Mehraufwand. So wurde sie bei den weiteren Untersuchungen der beiden Molekülsysteme ausgiebig
genutzt.
Schließlich wurden zwei Komplexsysteme eingehend analysiert und mit verschiedenen
Übergangsmetallkombinationen untersucht. Es wurde hierbei ein breites Spektrum an
Analysewerkzeugen eingesetzt, sodass die erhaltenen Ergebnisse nicht nur nähere Einsichten
zu den konkret untersuchten Komplexen liefern, sondern auch allgemein wertvolle
Erfahrungen zur Untersuchung der magnetischen Anisotropie an mehrkernigen
Übergangsmetallkomplexen mit DFT-Methoden liefern.
Zuerst wurden Komplexe mit Tripelsalenliganden untersucht, wie sie in der Arbeitsgruppe
von Thorsten Glaser synthetisiert und charakterisiert werden. Der Einfachheit halber
wurden zunächst Komplexe mit nur einer Tripelsaleneinheit untersucht, wobei als
Metallionen in der Tripelsaleneinheit Mn(III)-Ionen verwendet wurden. Hierbei zeigte sich,
dass durch die Substitution von Chrom(III) als Zentralion durch Wolfram(III) bzw.
Rhenium(IV) eine deutliche Steigerung der magnetischen Anisotropie erreicht werden kann.
In einem nächsten Schritt wurden diese Komplexe durch eine weitere Tripelsaleneinheit
erweitert. Diese Komplexe sind also von der Struktur, wie sie auch in der Arbeitsgruppe
Glaser synthetisiert werden. Es zeigt sich, dass die Mn6M-Komplexe eine sehr viel
geringere Anisotropie aufweisen. Dies liegt daran, dass bei den Mn3M-Komplexen die
oktaedrische Koordinationsumgebung um das Zentralion verzerrt ist, da nur 3 der 6
Cyanoliganden an der Koordination der Tripelsaleneinheit beteiligt sind. Durch diese
starke Verzerrung erreicht man bei Verwendung von 5d-Elementen hohe Anisotropien. Bei den
Mn6M-Komplexen ergibt sich die Anisotropie hauptsächlich aus den Beiträgen der
Manganionen.
Schließlich wurden Komplexe mit nur einer Tripelsaleneinheit untersucht, bei denen jedoch
in der Tripelsaleneinheit Mo(III)-Ionen koordiniert sind. Hier konnte gezeigt werden, dass
durch die Molybdänionen eine höhere Anisotropie erhalten werden kann als bei den
entsprechenden Mn3M-Komplexen. Bei den leichten Zentralionen (Chrom und Molybdän)
jedoch war die Anisotropie geringer als bei den Mn6M-Komplexen. Dies liegt daran,
dass bei diesen Komplexen der Beitrag der Ionen der Tripelsaleneinheit überwiegt und somit
macht sich bemerkbar, dass bei den Mn6M-Komplexen 6 Ionen in der Tripelsaleneinheit
koordiniert sind und bei den Mo3M-Komplexen lediglich 3.
Bei der Untersuchung der Tripelsalenkomplexe konnte ein Verfahren vorgestellt werden, wie
man sich die Symmetrie von Komplexen bei der Berechnung des anisotropen Austauschs zu
Nutze machen kann, um die Anzahl zu berechnender Konfigurationen zu minimieren.
Als zweites Komplexsystem wurde ein von Habib et al. vorgestellter Oximligand gewählt, der
als Chelatligand fungiert. Der ursprünglich synthetisierte Komplex enthielt zwei
Mn(IV)-Ionen außen und in der Mitte ein Mn(III)-Ion. Auch hier wurde die Kombination
verschiedener Metallzentren analysiert.
Für den Ursprungskomplex konnte ein axialer MAE-Parameter nachgewiesen, der im normalen
Rahmen für Mn(III)-Komplexe liegt. Die Substitution des zentralen Manganzentrums durch
schwerere Metallzentren führt hier jedoch nicht zum gewünschten Ergebnis. Der
Molybdänkomplex liefert eine sehr geringe Anisotropie und der Rheniumkomplex weist einen
positiven axialen MAE-Parameter auf. Für den Wolframkomplex hingegen erhält man
unplausible Ergebnisse.
Im nächsten Schritt wurden die äußeren Manganzentren durch Mo(III) ersetzt. Bei Molybdän,
Wolfram und Rhenium ergeben sich genau dieselben Probleme wie bei den Mn2M-Komplexen.
Der Mo2Mn-Komplex hingegen weist eine deutlich höhere Anisotropie auf als der
Mn2Mn-Komplex.
Untersuchungen zum langzeitbeständigen Kleben von organisch bandbeschichteten
Stahlfeinblechen mittels Labormethoden stoßen immer wieder auf
die Problematik der korrosiven Unterwanderung der Beschichtung. Diese ist
für die Bestimmung der Langzeitbeständigkeit der Klebung hinderlich, da sie in
der Realität in dieser Form selten vorzufinden ist. Ein Ziel dieser Arbeit war es,
eine angepasste Klebprobengeometrie zu entwickeln, welche eine korrosive
Unterwanderung der organischen Bandbeschichtung in zeitraffenden Laboralterungstests
verhindert. Es konnte gezeigt werden, dass mittels der entwickelten
Probengeometrie eine zuverlässige Bestimmung des Alterungsverhaltens
an geklebten Verbunden mit organisch bandbeschichteten Stahlfeinblechen
möglich ist.
Hierauf aufbauend zielten die weiteren Untersuchungen mit zweikomponentigen
Polyurethan-Klebstoffen auf einen Vergleich der zeitraffenden Laboralterungsmethoden
mit nicht zeitraffenden Alterungsverfahren wie der Freibewitterung
und der Auslagerung am Feldfahrzeug. Die Auslagerung am
Feldfahrzeug erfolgte an Kühlfahrzeugen, welche im Speditionsverkehr auf
europäischen Routen verkehrten. Hauptaugenmerk bei allen Alterungsversuchen
war es neben der Bestimmung der Restfestigkeit auch die klimatischen
Expositionsbedingungen zu dokumentieren. Dies ermöglichte zum einen die
Interpretation der Zugscherergebnisse. Zum anderen konnte hieraus eine
Korrelation und letztendlich ein mathematisches Modell zur Festigkeitsvorhersage
von reversiblen Alterungseffekten abgeleitet werden. Mit diesem Modell
ist eine Vorhersage der Zugscherfestigkeit auf Basis eines bekannten Expositionsprofils
möglich.
Accurate path tracking control of tractors became a key technology for automation in agriculture. Increasingly sophisticated solutions, however, revealed that accurate path tracking control of implements is at least equally important. Therefore, this work focuses on accurate path tracking control of both tractors and implements. The latter, as a prerequisite for improved control, are equipped with steering actuators like steerable wheels or a steerable drawbar, i.e. the implements are actively steered. This work contributes both new plant models and new control approaches for those kinds of tractor-implement combinations. Plant models comprise dynamic vehicle models accounting for forces and moments causing the vehicle motion as well as simplified kinematic descriptions. All models have been derived in a systematic and automated manner to allow for variants of implements and actuator combinations. Path tracking controller design begins with a comprehensive overview and discussion of existing approaches in related domains. Two new approaches have been proposed combining the systematic setup and tuning of a Linear-Quadratic-Regulator with the simplicity of a static output feedback approximation. The first approach ensures accurate path tracking on slopes and curves by including integral control for a selection of controlled variables. The second approach, instead, ensures this by adding disturbance feedforward control based on side-slip estimation using a non-linear kinematic plant model and an Extended Kalman Filter. For both approaches a feedforward control approach for curved path tracking has been newly derived. In addition, a straightforward extension of control accounting for the implement orientation has been developed. All control approaches have been validated in simulations and experiments carried out with a mid-size tractor and a custom built demonstrator implement.
In embedded systems, there is a trend of integrating several different functionalities on a common platform. This has been enabled by increasing processing power and the arise of integrated system-on-chips.
The composition of safety-critical and non-safety-critical applications results in mixed-criticality systems. Certification Authorities (CAs) demand the certification of safety-critical applications with strong confidence in the execution time bounds. As a consequence, CAs use conservative assumptions in the worst-case execution time (WCET) analysis which result in more pessimistic WCETs than the ones used by designers. The existence of certified safety-critical and non-safety-critical applications can be represented by dual-criticality systems, i.e., systems with two criticality levels.
In this thesis, we focus on the scheduling of mixed-criticality systems which are subject to certification. Scheduling policies cognizant of the mixed-criticality nature of the systems and the certification requirements are needed for efficient and effective scheduling. Furthermore, we aim at reducing the certification costs to allow faster modification and upgrading, and less error-prone certification. Besides certification aspects, requirements of different operational modes result in challenging problems for the scheduling process. Despite the mentioned problems, schedulers require a low runtime overhead for an efficient execution at runtime.
The presented solutions are centered around time-triggered systems which feature a low runtime overhead. We present a transformation to include event-triggered activities, represented by sporadic tasks, already into the offline scheduling process. Further, this transformation can also be applied on periodic tasks to shorten the length of schedule tables which reduces certification costs. These results can be used in our method to construct schedule tables which creates two schedule tables to fulfill the requirements of dual-criticality systems using mode changes at runtime. Finally, we present a scheduler based on the slot-shifting algorithm for mixed-criticality systems. In a first version, the method schedules dual-criticality jobs without the need for mode changes. An already certified schedule table can be used and at runtime, the scheduler reacts to the actual behavior of the jobs and thus, makes effective use of the available resources. Next, we extend this method to schedule mixed-criticality job sets with different operational modes. As a result, we can schedule jobs with varying parameters in different modes.
In this thesis, collision-induced dissociation (CID) studies serve to elucidate relative stabilities and to determine bond strengths within a given structure type of transition metal complexes. The infrared multi photon dissociation (IRMPD) spectroscopy combined with density functional theory (DFT) allow for structural analysis and provide insights into the coordination sphere of transition metal centers. The used combination of CID and IRMPD experiments is a powerful tool to obtain a detailed and comprehensive characterization and understanding of interactions between transition metals and organic ligands. The compounds’ spectrum comprises mono- or oligonuclear transition metal complexes containing iron, palladium, and ruthenium as well as lanthanide containing single molecule magnets (SMM). The presented investigations on the different transition metal complexes reveal manifold effects for each species leading to valuable results. A fundamental understanding of metal to ligand interactions is mandatory for the development of new and better organometallic complexes with catalytic, optical or magnetic properties.
Many tasks in image processing can be tackled by modeling an appropriate data fidelity term \(\Phi: \mathbb{R}^n \rightarrow \mathbb{R} \cup \{+\infty\}\) and then solve one of the regularized minimization problems \begin{align*}
&{}(P_{1,\tau}) \qquad \mathop{\rm argmin}_{x \in \mathbb R^n} \big\{ \Phi(x) \;{\rm s.t.}\; \Psi(x) \leq \tau \big\} \\ &{}(P_{2,\lambda}) \qquad \mathop{\rm argmin}_{x \in \mathbb R^n} \{ \Phi(x) + \lambda \Psi(x) \}, \; \lambda > 0 \end{align*} with some function \(\Psi: \mathbb{R}^n \rightarrow \mathbb{R} \cup \{+\infty\}\) and a good choice of the parameter(s). Two tasks arise naturally here: \begin{align*} {}& \text{1. Study the solver sets \({\rm SOL}(P_{1,\tau})\) and
\({\rm SOL}(P_{2,\lambda})\) of the minimization problems.} \\ {}& \text{2. Ensure that the minimization problems have solutions.} \end{align*} This thesis provides contributions to both tasks: Regarding the first task for a more special setting we prove that there are intervals \((0,c)\) and \((0,d)\) such that the setvalued curves \begin{align*}
\tau \mapsto {}& {\rm SOL}(P_{1,\tau}), \; \tau \in (0,c) \\ {} \lambda \mapsto {}& {\rm SOL}(P_{2,\lambda}), \; \lambda \in (0,d) \end{align*} are the same, besides an order reversing parameter change \(g: (0,c) \rightarrow (0,d)\). Moreover we show that the solver sets are changing all the time while \(\tau\) runs from \(0\) to \(c\) and \(\lambda\) runs from \(d\) to \(0\).
In the presence of lower semicontinuity the second task is done if we have additionally coercivity. We regard lower semicontinuity and coercivity from a topological point of view and develop a new technique for proving lower semicontinuity plus coercivity.
Dropping any lower semicontinuity assumption we also prove a theorem on the coercivity of a sum of functions.
Computational Homogenization of Piezoelectric Materials using FE² Methods and Configurational Forces
(2015)
Piezoelectric materials are electro-mechanically coupled materials. In these materials it is possible to produce an electric field by applying a mechanical load. This phenomenon is known as the piezoelectric effect. These materials also exhibit a mechanical deformation in response to an external electric loading, which is known as the inverse piezoelectric effect. By using these smart properties of piezoelectric materials, applications are possible in sensors and actuators. Ferroelectric or piezoelectric materials show switching behavior of the polarization in the material under an external loading. Due to this property, these materials are used to produce random access memory (RAM) for the non-volatile storage of data in computing devices. It is essential to understand the material responses of piezoelectric materials properly in order to use them in the engineering applications in innovative manners. Due to the growing interest in determining the material responses of smart material (e.g., piezoelectric material), computational methods are becoming increasingly important.
Many engineering materials possess inhomogeneities on the micro level. These inhomogeneities in the materials cause some difficulties in the determination of the material responses computationally as well as experimentally. But on the other hand, sometimes these inhomogeneities help the materials to render some good physical properties, e.g., glass or carbon fiber reinforced composites are light weight, but show higher strength. Piezoelectric materials also exhibit intense inhomogeneities on the micro level. These inhomogeneities are originating from the presence of domains, domain walls, grains, grain boundaries, micro cracks, etc. in the material. In order to capture the effects of the underlying microstructures on the macro quantities, it is essential to homogenize material parameters and the physical responses. There are several approaches to perform the homogenization. A two-scale classical (first-order) homogenization of electro-mechanically coupled materials using a FE²-approach is discussed in this work. The main objective of this work is to investigate the influences of the underlying micro structures on the macro Eshelby stress tensor and on the macro configurational forces. The configurational forces are determined in certain defect situations. These defect situations include the crack tip of a sharp crack in the macro specimen.
A literature review shows that the macro strain tensor is used to determine the micro boundary condition for the FE²-based homogenization in a small strain setting. This approach is capable to determine the consistent homogenized physical quantities (e.g., stress, strain) and the homogenized material quantities (e.g., stiffness tensor). But the application of these type of micro boundaries for the homogenization does not generate physically consistent macro Eshelby stress tensor or the macro configurational forces. Even in the absence of the micro volume configurational forces, this approach of the homogenization of piezoelectric materials produces unphysical volume configurational forces on the macro level. After a thorough investigation of the boundary conditions on the representative volume elements (RVEs), it is found that a displacement gradient driven micro boundary conditions remedy this issue. The use of the displacement gradient driven micro boundary conditions also satisfies the Hill-Mandel condition. The macro Eshelby stress tensor of a pure mechanical problem in a small deformation setting can be determined in two possible ways: by using the homogenized mechanical quantities (displacement gradient and stress tensor), or by homogenizing the Eshelby stress tensor on the micro level by volume averaging. The first approach does not satisfy the Hill-Mandel condition incorporating the Eshelby stress tensor in the energy term, on the other hand, the Hill-Mandel condition is satisfied in the second approach. In the case of homogenized Eshelby stress tensor determined from the homogenized physical quantities, the Hill-Mandel condition gives an additional energy term. A body in a small deformation setting is deformed according to the displacement gradient. If the homogenization is done using strain driven micro boundary conditions, the micro domain is deformed according to the macro strain, but the tiny vicinity around the corresponding Gauß point is deformed according to the macro displacement gradient. This implies that some restrictions are imposed at every Gauß point on the macro level. This situation helps the macro system to produce nonphysical volume configurational forces.
A FE²-based computational homogenization technique is also considered for the homogenization of piezoelectric materials. In this technique a representative volume element, which comprises of the micro structural features in the material, is assigned to every Gauß point of the macro domain. The macro displacement gradient and the macro electric field, or the macro stress tensor and the macro electric displacement are passed to the RVEs at every macro Gauß point. After determining boundary conditions on the RVEs, the homogenization process is performed. The homogenized physical quantities and the homogenized material parameters are passed back to macro Gauß points. In this work numerical investigations are carried out for two distinct situations of the microstructures of the piezoelectric materials regarding the evolution on the micro level: a) homogenization by using stationary microstructures, and b) homogenization by using evolving microstructures.
For the first case, the domain walls remain at fixed positions through out the simulations for the homogenization of piezoelectric materials. For a considerably large external loading, the real situation is different. But to understand the effects of the underlying microstructures on the macro configurational forces, to some extent it is sufficient to do the homogenization with fixed or stationary microstructures. The homogenization process is carried out for different microstructures and for different loading conditions. If the mechanical load is applied in the direction of the polarization, a smaller crack tip configurational force is observed in comparison to the configurational force determined for a mechanical loading perpendicular to the polarization. If the polarizations in the microstructures are parallel or perpendicular to the applied electric field and the applied displacement, configurational forces parallel to the crack ligament of the macro crack are observed only. In the case of inclined polarizations in the microstructures, configurational forces inclined to the crack ligament are obtained. The simulation results also reveal that an application of an external electric field to the material reduces the value of the nodal configurational forces at the crack tip.
In the second case, the interfaces of the micro structures are allowed to move from their initial positions at every step of the applied incremental external loading. Thus, at every step of the application of the external loading, the microstructures are changed when the external loading is larger than the coercive field. The movement of the interfaces is realized through the nodal configurational forces on the micro level. At every step of the application of the external loading, the nodal configurational forces per unit length on the domain walls are determined in the post-processing of the FE-simulation on the micro domain. With the help of the domain wall kinetics, the new positions of the domain walls are determined. Numerical results show that the crack tip region is the most affected area in the macro domain. For that reason a very different distribution of the macro electric displacement is observed comparing the same produced by using fixed microstructures. Due to the movement of the domain walls, the energy is dissipated in the system. As a result, a smaller configurational force appears at the crack tip on the macro level in the case of the homogenization by using evolving microstructures. By using the homogenization technique involving the evolution of the microstructures, it is possible to produce the electric displacement vs. electric field hysteresis loop on the macro level. The shape of the hysteresis loop depends on the value of the rate of application of the external electric loading. A faster deployment of the external electric field widens the hysteresis loop.
Motivated by the results of infinite dimensional Gaussian analysis and especially white noise analysis, we construct a Mittag-Leffler analysis. This is an infinite dimensional analysis with respect to non-Gaussian measures of Mittag-Leffler type which we call Mittag-Leffler measures. Our results indicate that the Wick ordered polynomials, which play a key role in Gaussian analysis, cannot be generalized to this non-Gaussian case. We provide evidence that a system of biorthogonal polynomials, called generalized Appell system, is applicable to the Mittag-Leffler measures, instead of using Wick ordered polynomials. With the help of an Appell system, we introduce a test function and a distribution space. Furthermore we give characterizations of the distribution space and we characterize the weak integrable functions and the convergent sequences within the distribution space. We construct Donsker's delta in a non-Gaussian setting as an application.
In the second part, we develop a grey noise analysis. This is a special application of the Mittag-Leffler analysis. In this framework, we introduce generalized grey Brownian motion and prove differentiability in a distributional sense and the existence of generalized grey Brownian motion local times. Grey noise analysis is then applied to the time-fractional heat equation and the time-fractional Schrödinger equation. We prove a generalization of the fractional Feynman-Kac formula for distributional initial values. In this way, we find a Green's function for the time-fractional heat equation which coincides with the solutions given in the literature.
The heterogeneity of today's access possibilities to wireless networks imposes challenges for efficient mobility support and resource management across different Radio Access Technologies (RATs). The current situation is characterized by the coexistence of various wireless communication systems, such as GSM, HSPA, LTE, WiMAX, and WLAN. These RATs greatly differ with respect to coverage, spectrum, data rates, Quality of Service (QoS), and mobility support.
In real systems, mobility-related events, such as Handover (HO) procedures, directly affect resource efficiency and End-To-End (E2E) performance, in particular with respect to signaling efforts and users' QoS. In order to lay a basis for realistic multi-radio network evaluation, a novel evaluation methodology is introduced in this thesis.
A central hypothesis of this thesis is that the consideration and exploitation of additional information characterizing user, network, and environment context, is beneficial for enhancing Heterogeneous Access Management (HAM) and Self-Optimizing Networks (SONs). Further, Mobile Network Operator (MNO) revenues are maximized by tightly integrating bandwidth adaptation and admission control mechanisms as well as simultaneously accounting for user profiles and service characteristics. In addition, mobility robustness is optimized by enabling network nodes to tune HO parameters according to locally observed conditions.
For establishing all these facets of context awareness, various schemes and algorithms are developed and evaluated in this thesis. System-level simulation results demonstrate the potential of context information exploitation for enhancing resource utilization, mobility support, self-tuning network operations, and users' E2E performance.
In essence, the conducted research activities and presented results motivate and substantiate the consideration of context awareness as key enabler for cognitive and autonomous network management. Further, the performed investigations and aspects evaluated in the scope of this thesis are highly relevant for future 5G wireless systems and current discussions in the 5G infrastructure Public Private Partnership (PPP).