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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.
We consider the multiscale model for glioma growth introduced in a previous work and extend it to account
for therapy effects. Thereby, three treatment strategies involving surgical resection, radio-, and
chemotherapy are compared for their efficiency. The chemotherapy relies on inhibiting the binding
of cell surface receptors to the surrounding tissue, which impairs both migration and proliferation.
A new solution approach for solving the 2-facility location problem in the plane with block norms
(2015)
Motivated by the time-dependent location problem over T time-periods introduced in
Maier and Hamacher (2015) we consider the special case of two time-steps, which was shown
to be equivalent to the static 2-facility location problem in the plane. Geometric optimality
conditions are stated for the median objective. When using block norms, these conditions
are used to derive a polygon grid inducing a subdivision of the plane based on normal cones,
yielding a new approach to solve the 2-facility location problem in polynomial time. Combinatorial algorithms for the 2-facility location problem based on geometric properties are
deduced and their complexities are analyzed. These methods differ from others as they are
completely working on geometric objects to derive the optimal solution set.
A nonlocal stochastic model for intra- and extracellular proton dynamics in a tumor is proposed.
The intracellular dynamics is governed by an SDE coupled to a reaction-diffusion
equation for the extracellular proton concentration on the macroscale. In a more general context
the existence and uniqueness of solutions for local and nonlocal
SDE-PDE systems are established allowing, in particular, to analyze the proton dynamics model both,
in its local version and the case with nonlocal path dependence.
Numerical simulations are performed
to illustrate the behavior of solutions, providing some insights into the effects of randomness on tumor acidity.
In this paper we propose a phenomenological model for the formation of an interstitial gap between the tumor and the stroma. The gap
is mainly filled with acid produced by the progressing edge of the tumor front. Our setting extends existing models for acid-induced tumor invasion models to incorporate
several features of local invasion like formation of gaps, spikes, buds, islands, and cavities. These behaviors are obtained mainly due to the random dynamics at the intracellular
level, the go-or-grow-or-recede dynamics on the population scale, together with the nonlinear coupling between the microscopic (intracellular) and macroscopic (population)
levels. The wellposedness of the model is proved using the semigroup technique and 1D and 2D numerical simulations are performed to illustrate model predictions and draw
conclusions based on the observed behavior.
Ausgehend von Andrew Moravcsiks liberaler Außenpolitiktheorie, nach
der gesellschaftliche Akteure rationale und risikoaverse bzw. kostensensitive
Präferenzen verfolgen, die konstitutiv für die Außen- und Sicherheitspolitik
von Staaten sind, untersucht die vorliegende Studie entlang
der Fallbeispiele Afghanistankrieg, Politik der gezielten Tötungen und
Schließung Guantánamos, inwiefern diese Präferenzen des amerikanischen
Demos die Politik der Terrorismusbekämpfung der Obama-
Administration maßgeblich beeinflussen, ob sich das politische System
gegenüber dem Präferenzmuster des Demos responsiv verhält und
unter welchen Kontextbedingungen gesellschaftliche Präferenzen von
Kongress und Präsident repräsentiert werden. Gestützt auf der Analyse
von Einstellungsmustern der amerikanischen Öffentlichkeit und dem
Kongruenzabgleich zwischen gesellschaftlichen Präferenzen und dem
Handeln von Kongress und Präsident ergibt sich ein widersprüchliches
Bild mit Blick auf die Grundannahmen der liberalen Außenpolitiktheorie.
Der amerikanische Demos favorisierte die gezielte Tötung von
Terroristen, lehnte aber die Schließung Guantánamos und die Intensivierung
des militärischen Einsatzes in Afghanistan ab. Der Kongress als
zentraler Transmissionsriemen gesellschaftlicher Präferenzen unterstützte
einerseits den Präsidenten seit 9/11 bei seiner Terrorismusbekämpfungsstrategie,
scheiterte aber trotz seiner verfassungsrechtlichen
Kompetenzen daran den exekutiven Dominanzanspruch beim Einsatz
militärischer Mittel in Frage zu stellen. Andererseits verweigerte der
Kongress fortlaufend die Schließung Guantánamos. Ziel dieser qualitativen
Studie ist es deshalb durch Verwendung von wirkmächtigen
intervenierenden Variablen wie Schwellenwertbedingung, Parteipolitische
Polarisierung und Politikfeldspezifizität - die argumentative und modelltheoretische
Erweiterung der liberalen Außenpolitiktheorie voranzutreiben
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.