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Cancer, a complex and multifaceted disease, continues to challenge the boundaries of biomedical research. In this dissertation, we explore the complexity of cancer genesis, employing multiscale modeling, abstract mathematical concepts such as stability analysis, and numerical simulations as powerful tools to decipher its underlying mechanisms. Through a series of comprehensive studies, we mainly investigate the cell cycle dynamics, the delicate balance between quiescence and proliferation, the impact of mutations, and the co-evolution of healthy and cancer stem cell lineages. The introductory chapter provides a comprehensive overview of cancer and the critical importance of understanding its underlying mechanisms. Additionally, it establishes the foundation by elucidating key definitions and presenting various modeling perspectives to address the cancer genesis. Next, cell cycle dynamics have been explored, revealing the temporal oscillatory dynamics that govern the progression of cells through the cell cycle.
The first half of the thesis investigates the cell cycle dynamics and evolution of cancer stem cell lineages by incorporating feedback regulation mechanisms. Thereby, the pivotal role of feedback loops in driving the expansion of cancer stem cells has been thoroughly studied, offering new perspectives on cancer progression. Furthermore, the mathematical rigor of the model has been addressed by deriving wellposedness conditions, thereby strengthening the reliability of our findings and conclusions. Then, expanding our modeling scope, we explore the interplay between quiescent and proliferating cell populations, shedding light on the importance of their equilibrium in cancer biology. The models developed in this context offer potential avenues for targeted cancer therapies, addressing perspective cell populations critical for cancer progression. The second half of the thesis focuses on multiscale modeling of proliferating and quiescent cell populations incorporating cell cycle dynamics and the extension thereof with mutation acquisition. Following rigorous mathematical analysis, the wellposedness of the proposed modeling frameworks have been studied along with steady-state solutions and stability criteria.
In a nutshell, this thesis represents a significant stride in our understanding of cancer genesis, providing a comprehensive view of the complex interplay between cell cycle dynamics, quiescence, proliferation, mutation acquisition, and cancer stem cells. The journey towards conquering cancer is far from over. However, this research provides valuable insights and directions for future investigation, bringing us closer to the ultimate goal of mitigating the impact of this formidable disease.
In this thesis, material removal mechanisms in grinding are investigated considering a gritworkpiece interaction as well as a grinding-wheel workpiece interaction. In grit-workpiece interaction in a micrometer scale, single grit scratch experiments were performed to investigate material removal mechanism in grinding namely rubbing, plowing, and cutting. Experiments performed were analyzed based on material removal, process forces and specific energy. A finite element model is developed to simulate a single-grit scratch process. As part of the development of the finite element scratch model a 2D and 3D model is developed. A 2D model is utilized to test
material parameters and test various mesh discretizational approaches. A 3D model undertaking the tested material parameters from the 2D model is developed and is tested against experimental results for various mesh discretization. The simulation model is validated based on process forces and ground topography from experiments. The model is also further scaled to simulate multiple grit-workpiece interaction validated against experimental results. As a final step, simulation models are developed to simulate material removal, due to the interaction of grinding wheel and workpiece. A developed virtual grinding wheel topographical model is employed to display
an approach, to upscale a grinding process from grit-workpiece interaction to wheel-workpiece
interaction. In conclusion, practical conclusions drawn and scope for future studies are derived
based on the developed simulation models.
Climate change will have severe consequences on Eastern Boundary Upwelling Systems (EBUS). They host the largest fisheries in the world supporting the life of millions of people due to their tremendous primary production. Therefore, it is of utmost importance to better understand predicted impacts like alternating upwelling intensities and light impediment on the structure and the trophic role of protistan plankton communities as they form the basis of the food web. Numerical models estimate the intensification of the frequency in eddy formation. These ocean features are of particular importance due to their influence on the distribution and diversity of plankton communities and the access to resources, which are still not well understood even to the present day. My PhD thesis entails two subjects conducted during large-scaled cooperation projects REEBUS (Role of Eddies in Eastern Boundary Upwelling Systems) and CUSCO (Coastal Upwelling System in a Changing Ocean).
Subject I of my study was conducted within the multidisciplinary framework REEBUS to investigate the influence of eddies on the biological carbon pump in the Canary Current System (CanCS). More specifically, the aim was to find out how mesoscale cyclonic eddies affect the regional diversity, structure, and trophic role of protistan plankton communities in a subtropical oligotrophic oceanic offshore region.
Samples were taken during the M156 and M160 cruises in the Atlantic Ocean around Cape Verde during July and December 2019, respectively. Three eddies with varying ages of emergence and three water layers (deep chlorophyll maximum DCM, right beneath the DCM and oxygen minimum zone OMZ) were sampled. Additional stations without eddy perturbation were analyzed as references. The effect of oceanic mesoscale cyclonic eddies on protistan plankton communities was analyzed by implementing three approaches. (i) V9 18S rRNA gene amplicons were examined to analyze the diversity and structure of the plankton communities and to infer their role in the biological carbon pump. (ii) By assigning functional traits to taxonomically assigned eDNA sequences, functional richness and ecological strategies (ES) were determined. (iii) Grazing experiments were conducted to assess abundance and carbon transfer from prokaryotes to phagotrophic protists.
All three eddies examined in this study differed in their ASV abundance, diversity, and taxonomic composition with the most pronounced differences in the DCM. Dinoflagellates were the most abundant taxa in all three depth layers. Other dominating taxa were radiolarians, Discoba and haptophytes. The trait-approach could only assign ~15% of all ASVs and revealed in general a relatively high functional richness. But no unique ES was determined within a specific eddy. This indicates pronounced functional redundancy, which is recognized to be correlated with ecosystem resilience and robustness by providing a degree of buffering capacity in the face of biodiversity loss. Elevated microbial abundances as well as bacterivory were clearly associated to mesoscale eddy features, albeit with remarkable seasonal fluctuations. Since eddy activity is expected to increase on a global scale in future climate change scenarios, cyclonic eddies could counteract climate change by enhancing carbon sequestration to abyssal depths. The findings demonstrate that cyclonic eddies are unique, heterogeneous, and abundant ecosystems with trapped water masses in which characteristic protistan plankton develop as the eddies age and migrate westward into subtropical oligotrophic offshore waters. Therefore, eddies influence regional protistan plankton diversity qualitatively and quantitatively.
Subject II of my PhD project contributed to the CUSCO field campaign to identify the influence of varying upwelling intensities in combination with distinct light treatments on the whole food web structure and carbon pump in the Humboldt Current System (HCS) off Peru. To accomplish such a task, eight offshore-mesocosms were deployed and two light scenarios (low light, LL; high light, HL) were created by darkening half of the mesocosms. Upwelling was simulated by injecting distinct proportions (0%, 15%, 30% and 45%) of collected deep-water (DW) into each of the moored mesocosms. My aim was to examine the changes in diversity, structure, and trophic role of protistan plankton communities for the induced manipulations by analyzing the V9 18S rRNA gene amplicons and performing short-term grazing experiments.
The upwelling simulations induced a significant increase in alpha diversity under both light conditions. In austral summer, reflected by HL conditions, a generally higher alpha diversity was recorded compared to the austral winter simulation, instigated by LL treatment. Significant alterations of the protistan plankton community structure could likewise be observed. Diatoms were associated to increased levels of DW addition in the mimicked austral winter situation. Under nutrient depletion, chlorophytes exhibited high relative abundances in the simulated austral winter scenario. Dinoflagellates dominated the austral summer condition in all upwelling simulations. Tendencies of reduced unicellular eukaryotes and increased prokaryotic abundances were determined under light impediment. Protistan-mediated mortality of prokaryotes also decreased by ~30% in the mimicked austral winter scenario.
The findings indicate that the microbial loop is a more relevant factor in the structure of the food web in austral summer and is more focused on the utilization of diatoms in austral winter in the HCS off Peru. It was evident that distinct light intensities coupled with multiple upwelling scenarios could lead to alterations in biochemical cycles, trophic interactions, and ecosystem services. Considering the threat of climate change, the predicted relocation of EBUS could limit primary production and lengthen the food web structure with severe socio-economic consequences.
Mixed Isogeometric Methods for Hodge–Laplace Problems induced by Second-Order Hilbert Complexes
(2024)
Partial differential equations (PDEs) play a crucial role in mathematics and physics to describe numerous physical processes. In numerical computations within the scope of PDE problems, the transition from classical to weak solutions is often meaningful. The latter may not precisely satisfy the original PDE, but they fulfill a weak variational formulation, which, in turn, is suitable for the discretization concept of Finite Elements (FE). A central concept in this context is the
well-posed problem. A class of PDE problems for which not only well-posedness statements but also suitable weak formulations are known are the so-called abstract Hodge–Laplace problems. These can be derived from Hilbert complexes and constitute a central aspect of the Finite Element Exterior Calculus (FEEC).
This thesis addresses the discretization of mixed formulations of Hodge-Laplace problems, focusing on two key aspects. Firstly, we utilize Isogeometric Analysis (IGA) as a specific paradigm for discretization, combining geometric representations with Non-Uniform Rational B-Splines (NURBS) and Finite Element discretizations.
Secondly, we primarily concentrate on mixed formulations exhibiting a saddle-point structure and generated from Hilbert complexes with second-order derivative operators. We go beyond the well-known case of the classical de Rham
complex, considering complexes such as the Hessian or elasticity complex. The BGG (Bernstein–Gelfand–Gelfand) method is employed to define and examine these second-order complexes. The main results include proofs of discrete well-posedness and a priori error estimates for two different discretization approaches. One approach demonstrates, through the introduction of a Lagrange multiplier, how the so-called isogeometric discrete differential forms can be reused.
A second method addresses the question of how standard NURBS basis functions, through a modification of the mixed formulation, can also lead to convergent procedures. Numerical tests and examples, conducted using MATLAB and the open-source software GeoPDEs, illustrate the theoretical findings. Our primary application extends to linear elasticity theory, extensively
discussing mixed methods with and without strong symmetry of the stress tensor.
The work demonstrates the potential of IGA in numerical computations, particularly in the challenging scenario of second-order Hilbert complexes. It also provides insights into how IGA and FEEC can be meaningfully combined, even for non-de Rham complexes.
The aim of this thesis is to introduce an equilibrium insurance market model and study its properties and possible applications in risk class management.
First, an insurance market model based on an equilibrium approach is developed. Depending on the premium, the insured will choose the amount of coverage they buy in order to maximize their expected utility. The behavior of the insurer in different market regimes is then compared. While the premiums in markets with perfect competition are calculated in order to make no profit at all, insurers try to maximize their margins in a monopolistic market.
In markets modeled in this way several phenomena become evident. Perhaps the most important one is the so-called push-out effect. When customers with different attributes are insured together, insurance might become so expensive for one type of customers that those agents are better off with buying no insurance at all. The push-out effect was already shown for theoretical examples in the literature. We present a comprehensive analysis of the equilibrium insurance market model and the push-out effect for different insurance products such as life, health and disability insurance contracts using real-life data from different sources. In a concluding chapter we formulate indicators when a push-out can be expected and when not.
Machine learning regression approaches such as neural networks have gained vast popularity in recent years. The exponential growth of computing power has enabled larger and more evolved networks that can perform increasingly complex tasks. In our feasibility study about the use of neural networks in the regression of equilibrium insurance premiums it is shown that this regression is quite robust and the risk of overfitting can almost be excluded -- as long as the regression is performed on at least a few thousand data points.
Grouping customers of different risk types into contracts is important for the stability and the robustness of an insurance market. This motivates the study of the optimal assignment of risk classes into contracts, also known as rating classes. We provide a theoretical framework that makes use of techniques from different mathematical fields such as non-linear optimization, convex analysis, herding theory, game theory and combinatorics. In addition, we are able to show that the market specifications have a large impact on the optimal allocation of risk classes to contracts by the insurer. However, there does not need to be an optimal risk class assignment for each of these specifications.
To address this issue, we present two different approaches, one more theoretical and another that can easily be implemented in practice. An extension of our model to markets with capacity constraints rounds off the topic and extends the applicability of our approach.
River ecosystems are being threatened by rising temperatures, aridity, and salinity due to climate change and increased water abstractions. These threats also put human well-being at risk, as people and rivers are closely connected, particularly in water-scarce regions. We aimed to investigate the relationship between human well-being and biological and physico-chemical river water quality using the arid Draa River basin as a case study. Physico-chemical water measurements, biological monitoring of aquatic macroinvertebrates, and household surveys were used to assess the state of the river water, ecosystem, and human well-being, as well as the associations between them. Salinity levels exceeded maximum permissible values for drinking water in 35 % and irrigation water in 12 % of the sites. Salinity and low flow were associated with low biological quality. Human satisfaction with water quantity and quality, agriculture, the natural environment, and overall life satisfaction were low particularly in the Middle Draa, where 89% of respondents reported emotional distress due to water salinity and scarcity. Drinking and irrigation water quality was generally rated lower in areas characterized by higher levels of water salinity and scarcity. The study found positive associations between the river water quality and biological quality indices, but no significant association between these factors and human satisfaction. These findings suggest that the relationship between human satisfaction and the biological and physicochemical river water quality is complex and that a more comprehensive approach to human well-being is likely needed to establish relationships.
Understanding human crowd behaviour has been an intriguing topic of interdisciplinary research in recent decades. Modelling of crowd dynamics using differential equations is an indispensable approach to unraveling the various complex dynamics involved in such interacting particle systems. Numerical simulation of pedestrian crowd via these mathematical models allows us to study different realistic scenarios beyond the limitations of studies via controlled experiments.
In this thesis, the main objective is to understand and analyse the dynamics in a domain shared by both pedestrians and moving obstacles. We model pedestrian motion by combining the social force concept with the idea of optimal path computation. This leads to a system of ordinary differential equations governing the dynamics of individual pedestrians via the interaction forces (social forces) between them. Additionally, a non-local force term involving the optimal path and desired velocity governs the pedestrian trajectory. The optimal path computation involves solving a time-independent Eikonal equation, which is coupled to the system of ODEs. A hydrodynamic model is developed from this microscopic model via the mean-field limit.
To consider the interaction with moving obstacles in the domain, we model a set of kinematic equations for the obstacle motion. Two kinds of obstacles are considered - "passive", which move in their predefined trajectories and have only a one-way interaction with pedestrians, and "dynamic", which have a feedback interaction with pedestrians and have their trajectories changing dynamically. The coupled model of pedestrians and obstacles is used to discern pedestrian collision avoidance behaviour in different computational scenarios in a long rectangular domain. We observe that pedestrians avoid collisions through route choice strategies that involve changes in speed and path. We extend this model to consider the interaction between pedestrians and vehicular traffic. We appropriately model the interactions of vehicles, following lane traffic, based on the car-following approach. We observe how the deceleration and braking mechanism of vehicles is executed at pedestrian crossings depending on the right of way on the roads.
As a second objective, we study the disease contagion in moving crowds. We consider the influence of the crowd motion in a complex dynamical environment on the course of infection of pedestrians. A hydrodynamic model for multi-group pedestrian flow is derived from the kinetic equations based on a social force model. It is coupled along with an Eikonal equation to a non-local SEIS contagion model for disease spread. Here, apart from the description of local contacts, the influence of contact times has also been modelled. We observe that the nature of the flow and the geometry of the domain lead to changes in density which affect the contact time and, consequently, the rate of spread of infection.
Finally, the social force model is compared to a variable speed based rational behaviour pedestrian model. We derive a hierarchy of the heuristics-based model from microscopic to macroscopic scales and numerically investigate these models in different density scenarios. Various numerical test cases are considered, including uni- and bi-directional flows and scenarios with and without obstacles. We observe that in low-density scenarios, collision avoidance forces arising from the behavioural heuristics give valid results. Whereas in high-density scenarios, repulsive force terms are essential.
The numerical simulations of all the models are carried out using a mesh-free particle method based on least square approximations. The meshfree numerical framework provides an efficient and elegant way to handle complex geometric situations involving boundaries and stationary or moving obstacles.
Lubricated tribological contact processes are important in both nature and in many technical applications. Fluid lubricants play an important role in contact processes, e.g. they reduce friction and cool the contact zone. The fundamentals of lubricated contact processes on the atomistic scale are, however, today not fully understood. A lubricated contact process is defined here as a process, where two solid bodies that are in close proximity and eventually in parts in direct contact, carry out a relative motion, whereat the remaining volume is submersed by a fluid lubricant. Such lubricated contact processes are difficult to examine experimentally. Atomistic simulations are an attractive alternative for investigating the fundamentals of such processes. In this work, molecular dynamics simulations were used for studying different elementary processes of lubricated tribological contacts. A simplified, yet realistic simulation setup was developed in this work for that purpose using classical force fields. In particular, the two solid bodies were fully submersed in the fluid lubricant such that the squeeze-out was realistically modeled. The velocity of the relative motion of the two solid bodies was imposed as a boundary condition. Two types of cases were considered in this work: i) a model system based on synthetic model substances, which enables a direct, but generic, investigation of molecular interaction features on the contact process; and ii) real substance systems, where the force fields describe specific real substances. Using the model system i), also the reproducibility of the findings obtained from the computer experiments was critically assessed. In most cases, also the dry reference case was studied. Both mechanical and thermodynamic properties were studied -- focusing on the influence of lubrication. The following properties were studied: The contact forces, the coefficient of friction, the dislocation behavior in the solid, the chip formation and the formation of the groove, the squeeze-out behavior of the fluid in the contact zone, the local temperature and the energy balance of the system, the adsorption of fluid particles on the solid surfaces, as well as the formation of a tribofilm. Systematic studies were carried out for elucidating the influence of the wetting behavior, the influence of the molecular architecture of the lubricant, and the influence of the lubrication gap height on the contact process. As expected, the presence of a fluid lubricant reduces the temperature in the vicinity of the contact zone. The presence of the lubricant is, moreover, found to have a significant influence on the friction and on the energy balance of the process. The presence of a lubricant reduces the coefficient of friction compared to a dry case in the starting phase of a contact process, while lubricant molecules remain in the contact zone between the two solid bodies. This is a result of an increased normal and slightly decreased tangential force in the starting phase. When the fluid molecules are squeezed out with ongoing contact time and the contact zone is essentially dry, the coefficient of friction is increased by the presence of a fluid compared to a dry case. This is attributed to an imprinting of individual fluid particles into the solid surface, which is energetically unfavorable. By studying the contact process in a wide range of gap height, the entire range of the Stribeck curve is obtained from the molecular simulations. Thereby, the three main lubrication regimes of the Stribeck curve and their transition regions are covered, namely boundary lubrication (significant elastic and plastic deformation of the substrate), mixed lubrication (adsorbed fluid layers dominate the process), and hydrodynamic lubrication (shear flow is set up between the surface and the asperity). The atomistic effects in the different lubrication regimes are elucidated. Notably, the formation of a tribofilm is observed, in which lubricant molecules are immersed into the metal surface. The formation of a tribofilm is found to have important consequences for the contact process. The work done by the relative motion is found to mainly dissipate and thereby heat up the system. Only a minor part of the work causes plastic deformation. Finally, the assumptions, simplifications, and approximations applied in the simulations are critically discussed, which highlights possible future work.
Reactive absorption with amines is the most important technique for the removal of CO2
from gas streams, e.g. from flue gas, natural gas or off-gas from the cement industry.
In this work a rigorous simulation model for the absorption and desorption of CO2 with
an amine-containing solvent is validated using data from pilot plants of various sizes.
This model was then coupled with a detailed simulation of a coal-fired power plant.
The power generation efficiency drop with CO2 capture was determined and process
parameters in the power plant and separation process were optimized. It was shown
that the high energy demand of CO2 separation significantly reduces power generation
efficiencies, which underlines the need for improvements. This can be achieved by better
solvents or by advanced process designs. In this work such improved CO2 separation
processes are described and evaluated by detailed simulation studies.
In order to develop detailed rigorous simulation models for reactive absorption with novel
solvent systems, a precise knowledge of the liquid phase reaction kinetics is necessary.
There are well established techniques for measuring species distributions in equilibirated
aqueous amine solutions by NMR spectrosopy. However, the existing NMR techniques
cannot be used for monitoring fast reactions in these solutions. Therefore, in this work
a novel temperature-controlled micro-reactor NMR probe head was developed which
enables studying reaction kinetics with time constants in the range of seconds.
On this basis, modern solvent systems for CO2 absorption can be characterized and
the scale-up of separation process for future plants can be accompanied using rigorous
process simulation.
Distributed Optimization of Constraint-Coupled Systems via Approximations of the Dual Function
(2024)
This thesis deals with the distributed optimization of constraint-coupled systems. This problem class is often encountered in systems consisting of multiple individual subsystems, which are coupled through shared limited resources. The goal is to optimize each subsystem in a distributed manner while still ensuring that system-wide constraints are satisfied. By introducing dual variables for the system-wide constraints the system-wide problem can be decomposed into individual subproblems. These resulting subproblems can then be coordinated by iteratively adapting the dual variables. This thesis presents two new algorithms that exploit the properties of the dual optimization problem. Both algorithms compute a quadratic surrogate function of the dual function in each iteration, which is optimized to adapt the dual variables. The Quadratically Approximated Dual Ascent (QADA) algorithm computes the surrogate function by solving a regression problem, while the Quasi-Newton Dual Ascent (QNDA) algorithm updates the surrogate function iteratively via a quasi-Newton scheme. Both algorithms employ cutting planes to take the nonsmoothness of the dual function into account. The proposed algorithms are compared to algorithms from the literature on a large number of different benchmark problems, showing superior performance in most cases. In addition to general convex and mixed-integer optimization problems, dual decomposition-based distributed optimization is applied to distributed model predictive control and distributed K-means clustering problems.