The diploma thesis deals with the loss of stability of thin-walled flue pipe at coal power plant. The problem of thin-walled structures is their propensity to the loss of stability due to abnormal conditions, for example a vacuum, effect of wind, snow and earthquake. The purpose is to find the optimal deployment of reinforcements so that the model meets the requirements of EN 1993-1-6. This required an evaluation of unreinforced version of the model to identify weaknesses and subsequently other configurations for the application of reinforcement. For determining the resistance the flue pipe to the loss of stability used primarily to access linear bifurcation analysis based on linear analysis, possibly materially nonlinear analysis. The result of work is a design of reinforcement of pipeline which under the requirements of the standard meets the latest model with a reserve of approximately 20 %.

Das Phänomen der Rissmusterentstehung stellt ein anschauliches und numerisch nachvollziehbares Beispiel für Strukturbildung dar. Im Rahmen der vorliegenden Arbeit gelang es, einfache dreidimensionale Rissmuster zu berechnen, die durch Zugspannungen infolge inhomogener Schrumpfung durch Wärme- oder Stofftransport wachsen. Aufgrund wechselseitiger Entlastung bleiben Risse oder Teile einer Rissfront zurück. Für idealisierte periodische Rissanordnungen wurden durch bruchmechanische Verzweigungsanalyse kritische Last- und Geometrieparameter berechnet, für die sich neben der Grundlösung eine weitere Lösung finden lässt. Dadurch konnte die Frage beantwortet werden, warum verschiedene Typen von Rissmustern entstehen. Untersucht wurden Rissmuster in trocknender Stärke und Basalt sowie Tunnelrisse zwischen Glasplatten. Dafür war es nötig, die Risskontur numerisch zu bestimmen, wofür ein neues iteratives Verfahren entwickelt wurde. Für Säulen bildende Rissmuster in Basalt und trocknender Stärke mit konstanten Säulendurchmessern, welche stationär getrieben wachsen, wurde gezeigt, dass entgegen der Annahme in der Literatur eine Verzweigungslösung für zusammenhängende dreidimensionale Rissmuster existiert. Die berechneten kritischen Rissabstände unterscheiden sich nur unwesentlich von denjenigen zweidimensionaler Rissmodelle. Zum Verständnis der Herausbildung der Basaltsäulen wurde ein neuer Mechanismus gefunden, für den die Wechselwirkung zwischen Riss- und Erstarrungsfront der flüssigen Lava eine wesentliche Rolle spielt. Mittels eines iterativen Verfahrens wurde dafür die Erstarrungsfront berechnet. Die Ergebnisse zeigen gute Übereinstimmung mit den experimentellen Werten. Die Durchmesser von Stärkesäulen und Basaltsäulen nehmen nach dem gleichen Potenzgesetz mit zunehmender Wachstumsgeschwindigkeit ab. Für Tunnelrisse sind die Rissabstände nahezu konstant, obwohl entgegen ursprünglicher Annahmen gezeigt werden konnte, dass diese instationär getrieben wachsen. Als Ursache dafür wurde die durch die Rissgeometrie stark reduzierte Wechselwirkung zwischen den Tunnelrissen erkannt. Der Rissabstand nimmt potenzartig mit Dicke der trocknenden Schicht zu. Mittels einer Skalenanalyse konnte gute Übereinstimmung mit den Experimenten gefunden werden. / Crack patterns are a remarkable example of pattern formation. In this work simple three-dimensional crack patterns where analysed that are driven by inhomogeneous shrinkage caused from heat or mass transport. Caused by mutual unloading, parts of the crack front stop to grow. For idealised periodically repeatable crack configurations through fracture mechanics bifurcation analysis critical loading or geometry parameters where calculated for which beside the basic solution (growth of all cracks) another solution is possible. Analysed were crack patterns in drying starch and basalt and tunnelling cracks between glass plates. For this it was necessary to calculate the crack contour numerically, for this a new iteration method was developed. For the connected crack patterns in starch and basalt that form columns, driven stationary, it could be shown that there exists a bifurcation solution. Agreement with experimental data was found by including the solidification front ahead the crack tips. For the tunnelling cracks the crack spacing is nearly constant also they are unsteadily driven. The crack spacing scales with a power law of the thickness of the drying layer. With a scaling analysis good agreement with experimental data was found.

Several technologies involve the coating of a bulk material with a thin layer made up of another material, so as to achieve enhanced mechanical properties for the composite system. The use of coated solids embraces a broad field of applications, so that a strong research effort has been devoted to these systems. From a mechanical point of view, a coating layer diffuses the load on an attached solid in a non-local way, thus introducing a characteristic length, and profoundly affects the mechanical response and failure mechanisms of the coated object. Therefore, the development of mechanical models to describe the behaviour of coated materials plays an important role in engineering design. In the framework of linear elasticity, the case of an elastic thin layer, perfectly bonded to an elastic disk, is analyzed in the present thesis by providing a mathematical tool with which to determine the mechanical response of the coating/bulk complex, which may find application in micro and nano technologies, for instance in the characterization of nanowires via nanoindentation. The coating is modelled by means of an Euler-Bernoulli curved rod, assumed to be perfectly bonded on the boundary of a circular elastic disk. The elastic rod acts as a coating for the disk and its axial inextensibility imposes an isoperimetric constraint on the internal disk, which is constrained to maintain its perimeter constant during the deformation process. The mechanical model for the coating/disk system is formulated for general loading, using the complex potential formalism. The elastic rod becomes equivalent to a Benveniste-Miloh interface characterized by the bending stiffness of the rod; in this way the problem can be solved entirely on the disk through the complex potential formalism and Kolosov- Muskhelishvili potentials. The kinematics and statics of the rod, together with its axial inextensibility, lead to the formulation of a 5th-order differential equation governing the mechanical state at every point on the boundary of the disk. The solution of this equation is obtained by means of a complex Fourier series expansion for the unknown fields on the boundary of the disk, when a particular distribution of the external load is prescribed. The complex variables method shows that the unknown complex coefficients involved in the series expansion depend only on the external load. Hence, all the elastic fields become known on the coating and on the boundary and within the disk. The analytical results are complemented with experiments related to a load distribution which models two equal and opposite concentrated forces. In this regard, two coated disks were designed and then manufactured (with a CNC engraving machine) from a single block of polymethyl methacrylate so that the bonding between the coating and disk was perfect and residual stresses were absent. The samples were tested in a circular polariscope and the results strongly supported the coated disk model, so the photoelastic fringes were very well captured by the elastic solution. Different situations were investigated in order to study the non-local stress diffusion of the coating. The limit case of an isoperimetric disk was also investigated by imposing a vanishing bending stiffness for the coating. This limit situation corresponded to a disk equipped with a device able to preserve the perimeter of the disk during the deformation. Exploiting the framework developed, the bifurcation problem of the coated disk was analyzed, assuming that the coating was subject to a radial pressure of three different types. A closed-form analytical solution was obtained for the bifurcation pressure and modes, showing that the presence of the disk profoundly changed the bifurcation landscape of the coating, forming a circular elastic rod. In fact, the circular rod admits only oval modes, while the coating/disk system displays high-frequency circumferential undulations. The experimental, analytical, and numerical results presented open new possibilities for the design of coated solids of cylindrical geometry, which may find applications in micro and nano technologies, for instance in the characterization of nanowires via nanoindentation.

Aufgrund ihrer niedrigen Stoffwechselrate zeigen Puppen des Atlasspinners Attacus atlas diskontinuierlichen Gasaustausch (DGC) mit drei deutlich getrennten Phasen. Andererseits sind einzelne Stigmen entweder vollständig offen oder vollständig geschlossen. In dieser Arbeit wurden Attacus Puppen mit künstlichen Gasmischungen perfundiert, um das Steady-State Verhalten der Stigmen zu ermitteln. Es konnte gezeigt werden, daß die Flutterphase des DGC kein Gleichgewichtsverhalten darstellt. Weiterhin deuteten die Ergebnisse der Perfusion auf eine mögliche Interaktion zweier unabhängiger Regelkreise hin. Zwei mathematische Modelle untersuchten daraufhin mögliche theoretische Mechanismen, welche die beobachteten Muster erzeugen könnten. Die Modelle erlaubten die Erzeugung der Gasaustauschmuster ohne komplexe neurale Integration und deuten auf eine rein passive Erzeugung des DGC. Die Diskussion erläutert mögliche biologische Mechanismen, welchen den abstrakten Beschreibungen der Modelle zu Grunde liegen könnten. Sowohl die Perfusion als auch die Modelle sind dabei mit der ''Emergent Property Hypothesis'' der DGC Entstehung vereinbar. Zusätzlich wird dargelegt, wie alternative Hypothesen widerspruchslos in das sich abzeichnende Bild eingefügt werden können. / While having low metabolic activity, pupae of the giant silk moth Attacus atlas show triphasic discontinuous gas exchange cycles (DGC). However, a single spiracle is either fully open or fully closed. In this study moth pupae have been perfused to assess the steady state behavior of the spiracles in response to the endotracheal gas composition. It could be shown that the flutter phase of DGC is not a steady state behavior. Moreover, the results from the perfusion hint to a possible interaction between two independent spiracle control mechanisms. Two mathematical models then explore possible mechanisms of generating the observed respiratory patterns during DGC. The models indicate that DGC might be generated without complex neural integration and might thus be a purely ''passive'' pattern. The discussion focuses on biological processes and mechanisms that may back the abstract descriptions in the models. Both, the perfusion and the two mathematical models, are consistent with the emergent property hypothesis of DGC origin. However, it is discussed that other existing hypotheses are not mutually exclusive and that they can be subsumed using the non-adaptive emergent property hypothesis as a framework.

By exploiting the concept of emergent network properties and the hierarchical nature of networks, we have constructed several levels of models facilitating the investigations of issues in the area of respiratory neural control. The first of such models is an intracellular second messenger pathway model, which has been shown to be an important contributor to intracellular calcium metabolism and mediate responses to neuromodulators such as serotonin. At the next level, we have constructed new single neuron models of respiratory-related neurons (e.g. the pre-Btzinger complex neuron and the Hypoglossal motoneuron), where the electrical activities of the neurons are linked to intracellular mechanisms responsible for chemical homeostasis. Beyond the level of individual neurons, we have constructed models of neuron populations where the effects of different component neurons, varying strengths and types of inter-neuron couplings, as well as network topology are investigated. Our results from these simulation studies at different structural levels are in line with experiment observations. The small-world topology, as observed in previous anatomical studies, has been shown here to support rhythm generation along with a variety of other network-level phenomena. The interactions between different inter-neuron coupling types simultaneously manifesting at time-scales orders of magnitude apart suggest possible explanations for variations in the outputs measured from the XII rootlet in experiments. In addition, we have demonstrated the significance of pacemakers, along with the importance of considering neuromodulations and second-messenger pathways in an attempt to understand important physiological functions such as breathing activities.

This work has been supported by the European FACETS-ITN project. Within the frameworkof this project, we contribute to the simulation of cortical cell types (employingexperimental electrophysiological data of these cells as references), using a specific VLSIneural circuit to simulate, at the single cell level, the models studied as references in theFACETS project. The real-time intrinsic properties of the neuromorphic circuits, whichprecisely compute neuron conductance-based models, will allow a systematic and detailedexploration of the models, while the physical and analog aspect of the simulations, as opposedthe software simulation aspect, will provide inputs for the development of the neuralhardware at the network level. The second goal of this thesis is to contribute to the designof a mixed hardware-software platform (PAX), specifically designed to simulate spikingneural networks. The tasks performed during this thesis project included: 1) the methodsused to obtain the appropriate parameter sets of the cortical neuron models that can beimplemented in our analog neuromimetic chip (the parameter extraction steps was validatedusing a bifurcation analysis that shows that the simplified HH model implementedin our silicon neuron shares the dynamics of the HH model); 2) the fully customizablefitting method, in voltage-clamp mode, to tune our neuromimetic integrated circuits usinga metaheuristic algorithm; 3) the contribution to the development of the PAX systemin terms of software tools and a VHDL driver interface for neuron configuration in theplatform. Finally, it also addresses the issue of synaptic tuning for future SNN simulation.

Des questions fondamentales concernant les propriétés de diffusion des systèmes biologiques dans des conditions isothermes et non-isothermes restent en suspens en raison de l’absence de techniques expérimentales capables de visualiser et de mesurer les phénomènes de diffusion avec une très bonne précision. Il existe en conséquence un besoin de développer de nouvelles techniques expérimentales permettant d’approfondir notre compréhension des phénomènes de diffusion. La convection naturelle en cavité tridimensionnelle inclinée est elle-aussi très peu étudiée. Cette inclinaison de la cavité peut correspondre à un léger défaut expérimental ou être imposée volontairement. Dans cette thèse, nous étudions les phénomènes de transport de chaleur et de masse en cavité parallélépipédique, nous intéressant particulièrement à la thermodiffusion en situation sans convection et à la convection naturelle en fluide pur (sans thermodiffusion). La diffusion de masse est étudiée à l’aide d’une nouvelle technique optique, tandis que la convection naturelle est tout d’abord étudiée en détails avec une méthode numérique sophistiquée, puis visualisée expérimentalement à l’aide du même système optique que pour les mesures de diffusion. Nous présentons l’interféromètre optique de haute précision développé pour les mesures de diffusion. Cet interféromètre comprend un interféromètre polarisé de Mach–Zehnder, un polariseur tournant, une caméra CCD et un algorithme de traitement d’images original. Nous proposons aussi une méthode pour déterminer le coefficient de diffusion isotherme en fonction de la concentration. Cette méthode, basée sur une analyse inverse couplée à un calcul numérique, permet de déterminer les coefficients de diffusion à partir des profils de concentration transitoires obtenus par le système optique. Mentionnons de plus que c’est la première fois que la thermodiffusion est visualisée dans des solutions aqueuses de protéines. La méthode optique proposée présente trois avantages principaux par rapport aux autres méthodes similaires : (i) un volume d’échantillon réduit, (ii) un temps de mesure court, (iii) une stabilité hydrodynamique améliorée. Toutes ces méthodes ont été validées par des mesures sur des systèmes de référence. La technique optique est d’abord utilisée pour étudier la diffusion isotherme dans des solutions de protéines : (a) dans des solutions binaires diluées, (b) dans des solutions binaires sur un large domaine de concentration, (c) dans des solutions ternaires diluées. Les résultats montrent que (a) le coefficient de diffusion isotherme dans les systèmes dilués décroit avec la masse moléculaire, comme prédit grossièrement par l’équation de Stokes-Einstein ; (b) la protéine BSA a un comportement diffusif de type sphère dure et la protéine lysozyme de type sphère molle ; (c) l’effet de diffusion croisée est négligeable dans les systèmes ternaires dilués. La technique optique est aussi utilisée (d) dans des solutions binaires diluées non-isothermes, révélant que les molécules d’aprotinin (6.5 kDa) et de lysozyme (14.3 kDa) sont, respectivement, thermophiliques et thermo-phobiques, quand elles sont en solutions aqueuses à température ambiante. Enfin, la technique optique est utilisée pour l’étude de la convection de Rayleigh-Bénard en cavité cubique horizontale. Puisque la convection peut aussi être étudiée de façon réaliste en utilisant les équations de Navier-Stokes, une analyse numérique de bifurcation est proposée, permettant une étude approfondie de la convection naturelle dans des cavités tridimensionnelles parallélépipédiques. Pour cela, une méthode de continuation a été développée à partir d’un code aux éléments finis spectraux. La méthode numérique proposée est particulièrement bien adaptée aux études de convection correspondant à des diagrammes de bifurcation complexes. [...] / Fundamental questions concerning the mass diffusion properties of biological systems under isothermal and non-isothermal conditions still remain due to the lack of experimental techniques capable of visualizing and measuring mass diffusion phenomena with a high accuracy. As a consequence, there is a need to develop new experimental techniques that can deepen our understanding of mass diffusion. Moreover, steady natural convection in a tilted three-dimensional rectangular enclosure has not yet been studied. This tilt can be a slight defect of the experimental device or can be imposed on purpose. In this dissertation, heat and mass transfer phenomena in parallelepiped enclosures are studied focusing on convectionless thermodiffusion and on natural convection of pure fluids (without thermodiffusion). Mass diffusion is studied with a novel optical technique, while steady natural convection is first studied in detail with an improved numerical analysis and then with the same optical technique initially developed for diffusion measurements. A construction of a precise optical interferometer to visualize and measure mass diffusion is described. The interferometer comprises a polarizing Mach–Zehnder interferometer, a rotating polariser, a CCD camera, and an original image-processing algorithm. A method to determine the isothermal diffusion coefficient as a function of concentration is proposed. This method uses an inverse analysis coupled with a numerical calculation in order to determine the diffusion coefficients from the transient concentration profiles measured with the optical system. Furthermore, thermodiffusion of protein molecules is visualized for the first time. The proposed method has three main advantages in comparison to similar methods: (i) reduced volume sample, (ii) short measurement time, and (iii) increased hydrodynamic stability of the system. These methods are validated by determining the thermophysical properties of benchmark solutions. The optical technique is first applied to study isothermal diffusion of protein solutions in: (a) dilute binary solutions, (b) binary solutions with a wide concentration range, and (c) dilute ternary solutions. The results show that (a) the isothermal diffusion coefficient in dilute systems decreases with molecular mass, as roughly predicted by the Stokes-Einstein equation; (b) BSA protein has a hard-sphere-like diffusion behaviour and lysozyme protein a soft sphere characteristic; and (c) the cross-term effect between the diffusion species in a dilute ternary system is negligible. The optical technique is then applied to (d) non-isothermal dilute binary solutions, revealing that that the aprotinin (6.5 kDa) and lysozyme (14.3 kDa) molecules are thermophilic and thermophobic, respectively, when using water as solvent at room temperature. Finally, the optical technique is applied to study Rayleigh-Bénard convection in a horizontal cubical cavity. Since natural convection can be studied in more depth by solving the Navier-Stokes equations, a bifurcation analysis is proposed to conduct a thorough study of natural convection in three-dimensional parallelepiped cavities. Here, a continuation method is developed from a three-dimensional spectral finite element code. The proposed numerical method is particularly well suited for the studies involving complex bifurcation diagrams of three-dimensional convection in rectangular parallelepiped cavities. This continuation method allows the calculation of solution branches, the stability analysis of the solutions along these branches, the detection and precise direct calculation of the bifurcation points, and the jump to newly detected stable or unstable branches, all this being managed by a simple continuation algorithm. This can be used to calculate the bifurcation diagrams describing the convection in tilted cavities. [...]

Philosophiae Doctor - PhD / Fitted Numerical Methods for Delay Di erential Equations Arising in Biology E.B.M. Bashier PhD thesis, Department of Mathematics and Applied Mathematics,Faculty of Natural Sciences, University of the Western Cape. This thesis deals with the design and analysis of tted numerical methods for some delay di erential models that arise in biology. Very often such di erential equations are very complex in nature and hence the well-known standard numerical methods seldom produce reliable numerical solutions to these problems. Ine ciencies of these methods are mostly accumulated due to their dependence on crude step sizes and unrealistic stability conditions.This usually happens because standard numerical methods are initially designed to solve a class of general problems without considering the structure of any individual problems. In this thesis, issues like these are resolved for a set of delay di erential equations. Though the developed approaches are very simplistic in nature, they could solve very complex problems as is shown in di erent chapters.The underlying idea behind the construction of most of the numerical methods in this thesis is to incorporate some of the qualitative features of the solution of the problems into the discrete models. Resulting methods are termed as tted numerical methods. These methods have high stability properties, acceptable (better in many cases) orders of convergence, less computational complexities and they provide reliable solutions with less CPU times as compared to most of the other conventional solvers. The results obtained by these methods are comparable to those found in the literature. The other salient feature of the proposed tted methods is that they are unconditionally stable for most of the problems under consideration.We have compared the performances of our tted numerical methods with well-known software packages, for example, the classical fourth-order Runge-Kutta method, standard nite di erence methods, dde23 (a MATLAB routine) and found that our methods perform much better. Finally, wherever appropriate, we have indicated possible extensions of our approaches to cater for other classes of problems. May 2009.

We first discuss some fundamental results such as equilibria, linearization, and stability of nonlinear dynamical systems arising in mathematical modeling. Next we study the dynamics in planar systems such as limit cycles, the Poincaré-Bendixson theorem, and some of its useful consequences. We then study the interaction between two and three different cell populations, and perform stability and bifurcation analysis on the systems. We also analyze the impact of immunotherapy on the tumor cell population numerically.

Работа содержит описание построения доверительной полосы стохастического хаоса и реализацию алгоритмов исследования n-мерных моделей. В работе рассматривается дискретная модель, представленная в виде нелинейной динамической системы разностных уравнений, которая описывает динамику взаимодействия потребителей. Выделяются две задачи, которые были поставлены и выполнены в рамках данной работы для расширения программного инструментария исследования динамических систем такого рода. Для двумерного случая осуществляется стохастический анализ чувствительности хаоса через построение доверительной области с использованием критических линий. Помимо этого, описывается разработанный и реализованный алгоритм построения внешней границы хаоса. Производится переход к n-мерному варианту модели (взаимодействие n потребителей). Выделяется 4 алгоритма для исследования n-мерной модели: 1. построение фазовой траектории, 2. построение бифуркационной диаграммы, 3. построение карты режимов, 4. построение показателей Ляпунова. Описывается реализация данных алгоритмов с уклоном в параллельные вычисления. Реализация алгоритмов выполнена на языке программирования C# (платформа .NET) в виде консольного приложения для запуска параллельных вычислений на вычислительном кластере УрФУ. / The work contains description of confidence band construction of a stochastic chaos and realization of algorithms for n-dimensional models studying. The thesis considers a discrete model presented in the form of a nonlinear dynamic system of difference equations, which describes the dynamic of consumer interaction. There are two task that were set and performed in this work to expand the software tools for research dynamic sys-tems of this kind. For the two-dimensional case, a stochastic analysis of the sensitivity of chaos is carried out through the construction of a confidence band using critical lines. In addition, there is description and implementation of algorithm, that can build outer boundary of chaos. A transition is made to the n-dimensional version of the model (interaction of n consumers). There are 4 algorithms for studying the n-dimensional model: 1. phase trajectory building, 2. bifurcation diagram building, 3. mode map building, 4. Lyapunov components building. Algorithm implementation is described with a bias in parallel computations. The algorithms are implemented with C# programming language (.NET platform) in the form of a console application for running parallel computations on the computing cluster of the Ural Federal University.