This diploma thesis deal with evaluation of the impact in the scale of uncertaintly stiffness on the tool deviation during grooving process. By the affect of the insufficient stiffness in each parts of the machine, there is presented a mechanical vibration during the cutting process which may cause a damage to the surface of the workpiece, to the tool or to the processing machine. The change of the stiffness is caused in the result of tool wear, impact of setted cutting conditions and many others. In the first part includes teoretical introduction to field of the uncertainty and choosing suitable methods for the solutions. Chosen methods are Monte Carlo and polynomial chaos expansion which are procced in the interface of MATLAB. Both of the methods are primery tested on the simple systems with the indefinited enters of the stiffness. These systems replace the parts of the stiffness characteristics of the each support parts. After that, the model is defined for the turning during the process of grooving with the 3 degrees of freedom. Then the analyses of the uncertainity and also sensibility analyses for uncertainity entering data of the stiffness are carried out again by both methods. At the end are both methods compared in the points of view by the time consuption and also by precission. Judging by gathered data it is clear that the change of the stiffness has significant impact on vibration in all degrees of freedome of the analysed model. As the example a maximum and a minimum calculated deviation of the workpiece stiffness was calculated via methode of Monte Carlo. The biggest impact on the finall vibration of the tool is found by stiffness of the ball screw. The solution was developed for the more stabile cutting process.

This master‘s thesis deals with analysis of electronic dynamical systems exhibiting chaotic solution. In introduction, some basic concepts for better understanding of dynamical systems are explained. After introduction, current knowledge from the world of circuits exhibiting chaotic solutions are discussed. The best-known chaotic systems are analyzed numerically in Matlab software. Numerical analysis and experimental verification were demonstrated at C class transistor amplifier, which confirmed the chaotic behavior and generation of a strange attractor.

This master's thesis deals with solving fractional-order ordinary differential equations by the Adomian decomposition method. A part of the work is therefore devoted to the theory of equations containing differential operators of non-integer order, especially the Caputo operator. The next part is devoted to the Adomian decomposition method itself, its properties and implementation in the case of Chen system. The work also deals with bifurcation analysis of this system, both for integer and non-integer case. One of the objectives is to clarify the discrepancy in the literature concerning the fractional-order Chen system, where experiments based on the use of the Adomian decomposition method give different results for certain input parameters compared with numerical methods. The clarification of this discrepancy is based on recent theoretical knowledge in the field of fractional-order differential equations and their systems. The conclusions are supported by numerical experiments, own code implementing the Adomian decomposition method on the Chen system was used.

This thesis aims at Multi-Objective Optimization under Uncertainty in structural design. We investigate Polynomial Chaos Expansion (PCE) surrogates which require extensive training sets. We then face two issues: high computational costs of an individual Finite Element simulation and its limited precision. From numerical point of view and in order to limit the computational expense of the PCE construction we particularly focus on sparse PCE schemes. We also develop a custom Latin Hypercube Sampling scheme taking into account the finite precision of the simulation. From the modeling point of view, we propose a multifidelity approach involving a hierarchy of models ranging from full scale simulations through reduced order physics up to response surfaces. Finally, we investigate multiobjective optimization of structures under uncertainty. We extend the PCE model of design objectives by taking into account the design variables. We illustrate our work with examples in sheet metal forming and optimal design of truss structures. / Doctorat en Sciences de l'ingénieur / info:eu-repo/semantics/nonPublished

Tunneling is one of the most prominent features of quantum mechanics. While the tunneling process in one-dimensional integrable systems is well understood, its quantitative prediction for systems with mixed phase space is a long-standing open challenge. In such systems regions of regular and chaotic dynamics coexist in phase space, which are classically separated but quantum mechanically coupled by the process of dynamical tunneling. We derive a prediction of dynamical tunneling rates which describe the decay of states localized inside the regular region towards the so-called chaotic sea. This approach uses a fictitious integrable system which mimics the dynamics inside the regular domain and extends it into the chaotic region. Excellent agreement with numerical data is found for kicked systems, billiards, and optical microcavities, if nonlinear resonances are negligible. Semiclassically, however, such nonlinear resonance chains dominate the tunneling process. Hence, we combine our approach with an improved resonance-assisted tunneling theory and derive a unified prediction which is valid from the quantum to the semiclassical regime. We obtain results which show a drastically improved accuracy of several orders of magnitude compared to previous studies. / Der Tunnelprozess ist einer der bedeutensten Effekte in der Quantenmechanik. Während das Tunneln in eindimensionalen integrablen Systemen gut verstanden ist, gestaltet sich dessen Beschreibung für Systeme mit gemischtem Phasenraum weitaus schwieriger. Solche Systeme besitzen Gebiete regulärer und chaotischer Bewegung, die klassisch getrennt sind, aber quantenmechanisch durch den Prozess des dynamischen Tunnelns gekoppelt werden. In dieser Arbeit wird eine theoretische Vorhersage für dynamische Tunnelraten abgeleitet, die den Zerfall von Zuständen, die im regulären Gebiet lokalisiert sind, in die sogenannte chaotische See beschreibt. Dazu wird ein fiktives integrables System konstruiert, das im regulären Bereich eine nahezu gleiche Dynamik aufweist und diese Dynamik in das chaotische Gebiet fortsetzt. Die Theorie zeigt eine ausgezeichnete Übereinstimmung mit numerischen Daten für gekickte Systeme, Billards und optische Mikrokavitäten, falls nichtlineare Resonanzketten vernachlässigbar sind. Semiklassisch jedoch bestimmen diese nichtlinearen Resonanzketten den Tunnelprozess. Daher kombinieren wir unseren Zugang mit einer verbesserten Theorie des Resonanz-unterstützten Tunnelns und erhalten eine Vorhersage,die vom Quanten- bis in den semiklassischen Bereich gültig ist. Ihre Resultate zeigen eine Genauigkeit, die verglichen mit früheren Theorien um mehrere Größenordnungen verbessert wurde.

We investigate systems with a mixed phase space, where regular and chaotic dynamics coexist. Classically, regions with regular motion, the regular islands, are dynamically not connected to regions with chaotic motion, the chaotic sea. Typically, this is also reflected in the quantum properties, where eigenstates either concentrate on the regular or the chaotic regions. However, it was shown that quantum mechanically, due to the tunneling process, a coupling is induced and flooding of regular islands may occur. This happens when the Heisenberg time, the time needed to resolve the discrete spectrum, is larger than the tunneling time from the regular region to the chaotic sea. In this case the regular eigenstates disappear. We study this effect by the time evolution of wave packets initially started in the chaotic sea and find increasing probability in the regular island. Using random matrix models a quantitative prediction is derived. We find excellent agreement with numerical data obtained for quantum maps and billiards systems. For open systems we investigate the phenomenon of flooding and disappearance of regular states, where the escape time occurs as an additional time scale. We discuss the reappearance of regular states in the case of strongly opened systems. This is demonstrated numerically for quantum maps and experimentally for a mushroom shaped microwave resonator. The reappearance of regular states is explained qualitatively by a matrix model. / Untersucht werden Systeme mit gemischtem Phasenraum, in denen sowohl reguläre als auch chaotische Dynamik auftritt. In der klassischen Mechanik sind Gebiete regulärer Bewegung, die sogenannten regulären Inseln, dynamisch nicht mit den Gebieten chaotischer Bewegung, der chaotischen See, verbunden. Dieses Verhalten spiegelt sich typischerweise auch in den quantenmechanischen Eigenschaften wider, so dass Eigenfunktionen entweder auf chaotischen oder regulären Gebieten konzentriert sind. Es wurde jedoch gezeigt, dass aufgrund des Tunneleffektes eine Kopplung auftritt und reguläre Inseln geflutet werden können. Dies geschieht wenn die Heisenbergzeit, das heißt die Zeit die das System benötigt, um das diskrete Spektrum aufzulösen, größer als die Tunnelzeit vom Regulären ins Chaotische ist, wobei reguläre Eigenzustände verschwinden. Dieser Effekt wird über eine Zeitentwicklung von Wellenpaketen, die in der chaotischen See gestartet werden, untersucht. Es kommt zu einer ansteigenden Wahrscheinlichkeit in der regulären Insel. Mithilfe von Zufallsmatrixmodellen wird eine quantitative Vorhersage abgeleitet, welche die numerischen Daten von Quantenabbildungen und Billardsystemen hervorragend beschreibt. Der Effekt des Flutens und das Verschwinden regulärer Zustände wird ebenfalls mit offenen Systemen untersucht. Hier tritt die Fluchtzeit als zusätzliche Zeitskala auf. Das Wiederkehren regulärer Zustände im Falle stark geöffneter Systeme wird qualitativ mithilfe eines Matrixmodells erklärt und numerisch für Quantenabbildungen sowie experimentell für einen pilzförmigen Mikrowellenresonator belegt.

Generic Hamiltonian systems have a mixed phase space, in which regular and chaotic motion coexist. In the chaotic sea the classical transport is limited by partial barriers, which allow for a flux \Phi given by the corresponding turnstile area. Quantum mechanically the transport is suppressed if Planck's constant is large compared to the classical flux, h >> \Phi, while for h << \Phi classical transport is recovered. For the transition between these limiting cases there are many open questions, in particular concerning the correct scaling parameter and the width of the transition. To investigate this transition in a controlled way, we design a kicked system with a particularly simple phase-space structure, consisting of two chaotic regions separated by one dominant partial barrier. We find a universal scaling with the single parameter \Phi/h and a transition width of almost two orders of magnitude in \Phi/h. In order to describe this transition, we consider several matrix models. While the numerical data is not well described by the random matrix model proposed by Bohigas, Tomsovic, and Ullmo, a deterministic 2x2-model, a channel coupling model, and a unitary model are presented, which describe the transitional behavior of the designed kicked system. This is also confirmed for the generic standard map, suggesting a universal scaling behavior for the quantum transition of a partial barrier. / Generische Hamilton'sche Systeme besitzen einen gemischten Phasenraum, in dem sowohl reguläre als auch chaotische Dynamik vorkommen. Der klassische Transport in der chaotischen See wird durch partielle Barrieren begrenzt, die nur einen Fluss \Phi hindurch lassen. Der quantenmechanische Transport ist stark unterdrückt, wenn die Planck'sche Konstante groß gegen den klassischen Fluss ist, h >> \Phi. Ist hingegen h << \Phi folgt die Quantenmechanik der klassischen Dynamik. Für den Übergangsbereich zwischen diesen Grenzfällen gibt es noch viele offene Fragen, insbesondere bezüglich des richtigen Skalierungsparameters und der Breite des Übergangs. Um gezielt diesen Übergang zu untersuchen, haben wir ein System mit einem besonders einfachen Phasenraum entworfen. Er besteht aus zwei chaotischen Gebieten, die durch eine dominante partielle Barriere getrennt sind. Es zeigt sich, dass das universelle Verhalten durch den Parameter \Phi/h beschrieben wird und der Übergang sich über zwei Größenordnungen erstreckt. Wir betrachten verschiedene Matrixmodelle um diesen Übergang zu verstehen. Die numerischen Daten werden nicht durch das Zufallsmatrixmodell von Bohigas, Tomsovic und Ullmo beschrieben. Ein deterministisches 2x2-Modell, eine Kanalkopplung und ein unitäres Matrixmodell beschreiben hingegen den Übergang des entworfenen gekickten Systems. Die Tatsache, dass auch die generische Standardabbildung diesem Verhalten folgt, spricht für ein universelles Verhalten des Quantenübergangs einer partiellen Barriere.

The dissertation consists of two parts: (1) the essay and (2) the composition. The essay elucidates the composer's creative process of the orchestral works, The Heidegger Collection. The Heidegger Collection has five movements. The titles of each movement are derived from the key philosophical concepts from Heidegger's most significant writing, Being and Time: (1) State-of-Mind, (2) Idle-Talk, (3) Moment-of-Vision, (4) Dread, and (5) Being-towards-the-End. The essay discusses the meanings of the five concepts, and explains how I express my reaction to Heidegger's thinking through music composition. The essay also discusses the essential musical language of The Heidegger Collection, such as interval cycles, polyrhythmic patterns, algorithmic elements, portamento effects, chaos theory, and oriental influence.

<div>Uncertainty quantification techniques based on the spectral approach have been studied extensively in the literature to characterize and quantify, at low computational cost, the impact that uncertainties may have on large-scale engineering problems. One such technique is the <i>generalized polynomial chaos</i> (gPC) which utilizes a time-independent orthogonal basis to expand a stochastic process in the space of random functions. The method uses a specific Askey-chaos system that is concordant with the measure defined in the probability space in order to ensure exponential convergence to the solution. For nearly two decades, this technique has been used widely by several researchers in the area of uncertainty quantification to solve stochastic problems using the spectral approach. However, a major drawback of the gPC method is that it cannot be used in the resolution of problems that feature strong nonlinear dependencies over the probability space as time progresses. Such downside arises due to the time-independent nature of the random basis, which has the undesirable property to lose unavoidably its optimality as soon as the probability distribution of the system's state starts to evolve dynamically in time.</div><div><br></div><div>Another technique is the <i>time-dependent generalized polynomial chaos</i> (TD-gPC) which utilizes a time-dependent orthogonal basis to better represent the stochastic part of the solution space (aka random function space or RFS) in time. The development of this technique was motivated by the fact that the probability distribution of the solution changes with time, which in turn requires that the random basis is frequently updated during the simulation to ensure that the mean-square error is kept orthogonal to the discretized RFS. Though this technique works well for problems that feature strong nonlinear dependencies over the probability space, the TD-gPC method possesses a serious issue: it suffers from the curse of dimensionality at the RFS level. This is because in all gPC-based methods the RFS is constructed using a tensor product of vector spaces with each of these representing a single RFS over one of the dimensions of the probability space. As a result, the higher the dimensionality of the probability space, the more vector spaces needed in the construction of a suitable RFS. To reduce the dimensionality of the RFS (and thus, its associated computational cost), gPC-based methods require the use of versatile sparse tensor products within their numerical schemes to alleviate to some extent the curse of dimensionality at the RFS level. Therefore, this curse of dimensionality in the TD-gPC method alludes to the need of developing a more compelling spectral method that can quantify uncertainties in long-time response of dynamical systems at much lower computational cost.</div><div><br></div><div>In this work, a novel numerical method based on the spectral approach is proposed to resolve the curse-of-dimensionality issue mentioned above. The method has been called the <i>flow-driven spectral chaos</i> (FSC) because it uses a novel concept called <i>enriched stochastic flow maps</i> to track the evolution of a finite-dimensional RFS efficiently in time. The enriched stochastic flow map does not only push the system's state forward in time (as would a traditional stochastic flow map) but also its first few time derivatives. The push is performed this way to allow the random basis to be constructed using the system's enriched state as a germ during the simulation and so as to guarantee exponential convergence to the solution. It is worth noting that this exponential convergence is achieved in the FSC method by using only a few number of random basis vectors, even when the dimensionality of the probability space is considerably high. This is for two reasons: (1) the cardinality of the random basis does not depend upon the dimensionality of the probability space, and (2) the cardinality is bounded from above by <i>M+n+1</i>, where <i>M</i> is the order of the stochastic flow map and <i>n</i> is the order of the governing stochastic ODE. The boundedness of the random basis from above is what makes the FSC method be curse-of-dimensionality free at the RFS level. For instance, for a dynamical system that is governed by a second-order stochastic ODE (<i>n=2</i>) and driven by a stochastic flow map of fourth-order (<i>M=4</i>), the maximum number of random basis vectors to consider within the FSC scheme is just 7, independent whether the dimensionality of the probability space is as low as 1 or as high as 10,000.</div><div><br></div><div>With the aim of reducing the complexity of the presentation, this dissertation includes three levels of abstraction for the FSC method, namely: a <i>specialized version</i> of the FSC method for dealing with structural dynamical systems subjected to uncertainties (Chapter 2), a <i>generalized version</i> of the FSC method for dealing with dynamical systems governed by (nonlinear) stochastic ODEs of arbitrary order (Chapter 3), and a <i>multi-element version</i> of the FSC method for dealing with dynamical systems that exhibit discontinuities over the probability space (Chapter 4). This dissertation also includes an implementation of the FSC method to address the dynamics of large-scale stochastic structural systems more effectively (Chapter 5). The implementation is done via a modal decomposition of the spatial function space as a means to reduce the number of degrees of freedom in the system substantially, and thus, save computational runtime.</div>

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