Lyapunov functions are functions with negative orbital derivative, whose existence guarantee the stability of an equilibrium point of an ODE. Moreover, sub-level sets of a Lyapunov function are subsets of the domain of attraction of the equilibrium. In this thesis, we improve an established numerical method to construct Lyapunov functions using the radial basis functions (RBF) collocation method. The RBF collocation method approximates the solution of linear PDE's using scattered collocation points, and one of its applications is the construction of Lyapunov functions. More precisely, we approximate Lyapunov functions, that satisfy equations for their orbital derivative, using the RBF collocation method. Then, it turns out that the RBF approximant itself is a Lyapunov function. Our main contributions to improve this method are firstly to combine this construction method with a new grid refinement algorithm based on Voronoi diagrams. Starting with a coarse grid and applying the refinement algorithm, we thus manage to reduce the number of collocation points needed to construct Lyapunov functions. Moreover, we design two modified refinement algorithms to deal with the issue of the early termination of the original refinement algorithm without constructing a Lyapunov function. These algorithms uses cluster centres to place points where the Voronoi vertices failed to do so. Secondly, we derive two verification estimates, in terms of the first and second derivatives of the orbital derivative, to verify if the constructed function, with either a regular grid of collocation points or with one of the refinement algorithms, is a Lyapunov function, i.e., has negative orbital derivative over a given compact set. Finally, the methods are applied to several numerical examples up to 3 dimensions.

This thesis is devoted to the study of multiphase jet fragmentation using Smoothed Particle Hydrodynamics (SPH). The theoretical aspects of three hydrodynamic instabilities, namely the Kelvin-Helmholtz instability (KHI), Rayleigh-Taylor instability (RTI), and Rayleigh Plateau instability (RPI) are reviewed. The linear growth rate of the combined KHI and RTI are derived by means of linear perturbation in chapter 2. The linear growth rate of the multiphase RPI is presented in chapter 7. An overview of the Smoothed Particle Hydrodynamics is given in chapter 3. A pseudo-consistent SPH scheme is presented for the simulation of multiphase flow problems. Additionally, two interface stabilisation models are presented: quasi-buoyancy model and gas-repulsion model. When used in combination with the pseudo-consistent SPH scheme, these models are found to be superior than those presented in the weakly-compressible SPH literature and allows for the simulations for density ratio up to three-magnitudes. The development of an idealised KHI and a KHI subjected to constant gravitational acceleration (stratified shear instability) is examined in chapter 5. The extracted linear growth rate are compared with the theoretical growth rate presented both in the literature and in chapter 2 for the purpose of validation. The development of a single- and multi-mode RTI are studied by means of SPH in chapter 6. Chapter 7 presents the results for the three-dimensional RPI occurring between two fluids. Based on the knowledge acquired in chapter 5-7, the multiphase jet fragmentation driven by the previously mentioned hydrodynamic instabilities are presented in chapter 8. Finally, the major research findings and recommendations are summarised in chapter 9.

Sobre as equações de Vertogen e de Vries no modelo de Dicke generalizado / On the Vertogen and de Vries equations in the generalized Dicke model

Neste trabalho apresentamos um estudo da dinâmica de subsistemas quânticos dando ênfase ao fenômeno da decoerência. Em particular, estudamos os efeitos da decoerência num sistema de dois níveis acoplado a um outro sistema cujo espectro de energia é um contínuo no contexto do fenômeno quântico de tunelamento. Investigamos também aspectos da relação entre a descrição do sistema de dois níveis e a descrição do modelo linear na aproximação de onda girante em termos de uma equação mestra na aproximação de Born-Markov e a dinâmica exata destes dois subsistemas. Estudamos também o experimento proposto por L. Davidovich e colaboradores para medir a decoerência de superposições mesoscópicas de estados coerentes dentro do contexto da computação quântica. Este experimento pode ser re interpretado como um computador quântico capaz de solucionar o problema de Deutsch. O grande problema deste experimento, do ponto de vista da computação quântica, é a presença da decoerência. Esta decoerência provoca o aparecimento de erros no resultado final. Assim, nós mostramos uma maneira de reduzi-la, diminuindo assim os eventuais erros que possam aparecer. Finalmente, mostramos através de dois exemplos a relação entre o efeito Zeno e a decoerência. / We study the quantum decoherence process in different contexts. In particular, we study decoherence effects in the dynamics of a two-level system coupled to another system which has a continuous energy spectrum. We also investigate when the master equation description, in a Born Markov approximation reproduces the exact solution. We extend this investigation to the model which describes the interaction between a harmonic oscillator and a thermal bath of harmonic oscillators in the rotating wave approximation. We also study the two atom correlation scheme originally proposed by Davidovich, Brune, Raimond and Haroche for measuring the decoherence of a mesoscopic superposition of coherent states of a QED cavity in the context of quantum computation. We show that this experiment is equivalent to a quantum computer solving Deutsch\'s problem. We find a way to reduce the errors due to decoherence. Finally, we show how the Zeno effect is related to the decoherence process.

Nesse trabalho estudamos dois tipos de modelos para a descrição dos sistemas magnéticos modulados. O primeiro é o modelo ANNNI (\"Axial Next-Nearest-Neighbor Ising\"). Dentro dos limites da aproximação de campo médio estudamos vários efeitos que surgem quando as interações intraplanares são menores que as interações interplanares. Investigamos em particular a ocorrência de fases desordenadas, assim denominadas por apresentar planos com magnetização nula. Estudamos ainda o comportamento dos pontos de acumulação de pontos de ramificação que determinam o limite entre as transições do tipo comensurável-comensurável e comensurável-incomensurável. Também investigamos dois variantes do modelo ANNNI, um com spin qualquer e outro com spin 1 e anisotropia de campo cristalino. O segundo tipo de modelo estudado, apropriado para sistemas apresentando modulações helicoidais das magnetizações, é um modelo planar clássico com interações competitivas entre primeiros e segundos vizinhos numa direção axial. Restringimos o nosso estudo ao zero absoluto de temperatura. Fizemos expansões das magnetizações em série de Fourier para campos baixos e altos. Também fizemos análise de estabilidade das fases comensuráveis, dentro de uma aproximação contínua, utilizando a teoria dos sólitons. Finalmente, estudamos numericamente o diagrama de fases do modelo aplicando o método dos potenciais efetivos. / In this work we have studied two types of models for the description of modulated magnetic systems. The first is the ANNNI (Axial NextNearest-Neighbor Ising) model. Within the mean-field approximation we have studied various effects which comes about when the intralayer interactions are weaker than the interlayer interactions. We have investigated in particular the possibility of partially disordered phases, characterized by the presence of layers with zero magnetization. We also have studied the behavior of the accumulation points of the branching points, which separates the commensurate-commensurate transitions from the commensurate-incommensurate transitions. We have also investigated two variants of the ANNNI model, one with an arbitrary spin and the other with spin 1 and a term of crystal-field anisotropy. The second type of the model we have studied, appropriate for systems presenting helicoidal modulation of the magnetization, is a classical planar model with nearest and next-nearest interactions along an axial direction. \\\'Ve have restricted our study to the absolute zero of the temperature. The magnetization is expanded in Fourier series for high and low fields. We also analyse the stability of the commensurate phases, within a continuum approximation, using the soliton theory. Finally, we construct numerically the phase diagram of the model using the method of effective potentials.

O interesse geral de nosso grupo é estudar sistemas moleculares em fase líquida. Para isto existem vários métodos de simulação. Dentre eles, os mais utilizados são o método Monte Carlo Matropolis e Dinâmica Molecular. Porém ambos necessitam do conhecimento prévio do potencial de interação molecular. Neste trabalho, estudamos um método de simulação recente que embora tenha sido idealizado em 1968 por Kaplow et al. só foi formalizado em 1988 por McGreevy et al. Este método é hoje chamado de Monte Carlo Reverso. A vantagem deste método é que ele não necessita do potencial de interação, mas sim do fator de estrutura ou da função de distribuição radial que podem ser obtidos experimentalmente por meio de espalhamento de raio-X ou de nêutrons. A desvantagem é que ainda existem algumas dúvidas sobre sua confiabilidade e aplicabilidade para sistemas líquidos. Neste trabalho, realizamos um estudo investigativo procurando adquirir conhecimento a respeito deste método para podermos obter nossas próprias experiências e conclusões. Inicialmente, desenvolvemos um código computacional utilizando o método Monte Carlo Reverso. Depois realizamos simulações de argônio líquido com a finalidade de testar nosso programa e posteriormente aplicarmos o método para o oxigênio líquido. Baseados nos resultados que obtivemos, concluímos que apesar do método Monte Carlo Reverso não ser capaz de descrever a termodinâmica do sistema, ele gera estruturas configuracionais com as mesmas características que as geradas com o Monte Carlo Metropolis. No futuro pretendemos analisar se as configurações geradas com o Monte Carlo Reverso são capazes de reproduzir as propriedades quânticas (dipolo, espectro de absorção, etc) da mesma maneira que o Monte Carlo Metropolis / Implementation and applications of the reverse Monte Carlo method for homogeneous liquids

The aging world-wide aviation fleet requires methods for accurately predicting the presence of structural flaws that compromise airworthiness in aircraft structures. Nondestructive Evaluation (NDE) provides the means to assess these structures quickly, quantitatively, and noninvasively. Ultrasonic guided waves, Lamb waves, are useful for evaluating the plate and shell structures common in aerospace applications. The amplitude and time-of-flight of Lamb waves depend on the material properties and thickness of a medium, and so they can be used to detect any areas of differing thickness or material properties which indicate flaws. By scanning sending and receiving transducers over an aircraft, large sections can be evaluated after a single pass. However, while this technique enables the detection of areas of structural deterioration, it does not allow for the quantification of the extent of that deterioration. Tomographic reconstruction with Lamb waves allows for the accurate reconstruction of the variation of quantities of interest, such as thickness, throughout the investigated region, and it presents the data as a quantitative map. The location, shape, and extent of any flaw region can then be easily extracted from this Tomographic image. Two Lamb wave tomography techniques using Parallel Projection tomography (PPT) and Cross Borehole tomography (CBT), are shown to accurately reconstruct flaws of interest to the aircraft industry. A comparison of the quality of reconstruction and practicality is then made between these two methods, and their limitations are discussed and shown experimentally. Higher order plate theory is used to derive analytical solutions for the scattering of the lowest order symmetric Lamb wave from a circular inclusion, and these solutions are used to explain the scattering effects seen in the Tomographic reconstructions. Finally, the means by which this scattering theory can be used to develop Lamb wave Tomographic algorithms that are more generally applicable in-the-field, is presented.

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No description available.

The work conducted in this project proposes a method to determine the location of fracture initiation for non-load carrying fillet welds based on continuous geometry measurements. Measurements were carried out on welded specimens using a laser line triangluator together with the weld quality evaluation software qWeld from Winteria®. The 119 specimens produced were after scanning fatigue tested until failure. The fracture surfaces have been investigated in order to find the most probable location(s) of fracture initiation. This data was then used to fit the proposed model parameters used to analytically predict the point(s) of fracture initiation. Local weld geometry measurements were extracted from the predicted fracture initiation location(s) in an effort to analyse the correlation between local weld geometry and fatigue life. This showed that fatigue life and leg length were positively correlated and that strong correlations exists between the individual geometrical parameters. New specimens were produced to test the fitted data of the proposed model, this data was however inconclusive as the new specimens had a high ratio of weld defects. / Arbetet i detta projekt föreslår en metod för att bestämma punkten/punkterna för sprickinitiering för icke-lastbärande kälsvetsar baserat på kontinuerliga mätmetoder. Mätningar utfördes på svetsade prover med hjälp av en laserlinjetriangluator tillsammans med utvärderingsprogrammet qWeld från Winteria®. De 119 provstavar som producerades testades efter mätningarna med en cyklisk belastning till de gick till brott. Brottytorna undersöktes sedan för att hitta platsen/platserna för de mest sannolika startpunkterna. Dessa punkter användes sedan för att passa de föreslagna modellparametrarna som användes för att förutsäga startpunkten. Lokal svetsgeometri extraherades från de förutsagda sprickinitieringsplatserna i ett försök att analysera korrelationen mellan lokal svetsgeometri och antalet cykler till brott. Detta visade att antalet cykler och katetlängden var positivt korrelerade och att det finns starka korrelationer mellan de enskilda geometriska parametrarna. Nya provbitar framställdes för att testa de anpassade parametrarna i den framtagna metoden, dessa gav dock inga tydliga resultat då en hög andel svetsdefekter upptäcktes efter att de provats.