This study uses the Phase-field method to simulate the transient thermal current of the metal surface heated and molten by a massing energy. The flow field uses a two-dimension module, considered with the mass conservation equation, momentum equation, energy equation and level-set equation, to solve for the distribution in whole domain, including the interface, of temperature, velocity, pressure and level-set number. We ignore the effect of concentration diffusion, but consider about the effect of heat translation on the flow field. Finally the results will display the flows of air around molten area forced by buoyancy which is caused by high temperature, and the flows in molten area forced by thermocapillary which is caused by temperature gradient.

The purpose of this thesis is to broaden the tools and knowledge available for understanding the behavior of metals under irradiation to aid in the pursuit of advanced materials for deployment in Generation IV (Gen-IV) nuclear reactor designs. A mean-field study is conducted on a body-centered cubic (BCC) A-B binary metal alloy system. The performance of the simulated metal system is measured by assessing the degree of segregation that occurs at the grain boundary (GB) in the center of the one-dimensional simulation box. This mean-field method was developed using rate theory equations to observe the diffusion of defects and solute atoms in the binary BCC alloy modeled after a section of planes in the <100> direction of α-iron. The method in this thesis is adapted from a previous radiation-induced segregation (RIS) study that was similarly validated against thermal segregation isotherms. This adapted simulation code was used to study RIS by varying the initial values and conditions across ranges relevant to Generation IV reactor designs. The simulations run with this code were centered around segregation energy and the diffusion coefficient relationships between defects and solute atoms. The most influential conditions applied to both the segregation energy and diffusion coefficient relationship test suites were the temperature and dose rate. The interplay of the various segregation energies, manipulated diffusion coefficients, temperatures, and dose rates is explored in this thesis. The code used in this thesis is presented as a modular framework for further parameter study with a clear direction for more complex alloys. / Master of Science / The growing electricity demand for more efficient, safe, reliable, and sustainable means of power generation requires research and subsequent implementation of advanced Generation IV (Gen-IV) nuclear reactor designs. These proposed designs operate under significantly more strenuous conditions from the perspective of materials used in constructing the reactor. Materials inside the reactor will experience temperatures, pressures, and radiation doses greatly exceeding those of previous generations: Gen II through III+. Metals are employed in almost every component inside a reactor and are particularly susceptible to the demanding conditions due to their tendency to lose their ductility under these stressors. This thesis presents a diffusion-based code that models a binary metal alloy under conditions similar to those expected in Gen-IV reactors. The results of the code give insight into the prevalence of a phenomenon known as radiation induced segregation (RIS) in metals under these Gen-IV relevant conditions. The values input into the code have significant effects on the resulting RIS behavior of the metal alloy. This thesis presents correlations between the initial parameters and the amount of segregation this alloy experiences. The results of this thesis allow a sort of mapping of material parameters and operating conditions so that materials can be designed for optimal performance over the lifespan of the next generation of nuclear reactors. The code in this thesis was developed with the expectation that its modularity would be expanded upon to apply to more complex alloys under a broader range of initial conditions.

Os nitretos do grupo III (BN, AIN,Gan e InN) e suas ligas ternárias Al-GaN e InGaN proporcionam, recentemente, um extraordinário avanço na fabricação de dispositivos opto-eletrônicos operando na região do espectro correspondente ao verde-azul-UV e na produção de dispositivos eletrônicos de alta frequência, alta temperatura e alta potência. Estes materiais semicondutores de gap largo atraíram enorme atenção dos pesquisadores nos últimos anos. O objetivo desta tese é o estudo das propriedades vibracionais dos nitretos do grupo III referente tanto ao cristal perfeito, quanto ao cristal com defeito. Utilizamos como base a Teoria Clássica do Crital Harmônico e o Método das Funções de Green. Com a Teoria Clássica do Cristal Harmônico, juntamente com o Método do Valence Force Filed e o Método da Soma de Ewald, que permitem gerar a matriz dinâmica do sistema, determinamos o comportamento vibracional dos nitretos binários e das ligas ternárias. A utilização destes métodos permitiu a obtenção do espectro de fônons dos nitretos binários, e o estudo do comportamento dos modos ópticos em para as ligas ternárias. A partir da Função de Green do cristal perfeito e da Função de Green do cristal com defeito, obtivemos as frequências e os modos vibracionais localizados e ressonantes introduzidos pela impureza de C e As em GaN. A partir das densidades de estados do cristal perfeito e do cristal com defeito, calculamos a entropia de formação da vacância de N em GaN. Os resultados obtidos foram usados na interpretação de dados experimentais disponíveis na literatura, relativos às propriedades vibracionais dos nitretos na estrutura wurtzita, e na predição e análise de dados experimentais obtidos pelo grupo do Laboratório de Novos Materiais Semicondutores do Instituto de Física da USP para os nitretos zincblende. / The group-III nitrides (BN, AIN, GaN and InN) and their ternary alloys AlGaN and InGaN generated recently an extraordirlary progress in the production of optoelectronic devices operating in the green-blue-UV region of the spectrum, and in the production of electronic devices of high frequency, high temperature and high power. These wide gap semiconductor materials attracted enormous attention in the last years. The objective of this Thesis was to study the vibrational properties of the bulk III nitrides, without and with defects. To accomplish this study we used the Classic Theory of the Harmonic Crystal and the Method of the Green\'s Functions. With the Classic Theory of the Harmonic Crystal, together with the Valence Force Field Method and the Method of the Ewald\'s Sum, that allow to generate the dynamic matrix of the system, we determined the vibrational behavior of the binary nitrides and of the ternary alloys. The use of these methods allowed us to obtain the phonon spectra of the binary nitrides and to study the behavior of the optical modes at of the ternary alloys. Starting from the Green\'s Function of the perfect crystal and the Green\'s Function of the crystal with defect, we obtained the frequencies and the localized and resonant vibrational modes introduced by the C and As impurities in GaN. Starting from the densities of states of the perfect crystal and of the crystal with defect, we calculated the formation entropy of the N vacancy in GaN. The obtained results were used in the interpretation of experimental data related to the vibrational properties of the wurtzite nitrides available in the literature, and in the prediction and analysis of experimental results obtained for zincblende nitrides by the group of the New Semiconductors Materials Laboratory of t11c Physics Institute at USP.

The error analysis is made for the simple annular domain with the circular boundaries having the same origin. The error bounds are derived, and the optimal convergence rates can be archived. For circular domains with circular boundaries, the same convergence rates can be achieved by strict proof. Moreover, the NFM algorithms and its conservative schemes can be applied to very small holes, which are difficult for other numerical methods to handle. Both the NFM and the collocation Trefftz method(CTM) are used for very small circular holes, the CTM is superior to the NFM in accuracy and stability.

The collocation Trefftz method (CTM) proposed in [36] is employed to annular shaped domains, and new error analysis is made to yield the optimal convergence rates. This popular method is then applied to the special case: the Dirichlet problems on circular domains with circular holes, and comparisons are made with the null field method (NFM) proposed , and new interior field method (IFM) proposed in [35], to find out that both errors and condition numbers are smaller. Recently, for circular domains with circular holes, the null fields method (NFM) is proposed by Chen and his groups. In NFM, the fundamental solutions (FS) with the source nodes Q outside of the solution domains are used in the Green formulas, and the FS are replaced by their series expansions. The Fourier expansions of the known or the unknown Dirichlet and Neumann boundary conditions on the circular boundaries are chosen, so that the explicit discrete equations can be easily obtained by means of orthogonality of Fourier functions. The NFM has been applied to elliptic equations and eigenvalue problems in circular domains with multiple holes, reported in many papers; here we cite those for Laplace¡¦s equation only (see [18, 19, 20]). For the boundary integral equation (BIE) of the first kind, the trigonometric functions are used in Arnold [4, 5], and error analysis is made for infinite smooth solutions, to derive the exponential convergence rates. In Cheng¡¦s Dissertation [21, 22], for BIE of the first kind, the source nodes are located outside of the solution domain, the linear combination of fundamental solutions are used, and error analysis is made only for circular domains. This fact implies that not only can the CTM be applied to arbitrary domains, but also a better numerical performance is provided. Since the algorithms of the CTM is simple and its programming is easy, the CTM is strongly recommended to replace the NFM for circular domains with circular holes in engineering problems.

Os nitretos do grupo III (BN, AIN,Gan e InN) e suas ligas ternárias Al-GaN e InGaN proporcionam, recentemente, um extraordinário avanço na fabricação de dispositivos opto-eletrônicos operando na região do espectro correspondente ao verde-azul-UV e na produção de dispositivos eletrônicos de alta frequência, alta temperatura e alta potência. Estes materiais semicondutores de gap largo atraíram enorme atenção dos pesquisadores nos últimos anos. O objetivo desta tese é o estudo das propriedades vibracionais dos nitretos do grupo III referente tanto ao cristal perfeito, quanto ao cristal com defeito. Utilizamos como base a Teoria Clássica do Crital Harmônico e o Método das Funções de Green. Com a Teoria Clássica do Cristal Harmônico, juntamente com o Método do Valence Force Filed e o Método da Soma de Ewald, que permitem gerar a matriz dinâmica do sistema, determinamos o comportamento vibracional dos nitretos binários e das ligas ternárias. A utilização destes métodos permitiu a obtenção do espectro de fônons dos nitretos binários, e o estudo do comportamento dos modos ópticos em para as ligas ternárias. A partir da Função de Green do cristal perfeito e da Função de Green do cristal com defeito, obtivemos as frequências e os modos vibracionais localizados e ressonantes introduzidos pela impureza de C e As em GaN. A partir das densidades de estados do cristal perfeito e do cristal com defeito, calculamos a entropia de formação da vacância de N em GaN. Os resultados obtidos foram usados na interpretação de dados experimentais disponíveis na literatura, relativos às propriedades vibracionais dos nitretos na estrutura wurtzita, e na predição e análise de dados experimentais obtidos pelo grupo do Laboratório de Novos Materiais Semicondutores do Instituto de Física da USP para os nitretos zincblende. / The group-III nitrides (BN, AIN, GaN and InN) and their ternary alloys AlGaN and InGaN generated recently an extraordirlary progress in the production of optoelectronic devices operating in the green-blue-UV region of the spectrum, and in the production of electronic devices of high frequency, high temperature and high power. These wide gap semiconductor materials attracted enormous attention in the last years. The objective of this Thesis was to study the vibrational properties of the bulk III nitrides, without and with defects. To accomplish this study we used the Classic Theory of the Harmonic Crystal and the Method of the Green\'s Functions. With the Classic Theory of the Harmonic Crystal, together with the Valence Force Field Method and the Method of the Ewald\'s Sum, that allow to generate the dynamic matrix of the system, we determined the vibrational behavior of the binary nitrides and of the ternary alloys. The use of these methods allowed us to obtain the phonon spectra of the binary nitrides and to study the behavior of the optical modes at of the ternary alloys. Starting from the Green\'s Function of the perfect crystal and the Green\'s Function of the crystal with defect, we obtained the frequencies and the localized and resonant vibrational modes introduced by the C and As impurities in GaN. Starting from the densities of states of the perfect crystal and of the crystal with defect, we calculated the formation entropy of the N vacancy in GaN. The obtained results were used in the interpretation of experimental data related to the vibrational properties of the wurtzite nitrides available in the literature, and in the prediction and analysis of experimental results obtained for zincblende nitrides by the group of the New Semiconductors Materials Laboratory of t11c Physics Institute at USP.

This thesis deals with safety verification analysis of collision avoidance systems for unmanned vehicles. The safety of the vehicle is dependent on collision avoidance algorithms and associated control laws, and it must be proven that the collision avoidance algorithms and controllers are functioning correctly in all nominal conditions, various failure conditions and in the presence of possible variations in the vehicle and operational environment. The current widely used exhaustive search based approaches are not suitable for safety analysis of autonomous vehicles due to the large number of possible variations and the complexity of algorithms and the systems. To address this topic, a new optimisation-based verification method is developed to verify the safety of collision avoidance systems. The proposed verification method formulates the worst case analysis problem arising the verification of collision avoidance systems into an optimisation problem and employs optimisation algorithms to automatically search the worst cases. Minimum distance to the obstacle during the collision avoidance manoeuvre is defined as the objective function of the optimisation problem, and realistic simulation consisting of the detailed vehicle dynamics, the operational environment, the collision avoidance algorithm and low level control laws is embedded in the optimisation process. This enables the verification process to take into account the parameters variations in the vehicle, the change of the environment, the uncertainties in sensors, and in particular the mismatching between model used for developing the collision avoidance algorithms and the real vehicle. It is shown that the resultant simulation based optimisation problem is non-convex and there might be many local optima. To illustrate and investigate the proposed optimisation based verification process, the potential field method and decision making collision avoidance method are chosen as an obstacle avoidance candidate technique for verification study. Five benchmark case studies are investigated in this thesis: static obstacle avoidance system of a simple unicycle robot, moving obstacle avoidance system for a Pioneer 3DX robot, and a 6 Degrees of Freedom fixed wing Unmanned Aerial Vehicle with static and moving collision avoidance algorithms. It is proven that although a local optimisation method for nonlinear optimisation is quite efficient, it is not able to find the most dangerous situation. Results in this thesis show that, among all the global optimisation methods that have been investigated, the DIviding RECTangle method provides most promising performance for verification of collision avoidance functions in terms of guaranteed capability in searching worst scenarios.

A sinterização do cobre ocorre através do transporte difusivo de matéria de regiões de alto para as de baixo potencial químico. Esse processo tem como força motriz a minimização da energia associada às interfaces. Na tentativa de quantificar o processo de sinterizaçao, diversos modelos analíticos foram desenvolvidos desde 1945. O presente trabalho teve como intuito implementar e validar um modelo matemático baseado no modelo de campo de fases (\"phase field method\") para simular o primeiro estágio do processo de sinterização. Para isso, um estudo termodinâmico detalhado foi realizado de modo a definir as equações a serem empregadas no modelo. Foi feita uma análise quantitativa (análise estatística) e qualitativamente (análise gráfica e pelo expoente do tempo) dos modelos analíticos teóricos comparando-os com os valores experimentais publicados em cinco artigos de grande relevância na área. A partir dos resultados estatísticos observou-se que o melhor mecanismo para descrever o processo de sinterização do cobre é o modelo combinado entre quatro dos seis principais modelos individuais. Os mecanismos de transporte de fase fictícia via evaporação-condensação e via difusão gasosa contribuem de maneira irrisória na sinterização do cobre, sendo por muitos autores desconsiderados. Foi verificado que a configuração inicial do metal, se na forma de esferas ou cilindros, modifica o processo de difusão dominante. Sendo que o efeito da difusão superficial é mais predominante nas esferas do que nos cilindros, consequentemente, o mecanismo combinado para a esferas inclui a difusão superficial, enquanto que o dos cilindros não. Na simulação em condições unidimensionais, o modelo foi capaz de impor as condições de equilíbrio termodinâmico local e de movimentar a interface no sentido contrário ao fluxo de lacunas. Sob condições bidimensionais, o mesmo impôs automaticamente a fração de lacunas de equilíbrio sob o efeito do raio de curvatura, responsável pela expansão ou retração do sólido/poro cilíndrico. Na simulação da formação do pescoço entre dois cilindros de cobre puro, observou-se um comportamento qualitativo consistente com o comportamento físico. A principal dificuldade encontrada na modelagem foi o tamanho da malha e o tempo de processamento computacional necessário. Para resolução destes dois aspectos, usou-se uma malha adaptativa e foi feita a paralelização em placa de vídeo do código computacional. / Copper sintering occurs through a diffusive transport of matter from regions with high chemical potential to regions of low chemical potential. The driving force of this process is the minimization of the energy associated with the interfaces of the system. In an attempt to quantify the sintering process, several analytical models have been developed since 1945. The aim of the present work was to implement and validate a mathematical model based on the phase field model to simulate the first stage of sintering. A very detailed thermodynamic study was done in order to define which equations should me used in the computational model. As well as, the use of a quantitative (statistical analysis) and a qualitative analysis (graphical analysis and by the exponent of time) to compare the theoretical models with the experimental values published in five articles of great relevance in the area. From the statistical results it was observed that the best mechanism to describe the copper sintering is the combined model between the main individual models (lattice diffusion from surface, lattice diffusion from grain boundary, surface diffusion and grain boundary diffusion). The mechanisms of gas-phase transport via evaporation-condensation and gas diffusion contribute in a negligible way in copper sintering, considered irrelevant by many authors . It has been found that the initial configuration of the metal, whether in the form of spheres or cylinders, modifies the dominant diffusion mechanism. Since the effect of surface diffusion is more predominant in the spheres than in the cylinders, therefore the combined mechanism for the spheres includes surface diffusion, while the cylinders do not. With the computational modeling, some important mechanisms that occur during stage I of sintering were simulated. Simulations performed under unidimensional conditions indicated that the model is able to impose local thermodynamic equilibrium conditions and to move the interface in the opposite direction of the vacancies flow. When used to simulate the transport of vacancies under two-dimensional conditions, the model automatically imposed the fraction of equilibrium vacancies under the effect of the radius of curvature. This fraction results in a flow that causes the expansion or retraction of the solid / cylindrical pore, which was reproduced by the implemented model. The main difficulties found in the computational modeling were the size of the mesh and the computational processing time required. To solve these two aspects, an adaptive mesh was used and the parallelization of the computational code was done, which resulted in a significant reduction in the simulation time.

Investigamos certas propriedades físicas da teoria de campos a temperatura finita. Mostramos que, na presença de campos externos bosonicos, o limite de altas temperaturas das funções de Green pode ser obtido supondo que todas as componentes dos momentos externos sao nulas, para todas as funções de Green 1PI em ordem de um laco e em ordem de dois laços para a auto-energia. No contexto do método do campo de fundo, este resultado implica que a lagrangiana efetiva estática de um campo externo bosonico pode ser obtida calculando a pressão na presença de um campo externo independente do espaço-tempo, no limite de altas temperaturas. Aplicamos este resultado para obter a lagrangiana efetiva estática de um campo de Maxwell interagindo com um plasma da QED em ordem de dois laços. Obtivemos também a lagrangiana efetiva estática de um campo gravitacional interagindo com um campo escalar real livre, e com um plasma da QED em ordem de dois laços. / We have investigated some physical proprieties of nite temperature field theory. We have shown that, in the presence of an external bosonic field, the high temperature limit of the static Green\'s functions can be obtained assuming that all components of the external momenta are zero. This was shown to be the case for all 1PI Green\'s functions, at one-loop order, as well as for the self-energy, at two-loop order. In the context of the background field method, this result implies that the static efective Lagrangian of an external field can be obtained computing the pressure in the presence of an external field which is independent of the space-time, in the high temperature limit. We have applied this result in order to obtain the efective Lagrangian of an external Maxwell field interacting with a QED plasma at two-loop order. We have also obtained the efective Lagrangian of the gravitational field interacting with a free neutral scalar field, as well as with a QED plasma at two-loop order.

O gás de Bose carregado foi estudado em duas e três dimensões, sendo que no caso bidimensional foram considerados dois tipos distintos de interação: l/r e ln(r). Aplicamos a esses sistemas o método do campo auto-consistente que leva em consideração a interação de curto alcance entre os bosons através de uma correção de campo local. Por meio de cálculos numéricos auto-consistentes determinamos o fator de estrutura S(&#8594k) em um amplo intervalo de densidades. A partir de S(&#8594k) obtivemos a função de correlação dos pares, a energia do estado fundamental que é essencialmente a energia de correlação, a pressão do gás e o espectro de excitações elementares. Calculamos ainda a densidade de blindagem induzida por uma impureza carregada fixada no gás. No limite de altas densidades nossos cálculos reproduzem os resultados da teoria de perturbação de Bogoliubov. Na região de densidades intermediárias em que os sistemas são fortemente correlacionados nossos resultados apresentam uma boa concordância com cálculos baseados na aproximação de HNC e no método de Monte Carlo. Nossos resultados são em várias situações confrontados com os de RPA demonstrando que o método que utilizamos é muito mais adequado para tratar o sistema. Os sistemas bidimensionais mostraram-se mais correlacionados que o tridimensional, sendo que o gás com interação l/r é mais correlacionado que o logarítmico a altas densidades, mas na região de densidades baixas essa situação se inverte. Finalmente calculamos as funções termodinâmicas dos sistemas bi e tridimensionais a temperaturas finitas próximas do zero absoluto baseando-nos nos espectros de excitação do gás a temperatura zero. / The two and three-dimensional charged Bose gas have been studied. In the bidimensional case two different types of interaction were considered: l/r and ln(r).We have applied to these systems the self-consistent-field method, which takes into account the short range correlations between the bosons through a local-field correction. By using self-consistent numerical calculations we determinate the structure factor S(&#8594k) in a wide range of densities. From S(&#8594k) we obtained the pair-correlation function, the ground-state energy, the pressure of the gas and the spectrum of elementary excitations. In addition we calculated the screening density induced by a fixed charged impurity. In the high-density limit our calculations reproduce the results given by Bogoliubov\'s perturbation theory. In the intermediate-density region, corresponding to the strongly coupled systems, our results are in very good agreement with calculations based on HNC approximation as well as Monte Carlo method. Our results are compared in several situations with RPA results showing that the self-consistent method is much more accurate. The two-dimensional systems showed to be more correlated than the three-dimensional one; the gas with interaction l/r is also more correlated than the logarithmic one at high densities, but it begins to be less correlated than this one in the low-density region. Finally we calculated the thermodynamic functions of the two and three-dimensional systems at finite temperatures near absolute zero, based upon the gas excitation spectra at zero temperature.