Collision avoidance is an essential component of autonomous shipping. As ships begin to advance towards autonomy, developing an advisory system is one of the first steps. An advisory system with a strong collision avoidance component can help the crew act more quickly and accurately in dangerous situations. One way to avoid colission is to make the vessel stop as fast as possible. In this work, two scenarios are studied, firstly, stopping along a predefined path, and secondly, stopping within a safe area defined by surrounding obstacles. The first scenario was further worked with to formulate a real-time solution. Movements of a vessel, described in three degrees of freedom with continuous dynamics, were simulated using mathematical models of the forces acting on the ship. Nonlinear optimal control problems were formulated for each scenario and solved numerically using discretization and a direct multiple shooting method. The results for the first problem showed that the vessel could stop without much deviation from the path. Paths with different curvatures were tested, and it was shown that a slightly longer distance was traveled when the curvature of the path was greater. The results for the second problem showed that the vessel stays within the safe area and chooses a relatively straight path as the optimal way of stoping. This results in a shorter distance traveled compared to the solution of the first problem. Two different real-time approaches were formulated, firstly a receding-horizon approach and secondly a lookup-based approach. Both approaches were solved with real-time feasibility, where the receding-horizon approach gave a better solution while lookup-based approach had a shorter computational time.

[pt] Seja F uma função definida de um espaço de Banach real X para um espaço de Banach real Y e g um ponto pertencente a Y. Descrevemos um algoritmo para calcular as soluções u da equação F de u igual a g. Inicialmente, o algoritmo parte de uma curva c no domínio, a qual é escolhida de modo a interceptar substancialmente o conjunto crítico de F. Calculamos através de métodos de continuação uma componente da imagem inversa de F de c e definimos essa componente de forma abstrata: grafo completamente espelhado. Claramente, os métodos de continuação padrão têm melhores chances de sucesso em diferentes pontos iniciais. Fornecemos argumentos geométricos para a abundância ocasional de soluções e uma busca estruturada dessas. Três exemplos são considerados detalhadamente. O primeiro é uma função do plano no plano, em que podemos validar os resultados com auxílio de um software. O segundo conjunto de exemplos é obtido a partir da discretização de um problema de Sturm-Liouville não linear com um número inesperado de soluções. Por último, calculamos as seis soluções aproximadas de um problema estudado por Solimini. / [en] Let F be a definite function from a real Banach space X to a real Banach space Y and g a point belonging to Y. We describe an algorithm for calculating the solutions u of the equation F of u equal to g. Initially, the algorithm starts from a curve c in the domain, which is chosen so as to substantially intercept the critical set of F. We calculate through continuation methods a component of the inverse image of F of c and define this component in an abstract way: graph completely mirrored. Clearly, standard continuation methods have better chances of success at different starting points. We provide geometric arguments for the occasional abundance of solutions and a structured search for these. Three examples are considered in detail. The first is a function of the plan in the plan, in which we can validate the results with the help of software. The second set of examples is obtained from the discretization of a non-linear Sturm-Liouville problem with an unexpected number of solutions. Finally, we calculate the six approximate solutions of a problem studied by Solimini.

[pt] Vigas flexíveis são encontradas com cada vez mais frequência em diferentes indústrias, uma vez que novos projetos têm buscado por estruturas mais longas e leves. Isso pode ser uma consequência direta das novas demandas estruturais nos projetos, ou uma simples consequência do engajamento das indústrias em programas de redução de custo (utilização de menos materiais). Em geral, vigas flexíveis são modeladas sob hipóteses de grandes deslocamentos, grandes rotações, mas com pequenas deformações. Essas hipóteses permitem que o equacionamento da dinâmica de vigas flexíveis seja feito através de elementos finitos co-rotacionais. A formulação co-rotacional decompõe o movimento das estruturas flexíveis em duas partes: uma contendo o movimento de corpo rígido e outra com uma (pequena) deformação elástica. Dessa forma, a não-linearidade geométrica causada pelos grandes deslocamentos e rotações das seções transversais das vigas podem ser computadas de forma eficiente. Uma das inovações dessa tese é o uso direto das equações de movimentos geradas pelos elementos finitos co-rotacionais no cálculo dos modos normais não-lineares (MNNs). Até agora, a maioria das análises dinâmicas com elementos finitos co-rotacionais foram restritas à integração das equações de movimento. O conhecimento de MNNs é útil na análise de sistemas não-lineares pois permitem um detalhado entendimento das vibrações nos regimes não-lineares. Com eles, pode-se, por exemplo, prever comportamentos de enrijecimento/relaxamento, localização de respostas, interação entre modos, existência de isolas, etc. A definição de Rosenberg sobre MNNs como sendo soluções periódicas (não necessariamente síncronas) do sistema é adotado na tese. Os métodos do Balanço Harmônico e do Tiro são apresentados e utilizados no cálculo de soluções periódicas de sistemas não-lineares. Um procedimento de continuação numérica é implementado para computar os MNN eficientemente para diferentes níveis de energia. Exemplos numéricos mostram a capacidade do método proposto quando aplicado aos elementos finitos co-rotacionais. / [en] Flexible beams are becoming ubiquitous in several industrial applications, as new projects often aim for lighter and longer structures. This fact is directly related to the new challenging demands on structural performances, or it is a simple consequence of the engagement of industries in cost reduction programs (usage of less material). Flexible beams are usually modeled under the assumption of large displacements, finite rotations, but with small strains. Such hypotheses allow the equation of motion to be built using co-rotational finite elements. The co-rotational formulation decomposes the total motion of a flexible structure into two parts: a rigid body displacement and an elastic (small) deformation. This way, the geometric nonlinearity caused by the large displacements and rotations of the beam s cross sections can be efficiently computed. One of the novelties of this thesis is the direct usage of the equation of motion generated by a co-rotational finite element formulation in the computation of nonlinear normal modes (NNM). So far, most of the dynamic analyses with co-rotation finite element models were restricted to numerical integrations of the equation of motion. The knowledge of NNMs can be beneficial in the analysis of any nonlinear structure since it allows a thoroughly understanding of the vibratory response in the nonlinear regime. They can be used, for example, to predict a hardening/softening behavior, a localization of the responses, the interactions between modes, the existence of isolas, etc. The Rosenberg s definition of NNM as periodic solutions (non-necessarily synchronous motion) is adopted here. The Harmonic Balance method and the Shooting methods are presented and used to compute periodic solutions of nonlinear systems. A numerical path continuation scheme is implemented to efficiently compute NNMs at different energy levels. Numerical examples show the capability of the proposed method when applied to co-rotational beam elements.

<p dir="ltr">Within the aerospace community in recent years, there has been a marked increase in interest in cislunar space. To this end, the study of the dynamics of this regime has flourished in both quantity and quality in recent years, spearheaded by the use of simplified dynamical models to gain insight into the dynamics and to generate viable mission concepts. The most popular and simple of these models, the Circular Restricted Three-Body Problem, has been thoroughly explored to meet these goals (even well-prior to the recent spike in interest). Much work has been done investigating periodic orbits within these models, and similarly has been performed on non-periodic transfers into periodic orbits. Studied less is the superposition of these two concepts, or using periodic orbits as a way to transit, for example, cislunar space. In this thesis, the development of periodic orbits amenable to transiting is accomplished. Beginning from periodic orbit families already present in the literature, this research finds a novel and useful family of periodic orbits, here dubbed the spatial 2:1-resonant orbit family. Within this newly-discovered family, multitudes of qualitative behaviors interesting to the astrodynamics community are found. Many family members seem accommadating to a diverse set of mission profiles, from purely-unstable family members best suited to use as transfers, to marginally stable ones best suited to longer-term use. This family as a whole is analyzed and catalogued with thorough descriptions of behavior, both quantitative and qualitative. While the Circular Restricted Three-Body Problem serves as an excellent starting point for analysis, trajectories found there must be generalized to higher-fidelity modeling. In this spirit, this thesis also focuses on demonstrating such generalization and putting it into practice using the more sophisticated Elliptic-Restricted Three-Body Problem. Documentation of the numerical tools necessary and helpful in accomplishing this generalization is included in this work. Prototypically, the truly 2:1 sidereally-resonant unstable member of the 2:1 family is transitioned into the elliptic problem, as is a nearly-stable L2 Halo orbit family member. This new trajectory is paired with a more classically-present example to show the validity of the methodology. To aid this analysis, symmetries present within the elliptic model are also explored and explained. With this analysis completed, this orbit family is demonstrated to be both interesting and useful, when considered under even more realistic modelling. Further work to mature this novel family of orbits is merited, both for use as the fundamental building block for transfers and for use for more-permanent habitation. More broadly, this work aims to achieve a further proliferation of the merger between transfer and orbit, concepts which seem distinct at first, but deserve more gradual consideration as different flavors of the same idea.</p>

Cette thèse porte sur l'étude de modèles non linéaires issus de la mécanique quantique relativiste.Dans la première partie, nous démontrons à l'aide d'une méthode de tir l'existence d'une infinité de solutions d'équations de Dirac non linéaires provenant d'un modèle de hadrons et d'un modèle de la physique des noyaux.Dans la seconde partie, nous prouvons par des méthodes variationnelles l'existence d'un état fondamental et d'états excités pour deux modèles de la physique des hadrons. Par la suite, nous étudions la transition de phase reliant les deux modèles grâce à la Gamma-convergence.La dernière partie est consacrée à l'étude d'un autre modèle de hadrons dans lequel les fonctions d'onde des quarks sont parfaitement localisées. Nous énonçons quelques résultats préliminaires que nous avons obtenus. / This thesis is devoted to the study of nonlinear models from relativistic quantum mechanics.In the first part, we show thanks to a shooting method, the existence of infinitely many solutions of nonlinear Dirac equations of two models from the physics of hadrons and the physics of the nucleus.In the second part, we prove thanks to variational methods the existence of a ground state and excited states for two models of the physics of hadrons. Next, we study the phase transition which links the models thanks to the $\Gamma$-convergence.The last part is devoted to the study of another model from the physics of hadrons in which the wave functions are perfectly confined. We give some preliminary results.

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