Neste trabalho estudamos alguns aspectos de geometria e topologia de variedades com o objetivo de aplicar tais resultados juntamente com dados observacionais para tentar determinar as variedades tridimensionais que possam servir de modelo para a parte espacial do universo. / In this work we study some aspects of geometry and topology of manifolds with the goal of applying such results with observational data to try to determine the tridimensional manifold that can serve as a model for the spatial part of the universe.

O objetivo deste trabalho é apresentar alguns métodos numéricos para resolver a equação da convecção-difusão, em um espaço uni-dimensional. Demonstrada a derivação desta equação, parte-se para a sua adaptação ao espaço uni-dimensional. A seguir, é considerado o fenômeno da difusão, isoladamente. Num segundo momento, é feito um estudo sobre o fenômeno da convecção, isoladamente. Posterionnente, considera-se o caso geral, em que os fenômenos da convecção e da difusão ocotrem simultaneamente. A importante questão da estabilidade em soluções numéricas também é analisada Ao final de cada uma das três situações mencionadas, são apresentados resultados práticos dos programas computacionais apl icados a elas. / The purpose of this work is show some numerical methods to solve diffusionconvection equation, in one dimension. Showed the deduction ofthis equation, discribring the problem, we present its adaptation to one dimension. After this, severa! steps are fulfi li: first, the di.tfusion phenomenon is considered isolated; second, the convection phenomenon is studied isolated; third, the general case, where convection and diffusion phenomena happened together. The major question about stability is also analysed. At the end ofthe three above situations, we give some praticai results of computational programs applied to them.

O objetivo deste trabalho é apresentar alguns métodos numéricos para resolver a equação da convecção-difusão, em um espaço uni-dimensional. Demonstrada a derivação desta equação, parte-se para a sua adaptação ao espaço uni-dimensional. A seguir, é considerado o fenômeno da difusão, isoladamente. Num segundo momento, é feito um estudo sobre o fenômeno da convecção, isoladamente. Posterionnente, considera-se o caso geral, em que os fenômenos da convecção e da difusão ocotrem simultaneamente. A importante questão da estabilidade em soluções numéricas também é analisada Ao final de cada uma das três situações mencionadas, são apresentados resultados práticos dos programas computacionais apl icados a elas. / The purpose of this work is show some numerical methods to solve diffusionconvection equation, in one dimension. Showed the deduction ofthis equation, discribring the problem, we present its adaptation to one dimension. After this, severa! steps are fulfi li: first, the di.tfusion phenomenon is considered isolated; second, the convection phenomenon is studied isolated; third, the general case, where convection and diffusion phenomena happened together. The major question about stability is also analysed. At the end ofthe three above situations, we give some praticai results of computational programs applied to them.

Neste trabalho estudamos alguns aspectos de geometria e topologia de variedades com o objetivo de aplicar tais resultados juntamente com dados observacionais para tentar determinar as variedades tridimensionais que possam servir de modelo para a parte espacial do universo. / In this work we study some aspects of geometry and topology of manifolds with the goal of applying such results with observational data to try to determine the tridimensional manifold that can serve as a model for the spatial part of the universe.

Neste trabalho, um problema de transferência de calor da dinâmica de gases rarefeitos, causado pela diferença de temperaturas nas superfícies de um canal, é abordado. O problema é formulado através dos modelos cinéticos BGK, S e Gross-Jackson da equação linearizada de Boltzmann e resolvido, de forma unificada, pelo método analítico de ordenadas discretas (método ADO). Resultados numéricos para as perturbações de densidade e temperatura e também para o fluxo de calor são apresentados e comparados, mostrando que não se pode dizer que algum dos três modelos seja uma melhor aproximação da solução aos resultados da equação linearizada de Boltzmann. / In this work, a heat transfer problem in the rarefied gas dynamics field, in a plane channel, is studied. In particular, the flow is induced by different temperatures at the wall surfaces. The formulation of the problem is based in an ”unified” kinetic equation which includes the BGK model, the S-model and the Gross-Jackson model of the linearized Boltzmann equation. An analytical version of the discrete-ordinates method (the ADO method) is used to develop the solution and to evaluate the density, temperature and heat-flow profiles. Numerical results are presented and used to establish comparisons with the linearized Boltzmann equation results. It is shown that, for an analysis based in all cases, it is not possible to say that one of the models is a better approximation of the solution of the linearized Boltzmann equation.

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No. of bitstreams: 2 Dissertação - Rafael Marques Paes Teixeira - 2017.pdf: 12639296 bytes, checksum: a0fc40914d28a4f117459a9314e27b7a (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2017-04-10 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / This dissertation presents the research that was focused on the study of solitons in nonintegrable systems governed by effective models of partial differential equations (PDEs) of (1+1) dimensions, that is, one spatial and one time dimension. In the main research line, we considered a model of two fields given by nonlinear Schrödinger equations with logarithmic nonlinearity and linear coupling, the main aspects of the collision dynamics of two Gaussian solitons (gaussons) were investigated, addressing the stability of the solutions, the interaction mechanisms, and the regularity of the scattering. The study of gaussons collisions in this model was performed via two approaches: semianalytic (variational/reduced model) and direct numerical simulation. For the system considered in this work, both approaches provided similar results which revealed the existence of a chaotic and fractal dynamics involving the initial conditions and the post-collisional properties of the solitons, which were verified by the analysis of the correlations between the input and output parameters. The fractal patterns are constituted by regularity windows that are positioned accordingly to a well defined rule, which was obtained via a linearization model that was found to be very similar in both approaches (i.e., the windows arrangement was well described by the variational model). In such windows the solitons collision dynamic turns out to be not chaotic (regular collision), it becomes predictable and very well defined, hence the width of these windows provides the conditions in which the control of the interaction is possible. Besides, by analyzing the dynamics of the variational parameters of the reduced model and also the dynamics of the most relevant physical quantities in the exact approach via PDEs (namely the total kinetic and potential energies), we attested that these windows arise due to a resonant mechanism of energy exchange between translational and vibrational soliton modes. By studying the linear stability of the solitary waves, we proved that the vibrational modes originate from shape oscillations induced by internal modes and also, possibly, by instability modes excited during the collisions. We verified that these oscillations are well defined in all regular collisions. The viability of the variational method in the system studied is due to the suitable description of the vibrational modes by the reduced model, and also the low influence of the emitted nonlinear radiation in the regular collision processes found in the direct simulations. / Essa dissertação apresenta o trabalho de pesquisa que teve como foco o estudo de sólitons em sistemas não-integráveis governados por modelos efetivos de equações diferenciais parciais (EDPs) de dimensão (1+1), isto é, uma dimensão espacial e uma temporal. Na linha de pesquisa principal, foi considerado um modelo de dois campos dado por equações de Schrödinger com não-linearidade logarítmica e acoplamento linear, em que foram investigados os principais aspectos da colisão de dois sólitons gaussianos (gaussons), considerando a estabilidade das soluções, seus mecanismos de interação e a regularidade do espalhamento. O estudo de colisões de gaussons nesse modelo foi realizado a partir de duas abordagens: semianalítica (modelo variacional/reduzido) e simulação numérica direta. Para o sistema considerado nesse trabalho, as duas abordagens renderam resultados semelhantes que revelam a existência de uma dinâmica caótica e fractal envolvendo as condições iniciais e as propriedades pós-colisionais dos sólitons, que foi constatada pela análise das correlações entre os parâmetros de entrada e saída. Os padrões fractais são constituídos por janelas de regularidade, cujo posicionamento segue uma regra bem definida que foi obtida via um modelo de linearização que se mostrou muito semelhante nas duas abordagens (i.e., a distribuição de janelas foi bem descrita variacionalmente). Em tais janelas a dinâmica de colisão dos sólitons deixa de ser caótica (colisão regular), tornando-se previsível e bem definida, por isso a largura dessas janelas fornece as condições em que o controle da interação é possível. Além disso, pela análise da dinâmica dos parâmetros variacionais do modelo reduzido e das quantidades físicas mais relevantes na abordagem exata via EDPs (energia total cinética e potencial), verificou-se que essas janelas surgem devido a um mecanismo ressonante de troca de energia entre os modos translacionais e vibracionais dos sólitons. Pelo estudo da estabilidade linear das ondas solitárias, verificou-se que os modos vibracionais decorrem de oscilações de forma induzidas por modos internos e também, possivelmente, por modos de instabilidade excitados durante as colisões. Constatou-se que essas oscilações são bem definidas em todas as colisões regulares. A viabilidade do método variacional no sistema estudado se deve à adequada descrição dos modos vibracionais pelo modelo reduzido e à baixa influência da radiação não-linear emitida sobre os processos de colisão regulares encontrados via simulações diretas.

Dielectric elastomers (DEs) are an important class of materials, currently employed in the design and realization of electrically-driven, highly deformable actuators and devices, which find application in several fields of technology and engineering, including aerospace, biomedical and mechanical engineering. Subject to a voltage, a membrane of a soft dielectric elastomer coated by compliant electrodes reduces its thickness and expands its area, possibly deforming in-plane well beyond 100%: this principle is exploited to conceive transducers for a broad range of applications, including soft robots, adaptive optics, Braille displays and energy harvesters. Soft dielectrics undergo finite strains, and their modelling requires a formulation based on the Mechanics of Solids at large deformations. A major problem that limits the widespread diffusion of such devices in everyday technology is the high voltage required to activate large strains, because of the low dielectric permittivity of typical materials (acrylic elastomers or silicones), in the order of few unities, which governs the electromechanical coupling. Composite materials (reinforcing a soft matrix with stiff and high-permittivity particles) provide a way to overcome these limitations, as suggested by some experiments. In addition, composites can display failure modes and instabilities not displayed by homogeneous specimens that must be thoroughly investigated. Commonly, instability phenomena are seen as a serious drawback, that should be predicted and avoided. However, in some cases they can be used to activate snap-through actuation, as in the case of buckling-like or highly-deformable balloon-like actuators. Soft dielectric elastomers display electrostrictive properties (permittivity depending on the deformation) and we show how to take into account such a phenomenon within the theory of electroelasticity. Original results regard the investigation of diffuse modes (buckling like instabilities etc.), surface mode instabilities and localized modes. New (analytical) solutions for band-localization instability are provided and then it has been investigated how such instabilities are related to electrostriction. Regarding DE composites, the goal is to evaluate in detail the behaviour of two-phase rank-1 laminates in terms of different types of actuation, geometric and mechanical properties of phases, applied boundary conditions, and instabilities phenomena, in order to establish precise ranges in which the performance enhancement is effective with respect to the homogeneous counterpart.

The aim of this thesis is to study renormalization of Wick polynomials of quantum Boson fields in locally covariant algebraic quantum field theory in curved spacetime. Vector fields are described as sections of natural vector bundles over globally hyperbolic spacetimes and quantized in a locally covariant framework through the known functorial machinery in terms of local *-algebras. These quantized fields may be defined on spacetimes with given classical background fields, also sections of natural vector bundles: The most obvious one is the metric of the spacetime itself, but we encompass also the case of generic spacetime tensors as background fields. In our framework also physical quantities like the mass of the field or the coupling to the curvature are viewed as background fields. Wick powers of the quantized vector field are then axiomatically defined imposing in particular local covariance, scaling properties and smooth dependence on smooth perturbation of the background fields. A general classification theorem is established for finite renormalization terms (or counterterms) arising when comparing different solutions satisfying the defining axioms of Wick powers. The result is then specialized to the case of spacetime tensor fields. In particular, the case of a vector Klein-Gordon field and the case of a scalar field renormalized together with its derivatives are discussed as examples. In each case, a more precise statement about the structure of the counterterms is proved. The finite renormalization terms turn out to be finite-order polynomials tensorially and locally constructed with the backgrounds fields and their covariant derivatives whose coefficients are locally smooth functions of polynomial scalar invariants constructed from the so-called marginal subset of the background fields. Our main technical tools are based on the Peetre-Slov\'ak theorem characterizing differential operators and on the classification of smooth invariants on representations of reductive Lie groups.

The purpose of this thesis is to focus on the aspect of passive devices allowing for WDM, routing, switching and filtering of optical signals, investigating novel routing concepts based on micro optical side coupled resonators to achieve large bandwidth by multiple cascading and/or multiple coupling (low group velocity) periodicity effects. We will describe some technical aspects necessary for the design and fabrication of some passive circuitry, and usually neglected in purely theoretical approaches, including optical routers based on racetrack resonators and novel SCISSOR and CROW devices.

Scientists are continuously fascinated by the high degree of sophistication found in natural materials, arising from evolutionary optimisation. In living organisms, nature provides a wide variety of materials, architectures, systems and functions, often based on weak constituents at the lower scales. One of the most extensively studied natural materials is spider silk, renowned for its outstanding mechanical properties, which include exceptional strength and toughness. Owing to its wide range of properties, which vary depending on factors such as the type of silk (up to seven) that each spider can produce, and the species of spider, it can be considered a class of semi-crystalline polymeric material. Indeed, spider silk cleverly combines, depending on the application required, the great deformability of an amorphous phase with the stiffness and strength conferred by pseudo-crystals consisting of specific secondary structures of some of the proteins constituting the material. Based on the countless studies conducted on spider silk, it is now also clear that its remarkable performance are the result of a sophisticated optimisation of the material's hierarchical structure. Nevertheless, many of the multiscale mechanisms that give rise to the striking macroscopic properties are still unclear. Many open problems are also related to the relevant effects of environmental conditions and in particular on temperature and humidity, strongly conditioning the mechanical performances. In this thesis, aimed at unveiling some of these open problems, we introduce a multiscale model for the thermo-hygro-mechanical response, starting with the influence of water molecules modifying the microstructure, up to the effects at the macroscopic scale, including softening, increase in elongation at break and supercontraction, i.e. the shortening (up to half the initial length) of the spider threads in wet environments. Thereafter, we describe how the supercontraction effect can be adopted to obtain humidity-driven actuators, and in particular, we determine the maximum actuation force depending on the silk properties at the molecular scale and on the constraining system representing other silk threads or the actuated device. The spider silk actuation properties turned out to be extraordinary, making spider silk potentially the best performing humidity-driven actuator known to date in terms of work density. As observed in many natural materials, spider silks are characterized by a strong variability in both chemical and structural organization, as for example described in the recently published experimental database of properties at different scales of about a thousand different spider silks, where evident correlations among quantities are scarce. This large variability makes the theoretical understanding of the observed material behavior, in relation of the complex hierarchical structure, particularly intriguing. To address this novel amount of experimental data without losing sight of theoretical analytical modelling, we propose a new data modelling methodology to obtain simple and interpretable relationships linking quantities at different scales. In particular, we employ a symbolic regression technique, known as 'Evolutionary Polynomial Regression', which integrates regression capabilities with the Genetic Programming paradigm, enabling the derivation of explicit analytical formulas, finally delivering a deeper comprehension of the analysed physical phenomenon. Eventually, we provide insights to improve our multiscale theoretical model accounting for the humidity effects on spider silks. This approach may represent a proof of concept for modelling in fields governed by multiscale, hierarchical differential equations. We believe that the analytical description of the macroscopic behaviour from microscale properties is of great value both for the full understanding of biological materials, as well as in the perspective of bioinspired materials and structures.