Dissertations / Theses on the topic 'Weakly nonlinear analysi'

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CNPq; INCT-FCx.
Nesta Tese são empregadas técnicas analíticas e numéricas para investigar o fenômeno de formação de dedos viscosos entre fluidos imiscíveis confinados quando um destes fluidos é um fluido magnético complexo. Diferentes tipos de esquemas geométricos efetivamente bidimensionais foram investigados. Duas situações distintas são tomadas com relação à natureza da amostra de fluido magnético: um fluido newtoniano usual, e um fluido magneto-reológico que apresenta um yield stress dependente da intensidade do campo magnético. Equações governantes adequadas são derivadas para cada um dos casos. Para obter um entendimento analítico dos estágios iniciais da evolução temporal da interface foi empregada uma análise fracamente não-linear de modos acoplados. Este tipo de análise acessa a estabilidade de uma interface inicialmente perturbada e também revela a morfologia dos dedos emergentes. Em algumas circunstâncias soluções estacionárias podem ser encontradas mesmo na ordem não-linear mais baixa. Nesta situação é feita uma comparação de algumas destas soluções com soluções estáticas totalmente não-lineares obtidas através de um formalismo de vortex-sheet na condição de equilíbrio. Em seguida foi desenvolvido um modelo de phase-field aplicado a fluidos magnéticos que é capaz de simular numericamente a dinâmica totalmente não-linear do sistema. O modelo consiste em introduzir uma função auxiliar que reproduz uma interface difusa de espessura finita. Utilizando esta ferramenta também é possível estudar um complexo problema de dedos viscosos de origem biológica: o fluxo de actina como um fluido ativo dentro de um fragmento lamelar.

In the first part of this dissertation, we produce and study a generalized mathematical model of solid combustion. Our generalized model encompasses two special cases from the literature: a case of negligible heat diffusion in the product, for example, when the burnt product is a foam-like substance; and another case in which diffusivities in the reactant and product are assumed equal. In addition to that, our model pinpoints the dynamics in a range of settings, in which the diffusivity ratio between the burned and unburned materials varies between 0 and 1. The dynamics of temperature distribution and interfacial front propagation in this generalized solid combustion model are studied through both asymptotic and numerical analyses. For asymptotic analysis, we first analyze the linear instability of a basic solution to the generalized model. We then focus on the weakly nonlinear case where a small perturbation of a neutrally stable parameter is taken so that the linearized problem is marginally unstable. Multiple scale expansion method is used to obtain an asymptotic solution for large time by modulating the most linearly unstable mode. On the other hand, we integrate numerically the exact problem by the Crank-Nicolson method. Since the numerical solutions are very sensitive to the derivative interfacial jump condition, we integrate the partial differential equation to obtain an integral-differential equation as an alternative condition. The result system of nonlinear algebraic equations is then solved by the Newton’s method, taking advantage of the sparse structure of the Jacobian matrix. By a comparison of our asymptotic and numerical solutions, we show that our asymptotic solution captures the marginally unstable behaviors of the solution for a range of model parameters. Using the numerical solutions, we also delineate the role of the diffusivity ratio between the burned and unburned materials. We find that for a representative set of this parameter values, the solution is stabilized by increasing the temperature ratio between the temperature of the fresh mixture and the adiabatic temperature of the combustion products. This trend is quite linear when a parameter related to the activation energy is close to the stability threshold. Farther from this threshold, the behavior is more nonlinear as expected. Finally, for small values of the temperature ratio, we find that the solution is stabilized by increasing the diffusivity ratio. This stabilizing effect does not persist as the temperature ratio increases. Competing effects produce a “cross-over” phenomenon when the temperature ratio increases beyond about 0.2. In the second part, we study the existence and decay rate of a transmission problem for the plate vibration equation with a memory condition on one part of the boundary. From the physical point of view, the memory effect described by our integral boundary condition can be caused by the interaction of our domain with another viscoelastic element on one part of the boundary. In fact, the three different boundary conditions in our problem formulation imply that our domain is composed of two different materials with one condition imposed on the interface and two other conditions on the inner and outer boundaries, respectively. These transmission problems are interesting not only from the point of view of PDE general theory, but also due to their application in mechanics. For our mathematical analysis, we first prove the global existence of weak solution by using Faedo-Galerkin’s method and compactness arguments. Then, without imposing zero initial conditions on one part of the boundary, two explicit decay rate results are established under two different assumptions of the resolvent kernels. Both of these decay results allow a wider class of relaxation functions and initial data, and thus generalize some previous results existing in the literature.

L'instabilité de flottement a été le sujet de nombreuses études depuis le milieu du vingtième siècle à cause de ses applications critiques en aéronautique. Elle est classiquement décrite comme un instabilité linéaire en écoulement potentiel, mais les effets visqueux et nonlinéaires du fluide peuvent avoir un impact crucial.La première partie de cette thèse est consacrée au développement de méthodes théoriques et numériques pour l'analyse linéaire et nonlinéaire de la dynamique d'une ``section typique aéroélastique'' --- une plaque montée sur des ressorts de flexion et torsion --- plongée dans un écoulement laminaire bidimensionnel modélisé par les équations de Navier--Stokes incompressibles.D'abord, on développe une analyse faiblement nonlinéaire pour étudier le régime basse amplitude, puis, une approche d'équilibrage harmonique, connue comme la Méthode Spectrale en Temps (TSM), de façon à capturer des solutions de flottement plus fortement nonlinéaires. Le défi de la résolution numérique des équations TSM est relevé grâce au développement d'une approche parallèle en temps de type Newton--Krylov, combinée à un préconditionneur spécialement développé, dit ``bloc-circulant''.La seconde partie de la thèse est dédiée à l'étude physique de la bifurcation de flottement. On commence par revisiter le problème de stabilité linéaire en mettant en lumière, en particulier, les effets de viscosité.On poursuit avec l'étude des effets nonlinéaires fluides: les structures légères et les hauts nombres de Reynolds favorisent des bifurcations sous-critiques.On achève cette partie en étudiant l'apparition de modulations de basse fréquence sur des solutions périodiques de flottement. On explique ce comportement par une instabilité linéaire (Floquet) de cycle limite.La dernière partie de la thèse vise à initier l'extension des différentes méthodes évoquées précédemment pour le cas de configurations tridimensionnelles à grande échelle. En guise de premier pas vers cet objectif à long terme, on développe un outil open-source massivement parallèle capable de réaliser l'analyse de stabilité linéaire hydrodynamique (structure figée) d'écoulements tridimensionnels possédant plusieurs dizaines de millions de degrés de liberté
The flutter instability has been the focus of numerous works since the middle of the twentieth century, due to its critical application in aeronautics. Flutter is classically described as a linear instability using potential flow models, but viscous and nonlinear fluid effects may both crucially impact this aeroelastic phenomenon.The first part of this thesis is devoted to the development of theoretical and numerical methods for analyzing the linear and nonlinear dynamics of a ``typical aeroelastic section'' --- a heaving and pitching spring-mounted plate --- immersed in a two-dimensional laminar flow modeled by the incompressible Navier--Stokes equations.First, we develop a semi-analytical weakly nonlinear analysis to efficiently study the small amplitude regime. Second, we develop a harmonic balance-type method, known as the Time Spectral Method (TSM), in order to tackle highly-nonlinear periodic flutter solutions. The challenging task of solving the TSM equations is tackled via a time-parallel Newton--Krylov approach in combination with a new, so-called block-circulant preconditioner.The second part of the thesis focuses on the physical investigation of the flutter bifurcation. We start by revisiting the linear stability problem using a Navier--Stokes fluid model allowing to highlight, in particular, the effect of viscosity.We continue our route on the flutter bifurcation by investigating the effect of fluid nonlinearities: low solid-to-fluid mass ratios and increasing Reynolds numbers foster subcritical bifurcations.We conclude our study by investigating the appearance of low-frequency amplitude modulations on top of a previously established periodic flutter solution. We explain this behavior by a (Floquet) linear instability of periodic solutions.The last part of the thesis aims at initiating the extension of the different methods previously evoked to large-scale three-dimensional configurations. As a first step towards this long-term goal, we develop an open-source massively parallel tool, able to perform hydrodynamic (the structure is fixed) linear stability analysis of three-dimensional flows possessing several tens of millions of degrees of freedom

L'objectif de cette thèse est d'analyser l'influence des non linéarités, du comportement rhéologique des fluides rhéofluidifiants, sur les conditions de stabilité et de transition vers la turbulence. Dans un premier temps, une analyse linéaire de stabilité avec une approche modale a été réalisée. Les résultats obtenus mettent clairement en évidence l'effet stabilisant de la rhéofluidification. Ensuite, une analyse faiblement non linéaire de stabilité a été menée en vue d'examiner l'influence de la perturbation de la viscosité sur la stabilité vis à vis de perturbations d'amplitude finie. L'analyse de la contribution des termes non linéaires d'inertie et visqueux montre que, contrairement aux termes d'inertie, les termes non linéaires visqueux ont tendance à accélérer l'écoulement et favoriser une bifurcation sur-critique. Les effets rhéofluidifiants tendent à réduire la dissipation visqueuse. Finalement, une analyse fortement non linéaire de stabilité a été conduite en utilisant les techniques de suivi de branches de solutions par des méthodes de continuation. Pour pouvoir traiter les termes visqueux fortement non linéaires, un code de calcul pseudo-spectral a été développé. Des solutions non linéaires d'équilibre ont été obtenues et caractérisées pour différentes valeurs des paramètres rhéologiques
The aim of this study is to understand the influence of the nonlinear rheological behaviour of the shear-thinning fluids on the flow stability and transition to turbulence. First, a linear stability analysis using modal approach was carried out. Results clearly highlight the stabilizing effect of shear-thinning. Then, as a first approach to take into account nonlinear effects of viscosity perturbation on the flow stability, a weakly nonlinear stability analysis is performed in the neighbourhood of the critical conditions. Results indicate that shear-thinning reduces the viscous dissipation and, in contrast to inertial terms, the nonlinear viscous terms tend to accelerate the flow and act in favour of supercritical bifurcation. Finally, a nonlinear stability analysis is done by following solution branches in the parameter space using continuation techniques. To deal with highly nonlinear viscous terms, a pseudo-spectral code is developed. Nonlinear equilibrium solutions was found and characterized for various values of the rheological parameters

Pattern formation and modulation is an active branch of mathematics, not only from the perspective of fundamental theory but also for its huge applications in many fields of physics, ecology, chemistry, biology, and other sciences. In this thesis, the occurrence of Turing and wave instabilities, giving rise to stationary and oscillatory patterns, respectively, is theoretically investigated by means of two-compartment reaction-transport hyperbolic systems. The goal is to elucidate the role of inertial times, which are introduced in hyperbolic models to account for the finite-time propagation of disturbances, in stationary and transient dynamics, in supercritical and subcritical regimes. In particular, starting from a quite general framework of reaction-transport model, three particular cases are derived. In detail, in the first case, the occurrence of stationary patterns is investigated in one-dimensional domains by looking for the inertial dependence of the main features that characterize the formation and stability process of the emerging patterns. In particular, the phenomenon of Eckhaus instability, in both supercritical and subcritical regimes, is studied by adopting linear and multiple-scale weakly-nonlinear analysis and the role played by inertia during the transient regime, where an unstable patterned state evolves towards a more favorable stable configuration through sequences of phase-slips, is elucidated. Then, in the second topic, the focus is moved to oscillatory periodic patterns generated by wave (or oscillatory Turing) instability. This phenomenon is studied by considering 1D two-compartment hyperbolic reaction-transport systems where different transport mechanisms of the species here involved are taken into account. In these cases, by using linear and weakly nonlinear stability analysis techniques, the dependence of the non-stationary patterns on hyperbolicity is underlined at and close to the criticality. In particular, it is proven that inertial effects play a role, not only during transient regimes from the spatially-homogeneous steady state toward the patterned state but also in altering the amplitude, the wavelength, the migration speed, and even the stability of the travelling waves. Finally, in the last case, the formation and stability of stationary patterns are investigated in bi-dimensional domains. To this aim, a general class of two-species hyperbolic reaction-transport systems is deduced following the guidelines of Extended Thermodynamics theory. To characterize the emerging Turing patterns, linear and weakly nonlinear stability analysis on the uniform steady states are addressed for rhombic and hexagonal planform solutions. In order to gain some insight into the above-mentioned dynamics, the previous theoretical predictions are corroborated by numerical simulations carried out in the context of dryland ecology. In this context, patterns become a relevant tool to identify early warning signals toward desertification and to provide a measure of resilience of ecosystems under climate change. Such ecological implications are discussed in the context of the Klausmeier model, one of the easiest two-compartment (vegetation biomass and water) models able to describe the formation of patterns in semi-arid environments. Therefore, it will be also here discussed how the experimentally-observed inertia of vegetation affects the formation and stability of stationary and oscillatory periodic vegetation patterns.
Pattern formation and modulation is an active branch of mathematics, not only from the perspective of fundamental theory but also for its huge applications in many fields of physics, ecology, chemistry, biology, and other sciences. In this thesis, the occurrence of Turing and wave instabilities, giving rise to stationary and oscillatory patterns, respectively, is theoretically investigated by means of two-compartment reaction-transport hyperbolic systems. The goal is to elucidate the role of inertial times, which are introduced in hyperbolic models to account for the finite-time propagation of disturbances, in stationary and transient dynamics, in supercritical and subcritical regimes. In particular, starting from a quite general framework of reaction-transport model, three particular cases are derived. In detail, in the first case, the occurrence of stationary patterns is investigated in one-dimensional domains by looking for the inertial dependence of the main features that characterize the formation and stability process of the emerging patterns. In particular, the phenomenon of Eckhaus instability, in both supercritical and subcritical regimes, is studied by adopting linear and multiple-scale weakly-nonlinear analysis and the role played by inertia during the transient regime, where an unstable patterned state evolves towards a more favorable stable configuration through sequences of phase-slips, is elucidated. Then, in the second topic, the focus is moved to oscillatory periodic patterns generated by wave (or oscillatory Turing) instability. This phenomenon is studied by considering 1D two-compartment hyperbolic reaction-transport systems where different transport mechanisms of the species here involved are taken into account. In these cases, by using linear and weakly nonlinear stability analysis techniques, the dependence of the non-stationary patterns on hyperbolicity is underlined at and close to the criticality. In particular, it is proven that inertial effects play a role, not only during transient regimes from the spatially-homogeneous steady state toward the patterned state but also in altering the amplitude, the wavelength, the migration speed, and even the stability of the travelling waves. Finally, in the last case, the formation and stability of stationary patterns are investigated in bi-dimensional domains. To this aim, a general class of two-species hyperbolic reaction-transport systems is deduced following the guidelines of Extended Thermodynamics theory. To characterize the emerging Turing patterns, linear and weakly nonlinear stability analysis on the uniform steady states are addressed for rhombic and hexagonal planform solutions. In order to gain some insight into the above-mentioned dynamics, the previous theoretical predictions are corroborated by numerical simulations carried out in the context of dryland ecology. In this context, patterns become a relevant tool to identify early warning signals toward desertification and to provide a measure of resilience of ecosystems under climate change. Such ecological implications are discussed in the context of the Klausmeier model, one of the easiest two-compartment (vegetation biomass and water) models able to describe the formation of patterns in semi-arid environments. Therefore, it will be also here discussed how the experimentally-observed inertia of vegetation affects the formation and stability of stationary and oscillatory periodic vegetation patterns.

The thesis studies linear and semilinear Dirichlet problems driven by different fractional Laplacians. The boundary data can be smooth functions or also Radon measures. The goal is to classify the solutions which have a singularity on the boundary of the prescribed domain. We first remark the existence of a large class of harmonic functions with a boundary blow-up and we characterize them in terms of a new notion of degenerate boundary trace. Via some integration by parts formula, we then provide a weak theory of Stampacchia's sort to extend the linear theory to a setting including these functions: we study the classical questions of existence, uniqueness, continuous dependence on the data, regularity and asymptotic behaviour at the boundary. Afterwards we develop the theory of semilinear problems, by adapting and generalizing some sub- and supersolution methods. This allows us to build the fractional counterpart of large solutions in the elliptic PDE theory of nonlinear equations, giving sufficient conditions for the existence. The thesis is concluded with the definition and the study of a notion of nonlocal directional curvatures.

Over the last decades, there has been considerable attention in the accurate mathematical modeling and numerical simulations of free surface wave propagation in near-shore environments. A physical correct description of the large scale phenomena, which take place in the shallow water region, must account for strong nonlinear and dispersive effects, along with the interaction with complex topographies. First, a study on the behavior in nonlinear regime of different Boussinesq-type models is proposed, showing the advantage of using fully-nonlinear models with respect to weakly-nonlinear and weakly dispersive models (commonly employed). Secondly, a new flexible strategy for solving the fully-nonlinear and weakly-dispersive Green-Naghdi equations is presented, which allows to enhance an existing shallow water code by simply adding an algebraic term to the momentum balance and is particularly adapted for the use of hybrid techniques for wave breaking. Moreover, the first discretization of the Green-Naghdi equations on unstructured meshes is proposed via hybrid finite volume/ finite element schemes. Finally, the models and the methods developed in the thesis are deployed to study the physical problem of bore formation in convergent alluvial estuary, providing the first characterization of natural estuaries in terms of bore inception
Ces dernières décennies, une attention particulière a été portée sur la modélisation mathématique et la simulation numérique de la propagation de vagues en environnements côtiers. Une description physiquement correcte des phénomènes à grande échelle, qui apparaissent dans les régions d'eau peu profonde, doit prendre en compte de forts effets non-linéaires et dispersifs, ainsi que l'interaction avec des bathymétries complexes. Dans un premier temps, une étude du comportement en régime non linéaire de différents modèles de type Boussinesq est proposée, démontrant l'avantage d'utiliser des modèles fortement non-linéaires par rapport à des modèles faiblement non-linéaires et faiblement dispersifs (couramment utilisés). Ensuite, une nouvelle approche flexible pour résoudre les équations fortement non-linéaires et faiblement dispersives de Green-Naghdi est présentée. Cette stratégie permet d'améliorer un code "shallow water" existant par le simple ajout d'un terme algébrique dans l'équation du moment et est particulièrement adapté à l'utilisation de techniques hybrides pour le déferlement des vagues. De plus, la première discrétisation des équations de Green-Naghdi sur maillage non structuré est proposée via des schémas hybrides Volume Fini/Élément Fini. Finalement, les modèles et méthodes développés dans la thèse sont appliqués à l'étude du problème physique de la formation du mascaret dans des estuaires convergents et alluviaux. Cela a amené à la première caractérisation d'estuaire naturel en terme d'apparition de mascaret

[EN] Fiber grating sensors and devices have demonstrated outstanding capabilities in both telecommunications and sensing areas, due to their well-known advantageous characteristics. Therefore, one of the most important motivations lies in the potential of customized fiber gratings to be suitably employed for improving the interrogation process of optical fiber sensors and systems. This Ph.D. dissertation is focused on the study, design, fabrication and performance evaluation of customized fiber Bragg gratings (FBGs) and long period gratings (LPGs) with the double aim to present novel sensing technologies and to enhance the response of existing sensing systems. In this context, a technique based on time-frequency domain analysis has been studied and applied to interrogate different kind of FBGs-based sensors. The distribution of the central wavelength along the sensing structures has been demonstrated, based on a combination of frequency scanning of the interrogating optical pulse and optical time-domain reflectometry (OTDR), allowing the detection of spot events with good performance in terms of measurand resolution. Moreover, different customized FBGs have been interrogated using a technology inspired on the operation principle of microwave photonics (MWP) filters, enabling the detection of spot events using radio-frequency (RF) devices with modest bandwidth. The sensing capability of these technological platforms has been fruitfully employed for implementing a large scale quasi-distributed sensor, based on an array of cascaded FBGs. The potentiality of LPGs as fiber optic sensors has also been investigated in a new fashion, exploiting the potentials of MWP filtering techniques. Besides, a novel approach for simultaneous measurements based on a half-coated LPG has been proposed and demonstrated. Finally, the feasibility of FBGs as selective wavelength filters has been exploited in sensing applications; an alternative approach to improve the response and performance of Brillouin distributed fiber sensors has been studied and validated via experiments. The performance of the reported sensing platforms have been analyzed and evaluated so as to characterize their impact on the fiber sensing field and to ultimately identify the use of the most suitable technology depending on the processing task to be carried out and on the final goal to reach.
[ES] Los sensores y dispositivos en fibra basados en redes de difracción han mostrado excepcionales capacidades en el ámbito de las telecomunicaciones y del sensado, gracias a sus excelentes propiedades. Entre las motivaciones más estimulantes destaca la posibilidad de fabricar redes de difracción ad-hoc para implementar y/o mejorar las prestaciones de los sensores fotónicos. Esta tesis doctoral se ha enfocado en el estudio, diseño, fabricación y evaluación de las prestaciones de redes de difracción de Bragg (FBGs) y de redes de difracción de periodo largo (LPGs) personalizadas con el fin de desarrollar nuevas plataformas de detección y a la vez mejorar la respuesta y las prestaciones de los sensores fotónicos ya existentes. En este contexto, una técnica basada en el análisis tiempofrecuencia se ha estudiado e implementado para la interrogación de sensores en fibra basados en varios tipos y modelos de FBGs. Se ha analizado la distribución de la longitud de onda central a lo largo de la estructura de sensado, gracias a una metodología que conlleva el escaneo en frecuencia del pulso óptico incidente y la técnica conocida como reflectometria óptica en el dominio del tiempo (OTDR). De esta manera se ha llevado a cabo la detección de eventos puntuales, alcanzando muy buenas prestaciones en términos de resolución de la magnitud a medir. Además, se han interrogado varias FBGs a través de una técnica basada en el principio de operación de los filtros de fotónica de microondas (MWP), logrando así la detección de eventos puntuales usando dispositivos de radio-frecuencia (RF) caracterizados por un moderado ancho de banda. La capacidad de sensado de estas plataformas tecnológicas ha sido aprovechada para la realización de un sensor quasi-distribuido de gran alcance, formado por una estructura en cascada de muchas FBGs. Por otro lado, se han puesto a prueba las capacidades de las LPGs como sensores ópticos según un enfoque novedoso; para ello se han aprovechados las potencialidades de los filtros de MWP. Asimismo, se ha estudiado y demostrado un nuevo método para medidas simultáneas de dos parámetros, basado en una LPG parcialmente recubierta por una película polimérica. Finalmente, se ha explotado la viabilidad de las FBGs en cuanto al filtrado selectivo en longitud de onda para aplicaciones de sensado; para ello se ha propuesto un sistema alternativo para la mejora de la respuesta y de las prestaciones de sensores ópticos distribuidos basados en el scattering de Brillouin. En conclusión, se han analizado y evaluado las prestaciones de las plataformas de sensado propuestas para caracterizar su impacto en el ámbito de los sistemas de detección por fibra y además identificar el uso de la tecnología más adecuada dependiendo de la tarea a desarrollar y del objetivo a alcanzar.
[CAT] Els sensors i dispositius en fibra basats en xarxes de difracció han mostrat excepcionals capacitats en l'àmbit de les telecomunicacions i del sensat, gràcies a les seus excel¿lents propietats. Entre les motivacions més estimulants destaca la possibilitat de fabricar xarxes de difracció ad-hoc per a implementar i/o millorar les prestacions de sensors fotònics. Esta tesi doctoral s'ha enfocat en l'estudi, disseny, fabricació i avaluació de les prestacions de xarxes de difracció de Bragg (FBGs) i de xarxes de difracció de període llarg (LPGs) personalitzades per tal de desenvolupar noves plataformes de detecció i al mateix temps millorar la resposta i les prestacions dels sensors fotònics ja existents. En este context, una tècnica basada en l'anàlisi temps-freqüència s'ha estudiat i implementat per a la interrogació de sensors en fibra basats en diversos tipus i models de FBGs. S'ha analitzat la distribució de la longitud d'ona central al llarg de l'estructura de sensat, gràcies a una metodologia que comporta l'escaneig en freqüència del pols òptic incident i la tècnica coneguda com reflectometria òptica en el domini del temps (OTDR). D'esta manera s'ha dut a terme la detecció d'esdeveniments puntuals, aconseguint molt bones prestacions en termes de resolució de la magnitud a mesurar. A més, s'han interrogat diverses FBGs a través d'una tècnica basada en el principi d'operació dels filtres de fotònica de microones (MWP), aconseguint així la detecció d'esdeveniments puntuals utilitzant dispositius de ràdio-freqüència (RF) caracteritzats per un moderat ample de banda. La capacitat de sensat d'aquestes plataformes tecnològiques ha sigut aprofitada per a la realització d'un sensor quasi-distribuït a llarga escala, format per una estructura en cascada de moltes FBGs. D'altra banda, s'han posat a prova les capacitats de les LPGs com a sensors òptics segons un enfocament nou; per a això s'han aprofitat les potencialitats dels filtres de MWP. Així mateix, s'ha estudiat i demostrat un nou mètode per a mesures simultànies de dos paràmetres, basat en una LPG parcialment recoberta per una pel¿lícula polimèrica. Finalment, s'ha explotat la viabilitat de les FBGs pel que fa al filtrat selectiu en longitud d'ona per a aplicacions de sensat; per això s'ha proposat un sistema alternatiu per a la millora de la resposta i de les prestacions de sensors òptics distribuïts basats en el scattering de Brillouin. S'han analitzat i avaluat les prestacions de les plataformes de sensat propostes per a caracteritzar el seu impacte en l'àmbit dels sistemes de detecció per fibra i a més identificar l'ús de la tecnologia més adequada depenent de la tasca a desenvolupar i de l'objectiu a assolir.
Ricchiuti, AL. (2016). Design and fabrication of customized fiber gratings to improve the interrogation of optical fiber sensors [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/66343
TESIS
Premiado

Dottorato di Ricerca in Scienze e Tecnologie Fisiche, Chimiche e dei Materiali. Ciclo XXX
The local emergency of Beltrami ows is a fundamental characteristic of the uid turbulence dynamics (Navier-Stokes equations), where the formation of singularities starting from smooth initial data, i.e. the breakdown of regularity in the solutions, can individuate the onset of the turbulent behaviour. This property of nonlinear interactions has been used as a basic ingredient in the formal proof of Onsager conjecture, about the existence of weak solutions of Euler equations which do not conserve kinetic energy of the ow. The breakdown from smooth to weak solutions and the energy dissipation phenomenon can be possibly found also in magnetohydrodynamics (MHD) when progressively increasing Reynolds and magnetic Reynolds numbers. Thus a deep study of these phenomena of local formation of strong correlations between the dynamical variables of the systems could give important elements for understanding which mathematical conditions characterise the singularity emergence in weak solutions of MHD ideal case. In order to deal with these problems a multidisciplinary approach, embedding experimental data analysis and mathematical rigorous study, is needed. In this thesis both approaches have been carried out. An ad hoc data analysis have been identi ed for investigating the dynamics described by particular nonlinear partial di erential equations that can generates wide modes cascades and thus turbulence (MHD equations and Hasegawa-Mima equation). In addition the problem of investigating the second order regularity of solutions to particular degenerate nonlinear elliptic equations has been discussed
Università della Calabria

Inverse problems occur in a wide variety of applications and are an active area of research in many disciplines. We consider inverse problems for a broad class of nonlinear systems of partial differential equations (PDEs). We develop collage-based approaches for solving inverse problems for nonlinear PDEs of elliptic, parabolic and hyperbolic type. The original collage method for solving inverse problems was developed in [29] with broad application, in particular to ordinary differential equations (ODEs). Using a consequence of Banach’s fixed point theorem, the collage theorem, one can bound the approximation error above by the so-called collage distance, which is more readily minimizable. By minimizing the collage distance the approximation error can be controlled. In the case of nonlinear PDEs we consider the weak formulation of the PDE and make use of the nonlinear Lax-Milgram representation theorem and Galerkin approximation theory in order to develop a similar upper-bound on the approximation error. Supporting background theory, including weak solution theory,is presented and example problems are solved for each type of PDE to showcase the methods in practice. Numerical techniques and considerations are discussed and results are presented. To demonstrate the practical applicability of this work, we study two real-world applications. First, we investigate a model for the migration of three fish species through floodplain waters. A development of the mathematical model is presented and a collage-based method is applied to this model to recover the diffusion parameters. Theoretical and numerical particulars are discussed and results are presented. Finally, we investigate a model for the “Gao beam”, a nonlinear beam model that incorporates the possibility of buckling. The mathematical model is developed and the weak formulation is discussed. An inverse problem that seeks the flexural rigidity of the beam is solved and results are presented. Finally, we discuss avenues of future research arising from this work.
Natural Sciences and Engineering Research Council of Canada, Department of Mathematics & Statistics

This Ph.D. Thesis is devoted to boundary value problems associated to some classes of second order nonlinear elliptic PDEs in bounded open subsets of R^N. More precisely, we consider, first, two classes of noncoercive Dirichlet problems and we study the regularizing effect of a lower order term of power type on the summability properties of solutions. Then, for one class, we investigate the asymptotic behaviour of solutions as the power goes to infinity, while, for the other, we analyse local properties of solutions depending on local properties of data (with and without the lower order term of power type). Finally, the last topic is also studied for a class of nonlinear elliptic Dirichlet problems with a singular nonlinearity.

A main thrust of this thesis is to develop and explore linearization-based numeric-analytic integration techniques in the context of stochastically driven nonlinear oscillators of relevance in structural dynamics. Unfortunately, unlike the case of deterministic oscillators, available numerical or numeric-analytic integration schemes for stochastically driven oscillators, often modelled through stochastic differential equations (SDE-s), have significantly poorer numerical accuracy. These schemes are generally derived through stochastic Taylor expansions and the limited accuracy results from difficulties in evaluating the multiple stochastic integrals. We propose a few higher-order methods based on the stochastic version of transversal linearization and another method of linearizing the nonlinear drift field based on a Girsanov change of measures. When these schemes are implemented within a Monte Carlo framework for computing the response statistics, one typically needs repeated simulations over a large ensemble. The statistical error due to the finiteness of the ensemble (of size N, say)is of order 1/√N, which implies a rather slow convergence as N→∞. Given the prohibitively large computational cost as N increases, a variance reduction strategy that enables computing accurate response statistics for small N is considered useful. This leads us to propose a weak variance reduction strategy. Finally, we use the explicit derivative-free linearization techniques for state and parameter estimations for structural systems using the extended Kalman filter (EKF). A two-stage version of the EKF (2-EKF) is also proposed so as to account for errors due to linearization and unmodelled dynamics. In Chapter 2, we develop higher order locally transversal linearization (LTL) techniques for strong and weak solutions of stochastically driven nonlinear oscillators. For developing the higher-order methods, we expand the non-linear drift and multiplicative diffusion fields based on backward Euler and Newmark expansions while simultaneously satisfying the original vector field at the forward time instant where we intend to find the discretized solution. Since the non-linear vector fields are conditioned on the solution we wish to determine, the methods are implicit. We also report explicit versions of such linearization schemes via simple modifications. Local error estimates are provided for weak solutions. Weak linearized solutions enable faster computation vis-à-vis their strong counterparts. In Chapter 3, we propose another weak linearization method for non-linear oscillators under stochastic excitations based on Girsanov transformation of measures. Here, the non-linear drift vector is appropriately linearized such that the resulting SDE is analytically solvable. In order to account for the error in replacing of non-linear drift terms, the linearized solutions are multiplied by scalar weighting function. The weighting function is the solution of a scalar SDE(i.e.,Radon-Nikodym derivative). Apart from numerically illustrating the method through applications to non-linear oscillators, we also use the Girsanov transformation of measures to correct the truncation errors in lower order discretizations. In order to achieve efficiency in the computation of response statistics via Monte Carlo simulation, we propose in Chapter 4 a weak variance reduction strategy such that the ensemble size is significantly reduced without seriously affecting the accuracy of the predicted expectations of any smooth function of the response vector. The basis of the variance reduction strategy is to appropriately augment the governing system equations and then weakly replace the associated stochastic forcing functions through variance-reduced functions. In the process, the additional computational cost due to system augmentation is generally far less besides the accrued advantages due to a drastically reduced ensemble size. The variance reduction scheme is illustrated through applications to several non-linear oscillators, including a 3-DOF system. Finally, in Chapter 5, we exploit the explicit forms of the LTL techniques for state and parameters estimations of non-linear oscillators of engineering interest using a novel derivative-free EKF and a 2-EKF. In the derivative-free EKF, we use one-term, Euler and Newmark replacements for linearizations of the non-linear drift terms. In the 2-EKF, we use bias terms to account for errors due to lower order linearization and unmodelled dynamics in the mathematical model. Numerical studies establish the relative advantages of EKF-DLL as well as 2-EKF over the conventional forms of EKF. The thesis is concluded in Chapter 6 with an overall summary of the contributions made and suggestions for future research.

Title: Mathematical analysis and computer simulations of deformation of nonlinear elastic bodies in the small strain range. Author: Vojtěch Kulvait Department: Mathematical Institute of Charles University Supervisor: prof. RNDr. Josef Málek, CSc., Dsc. Abstract: Implicit constitutive theory provides a suitable theoretical framework for elastic materials that exhibit a nonlinear relationship between strain and stress in the range of small strains. We study a class of power-law models, where the nonlinear dependence of strain on the deviatoric part of the stress tensor and its trace are mutually separated. We show that these power-law models are capable to describe the response of a wide variety of beta phase titanium alloys in the small strain range and that these models fit available experimental data exceedingly well. We also develop a mathematical theory regarding the well-posedness of boundary value problems for the considered class of power-law solids. In particular, we prove the existence of weak solutions for power law exponents in the range (1, ∞). Finally, we perform computer simulations for these problems in the anti-plane stress setting focusing on the V-notch type geometry. We study the dependence of solutions on the values of power law exponents and on the V-notch opening angle. We achieve...