L'objet de cette thèse est l'étude des équations de Schrödinger et des ondes, à la fois linéaires et non linéaires, dans des domaines extérieurs. Nous nous intéressons en particulier aux inégalités dites de Strichartz, qui sont une famille d'estimations dispersives mesurant la décroissance du flot linéaire, particulièrement utiles à l'étude des problèmes non linéaires correspondants. Dans des géométries dites non-captantes, c'est à dire où tous les rayons de l'optique géométrique partent à l'infini, de nombreux résultats montrent que de telles estimations sont aussi bonnes que dans l'espace libre. D'autre part, la présence d'une trajectoire captive induit nécessairement une perte au niveau d'une autre famille d'estimations à priori, les estimations d'effet régularisant et de décroissance locale de l'énergie, respectivement pour Schrödinger et pour les ondes. En contraste de quoi, nous montrons des estimations de Strichartz sans perte dans une géométrie captante instable : l'extérieur de plusieurs obstacles strictement convexes vérifiant la condition d'Ikawa. La seconde partie de cette thèse est dédiée à l'étude du comportement en temps long des équations non-linéaires sous-jacentes. Lorsque le domaine dans lequel elles vivent n'induit pas trop de concentration de l'énergie, on s'attend à ce qu'elles diffusent, c'est à dire se comportent de manière linéaire asymptotiquement en temps. Nous montrons un tel résultat pour les ondes non linéaires critiques à l'extérieur d'une classe d'obstacles généralisant la notion d'étoilé. A l'extérieur de deux obstacles strictement convexes, nous obtenons un résultat de rigidité concernant les solutions à flot compact, premier pas vers un résultat général. Enfin, nous nous intéressons à l'équation de Schrödinger non linéaire, dans l'espace libre, mais avec un potentiel. Nous montrons que les solutions diffusent si l'on prend un potentiel répulsif, ainsi qu'une somme de deux potentiels répulsifs ayant des surfaces de niveau convexes, ce qui fournit un exemple de diffusion dans une géométrie captante analogue à l'extérieur de deux convexes stricts. / We are concerned with Schrödinger and wave equations, both linear and non linear, in exterior domains. In particular, we are interested in the so-called Strichartz estimates, which are a family of dispersive estimates measuring decay for the linear flow. They turn out to be particularly useful in order to study the corresponding non linear equations. In non-captive geometries, where all the rays of geometrical optics go to infinity, many results show that Strichartz estimates hold with no loss with respect to the flat case. Moreover, the local smoothing estimates for the Schrödinger equation, respectively the local energy decay for the wave equation, which are another family of dispersive estimates, are known to fail in any captive geometry. In contrast, we show Strichartz estimates without loss in an unstable captive geometry: the exterior of many strictly convex obstacles verifying Ikawa's condition. The second part of this thesis is dedicated to the study of the long time asymptotics of the corresponding non linear equations. We expect that they behave linearly in large times, or scatter, when the domain they live in does not induce too much concentration effect. We show such a result for the non linear critical wave equation in the exterior of a class of obstacles generalizing star-shaped bodies. In the exterior of two strictly convex obstacles, we obtain a rigidity result concerning compact flow solutions, which is a first step toward a general result. Finally, we consider the non linear Schrödinger equation in the free space but with a potential. We prove that solutions scatter for a repulsive potential, and for a sum of two repulsive potentials with strictly convex level surfaces. This provides a scattering result in a framework similar to the exterior of two strictly convex obstacles.

<p>This thesis project addresses the need for an interactive, real-time water animation tech-<br> nique that can showcase visually convincing effects such as splashes and breaking waves while<br> being computationally inexpensive. Our method couples SPH and wave equations in a one-<br> way manner to simulate the behavior of water in real-time, leveraging OpenGL’s Compute<br> Shaders for interactive performance and a novel Uniform Grid implementation. Through a<br> review of related literature on real-time simulation methods of fluids, and water animation,<br> this thesis presents a feasible algorithm, animations to showcase interesting water effects,<br> and a comparison of computational costs between SPH, wave equations, and the coupled<br> approach. The program renders a water body with a planar surface and discrete particles.<br> This project aims to provide a solution that can meet the needs of various water animation<br> use-cases, such as games, and movies, by offering a computationally efficient technique that<br> can animate water to behave plausibly and showcase essential effects in real-time.</p>

This dissertation studies immersed finite elements (IFE) for a second order elliptic operator and their applications to interface problems of related partial differential equations. We start with the immersed finite element methods for the second order elliptic operator with a discontinuous coefficient associated with the elliptic interface problems. We introduce an energy norm stronger than the one used in [111]. Then we derive an estimate for the IFE interpolation error with this energy norm using patches of interface elements. We prove both the continuity and coercivity of the bilinear form in a partially penalized IFE (PPIFE) method. These properties allow us to derive an error bound for the PPIFE solution in the energy norm under the standard piecewise $H^2$ regularity assumption instead of the more stringent $H^3$ regularity used in [111]. As an important consequence, this new estimation further enables us to show the optimal convergence in the $L^2$ norm which could not be done by the analysis presented in [111]. Then we consider applications of IFEs developed for the second order elliptic operator to wave propagation and diffusion interface problems. The first application is for the time-harmonic wave interface problem that involves the Helmholtz equation with a discontinuous coefficient. We design PPIFE and DGIFE schemes including the higher degree IFEs for Helmholtz interface problems. We present an error analysis for the symmetric linear/bilinear PPIFE methods. Under the standard piecewise $H^2$ regularity assumption for the exact solution, following Schatz's arguments, we derive optimal error bounds for the PPIFE solutions in both an energy norm and the usual $L^2$ norm provided that the mesh size is sufficiently small. {In the second group of applications, we focus on the error analysis for IFE methods developed for solving typical time-dependent interface problems associated with the second order elliptic operator with a discontinuous coefficient.} For hyperbolic interface problems, which are typical wave propagation interface problems, we reanalyze the fully-discrete PPIFE method in [143]. We derive the optimal error bounds for this PPIFE method for both an energy norm and the $L^2$ norm under the standard piecewise $H^2$ regularity assumption in the space variable of the exact solution. Simulations for standing and travelling waves are presented to corroborate the results of the error analysis. For parabolic interface problems, which are typical diffusion interface problems, we reanalyze the PPIFE methods in [113]. We prove that these PPIFE methods have the optimal convergence not only in an energy norm but also in the usual $L^2$ norm under the standard piecewise $H^2$ regularity. / Doctor of Philosophy / This dissertation studies immersed finite elements (IFE) for a second order elliptic operator and their applications to a few types of interface problems. We start with the immersed finite element methods for the second order elliptic operator with a discontinuous coefficient associated with the elliptic interface problem. We can show that the IFE methods for the elliptic interface problems converge optimally when the exact solution has lower regularity than that in the previous publications. Then we consider applications of IFEs developed for the second order elliptic operator to wave propagation and diffusion interface problems. For interface problems of the Helmholtz equation which models time-Harmonic wave propagations, we design IFE schemes, including higher degree schemes, and derive error estimates for a lower degree scheme. For interface problems of the second order hyperbolic equation which models time dependent wave propagations, we derive better error estimates for the IFE methods and provides numerical simulations for both the standing and traveling waves. For interface problems of the parabolic equation which models the time dependent diffusion, we also derive better error estimates for the IFE methods.

Neste trabalho estudamos uma classe de equações de ondas da forma &#8739;&part;tu&#8739;p &part; ttu - &Delta;&part;ttu - &alpha;u + &#8747;&infin;0&micro;(s)&Delta;u(t - s)ds +F(u) = h, definida num domínio limitado de R3, com condição de fronteira de Dirichlet e parâmetros &alpha;, &rho; &gt;0. Tais equações modelam problemas de viscoelasticidade não linear e têm sido estudados por diversos autores. Aqui, apresentamos um teorema de existência, unicidade e dependência contínua em relação aos dados iniciais, para soluções fracas, como discutido por Conti, Marchini & Pata (2014). Em seguida provamos um teorema novo sobre a existência de atratores globais para o sistema dinâmico associado ao problema, explorando tão somente a dissipação dada pelo termo de memória. Tal resultado generaliza substancialmente o trabalho pioneiro de Araújo, Ma & Qin (2013). / In this work we study a class of wave equations of the form &#8739;&part;tu&#8739;p &part; ttu - &alpha;&Delta;u + &#8747;&infin;0&micro;(s)&Delta;u(t - s)ds +f(u) = h, defined in a bounded domain of R3, with Dirichlet boundary condition and parameters &alpha;, &rho; &gt; 0. Such equations model problems from nonlinear visco-elasticity and have been considered by several authors. Here, we prove the well-posedness of the problem, as discussed by Conti, Marchini & Pata (2014). Next, we prove a new result on the existence of global attractors for the dynamical system generated by the problem, by exploring the dissipation the memory term only. The result extends substantially the pioneering work by Araújo, Ma & Qin (2013).

Um novo procedimento para construir os estados coerentes (CS) e os estados semiclássicos (SS) no campo de um magnético-solenóide é proposto. A idéia principal é baseada sobre o fato de que o AB solenóide quebra a simetria translacional no plano-xy, isto apresenta um efeito topológico tal que surgem dois tipos de trajetórias, aquelas que circundam e aquelas que não circundam o solenóide. Devido a este fato, deve-se construir dois tipos diferentes dos CS/SS, os quais correspondem as referidas trajetórias no limite semiclássico. Seguindo esta idéia, construímos os CS em duas etapas, primeiro os CS instantâneos (ICS) e os CS/SS dependentes do tempo como uma evolução dos ICS. A construção é realizada para partículas não-relativísticas e relativísticas, de spin-zero e com spin ambas em (2 + 1)- e (3 + 1)- dimensões e gera um exemplo não-trivial de SS/CS para sistemas com uma Hamiltoniana não-quadrática. É enfatizado que os CS dependendo dos seus parâmetros (números quânticos), descrevem ambos os estados puramente quânticos e semiclássicos. Uma análise é representada de modo que classifica os parâmetros dos CS em tal relação. Tal classificação é usada para as decomposições semiclásicas de diversas quantidades físicas. / A new approach to constructing coherent states (CS) and semiclassical states (SS) in magnetic-solenoid field is proposed. The main idea is based on the fact that the AB solenoid breaks the translational symmetry in the xy-plane, this has a topological effect such that there appear two types of trajectories which embrace and do not embrace the solenoid. Due to this fact, one has to construct two different kinds of CS/SS, which correspond to such trajectories in the semiclassical limit. Following this idea, we construct CS in two steps, first the instantaneous CS (ICS) and the time dependent CS/SS as an evolution of the ICS. The construction is realized for nonrelativistic and relativistic, spinning and spinless particles both in (2 + 1)- and (3 + 1)- dimensions and gives a non-trivial example of SS/CS for systems with a nonquadratic Hamiltonian. It is stressed that CS depending on their parameters (quantum numbers) describe both pure quantum and semiclassical states. An analysis is presented that classifies parameters of the CS in such respect. Such a classification is used for the semiclassical decompositions of various physical quantities.

La thèse porte essentiellement sur la stabilisation indirecte de certains systèmes d’équations couplées moyennant un seul contrôle agissant localement à l’intérieur ou sur le bord du domaine. La nature du système ainsi couplé dépend du couplage des équations et du type de l’amortissement, et ceci donne divers résultats de stabilisation (exponentielle ou polynômiale) des systèmes étudiés. D’abord, dans le cas de la stabilisation d’un système de Bresse formé de trois équations d’ondes couplées, un amortissement local de type chaleur est appliqué à une seule équation. Par une méthode fréquentielle combinée avec une méthode de multiplicateurs par morceau la décroissance exponentielle de l’énergie du système est établie sous la condition d’égalité de vitesses de propagation des ondes. Dans le cas contraire, une décroissance polynomiale est assurée. Ensuite, un système de deux équations d’ondes couplées sous l’effet d’un seul amortissement frontière appliqué à une seule équation est considéré. Dans ce cas, la stabilité du système est influencée par la nature algébrique du terme de couplage ainsi que par la nature arithmétique du quotient de vitesses de propagation des ondes. Par conséquence, différents résultats de stabilité exponentielle ou polynomiale sont établis. Une étude spectrale conduit à l’optimalité des résultats obtenus. Finalement, dans le cas de la stabilisation d’un système de deux équations d’ondes couplées, un amortissement localement distribué de type Kelvin-Voight est appliqué à une seule équation. D’abord, d’après un théorème de Hormander, un résultat d’unicité est montré et par conséquent la stabilité forte du système est assurée. Ensuite, une décroissance polynomiale de l’énergie du système est établie. / Résumé en anglais non disponible

In this thesis we analyse the high-order in time discontinuous Galerkin nite element method (DGFEM) for second-order in time linear abstract wave equations. Our abstract approximation analysis is a generalisation of the approach introduced by Claes Johnson (in Comput. Methods Appl. Mech. Engrg., 107:117-129, 1993), writing the second order problem as a system of fi rst order problems. We consider abstract spatial (time independent) operators, highorder in time basis functions when discretising in time; we also prove approximation results in case of linear constraints, e.g. non-homogeneous boundary data. We take the two steps approximation approach i.e. using high-order in time DGFEM; the discretisation approach in time introduced by D Schötzau (PhD thesis, Swiss Federal institute of technology, Zürich, 1999) to fi rst obtain the semidiscrete scheme and then conformal spatial discretisation to obtain the fully-discrete formulation. We have shown solvability, unconditional stability and conditional a priori error estimates within our abstract framework for the fully discretized problem. The skew-symmetric spatial forms arising in our abstract framework for the semi- and fully-discrete schemes do not full ll the underlying assumptions in D. Schötzau's work. But the semi-discrete and fully discrete forms satisfy an Inf-sup condition, essential for our proofs; in this sense our approach is also a generalisation of D. Schötzau's work. All estimates are given in a norm in space and time which is weaker than the Hilbert norm belonging to our abstract function spaces, a typical complication in evolution problems. To the best of the author's knowledge, with the approximation approach we used, these stability and a priori error estimates with their abstract structure have not been shown before for the abstract variational formulation used in this thesis. Finally we apply our abstract framework to the acoustic and an elasto-dynamic linear equations with non-homogeneous Dirichlet boundary data.

Le secteur de l’énergie houlomotrice s’appuie fortement sur la modélisation mathématique et la simulation d’expériences physiques mettant en jeu les interactions entre les ondes et les corps. Dans ce travail, nous avons développé un modèle d’interaction de fidélité moyenne vague-corps pour la simulation de structures tronquées flottantes fonctionnant en mouvement vertical. Ce travail concerne l’ingénierie de l’énergie marine, pour des applications telles que les convertisseurs d’énergie de vague (WEC) à absorption ponctuelle, même si ses applications peuvent aussi être utilisées en ingénierie maritime et navale. Les motivations de ce travail reposent sur les méthodes standard actuelles pour décrire l’interaction corps-vague. Cellesci sont basées sur des modèles résolvant le flux de potentiel linéaire (LPF), du fait de leur grande efficacité. Cependant, les modèles LPF sont basés sur l’hypothèse de faible amplitude et ne peuvent pas répresenter les effets hydrodynamiques non linéaires, importants pour le WEC opérant dans la région de résonance ou dans les régions proches du rivage. En effet, il a été démontré que les modèles LFP prédisent de manière excessive la production de puissance, sauf si des coefficients de traînée sont calibrés. Plus récemment, des simulations Reynolds Averaged Navier-Stokes (RANS) ont été utilisées pour les WEC. RANS est un modèle complet et précis, mais très coûteux en calcul. Il n’est ni adapté à l’optimisation d’appareils uniques ni aux parcs énergétiques. Nous avons donc proposé un modèle de fidélité moyenne basé sur des équations de type Boussinesq, afin d’améliorer le compromis entre précision et efficacité. Les équations de type Boussinesq sont des modèles d’ondes intégrées en profondeur et ont été un outil d’ingénierie standard pour la simulation numérique de la propagation d’ondes non linéaires dans les eaux peu profondes et les zones côtières. Grâce à l’élimination de la dimension verticale, le modèle résultant est très efficace et évite la description temporelle de la limite entre la surface libre et l’air. Jiang (2001) a proposé un modèle de Boussinesq unifié, décomposant le problème en deux domaines : surface libre et corps. Dans cette méthode, le domaine du corps est également modélisé par une approche intégrée en profondeur - d’où le terme unifié. Récemment, Lannes (2016) avait analysé de manière rigoureuse une configuration similaire dans une équation non linéaire en eaux peu profondes, en déduisant une solution exacte et semi-analitique pour des corps en mouvement. Suivant la même approche, Godlewski et al. (2018) a élaboré un modèle de flux d’eau peu profonde encombrée. [...] Dans cette thèse, nous développons les résultats présentés par Eskilsson et al. (2016) et Bosi et al. (2019). Le modèle est étendu à deux dimensions horizontales. Le modèle 1D est vérifié à l’aide de solutions fabriquées et validé par rapport aux résultats publiés sur l’interaction vague-corps en 1D pour les pontons fixes et corps en mouvement de soulèvement forcé et libre. Les résultats des preuves de concept de la simulation de plusieurs corps sont présentés. Nous validons et vérifions le modèle 2D en suivant des étapes similaires. Enfin, nous mettons en oeuvre la technique de verrouillage, une méthode de contrôle de mouvement du corps pour améliorer la réponse au mouvement des vagues. Il est démontré que le modèle possède une excellente précision, qu’il est pertinent pour les applications d’ondes en interaction avec des dispositifs à énergie houlomotrice et qu’il peut être étendu pour simuler des cas plus complexes. / The wave energy sector relies heavily on mathematical modelling and simulation of the interactions between waves and floating bodies. In this work, we have developed a medium-fidelity wave-body interaction model for the simulation of truncated surface piercing structures operating in heave motion, such as point absorbers wave energy converters (WECs). The motivation of the work lies in the present approach to wave-body interaction. The standard approach is to use models based on linear potential flow (LPF). LPF models are based on the small amplitude/ small motion assumption and, while highly computational efficient, cannot account for nonlinear hydrodynamic effects (except for Morison-type drag). Nonlinear effects are particularly important for WEC operating in resonance, or in nearshore regions where wave transformations are expected. More recently, Reynolds Averaged Navier-Stokes (RANS) simulations have been employed for modelling WECs. RANS is a complete and accurate model but computationally very costly. At present RANS models are therefore unsuited for the optimization of single devices, not to mention energy farms. Thus, we propose a numerical model based built on Boussinesq-type equations to include wave-wave interaction as well as finite body motion in a computationally efficient formulation. Boussinesq-type equations are depth-integrated wave models and are standard engineering tool for numerical simulation of propagation of nonlinear wave in shallow water and coastal areas. Thanks to the elimination of the vertical dimension and the avoidance of a time-dependent computational the resulting model is very computational efficient. Jiang (Jiang, 2001) proposed a unified Boussinesq model, decomposing the problem into free surface and body domains. Notably, in Jiang’s methodology also the body domain is modeled by a depth-integrated approach –hence the term unified. As all models based on Boussinesq-type equations, the model is limited to shallow and intermediate depth regimes. We consider the Madsen and Sørensen model, an enhanced Boussinesq model, for the propagation of waves. We employ a spectral/hp finite element method (SEM) to discretize the governing equations. The continuous SEM is used inside each domain and flux-based coupling conditions are derived from the discontinuous Galerkin method. The use of SEM give support for the use of adaptive meshes for geometric flexibility and high-order accurate approximations makes the scheme computationally efficient. In this thesis, we present 1D results for the propagation and interaction of waves with floating structures. The 1D model is verified using manufactured solutions. The model is then validated against published results for wave-body interaction. The hydrostatic cases (forced motion and decay test) are compared to analytical and semi-analytical solutions (Lannes, 2017), while the non-hydrostatic tests (fixed pontoon and freely heaving bodies) are compared to RANS reference solutions. The model is easily extended to handle multiple bodies and a proof-of-concept result is presented. Finally, we implement the latching technique, a method to control the movement of the body such that the response to the wave movement is improved. The model is extended to two horizontal dimensions and verified and validated against manufactured solutions and RANS simulations. The model is found to have a good accuracy both in one and two dimensions and is relevant for applications of waves interacting with wave energy devices. The model can be extended to simulate more complex cases such as WEC farms/arrays or include power generation systems to the device.

Made available in DSpace on 2016-12-12T20:15:50Z (GMT). No. of bitstreams: 1 Karla Weber.pdf: 792478 bytes, checksum: 5ad5bec34e3e16a7e56322ccf40af568 (MD5) Previous issue date: 2014-03-18 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work we will show that the wave equations for any twistor fields carrying a single index, which occur in the frameworks of the Infeld-van der Waerden &#947; formalisms for General Relativity, must be formally the same. This result stems mainly from the fact that the spinor transcription of the traditional conformal Killing equation provides twistor equations of the same form. A consequence of this result is that the usual &#947; formalism covariant differential devices for controlling valences of spinor-index configurations turn out to be inapplicable as regards the speciation of the formal patterns for the wave equations under consideration here. / Mostraremos neste trabalho que as equações de onda para quaisquer campos twistoriais de um único índice, as quais ocorrem no contexto dos formalismos espinoriais &#947; de Infeld e van der Waerden para a Relatividade Geral, devem ser formalmente as mesmas. Este resultado decorre essencialmente do fato que a transcrição espinorial da tradicional equação conforme de Killing fornece equações twistoriais da mesma forma. Uma conseqüência deste resultado e que os dispositivos diferenciais covariantes do formalismo &#947;, os quais usualmente servem para controlar valências de conjurações indiciais, tornam-se inaplicáveis no que concerne a obtenção dos padrões formais das equações de onda sob consideração aqui.

Um novo procedimento para construir os estados coerentes (CS) e os estados semiclássicos (SS) no campo de um magnético-solenóide é proposto. A idéia principal é baseada sobre o fato de que o AB solenóide quebra a simetria translacional no plano-xy, isto apresenta um efeito topológico tal que surgem dois tipos de trajetórias, aquelas que circundam e aquelas que não circundam o solenóide. Devido a este fato, deve-se construir dois tipos diferentes dos CS/SS, os quais correspondem as referidas trajetórias no limite semiclássico. Seguindo esta idéia, construímos os CS em duas etapas, primeiro os CS instantâneos (ICS) e os CS/SS dependentes do tempo como uma evolução dos ICS. A construção é realizada para partículas não-relativísticas e relativísticas, de spin-zero e com spin ambas em (2 + 1)- e (3 + 1)- dimensões e gera um exemplo não-trivial de SS/CS para sistemas com uma Hamiltoniana não-quadrática. É enfatizado que os CS dependendo dos seus parâmetros (números quânticos), descrevem ambos os estados puramente quânticos e semiclássicos. Uma análise é representada de modo que classifica os parâmetros dos CS em tal relação. Tal classificação é usada para as decomposições semiclásicas de diversas quantidades físicas. / A new approach to constructing coherent states (CS) and semiclassical states (SS) in magnetic-solenoid field is proposed. The main idea is based on the fact that the AB solenoid breaks the translational symmetry in the xy-plane, this has a topological effect such that there appear two types of trajectories which embrace and do not embrace the solenoid. Due to this fact, one has to construct two different kinds of CS/SS, which correspond to such trajectories in the semiclassical limit. Following this idea, we construct CS in two steps, first the instantaneous CS (ICS) and the time dependent CS/SS as an evolution of the ICS. The construction is realized for nonrelativistic and relativistic, spinning and spinless particles both in (2 + 1)- and (3 + 1)- dimensions and gives a non-trivial example of SS/CS for systems with a nonquadratic Hamiltonian. It is stressed that CS depending on their parameters (quantum numbers) describe both pure quantum and semiclassical states. An analysis is presented that classifies parameters of the CS in such respect. Such a classification is used for the semiclassical decompositions of various physical quantities.