Neste trabalho estudamos a equação de ondas do tipo p-Laplaciano \'u IND. tt\' - \'DELTA\' IND.p u + \'(- \'DELTA\' POT. alpha\' u IND. t\' = \' [u] POT.q - 2 u, definida num domínio limitado limitado do \'R POT. n\', com 2 \' > ou = \' p < q e 0 < \' alpha\' < 1. Utilizando o método de Faedo-Galerkin provamos a existência de soluções fracas globais para dados iniciais pequenos. Para essas soluções estudamos também o decaimento polinomial da energia associada. A questão da não existência de soluções globais é considerada para o caso em que a energia inicial do sistema é negativa / In this work we study the p-Laplacian wave equation \'u IND. tt\' - \' DELTA\' IND p u + \'(- \'DELTA\' POT. \'alpha\' \' u IND. t\' = \'[u] POT. q - 2 u, defined in a bounded domain of \'R POT n\', with 2 \'> or =\' p < q and 0 < \' alpha\' < 1. By using the Faedo-Galerkin method we prove the existence of weak global solutions for small initial data. We also study the polynomial decay of the associate energy. The blow-up of solutions in finite time is considered for negative initial energy

In dieser Arbeit entwickeln und untersuchen wir ein numerisches Schema für die eindimensionale Wellengleichung mit Hysterese für unterschiedliche Arten von Randbedingungen. Diese Gleichung ist ein Modell für die Longitudinal- oder Torsionsschwingungen eines homogenen Stabes unter dem Einfluß einer uniaxialen äußeren Kraftdichte, wobei wir ein elastoplastisches Materialgesetz annehmen. Hysterese-Operatoren sind ratenunabhängige Volterra-Operatoren, die Zeitfunktionen in Zeitfunktionen abbilden. Mit ihnen lassen sich Gedächtniseffekte modellieren, wie sie zum Beispiel in der Elastoplastizität oder im Ferromagnetismus auftauchen. Zunächst führen wir Hysterese-Operatoren allgemein ein und analysieren dann eine spezielle Klasse von Hysterese-Operatoren, die Prandtl-Ishlinskii-Operatoren. Wir untersuchen ihre Gedächtnisstruktur und erklären, wie sich die Operatoren numerisch auswerten lassen. Dazu stellen wir zwei verschiedene Approximationsansätze vor. Wir führen aus, wie sich die approximierenden Operatoren implementieren lassen und leiten lineare und quadratische Fehlerabschätzungen her. Zur numerischen Lösung des gekoppelten Systems aus der Wellengleichung mit einem Hysterese-Operator führen wir ein implizites Differenzenschema mit Gedächtnis ein. Für eine Klasse von Hysterese-Operatoren zeigen wir die Existenz und Eindeutigkeit der Lösung des numerischen Schemas, beweisen mit Hilfe von Kompaktheitsschlüssen und einem Monotonieargument die Konvergenz des Verfahrens und leiten eine Fehlerabschätzung der Ordnung 1/2 her. Wir diskutieren, wie das vorgestellte Verfahren auf die Prandtl-Ishlinskii-Operatoren angewendet werden kann. / In this thesis we develop and investigate a numerical scheme for the one-dimensional wave equation with hysteresis for different kinds of boundary conditions. This equation can be regarded as a model for the longitudinal or torsional oscillations of a homogeneous bar under the influence of an uniaxial external force density assuming an elastoplastic material law. Hysteresis operators are rate-independent Volterra operators mapping time functions to time functions. This kind of operator can be used to model memory effects as they appear in elastoplasticity or ferromagnetism, for example. We first give an introduction to the general concept of hysteresis operators before we analyze a special class of hysteresis operators called Prandtl-Ishlinskii operators. We investigate their memory structure and explain how the operators can be evaluated numerically. To that end we present two different kinds of approximation schemes. We point out how the approximating operators can be implemented and we derive linear and quadratic error estimates. For the numerical solution of the coupled system of the wave equation with a hysteresis operator we introduce an implicit difference scheme with memory. For a class of hysteresis operators we show the existence and uniqueness of the numerical solution. We prove the convergence of the scheme by compactness and monotonicity arguments. We derive an error estimate of order 1/2. We discuss the application of the method presented to Prandtl-Ishlinskii operators.

Este trabalho é dedicado ao estudo do comportamento a longo prazo de uma classe de equações viscoelásticas não lineares com memória, da forma |\'upsilon IND. t\'| POT. ho\' \'upsilon IND. tt\' - DELTA \'upsilon\' - \'DELTA upsilon IND. tt\' + \'INT. SUP. t INF. \\tau\' upsilon (t- s) \'DELTA epsilon\' (s) ds = h, \'\\tau\' > 0, definida num domínio limitado de \'R POT. N\'. Tal classe de problemas foi estudada por diversos autores desde 2001, com \'\\tau = 0. Os resultados existentes são principalmente devotados à existência de soluções globais, decaimento da energia, com ou sem dissipações adicionais, existência com dados pequenos, entre outros. Entretanto, a questão da unicidade de soluções e existência de atratores globais não foram discutidas em trabalhos anteriores. No presente trabalho, apresentamos resultados de unicidade e existência de atratores globais para essa classe de problemas num contexto mais geral, incluindo o caso em que \'\\tau\' = -\'INFINITO\'. Além disso, incluímos um problema complementar, de quarta ordem onde estudamos a existência de atratores exponenciais / This work is concerned with the long-time behaviour of a class nonlinear viscoelastic equations of the form |\'upsilon IND. t\'| POT. ho\' \'upsilon IND. tt\' - DELTA \'upsilon\' - \'DELTA upsilon IND. tt\' + \'INT. SUP. t INF. \\tau\' upsilon (t- s) \'DELTA epsilon\' (s) ds = h, \'ho\' > 0, defined in a bounded domain of \'R POT. N\'. Such class of problems was studied by several authors since 2001, with \'\\tau\' = 0. Existing results are mainly devoted to global existence, energy decay, with or without additional dampings, existence with small data, among others. However, uniqueness and existence of global attractors were not considered previously. In the present work, we establish some results on the uniqueness of solutions and existence of global attractors in a more general setting, including \'\\tau\' = - \'INFINITY\'. In addition, we have added a second problem concerned with a fourth order equation where we study the existence of exponential attractors

Cette thèse porte sur le contrôle acoustique actif (ANC) dans une cavité. L’objectif est d’atténuer l’effet d’une onde sonore perturbatrice en des points ou dans un volume. Ceci est réalisé à l’aide d’un contre-bruit généré, par exemple, par un haut-parleur. Cette étude requiert l’utilisation de modèles dynamiques rendant compte de l’évolution des pressions aux points d’intérêt en fonction des bruits exogènes. Ce modèle peut être obtenu par une identification fréquentielle des réponses point-à-point ou en utilisant le modèle physique sous jacent (équation des ondes). Dans ce dernier cas, la recherche d'un modèle de dimension finie est souvent un préalable à l’étude conceptuelle d'un système d’ANC. Les contributions de cette thèse portent donc sur l’élaboration de différents modèles simplifiés paramétrés par la position pour les systèmes acoustiques et sur la conception de lois de commande pour l’ANC. Le premier volet de la thèse est dédié à l’élaboration de différents modèles simplifiés de système de propagation acoustique au sein d’une cavité. Pour cela, les simplifications envisagées peuvent être de nature spatiale autant que fréquentielle. Nous montrons notamment qu'il est possible, sous certaines conditions, d’approximer le système 3D par un système 1D. Ceci a été mis en évidence expérimentalement sur le banc d’essai LS2NBox. Le second volet porte sur la conception de lois de commande. En premier lieu, les stratégies de commandes couramment utilisées pour l’ANC sont comparées. L'effet dela commande multi-objectif H en différents points voisins des points d'atténuation est analysé. La possibilité d’une annulation parfaitedu bruit en un point est aussi discutée. / This thesis deals with active noise control (ANC) in a cavity. The aim is to mitigate the effect of a disturbing sound wave at some points or in a volume. This is achieved using ananti-noise generated, for example, by a loudspeaker. This study requires the use of dynamic models that report changes in pressure at points of interest in response to exogenous noises. Such models can be obtained by frequency identification of point-to-point responses or by using the underlying physical model (wave equation). In the latter case, the search for a low-complexity model (finite dimensional model) is often a prerequisite for the conceptual study of an active control system. The contributions of this thesis concern the development of different simplified models parameterized by the spatial position for acoustic systems, and the design of control laws for noise attenuation. The first part of the thesis is dedicated to the development of various simplified models of acoustic propagation system within a cavity. For that, the simplifications envisaged can be of spatial nature as much as frequential. We show in particular that it is possible, under certain conditions, to approximate the 3D system by a 1D system. This has been demonstrated experimentally on the prototype system, LS2NBox. The second part of the thesis deals with the design of control laws. First, the control strategies commonly used for ANC are compared. The effect of multi-objective H control at different spatial positions close to the attenuation points is analyzed. The possibility of perfect noise cancellation at one point is also discussed.

Mes travaux de thèse portent sur le développement de schémas numériques d'ordre élevé en temps et en espace pour la simulation de propagation des ondes. Nous avons proposé de discrétiser dans un premier temps l'équation des ondes par rapport au temps, en utilisant une technique de type équation modifiée. Puis nous avons utilisé une méthode d'éléments finis de type Galerkine discontinue pour la discrétisation en espace. En modifiant l'ordre de la discrétisation, nous avons construit des schémas tout aussi précis que ceux déjà existants pour un coût de mise en oeuvre très intéressant. Après avoir validé numériquement la nouvelle méthode, nous nous sommes intéressés à sa stabilité ainsi qu'à son adaptivité en temps et en espace. Pour arriver à cela, nous avons dû faire une étude précise de la stabilité de la méthode de Galerkine discontinue et nous avons proposé des améliorations à cette technique entraînant des gains de temps significatifs. / My work consists in developing some high order numerical schemes in time and space for the modeling of the wave propagation. We have proposed to first discretize the wave equation with respect to the time using the so called Modified Equation Technique. Then, we have used a Discontinuous Galerkine Finite Element method for the space discretization. Switching the classical discretization process, we have constructed schemes as accurate as the classical ones with a numerical cost very interesting. After the numerical validation of this method, we have focused on its stability and on its adaptibility in time and space. To reach these objectives, we have performed a stability analysis of the Discontinuous Galerkin method and we have proposed some improvements to this technique which imply very important gain in terms of computationnal time.

Mes travaux de thèse portent sur la construction de conditions aux limites absorbantes (CLAs) pour des problèmes de propagation d'ondes posés dans des milieux limités par des surfaces régulières. Ces conditions sont nouvelles car elles prennent en compte non seulement les ondes proagatives (comme la plupart des CLAs existantes) mais aussi les ondes évanescentes et rampantes. Elles sont donc plus performantes que les conditions existantes. De plus, elles sont facilement implémentables dans un schéma d'éléments finis de type Galerkine Discontinu (DG) et ne modifie pas la condition de stabilité de Courant-Friedrichs-Lewy (CFL). Ces CLAs ont été implémentées dans un code simulant la propagation des ondes acoustiques ainsi que dans un code simulant la propagation des ondes en régime harmonique. Les comparaisons réalisées entre les nouvelles conditions et celles qui sont les plus utilisées dans la littérature montrent que prendre en compte les ondes évanescentes et les ondes rampantes permet de diminuer les réflexions issues de la frontière artificielle et donc de rapprocher la frontière artificielle du bord de l'obstacle. On limite ainsi les coûts de calcul, ce qui est un des avantages de mes travaux. De plus, compte tenu du fait que les nouvelles CLAs sont écrites pour des frontières quelconques, elles permettent de mieux adapter le domaine de calcul à la forme de l'obstacle et permettent ainsi de diminuer encore plus les coûts de calcul numérique. / In my PhD, I have worked on the construction of absorbing boundary conditions (ABCs) designed for wave propagation problems set in domains bounded by regular surfaces. These conditions are new since they take into account not only propagating waves (as most of the existing ABCs) but also evanescent and creeping waves. Therefore, they outperform the existing ABCs. Moreover, they can be easily implemented in a discontinuous Galerkin finite element scheme and they do not change the Courant-Friedrichs-Lewy stability condition. These ABCs have been implemented in two codes that respectively simulate the propagation of acoustic waves and harmonic waves. The comparisons performed between these ABCs and the ABCs mostly used in the litterature show that when we take into account evanescent and creeping waves, we reduce the reflections coming from the artificial boundary. Therefore, thanks to these new ABCs, the artificial boundary can get closer to the obstacle. Consequently, we reduce the computational costs which is one of the advantages of my work. Moreover, since these new ABCs are written for any kind of boundary, we can adapt the shape of the computational domain and thus we can reduce again the computational costs.

Neste trabalho, é apresentada uma formulação apropriada à análise de guias de ondas eletromagnéticos, cobrindo do espectro de microondas até o da óptica. Nas regiões a partir do ultravioleta, os comprimentos de onda são equivalentes às dimensões atômicas e a formulação necessita de uma abordagem quântica, que não é considerada neste estudo. A formulação é fundamentada nos métodos vetorial magnético e dos elementos finitos (MEF), em meios não homogêneos, anisotrópicos e não dissipativos, embora a dissipação possa ser facilmente introduzida na análise. Deu-se preferência à formulação com o campo magnético em vez do elétrico, pelo fato do campo magnético ignorar descontinuidades elétricas. Ele é contínuo em regiões de permeabilidade homogênea, propriedade dos meios dielétricos em geral (&#61549; = &#61549;0), independente da permissividade dos respectivos meios, conquanto os campos elétricos sejam descontínuos entre regiões de permissividades diferentes. / This work presents a suitable formulation to the analysis of electromagnetic waveguide, covering the spectrum of the microwave to optics. In regions from the ultraviolet, the wavelengths are equivalent to atomic dimensions and the design needs a quantum approach, which is not considered in this study. The formulation is based on the magnetic vector and the finite element methods (FEM), in non-homogeneous, anisotropic and non-dissipative dielectric materials, while the dissipation can be easily introduced in the analysis. Preference was given to the formulation with the magnetic field, because the magnetic field ignores electrical discontinuities. It is continuous in regions of homogeneous permeability, property of all dielectric materials (&#61549;=&#61549;0), independent of the permissiveness of respective regions, while electric fields are discontinuous between regions of different permittivities.

Structures acting as waveguides are quite common withexamples being, construction beams, fluid filled pipes, railsand extruded aluminium profiles. Curved structures like cartyres and pipe-bends may also be considered as waveguides. Wavesolutions in such structures may be found by a method calledthe Waveguide Finite Element Method or WFEM. This method uses afinite element approach on the cross-section of a waveguide tomodel the vibro-acoustic response as a set of linear, coupled,one dimensional, wave-equations. In this thesis six novel waveguide finite elements arederived and validated. These elements are, straight and curvedpre-stressed, orthotropic or anisotropic shell elements,straight and curved fluid elements, and straight and curvedfluid-shell coupling elements. Forced response and input power calculations for infiniteand periodic waveguides are presented. The assembled waveguidemodels can also serve as input for the Super Spectral FiniteElement Method, which enables forced response calculations formore complex boundaries. Furthermore, several properties ofdamped and undamped wave solutions are investigated. Finally, a car tyre model, encompassing for the highlyanisotropic material and the air cavity inside the tyre is setforth. A number of forced response calculations for this modelare presented and compared with measurements with goodagreement. Keywords:wave equation, wave solution, waveguide,finite element, spectral finite element, tyre noise, tyrevibration, input power, shells, pre-stress, fluid-shellcoupling axi-symmetric, two-and-half-dimensional

Ce manuscrit se décompose en deux parties principales, associées à deux types d'applications assez différentes. Dans la première partie qui comprend les deux premiers chapitres, je m'intéresse à des systèmes issus de problèmes de contrôle et d'estimation en physique quantique; dans la deuxième partie (troisième chapitre du manuscrit), j'étudie la propagation d'ondes électriques le long des fils classiques dans un réseau de lignes de transmission et je considère certains problèmes d'estimation de paramètres. Dans le premier chapitre nous étudions le problème de la planification de trajectoires pour des systèmes quantiques fermés modélisés par des équations de Schrödinger bilinéaire. Nous démontrons alors des résultats de la stabilisation approchée pour le cas d'une boite quantique infinie ainsi que pour le cas d'un potentiel décroissant. Dans les deux cas, le manque de pré-compacité des trajectoires dans des espaces fonctionnels appropriés nous oblige à proposer des méthodes de Lyapunov qui évitent des phénomènes de perte de masse à l'infini. Dans le deuxième chapitre nous étudions le problème de stabilisation de systèmes quantiques en observation. Cette observation nécessite l'ouverture du système à son environnement. Les modèles pertinents pour l'évolution de ce type de systèmes sont des modèles stochastiques basés sur des trajectoires de Monte-Carlo quantiques. Nous étudions alors certains problèmes de stabilisation qui parviennent de vraies expériences physiques. Enfin, dans le chapitre 3 nous considérons le problème d'estimation de paramètres pour un réseau de fils de câblage électrique. Dans ce but, nous étudions deux approches : l'approche temporelle et l'approche fréquentielle. Dans l'approche temporelle, nous considérons le réseau le plus simple qui consiste d'une seule ligne de transmission et nous proposons un algorithme d'identification pour l'équation d'onde associé qui est basé sur l'application des observateurs asymptotiques. Dans l'approche fréquentielle, nous considérons un réseau plus compliqué de la forme étoile. Nous proposons alors des résultats d'identifiabilité basés sur des techniques de l'inverse scattering.

A systematic research on aeroacoustics conducted around the world for the last few decades has revealed various inherent characteristics of the jet noise radiation. However, a lot more needs to be done for the theoretical as well as experimental predictions of various jet noise features based on actual flow details. The work reported in the present thesis is an attempt in this direction. A critical study of existing literature on jet noise shows that none of the general wave equations lends itself easily for predictions of all the jet noise features. It is shown that while LighthilPs classical acoustic analogy approach, with some reasonable approximations, can be used to yield most of the information needed by the engineers, the convected wave equations of Phillips and Lilley are required to study the acoustic radiation in what has come to be known as "Refraction valley" or "Cone of relative silence". The characteristics of the sound field of underexpanded cold jet impingement flows were studied by measuring the noise emanating from two convergent nozzles of throat diameter 2.5 mm and 5 mm each and a convergent - divergent nozzle of exit diameter of 6.49 mm, when the jet impinges on a flat plate kept perpendicular to the direction of the jet. The measurements were conducted upstream of the nozzle over an extensive envelope of jet operating conditions such as chamber stagnation pressure, mass flow rate through the nozzle and diameter of the nozzle. The source strength at the jet boundary was obtained by measuring acoustic pressure amplitude close to the jet contour assuming it as locally cylindrical. Particular attention was focussed on backward projection of the sound field on to a cylindrical surface. This is the application of acoustic holography to study the sound radiation in the audio frequency region. With the help of FFT and software developed for this purpose, the theoretical predictions using data from several cylindrical surfaces were compared. A detailed analysis of noise radiation from a cold sonic and supersonic free jet was also carried out. The experimental work involved the measurement of noise field from a 2.5 mm, 5 mm convergent and a convergent - divergent nozzle of exit diameter of 6.49 mm and area ratio 1.687 for designed Mach number of two. The experimental setup consisted essentially of a pressure chamber made of mild steel, designed to withstand 50 bar pressure. This chamber is a cylinder with dia 0.421 m and length 0.85 m. The nozzles were made of mild steel. Compressed air approximately at room temperature is supplied to the nozzle via a control valve. The measuring and recording instruments consists of B & K Microphones, Preamplifiers, Conditioning amplifier and a Mediator, which measure a Sound Pressure Level at a point. The nozzles were operated at pressure ratio upto 25 bar. The noise signal was processed through 12 channel data acquisition system. Acoustic pressure and SPL were" calculated using theoretical relations and software developed. Using this software Fast Fourier Transformations of raw signal was obtained from 20 Hz to 20 kHz. Also constant SPL contour graphs were obtained. Source strength distribution at the jet boundary has been obtained by the principle of acoustic holography. Experimental values are closely matching with the results obtained by acoustic holography. The percentage error for acoustic pressure and SPL were less than 12%. The experimental results were used to obtain the source distribution in terms of gross jet parameters.