在這篇論文中，我們討論具有初始值條件的二階微分方程 □□□□□ 和系統微分方程 □□□□□等問題。 我們利用能量方法（即能量是常數的特性）來探討上述方程解之特性。例如：生成時間、爆破和爆破速率以及局部解的漸進行為。 / In this paper we shall consider the initial value problem for second order differential equation of the form □□□□□ and the system □□□□□ . We shall discuss the blow-up properties, such as the life-span, the blow-up rate and blow-up constants, and the asymptotic behavior of the global solution by using the energy method.

Nowadays disabled persons are actively integrated into social life. Different compensatory equipment allows them to work and travel independently and one of such means is the wheelchair. Not every disabled person has possibilities to travel by his own car, it is more convenient to use public transport facilities. Transportation safety of the wheelchair users is one of the most important problems facing engineers and transit providers, becouse improperly or totally unsecured wheelchair can lose the stability and tip over during the emergency driving situations. The main object of the scientific research work is complex dynamic system "Man - Wheelchair - Vehicle", whis is under action of environmental factors (road roughness, motion oscillations of vehicle). The main tasks of the work are to form and research nonlinear model of dynamic system considered and to define system's stability limits, providing means for safe travel; to determine main characteristics of the external action and analyze its influence on to dynamic system; to build engineering computation methodology for estimation of the rational parameters to fasten the wheelchair to the vehicle.

Les joints de porte et de vitrage des voitures jouent un rôle très important dans la réduction du bruit d’origine aérodynamique, à la fois par transmission directe et de par son rôle en tant que condition limite ees autres éléments transmetteurs (portes et fenêtres). Par conséquent, sa conception est fondamentale pour l’optimisation du confort de passager. Néanmoins, la méthode traditionnelle pour sa conception, basée sur une approche par tâtonnement de tests en soufflerie, est très coûteuse et insuffisante. Une approche différente est envisagée dans ce manuscrit, par la création d’un modèle capable de prédire la transmission du bruit jusqu’`a la cavité du véhicule, qui puisse être appliqué ultérieurement dans une routine d’optimisation. La modélisation des joints de vitrage et des fenêtres fait face à plusieurs difficultés. La fermeture de la porte subie par le joint de porte avant d’être soumis à l’excitation acoustique, ainsi que le comportement hyperélastique du caoutchouc, mènent à des déformations non-linéaires. Ce comportement change les propriétés (telles que la rigidité) du joint comprimé lors qu’il est soumis à l’excitation acoustique. De plus, l’interaction du son transmis par les joints avec la cavité du véhicule doit être prise en compte. Néanmoins, la taille réduite et la géométrie complexe du joint appellent à une approche telle que la méthode EF, tandis que la grande taille de la cavité véhicule nécessite d’une approche plus grossière, pour ne pas aboutir sur un modèle trop lourd. La solution proposée dans ce manuscrit implique la création d’un modèle hybride capable de modéliser le joint et la cavité séparément, avec l’approche la plus adaptée `a chaque cas, et de les coupler dans un seul modèle. Les comportements hyperélastique et viscoélastique des joints, avant et durant l’excitation acoustique, sont modélisés à l’aide du code commercial ABAQUS, tandis qu’une méthode énergétique appelée Méthode Energétique Simplifiée est utilisée pour la propagation ´ du son depuis les joints jusqu’au reste de la cavité. Cette méthode, adaptée aux besoins de l’application souhaitée, et couplée aux résultats du modèle EF, permet l’obtention rapide et locale du niveau de pression acoustique en n’importe quel point de la cavité. Finalement, des campagnes expérimentales sont mises en œuvre pour la validation des modèles. Les mises en place et les résultats sont détaillés dans ce manuscrit. / Car door and window seals have been proven to be of utmost importance to reduce aerodynamic noise, both through direct transmission and through their role as boundary conditions of the other transmitting elements (car doors and windows). As consequence, their design has become of great relevance when it comes to passenger comfort optimization. However, the traditional method for their conception, based on a trial and error approach through wind-tunnel testing, has been found to be insufficient and costly. A different approach is contemplated in this dissertation, through the development of a model capable of predicting sound transmission through seals and into the vehicle cavity, for its subsequent application into an optimization procedure. Several difficulties arise from the modeling of car door and window seals. Indeed, the door closure imposed on the door seal before any acoustic excitation, as well as the hyperelasticity of the rubber lead to a non-linear deformation behavior. This behavior changes the seal properties (e.g. stiffness) which have to be modeled under acoustic excitation. Additionally, the interaction of the transmitted sound with the vehicle cavity must be taken into account. However, the small, precise geometry of the seal would call for an approach such as FE method, whereas the big dimensions of a vehicle cavity demand a much coarser approach so that the problem doesn’t become unmanageable in size. The solution that is proposed in this dissertation, implies the creation of an hybrid model capable of modeling the seal and the vehicle cavity separately, with the most adequate approach to each case, and coupling them afterward into a single model. As consequence, the hyperelastic and viscoelastic behaviors of the seals, prior to and during the acoustic excitation, are modeled through FE software ABAQUS, whereas an energy method called Méthode Energétique Simplifiée is used for the propagation ´ of the sound from the seal to the rest of the cavity. This method, improved to better suit the requirements of the discussed application, and coupled to the results of the FE model, allows a fast and local computation of the sound pressure level at any point inside the cavity. Finally, some experimental tests are put in place for the validation of the models. The different setups and results are detailed in this dissertation.

In this thesis we look at acoustic wave propagation under water. We look in particular at waves generated by a point source and what happens with the propagation when we model the bottom as flat or as curvilinear. We assume the source to be working at a certain frequency and therefore we model this problem by solving the Helmholtz equation. Since Helmholtz equation has some unwanted numerical properties we are interested in finding new numerical methods that could accelerate the solver. In this thesis we use the Waveholtz iteration, which solves Helmholtz equation by connecting it to the time-dependent wave equation. We use finite differences and the SBP-SAT method to approximate the spatial problem numerically and for modelling the sea bottom we use curvilinear coordinates.  To compare the Waveholtz iteration we also solve Helmholtz equation with a naive solver. The naive solver consists of approximating the equation with finite differences and then solving the linear system of equation by some iterative solver, which for our tests will be GMRES. The results show that the Waveholtz iteration converges in less iterations than our naive solver. It also shows that the number of iterations stays unchanged when changing our discretization, which otherwise is a big problem for our naive solver. This allows us to increase the accuracy of our numerical solution without changing the computation time too much.  We show that the number of iterations increases according to theory for an increasing frequency, and that for open problems we even see a smaller increase. For certain resonant frequencies in Helmholtz equation we do not expect the Waveholtz iteration to converge. In the neighbourhood of these frequencies the convergence becomes slow and we need many iterations for a solution of a certain accuracy. By reformulating the Waveholtz iteration as a Krylov solution we can see that resonances in Helmholtz equation have a smaller impact of the convergence. / I detta examensarbete undersöker vi akustisk vågutbredning i vatten. Vi kollar specifikt på vågor som genereras av en punktkälla och vad som sker när vi modellerar botten som plan eller som kurvlinjär. Då vi antar att punktkällan arbetar vid en bestämd frekvens, kommer vi modellera det fysikaliska problemet genom att lösa Helmholtz ekvation. Helmholtz ekvation har dock några numeriska egenskaper som är oönskade, och därför finns ett intresse av att hitta nya numeriska metoder som löser ekvationen. I detta examensarbete undersöker vi Waveholtz iteration, som löser Helmholtz ekvation genom att koppla den till den tidsberoende vågekvationen. Vi använder finita differenser och SBP-SAT metoden för att approximera det rumsliga problemet numeriskt. För att ge en detaljerad beskrivning av botten använder vi kurvlinjära koordinater. För att jämföra Waveholtz iterationen med något löser vi även Helmholtz med hjälp av en naiv lösare. Den naiva lösaren består av att approximera problemet med finita differenser och sedan lösa det linjära systemet rakt av med en iterativ lösare (vilket för våra fall kommer vara GMRES). Resultatet visar att Waveholtz iteration konvergerar på ett lägre antal iterationer än vår naiva lösare. Det visar även att antalet iterationer inte förändras när vi ändrar diskretisering, vilket annars är ett problem för vår naiva lösare. Detta innebär att vi kan få en högre noggrannhet utan att förlänga beräkningstiden alltför mycket.  Vi visar även att antalet iterationer ökar som förväntat med en ökad frekvens, samt att för öppna problem så ökar antalet iteration mindre än enligt teorin. Vid vissa resonanta frekvenser i Helmholtz ekvation förväntar vi oss att Waveholtz iteration inte kommer konvergerar. I närheten av dessa frekvenser blir konvergensen långsam och vi behöver många iterationer för att lösa problemet. Genom att formulera Waveholtz iteration som en Krylov lösning kommer resonanser i Helmholtz ekvation ge en mindre negativ effekt på konvergensen än om den är formulerad som en fixpunkts iteration.