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Proposed designs for Very Large Floating Structures motivate us to understand water-wave propagation through arrays of hundreds, or possibly thousands, of floating structures. The water-wave problems we study are each formulated under the usual conditions of linear wave theory. We study the frequency-domain problem of water-wave propagation through a periodically arranged array of structures, which are solved using a variety of methods. In the first instance we solve the problem for a periodically arranged infinite array using the method of matched asymptotic expansions for both shallow and deep water; the structures are assumed to be small relative to the wavelength and the array periodicity, and may be fixed or float freely. We then solve the same infinite array problem using a numerical approach, namely the Rayleigh-Ritz method, for fixed cylinders in water of finite depth and deep water. No limiting assumptions on the size of the structures relative to other length scales need to be made using this method. Whilst we aren t afforded the luxury of explicit approximations to the solutions, we are able to compute diagrams that can be used to aid an investigation into negative refraction. Finally we solve the water-wave problem for a so-called strip array (that is, an array that extends to infinity in one horizontal direction, but is finite in the other), which allows us to consider the transmission and reflection properties of a water-wave incident on the structures. The problem is solved using the method of multiple scales, under the assumption that the evolution of waves in a horizontal direction occurs on a slower scale than the other time scales that are present, and the method of matched asymptotic expansions using the same assumptions as for the infinite array case.

In this thesis, we use asymptotic methods to solve problems of wave propagation through infinite and finite (only consider those that are finite in one direction) arrays of scatterers. Both two- and three-dimensional arrays are considered. We always assume the scatterer size is much smaller than both the wavelength and array periodicity. Therefore a small parameter is involved and then the method of matched asymptotic expansions is applicable. When the array is infinite, the elastic wave scattering in doubly-periodic arrays of cavity cylinders and acoustic wave scattering in triply-periodic arrays of arbitrary shape rigid scatterers are considered. In both cases, eigenvalue problems are obtained to give perturbed dispersion approximations explicitly. With the help of the computer-algebra package Mathematica, examples of explicit approximations to the dispersion relation for perturbed waves are given. In the case of finite arrays, we consider the multiple resonant wave scattering problems for both acoustic and elastic waves. We use the methods of multiple scales and matched asymptotic expansions to obtain envelope equations for infinite arrays and then apply them to a strip of doubly or triply periodic arrays of scatterers. Numerical results are given to compare the transmission wave intensity for different shape scatterers for acoustic case. For elastic case, where the strip is an elastic medium with arrays of cavity cylinders bounded by acoustic media on both sides, we first give numerical results when there is one dilatational and one shear wave in the array and then compare the transmission coefficients when one dilatational wave is resonated in the array for normal incidence. Key words: matched asymptotic expansions, multiple scales, acoustic scattering, elastic scattering, periodic structures, dispersion relation.

We consider the motion of a Brownian particle trapped in an arbitrary bounded two or three-dimensional domain, whose boundary is reflecting except for a small absorbing window through which the particle can escape. We use the method of matched asymptotic expansions to calculate the mean first passage time, defined as the time taken for the Brownian particle to escape from the domain through the absorbing window. This is known as the narrow escape problem. Since the mean escape time diverges as the window shrinks, the calculation is a singular perturbation problem. We extend our results to include N absorbing windows of varying length in two dimensions and varying radius in three dimensions. We present findings in two dimensions for the unit disk, unit square and ellipse and in three dimensions for the unit sphere. The narrow escape problem has various applications in many fields including finance, biology, and statistical mechanics.

Using the method of matched asymptotic expansions the reflection and transmission coefficients are calculated for scattering of oblique water waves by a vertical barrier. Here an assumption is made that the barrier is small compared to the wavelength and the depth of water. A number of sloshing problems are considered. The eigenfrequencies are calculated when a body is placed in a rectangular tank. Here the bodies considered are a vertical surface-piercing or bottom-mounted barrier, and circular and elliptic cylinders. When the body is a vertical barrier, the eigenfunction expansion method is applied. When the body is either a circular or elliptic cylinder, and the motion is two-dimensional, the boundary element method is applied to calculate the eigenfrequencies. For comparison, two approximations, "a wide-spacing", and "a small-body" are used for a vertical barrier and circular cylinder. In the wide-spacing approximation, the assumption is made that the wavelength is small compared with the distance between the body and walls. The small-body approximation means that a typical dimension of the body is much larger than the cross-sectional length scale of the fluid motion. For an elliptic cylinder, the method of matched asymptotic expansions is used and compared with the result of the boundary- element method. Also a higher-order solution is obtained using the method of matched asymptotic expansions, and it is compared with the exact solution for a surface-piercing barrier. Again the assumption is made that the length scale of the motion is much larger than a typical body dimension. Finally, the drift force on multiple bodies is considered the ratio of horizontal drift force in the direction of wave advance on two cylinders to that on an isolated cylinder is calculated. The method of matched asymptotic expansions is used under the assumption that the wavelength is much greater than the cylinder spacing.

We consider the motion of a Brownian particle trapped in an arbitrary bounded two or three-dimensional domain, whose boundary is reflecting except for a small absorbing window through which the particle can escape. We use the method of matched asymptotic expansions to calculate the mean first passage time, defined as the time taken for the Brownian particle to escape from the domain through the absorbing window. This is known as the narrow escape problem. Since the mean escape time diverges as the window shrinks, the calculation is a singular perturbation problem. We extend our results to include N absorbing windows of varying length in two dimensions and varying radius in three dimensions. We present findings in two dimensions for the unit disk, unit square and ellipse and in three dimensions for the unit sphere. The narrow escape problem has various applications in many fields including finance, biology, and statistical mechanics.

Cette thèse est consacrée à l'étude de l'influence d'une fine couche hétérogène sur le comportement élastique linéaire d'une structure tridimensionnelle.Deux types d'hétérogénéités sont pris en compte : des cavités et des inclusions élastiques. Une étude complémentaire, dans le cas d'inclusions de grande rigidité, a été réalisée en considérant un problème de conduction thermique.Une analyse formelle par la méthode des développements asymptotiques raccordés conduit à un problème d'interface qui caractérise le comportement macroscopique de la structure. Le comportement microscopique de la couche est lui déterminé sur une cellule de base. Le modèle asymptotique obtenu est ensuite implémenté dans un code éléments finis. Une étude numérique permet de valider les résultats de l'analyse asymptotique. / This thesis is devoted to the study of the influence of a thin heterogeneous layeron the linear elastic behavior of a three-dimensional structure. Two types of heterogeneties are considered : cavities and elastic inclusions. For inclusions of high rigidty a further study was performed in the case of a heat conduction problem.A formal analysis using the matched asymptotic expansions method leads to an interface problem which characterizes the macroscopic behavior of the structure. The microscopic behavior of the layer is determined in a basic cell.The asymptotic model obtained is then implemented in a finite element software.A numerical study is used to validate the results of the asymptotic analysis.

We consider the motion of a Brownian particle trapped in an arbitrary bounded two or three-dimensional domain, whose boundary is reflecting except for a small absorbing window through which the particle can escape. We use the method of matched asymptotic expansions to calculate the mean first passage time, defined as the time taken for the Brownian particle to escape from the domain through the absorbing window. This is known as the narrow escape problem. Since the mean escape time diverges as the window shrinks, the calculation is a singular perturbation problem. We extend our results to include N absorbing windows of varying length in two dimensions and varying radius in three dimensions. We present findings in two dimensions for the unit disk, unit square and ellipse and in three dimensions for the unit sphere. The narrow escape problem has various applications in many fields including finance, biology, and statistical mechanics. / Science, Faculty of / Mathematics, Department of / Graduate

The focus of this study is the prediction of trajectories of solid particles injected into either a cylindrically- shaped solid rocket motor (SRM) or a bidirectional vortex chamber (BV). The Lagrangian particle trajectory is assumed to be governed by drag, virtual mass, Magnus, Saffman lift, and gravity forces in a Stokes flow regime. For the conditions in a solid rocket motor, it is determined that either the drag or gravity forces will dominate depending on whether the sidewall injection velocity is high (drag) or low (gravity). Using a one-way coupling paradigm in a solid rocket motor, the effects of particle size, sidewall injection velocity, and particle-to-gas density ratio are examined. The particle size and sidewall injection velocity are found to have a greater impact on particle trajectories than the density ratio. Similarly, for conditions associated with a bidirectional vortex engine, it is determined that the drag force dominates. Using a one-way particle tracking Lagrangian model, the effects of particle size, geometric inlet parameter, particle-to-gas density ratio, and initial particle velocity are examined. All but the initial particle velocity are found to have a significant impact on particle trajectories. The proposed models can assist in reducing slag retention and identifying fuel injection configurations that will ensure proper confinement of combusting droplets to the inner vortex in solid rocket motors and bidirectional vortex engines, respectively.

De cette thèse, une nouvelle méthode éléments finis hybride MAX-FEM dédiée à la modélisation thermomécanique des structures avec couches minces a été développée. Cette nouvelle approche se base sur un couplage analytique-numérique de deux méthodes : les Développements Asymptotiques Raccordés (MAE) et la Partition de l'Unité (PUM). Ce couplage consiste à construire l'enrichissement de la PUM par MAE est mène à une forme corrigée de la méthode des éléments finis classique (FEM). Cette correction est obtenue à travers des matrices de correction contenant les informations géométriques et caractéristiques du matériau de la couche mince. Les matrices introduites par l'approche MAX-FEM simplifient son implémentation numérique sous différents codes de calculs (MATLAB, ABAQUS, ...) et permettent l'obtention de la solution globale en un seul calcul. Les résultats obtenus par la MAX-FEM pour des applications 1D et 2D thermomécaniques montrent une très bonne précision avec un temps de calcul minimal et sans raffinement de maillage. De plus, la MAX-FEM surmonte les limitations de la MAE ainsi que celle de la PUM en termes de nombre de calculs, de la sensibilité aux propriétés des matériaux, des conditions aux limites ainsi que l'intégration numérique. Finalement, l'approche MAX-FEM est exploitée pour le développement d'un nouveau protocole expérimental dédié à la caractérisation thermique des couches minces. Ce protocole vise l'identification, de manière simple, de la conductivité thermique de la couche mince après son élaboration et sous les deux régimes transitoire et permanent. L'approche consiste à confronter la nature du transfert thermique d'une éprouvette homogène à une contenant une couche mince. La différence relevée est directement liée à la conductivité thermique de la couche mince. Les résultats obtenus, après réalisation du banc d'essais, montrent une bonne précision de l'approche avec une méthodologie de mesure simple à mettre en oeuvre / This work introduces a new simplified finite elements method MAX-FEM based on hybrid analytical-numerical coupling. This method is intended to the multi-scales analysis of transient thermomechanical behavior of mediums containing thin layers such as bounded and coated structures. The MAX-FEM consists in correcting the classical Finite Elements Method (FEM) by correction matrices taking into account the presence of thin layers without any mesh refinement. The proposed correction is based on the analytical approach of Matched Asymptotic Expansions (MAE) and the numerical method of Partition of Unity Method (PUM). The developed approach can easily implemented under different numerical codes (MATLAB, ABAQUS, ...) and can be used to perform mechanical, thermal and thermomechanical analyses of 1D and 2D bounded and coated structures. The obtained results show a good accuracy with short computation time, and without any required mesh refinement. Also, the developed method overcomes the limitation of the MAE and PUM methods by exploiting the advantages of their coupling. Finally, the MAX-FEM approach was also used to develop an experimental test bench intended to the thermal characterization of thin layers. Indeed, a simple confrontation between the heat transfer in an homogeneous structure and a second structure with thin layer allows identifying the thermal conductivity in both transient and stationary regimes. The test bench is simple to release and the obtained results for brazed structure show a good accuracy of the developed approach.