We develop well-posedness theory and analytical and numerical solution techniques for Boussinesq-type equations. Firstly, we consider the Cauchy problem for a generalized Boussinesq equation. We show that under suitable conditions, a global solution for this problem exists. In addition, we derive sufficient conditions for solution blow-up in finite time. / Secondly, a generalized Jacobi/exponential expansion method for finding exact solutions of non-linear partial differential equations is discussed. We use the proposed expansion method to construct many new, previously undiscovered exact solutions for the Boussinesq and modified Korteweg-de Vries equations. We also apply it to the shallow water long wave approximate equations. New solutions are deduced for this system of partial differential equations. / Finally, we develop and validate a numerical procedure for solving a class of initial boundary value problems for the improved Boussinesq equation. The finite element method with linear B-spline basis functions is used to discretize the equation in space and derive a second order system involving only ordinary derivatives. It is shown that the coefficient matrix for the second order term in this system is invertible. Consequently, for the first time, the initial boundary value problem can be reduced to an explicit initial value problem, which can be solved using many accurate numerical methods. Various examples are presented to validate this technique and demonstrate its capacity to simulate wave splitting, wave interaction and blow-up behavior.

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In the present thesis, computational methods for theanalysis and design of photonic bandgap structure areconsidered. Many numerical methods have been used to study suchstructures. Among them, the plane wave expansion method is veryoften used. Using this method, we show that inclusions ofelliptic air holes can be used effectively to obtain a largercomplete band gap for two-dimensional (2D) photonic crystals.An optimal design of a 2D photonic crystal is also consideredin the thesis using a combination of the plane wave expansionmethod and the conjugate gradient method. We find that amaximum complete 2D band gap can be obtained by connectingdielectric rods with veins for a photonic crystal with a squarelattice of air holes in GaAs. For some problems, such as defect modes, the plane waveexpansion method is extremely time-consuming. It seems that thefinite-difference time-domain (FDTD) method is promising, sincethe computational time is proportional to the number of thediscretization points in the computation domain (i.e., it is oforderN). A FDTD scheme in a nonorthogonal coordinate systemis presented in the thesis to calculate the band structure of a2D photonic crystal consisting of askew lattice. The algorithmcan easily be used for any complicated inclusion configuration,which can have both the dielectric and metallic constituents.The FDTD method is also applied to calculate the off-plane bandstructures of 2D photonic crystals in the present thesis. Wealso propose a numerical method for computing defect modes in2D crystals (with dielectric or metallic inclusions). Comparedto the FDTD transmission spectra method, our method reduces thecomputation time and memory significantly, and finds as manydefect modes as possible, including those that are not excitedby an incident plane wave in the FDTD transmission spectramethod. The FDTD method has also been applied to calculateguided modes and surface modes in 2D photonic crystals using acombination of the periodic boundary condition and theperfectly matched layer for the boundary treatment. Anefficient FDTD method, in which only real variables are used,is also proposed for the full-wave analysis of guided modes inphotonic crystal fibers. / QC 20100629

Submitted by Cássia Santos (cassia.bcufg@gmail.com) on 2015-01-30T10:49:22Z No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Dissertação_Eduardo Koiti Hirota - 2014.pdf: 894506 bytes, checksum: 39a1f1c9a2e91954ecfdd1ef0513c5c0 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2015-01-30T13:05:19Z (GMT) No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Dissertação_Eduardo Koiti Hirota - 2014.pdf: 894506 bytes, checksum: 39a1f1c9a2e91954ecfdd1ef0513c5c0 (MD5) / Made available in DSpace on 2015-01-30T13:05:19Z (GMT). No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Dissertação_Eduardo Koiti Hirota - 2014.pdf: 894506 bytes, checksum: 39a1f1c9a2e91954ecfdd1ef0513c5c0 (MD5) Previous issue date: 2014-10-09 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Phenomenon that occur in the nature are essentially nonlinear and the dynamical systems theory aims to obtain a mathematical model that best represents the real physical systems, then nothing more coherent than the description or analysis of these natural phenomenon using models and techniques. In this dissertation, the technique of direct expansion for the development of two differential equations order to solve a nonlinear equation and the approximate determination of the roots of order algebraic equation higher or equal to two, was used. For this purpose, it was initially shown the development of a differential equation of motion subjected to a nonlinear damping, which is represented by the equation of Duffing – Van der Pol. Generally, it’s not easy to obtain an approximated analytical solution for this type equation, but this study was done with the purpouse of illustrating the technique used in the work, solving type solving a problem in which these techniques are routinely used to obtain a solution. Studied for application in basic education, it presents a way to obtain the approximate roots of equations of second and third degrees, using the technique of direct expansion for the sake of comparison. Since there are formulas for resolving this, It was proved that is possible to determine the roots of high-order equations by using the same technique. / Os fenômenos que ocorrem na natureza são essencialmente não lineares e a teoria de sistemas dinâmicos tem como objetivo obter um modelo matemático que represente melhor os sistemas físicos reais, então nada mais coerentes que a descrição ou análise desses fenômenos naturais usando modelos e técnicas não lineares. Nesta dissertação, foi utilizada a técnica da expansão direta para o desenvolvimento de equações diferenciais de ordem dois para resolução de uma equação não linear e na determinação aproximada de raízes de equações algébricas de ordem maior ou igual a dois. Com esse intuito, mostrou-se, inicialmente, o desenvolvimento de uma equação diferencial do movimento sujeito a um amortecimento não linear, que é representado pela equação de Duffing – Van der Pol. Geralmente, não é fácil obter uma solução analítica aproximada para esse tipo de equação, porém, este estudo é feito com a finalidade de ilustrar a técnica empregada no trabalho, resolvendo um tipo de problema no qual essas técnicas são corriqueiramente utilizadas para obter uma solução. Visando a aplicabilidade no ensino básico, apresenta-se uma forma de se obter as raízes aproximadas de equações do segundo e terceiro graus usando a técnica da expansão direta para efeito de comparação uma vez que existem fórmulas resolutivas para isso, provouse que é possível determinar as raízes de equações de ordem maior por meio da mesma técnica.

Ce manuscrit est consacré à l’étude de la méthode du développement en singularités (SEM) appliquée aux antennes. Dans la première partie de ce travail, trois méthodes d’extraction des pôles de résonance sont présentées et comparées : les méthodes de Prony et Matrix Pencil dans le domaine temporel et la méthode de Cauchy dans le domaine fréquentiel. Une procédure est établie pour optimiser l’extraction avec chaque méthode et une étude de robustesse montre que la méthode Matrix Pencil permet d’obtenir plus de pôles et avec une meilleure précision que les deux autres méthodes en présence de bruit. Dans un second temps, la méthode Matrix Pencil est appliquée sur des réponses d’antennes, obtenues en rayonnement et en Surface Equivalente Radar (SER), et les pôles de résonance extraits sont identiques pour les deux approches. Cette étude valide donc la possibilité d’extraire les pôles de résonance d’une antenne directement à partir de sa SER. La variation de la position des pôles de résonance en fonction des dimensions et de la charge de deux antennes est ensuite étudiée et met en évidence le lien entre l’impédance d’entrée de l’antenne et ses pôles de résonance. Enfin, les mesures de la SER de trois antennes valident expérimentalement l’extraction des pôles de résonance à partir de la SER d’une antenne. Ce travail pose donc les bases de la caractérisation d’antennes à l’aide de la SEM appliquée à la SER de l’antenne. / This manuscript deals with the Singularity Expansion Method (SEM) applied to antenna characterization. In the first part of this work, three resonant poles extraction methods are presented and compared: the Prony and Matrix Pencil methods in the transient domain and the Cauchy method in the frequency domain. A procedure is defined to optimize the extraction with each method and a robustness study shows that Matrix Pencil method allows obtaining more physical poles with a better accuracy than the two other methods in presence of noise. In a second part, the Matrix Pencil algorithm is applied on radiated and backscattered antenna responses. Extracted resonant poles from both responses are exactly the same. This study validates the possibility to extract poles directly from its backscattered response. The position of resonant poles is analyzed with respect to antenna’s dimension and its load for two different cases. This emphasizes the link between antenna poles and antenna input impedance. Finally, RCS measurements of three antennas validate antenna poles extraction directly from its RCS. This work lays the foundations of antenna characterization using the SEM applied to RCS measurements.

This thesis presents a nonadiabatic (4-body) description of the hydrogen-antihydrogen system at a nonrelativistic level. The properties of the system, the rearrangement processes and the possible existence of resonance states are investigated by using a variational method for coupled arrangement channels, the Gaussian Expansion Method, and the stabilization method. The 4-body basis set is optimized by means of prediagonalization of 2-body fragments. In paper I, a mass-scaling procedure of the Born-Oppenheimer potential is introduced for the description of the relative motion between hydrogen and antihydrogen. The nonadiabaticity of the system is investigated in paper II.

Ce mémoire traite de la mise au point et des applications d’un fluxmètre gazeux à pression constante, instrument de référence primaire pour la mesure de très faibles débits gazeux, couramment utilisé par les Laboratoires nationaux de métrologie. Il intervient dans la traçabilité des basses pressions absolues, via la méthode d’expansion continue, et celle des fuites d’hélium, liées aux applications dans le domaine du vide. De plus, nous avons montré que le fluxmètre à pression constante du Laboratoire commun de métrologie (LCM) permettait le raccordement des mesures de micro-débits, sous-domaine de la débitmétrie. Outre les points clés de la conception et la caractérisation métrologique, ce mémoire décrit l’étude de l’expansion continue ainsi que les travaux de comparaison du fluxmètre gazeux à pression constante avec les méthodes de référence employées au LCM, en particulier la méthode de gravimétrie dynamique. / This dissertation concerns the development and applications of a constant pressure gas flowmeter, the primary reference instrument used by National metrology laboratories to measure very low gas flows. It guarantees the traceability of low absolute pressures, via the continuous expansion method, and that of helium leaks, both related to applications in the field of vacuum. In addition, we have shown that the Laboratoire commun de métrologie (LCM) constant pressure flowmeter is well suited to micro-flow measurements, a sub-field of flow metering. Besides key points of the design and metrological characterization, this document describes the study of the continuous expansion method and work on comparisons of the constant pressure gas flowmeter with reference methods used at LCM, in particular the dynamic gravimetric method.

The solution of large and sparse eigenvalue problems arising from the discretization of the diffusion equation is considered. The multigroup<p>diffusion equation is discretized by means of the Nodal expansion Method (NEM) [9, 10]. A new formulation of the higher order NEM variants revealing the true nature of the problem, that is, a generalized eigenvalue problem, is proposed. These generalized eigenvalue problems are solved using the Jacobi-Davidson (JD) method<p>[26]. The most expensive part of the method consists of solving a linear system referred to as correction equation. It is solved using Krylov subspace methods in combination with aggregation-based Algebraic Multigrid (AMG) techniques. In that context, a particular<p>aggregation technique used in combination with classical smoothers, referred to as oblique geometric coarsening, has been derived. Its particularity is that it aggregates unknowns that<p>are not coupled, which has never been done to our<p>knowledge. A modular code, combining JD with an AMG preconditioner, has been developed. The code comes with many options, that have been tested. In particular, the instability of the Rayleigh-Ritz [33] acceleration procedure in the non-symmetric case has been underlined. Our code has also been compared to an industrial code extracted from ARTEMIS. / Doctorat en Sciences de l'ingénieur / info:eu-repo/semantics/nonPublished

The presented diploma thesis deals with the problem of determining the stress singularity exponent of the V-notch. This task can be divided into two parts. The first deals with the theoretical background, that means the basic relations of mechanics and the basic concepts of fracture mechanics. The second part deals with the elaboration of the Williams method and the creation of a program for calculating the stress singularity exponent.

Although it is possible to directly solve an entire flow-acoustics problem in one computation, this approach remains prohibitively large in terms of the computational resource required for most practical applications. Aeroacoustic problems are therefore usually split into two parts; one consisting of the source computation and one of the source propagation. Although both these parts entail great challenges on the computational method, in terms of accuracy and efficiency, it is still better than the direct solution alternative. The source usually consists of highly turbulent flows, which for most cases will need to be, at least partly, resolved. Then, acoustic waves generated by these sources often have to be propagated for long distances compared to the wavelength and might be subjected to scattering by solid objects or convective effects by the flow. Numerical methods used solve these problems therefore have to possess low dispersion and dissipation error qualities for the solution to be accurate and resource efficient. The wave expansion method (WEM) is an efficient discretization technique, which is used for wave propagation problems. The method uses fundamental solutions to the wave operator in the discretization procedure and will thus produce accurate results at two to three points per wavelength. This thesis presents a method that uses the WEM in an aeroacoustic context. Addressing the propagation of acoustic waves and transfer of sources from flow to acoustic simulations. The proposed computational procedure is applied to a co-rotating vortex pair and a cylinder in cross-flow. Overall, the computed results agree well with analytical solutions. Although the WEM is efficient in terms of the spatial discretization, the procedure requires that a Moore-Penrose pseudo-inverse is evaluated at each unique node-neighbour stencil in the grid. This evaluation significantly slows the procedure. In this thesis, a method with a regular grid is explored to speed-up this process. / <p>QC 20161121</p>