Dissertations / Theses on the topic 'Fourier modal method'

La production de carburants est possible à partir d’eau, d’énergie solaire et de CO2 par la voie de la photosynthèse artificielle. L’optimisation de ce processus est un thème de recherche de l’Institut Pascal. À la petite échelle contrôlant ce procédé, il est indispensable de déterminer les propriétés radiatives des microalgues photosynthétiques pour résoudre l’équation de transfert radiatif au sein des photobioréacteurs. La grande variété des micro-organismes liée à la forme, à l’élongation et aux paramètres de taille fait que la mise en œuvre des méthodes numériques existantes échoue pour des raisons de précision ou de capacité mémoire. De nombreuses communautés scientifiques se heurtent à ce problème d’électromagnétisme non encore résolu surtout pour les particules de grands paramètres de taille. Les travaux réalisés dans le cadre de cette thèse ont consisté à résoudre ce problème par la méthode modale de Fourier, une méthode numérique a priori développée et optimisée pour modéliser les problèmes de l’optique électromagnétique. Dans cette méthode, chaque micro-organisme est approché par un empilement de couches ce qui revient à approcher son profil par des marches d’escalier. L’approche proposée a été validée par comparaison avec les résultats disponibles dans la littérature. Une validation expérimentale des calculs théoriques a également été faite dans le domaine des micro-ondes grâce à une collaboration avec l’équipe HIPE de l’Institut Fresnel (Marseille, UMR 7249). Les résultats obtenus montrent la pertinence de la méthode développée
The production of fuels is possible from water, solar energy and CO2 through artificial photosynthesis. The optimization of this process is a research topic of Pascal Institute. At a small scale controlling this process, it is essential to determine the radiative properties of photosynthetic microalgae to solve the radiative transfer equation in photobioreactors. The wide variety of microorganisms related to the form, the elongation and size parameters make that the implementation of existing numerical methods fails because of lack of accuracy or memory. Many scientific communities face this problem of electromagnetism unresolved especially for particles of large size settings. The work achieved in this research is aimed at solving this problem by the Fourier modal method which is a numerical method first developed and optimized for modelling the electromagnetic optics problems. In this method, each microorganism is approached by a stack of layers which leads to replace the profile by the staircase approximation. The proposed approach was validated by comparison with results available in the literature. An experimental validation of theoretical calculations was also made in the microwave spectrum thanks to a collaboration with the HIPE team from Fresnel Institute (Marseille, UMR 7249). The results show the accomodation of the developed method

Les propriétés optiques des nanomatériaux, tels que les cristaux photoniques ou les métamatériaux, ont reçu beaucoup d’attention dans les dernières années [1–9]. La dérivation numérique de ces propriétés se révèle pourtant très compliquée, en particulier dans le cas des structures métallo-diélectriques, qui comportent des résonances plasmoniques. C’est pourquoi des méthodes numériques avancées et des modèles semi-analytiques sont nécessaires. Dans cette thèse, nous montrerons que le formalisme de la matrice de diffraction peut satisfaire ces deux aspects. La méthode de la matrice de diffraction est un concept très général en physique. Dans le cas des structures périodiques, on peut dériver la matrice de diffraction à l’aide de la méthode modale de Fourier [10]. Pour la description exacte des géométries planes, nous avons développé la méthode des coordonnées adaptées [11], qui nous donne un nouveau système de coordonnées, dans lequel les interfaces des matériaux sont des surfaces de coordonnées constantes. En combinaison avec la méthode de la résolution spatiale adaptative, la méthode des coordonnées adaptées permet d’améliorer considérablement la convergence de la méthode modale de Fourier, de telle sorte qu’on peut calculer des structures métalliques compliquées très efficacement. Si on utilise la matrice de diffraction, il est non seulement possible de dériver les propriétés optiques en illumination de champ lointain, comme la transmission, la réflexion, l’absorption, et le champ proche, mais aussi de décrire l’émission d’un objet à l’intérieur d’une structure et d’obtenir les résonances optiques d’un sytème. Dans cette thèse, nous présenterons une méthode efficace pour la dérivation des résonances optiques tridimensionnelles, utilisant directement la matrice de diffraction [14]. Si on connaît les résonances d’un système isolé, il est aussi possible d’obtenir une approximation des résonances dans le cas d’un système combiné à l’aide de notre méthode du couplage des résonances [15, 16]. Cette méthode permet de décrire le régime de couplage des champs lointain et proche, y compris le couplage fort avec les résonances Fabry-Perot, pour des systèmes qui se composent d’un empilement de deux structures planes et périodiques. Pour cette raison, on peut étudier efficacement le couplage de ces systèmes. Cette thèse est écrite de manière à donner une idée d’ensemble du formalisme de la matrice de diffraction et de la méthode modale de Fourier. En outre, nous décrivons notre généralisation de ces méthodes et nous montrons la validité de nos approches pour différents exemples
The optical properties of nanostructures such as photonic crystals and metamaterials have drawn a lot of attention in recent years [1–9]. The numerical derivation of these properties, however, turned out to be quite complicated, especially in the case of metallo-dielectric structures with plasmonic resonances. Hence, advanced numerical methods as well as semi-analytical models are required. In this work, we will show that the scattering matrix formalism can provide both. The scattering matrix approach is a very general concept in physics. In the case of periodic grating structures, the scattering matrix can be derived by the Fourier modal method [10]. For an accurate description of non-trivial planar geometries, we have extended the Fourier modal method by the concept of matched coordinates [11], in which we introduce a new coordinate system that contains the material interfaces as surfaces of constant coordinates. In combination with adaptive spatial resolution [12,13], we can achieve a tremendously improved convergence behavior which allows us to calculate complex metallic shapes efficiently. Using the scattering matrix, it is not only possible to obtain the optical properties for far field incidence, such as transmission, reflection, absorption, and near field distributions, but also to solve the emission from objects inside a structure and to calculate the optical resonances of a system. In this work, we provide an efficient method for the ab initio derivation of three-dimensional optical resonances from the scattering matrix [14]. Knowing the resonances in a single system, it is in addition possible to obtain approximated resonance positions for stacked systems using our method of the resonant mode coupling [15, 16]. The method allows describing both near field and far field regime for stacked two-layer systems, including the strong coupling to Fabry-Perot resonances. Thus, we can study the mutual coupling in such systems efficiently. The work will provide the reader with a basic understanding of the scattering matrix formalism and the Fourier modal method. Furthermore, we will describe in detail our extensions to these methods and show their validity for several examples

Nous présentons un modèle théorique et numérique pour simuler la diffraction d'ondes électromagnétiques par des lentilles artificielles métalliques. Le premier chapitre présente les radars anti-collision dans le contexte automobile, le système d'antenne est composé d'une source primaire ponctuelle et d'une lentille artificielle. Cette dernière est réalisée de façon très simple en assemblant des lames métalliques minces sur des morceaux de mousse. Une méthode approchée permet d'obtenir rapidement le champ rayonné à travers une lentille par une source ponctuelle à l'aide des concepts d'optique géométrique et d'optique physique. Dans le second chapitre, deux variantes de la méthode modale sont proposées pour l'étude de la diffraction par des réseaux de lames parfaitement conductrices infiniment minces, une dite classique, décrit le champ à l'intérieur des guides parfaitement conducteurs à l'aide des modes de ces derniers, et l'autre considère que les guides forment un milieu inhomogène par morceaux. Les parois des guides sont vues comme des matériaux d'épaisseur très fine et très conducteurs. Numériquement, cet artifice est possible grâce à la technique de résolution spatiale adaptative aussi appelée formulation paramétrique. Dans le chapitre 3, l'ensemble des techniques présentées précédemment est appliqué aux lentilles. Un modèle numérique et électromagnétique est présenté où la lentille métallique est vue comme un empilement de réseaux lamellaires. Le champ global est obtenu en raccordant les modes de chaque couche. Une autre extension qui permet de modéliser des objets non périodiques est introduite : il s'agit d'un changement de coordonnées complexes qui produit des conditions aux limites absorbantes aux bords du domaine de calcul. Dans le chapitre 4, l'ensemble des techniques numériques développées plus haut est mis en oeuvre sur des cas concrets de lentilles artificielles et des comparaisons avec le modèle simplifié du chapitre 1 sont effectuées. Le chapitre 5 est également consacré à l'étude de lentilles. Cependant le domaine de longueur d'onde envisagé n'est plus le même puisqu'on passe dans le domaine optique. La notion de métal perd le sens qu'on lui donne habituellement. Le métal est caractérisé par une permittivité complexe dont la partie réelle peut être négative. Des modes nouveaux apparaissent. La méthode d'analyse retenue est encore une méthode modale. Pour tenir compte des profils d'entrée et de sortie de la lentille, on effectue un changement de coordonnée grâce auquel ces derniers deviennent des surfaces de coordonnées.

The fast Fourier transform (FFT), even though it has been widely applicable in Physics and Engineering, it has become attractive in Finance as well for it’s enhancement of computational speed. Carr and Madan succeeded in implementing the FFT for pricing of an option. This project, inspired by Carr and Madan’s paper, attempts to elaborate and connect the various mathematical and theoretical concepts that are helpful in understanding of the derivation. Further, we derive the characteristic function of the risk neutral probability for the logarithmic terminal stock price. The Black-Scholes-Merton (BSM) model is also revised including derivation of the partial deferential equation and the formula. Finally, comparison of the BSM numerical implementation with and without the FFT method is done using MATLAB.

Thesis (M.S. in Meterology and Physical Oceanography)--Naval Postgraduate School, September 2005.
Thesis Advisor(s): D. Benjamin Reeder, John A. Colosi. Includes bibliographical references (p. 73-75). Also available online.

Current transformer saturation will cause the secondary current distortion. When saturation occurs, the secondary current will not be linearly proportional to the primary current, which may lead to maloperation of protection devices. This thesis researches and tests two detecting methods: Fast Fourier Transform (FFT) and Wavelet Transform based methods. Comparing these two methods, FFT has a better performance in steady state saturation, and Wavelet Transform can determine singularity to provide the moment of distortion. The Jiles-Atherton (J-A) theory of ferromagnetic hysteresis is one approach used in electromagnetics transient modeling. With decades of development, the J-A model has evolved into different versions. The author summarizes the different models and implements J-A model in both MATLAB and Simulink.

This thesis discusses about possibilities of construction colour DNA spectrograms and about patterns which can be detected there. Spectrograms as tools of spectral analysis give us a simultaneous view of the local frequency throughout the nucleotide sequence. They are suitable for gene identification or gene regions identification, determination of global character about whole chromosomes and also give us a chance for the discovery of yet unknown regions of potential significance. For purpose of this kind of DNA analysis is possible to use digital signal processing methods. We can apply them on only after conversion of DNA sequence to numerical representation. Selection of correct numerical representation affects how well will be reflected biological features in numerical record which we need for another use in digital signal analysis.

A novel analytical method is developed to examine the chemical composition of plant oils by electrospray ionisation high-resolution Fourier transform ion cyclotron resonance mass spectrometry in both positive- and negative-ion modes. To date, this is the first reported application of this technique for the study of macadamia nut oil. Samples of macadamia nut oil from the Macadamia Integrifolia- Proteaceae family (smooth shell) are examined. The fatty acid profile of the oil is obtained by this mass spectrometric examination of the transesterified and hydrolysed oil samples. The Fourier transform ion-cyclotron resonance mass spectrometry results are compared to those obtained from similar samples using gas chromatography-mass spectrometry techniques. High performance liquid chromatography and Fourier transform ion cyclotron resonance mass spectrometry are used to separate and assign the isomers present in the methanol extract of the oils in separate experiments. Significant results in this study include: - The first observation and identity of a number of oxidised triacylglycerols in macadamia oil samples. - The first observation of oxidised and free fatty acids, measured directly in hydrolysed oil and in the methanol extract of macadamia oil. - High resolution Fourier transform ion cyclotron resonance mass spectrometry in broadband mode which enables isobars to be observed. - Esterified oil Fourier transform ion cyclotron resonance mass spectrometry results are consistent with our gas chromatography-mass spectrometry results and with the results of similar studies on macadamia oil in the literature. - A number of fatty acids with odd number of carbon atoms are observed in the oil. - In electrospray ionisation Fourier transform ion cyclotron resonance mass spectrometry of oils, the sample preparation is straightforward. The sample is dissolved in methanol or acetonitrile and the solution is introduced to the electrospray source directly. Introducing oil samples to the gas chromatograph-mass spectrometer needs the oils to be esterified prior to the analysis. - In this work, state-of-the-art mass spectrometry demonstrates distinct advantages in comparison to gas chromatography measurements such as direct identification of free fatty acids in oil samples, whereas this is not possible in gas chromatography-mass spectrometry due to the required esterification step prior to the analysis. - High performance liquid chromatography fraction collection is combined with Fourier transform ion cyclotron resonance mass spectrometry in off-line mode and found to improve the sensitivity, selectivity and signal to noise levels due to the lower number of compounds in each high performance liquid chromatography fraction compared to the methanol extract of macadamia oil sample. Also isomers of monoacylglycerols have been resolved using the high performance liquid chromatography technique.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
Modern engineering increasingly requires the comprehension of phenomena related to combustion, aeroacustics, turbulence transition, among others. For these purposes the Computational Fluids Dynamics (CFD) requires the used high order methods. One of these methods is the Fourier pseudo-spectral method, that provides an excellent numerical accuracy, and with the use of the Fast Fourier Transform algorithm (FFT), it presents a low computational cost in comparison to anothers high-order methods. Another important issue is the projection method of the pression term, which does not require the pressure computation from the Navier-Stokes equations. The procedure to calculate the pression field is usually the most onerous in classical methodologies. Nevertheless, the pseudo-spectral method can be only applied to periodic boundary flows, thus limiting its use. Aiming to solve this restriction, a new methodology is proposed at the present work, which has the objective of simulating nonperiodic flows using the Fourier pseudo-spectral method. For this purpose the immersed boundary method, that represents the boundary conditions through a force field imposed at Navier-Stokes equations is used. As a test to this new methodology, a classic problem of Computational Fluid Dynamics, The Lid Driven Cavity was simulated. The obtained results are promising and demonstrate the possibility to simulating non-periodic flows making use of the Fourier pseudo-spectral method.
Para compreender fenômenos relacionados à combustão, aeroacústica, transição a turbulência entre outros, a Dinâmica de Fluídos Computacional (CFD) utiliza os métodos de alta ordem. Um dos mais conhecidos é o método pseudo-espectral de Fourier, o qual alia: alta ordem de precisão na resolução das equações, com um baixo custo computacional. Este está ligado à utilização da FFT e do método da projeção do termo da pressão, o qual desvincula os cálculos da pressão da resolução das equações de Navier-Stokes. O procedimento de calcular o campo de pressão, normalmente é o mais oneroso nas metodologias convencionais. Apesar destas vantagens, o método pseudo-espectral de Fourier só pode ser utilizado para resolver problemas com condições de contorno periódicas, limitando o seu uso no campo da dinâmica de fluídos. Visando resolver essa restrição uma nova metodologia é proposta no presente trabalho, que tem como objetivo simular escoamentos não-periódicos utilizando o método pseudo-espectral de Fourier. Para isso, é utilizada a metodologia da Fronteira Imersa, a qual representa as condições de contorno de um escoamento através de um campo de força imposto nas equações de Navier-Stokes. Como teste, para essa nova metodologia, foi simulada uma cavidade com tampa deslizante (Lid Driven Cavity), problema clássico da mecânica de fluídos, que objetiva validar novas metodologias e códigos computacionais. Os resultados obtidos são promissores e demostram que é possível simular um escoamento não-periódico com o método pseudo-espectral de Fourier.
Mestre em Engenharia Mecânica

Traditionally, communication systems have been dominated by voice applications. Today with the emergence of smartphones, focus has shifted towards packet switched networks. The Internet provides a wide variety of services such as video streaming, web browsing, e-mail etc, and IP trac models are needed in all stages of product development, from early research to system tests. In this thesis, we propose a multi-level model of IP traffic where the user behavior and the actual IP traffic generated from different services are considered as being two independent random processes. The model is based on observations of IP packet header logs from live networks. In this way models can be updated to reflect the ever changing service and end user equipment usage. Thus, the work can be divided into two parts. The first part is concerned with modeling the traffic from different services. A subscriber is interested in enjoying the services provided on the Internet and traffic modeling should reflect the characteristics of these services. An underlying assumption is that different services generate their own characteristic pattern of data. The FFT is used to analyze the packet traces. We show that the traces contains strong periodicities and that some services are more or less deterministic. For some services this strong frequency content is due to the characteristics of cellular network and for other it is actually a programmed behavior of the service. The periodicities indicate that there are strong correlations between individual packets or bursts of packets. The second part is concerned with the user behavior, i.e. how the users access the different services in time. We propose a model based on a Markov renewal process and estimate the model parameters. In order to evaluate the model we compare it to two simpler models. We use model selection, using the model's ability to predict future observations as selection criterion. We show that the proposed Markov renewal model is the best of the three models in this sense. The model selection framework can be used to evaluate future models.

Cette thèse se focalise sur le développement d’un élément continu de coque axisymétrique raidie de type Reissner/Mindlin. La démarche consiste à utiliser les éléments de coques axisymétriques développés aux LISMMA pour lesquels une méthode de prise en compte de chargements repartis a été définie. L'introduction des raidisseurs longitudinaux est menée à partir d'éléments de poutres droites couplés à la coque. L'introduction des raidisseurs circonférentiels nécessite quant à elle le développement d'éléments de poutres circulaires selon une formulation analogue à celle des coques axisymétriques. Cette formulation appelée méthode des éléments continus est basée,entre autre, sur le développement en série de Fourier des champs inconnus selon la dimension circonférentielle et sur la détermination de la matrice de rigidité dynamique de la structure étudiée.Deux configurations de couplage ont été envisagées: Le couplage de coques axisymétriques avec des éléments d'anneaux circulaires de Timoshenko agissant comme raidisseurs circonférentiels puis avec des poutres droites de Timoshenko agissant comme raidisseurs longitudinaux. Des analyses harmoniques sont ensuite menées de manière à valider les formulations présentées par comparaison avec les résultats issus de modélisations éléments finis
This thesis focuses on the development of a continuous stiffened axisymmetric shell element of type Reissner/Mindlin. The approach consist in using the axisymmetric shell element developed in LISMMA for which distributed loads were applied on it.The introduction of longitudinal stiffeners is achieved by using a straight beam element coupled to the shell. The introduction of circumferential stiffeners requires the development of circular beam element according to a formulation similar to that used for the axisymmetric shell.In fact, this continuous element method is based on the development of the unknown fields on the Fourier series according to the circumferential dimensions and on the determination of the dynamic stiffness matrix of the studied structure.Two coupling configuration were considered: First of all the coupling of axisymmetric shell with circular Timoshenko beam element acting as circumferential stiffeners, then with straight Timoshenko beam element acting as longitudinal stiffener. Harmonic analyzes are conducted in order to validate the formulations presented in comparison with the results obtained from finite element model

La réduction du temps de calcul lors de la résolution de problèmes d’évolution dans le cadre du calcul de structure constitue un enjeu majeur pour, par exemple, la mise en place de critères de rupture des pièces dans le secteur de l’aéronautique et de l’automobile. En particulier, la prédiction du cycle stabilisé des polymères sollicités sous chargement cyclique nécessite de résoudre un problème thermo-viscoélastique à grand nombre de cycles. La présence de différentes échelles de temps telles que le temps de relaxation (viscosité), le temps caractéristique associé au problème thermique et le temps du cycle de chargement conduit à un temps de calcul significatif lorsqu’un schéma incrémental est utilisé comme c’est le cas avec la méthode des éléments finis (MEF). De plus, un nombre important de données doit être stocké (au moins à chaque cycle). L’objectif de cette thèse est de proposer de nouvelles méthodes ainsi que d’étendre des méthodes existantes. Il est choisi de résoudre un problème thermique transitoire cyclique impliquant différentes échelles de temps avec l’objectif de réduire le temps de calcul réduit. Les méthodes proposées font partie des méthodes de réduction de modèles. Tout d’abord, la méthode de décomposition propre généralisée(PGD) a été étendue à un problème transitoire cyclique 3D non linéaire, la non-linéarité a été traitée en combinant la méthode PGD à la Méthode d’interpolation empirique discrète (DEIM), stratégie numérique déjà proposée dans la littérature. Les résultats ont montré l’efficacité de la PGD pour générer des résultats précis par rapport à la solution FEM avec une erreur relative inférieure à (1%). Ensuite, afin de réduire le temps de calcul, une autre approche alternative a été développée. Cette approche est basée sur l’utilisation d’une collection de modes, les modes les plus significatifs, issus de solutions PGD pour différentes échelles de temps et différentes valeurs de paramètres. Un dictionnaire regroupant ces modes est alors utilisé pour construire des solutions pour différents temps caractéristiques et différentes conditions aux limites, uniquement par projection de la solution sur les modes du dictionnaire. Cette approche a été adaptée pour traiter un problème faiblement couplé diffuso-thermique. La nouveauté de cette approche est de considérer un dictionnaire composé de bases spatio-temporelles et non pas uniquement de bases spatiales comme dans la fameuse méthode POD. Les résultats obtenus avec cette approche sont précis et permettent une réduction notable du temps de calcul on line. Néanmoins, lorsque différents temps de cycles sont considérés, le nombre de modes dans le dictionnaire augmente, ce qui en limite son utilisation. Afin de pallier cette limitation,une troisième stratégie numérique est proposée dans cette thèse. Elle consiste à considérer comme a priori connues des bases temporelles, elle est appelée stratégie mixte. L’originalité dans cette approche réside dans la construction de la base temporelle a prior basée sur l’analyse de Fourier de différentes simulations pour différents temps et différentes valeurs des paramètres. Une fois cette étude réalisée, une expression analytique des bases temporelles fonction des paramètres tels que le temps caractéristique et le temps du cycle est proposée. Les bases spatiales associées sont calculées à l’aide d’un algorithme type PGD. Cette méthode est ensuite testée pour la résolution de problèmes thermiques 3D sous chargement cyclique linéaires et non linéaires et un problème faiblement couplé thermo-diffusion
In the framework of structural calculation, the reduction of computation time plays an important rolein the proposition of failure criteria in the aeronautic and automobile domains. Particularly, the prediction of the stabilized cycle of polymer under cyclic loading requires solving of a thermo-viscoelastic problem with a high number of cycles. The presence of different time scales, such as relaxation time (viscosity), thermal characteristic time (thermal), and the cycle time (loading) lead to a huge computation time when an incremental scheme is used such as with the Finite Element Method (FEM).In addition, an allocation of memory will be used for data storage. The objective of this thesis isto propose new techniques and to extend existent ones. A transient thermal problem with different time scales is considered in the aim of computation time reduction. The proposed methods are called model reduction methods. First, the Proper Generalized Decomposition method (PGD) was extended to a nonlinear transient cyclic 3D problems. The non-linearity was considered by combining the PGD method with the Discrete Empirical Interpolation Method (DEIM), a numerical strategy used in the literature. Results showed the efficiency of the PGD in generating accurate results compared to the FEM solution with a relative error less than 1%. Then, a second approach was developed in order to reduce the computation time. It was based on the collection of the significant modes calculated from the PGD method for different time scales. A dictionary assembling these modes is then used to calculate the solution for different characteristic times and different boundary conditions. This approach was adapted in the case of a weak coupled diffusion thermal problem. The novelty of this method is to consider a dictionary composed of spatio-temporal bases and not spatial only as usedin the POD. The results showed again an exact reproduction of the solution in addition to a huge time reduction. However, when different cycle times are considered, the number of modes increases which limits the usage of the approach. To overcome this limitation, a third numerical strategy is proposed in this thesis. It consists in considering a priori known time bases and is called the mixed strategy. The originality in this approach lies in the construction of a priori time basis based on the Fourier analysis of different simulations for different time scales and different values of parameters.Once this study is done, an analytical expression of time bases based on parameters such as the characteristic time and the cycle time is proposed. The related spatial bases are calculated using the PGD algorithm. This method is then tested for the resolution of 3D thermal problems under cyclic loading linear and nonlinear and a weak coupled diffusion thermal problem

Cette thèse est consacrée à démontrer l'existence des solutions globales en temps pour certaines équations non-linéaires, y compris les équations d'ondes et de Klein-Gordon et les équations de Dirac. Pour les équations d'ondes qui sont basées sur la méthode de la feuilletage hyperboloïde, nous établissons plusieurs résultats de stabilité globale, explorons les comportements asymptotiques de ces solutions, et étudions comment les solutions sont affectées quand des paramètres de masse atteignent certaines limites. Comme application, nous prouvons que plusieurs modèles physiques sont globalement stables : le modèle de Dirac-Klein-Gordon, le modèle de Dirac-Proca, le modèle de Klein-Gordon-Zakharov, le modèle U(1) des interactions électro-faibles etc.. Dans la partie I, nous étudions le modèle standard électro-faible. Nous prouvons la stabilité globale non-linéaire du modèle U(1), dans laquelle nous obtenons des bornes uniformes (modulo une croissance lente de logarithme). Puis, pour le modèle standard complet, nous obtenons la stabilité globale dans certains cas spéciaux. Dans la partie II, nous analysons une sorte d'équations d'ondes et de Klein-Gordon avec non-linéalités critiques, et prouvons l'existence des solutions globales qui possède une propriété de décroissance pointue. D'ailleurs, nous étudions une sorte d'equation de Klein-Gordon avec une masse qui pourrait s'annuler, et prouvons la décroissance qui est uniforme en terme de sa masse. Dans la partie III, nous examinons principalement la transformation (hyperboloïde) de Fourier, et dérivons une nouvelle estimation de type-L2 pour les ondes
This thesis is devoted to showing the existence of global-in-time solutions to some nonlinear equations, including the wave-Klein-Gordon equations and the Dirac equations. For the wave-Klein-Gordon-Dirac equations, based on the hyperboloidal foliation method we establish several global stability results, explore the asymptotic behaviors of these solutions, and study how the solutions are affected when some mass parameters go to certain limits. As an application, we can prove that several physical models are globally stable: the Dirac-Klein-Gordon model, the Dirac-Proca model, the Klein-Gordon-Zakharov model, the U(1) model of electroweak interactions and so on. In Part I, we study the electroweak standard model. We first prove the global nonlinear stability results for the U(1) model, where we obtain uniform energy bounds (modulo a slow logarithm growth). Next we move to the full standard model, and get global stability results in some special cases. In Part II, we analyse a class of coupled wave-Klein-Gordon equations with critical nonlinearities, and prove the existence of global-in-time solutions which enjoy sharp pointwise decay property. Besides we study a class of Klein-Gordon equations with possibly vanishing mass, and prove sharp pointwise decay results which are uniform in its mass. In Part III, we mainly investigate the hyperbolic Fourier transform, and derive a new L2-type estimate for waves

In my research work, I study diffraction by planar periodic structures (diffraction grating). In the first chapter, I try to familiarize the importance of diffraction gratings in spectroscopy. The second chapter explains the basic principle of diffraction without involving difficult mathematical tools. In the third chapter, I deduce Fourier modal method from Maxwell equations, a modern mathematical approach used for calculating the electromagnetic response of diffraction. The most important part of my work consists of the fourth and fifth chapter. At first, I try to utilize Fourier modal method for deriving formulas involving just 0th and (-1)st diffraction orders. Because of the unreasonable difficulty of uncovered formulas, I decided to consider only a special case - so-called Littrow configuration, in which the solution can be superposed from symmetric and antisymmetric couples of rays. I further develop so-called Local modal method - very inaccurate, but mathematically fairly simple, and discuss its physical limitations. Whole work is finished by the sixth chapter, which compares accurate computations gained from simulation and derived analytical formulas for both methods.

No
This paper demonstrates a non-Fourier prediction methodology of triple-layer human skin tissue for determining skin burn injury with non-ideal properties of tissue, metabolism and blood perfusion. The dual-phase lag (DPL) bioheat model is employed and solved using joint integral transform (JIT) through Laplace and Fourier transforms methods. Parametric studies on the effects of skin tissue properties, initial temperature, blood perfusion rate and heat transfer parameters for the thermal response and exposure time of the layers of the skin tissue are carried out. The study demonstrates that the initial tissue temperature, the thermal conductivity of the epidermis and dermis, relaxation time, thermalisation time and convective heat transfer coefficient are critical parameters to examine skin burn injury threshold. The study also shows that thermal conductivity and the blood perfusion rate exhibits negligible effects on the burn injury threshold. The objective of the present study is to support the accurate quantification and assessment of skin burn injury for reliable experimentation, design and optimisation of thermal therapy delivery.

Yes
This paper demonstrates a non-Fourier prediction methodology of triple-layer human skin tissue for determining skin burn injury with non-ideal properties of tissue, metabolism and blood perfusion. The dual-phase lag (DPL) bioheat model is employed and solved using joint integral transform (JIT) through Laplace and Fourier transforms methods. Parametric studies on the effects of skin tissue properties, initial temperature, blood perfusion rate and heat transfer parameters for the thermal response and exposure time of the layers of the skin tissue are carried out. The study demonstrates that the initial tissue temperature, the thermal conductivity of the epidermis and dermis, relaxation time, thermalisation time and convective heat transfer coefficient are critical parameters to examine skin burn injury threshold. The study also shows that thermal conductivity and the blood perfusion rate exhibits negligible effects on the burn injury threshold. The objective of the present study is to support the accurate quantification and assessment of skin burn injury for reliable experimentation, design and optimisation of thermal therapy delivery.
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碩士
臺灣大學
光電工程學研究所
98
In this thesis, the finite-difference modal method (FDMM) with step-index formulation for simulating grating structures is proposed and compared with rigorous coupled-wave analysis (RCWA), also called Fourier modal method (FMM). It is verified that FDMM has better convergence and accuracy than RCWA for TE polarization in almost all cases and TM polarization for high conductive and lossless metallic materials. In the FDMM, the relation of interface conditions to arbitrary high orders is considered and combines with Taylor series expansion. The generalized Douglas (GD) scheme is also adopted to enhance the convergence order without considering more sampled points. With the techniques mentioned above, the sparse matrix of eigenvalue problem could be constructed to solve the fields and the propagation constants of modes inside each layer. In addition, the enhanced transmittance matrix approach proposed by Moharam emph{et al.} for RCWA is used to make matrix manipulation stable for multi-layer or even single layer gratings. The diffraction properties of gratings, such as accuracy, convergence, dependence of diffraction efficiencies on incident angle, thickness, duty cycle, etc, will be discussed for numerical assessment of FDMM. Moreover, two-dimensional finite-difference methods combined with periodic boundary conditions and absorbing boundary conditions will be executed for comparison.

For some alloys martensitic transformation is responsible for the so-called shape memory effect and pseudoelasticity. These properties are used in a wide range of industry applications. Each of these materials is transformed to the shape it was manufactured in when heated to its critical temperature (austenite phase) no matter how seriously it was deformed at lower temperatures (martensite phase). Looking at the microstructure, one can observe significant change of crystalographic lattice depending on temperature and deformation. This the- sis focuses on modelling the evolution of microstructure during deformation for materials in the martensite phase. In this case, the creation of multiple variants of martensite is observed, divided by interfaces where a part of energy is stored. This behaviour can be described by the phase-field model. The numerical im- plementation of this model using the standard finite element method requires large computational costs. The aim of this thesis is to implement this model in MATLAB using a spectral method based on the fast Fourier transform, which is suitable for solving problems on a periodic domain. It is interesting to com- pare the computation using spectral method on a conventional PC with the computation written in FEniCS computed on a cluster. However, the...