<p>This thesis concerns the numerical simulation of creeping rays and their contribution to high frequency scattering problems.</p><p>Creeping rays are a type of diffracted rays which are generated at the shadow line of the scatterer and propagate along geodesic paths on the scatterer surface. On a perfectly conducting convex body, they attenuate along their propagation path by tangentially shedding diffracted rays and losing energy. On a concave scatterer, they propagate on the surface and importantly, in the absence of dissipation, experience no attenuation. The study of creeping rays is important in many high frequency problems, such as design of sophisticated and conformal antennas, antenna coupling problems, radar cross section (RCS) computations and control of scattering properties of metallic structures coated with dielectric materials.</p><p>First, assuming the scatterer surface can be represented by a single parameterization, we propose a new Eulerian formulation for the ray propagation problem by deriving a set of <i>escape </i>partial differential equations in a three-dimensional phase space. The equations are solved on a fixed computational grid using a version of fast marching algorithm. The solution to the equations contain information about all possible creeping rays. This information includes the phase and amplitude of the ray field, which are extracted by a fast post-processing. The advantage of this formulation over the standard Eulerian formulation is that we can compute multivalued solutions corresponding to crossing rays. Moreover, we are able to control the accuracy everywhere on the scatterer surface and suppress the problems with the traditional Lagrangian formulation. To compute all possible creeping rays corresponding to all shadow lines, the algorithm is of computational order O(<i>N</i>³ log <i>N</i>), with<i> N</i>³ being the total number of grid points in the computational phase space domain. This is expensive for computing the wave field for only one shadow line, but if the solutions are sought for many shadow lines (for many illumination angles), the phase space method is more efficient than the standard methods such as ray tracing and methods based on the eikonal equation.</p><p>Next, we present a modification of the single-patch phase space method to a multiple-patch scheme in order to handle realistic problems containing scatterers with complicated geometries. In such problems, the surface is split into multiple patches where each patch has a well-defined parameterization. The escape equations are solved in each patch, individually. The creeping rays on the scatterer are then computed by connecting all individual solutions through a fast post-processing.</p><p>We consider an application to mono-static radar cross section problems where creeping rays from all illumination angles must be computed. The numerical results of the fast phase space method are presented.</p>

Accurate modeling of current flow and network parameter extraction in 2-D and 3-D conductors has an important application in signal integrity of high-speed interconnects. In this thesis, we propose a new rigorous single-source Surface-Volume-Surface Electric Field Integral Equation (SVS-EFIE) for magnetostatic analysis of 2-D transmission lines and broadband resistance and inductance extraction in 3-D interconnects. Furthermore, the novel integral equation can be used for the solution of full-wave scattering problems on penetrable 2-D cylinders of arbitrary cross-section under transverse magnetic polarization. The new integral equation is derived from the classical Volume Electric Field Integral Equation (V-EFIE) by representing the electric field inside a conductor or a scatterer as a superposition of the cylindrical waves emanating from the conductor’s surface. This converts the V-EFIE into a surface integral equation involving only a single unknown function on the surface. The novel equation features a product of integral operators mapping the field from the conductor surface to its volume and back to its surface terming the new equation the Surface-Volume-Surface EFIE. The number of unknowns in the proposed SVS-EFIE is approximately the square root of the number of degrees of freedom in the traditional V-EFIE; therefore, it allows for substantially faster network parameter extraction and solutions to 2-D scattering problems without compromising the accuracy. The validation and benchmark of the numerical implementation of the Method of Moment discretization of the novel SVS-EFIE has been done via comparisons against numerical results obtained by using alternative integral equations, data found in literature, simulation results acquired from the CAD software, and analytic formulas.

Compressive Sensing (CS) is a novel signal-processing paradigm that allows sampling of sparse or compressible signals at lower than Nyquist rate. The past decade has seen substantial research on imaging applications using compressive sensing. In this thesis, CS is combined with the commercial electromagnetic (EM) simulation software newFASANT to improve its efficiency in solving EM scattering problems such as Radar Cross Section (RCS) of complex targets at GHz frequencies. This thesis proposes a CS-RCS approach that allows efficient and accurate recovery of under-sampled RCSs measured from a random set of incident angles using an accelerated iterative soft thresh-holding reconstruction algorithm. The RCS results of a generic missile and a Canadian KingAir aircraft model simulated using Physical Optics (PO) as the EM solver at various frequencies and angular resolutions demonstrate good efficiency and accuracy of the proposed method.

京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第20513号 / 情博第641号 / 新制||情||111(附属図書館) / 京都大学大学院情報学研究科複雑系科学専攻 / (主査)教授 西村 直志, 教授 磯 祐介, 准教授 吉川 仁 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

This thesis concerns the numerical simulation of creeping rays and their contribution to high frequency scattering problems. Creeping rays are a type of diffracted rays which are generated at the shadow line of the scatterer and propagate along geodesic paths on the scatterer surface. On a perfectly conducting convex body, they attenuate along their propagation path by tangentially shedding diffracted rays and losing energy. On a concave scatterer, they propagate on the surface and importantly, in the absence of dissipation, experience no attenuation. The study of creeping rays is important in many high frequency problems, such as design of sophisticated and conformal antennas, antenna coupling problems, radar cross section (RCS) computations and control of scattering properties of metallic structures coated with dielectric materials. First, assuming the scatterer surface can be represented by a single parameterization, we propose a new Eulerian formulation for the ray propagation problem by deriving a set of escape partial differential equations in a three-dimensional phase space. The equations are solved on a fixed computational grid using a version of fast marching algorithm. The solution to the equations contain information about all possible creeping rays. This information includes the phase and amplitude of the ray field, which are extracted by a fast post-processing. The advantage of this formulation over the standard Eulerian formulation is that we can compute multivalued solutions corresponding to crossing rays. Moreover, we are able to control the accuracy everywhere on the scatterer surface and suppress the problems with the traditional Lagrangian formulation. To compute all possible creeping rays corresponding to all shadow lines, the algorithm is of computational order O(N3 log N), with N3 being the total number of grid points in the computational phase space domain. This is expensive for computing the wave field for only one shadow line, but if the solutions are sought for many shadow lines (for many illumination angles), the phase space method is more efficient than the standard methods such as ray tracing and methods based on the eikonal equation. Next, we present a modification of the single-patch phase space method to a multiple-patch scheme in order to handle realistic problems containing scatterers with complicated geometries. In such problems, the surface is split into multiple patches where each patch has a well-defined parameterization. The escape equations are solved in each patch, individually. The creeping rays on the scatterer are then computed by connecting all individual solutions through a fast post-processing. We consider an application to mono-static radar cross section problems where creeping rays from all illumination angles must be computed. The numerical results of the fast phase space method are presented. / QC 20101119

京都大学 / 新制・課程博士 / 博士(情報学) / 甲第24738号 / 情博第826号 / 新制||情||138(附属図書館) / 京都大学大学院情報学研究科先端数理科学専攻 / (主査)教授 磯 祐介, 准教授 吉川 仁, 准教授 藤原 宏志, 教授 西村 直志(京都大学 名誉教授) / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

Les problèmes de diffusion de particules, à deux et à trois corps, ont une importance cruciale en physique atomique, car ils servent à décrire différents processus de collisions. Actuellement, le cas de deux corps peut être résolu avec une précision numérique désirée. Les problèmes de diffusion à trois particules chargées sont connus pour être bien plus difficiles mais une déclaration similaire peut être affirmée. L’objectif de ce travail est de contribuer, d’un point de vue analytique, à la compréhension des processus de diffusion Coulombiens à trois corps. Ceci a non seulement un intérêt fondamental, mais est également utile pour mieux maîtriser les approches numériques en cours d’élaboration au sein de la communauté de collisions atomiques. Pour atteindre cet objectif, nous proposons d’approcher la solution du problème avec des développements en séries sur des ensembles de fonctions appropriées et possédant une expression analytique. Nous avons ainsi développé un nombre d’outils mathématiques faisant intervenir des fonctions Coulombiennes, des équations différentielles de second ordre homogènes et non-homogènes, et des fonctions hypergéométriques à une et à deux variables / Two and three-body scattering problems are of crucial relevance in atomic physics as they allow to describe different atomic collision processes. Nowadays, the two-body cases can be solved with any degree of numerical accuracy. Scattering problem involving three charged particles are notoriously difficult but something similar -- though to a lesser extent -- can be stated. The aim of this work is to contribute to the understanding of three-body Coulomb scattering problems from an analytical point of view. This is not only of fundamental interest, it is also useful to better master numerical approaches that are being developed within the collision community. To achieve this aim we propose to approximate scattering solutions with expansions on sets of appropriate functions having closed form. In so doing, we develop a number of related mathematical tools involving Coulomb functions, homogeneous and non-homogeneous second order differential equations, and hypergeometric functions in one and two variables

Nous sommes intéressés dans cette thèse par l'analyse de la diffraction directe et inverse des ondes par des couches infinies périodiques localement perturbées à une fréquence fixe. Ce problème a des connexions avec le contrôle non destructif des structures périodiques telles que des structures photoniques, des fibres optiques, des réseaux, etc. Nous analysons d'abord le problème direct et établissons certaines conditions sur l'indice de réfraction pour lesquelles il n'existe pas de modes guidés. Ce type de résultat est important car il montre les cas pour lesquels les mesures peuvent être effectuées par exemple sur une couche au dessus de la structure périodique sans perdre des informations importantes dans la partie propagative de l'onde. Nous proposons ensuite une méthode numérique pour résoudre le problème de diffraction basée sur l'utilisation de la transformée de Floquet-Bloch dans les directions de périodicité. Nous discrétisons le problème de manière uniforme dans la variable de Floquet-Bloch et utilisons une méthode spectrale dans la discrétisation spatiale. La discrétisation en espace exploite une reformulation volumétrique du problème dans une cellule (équation intégrale de Lippmann-Schwinger) et une périodisation du noyau dans la direction perpendiculaire à la périodicité. Cette dernière transformation permet d'utiliser des techniques de type FFT pour accélérer le produit matrice-vecteur dans une méthode itérative pour résoudre le système linéaire. On aboutit à un système d'équations intégrales couplées (à cause de la perturbation locale) qui peuvent être résolues en utilisant une décomposition de Jacobi. L'analyse de la convergence est faite seulement dans le cas avec absorption et la validation numérique est réalisées sur des exemples 2D. Pour le problème inverse, nous étendons l'utilisation de trois méthodes d'échantillonnage pour résoudre le problème de la reconstruction de la géométrie du défaut à partir de la connaissance de données mutistatiques associées à des ondes incidentes planes en champ proche (c.à.d incluant certains modes évanescents). Nous analysons ces méthodes pour le problème semi-discrétisée dans la variable Floquet-Bloch. Nous proposons ensuite une nouvelle méthode d'imagerie capable de visualiser directement la géométrie du défaut sans savoir ni les propriétés physiques du milieux périodique, ni les propriétés physiques du défaut. Cette méthode que l'on appelle imagerie-différentielle est basée sur l'analyse des méthodes d'échantillonnage pour un seul mode de Floquet-Bloch et la relation avec les solutions de problèmes de transmission intérieurs d'un type nouveau. Les études théoriques sont corroborées par des expérimentations numériques sur des données synthétiques. Notre analyse est faite d'abord pour l'équation d'onde scalaire où le contraste est sur le terme d'ordre inférieur de l'opérateur de Helmholtz. Nous esquissons ensuite l'extension aux cas où la le contraste est également présent dans l'opérateur principal. Nous complémentons notre travail par deux résultats sur l'analyse du problème de diffraction pour des matériaux périodiques ayant des indices négatifs. Nous établissons en premier le caractère bien posé du problème en 2D dans le cas d'un contraste est égal à -1. Nous montrons également le caractère Fredholm de la formulation Lipmann-Schwinger du problème en utilisant l'approche de T-coercivité dans le cas d'un contraste différent de -1. / We are interested in this thesis by the analysis of scattering and inverse scattering problems for locally perturbed periodic infinite layers at a fixed frequency. This problem has connexions with non destructive testings of periodic media like photonics structures, optical fibers, gratings, etc. We first analyze the forward scattering problem and establish some conditions under which there exist no guided modes. This type of conditions is important as it shows that measurements can be done on a layer above the structure without loosing substantial informations in the propagative part of the wave. We then propose a numerical method that solves the direct scattering problem based on Floquet-Bloch transform in the periodicity directions of the background media. We discretize the problem uniformly in the Floquet-Bloch variable and use a spectral method in the space variable. The discretization in space exploits a volumetric reformulation of the problem in a cell (Lippmann-Schwinger integral equation) and a periodization of the kernel in the direction orthogonal to the periodicity. The latter allows the use of FFT techniques to speed up Matrix-Vector product in an iterative to solve the linear system. One ends up with a system of coupled integral equations that can be solved using a Jacobi decomposition. The convergence analysis is done for the case with absorption and numerical validating results are conducted in 2D. For the inverse problem we extend the use of three sampling methods to solve the problem of retrieving the defect from the knowledge of mutistatic data associated with incident near field plane waves. We analyze these methods for the semi-discretized problem in the Floquet-Bloch variable. We then propose a new method capable of retrieving directly the defect without knowing either the background material properties nor the defect properties. This so-called differential-imaging functional that we propose is based on the analysis of sampling methods for a single Floquet-Bloch mode and the relation with solutions toso-called interior transmission problems. The theoretical investigations are corroborated with numerical experiments on synthetic data. Our analysis is done first for the scalar wave equation where the contrast is the lower order term of the Helmholtz operator. We then sketch the extension to the cases where the contrast is also present in the main operator. We complement our thesis with two results on the analysis of the scattering problem for periodic materials with negative indices. Weestablish the well posedness of the problem in 2D in the case of a contrast equals -1. We also show the Fredholm properties of the volume potential formulation of the problem using the T-coercivity approach in the case of a contrast different from -1.

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