The Finite Element Tearing and Interconnecting (FETI) and its variants are probably the most celebrated domain decomposition algorithms for partial differential equation (PDE) scientific computations. In electromagnetics, such methods have advanced research frontiers by enabling the full-wave analysis and design of finite phased array antennas, metamaterials, and other multiscale structures. Recently, closer scrutiny of these methods have revealed robustness and numerical scalability problems that prevent the most memory and time efficient variants of FETI from gaining widespread acceptance. This work introduces a new class of FETI methods and preconditioners that lead to exponential iterative convergence for a wide class of problems, are robust and numerically scalable. First, a two Lagrange multiplier (LM) variant of FETI with impedance transmission conditions, the FETI-2λ, is introduced to facilitate the symmetric treatment of non-conforming grids while avoiding matrix singularites that occur at the interior resonance frequencies of the domains. A thorough investigation on the approximability and stability of the Lagrange multiplier discrete space is carried over to identify the correct LM space basis. The resulting method, although accurate and flexible, exhibits unreliable iterative convergence. To accelerate the iterative convergence, the Locally Exact Algebraic Preconditioner (LEAP), which is responsible for improving the information transfer between neighboring domains is introduced. The LEAP was conceived by carefully studying the properties of the Dirichlet-to-Neumann (DtN) map that is involved in the sub-structuring process of FETI. LEAP proceeds in a hierarchical way and directly factorizes the signular and near-singular interactions of the DtN map that arise from domain-face, domain-edge and domain-vertex interactions. For problems with small number of domains LEAP results in scalable implementations with respect to the discretization. On problems with large domain numbers, the numerical scalability can only be obtained through ``global'' preconditioners that directly convey information to remotely separated domains at every DDM iteration. The proposed ``global" preconditiong stage is based on the new Multigrid FETI (MG-FETI) method. This method provides a coarse grid correction mechanism defined in the dual space. Macro-basis functions, that satisfy thecurl-curl equation on each interface are constructed to reduce the size of the coarse problem, while maintaining a good approximation of the characteristic field modes. Numerical results showcase the performance of the proposed method on one-way, 2D and 3D decomposed problems, with structured and unstructured partitioning, conforming and non-conforming interface triangulations. Finally, challenging, real life computational examples showcase the true potential of the method.

Accurate modeling and simulation of wave propagation in dispersive dielectrics such as water, human tissue and sand, among others, has a variety of applications. For example in medical imaging, electromagnetic waves are used to interrogate human tissue in a non-invasive manner to detect anomalies that could be cancerous. In non-destructive evaluation of materials, such interrogation is used to detect defects in these materials. In this thesis we present the construction and analysis of two novel operator splitting methods for Maxwell's equations in dispersive media of Debye type which are used to model wave propagation in polar materials like water and human tissue. We construct a sequential and a symmetrized operator splitting scheme which are first order, and second order, respectively, accurate in time. Both schemes are second order accurate in space. The operator splitting methods are shown to be unconditionally stable via energy techniques. Their accuracy and stability properties are compared to established schemes like the Yee or FDTD scheme and the Crank-Nicolson scheme. Finally, results of numerical simulations are presented that confirm the theoretical analysis. / Graduation date: 2012 / Access restricted to the OSU Community at author's request from June 20, 2012 - Dec. 20, 2012

Thesis (Ph. D.)--Ohio State University, 2007. / Title from first page of PDF file. Includes bibliographical references (p. 127-134).

Made available in DSpace on 2016-12-12T20:15:53Z (GMT). No. of bitstreams: 1 Resumo - Denilso.pdf: 6952 bytes, checksum: d64faf1cec322aeb51d49ed61bf9358e (MD5) Previous issue date: 2009-05-29 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work the two-component spinor formulation of the classical theory of electromagnetic fields is presented. In particular, we obtain explicitly the wave equa-tion for photons of both helicities. For this purpose, we present first the formulation of the theory in Minkowski spacetime together with the homomorphism between SL(2;C) and the restricted Lorentz group. / Neste trabalho apresentaremos a formulação da teoria eletromagnética clássica em termos de espinores de duas componentes. Em particular, obteremos explicitamente as equações de onda para fotons de ambas helicidades. Para isso, primeiro trataremos explicitamente da formulação covariante da teoria eletromagnética clássica. Explicitaremos também o homomorfismo entre o grupo SL(2,C) e o grupo de Lorentz restrito.

ork tie together on first project, in which the was designed apparatus for tests wireless transmission energy. Antenna was synthesized and theoretically optimized for experimental operation, in which the will performed series measuring. In first part is practical description of realization proposal. It consists of circuital solution, proposal measuring workplace and mechanical construction with illustration photographs of arrangement. Second part includes results of performed measuring. These measuring will divided by three basic groups: in first group will by testing feature one’s antennae, in second group will series transmission measuring power gain, and in third group will photographed shape of field, and comparison with simple simulation finite difference method. Third part summarises results from measuring, and prepares consecutive balancing visualisation project, which is of thematic bent on chosen aspects theoretical hypothesis and effected experiments.

The Maxwell equations of electromagnetism form the foundation of classical electromagnetism, and are of interest to mathematicians, physicists, and engineers alike. The first part of this thesis concerns boundary value problems for the anisotropic Maxwell equations in Lipschitz domains. In this case, the material parameters that arise in the Maxwell system are matrix valued functions. Using methods from functional analysis, global in time solutions to initial boundary value problems with general nonzero boundary data and nonzero current density are obtained, only assuming the material parameters are bounded and measurable. This problem is motivated by an electromagnetic inverse problem, similar to the classical Calder\'on inverse problem in Electrical Impedance Tomography. The second part of this thesis deals with materials having negative refractive index. Materials which possess a negative refractive index were postulated by Veselago in 1968, and since 2001 physicists were able to construct these materials in the laboratory. The research on the behavior of these materials, called metamaterials, has been extremely active in recent years. We study here refraction problems in the setting of Negative Refractive Index Materials (NIMs). In particular, it is shown how to obtain weak solutions (defined similarly to Brenier solutions for the Monge-Amp\`ere equation) to these problems, both in the near and the far field. The far field problem can be treated using Optimal Transport techniques; as such, a fully nonlinear PDE of Monge-Amp\`ere type arises here. / Mathematics

Thesis (MScEng (Electrical and Electronic Engineering))--University of Stellenbosch, 2007. / The finite element tearing and interconnect(FETI) domain decomposition(DD) method is investigated in terms of the 2D transverse electric(TEz) finite element method(FEM). The FETI is for the first time rigorously derived using the weighted residual framework from which important insights are gained. The FETI is used in a novel way to implement a total-/scattered field decomposition and is shown to give excellent results. The FETI is newly formulated for the time domain(FETI-TD), its feasibility is tested and it is further formulated and tested for implementation on a distributed computer architecture.

In this research, a new technique for modeling periodic structures in the finite-difference time-domain (FDTD) method is developed, and the technique is applied to the study of structural color in insects. Various recent supplements to the FDTD method, such as a nearly-perfect plane-wave injector and convolutional perfectly matched layer boundary condition, are used. A method for implementing the FDTD method on a parallel, distributed-memory computer cluster is given. To model a periodic structure, a single periodic cell is terminated by periodic boundary conditions (PBCs). A new technique for incorporating PBCs into the FDTD method is presented. The simplest version of the technique is limited to two-dimensional, singly-periodic geometries. The accuracy is demonstrated by comparing to independent results calculated with a frequency-domain, mode-matching method. The periodic FDTD method is then extended to the more general case of three-dimensional, doubly-periodic problems. This extension requires additional steps and imposes new limitations. The computational cost and limitations of the method are presented. Certain species of butterflies exhibit structural color, which is caused by quasi-periodic structures on the scales covering the wings. Numerical experiments are performed to develop a technique for modeling quasi-periodic structures using the periodic FDTD method. The observed structural color of butterflies is then calculated from the electromagnetic data using colorimetric theory. Three types of butterflies are considered. The first type are from the Morpho genus. These are typically a brilliant, almost metallic, blue color. The second type is the Troides magellanus, which exhibits an interplay of structural and pigmentary color, but the structural color is only visible near grazing incidence. The final type is the Ancyluris meliboeus, which exhibits iridescence on the ventral side. For all cases, the effects of changing the dimensions of various structural elements are considered. Finally, some earlier work on the design of TEM horn antennas is presented. The TEM horn is a simple and popular antenna, but only limited design information is available in the literature. A parametric study was performed, and the results are given. A complete derivation of the characteristic impedance of the basic antenna is also presented.

V klasické fyzice m·že být ustavena statická rovnováha v soustavě, která obsahuje extrémně nabité zdroje gravitačního a elektromagnetického pole. Udivujícím faktem je, že tato situace m·že nastat i pro černé díry v relativis- tické fyzice. Tato práce vyšetřuje speciální případ nekonečně dlouhé, extrémně nabité struny, zkoumá geometrii prostoročasu, elektrogeodetiky, vlastnosti zdroje a srovnává řešení se situací v klasické fyzice. Dále se zabýváme analogickou situací v dynamickém prostoročase s kosmologickou konstantou, a řešení porovnáváme s jeho statickou verzí. Nakonec zkoumáme periodické řešení Laplaceovy rovnice, které odpovídá nekonečně mnoha extremálním bodovým zdroj·m rozloženým v pravidelném rozestupu podél přímky. Vyšetřujeme vlastnosti elektrostatického potenciálu a ukazujeme, že v limitě velké vzdálenosti od osy tvořené zdroji pře- chází toto řešení v nabitou strunu. 1

Mes travaux concernent deux systèmes d’équations utilisés dans la modélisation mathématique de semi-conducteurs et de plasmas : le système d’Euler-Poisson et le système d’Euler-Maxwell. Le premier système est constitué des équations d’Euler pour la conservation de la masse et de la quantité de mouvement couplées à l’équation de Poisson pour le potentiel électrostatique. Le second système décrit le phénomène d’électro-magnétisme. C’est un système couplé, qui est constitué des équations d’Euler pour la conservation de la masse et de la quantité de mouvement et les équations de Maxwell, aussi appelées équations de Maxwell-Lorentz. Les équations de Maxwell sont dues aux lois fondamentales de la physique. Elles constituent les postulats de base de l’électromagnétisme, avec l’expression de la force électromagnétique de Lorentz. En utilisant une technique de développement asymptotique, nous étudions les limites en zéro du système d’Euler-Poisson dans les modèles unipolaire et bipolaire. Il est bien connu que la limite formelle du système d’Euler-Poisson est gouvernée par les équations de dérive-diffusion lorsque le temps de relaxation tend vers zéro. Par des estimations d’énergie aux systèmes hyperboliques symétriques, nous justifions rigoureusement cette limite lorsque les conditions initiales sont bien préparées. Le phénomène des conditions initiales mal préparées est interprété par l’apparition de couches initiales. Dans ce cas, nous faisons une analyse mathématique de ces couches initiales en ajoutant des termes de correction dans le développement asymptotique. En utilisant les techniques itératives des systèmes hyperboliques symétrisables et la technique de développement asymptotique, nous étudions la limite de relaxation en zéro du système d’Euler-Maxwell, avec des conditions initiales bien préparées ainsi que l’étude des couches initiales, dans le modèle évolutif bipolaire et unipolaire. / My work is concerned with two different systems of equations used in the mathematical modeling of semiconductors and plasmas : the Euler-Poisson system and the Euler-Maxwell system. The first is given by the Euler equations for the conservation of the mass and momentum, with a Poisson equation for the electrostatic potential. The second system describes the phenomenon of electromagnetism. It is given by the Euler equations for the conservation of the mass and momentum, with a Maxwell equations for the electric field and magnetic field which are coupled to the electron density through the Maxwell equations and act on electrons via the Lorentz force. Using an asymptotic expansion method, we study the zero relaxation limit of unipolar Euler-Poisson system and of two-fluid multidimensional Euler-Poisson equations, we prove the existence and uniqueness of profiles to the asymptotic expansion and some error estimate. By employing the classical energy estimate for symmetrizable hyperbolic equations, we justify rigorously the convergence of Euler-Poisson system with well-prepared initial data. For ill-prepared initial data, the phenomenon of initial layers occurs. In this case, we also add the correction terms in the asymptotic expansion. Using an iterative method of symmetrizable hyperbolic systems and asymptotic expansion method, we study the zero-relaxation limit of unipolar and bipolar Euler-Maxwell system. For well-prepared initial data, we construct an approximate solution by an asymptotic expansion up to any order. For ill-prepared initial data, we also construct initial layer corrections in the asymptotic expansion.