Development and Validation of a Method of Moments approach for modeling planar antenna structures

In this dissertation, a Method of Moments (MoM) Volume Integral Equation (VIE)-based modeling approach suitable for a patch or slot antenna on a thin finite dielectric substrate is developed and validated. Two new key features of this method are the use of proper dielectric basis functions and proper VIE conditioning, close to the metal surface, where the surface boundary condition of the zero tangential-component must be extended into adjacent tetrahedra. The extended boundary condition is the exact result for the piecewise-constant dielectric basis functions. The latter operation allows one to achieve a good accuracy with one layer of tetrahedra for a thin dielectric substrate and thereby greatly reduces computational cost. The use of low-order basis functions also implies the use of low-order integration schemes and faster filling of the impedance matrix. For some common patch/slot antennas, the VIE-based modeling approach is found to give an error of about 1% or less in the resonant frequency for one-layer tetrahedral meshes with a relatively small number of unknowns. This error is obtained by comparison with fine finite- element method (FEM) simulations, or with measurements, or with the analytical mode matching approach. Hence it is competitive with both the method of moments surface integral equation approach and with the FEM approach for the printed antennas on thin dielectric substrates. Along with the MoM development, the dissertation also presents the models and design procedures for a number of practical antenna configurations. They in particular include: i. a compact linearly polarized broadband planar inverted-F antenna (PIFA); ii. a circularly polarized turnstile bowtie antenna. Both the antennas are designed to operate in the low UHF band and used for indoor positioning/indoor geolocation

Fast boundary element methods for the simulation of wave phenomena

This thesis is concerned with the efficient implementation of boundary element methods (BEM) for their application in wave problems. BEM present a particularly useful tool, since they reduce the dimension of the problems by one, resulting in much fewer unknowns. However, this comes at the cost of dense system matrices, whose entries require the integration of singular kernel functions over pairs of boundary elements. Because calculating these four-dimensional integrals by cubature rules is expensive, a novel approach based on singularity cancellation and analytical integration is proposed. In this way, the dimension of the integrals is reduced and closed formulae are obtained for the most challenging cases. This allows for the accurate calculation of the matrix entries while requiring less computational work compared with conventional numerical integration. Furthermore, a new algorithm based on hierarchical low-rank approximation is presented, which compresses the dense matrices and improves the complexity of the method. The idea is to collect the matrices corresponding to different time steps in a third-order tensor and to approximate individual sub-blocks by a combination of analytic and algebraic low-rank techniques. By exploiting the low-rank structure in several ways, the method scales almost linearly in the number of spatial degrees of freedom and number of time steps. The superior performance of the new method is demonstrated in numerical examples.Diese Arbeit befasst sich mit der effizienten Implementierung von Randelementmethoden (REM) für ihre Anwendung auf Wellenprobleme. REM stellen ein besonders nützliches Werkzeug dar, da sie die Dimension der Probleme um eins reduzieren, was zu weit weniger Unbekannten führt. Allerdings ist dies mit vollbesetzten Matrizen verbunden, deren Einträge die Integration singulärer Kernfunktionen über Paare von Randelementen erfordern. Da die Berechnung dieser vierdimensionalen Integrale durch Kubaturformeln aufwendig ist, wird ein neuer Ansatz basierend auf Regularisierung und analytischer Integration verfolgt. Auf diese Weise reduziert sich die Dimension der Integrale und es ergeben sich geschlossene Formeln für die schwierigsten Fälle. Dies ermöglicht die genaue Berechnung der Matrixeinträge mit geringerem Rechenaufwand als konventionelle numerische Integration. Außerdem wird ein neuer Algorithmus beruhend auf hierarchischer Niedrigrangapproximation präsentiert, der die Matrizen komprimiert und die Komplexität der Methode verbessert. Die Idee ist, die Matrizen der verschiedenen Zeitpunkte in einem Tensor dritter Ordnung zu sammeln und einzelne Teilblöcke durch eine Kombination von analytischen und algebraischen Niedrigrangverfahren zu approximieren. Durch Ausnutzung der Niedrigrangstruktur skaliert die Methode fast linear mit der Anzahl der räumlichen Freiheitsgrade und der Anzahl der Zeitschritte. Die überlegene Leistung der neuen Methode wird anhand numerischer Beispiele aufgezeigt

Simulation de la propagation d'ondes électromagnétiques en nano-optique par une méthode Galerkine discontinue d'ordre élevé

The goal of this thesis is to develop a discontinuous Galerkin time-domain method to be able to handle realistic nanophotonics computations. During the last decades, the evolution of lithography techniques allowed the creation of geometrical structures at the nanometer scale, thus unveiling a variety of new phenomena arising from light-matter interactions at such levels. These effects usually occur when the device is of comparable size or (much) smaller than the wavelength of the incident field. This work relies on the development and implementation of appropriate models for dispersive materials (mostly metals), as well as on a large panel of classical computational techniques. Two major methodological developments are presented and studied in details: (i) curvilinear elements, and (ii) local order of approximation. This work is complemented with several physical studies of real-life nanophotonics applications.L’objectif de cette thèse est de développer une méthode Galerkine discontinue d’ordre élevé capable de prendre en considération des simulations réalistes liées à la nanophotonique. Au cours des dernières décennies, l’évolution des techniques de lithographie a permis la création de structure géométriques de tailles nanométriques, révélant ainsi une large gamme de phénomènes nouveaux nés de l’interaction lumière-matière à ces échelles. Ces effets apparaissent généralement pour des objets de taille égale ou (très) inférieure à la longueur d’onde du champ incident. Ce travail repose sur le développement et l’implémentation de modèles de dispersion appropriés (principalement pour les métaux), ainsi que sur un large éventail de méthodes computationnelles classiques. Deux développements méthodologiques majeurs sont présentés et étudiés en détails: (i) les éléments courbes, et (ii) l’ordre d’approximation local. Ces études sont accompagnées de plusieurs cas-tests réalistes tirés de la nanophotonique

Optimal higher order modeling methodology based on method of moments and finite element method for electromagnetics

2011 Fall.Includes bibliographical references.General guidelines and quantitative recipes for adoptions of optimal higher order parameters for computational electromagnetics (CEM) modeling using the method of moments and the finite element method are established and validated, based on an exhaustive series of numerical experiments and comprehensive case studies on higher order hierarchical CEM models of metallic and dielectric scatterers. The modeling parameters considered are: electrical dimensions of elements (subdivisions) in the model (h-refinement), polynomial orders of basis and testing functions (p-refinement), orders of Gauss-Legendre integration formulas (numbers of integration points - integration accuracy), and geometrical orders of elements (orders of Lagrange-type curvature) in the model. The goal of the study, which is the first such study of higher order parameters in CEM, is to reduce the dilemmas and uncertainties associated with the great modeling flexibility of higher order elements, basis and testing functions, and integration procedures (this flexibility is the principal advantage but also the greatest shortcoming of the higher order CEM), and to ease and facilitate the decisions to be made on how to actually use them, by both CEM developers and practitioners. The ultimate goal is to close the large gap between the rising academic interest in higher order CEM, which evidently shows great numerical potential, and its actual usefulness and application to electromagnetics research and engineering applications

An unsteady, accelerated, high order panel method with vortex particle wakes

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2006.Includes bibliographical references (leaves 125-138).Potential flow solvers for three dimensional aerodynamic analysis are commonly used in industrial applications. The limitation on the number of discretization elements and the user expertise and effort required to specify the wake location are two significant drawbacks preventing the even more widespread use of these codes. These drawbacks are addressed by the hands off, accelerated, unsteady, panel method with vortex particle wakes which is described. In the thesis, an unsteady vortex particle representation of the domain vorticity is coupled to several boundary element method potential flow formulations. Source-doublet, doublet-Neumann membrane (doublet lattice), and source-Neumann boundary integral equation formulations are implemented. A precorrected-FFT accelerated Krylov subspace iterative solution technique is implemented to efficiently solve the boundary element method linear system of equations. Similarly, a Fast Multipole Tree algorithm is used to accelerate the vortex particle interactions. Additional simplification of the panel method setup is achieved through the introduction of a body piercing wake discretization for lifting bodies with thickness.(cont.) Linear basis functions on flat panel surface triangulations are implemented in the accelerated potential flow framework. The advantages of linear order basis functions outweigh the increased complexity of the implementation when compared with traditional constant collocation approaches. Panel integration approaches for the curved panel, double layer self term are presented. A quadratic curved panel, quadratic basis function, Green's theorem direct potential flow solver is presented.by David Joe Willis.Ph.D

Electromagnetic analysis of 2.5D structures in open layered media

This thesis presents a specialization of the integral equation (IE) method for the analysis of three-dimensional metallic and dielectric structures embedded in laterally unbounded (open) layered media. The method remains basically spatial but makes use of extensive analytical treatment of the vertical dependence of the problem in the 2D Fourier-transformed domain. The analytical treatment restricts somewhat the class of structures that can be analyzed. Still, the field of applicability remains very large, and includes most printed circuit and integrated circuit structures. The method is developed in full numerical detail, from first principles down to the properties of new Green's functions and the computation of particular types of convolution integrals. We show how the memory and time complexity are considerably reduced when compared to the requirements of the analysis of general 3D structures. With the newly developed tool, it is possible to deal with some peculiar characteristics of microwave and millimeter-wave circuits and antennas. Most noteworthy among these is the presence of thick metallizations (either electrically, or relative to circuit features). A novel full-wave analysis of arbitrarily shaped apertures in thick metallic screens is presented. This is compared to other methods, both full-wave and approximate, and demonstrated to offer excellent accuracy. Comparison with measured data, obtained from specially constructed prototypes, further validates the new technique. A second application is to the analysis of airbridges in coplanar waveguide (CPW) and slotline (SL) circuits. Comparison of measured and simulated data validates again our technique and provides valuable information about the behavior of CPW-fed slot loop antennas. Among the more specific applications, particular attention is devoted to the analysis and design of submillimeter-wave integrated dielectric lens feeds. These were object of study in the frame of a European Space Agency project, Integrated Front-End Receivers (IFER), which our Laboratory carried out in cooperation with a team at University of Toronto. The analysis method developed in this work encompasses and extends all previous work done at our Laboratory (LEMA) related with the analysis of this kind of feed. Together with the advanced 3D ray-tracing code developed at University of Toronto, it is possible to gain a high degree of insight into the behavior of these integrated receivers

Fast, Parallel Techniques for Time-Domain Boundary Integral Equations

This work addresses the question of the efficient numerical solution of time-domain boundary integral equations with retarded potentials arising in the problems of acoustic and electromagnetic scattering. The convolutional form of the time-domain boundary operators allows to discretize them with the help of Runge-Kutta convolution quadrature. This method combines Laplace-transform and time-stepping approaches and requires the explicit form of the fundamental solution only in the Laplace domain to be known. Recent numerical and analytical studies revealed excellent properties of Runge-Kutta convolution quadrature, e.g. high convergence order, stability, low dissipation and dispersion. As a model problem, we consider the wave scattering in three dimensions. The convolution quadrature discretization of the indirect formulation for the three-dimensional wave equation leads to the lower triangular Toeplitz system of equations. Each entry of this system is a boundary integral operator with a kernel defined by convolution quadrature. In this work we develop an efficient method of almost linear complexity for the solution of this system based on the existing recursive algorithm. The latter requires the construction of many discretizations of the Helmholtz boundary single layer operator for a wide range of complex wavenumbers. This leads to two main problems: the need to construct many dense matrices and to evaluate many singular and near-singular integrals. The first problem is overcome by the use of data-sparse techniques, namely, the high-frequency fast multipole method (HF FMM) and H-matrices. The applicability of both techniques for the discretization of the Helmholtz boundary single-layer operators with complex wavenumbers is analyzed. It is shown that the presence of decay can favorably affect the length of the fast multipole expansions and thus reduce the matrix-vector multiplication times. The performance of H-matrices and the HF FMM is compared for a range of complex wavenumbers, and the strategy to choose between two techniques is suggested. The second problem, namely, the assembly of many singular and nearly-singular integrals, is solved by the use of the Huygens principle. In this work we prove that kernels of the boundary integral operators $w_n^h(d)$ ($h$ is the time step and $t_n=nh$ is the time) exhibit exponential decay outside of the neighborhood of $d=nh$ (this is the consequence of the Huygens principle). The size of the support of these kernels for fixed $h$ increases with $n$ as $n^a,a<1$, where $a$ depends on the order of the Runge-Kutta method and is (typically) smaller for Runge-Kutta methods of higher order. Numerical experiments demonstrate that theoretically predicted values of $a$ are quite close to optimal. In the work it is shown how this property can be used in the recursive algorithm to construct only a few matrices with the near-field, while for the rest of the matrices the far-field only is assembled. The resulting method allows to solve the three-dimensional wave scattering problem with asymptotically almost linear complexity. The efficiency of the approach is confirmed by extensive numerical experiments