Higher-order basis functions are widely used to model currents and fields in numerical simulations of electromagnetics problems because of the greater accuracy and computational efficiency they can provide. Different problem formulations, such as method of moments (MoM) and the finite element method (FEM) require different constraints on basis functions for optimal performance, such as normal or tangential continuity between cells. In this thesis, a method of automatically generating bases that satisfy the desired basis constraints is applied to a MoM formulation for scattering problems using surface integral equations. Numerical results demonstrate the accuracy of this approach, and show good system matrix conditioning when compared to other higher-order bases.

[pt] O objetivo principal deste trabalho é o desenvolvimento de modelos computacionais para analisar a resposta eletromagnética de ferramentas de perfilagem LWD/MWD em formações geofísicas arbitrárias. Essa modelagem envolve a determinação precisa de campos eletromagnéticos em regiões tridimensionais (3D) complexas e, conseqüentemente, a solução de sistemas lineares não-hermitianos de larga escala. A modelagem numérica é realizada através da aplicação do método dos volumes finitos (FVM) no domínio da freqüência. Desenvolvem-se dois modelos computacionais, o primeiro válido em regiões isotrópicas e o segundo considerando a presença de anisotropias no meio. As equações de Maxwell são resolvidas através de duas formulações distintas: formulação por campos e formulação por potenciais vetor e escalar. A discretização por volumes finitos utiliza um esquema de grades entrelaçadas em coordenadas cilíndricas para evitar erros de aproximação de escada da geometria da ferramenta. Os modelos desenvolvidos incorporam quatro técnicas numéricas para aumentar a eficiência computacional e a precisão do método. As formulações por campos e por potenciais vetor e escalar são comparadas em termos da taxa de convergência e do tempo de processamento em cenários tridimensionais. Os modelos foram validados e testados em cenários tridimensionais complexos, tais como: (i) poços horizontais ou direcionais; (ii) formações não homogêneas com invasões de fluído de perfuração; (iii) formações anisotrópicas e (iv) poços excêntricos. Motivado pela flexibilidade dos modelos e pelos resultados numéricos obtidos em diferentes cenários tridimensionais, estende-se a metodologia para analisar a resposta de ferramentas LWD que empregam antenas inclinadas em relação ao eixo da ferramenta. Tais ferramentas podem prover dados com sensibilidade azimutal, assim como estimativas da anisotropia da formação, auxiliando o geodirecionamento de poços direcionais e horizontais. / [en] The main objective of this work is to develop computational models to analyze electromagnetic logging-while-drilling tool response in arbitrary geophysical formations. This modeling requires the determination of electromagnetic fields in three- dimensional (3-D) complex regions and consequently, the solution of large scale non-hermitian systems. The numerical modeling is done by using Finite Volume Methods (FVM) in the frequency domain. Both isotropic and anisotropic models are developed. Maxwell's equations are solved by using both the field formulation and the coupled vector-scalar potentials formulation. The proposed FVM technique utilizes an edge-based staggered-grid scheme in cylindrical coordinates to avoid staircasing errors on the tool geometry. Four numerical techniques are incorporated in the models in order to increase the computational efficiency and the accuracy of the method. The field formulation and the coupled vector-scalar potentials formulation are compared in terms of their accuracy, convergence rate, and CPU time for three-dimensional environments. The models were validated and tested in 3-D complex environments, such as:(i) horizontal and directional boreholes; (ii) multilayered geophysical formations including mud-filtrate invasions; (iii) anisotropic formations and (iv)eccentric boreholes. The methodology is extended to analyze LWD tools that are constructed with the transmitters and/or receivers tilted with respect to the axis of the drill collar. Such tools can provide improved anisotropy measurements and azimuthal sensitivity to benefit geosteering.

The phenomenon of polarization interaction in certain nonlinear materials is presented, and the design of an all-optical logic device based on this concept is described. An efficient two-dimensional finite-difference time-domain code for studying third-order nonlinear optical phenomena is discussed, in which both the slowly varying and the rapidly varying components of the electromagnetic fields are considered. The algorithm solves the vector form Maxwell s equations for all field components and uses the nonlinear constitutive relation in matrix form as the equations required to describe the nonlinear system. The stability of the code is discussed and its accuracy demonstrated through the simulation of the self-phase modulation effect observed in Kerr media. Finally, the code is used to simulate polarization mixing in photonic crystal-based line defect and coupled resonator optical waveguides.

In the thesis, we propose a newly-developed method called the method of Connected Local Fields (CLF) to analyze opto-electromagnetic passive devices. The method of CLF somewhat resembles a hybrid between the finite difference and pseudo-spectral methods. For opto-electromagnetic passive devices, our primary concern is their steady state behavior, or narrow-band characteristics, so we use a frequency-domain method, in which the system is governed by the Helmholtz equation. The essence of CLF is to use the intrinsic general solution of the Helmholtz equation to expand the local fields on the compact stencil. The original equation can then be transformed into the discretized form called LFE-9 (in 2-D case), and the intrinsic reconstruction formulae describing each overlapping local region can be obtained. Further, we present rigorous analysis of the numerical dispersion equation of LFE-9, by means of first-order approximation, and acquire the closed-form formula of the relative numerical dispersion error. We are thereby able to grasp the tangible influences brought both by the sampling density as well as the propagation direction of plane wave on dispersion error. In our dispersion analysis, we find that the LFE-9 formulation achieves the sixth-order accuracy: the theoretical highest order for discretizing elliptic partial differential equations on a compact nine-point stencil. Additionally, the relative dispersion error of LFE-9 is less than 1%, given that sampling density greater than 2.1 points per wavelength. At this point, the sampling density is nearing that of the Nyquist-Shannon sampling limit, and therefore computational efforts can be significantly reduced.

Optical whispering-gallery mode (WGM) cavities, which exhibit extraordinary spatial and temporal confinement of light, are one of the leading transducers for examining molecular recognition at low particle counts. With the advent of hybrid photonic-plasmonic and increasingly sophisticated forms of these resonators, the importance of supporting numerical methods has correspondingly become evident. In response, we adopt a full-vector finite difference approximation in order to solve for WGM's in terms of their field distributions, resonant wavelengths, and quality factors in the context of naturally discontinuous permittivity structure. A segmented Taylor series and alignment/rotation operator are utilized at such singularities in conjunction with arbitrarily spaced grid points. Simulations for microtoroids, with and without dielectric nanobeads, and plasmonic microdisks are demonstrated for short computation times and shown to be in agreement with data in the literature. Constricted surface plasmon polariton (SPP) WGM's are also featured within this document. The module of this thesis is devised as a keystone for composite WGM models that may guide experiments in the field. / Graduate

The Method of Moments (MoM) has been widely used for the full-wave electromagnetic analysis of planar multilayered media. However, the MoM is a computationally intensive process and requires considerable computer resources to perform the analysis. Thus, several efficient numerical techniques both in the spectral domain and spatial domain are investigated and further developed in this research. Two fitting procedures, i.e., the Rational Function Fitting Method (RFFM) and the Discrete Complex Image Method (DCIM), are investigated and developed in order to obtain closed-form spatial-domain Green's functions (GFs). Because the subtraction of the surface-wave pole contribution plays an important role for the accurate estimation of the spatial-domain GFs via DCIM, an efficient and accurate surface-wave pole location method is developed to find all the surface-wave poles for general multilayered media. The RFFM can be realized through either the Total Least Square Algorithm (TLSA) or the Vector Fitting (VECTFIT) method. Both the RFFM using VECTFIT and DCIM are detailed in step by step procedures. An efficient and low cost algorithm combining the advantages of DCIM and TLSA is also developed to evaluate the closed-form Green's functions for general multilayered media. A prototype version of the Full-Wave Layered-Interconnect Simulator (UA-FWLIS) was developed by analytically calculating the MoM reaction elements via Cauchy's residue theorem and the Complementary Incomplete Lipschitz-Hankel Integrals in stripline structures. After applying RFFM via VECTFIT to the G-functions, which are directly related to the spectral-domain GFs for microstrip structures, a procedure that is similar to the one used in the previously developed UA-FWLIS for stripline structures can be applied to calculate the MoM reaction elements analytically when the two reaction cells are close (< 0.1λ₀) in the spectral domain via the Electrical Field Integral Equation (EFIE). When the two reaction cells are far enough away (> 0.10.1λ₀), a simple expression for the reaction element can be obtained in the spatial domain via the Mixed Potential Integral Equation (MPIE) by a summation of a few complex image terms for the evaluation of the vector and scalar GFs. An efficient hybrid spectral-spatial method is thus developed to extend UA-FWLIS to microstrip structures. The method is validated by several numerical examples.

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.

Maxwell's equations are a system of partial differential equations of vector fields. Imposing the constitutive relations for material properties yields equations for the curl and divergence of the electric and magnetic fields. The curl and divergence equations must be solved simultaneously, which is not the same as solving three separate scalar problems in each component of the vector field. This thesis describes a new method for solving partial differential equations of vector fields using the finite element method. New basis functions are used to solve the curl equation while allowing the divergence to be set as a constraint. The basis functions are defined on a mesh of bricks and the method is applicable for geometries that conform to a Cartesian coordinate system. The basis functions are a combination of cubic Hermite splines and second order Lagrange interpolation polynomials. The method yields a linearly independent set of constraints for the divergence, which is modelled to second order accuracy within each brick. Mesh refinement is accomplished by dividing selected bricks into $2\times 2\times 2$ smaller bricks of equal size. The change in the node pattern at an interface where mesh refinement occurs necessitates a modified implementation of the divergence constraints as well as additional constraints for hanging nodes. The mesh can be refined to an arbitrary number of levels. The basis functions can exactly model the discontinuity in the normal component of the field at a planar interface. The method is modified to solve problems with singularities at material boundaries that form $90^{\circ}$ edges and corners. The primary test problem of the new basis functions is to obtain the resonant frequencies and fields of three-dimensional cavities. The new basis functions can resolve physical solutions and non-physical, spurious modes. The eigenvalues obtained with the new method are in good agreement with exact solutions and experimental values in cases where they exist. There is also good agreement with results from second-order edge elements that are obtained with the software package HFSS. Finally, the method is modified to solve problems in cylindrical coordinates provided the domain does not contain the coordinate axis.

Maxwell's equations are a system of partial differential equations of vector fields. Imposing the constitutive relations for material properties yields equations for the curl and divergence of the electric and magnetic fields. The curl and divergence equations must be solved simultaneously, which is not the same as solving three separate scalar problems in each component of the vector field. This thesis describes a new method for solving partial differential equations of vector fields using the finite element method. New basis functions are used to solve the curl equation while allowing the divergence to be set as a constraint. The basis functions are defined on a mesh of bricks and the method is applicable for geometries that conform to a Cartesian coordinate system. The basis functions are a combination of cubic Hermite splines and second order Lagrange interpolation polynomials. The method yields a linearly independent set of constraints for the divergence, which is modelled to second order accuracy within each brick. Mesh refinement is accomplished by dividing selected bricks into $2\times 2\times 2$ smaller bricks of equal size. The change in the node pattern at an interface where mesh refinement occurs necessitates a modified implementation of the divergence constraints as well as additional constraints for hanging nodes. The mesh can be refined to an arbitrary number of levels. The basis functions can exactly model the discontinuity in the normal component of the field at a planar interface. The method is modified to solve problems with singularities at material boundaries that form $90^{\circ}$ edges and corners. The primary test problem of the new basis functions is to obtain the resonant frequencies and fields of three-dimensional cavities. The new basis functions can resolve physical solutions and non-physical, spurious modes. The eigenvalues obtained with the new method are in good agreement with exact solutions and experimental values in cases where they exist. There is also good agreement with results from second-order edge elements that are obtained with the software package HFSS. Finally, the method is modified to solve problems in cylindrical coordinates provided the domain does not contain the coordinate axis.

Thesis (MEng)--Stellenbosch University, 2015. / ENGLISH ABSTRACT: The Method of Moments (MoM) is routinely used for the numerical solution of electromagnetic surface integral equations. Solution errors are inherent to any numerical computational method, and error estimators can be effectively employed to reduce and control these errors. In this thesis, gradient recovery techniques of the Finite Element Method (FEM) are formulated within the MoM context, in order to recover a higher-order charge of a Rao-Wilton-Glisson (RWG) MoM solution. Furthermore, a new recovery procedure, based specifically on the properties of the RWG basis functions, is introduced by the author. These recovered charge distributions are used for a posteriori error estimation of the charge. It was found that the newly proposed charge recovery method has the highest accuracy of the considered recovery methods, and is the most suited for applications within recovery based error estimation. In addition to charge recovery, the possibility of recovery procedures for the MoM solution current are also investigated. A technique is explored whereby a recovered charge is used to find a higher-order divergent current representation. Two newly developed methods for the subsequent recovery of the solenoidal current component, as contained in the RWG solution current, are also introduced by the author. A posteriori error estimation of the MoM current is accomplished through the use of the recovered current distributions. A mixed second-order recovered current, based on a vector recovery procedure, was found to produce the most accurate results. The error estimation techniques developed in this thesis could be incorporated into an adaptive solver scheme to optimise the solution accuracy relative to the computational cost. / AFRIKAANSE OPSOMMING: Die Moment Metode (MoM) vind algemene toepassing in die numeriese oplossing van elektromagnetiese oppervlak integraalvergelykings. Numeriese foute is inherent tot die prosedure: foutberamingstegnieke is dus nodig om die betrokke foute te analiseer en te reduseer. Gradiënt verhalingstegnieke van die Eindige Element Metode word in hierdie tesis in die MoM konteks geformuleer. Hierdie tegnieke word ingespan om die oppervlaklading van 'n Rao-Wilton-Glisson (RWG) MoM oplossing na 'n verbeterde hoër-orde voorstelling te neem. Verder is 'n nuwe lading verhalingstegniek deur die outeur voorgestel wat spesifiek op die eienskappe van die RWG basis funksies gebaseer is. Die verhaalde ladingsverspreidings is geïmplementeer in a posteriori fout beraming van die lading. Die nuut voorgestelde tegniek het die akkuraatste resultate gelewer, uit die groep verhalingstegnieke wat ondersoek is. Addisioneel tot ladingsverhaling, is die moontlikheid van MoM-stroom verhalingstegnieke ook ondersoek. 'n Metode vir die verhaling van 'n hoër-orde divergente stroom komponent, gebaseer op die verhaalde lading, is geïmplementeer. Verder is twee nuwe metodes vir die verhaling van die solenodiale komponent van die RWG stroom deur die outeur voorgestel. A posteriori foutberaming van die MoM-stroom is met behulp van die verhaalde stroom verspreidings gerealiseer, en daar is gevind dat 'n gemengde tweede-orde verhaalde stroom, gebaseer op 'n vektor metode, die beste resultate lewer. Die foutberamingstegnieke wat in hierdie tesis ondersoek is, kan in 'n aanpasbare skema opgeneem word om die akkuraatheid van 'n numeriese oplossing, relatief tot die berekeningskoste, te optimeer.