Computer simulation of electromagnetic behaviour of a device is a common practice in modern engineering. Maxwell's equations are solved on a computer with help of numerical methods. Contemporary devices constantly grow in size and complexity. Therefore, new numerical methods should be highly efficient. Many industrial and research applications of numerical methods need to account for the frequency dependent materials. The Finite-Difference Time-Domain (FDTD) method is one of the most widely adopted algorithms for the numerical solution of Maxwell's equations. A major drawback of the FDTD method is the interdependence of the spatial and temporal discretisation steps, known as the Courant-Friedrichs-Lewy (CFL) stability criterion. Due to the CFL condition the simulation of a large object with delicate geometry will require a high spatio-temporal resolution everywhere in the FDTD grid. Application of subgridding increases the efficiency of the FDTD method. Subgridding decomposes the simulation domain into several subdomains with different spatio-temporal resolutions. The research project described in this dissertation uses the Huygens Subgridding (HSG) method. The frequency dependence is included with the Auxiliary Differential Equation (ADE) approach based on the one-pole Debye relaxation model. The main contributions of this work are (i) extension of the one-dimensional (1D) frequency-dependent HSG method to three dimensions (3D), (ii) implementation of the frequency-dependent HSG method, termed the dispersive HSG, in Fortran 90, (iii) implementation of the radio environment setting from the PGM-files, (iv) simulation of the electromagnetic wave propagating from the defibrillator through the human torso and (v) analysis of the computational requirements of the dispersive HSG program.

Many systems designed by electrical & computer engineers rely on electromagnetic (EM) signals to transmit, receive, and extract either information or energy. In many cases, these systems are large and complex. Their accurate, cost-effective design requires high-fidelity computer modeling of the underlying EM field/material interaction problem in order to find a design with acceptable system performance. This modeling is accomplished by projecting the governing Maxwell equations onto finite dimensional subspaces, which results in a large matrix equation representation (Zx = b) of the EM problem. In the case of integral equation-based formulations of EM problems, the M-by-N system matrix, Z, is generally dense. For this reason, when treating large problems, it is necessary to use compression methods to store and manipulate Z. One such sparse representation is provided by so-called H^2 matrices. At low-to-moderate frequencies, H^2 matrices provide a controllably accurate data-sparse representation of Z. The scale at which problems in EM are considered ``large'' is continuously being redefined to be larger. This growth of problem scale is not only happening in EM, but respectively across all other sub-fields of computational science as well. The pursuit of increasingly large problems is unwavering in all these sub-fields, and this drive has long outpaced the rate of advancements in processing and storage capabilities in computing. This has caused computational science communities to now face the computational limitations of standard linear algebraic methods that have been relied upon for decades to run quickly and efficiently on modern computing hardware. This common set of algorithms can only produce reliable results quickly and efficiently for small to mid-sized matrices that fit into the memory of the host computer. Therefore, the drive to pursue larger problems has even began to outpace the reasonable capabilities of these common numerical algorithms; the deterministic numerical linear algebra algorithms that have gotten matrix computation this far have proven to be inadequate for many problems of current interest. This has computational science communities focusing on improvements in their mathematical and software approaches in order to push further advancement. Randomized numerical linear algebra (RandNLA) is an emerging area that both academia and industry believe to be strong candidates to assist in overcoming the limitations faced when solving massive and computationally expensive problems. This thesis presents results of recent work that uses a random sampling method (RSM) to implement algebraic operations involving multiple H^2 matrices. Significantly, this work is done in a manner that is non-invasive to an existing H^2 code base for filling and factoring H^2 matrices. The work presented thus expands the existing code's capabilities with minimal impact on existing (and well-tested) applications. In addition to this work with randomized H^2 algebra, improvements in sparse factorization methods for the compressed H^2 data structure will also be presented. The reported developments in filling and factoring H^2 data structures assist in, and allow for, the further pursuit of large and complex problems in computational EM (CEM) within simulation code bases that utilize the H^2 data structure.

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The main objective of this research is to model and analyse complex electromagnetic problems by means of a new hybridised computational technique combining the frequency domain Method of Moments (MoM), Finite-Difference Time-Domain (FDTD) method and a subgridded Finite-Difference Time-Domain (SGFDTD) method. This facilitates a significant advance in the ability to predict electromagnetic absorption in inhomogeneous, anisotropic and lossy dielectric materials irradiated by geometrically intricate sources. The Method of Moments modelling employed a two-dimensional electric surface patch integral formulation solved by independent linear basis function methods in the circumferential and axial directions of the antenna wires. A similar orthogonal basis function is used on the end surface and appropriate attachments with the wire surface are employed to satisfy the requirements of current continuity. The surface current distributions on structures which may include closely spaced parallel wires, such as dipoles, loops and helical antennas are computed. The results are found to be stable and showed good agreement with less comprehensive earlier work by others. The work also investigated the interaction between overhead high voltage transmission lines and underground utility pipelines using the FDTD technique for the whole structure, combined with a subgridding method at points of interest, particularly the pipeline. The induced fields above the pipeline are investigated and analysed. FDTD is based on the solution of Maxwell's equations in differential form. It is very useful for modelling complex, inhomogeneous structures. Problems arise when open-region geometries are modelled. However, the Perfectly Matched Layer (PML) concept has been employed to circumvent this difficulty. The establishment of edge elements has greatly improved the performance of this method and the computational burden due to huge numbers of time steps, in the order of tens of millions, has been eased to tens of thousands by employing quasi-static methods. This thesis also illustrates the principle of the equivalent surface boundary employed close to the antenna for MoM-FDTD-SGFDTD hybridisation. It depicts the advantage of using hybrid techniques due to their ability to analyse a system of multiple discrete regions by employing the principle of equivalent sources to excite the coupling surfaces. The method has been applied for modelling human body interaction with a short range RFID antenna to investigate and analyse the near field and far field radiation pattern for which the cumulative distribution function of antenna radiation efficiency is presented. The field distributions of the simulated structures show reasonable and stable results at 900 MHz. This method facilitates deeper investigation of the phenomena in the interaction between electromagnetic fields and human tissues.

Thesis (PhD)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: In this work the Method of Manufactured Solutions (MMS) is introduced for the code veri cation of full-wave frequency dependent electromagnetic computational software. At rst the method is sketched in the context of the veri cation and validation process and the need for proper code veri cation is highlighted. Subsequently, the MMS is investigated in its natural context: the Finite Element Method, speci cally for the E- eld Vector Wave Equation. The usefulness of the method to detect error in a computational code is demonstrated. The selection of Manufactured Solutions is discussed and it is demonstrated how it can be used to nd the probable cause of bugs. Mutation testing is introduced and used to show the ability to detect errors present in code. The MMS is nally applied in a novel manner to a Method of Moments (MoM) code. The challenges of numerical integration associated with the application of the operator is discussed and correct integration is successfully demonstrated. Subsequently the MMS is demonstrated to be successfully applied to the MoM and mutation testing is used to demonstrate the practical e cacy of the method. The application of the MMS to the MoM is the main contribution of this work. / AFRIKAANSE OPSOMMING: Die Metode van Vervaardigde Oplossings (MVO) word hier bekend gestel vir die veri kasie van numeriese volgolf frekwensie-afhanklike elektromagnetise kode. Die metode word eerstens in die bre e konteks van algemene veri kasie en validasie geplaas en gevolglik word die noodsaaklikheid van kode veri kasie beklemtoon. Daarna, word die toets-metode in die konteks van die Eindige Element Metode vir die E-veld vektorgolf vergelyking bestudeer. Die MVO is oorspronklik ontwikkel in die di erentiaalvergelyking omgewing. Die bruikbaarheid van die metode vir elektromagnetiese simulasies word prakties gedemonstreer deur die opsporing van werklike foute. Die metode word ook verder ondersoek vir die oorsprong van foute. Mutasietoetsing word bekendgestel en word gebruik om die metode verder prakties te veri eer. Die MVO word laastens in 'n nuwe manier gebruik om 'n Moment Metode kode te veri eer. Die praktiese probleme betrokke by numeriese integrasie word ondersoek en die korrekte toepassing van die integraal operator word prakties gedemonstreer. Daarna, word die MVO in hierdie konteks gedemonstreer deur verskeie voorbeelde te ondersoek. Mutasietoetsing word weereens gebruik om na die e ektiewiteit van die MVO te kyk om 'n Moment Metode kode te toets. Die toepassing van die MVO op 'n Moment Metode kode is die hoof bydrae van hierdie werk.

Thesis (PhD)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: The parallel FDTD method as used in computational electromagnetics is implemented on a variety of different high performance computing platforms. These parallel FDTD implementations have regularly been compared in terms of performance or purchase cost, but very little systematic consideration has been given to how much effort has been used to create the parallel FDTD for a specific computing architecture. The deployment effort for these platforms has changed dramatically with time, the deployment time span used to create FDTD implementations in 1980 ranging from months, to the contemporary scenario where parallel FDTD methods can be implemented on a supercomputer in a matter of hours. This thesis compares the effort required to deploy the parallel FDTD on selected computing platforms from the constituents that make up the deployment effort, such as coding complexity and time of coding. It uses the deployment and performance of the serial FDTD method on a single personal computer as a benchmark and examines the deployments of the parallel FDTD using different parallelisation techniques. These FDTD deployments are then analysed and compared against one another in order to determine the common characteristics between the FDTD implementations on various computing platforms with differing parallelisation techniques. Although subjective in some instances, these characteristics are quantified and compared in tabular form, by using the research information created by the parallel FDTD implementations. The deployment effort is of interest to scientists and engineers considering the creation or purchase of an FDTD-like solution on a high performance computing platform. Although the FDTD method has been considered to be a brute force approach to solving computational electromagnetic problems in the past, this was very probably a factor of the relatively weak computing platforms which took very long periods to process small model sizes. This thesis will describe the current implementations of the parallel FDTD method, made up of a combination of several techniques. These techniques can be easily deployed in a relatively quick time frame on computing architectures ranging from IBM’s Bluegene/P to the amalgamation of multicore processor and graphics processing unit, known as an accelerated processing unit. / AFRIKAANSE OPSOMMING: Die parallel Eindige Verskil Tyd Domein (Eng: FDTD) metode word gebruik in numeriese elektromagnetika en kan op verskeie hoë werkverrigting rekenaars geïmplementeer word. Hierdie parallele FDTD implementasies word gereeld in terme van werkverrigting of aankoop koste vergelyk, maar word bitter min sistematies oorweeg in terme van die hoeveelheid moeite wat dit geverg het om die parallele FDTD vir 'n spesifieke rekenaar argitektuur te skep. Mettertyd het die moeite om die platforms te ontplooi dramaties verander, in the 1980's het die ontplooings tyd tipies maande beloop waarteenoor dit vandag binne 'n kwessie van ure gedoen kan word. Hierdie tesis vergelyk die inspanning wat nodig is om die parallelle FDTD op geselekteerde rekenaar platforms te ontplooi deur te kyk na faktore soos die kompleksiteit van kodering en die tyd wat dit vat om 'n kode te implementeer. Die werkverrigting van die serie FDTD metode, geïmplementeer op 'n enkele persoonlike rekenaar word gebruik as 'n maatstaf om die ontplooing van die parallel FDTD met verskeie parallelisasie tegnieke te evalueer. Deur hierdie FDTD ontplooiings met verskillende parallelisasie tegnieke te ontleed en te vergelyk word die gemeenskaplike eienskappe bepaal vir verskeie rekenaar platforms. Alhoewel sommige gevalle subjektief is, is hierdie eienskappe gekwantifiseer en vergelyk in tabelvorm deur gebruik te maak van die navorsings inligting geskep deur die parallel FDTD implementasies. Die ontplooiings moeite is belangrik vir wetenskaplikes en ingenieurs wat moet besluit tussen die ontwikkeling of aankoop van 'n FDTD tipe oplossing op 'n höe werkverrigting rekenaar. Hoewel die FDTD metode in die verlede beskou was as 'n brute krag benadering tot die oplossing van elektromagnetiese probleme was dit waarskynlik weens die relatiewe swak rekenaar platforms wat lank gevat het om klein modelle te verwerk. Hierdie tesis beskryf die moderne implementering van die parallele FDTD metode, bestaande uit 'n kombinasie van verskeie tegnieke. Hierdie tegnieke kan maklik in 'n relatiewe kort tydsbestek ontplooi word op rekenaar argitekture wat wissel van IBM se BlueGene / P tot die samesmelting van multikern verwerkers en grafiese verwerkings eenhede, beter bekend as 'n versnelde verwerkings eenheid.

Thesis (PhD (Electrical and Electronic Engineering))--University of Stellenbosch, 2011. / ENGLISH ABSTRACT: This work considers the acceleration of matrix-based computational electromagnetic (CEM) techniques using graphics processing units (GPUs). These massively parallel processors have gained much support since late 2006, with software tools such as CUDA and OpenCL greatly simplifying the process of harnessing the computational power of these devices. As with any advances in computation, the use of these devices enables the modelling of more complex problems, which in turn should give rise to better solutions to a number of global challenges faced at present. For the purpose of this dissertation, CUDA is used in an investigation of the acceleration of two methods in CEM that are used to tackle a variety of problems. The first of these is the Method of Moments (MOM) which is typically used to model radiation and scattering problems, with the latter begin considered here. For the CUDA acceleration of the MOM presented here, the assembly and subsequent solution of the matrix equation associated with the method are considered. This is done for both single and double precision oating point matrices. For the solution of the matrix equation, general dense linear algebra techniques are used, which allow for the use of a vast expanse of existing knowledge on the subject. This also means that implementations developed here along with the results presented are immediately applicable to the same wide array of applications where these methods are employed. Both the assembly and solution of the matrix equation implementations presented result in signi cant speedups over multi-core CPU implementations, with speedups of up to 300x and 10x, respectively, being measured. The implementations presented also overcome one of the major limitations in the use of GPUs as accelerators (that of limited memory capacity) with problems up to 16 times larger than would normally be possible being solved. The second matrix-based technique considered is the Finite Element Method (FEM), which allows for the accurate modelling of complex geometric structures including non-uniform dielectric and magnetic properties of materials, and is particularly well suited to handling bounded structures such as waveguide. In this work the CUDA acceleration of the cutoff and dispersion analysis of three waveguide configurations is presented. The modelling of these problems using an open-source software package, FEniCS, is also discussed. Once again, the problem can be approached from a linear algebra perspective, with the formulation in this case resulting in a generalised eigenvalue (GEV) problem. For the problems considered, a total solution speedup of up to 7x is measured for the solution of the generalised eigenvalue problem, with up to 22x being attained for the solution of the standard eigenvalue problem that forms part of the GEV problem. / AFRIKAANSE OPSOMMING: In hierdie werkstuk word die versnelling van matriksmetodes in numeriese elektromagnetika (NEM) deur die gebruik van grafiese verwerkingseenhede (GVEe) oorweeg. Die gebruik van hierdie verwerkingseenhede is aansienlik vergemaklik in 2006 deur sagteware pakette soos CUDA en OpenCL. Hierdie toestelle, soos ander verbeterings in verwerkings vermoe, maak dit moontlik om meer komplekse probleme op te los. Hierdie stel wetenskaplikes weer in staat om globale uitdagings beter aan te pak. In hierdie proefskrif word CUDA gebruik om ondersoek in te stel na die versnelling van twee metodes in NEM, naamlik die Moment Metode (MOM) en die Eindige Element Metode (EEM). Die MOM word tipies gebruik om stralings- en weerkaatsingsprobleme op te los. Hier word slegs na die weerkaatsingsprobleme gekyk. CUDA word gebruik om die opstel van die MOM matriks en ook die daaropvolgende oplossing van die matriksvergelyking wat met die metode gepaard gaan te bespoedig. Algemene digte lineere algebra tegnieke word benut om die matriksvergelykings op te los. Dit stel die magdom bestaande kennis in die vagebied beskikbaar vir die oplossing, en gee ook aanleiding daartoe dat enige implementasies wat ontwikkel word en resultate wat verkry word ook betrekking het tot 'n wye verskeidenheid probleme wat die lineere algebra metodes gebruik. Daar is gevind dat beide die opstelling van die matriks en die oplossing van die matriksvergelyking aansienlik vinniger is as veelverwerker SVE implementasies. 'n Verselling van tot 300x en 10x onderkeidelik is gemeet vir die opstel en oplos fases. Die hoeveelheid geheue beskikbaar tot die GVE is een van die belangrike beperkinge vir die gebruik van GVEe vir groot probleme. Hierdie beperking word hierin oorkom en probleme wat selfs 16 keer groter is as die GVE se beskikbare geheue word geakkommodeer en suksesvol opgelos. Die Eindige Element Metode word op sy beurt gebruik om komplekse geometriee asook nieuniforme materiaaleienskappe te modelleer. Die EEM is ook baie geskik om begrensde strukture soos golfgeleiers te hanteer. Hier word CUDA gebruik of om die afsny- en dispersieanalise van drie gol eierkonfigurasies te versnel. Die implementasie van hierdie probleme word gedoen deur 'n versameling oopbronkode wat bekend staan as FEniCS, wat ook hierin bespreek word. Die probleme wat ontstaan in die EEM kan weereens vanaf 'n lineere algebra uitganspunt benader word. In hierdie geval lei die formulering tot 'n algemene eiewaardeprobleem. Vir die gol eier probleme wat ondersoek word is gevind dat die algemene eiewaardeprobleem met tot 7x versnel word. Die standaard eiewaardeprobleem wat 'n stap is in die oplossing van die algemene eiewaardeprobleem is met tot 22x versnel.

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.