Dissertation (PhD)--University of Stellenbosch, 2002. / ENGLISH ABSTRACT: This document presents a Galerkin FE formulation for the full-wave, frequency domain,
electromagnetic analysis of three dimensional structures relevant to microwave engineering,
together with the investigation of two techniques to enhance the formulation's computational
efficiency. The first technique considered is the fast multi pole method (FMM) and the second
technique is adaptive refinement of the discretization, based on a posteriori error estimation.
Thus, the motivation for the work presented in this document is to increase the computational
efficiency of the FE formulation considered.
The FE formulation considered is widely used within the microwave engineering, finite element
community. Tetrahedral, rectilinear, curl-conforming, mixed- and full order, hierarchical
vector elements are used. The formulation is extended to incorporate a cavity backed
aperture employing the appropriate half-space Green function within a BI boundary condition,
which represents a specific member of a large class of hybrid FE-BI formulations. The
formulation is also extended to model coaxial ports via a Neumann boundary condition, using
a priori knowledge of the dominant modal fields. Results are presented in support of the
formulation and its extensions, including novel results on the coupling between microstrip
patch antennas on a perforated substrate.
The FMM is investigated first, with the purpose of optimizing the non-local BI component
of the cavity FE-BI formulation, in light of its coupling with the differential equation based,
sparse FEM. The FMM results in a partly sparse factorization of the BI contribution to
the system matrix. Error control schemes for the FMM are thoroughly reviewed and an
additional, novel scheme is empirically devised.
The second technique investigated, which is more directly related to the FEM and larger in
scope, is the use of a posteriori error estimation, in order to optimize the FE discretization
through adaptive refinement. A overview of available a posteriori error estimation techniques
in the general FE literature is given as well as a survey of available techniques that are
specifically tailored to Maxwell's equations. Two known approaches within the applied
mathematics literature are adapted to the FE formulation at hand, resulting in two novel,
residual based error estimation procedures for this FE formulation - one explicit in nature
and the other implicit. The two error estimators are then used to drive a single p adaptive
analysis cycle of the FE formulation, experimentally demonstrating their effectiveness. A
quasi-static condition is introduced and successfully used to enhance the adaptive algorithm's
effectiveness, independently of the error estimation procedure employed. The novel error
estimation schemes and adaptive results represent the main research contributions of this
study. / AFRIKAANSE OPSOMMING: Hierdie dokument beskryf 'n Galerkin eindige element (EE) formulering vir die volgolf, frekwensiegebied,
elektromagnetiese analise van driedimensionele strukture relevant vir mikrogolfingenieurwese,
saam met die ondersoek van twee tegnieke om die numeriese effektiwiteit
van die formulering te verbeter. Die eerste tegniek wat ondersoek word, is die vinnige multipooi
metode (VMM) en die tweede is die aanpasbare verfyning van die EE diskretisering,
gebaseer op a posteriori foutberaming. Dus, die motivering vir hierdie werk is om die numeriese
effektiwiteit van die genoemde EE formulering te verbeter.
Die bogenoemde EE formulering word algemeen gebruik deur die mikrogolfingenieurswese,
eindige element-gemeenskap. Tetrahedriese, reglynige, curl-ondersteunende, hierargiese vektorelemente
van gemengde- en volledige ordes word gebruik. Die formulering word uitgebrei
om holtes in 'n oneindige grondvlak te kan hanteer, deur gebruik te maak van die toepaslike
Green funksie binne 'n grensintegraal (GI) grensvoorwaarde, wat 'n spesifieke lid is van 'n
groot klas, hibriede, EE-GI formulerings. Die formulering word ook uitgebrei om koaksiale
poorte to modelleer via 'n Neumann grensvoorwaarde, deur die gebruik van a priori kennis
van die koaksiale, dominante modus-velde. Resultate word gelewer om die formulering, saam
met die uitbreidings daarvan, te ondersteun, insluitende oorspronklike resultate in verband
met die koppeling tussen mikrostrook plakantennes op 'n geperforeerde substraat.
Die VMM word eerste ondersoek, met die doelom die nie-lokale, GI komponent van die EEGI
formulering vir holtes te optimeer, weens die koppeling daarvan met die yl, differensiaalvergelyking-
gebaseerde, eindige element-metode. Die VMM lei tot 'n gedeeltelik-yl faktorisering
van die GI bydrae tot die algehele matriksvergelyking. Skemas om die VMM fout
te beheer word deeglik ondersoek en 'n addisionele, oorspronklike skema word empiries ontwikkel.
Die tweede tegniek wat ondersoek word, wat meer direk verband hou met die eindige elementmetode,
en van groter omvang is, is die gebruik van a posteriori foutberaming om die EE
diskretisasie te optimeer deur middel van aanpasbare verfyning. 'n Oorsig van beskikbare, a
posteriori foutberamingstegnieke in die algemene EE literatuur word gegee, asook 'n opname
van beskikbare tegnieke wat spesifiek gerig is op Maxwell se vergelykings. Twee bekende
benaderings binne die toegepaste wiskunde-literatuur word aangepas by die bogenoemde
EE formulering, wat lei tot twee oorspronklike residu-gebaseerde foutberamingstegnieke vir
hierdie formulering - een van 'n eksplisiete aard en die ander implisiet. Die twee foutberamingstegnieke
word gebruik om 'n enkel, p-aanpasbare analisesiklus aan te dryf, wat die
effektiwiteit van die foutberamingstegnieke eksperimenteel demonstreer. 'n Kwasi-statiese
vereiste word beskryf en suksesvol gebruik om die aanpasbare algoritme se effektiwiteit te
verhoog, onafhanklik van die foutberamingstegniek wat gebruik word. Die oorspronklike
foutberamingstegnieke en aanpasbare algoritme-resultate verteenwoordig die hoof navorsingsbydraes
van hierdie studie.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/52818 |
| Date | 09 1900 |
| Creators | Botha, Matthys Michiel |
| Contributors | Davidson, D. B., Stellenbosch University. Faculty of Engineering. Dept. of Electrical and Electronic Engineering. |
| Publisher | Stellenbosch : Stellenbosch University |
| Source Sets | South African National ETD Portal |
| Language | en_ZA |
| Detected Language | Unknown |
| Type | Thesis |
| Format | 184 p. : ill. |
| Rights | Stellenbosch University |
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