# 2D Edge-based finite elements for guided and scattered wave problems

Hansmann, Riana Helena (1999-03)

Thesis (MSc)--Stellenbosch University, 1999.

Thesis

ENGLISH ABSTRACT: This thesis may be divided into two parts: the first describes the Finite Element Method (FEM) and its application to guided wave problems. The second part is devoted to scattering configurations, specifically the use of the Boundary Element Method (BEM) and the hybrid Finite Element Method-Boundary Element Method (FEM-BEM) to obtain solutions for scattering problems. The formulations are restricted to two dimensions throughout the thesis. A variational formulation is introduced and the implementation of boundary conditions is described. The use of vector approximation functions for the Finite Element Method is explained and the advantages highlighted. The properties of these functions are derived and graphical representations are given. A comparison between a lower order and higher order approximation is made. This is applied to problems which demonstrate the capabilities of the Finite Element Method such as ridged waveguides and circular waveguides containing eccentric dielectric rods. Results obtained compare well to analytic solutions, in the cases where these are available. An integral equation for scattering problems is derived. This relates the tangential field components on a contour enclosing a scattering object to the scattered fields and enables a solution to be obtained when the tangential components on the contour are known. It is shown how the interior region enclosed by the contour is discretised and how the Finite Element Method can be coupled with the Boundary Element Method by imposing continuity conditions on the enclosing contour. The resulting system of equations obtained may be solved. Solutions for scattering from perfectly conducting cylinders are obtained and compare well to analytic results.

AFRIKAANSE OPSOMMING: Hierdie tesis bestaan uit twee dele: die eerste beskryf die toepassing van die Eindige Element Metode (EEM) vir golfieier probleme. Die tweede deel handel oor strooiings probleme en die toepassing van die hibriede Eindige Element-Randelement Metode om numeriese oplossings vir hierdie tipe probleme te vind. Die formulerings in hierdie tesis is deurgaans tweedimensioneel van aard. 'n Variasionele formulering word beskryf saam met die geassosieerde randvoorwaardes. Die gebruik van vektor basis funksies vir die Eindige Element Metode word beskryf en die voordele uitgelig. Die eienskappe van hierdie vektor funksies word grafies voorgestel en herlei. Verskillende orde benaderings is ge1mplimenteer en die akkuraatheid van die benaderings word ondersoek. Probleme wat die vermoe van die metode ten toon stel is gekies, byvoorbeeld golfleiers met riwwe en ronde golfleiers wat dielektriese silinders bevat. In die gevalle waar analitiese oplossing beskikbaar is, vergelyk die Eindige Element resultate goed met die analitiese oplossings. In die tweede deel van die tesis word 'n integraal vergelyking wat van toepassing is op strooiingsprobleme, afgelei. Die tangensiale veld-komponente op 'n kontoer wat 'n voorwerp omsluit word in verband gebring met die verstrooide veldkomponente elders in die omgewing rondorn die voorwerp. Die diskretisasie van die ornsluite gebied en die koppeling van die Eindige Element Metode met die Rand-Element Metode deur kontinulteitsvoorwaardes op die kontoer word bespreek. 'n Stelsel van vergelykings word gevorm wat opgelos kan word om die veldkomponente op die kontoer en sodoende die strooiingseffek van 'n voorwerp te bepaal. Die strooiingseffek van perfek geleidende silinders is bepaal en vergelyk goed met analitiese oplossings.

Please refer to this item in SUNScholar by using the following persistent URL: http://hdl.handle.net/10019.1/51089

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