Thesis (PhD (Mathematical Sciences. Applied Mathematics))--University of Stellenbosch, 2008. / This dissertation contains a detailed report on the results of a research project on the behaviour of a dynamical system consisting of a hoop to which a heavy particle is fixed at the rim. This loaded hoop rolls on a rough surface while remaining in the vertical plane. The motion of the hoop consists of various, possibly alternating, phases consisting of rolling without slipping, spinning or skidding motion and in some cases ends by hopping off the surface. A general mathematical model is developed, consisting of a system of second order ordinary differential equations, one for each of the three degrees of freedom. Analytic solutions are obtained in some cases; otherwise numerical solutions are used. Three specific applications of the general model are dealt with. In the first application the problem of massless hoops is investigated. The main emphasis is on the somewhat controversial question of what happens after the normal reaction becomes zero in a position where the particle is still moving downwards. A new result shows that the hoop can continue to move horizontally in a motion defined as skimming. The second application deals with rigid hoops and a large number of detailed results are presented. Classification schemes for the different types of behaviour are introduced and summarised in the form of phase diagrams. Some emphasis is placed on the rather amazing number of different patterns of motion that can be obtained by varying the parameters. In the third application two elastic models are analysed, with the primary purpose of explaining one aspect of the reported behaviour of experimental hoops, namely hopping while the particle is moving downwards. A chapter on experimental models rounds off the project.

Thesis (PhD (Mathematical Sciences. Applied Mathematics))--University of Stellenbosch, 2006. / This thesis addresses the problem of bulding a video-based traffic monitoring system. We employ clustering, trackiing and three-dimensional reconstruction of moving objects over a long image sequence. We present an algorithms that robustly recovers the motion and reconstructs three-dimensional shapes from a sequence of video images, Magaia et al [91]. The problem ...

Thesis (PhD)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: See full text for abstract / AFRIKAANSE OPSOMMING: Sien volteks vir opsomming

Thesis (PhD)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: Different generalized Newtonian fluids (where the normal stresses were neglected) were considered in this study. Analytical expressions were derived for time independent, fully developed velocity profiles of Herschel-Bulkley fluids (including the simplifications thereof: Newtonian, power law and Bingham plastic fluids) and Casson fluids through open channel sections. Both flow through cylindrical pipes (Hagen-Poiseuille flow) and parallel plates (plane Poiseuille flow) were brought under consideration. Equations were derived for the wall shear stresses in terms of the average channel velocities. These expressions for plane Poiseuille flow were then utilized in the modelling of flow through homogeneous, isotropic porous media. Flow through parallel plates was extended and a possibility of a moving lower wall (plane Couette-Poiseuille flow) was included for Herschel-Bulkley fluids (and the simplifications thereof). The velocity of the wall was assumed to be opposite to the pressure gradient (thus in the streamwise direction) yielding three different possible flow scenarios. These equations were again revisited in the study on flow over porous structures. Averaging of the microscopic momentum transport equation was carried out by means of volume averaging over an REV (Representative Elementary Volume). Flow through parallel plates enclosing a homogeneous porous medium (assumed homogeneous up to the external boundary) was studied at the hand of Brinkman’s equation. It was as- sumed (also for non-Newtonian fluids) that the term dominating outside the external boundary layer area is directly proportional to the superficial velocity that is, since only the viscous flow regime was considered, referred to as the ‘Darcy’ velocity if the diffusive Brinkman term is completely neglected. For a shear thinning or shear thickening fluid, the excess superficial velocity term was included in the proportionality coefficient that is constant for a particular fluid traversing a particular porous medium subjected to a specific pressure gradient. For such fluids only the inverse functions could be solved. If the ‘Darcy’ velocity is not reached within the considered domain, Gauss’s hypergeo- metric function had to be utilized. For Newtonian and Bingham plastic fluids, direct solutions were obtained. The effect of the constant yield stress was embedded in the proportionality coefficient. For linear flow, the proportionality coefficient consists of both a Darcy and a Forch- heimer term applicable to the viscous and inertial flow regimes respectively. Secondary averaging for different types of porous media was accomplished by using an RUC (Representative Unit Cell) to estimate average interstitial properties. Only homoge- neous, isotropic media were considered. Expressions for the apparent permeability as well as the passability in the Forchheimer regime (also sometimes referred to as the non-Darcian permeability) were derived for the various fluid types. Finally fluid flow in a domain consisting of an open channel adjacent to an infinite porous domain is considered. The analytically derived velocity profiles for both plane Couette- Poiseuille flow and the Brinkman equation were matched by assuming continuity in the shear stress at the porosity jump between the two domains. An in-house code was developed to simulate such a composite domain numerically. The difference between the analytically assumed constant apparent permeability in a macro- scopic boundary layer region as opposed to a dependency of the varying superficial velocity was discussed. This code included the possibility to alter the construction of the domain and to simulate axisymmetrical flow in a cylinder. / AFRIKAANSE OPSOMMING: Verskeie veralgemeende Newtoniese vloeistowwe (waarvan die normaalspannings ignoreer- baar is) word in hierdie studie beskou. Analitiese uitdrukkings vir tyd-onafhanklike, ten volle ontwikkelde snelheidsprofiele vir Herschel-Bulkley vloeistowwe (wat die vereen- voudigde weergawes daarvan insluit: Newtoniese, magswet- en Bingham-plastiek vloei- stowwe), sowel as Casson vloeistowwe, is afgelei vir vloei deur ‘n oop kanaal. Beide vloei deur silindriese pype (Hagen-Poiseuille vloei) en parallelle plate (vlak-Poiseuille vloei) is oorweeg. Vergelykings vir die skuifspannings op ‘n wand in terme van die gemiddelde snelhede is afgelei. Hierdie uitdrukking wat vir vlak-Poiseuille vloei verkry is, is in die modellering van vloei deur homogene, isotropiese poreuse media ook gebruik. Vloei deur parallelle plate is uitgebrei en die moontlikheid van ‘n bewegende onderste wand (vlak-Couette-Poiseuille vloei) is ondersoek vir Herschel-Bulkley vloeistowwe (en die vereenvoudigings daarvan). Dit word aangeneem dat die snelheid van die wand in die teenoorgestelde rigting as die drukgradiënt georiënteer is (dus in die stroomgewyse rigting) wat dan tot drie verskillende moontlike vloeigevalle lei. Hierdie vergelykings is weer in die studie van vloei oor poreuse strukture gebruik. Die gemiddelde van die mikroskopiese momentum transportvergelyking is bereken oor die volume van ‘n REV (“Representative Elementary Volume”). Vloei deur parallelle plate wat ‘n homogene poreuse medium omsluit (waar die medium homogeen aanvaar word tot by die eksterne grens) is bestudeer aan die hand van Brinkman se vergelyking. Daar is aanvaar (ook vir nie-Newtoniese vloeistowwe) dat die dominante term buite die eksterne grenslaaggebied direk eweredig is aan die oppervlaksnelheid en, aangesien slegs vloei in die viskeuse gebied oorweeg word, daarna verwys word as die “Darcy”- snelheid, indien die diffusiewe Brinkman-term heeltemal weglaatbaar is. Vir ‘n span-ningsverdunnende of -verdikkende vloeistof, word die oortollige oppervlaksnelheidsterm ingesluit by die proporsionaliteitskoëffisiënt wat konstant is vir ‘n spesifieke vloeistof wat deur ‘n sekere poreuse medium, onderhewig aan ‘n spesifieke drukgradiënt, vloei. Vir sulke vloeistowwe kon slegs die inverse funksies opgelos word. As die “Darcy”- snelheid nie binne die betrokke gebied bereik word nie, is daar van Gauss se hipergeometriese funksie gebruik gemaak. Vir Newtoniese en Bingham-plastiek vloeistowwe is egter direkte oplossings verkry. Die effek van die konstante toegeespanning is ingebed in die proporsionaliteitskoëffisiënt. Vir lineêre vloei bestaan die proporsionaliteitskoëffisiënt uit beide ‘n Darcy- en ‘n Forch- heimer-term wat van toepassing is in die viskeuse- en traagheidsvloeigebiede onder- skeidelik. Sekondˆere gemiddeldes vir verskillende tipes poreuse media is verkry; deur gebruik te maak van ‘n RUC (“Representative Unit Cell”) kan interstisiële gemiddelde eienskappe geskat word. Slegs homogene, isotrope media is in oorweging gebring. Uit- drukkings vir die o¨enskynlike deurlaatbaarheid sowel as die deurdringbaarheid in die Forchheimer-gebied (ook soms na verwys as die nie-Darcy deurlaatbaarheid) is afgelei vir die verskillende vloeistoftipes. Ten slotte is vloeistofvloei in ‘n gebied wat bestaan uit ‘n oop kanaal aangrensend aan ‘n oneindige poreuse domein ondersoek. Die analities-afgeleide snelheidsprofiele vir beide vlak-Couette-Poiseuille vloei en die Brinkman-vergelyking is gekoppel deur ‘n kontinu¨ıteit in die skuifspanning by die poreuse-sprong tussen die twee gebiede te aanvaar. ‘n Interne numeriese kode is ontwikkel om so ‘n saamgestelde domein numeries te simuleer. Die verskil tussen die analities konstant-aanvaarde deurlaatbaarheid in ‘n makroskopiese grenslaagstreek, eerder as ‘n afhanklikheid met die veranderende opper- vlaksnelheid, is bespreek. Hierdie kode sluit ook die moontlikheid in om die domein te herkonstrueer, asook om die simulasie van aksiaal-simmetriese vloei in ‘n silinder te ondersoek.

Thesis (MSc (Mathematical Sciences. Applied Mathematics))--University of Stellenbosch, 2007. / South Africa is involved in huge mining operations deep in the earth's crust. Stresses induced by these mining operations may cause seismic events or rockbursts to occur, which could damage infrastructure and put miners' lives at risk. The effect of different mining layouts are modelled and used by engineers to make design decisions. The frequency at which models are updated and integrated with the decision making process is not optimal. These large mining layouts can not be modelled adequately using domain methods, but they are particularly well suited for the boundary element method (BEM). This work focuses on the theory and background needed for creating a linear elastic static stress boundary element solver suited to South African mining layouts. It starts with linear elastic theory and subsequently describes the physical continuum, governing equations and the fundamental solutions which are an integral part of the BEM. Kelvin's solution cannot be applied to crack-like excavations, therefore the displacement discontinuity kernels, which are very well suited to model fractures, are derived. The derivation is approached from both the direct and indirect BEM's perspectives. The problem is cast as a boundary integral equation which can be solved using the BEM. Some of the different specializations of the BEM are discussed. The major drawback of the BEM is that it produces a dense influence matrix which quickly becomes intractable on desktop computers. Generally a mining layout requires a large amount of boundary elements, even for coarse discretization, therefore different techniques of representing the influence matrix are discussed, which, combined with an iterative solver like GMRES or Bi-CG, allows solving linear elastic static stress models.

Thesis (MSc (Applied Mathematics))--Stellenbosch University, 2008. / A popular method for interpolating multidimensional scattered data is using radial basis functions. In this thesis we present the basic theory of radial basis function interpolation and also regard the solvability and stability of the method. Solving the interpolant directly has a high computational cost for large datasets, hence using numerical methods to approximate the interpolant is necessary. We consider some recent numerical algorithms. Software to implement radial basis function interpolation and to display the 3D interpolants obtained, is developed. We present results obtained from using our implementation for radial basis functions on GIS and 3D face data as well as an image warping application.

Thesis (MSc (Mathematical Sciences. Applied Mathematics))--University of Stellenbosch, 2006. / In this work we present a novel method to do measurement correlation between target observations made by two ormore radar systems. Some of the most common radar sensors available are those measuring only range (distance to the target) and bearing (azimuth angle). We use these measurements to determine the correlation between two di¤erent sensors observing the same target. As a by-product of the correlation algorithm, we nd a way to estimate the target height for a target observed by at least two radar sensors. The correlation method is expounded upon, where we discuss measurement correlation for moving targets. Targets are tracked using a Kalman Filter, and correlation is done between new observations and existing target tracks. Finally, the correlation algorithm is implemented in an interactive 3D computer simulation. Results obtained indicate a high success rate, with false correlations only obtained where sensor accuracy is the limiting factor.

Thesis (MScEng (Mathematical Sciences. Applied Mathematics))--University of Stellenbosch, 2006. / An analytical modelling procedure for predicting the streamwise pressure gradient for steady laminar incompressible flow of a Newtonian fluid through homogeneous isotropic granular porous media is introduced. The modelling strategy involves the spatial volume averaging of a statistical representative portion of the porous domain to obtain measurable macroscopic quantities from which macroscopic transport equations can be derived. A simple pore-scale model is introduced to approximate the actual complex granular porous microstructure through rectangular cubic geometry. The sound physical principles on which the modelling procedure is based avoid the need for redundant empirical coefficients. The model is generalized to predict the rheological flow behaviour of non-Newtonian purely viscous power law fluids by introducing the dependence of the apparent viscosity on the shear rate through the wall shear stress. The field of application of the Newtonian model is extended to predict the flow behaviour in fluidized beds by adjusting the Darcy velocity to incorporate the relative velocity of the solid phase. The Newtonian model is furthermore adjusted to predict fluid flow through Fontainebleau sandstone by taking into account the effect of blocked throats at very low porosities. The analytical model as well as the model generalizations for extended applicability is verified through comparison with other analytical and semi-empirical models and a wide range of experimental data from the literature. The accuracy of the predictive analytical model reveals to be highly acceptable for most engineering designs.

Thesis (MSc (Mathematical Sciences. Applied Mathematics))--University of Stellenbosch, 2007. / Image morphing is a popular technique used to create spectacular visual effects, by gradually transforming one image into another. This thesis explains what exactly is meant by the terms “image morphing” / “warping”, where it is used and how it is done. A few existing morphing techniques are described and finally an implementation using Delaunay triangulation and texture mapping is presented.

Thesis (MSc (Mathemathical Sciences))--Stellenbosch University, 2009. / We have the familiar Kuratowski-Mr owka theorem in topology, where compactness is characterised by a closure and a projection-map (X is compact i p : X Y ! Y is a closed mapping, for any space Y , i.e. p(A) = p(A) A X Y ). Using this as our starting point, we generalise compactness to a categorical setting. We then generalise even further to "asymmetric" compactness. Then we discuss a functional approach to compactness, where we do not explicitly mention closure operators. All this provides economical proofs as well as applications in di erent areas of mathematics.