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Thesis (Ph. D.)--University of Washington, 2000. / Vita. Includes bibliographical references (leaves 139-148).

This thesis consists of an introduction and four research papers related to free boundary problems and systems of fully non-linear elliptic equations. Paper A and Paper B prove optimal regularity of solutions to general elliptic and parabolic free boundary problems, where the operators are fully non-linear and convex. Furthermore, it is proven that the free boundary is continuously differentiable around so called "thick" points, and that the free boundary touches the fixed boundary tangentially in two dimensions. Paper C analyzes singular points of solutions to perturbations of the unstable obstacle problem, in three dimensions. Blow-up limits are characterized and shown to be unique. The free boundary is proven to lie close to the zero-level set of the corresponding blow-up limit. Finally, the structure of the singular set is analyzed. Paper D discusses an idea on how existence and uniqueness theorems concerning quasi-monotone fully non-linear elliptic systems can be extended to systems that are not quasi-monotone. / <p>QC 20151210</p>

The work of this thesis was undertaken as a C.A.S.E. award project in collaboration with Rolls-Royce to examine compressible laminar boundary layers with sharp adverse pressure-gradients. Much of the work is devoted to the solution of two important particular problems. The first flow considered is that along a semi-infinite flat plate with uniform pressure when X < X0 and with the pressure for X > X0 being so chosen that the boundary layer is just on the point of separation for all X > X0. Immediately downstream of X0 there is a sharp pressure rise to which the flow reacts mainly in a thin inner sublayer; so inner and outer asymptotic expansions are derived and matched for the stream function and a function of the temperature. Throughout the thesis the ratio of the viscosity to the absolute temperature is taken to be a function of x, the distance along the wall, alone, and the Illingworth-Stewartson transformation is applied. The Prandtl number, σ, is taken to be of order unity and detailed results are presented for σ= 1 and 0.72. The second flow considered is that along a finite flat plate where the transformed external velocity U1(X) is chosen such that U1(X) = u0(-X/L)[super]ε, where O< ε <<1, is the transformed length of the plate and X represents transformed distance downstream from the trailing edge. The skin friction, position of separation and heat transfer right up to separation are determined. On the basis of these two solutions, another solution which is not presented in detail, and a solution (due to Curie) to a fourth sharp pressure gradient problem, a general Stratford-type method for computing compressible boundary layers is derived, which may be used to predict the position of separation, skin friction, heat transfer, displacement and momentum thicknesses for a compressible boundary layer with an unfavourable pressure gradient. In all this work techniques of series analysis are used to good effect. This led us to look at another boundary-layer problem in which such techniques could be used, one in which two parallel infinite disks are initially rotating with angular velocity Ω about a common axis in incompressible fluid, the appropriate Reynolds number being very large. Suddenly the angular velocity of one of the disks is reversed. A new examination of this problem is presented in the appendix to the thesis.

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Abstract This thesis summarizes certain boundary element methods applied to some initial and boundary value problems. Our model problem is the two-dimensional homogeneous heat conduction problem with vanishing initial data. We use the heat potential representation of the solution. The given boundary conditions, as well as the choice of the representation formula, yield various boundary integral equations. For the sake of simplicity, we use the direct boundary integral approach, where the unknown boundary density appearing in the boundary integral equation is a quantity of physical meaning. We consider two different sets of boundary conditions, the Dirichlet problem, where the boundary temperature is given and the Neumann problem, where the heat flux across the boundary is given. Even a nonlinear Neumann condition satisfying certain monotonicity and growth conditions is possible. The approach yields a nonlinear boundary integral equation of the second kind. In the stationary case, the model problem reduces to a potential problem with a nonlinear Neumann condition. We use the spaces of smoothest splines as trial functions. The nonlinearity is approximated by using the L2-orthogonal projection. The resulting collocation scheme retains the optimal L2-convergence. Numerical experiments are in agreement with this result. This approach generalizes to the time dependent case. The trial functions are tensor products of piecewise linear and piecewise constant splines. The proposed projection method uses interpolation with respect to the space variable and the orthogonal projection with respect to the time variable. Compared to the Galerkin method, this approach simplifies the realization of the discrete matrix equations. In addition, the rate of the convergence is of optimal order. On the other hand, the Dirichlet problem, where the boundary temperature is given, leads to a single layer heat operator equation of the first kind. In the first approach, we use tensor products of piecewise linear splines as trial functions with collocation at the nodal points. Stability and suboptimal L2-convergence of the method were proved in the case of a circular domain. Numerical experiments indicate the expected quadratic L2-convergence. Later, a Petrov-Galerkin approach was proposed, where the trial functions were tensor products of piecewise linear and piecewise constant splines. The resulting approximative scheme is stable and convergent. The analysis has been carried out in the cases of the single layer heat operator and the hypersingular heat operator. The rate of the convergence with respect to the L2-norm is also here of suboptimal order.

[pt] Em análises efetuadas através de Métodos de Elementos de Contorno, o procedimento de integração exerce um papel fundamental, já que faz-se necessária a avaliação de integrais singulares e quase-singulares que introduzem erros numéricos nos resultados quando não são devidamente avaliadas. Nos últimos anos, vários pesquisadores sugeriram diferentes técnicas de integração para tratar os problemas de integração de uma forma adequada. Este trabalho inicia apresentando alguns conceitos básicos e uma revisão bibliográfica das principais técnicas sugeridas. Em seguida, apresenta-se uma técnica de integração unificada, que possui uma forma simples e oferece resultados com excelente precisão. A técnica proposta foi aplicada para integrais singulares ou quase- singulares possuindo pólos simples ou múltiplos, sendo que tanto integrais unidimensionais quanto integrais bidimensionais foram consideradas.Paralelamente ao estudo das integrais, foi desenvolvido um programa computacional em linguagem orientada a objetos (C++), destinado a implementar simultaneamente a formulação convencional e as formulações híbridas dos Métodos de Elementos de Contorno. Da forma em que foi planejado, a implementação suporta diferentes aplicações de análises de engenharia. Este programa resultou de um trabalho conjunto realizado com pesquisadores da Universidade de Stuttgart. Por fim, apresentam-se diversos exemplos numéricos e resultados de análises, ressaltando o bom desempenho da técnica proposta e a influência do procedimento de integração em análises através de Métodos de Elementos de Contorno. / [en] In Boundary Element Method analysis, the integration procedure is one of the most important tasks, since one has to deal with singular and quasi-singular integrals which introduce numerical errors in the results, if they are not evaluated adequately. In the last years, several researchers have suggested different techniques with the aim of handling the problem adequately. This work begins presenting some basic concepts and a review of the most important work published before. Following, it introduces a unified integration technique which has a simple form and provides highly accurate results. The proposed scheme also deals with one- or two-dimensional singular or quasi- singular integrals having single or multiple poles. Besides the study of the integrals, a computational code was developed using an objectoriented computer language (C++). This code takes into account the conventional and hybrid formulations of the Boundary Element Method and supports different types of engineering analysis. This computer program was developed in a frame of a joint project with some researchers from the University of Stuttgart.Finally, several numerical examples and analysis results are displayed, showing the good performance of the proposed technique and the influence of the integration task in analysis using Boundary Element Methods.

Magister Artium - MA / Ruacana is a town in northern Namibia, located on the border with Angola on the Kunene River. It is about 150 kilometres north of Oshakati. The town was established in the early 1970s by the South West Africa Water and Electricity Commission, to provide accommodation for the Ruacana Hydropower station staff. Having been established without forced removals, Ruacana was an ideal ‘apartheid town’ as only ‘white’ staff lived in the wall-fenced –off town. The ‘black’ staff, soldiers as well as those that provided services in the town, were accommodated in a nearby township known as Oshifo, A few years later, the South African colonial government established one of its largest army bases in Owambo ‘district’ to safeguard the hydropower station from possible guerrilla attacks. However, the town is rarely documented in academic or even South African colonial government publications. It is rather the hydropower complex that is well documented, where Ruacana is represented through its projects of modernization. Also, other than claims to natural heritage and a heritage of ethnicity, Ruacana town lacks formal invocations of heritage. Thus it is argued that Ruacana points to a different pattern of heritage production, as the future itself was planned as heritage. This study is an attempt to analyse how Ruacana became a place of a heritage of development, even though heritage is not formally acknowledged in the institutional structures.

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A mini-lidar system, Vaisala's CL31 ceilometer, was installed within a suburban area of Vancouver, BC, for the purpose of making continuous observations of the boundary layer over a period of 11 months. Initial validation of the ceilometer for use in boundary layer observations was undertaken. This involved the comparison of ceilometer data with eight months of ground-level particulate matter measurements, as well as with 16 vertical profiles of particulate matter and meteorological data. Once a variety of persistent noise structures within the data were accounted for, it was found that the ceilometer data showed good agreement with the particulate matter data, suggesting its usefulness for assessing air-quality throughout the bottom 1km of the atmosphere. Additionally, two algorithms were developed in order to estimate the height of the convective boundary layer, or the mixed-layer height, from the ceilometer data. One involved the fitting of an ideal-profile to the measured data, while the other involved the location of a minimum-gradient in the backscatter profile. The performance of these two techniques were assessed and compared, and it was found that the ideal-profile method was the more robust of the two. Finally, mixed-layer heights were estimated for fair weather, convectively active days. In order to isolate such conditions, an automatic flagging algorithm was developed. However, additional manual assessment was needed to avoided unsuitable conditions. Mixed-layer heights were estimated for 19 days over an 11 month period. the estimates presented here were found to agree with previous observations. Daily maximum mixed-layer heights ranged from 650m in July to 350m in December, indicating that the height of the convective boundary layer within Vancouver is significantly suppressed due to the city's coastal location. / Arts, Faculty of / Geography, Department of / Graduate