Geodetic models suggest that much of the convergence across the Himalaya (~20 mm yr-1) is taken up on the Main Himalayan Thrust, the main decollement beneath the Himalayan orogenic wedge. In Central Nepal and the majority of Northwest India, several geomorphic, geophysical and seismological datasets indicate that this decollement has a mid-crustal ramp that continues uninterrupted for hundreds of kilometers along strike from Nepal in the east to Uttarakhand in the west. In this study, I use spatial analyses of elevation, relief, channel steepness indices, and basin-wide erosion rates from cosmogenic 10-Be concentrations to outline a potential large-scale change in the active fault configuration between the Main Himalayan Thrust and Main Boundary Thrust near longitude 77°E in the Northwestern Indian Himalaya. The physiography in the areas to the east of 77ºE appears similar to that observed along much of the Himalaya where topographic relief, erosion rates, and river channel steepness (ksn <200) remain relatively low in the areas to the south of a line known as the Physiographic Transition-2. North of the Physiographic Transition-2, these metrics increase sharply within a 30-km zone due to higher rock uplift rates above a mid-crustal ramp on the decollement or an unidentified out-of-sequence thrust fault that soles to the decollement. Either of these models are perceivable with a duplex growing by underplating of the Indian plate into the Himalayan orogenic wedge contributing to higher rock uplift rates north of the Physiographic Transition-2. To the west of 77ºE, however, the landscape morphology indicates the Main Boundary Thrust makes a northward bend coinciding with the along-strike termination of the Physiographic Transition-2 and an arc-perpendicular Bouguer gravity anomaly reflecting a trough on the Indian plate near longitude 77°E. These data suggest that the Main Boundary Thrust merges along strike with the ramp or with an emergent fault soling into the Main Himalayan Thrust at this location, potentially marking a significant change in tectonic configuration along the Himalayan arc. / Graduate

The work carried out for this thesis forms part of the air-sea interaction program, which has been under way since 196l at the Institute of Oceanography of the University of British Columbia. Measurements of fluctuations in the vertical and horizontal components of air velocity were made using hot wires in an X-array, in order to study the spectra of the fluctuations, and their co-spectrum over a range of mean wind speeds from 140 – 1000 cm./sec. in the boundary layer over the sea. In order to use the X-wire probe properly in the field, special techniques were developed to mount and calibrate the wires, and to measure directly their responses to the two velocity fluctuations. Analog techniques were developed to analyze the hot wire signals, and final calculations were made by digital computer. Single (U-wire) hot wire probes were used to measure the horizontal velocity fluctuations to check the behaviour of X-wires, and to provide additional checks on the similarity theory of turbulence. Measurements showed that X-wire techniques can be used successfully to measure velocity fluctuations in two directions in the field. Hot wires have responses which give spectral levels which are accurate only within about 35%, but comparison of the horizontal velocity spectrum measured simultaneously with the X- and U-wire probes showed that their spectral shapes were similar, giving confidence in the X-wire measurements. In the high frequency range, the observed spectra of the two velocity fluctuations did not conform to the theoretical predictions. The observed behaviour is believed to be real. The cospectrum gives a direct estimate of contributions to the Reynolds' stress by fluctuations in small ranges of frequency. The stress observed between the frequency limits O.Ol6 to 10 Hz had significant contributions over about one frequency decade, which apparently lies entirely within these extremes. Estimates of the frequencies of dominant waves at the experimental site fell between about 0.2 to 0.5 Hz. Significant stress was present in this interval, but the largest proportion of the observed stress was present at lower frequencies. Ten direct estimates of stress were obtained with the X-wire. Values estimated indirectly from the wind profiles tended to give low estimates and were poorly correlated with the direct estimates. Values determined indirectly using the inertial subrange appeared to be consistently related to the directly estimated stress, but overestimated it by about 40%. Drag coefficients corrected to the 5m height were near 1.5 x 10⁻³ for wind speeds between 1.4 and 10m.sec-¹. Measurements by three U-wires spaced vertically, provided confirmation of the validity of the Monin-Obukhov similarity theory at heights below about 5m. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

In the numerical solution of the two point "boundary value problem, [ equation omitted ] (1) the usual method is to approximate the problem by a finite difference analogue of the form [ equation omitted ] (2) with k = 2, and the truncation error T.E. = O(h⁴) or O(h⁶), where h is the step-size. Varga (1962) has obtained error bounds for the former when the problem (1) is linear and of class M . In this thesis, more accurate finite difference methods are considered. These can be obtained in essentially two different ways, either by increasing the value k in difference equations (2), or by introducing higher order derivatives. Several methods of both types have been derived. Also, it is shown how the initial value problem y' = ϕ(x,y) can be formulated as a two point boundary value problem and solved using the latter approach. Error bounds have been derived for all of these methods for linear problems of class M . In particular, more accurate bounds have been derived than those obtained by Varga (1962) and Aziz and Hubbard (1964). Some error estimates are suggested for the case where [ equation omitted ], but these are not accurate bounds, especially when [ equation omitted ] not a constant. In the case of non-linear differential equations, sufficient conditions are derived for the convergence of the solution of the system of equations (2) by a generalized Newton's method. Some numerical results are included and the observed errors compared with theoretical error bounds. / Science, Faculty of / Mathematics, Department of / Graduate

The theme of the research is on the use of the hypersingular boundary element method for the modelling of surface waves. Surface waves in solids are known to be partially reflected & transmitted and mode converted into body waves at stress discontinuities, which suggests that a formulation continuous in stress and strain might prove beneficial for modelling purposes. Such continuity can be achieved with a subparametric approach where the geometry is approximated using linear elements and the field variables, displacement and traction, are approximated using cubic Hermitian and linear shape functions respectively. The higher order polynomial for approximating displacement is intended to be a more accurate representation of the physics relating to surface wave phenomena, especially at corners, and thus, is expected to capture this behaviour with greater accuracy than the standard isoparametric approach. The subparametric approach affords itself to continuity in stress and strain by imposing a smoothness in the elements, which is not available to the isoparametric approach. As the attention is focused primarily on the modelling of surface waves on the boundary of a medium rather than the interior, the boundary element method lends itself appropriately to this end.A 2D semi analytical integration scheme is employed to evaluate the integrals appearing in the hypersingular boundary integral formulation. The integration scheme is designed to reduce the errors incurred when integrals with singular integrands are evaluated numerically. The scheme involves the application of Taylor expansions to formulate the integrals into two parts. One part is regular and is evaluated numerically and the other part is singular but sufficiently simple to be evaluated analytically. The scheme makes use of the aforementioned subparametric approach and is applied to linear elements for the use in steady state elastodynamic boundary element method problems. The steady state problem is used as it is a simplified problem and is sufficient to permit the investigation of surface vibration at a constant motion. The 2D semi analytical integration scheme presented can be naturally extended to 3D.A particular focus and novelty of the work is the application of different limiting approaches to determine the free terms common to boundary integral methods. The accurate numerical solution of hypersingular boundary integral equations necessitates the precise evaluation of free terms, which are required to counter discontinuous and often unbounded behaviour of hypersingular integrals at a boundary. The common approach for the evaluation of free terms involves integration over a portion of a circular/spherical shaped surface centred at a singularity and allowing the radius of the circle/sphere to tend to zero. This approach is revisited in order to ascertain whether incorrect results are possible as a consequence of shape dependency, which is a recognised issue for hypersingular integrals.Two alternative methods, which are shape invariant, are proposed and investigated for the determination of free terms. The first approach, the point limiting method, involves moving a singularity towards a shrinking integration domain at a faster rate than the domain shrinks. Issues surrounding the choice of approach, shrinkage rates and path dependency are examined. A related and second approach, the boundary limiting method, involves moving an invariant, but shrinking, boundary toward the singularity, again at a faster rate than the shrinkage of the domain. The latter method can be viewed as a vanishing exclusion zone approach but the actual boundary shape is used for the boundary of the exclusion zone. Both these methods are shown to provide consistent answers and can be shown to be directly related to the result obtained by moving a singularity towards a boundary, that is, by comparison with the direct method. Unlike the circular/spherical approach the two methods involve integration over the actual boundary shape and consequently shape dependency is not an issue. A particular highlight of the point limiting approach is the ability to obtain free terms in mixed formulation, which is not available to the circular/spherical approach.There are three numerical problems considered in this research. The first problem considers the longitudinal vibration of a square plate. This is a problem for which a known analytical solution exists and is used to verify the equation formulation and integration scheme adopted for the isoparametric and subparametric formulations. Both formulations are as accurate as each other and produce results that are in keeping with the analytical solution, thus instilling confidence in their predictions.The second problem considers the simulation of surface waves on a square plate. Various boundaries of a square plate have displacement conditions imposed on them as a result of surface wave propagation. The results indicate that the surface wave behaviour is not captured. However, the analytical solution does not make any consideration for the effects from corners; the analytical solution is for a Rayleigh wave propagating upon a planar surface. It does not take into account the wave phenomena encountered at corners. Therefore, these results cannot be used to validate the predictions obtained on the boundary of the problem considered. The purpose of this problem is to illustrate the impact of corners on the surface wave propagation. Sensitivity studies are conducted to illustrate the effect of corners on the computed solution at the boundary.The final problem considers the simulation of surface waves on a circular plate. Various portions of the boundary of the circular plate have displacement conditions imposed on them as a result of surface wave propagation on curved surfaces. The results indicate that the isoparametric and subparametric predictions are similar to one another. However, both displacement profiles predict the presence of other waves. Given the multi faceted nature of the mesh, the computed solution is picking up mode conversion and partial reflection & transmission of surface waves. In reality, this is not expected as the surface of the boundary is smooth. However, due to the discretisation there are many corners in this problem.

This thesis investigates the viability of two boundary element methods for solving steady state problems, the continuous least squares method and the Galerkin minimization technique. In conventional boundary element methods, the singularities of the fundamental solution involved are usually located at fixed points on the boundary of the problem's domain or on an auxiliary boundary. This leads to some difficulties: when the singularities are located on the problem domain's boundary, it is not easy to evaluate the solution for points on or near that boundary whereas if the singularities are placed on an auxiliary boundary, this auxiliary boundary would have to be carefully chosen. Hence the methods studied here allow the singularities, initially located at some auxiliary boundary, to move until the best positions are found. These positions are determined by attempting to minimize the error via the least squares or the Galerkin technique. This results in a highly accurate, adaptive, but nonlinear method. We study various methods for solving systems of nonlinear equations resulting from the Galerkin technique. A hybrid method has been implemented, which involves the objective function from the least squares method while the gradient is due to the Galerkin method. Numerical examples involving Laplace's equation in two dimensions are presented and results using the discrete least squares method, the continuous least squares method and the Galerkin method are compared and discussed. The continuous least squares method appears to give the best results for the sample problems tried. / Science, Faculty of / Computer Science, Department of / Graduate

A one-dimensional, single phase Stefan Problem is considered. This problem is shown to have a unique solution which depends continuously on the boundary data. In addition two algorithms are formulated for its approximate numerical solution. The first algorithm (the Similarity Algorithm), which is based on Similarity, is shown to converge with order of convergence between one half and one. Moreover, numerical examples illustrating various aspects of this algorithm are presented. In particular, modifications to the algorithm which are suggested by the proof of convergence are shown to improve the numerical results significantly. Furthermore, a brief comparison is made between the algorithm and a well-known difference scheme. The second algorithm (a Collocation Scheme) results from an attempt to reduce the problem to a set of ordinary differential equations. It is observed that this set of ordinary differential equations is stiff. Moreover, numerical examples indicate that this is a high order scheme capable of achieving very accurate approximations. It is observed that the apparent stiffness of the system of ordinary differential equations renders this second algorithm relatively inefficient. / Science, Faculty of / Statistics, Department of / Graduate

In the present investigation, zinc was plated onto a cadmium crystal to determine the effect on the scattering parameter.

B.URP / Department of Urban and Regional Planning

NR 20140805

One of the basic goals in engineering is to generate models which will provide a means for analytically predicting observed phenomenon. Such a model is often modified several times to obtain better results. The purpose of this study was to generate a model for an equilibrium turbulent boundary layer for steady flow over a flat plate and compare the results obtained by using the model with experimental data. Part of the objective was to also suggest ways in which the model could be modified to obtain better results.