Applications of finite element computer modelling and thermal infrared remote sensing to the study of geothermal anomalies

Buried heat sources can be investigated by examining thermal infrared images and comparing these with the results of theoretical models which predict the thermal anomaly a given heat source may generate. Key factors influencing surface temperature include the geometry and temperature of the heat source, the surface meteorological environment, and the thermal conductivity and anisotropy of the rock. In general, a geothermal heat flux of greater than 2% of solar insolation is required to produce a detectable thermal anomaly in a thermal infrared image. A heat source of, for example, 2-300K greater than the average surface temperature must be a t depth shallower than 50m for the detection of the anomaly in a thermal infrared image, for typical terrestrial conditions. Atmospheric factors are of critical importance. While the mean atmospheric temperature has little significance, the convection is a dominant factor, and can act to swamp the thermal signature entirely. Given a steady state heat source that produces a detectable thermal anomaly, it is possible to loosely constrain the physical properties of the heat source and surrounding rock, using the surface thermal anomaly as a basis. The success of this technique is highly dependent on the degree to which the physical properties of the host rock are known. Important parameters include the surface thermal properties and thermal conductivity of the rock. Modelling of transient thermal situations was carried out, to assess the effect of time dependant thermal fluxes. One-dimensional finite element models can be readily and accurately applied to the investigation of diurnal heat flow, as with thermal inertia models. Diurnal thermal models of environments on Earth, the Moon and Mars were carried out using finite elements and found to be consistent with published measurements. The heat flow from an injection of hot lava into a near surface lava tube was considered. While this approach was useful for study, and long term monitoring in inhospitable areas, it was found to have little hazard warning utility, as the time taken for the thermal energy to propagate to the surface in dry rock (several months) in very long. The resolution of the thermal infrared imaging system is an important factor. Presently available satellite based systems such as Landsat (resolution of 120m) are inadequate for detailed study of geothermal anomalies. Airborne systems, such as TIMS (variable resolution of 3-6m) are much more useful for discriminating small buried heat sources. Planned improvements in the resolution of satellite based systems will broaden the potential for application of the techniques developed in this thesis. It is important to note, however, that adequate spatial resolution is a necessary but not sufficient condition for successful application of these techniques.

Uniformly convergent finite element and finite difference methods for singularly perturbed ordinary differential equations

This thesis is concerned with uniformly convergent finite element and finite difference methods for numerically solving singularly perturbed two-point boundary value problems. We examine the following four problems: (i) high order problem of reaction-diffusion type; (ii) high order problem of convection-diffusion type; (iii) second order interior turning point problem; (iv) semilinear reaction-diffusion problem. Firstly, we consider high order problems of reaction-diffusion type and convection-diffusion type. Under suitable hypotheses, the coercivity of the associated bilinear forms is proved and representation results for the solutions of such problems are given. It is shown that, on an equidistant mesh, polynomial schemes cannot achieve a high order of convergence which is uniform in the perturbation parameter. Piecewise polynomial Galerkin finite element methods are then constructed on a Shishkin mesh. High order convergence results, which are uniform in the perturbation parameter, are obtained in various norms. Secondly, we investigate linear second order problems with interior turning points. Piecewise linear Galerkin finite element methods are generated on various piecewise equidistant meshes designed for such problems. These methods are shown to be convergent, uniformly in the singular perturbation parameter, in a weighted energy norm and the usual L2 norm. Finally, we deal with a semilinear reaction-diffusion problem. Asymptotic properties of solutions to this problem are discussed and analysed. Two simple finite difference schemes on Shishkin meshes are applied to the problem. They are proved to be uniformly convergent of second order and fourth order respectively. Existence and uniqueness of a solution to both schemes are investigated. Numerical results for the above methods are presented.

Uniformly convergent finite element methods for singularly perturbed parabolic partial differential equations

This thesis is concerned with uniformly convergent finite element methods for numerically solving singularly perturbed parabolic partial differential equations in one space variable. First, we use Petrov-Galerkin finite element methods to generate three schemes for such problems, each of these schemes uses exponentially fitted elements in space. Two of them are lumped and the other is non-lumped. On meshes which are either arbitrary or slightly restricted, we derive global energy norm and L2 norm error bounds, uniformly in the diffusion parameter. Under some reasonable global assumptions together with realistic local assumptions on the solution and its derivatives, we prove that these exponentially fitted schemes are locally uniformly convergent, with order one, in a discrete L∞norm both outside and inside the boundary layer. We next analyse a streamline diffusion scheme on a Shishkin mesh for a model singularly perturbed parabolic partial differential equation. The method with piecewise linear space-time elements is shown, under reasonable assumptions on the solution, to be convergent, independently of the diffusion parameter, with a pointwise accuracy of almost order 5/4 outside layers and almost order 3/4 inside the boundary layer. Numerical results for the above schemes are presented. Finally, we examine a cell vertex finite volume method which is applied to a model time-dependent convection-diffusion problem. Local errors away from all layers are obtained in the l2 seminorm by using techniques from finite element analysis.

High-current inductors for high-power automotive DC-DC converters

This thesis is focused on the investigation of magnetic materials for high-power dcdc converters in hybrid and fuel cell vehicles and the development of an optimized high-power inductor for a multi-phase converter. The thesis introduces the power system architectures for hybrid and fuel cell vehicles. The requirements for power electronic converters are established and the dc-dc converter topologies of interest are introduced. A compact and efficient inductor is critical to reduce the overall cost, weight and volume of the dc-dc converter and optimize vehicle driving range and traction power. Firstly, materials suitable for a gapped CC-core inductor are analyzed and investigated. A novel inductor-design algorithm is developed and automated in order to compare and contrast the various magnetic materials over a range of frequencies and ripple ratios. The algorithm is developed for foil-wound inductors with gapped CC-cores in the low (10 kHz) to medium (30 kHz) frequency range and investigates the materials in a natural-convection-cooled environment. The practical effects of frequency, ripple, air-gap fringing, and thermal configuration are investigated next for the iron-based amorphous metal and 6.5 % silicon steel materials. A 2.5 kW converter is built to verify the optimum material selection and thermal configuration over the frequency range and ripple ratios of interest. Inductor size can increase in both of these laminated materials due to increased airgap fringing losses. Distributing the airgap is demonstrated to reduce the inductor losses and size but has practical limitations for iron-based amorphous metal cores. The effects of the manufacturing process are shown to degrade the iron-based amorphous metal multi-cut core loss. The experimental results also suggest that gap loss is not a significant consideration in these experiments. The predicted losses by the equation developed by Reuben Lee and cited by Colonel McLyman are significantly higher than the experimental results suggest. Iron-based amorphous metal has better preformance than 6.5 % silicon steel when a single cut core and natural-convection-cooling are used. Conduction cooling, rather than natural convection, can result in the highest power density inductor. The cooling for these laminated materials is very dependent on the direction of the lamination and the component mounting. Experimental results are produced showing the effects of lamination direction on the cooling path. A significant temperature reduction is demonstrated for conduction cooling versus natural-convection cooling. Iron-based amorphous metal and 6.5% silicon steel are competitive materials when conduction cooled. A novel inductor design algorithm is developed for foil-wound inductors with gapped CC-cores for conduction cooling of core and copper. Again, conduction cooling, rather than natural convection, is shown to reduce the size and weight of the inductor. The weight of the 6.5 % silicon steel inductor is reduced by around a factor of ten compared to natural-convection cooling due to the high thermal conductivity of the material. The conduction cooling algorithm is used to develop high-power custom inductors for use in a high power multi-phase boost converter. Finally, a high power digitally-controlled multi-phase boost converter system is designed and constructed to test the high-power inductors. The performance of the inductors is compared to the predictions used in the design process and very good correlation is achieved. The thesis results have been documented at IEEE APEC, PESC and IAS conferences in 2007 and at the IEEE EPE conference in 2008.