Includes bibliographical references (leaves 142-149). / In this document a mathematical model for the Czochralski crystal growth process is developed. The trend in current research involves developing cumbersome numerical simulations that provide little or no understanding of the underlying physics. We attempt to review previous research methods, mainly devoted to silicon, and develop a novel analytical tool for indium antimonide (lnSb) crystal growth. This process can be subdivided into two categories: solidification and fluid mechanics. Thus far, crystal solidification of the Czochralski process has been described in the literature mainly qualitatively. There has been little work in calculating actual solidification dynamics. Czochralski crystal growth is a very sensitive process, particularly for lnSb, so it is crucial to describe the system as accurately as possible. A novel ID quasi-steady method is proposed for the shape and temperature field of an lnSb crystal, incorporating the effects of the melt. The fluid mechanics of the Czochralski melt have been modelled by numerous researchers,with calculations performed using commercial software. However, a descriptionof the buoyancy and rotation interaction in the melt has not been adequatelyperformed. Many authors have presented flow patterns but none have indicated either: melt conditions preferential for crystal growth or at least a description of a typical melt structure. In this work, a scale analysis is performed that implies an idealized flow structure. An asymptotic model is then derived based on this order of magnitude analysis, resulting in a fast and efficient fluid flow calculation. The asymptotic model is validated against a numerical solution to ensure that the macroscopic features of the flow structure are present. The asymptotic model does not show exact agreement, but does provide an estimate of the melt heat flux that is necessary for the solidification calculation. The asymptotic model is also used to predict macroscopic changes in the melt due to rotation.

The finite element method (FEM) is widely used for the simulation of metal forming processes and has been successfully used in contact problems which arise in processes such as deep-drawing, punching, extrusion and rolling. All these processes involve friction between the contact surfaces: the sheet-metal workpiece and the toolpieces. The model of friction is thus an important part of any simulation of metal forming processes. Most FEM codes use a friction model that assumes that the contact surface is a plane. Attempts to address this problem have focused on the convective description of deformation, which has the advantage of being naturally extended to numerical methods like the FEM at the expense of additional computation and numerical complexity. The convective description is used in this work, which focuses on the numerical implementation of the objective measure. The effects of the rotation of the material contact point is taken into account by including objective time derivatives of the slipping (tangential) direction function. The objective rate of the direction function includes the surface spin induced by the rigid motion of a contact point sliding over the tool surface, and the material spin occurring during the elastic-plastic deformation of the blank. This is introduced by adapting the incremental relations of the friction slip. This thesis presents the results of numerical experiment to determine the influence that the rotation and convection of contact points has on the frictional stresses and slipping energy. Four different friction models are implemented within the finite element program ABAQUS and applied to simulations of standardmetal forming benchmark processes: the square-cup and s-rail deep drawing benchmarks of the Numisheet conferences, for which several experimental and numerical results are available to compare with the solution of a finite element simulation. The results for each metal-forming simulation are calculated for different friction models, and are compared and a choice made as to which is the “best” friction model for the process. Further, the reverse problem of determining the values of friction parameters by comparison of simulation and experimental results is performed for these benchmark problems. As there is yet no ideal friction model for all processes that are modelled, finding the most appropriate friction model by numerical means is proposed to improve the quality of a simulation.

Cosmology is emerging into a new and exciting period thanks to a wealth of ongoing and planned massive surveys which will deliver exponentially growing volumes of data over the next two decades. As a result of this rapid growth, which exhibits erce competition between di erent surveys due to the spiralling costs, forecasting and optimization have become critical to help best use and bene t from this new boon. In this thesis various aspects of forecasting and optimization are explored, with particular emphasis on, but not limited to, cosmology. We introduce a new optimization algorithm which signi cantly outperforms all standard algorithms, especially in higher dimensions where the improvement is remarkable. The new algorithm, Hybrid-MTM, should provide a powerful new tool in addressing high-dimensional optimization problems. We then forecast the prospects for detecting dynamics in tracking dark energy models. We show that Big Bang Nucleosynthesis and Cosmic Microwave Background constraints in these models are extremely di cult to match with existing data. As a result it is unlikely that a detectable deviations from the cosmological constant for these models is possible before the Stage-IV DETF experiments, which will only come on-line post-2015. Finally we present new results on Fisher matrix forecasts for cosmology produced using the Fisher4Cast code. Fisher4Cast allows novel insights into the nature of how information is gained from cosmological experiments and the interplay between the measurements of Hubble, distance and growth in constraining cosmological parameters. In the nal chapter we provide a detailed overview of the code structure in Fisher4Cast and its Graphical User Interface together with its unique features including a LATEXreporting module which breaks new ground in the automated generation of publication-quality scienti c research.

In this thesis, we study gravitational theories for which the natural choice of an affine connection is metric compatible while not being symmetric. More specifically, we study gravitational theories constructed on the Riemann-Cartan and Weitzenböck space-times. Firstly, we outline the mathematical notions needed to construct a definition of space-time. Following this, we introduce the space-time definitions to be made use of throughout this thesis. We then discuss the notions of extremal and auto-parallel curves on the Riemann-Cartan space-time. It is noted that test particles follow extremal curves which are auto-parallel curves of the LeviCivita connection. Therefore, one must turn to the standard, torsion-free Raychaudhuri equation when studying the focusing conditions that arise in theories constructed on the Riemann-Cartan or Weitzenböck space-times. Once we have introduced the definitions of the relevant space-times, we move on to review some of the gravitational theories that involve non-vanishing torsion. We first review the Einstein-Cartan theory and two of its modifications. We then review the so-called f(T) theories of gravity before discussing the focusing conditions that arise in this context. By making use of the f(T) field equations together with the torsion-free Raychaudhuri equation, we derive for the first time the f(T) focusing conditions for a one-parameter dependent congruence of timelike auto-parallel curves of the LeviCivita connection. We then study these focusing conditions for three bi-parametric cosmological models. Finally, we turn our attention back to the Einstein-Cartan theory and derive the Arnowitt-DeserMisner formulation of this theory. By making use of this formulation, we derive for the first time the Generalised-Baumgarte-Shapiro-Shibata-Nakamura formulation of the Einstein-Cartan theory. We then consider the case of a vacuum in spherical symmetry and construct a 1-dimensional code to evolve the system numerically. We leave the inclusion of torsion into this code as the subject for future work.