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This paper will demonstrate the principles and important facts of the randomized Kaczmarz algorithm as well as its extended version proposed by Zouzias and Ferris. Through the analysis made by Strohmer and Vershynin as well as Needell, it can be shown that the randomized Kaczmarz method is theoretically applicable in solving over-determined linear systems with or without noise. The extension of the randomized Kaczmarz algorithm further applies to the linear systems with non-unique solutions. In the experiment section of this paper, we compare the accuracies of the algorithms discussed in the paper in terms of making real-world macroeconomic analyses and predictions. The extended randomized Kaczmarz method outperforms both the randomized Kaczmarz method and the randomized Gauss-Seidel method on our data sets.

published_or_final_version / Civil Engineering / Doctoral / Doctor of Philosophy

published_or_final_version / abstract / Physics / Doctoral / Doctor of Philosophy

Building pounding is a recognised phenomenon where adjacent buildings collide under lateral loading due to insufficient provision of building separation. The consequences of this interaction are known to be complex, and both buildings’ responses can be significantly affected. In the absence of extensive experimental data, numerical modelling has been frequently adopted as a means of evaluating building pounding risk during earthquakes. In performing numerical analysis, it becomes necessary to create specialised ‘contact’ elements to simulate building contact. While many contact elements have been previously proposed, detailed consideration of their inherent assumptions has frequently been overlooked. This thesis considers the significance and consequences of using the Kelvin contact element for a variety of pounding situations and with varying levels of model detail. Pounding between two adjacent floors (floor/floor collision) is considered as a one dimensional wave propagation problem. By modelling each floor as a flexible rod (termed distributed mass modelling), theoretical relationships for collision force, collision duration and post-collision velocity are derived. This theory is then compared to the predictions made when using the traditionally adopted assumptions of fully rigid colliding floors (termed lumped mass modelling). The post-collision velocities obtained from each method are found to agree only when the axial period of both floors is identical. Relationships between lumped mass and distributed mass models are formed, and an ‘equivalent lumped mass’ method is developed where distributed mass effects can be emulated without explicit modelling of floor flexibility. The theoretical solution method is then adapted for use in Non-Linear Time History Analysis (NLTHA) software to model specific pounding situations. Numerical modelling of a single collision is performed to compare these results to the theoretical predictions. Good agreement is found, and the model’s complexity is simplified until a sufficiently accurate simulation is performed without overly onerous computational requirements. Five methods are detailed that incorporate energy loss during collision into the distributed mass models and a calibration method is developed that enables researchers to define the level of energy loss that occurs during a single collision. Using the developed modelling methods, the pounding response of two existing Wellington buildings is predicted. This is first performed using 2D analysis of the stiffest frame from each building. The predicted building pounding damage is categorised into local damage (damageresulting from the magnitude of the force applied during contact) and global damage (damage due to the change in dynamic building properties resulting from momentum transfer during collision). Local and global damage effects are found to be fundamentally different consequences of collision, with the two categories responding differently to changes in the modelled system. The effects of building separation, scaling of input motion, modelling of soil-structure-interaction, collision damping, and floor rigidity are investigated for the considered system. 3D analysis of the building configuration is then investigated. Additional complications arising from the transverse movement of buildings prior to and during collision are identified and refined modelling methods are developed. The 3D configuration of these buildings causes torsional interaction, despite both buildings being perfectly symmetrical. This torsion is due to the eccentric positioning of the buildings relative to each other, which causes an eccentric contact load when pounding occurs. The 3D models are used to test the effects of building separation, 2D vs. 3D modelling, collision damping, floor rigidity, and the significance of the torsional interactions. Attention is then focused on collisions between a building’s floors and an adjacent building’s columns (floor/column collision). Due to the high frequency content of pounding impacts, the significance of using Timoshenko beam theory instead of Euler-Bernoulli theory is assessed. The shear stiffness in the Timoshenko formulation is found to significantly affect the columns’ predicted performance, and is used in subsequent modelling. An appropriately accurate method of modelling that minimises computational effort is then developed. The simplified model is used to predict the performance of two three-storey buildings that experience floor/column collision. The effects of floor/column impact are predicted for collisions at mid-height, and near the support of the impacted column. Each of these scenarios investigates the effect of building separation on local damage and global damage. Finally, a method to model collision between two adjacent walls that collide out-of-plane is developed (wall/wall contact). The adopted contact element properties are selected using analogous situations that have been previously investigated. The method is used to investigate a single collision between two different wall configurations. In the conclusions, the developed modelling methods from all the considered collision configurations are collected and presented in a summary table. It is intended that these recommendations will assist other researchers in selecting appropriate building pounding modelling properties.

An adaptive refinement procedure combining the ideas of solving a modified difference equation and of adaptive mesh refinement is introduced. The numerical solution on a fixed grid is improved by inclusion of approximated truncation error computed from local subgrid refinement. Following this procedure, a reliable scheme has been developed for refined computations of the flow past a wedge at transonic speeds. Effects of the truncation error on the pressure, wave drag, sonic line, and shock position are investigated. By comparing the pressure drag on the wedge and the wave drag due to the shocks, the existence of a supersonic-to-supersonic shock originating from the wedge shoulder is confirmed.

No description available.

No description available.

The thesis investigates the application of ice thermal storage technology to situations where the price of electricity varies continuously with instantaneous network demand. A central hypothesis is postulated in chapter 1, which states: "A variable electricity pricing structure, in which unit price continuously varies in response to instantaneous network demand, enhances the opportunities and benefits of ice thermal storage. The benefits both financial and environmental are dependent on the establishment of control and design strategies which optimise performance by matching refrigeration load with the instantaneous electricity price. " For ease of reference, the form of pricing described above is referred to in the thesis as 'real time' electricity pricing. The 'pool price' which is used to facilitate the competitive electricity awkct in England and Wales, is one of the foremost examples of real time pricing. The thesis therefore uses the electricity supply industry in the UK as its research vehicle. Notwithstanding this, the work contained in the thesis can be applied to any country which applies real time electricity pricing mechanisms. The validity of the hypothesis is assessed in the thesis through the development of a variety of numerical and computer models. These models fall into two distinct categories; those concerned with predicting and optimising the financial benefits of ice thermal storage, and those concerned with predicting and optimising the environmental benefits of ice thermal storage. Chapters 2,3 and 4 should be treated as support chapters, which equip the reader with the prerequisite knowledge necessary to understand the research work contained in the later chapters. As such, these chapters contain, respectively, a description of the electricity supply industry in the UK, a discussion of demand side management in the UK, and a description of the technology involved in ice thermal storage. The parametric study contained in chapter 4 is however an original piece of research work by the author. The models developed to evaluate and optimise the economic benefits of ice thermal storage are presented in chapters 5 and 6, and are applied to contrasting theoretical case study applications, namely an office building and a dairy. In chapter 5a 'long hand' numerical analysis technique is used. In chapter 6 this technique is rationalised and developed into a computer model for optimising both the design and control of ice storage installations in real time electricity pricing applications. The environmental studies are presented in chapter 7. These concentrate on the ability of ice thermal storage to reduce carbon dioxide emissions. Although the overall objective of the chapter is to evaluate the carbon dioxide emissions associated with ice thermal storage, the bulk of the chapter is concerned with the development of a model for predicting the carbon dioxide emissions per kWh of delivered electrical energy in England and Wales on a time related basis. The development of this 'time of day' carbon dioxide model is one of the main objectives of the thesis. Having established this model, it is then used to analyse the carbon dioxide emissions associated with the dairy case study.

In this dissertation we obtain an e cient hybrid numerical method for the solution of stochastic di erential equations (SDEs). Speci cally, our method chooses between two numerical methods (Euler and Milstein) over a particular discretization interval depending on the value of the simulated Brownian increment driving the stochastic process. This is thus a new1 adaptive method in the numerical analysis of stochastic di erential equation. Mauthner (1998) and Hofmann et al (2000) have developed a general framework for adaptive schemes for the numerical solution to SDEs, [30, 21]. The former presents a Runge-Kutta-type method based on stepsize control while the latter considered a one-step adaptive scheme where the method is also adapted based on step size control. Lamba, Mattingly and Stuart, [28] considered an adaptive Euler scheme based on controlling the drift component of the time-step method. Here we seek to develop a hybrid algorithm that switches between euler and milstein schemes at each time step over the entire discretization interval, depending on the outcome of the simulated Brownian motion increment. The bias of the hybrid scheme as well as its order of convergence is studied. We also do a comparative analysis of the performance of the hybrid scheme relative to the basic numerical schemes of Euler and Milstein. / Mathematical Sciences / M.Sc. (Applied Mathematics)