This dissertation presents a duality theorem of the Aubin-Nitsche type for discontinuous Petrov Galerkin (DPG) methods. This explains the numerically observed higher convergence rates in weaker norms. Considering the specific example of the mild-weak (or primal) DPG method for the Laplace equation, two further results are obtained. First, for triangular meshes, the DPG method continues to be solvable even when the test space degree is reduced, provided it is odd. Second, a non-conforming method of analysis is developed to explain the numerically observed convergence rates for a test space of reduced degree. Finally, for rectangular meshes, the test space is reduced, yet the convergence is recovered regardless of parity.

Magnetic nanoparticles (NPs) with sizes ranging from 2 to 20 nm in diameter represent an important class of artificial nanostructured materials, since the NP size is comparable to the size of a magnetic domain. They have potential applications in data storage, catalysis, permanent magnetic nanocomposites, and biomedicine. To begin with, a brief overview on the background of Fe-based bimetallic NPs and their applications for data-storage and catalysis was presented in Chapter 1. In Chapter 2, L10-ordered FePt NPs with high coercivity were directly prepared from a novel bimetallic acetylenic alternating copolymer P3 by a one-step pyrolysis method without post-thermal annealing. The chemical ordering, morphology and magnetic properties were studied. Magnetic measurements showed that a record coercivity of 3.6 T (1 T = 10 kOe) was obtained in FePt NPs. By comparison of the resultant FePt NPs synthesized under Ar and Ar/H2, the characterization proved that the incorporation of H2 would affect the nucleation and promote the growth of FePt NPs. The L10 FePt NPs were also successfully patterned on Si substrate by nanoimprinting lihthography (NIL). The highly ordered ferromagnetic arrays on a desired substrate for bit-patterned media (BPM) were studied and promised bright prospects for the progress of data-storage. Furthermore, we also reported a new FePt-containing metallopolymer P4 as the single-source precursor for metal alloy NPs synthesis, where the metal fractions were on the side chain and the ratio could be easily controlled. This polymer was synthesized from random copolymer poly(styrene-4-ethynylstyrene) PES-PS and bimetallic precursor TPy-FePt ([Pt(4’-ferrocenyl-(N^N^N))Cl]Cl) by Sonogashira coupling reaction. After pyrolysis of P4, the stoichiometry of Fe and Pt atoms in the synthesized NPs (NPs) is nearly close to 1:1, which is more precise than using TPy-FePt as precursor. Polymer P4 was also more favorable for patterning by high throughout NIL as compared to TPy-FePt. Ferromagnetic nanolines, potentially as bit-patterned magnetic recording media, were successfully fabricated from P4 and fully characterized. In Chapter 3, a novel organometallic compound TPy-FePd-1 [4’-ferrocenyl-(N^N^N)PdOCOCH3] was synthesized and structurally characterized, whose crystal structure showed a coplanar Pd center and Pd-Pd distance (3.17 Å). Two metals Fe and Pd were evenly embedded in the molecular dimension and remained tightly coupled between each other benefiting to the metalmetal (Pd-Pd) and ligand ππ stacking interactions, all of which made it facilitate the nucleation without sintering during preparing the FePd NPs. Ferromagnetic FePd NPs of ca. 16.2 nm in diameter were synthesized by one-pot pyrolysis of the single-source precursor TPy-FePd-1 under getter gas with metal-ion reduction and minimal nanoparticle coalescence, which have a nearly equal atomic ratio (Fe/Pd = 49/51) and exhibited coercivity of 4.9 kOe at 300 K. By imprinting the mixed chloroform solution of TPy-FePd-1 and polystyrene (PS) on Si, reproducible patterning of nanochains was formed due to the excellent self-assembly properties and the incompatibility between TPy-FePd-1 and PS under the slow evaporation of the solvents. The FePd nanochains with average length of ca. 260 nm were evenly dispersed around the PS nanosphere by self-assembly of TPy-FePd-1. In addition, the orientation of the FePd nanochains could also be controlled by tuning the morphology of PS, and the length was shorter in confined space of PS. Orgnic skeleton in TPy-FePd-1 and PS were carbonized and removed by pyrolysis under Ar/H2 (5 wt%) and only magnetic FePd alloy nanochains with domain structure were left. Besides, a bimetallic complex TPy-FePd-2 was prepared and used as a single-source precursor to synthesize ferromagnetic FePd NPs by one-pot pyrolysis. The resultant FePd NPs have a mean size of 19.8 nm and show the coercivity of 1.02 kOe. In addition, the functional group (-NCMe) in TPy-FePd-2 was easily substituted by a pyridyl group. A random copolymer PS-P4VP was used to coordinate with TPy-FePd-2, and the as-synthesized polymer made the metal fraction disperse evenly along the flexible chain. Fabrication of FePd NPs from the polymers was also investigated, and the size could be easily controlled by tuning the metal fraction in polymer. FePd NPs with the mean size of 10.9, 14.2 and 17.9 nm were prepared from the metallopolymer with 5 wt%, 10 wt% and 20wt% of metal fractions, respectively. In Chapter 4, molybdenum disulfide (MoS2) monolayers decorated with ferromagnetic FeCo NPs on the edges were synthesized through a one-step pyrolysis of precursor molecules in an argon atmosphere. The FeCo precursor was spin coated on the MoS2 monolayer grown on Si/SiO2 substrate. Highly-ordered body-centered cubic (bcc) FeCo NPs were revealed under optimized pyrolysis conditions, possessing coercivity up to 1000 Oe at room temperature. The FeCo NPs were well-positioned along the edge sites of MoS2 monolayers. The vibration modes of Mo and S atoms were confined after FeCo NPs decoration, as characterized by Raman shift spectroscopy. These MoS2 monolayers decorated with ferromagnetic FeCo NPs can be used for novel catalytic materials with magnetic recycling capabilities. The sizes of NPs grown on MoS2 monolayers are more uniform than from other preparation routines. Finally, the optimized pyrolysis temperature and conditions provide receipts for decorating related noble catalytic materials. Finally, Chapters 5 and 6 present the concluding remarks and the experimental details of the work described in Chapters 2-4.

This thesis presents an investigation into sources of variability in activating and processing numerical information. Chapter 1 provides an overview of research literature exploring the ways in which magnitude information can be represented, and how models relating to number information have developed. These theoretical models are addressed in relation to the neural representation of number, and the range of behavioural markers which suggest an association between spatial and numerical processing. Chapter 2 using a dual-task paradigm investigated whether magnitude information is accessed on perceiving numbers, or if this information is linked to response selection or execution. Previous research studies investigating this question produced inconsistent findings (Oriet, Tombu & Jolicoeur, 2005; Sigman & Dehaene, 2005) with regard to the locus of magnitude processing; the findings of Experiments 1-3 reliably support access to magnitude information during response selection. Chapter 3 explored the activation of spatial-numerical response associations, where response-irrelevant magnitude information was not represented by a single stimulus (i.e. an Arabic digit) but by a numerosity representation. Experiments 4-7 found a strong association between spatial-orientation processing and numerical magnitude, but no association with perceptual-colour processing, extending previous work by Fias, Lammertyn and Lauwereyns (2001) regarding the neural overlap between the attended and irrelevant stimulus dimensions. However the strength of this association was found to be inconsistent across the number range. Chapter 4 investigated the impact of healthy aging on the presence of neural-overlap in processing spatial-numerical information, further developing the paradigm used in Chapter 3, and addressed direct predictions from the literature as to how age should influence these associations (Wood, Willmes, Nuerk & Fischer, 2008). Experiments 8-11 found evidence for spatial-numerical associations across the lifespan, but that the strength of these effects were moderated by 5 task instruction. Chapter 5 was designed to assess aging differences in numerical and spatial processing with a battery of tests and the extent to which other sources of individual difference (sex, embodiment) have a measureable impact. A range of standardised measures were used to assess verbal ability, mathematical processing, and spatial working memory alongside behavioural measures of spatial numerical associations. Experiment 12 provided evidence of aging and sex differences in different cognitive tasks and a marginal impact of embodiment on spatial-numerical processing; however the effect of embodiment was not supported in a larger more homogenous sample in Experiment 13. Chapter 6 reflects on the current findings and provides contextual information on how they align with previous research, outlining how evidence from the thesis extend current research paradigms and provides new evidence regarding the maintenance of spatial-numerical associations in healthy aging. Methodologies developed in the thesis are considered with relation to how they may be applied to assess individual differences in early number acquisition in children. Finally the discussion outlines methods and controversies within the field of numerical cognition, with consideration of new methods for measuring the strength of spatial-numerical associations (Pinhas, Tzelgov, & Ganor-Stern, 2012), alongside the potential application of modelling techniques to investigate individual differences in task performance.

published_or_final_version / Mechanical Engineering / Doctoral / Doctor of Philosophy

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No description available.

The present thesis consists of two parts. The first part is devoted to the analysis of discretizations of a class of basic electronic density functionals. In the second part we suggest and analyze Quasicontinuum Methods for an atomistic interaction potential that is based on a field. We begin by formulating and analyzing a model for the study of finite clusters of atoms or localized defects in infinite crystals based on a version of the classical Thomas{Fermi{Dirac{von Weizs?acker density functional. We show that the resulting constrained optimization problem has a minimizer and we provide a careful analysis of the solvability of the associated Euler{Lagrange equation. Based on these results, and using tools from saddle-point theory and nonlinear analysis, we then show that a Galerkin discretization has a solution that converges to the correct limit (in the case of Dirichlet as well as periodic boundary conditions). Furthermore, we investigate the issue of optimal convergence rates. Using appropriate dual problems, we can show faster convergence for the energy, the Lagrange multiplier of the underlying minimization problem, and the L2-errors of the solutions. We also look at the dependence of the density functional on the nucleus coordinates and show a convergence result for minimizing nucleus configurations. These results are subsequently generalized to the case of discretizations with numerical integration. Existence and convergence of solutions, as well as optimal convergence rates can be established if quadrature rules of sufficiently high order are applied. In the second part of the thesis we consider an atomistic interaction potential in one dimension given through a minimization problem, which gives rise to a field. The forces on atoms are in this case given by local expressions involving this field. A convenient feature of this model is the existence of a weak formulation for the forces, which provides a natural connection point for the coupling with a continuum model. We suggest Quasicontinuum-like coupling mechanisms that are based on a decomposition of the domain into an atomistic and a continuum region. In the continuum region we use an approximation based on the Cauchy{ Born rule. In the atomistic subdomain a version of the atomistic model with Dirichlet boundary conditions is applied. Special attention has to be paid to the dependence of the atomistic subproblem on the boundary and the boundary conditions. Applying concepts from nonlinear analysis we show existence and convergence of solutions to the Quasicontinuum approximation.

The theory and the formulation of the special wave finite elements are discussed, and the special integration schemes for the elements are developed. Then the special wave infinite elements, a new concept of the mapped wave infinite elements with multiple wave directions, are developed. Computational models using these elements coupled together are tested by the applications of wave problems. In the special wave finite elements, the potential at each node is expanded in a discrete series of approximating plane waves propagating in different directions. Because of this a single element can contain many wavelengths, unlike the standard finite elements. This is a great advantage in the reduction of the degree of freedom of the problem, however the computational cost of the numerical integration over an element becomes high due to the oscillatory shape functions. Therefore the special semi-analytical integration schemes for the special wave finite elements are developed. The schemes are independent of wavenumber and efficient for short waves problems. In many cases of wave problems, it is practical to consider the domain as being infinite. However the finite element method can not deal with infinite domains. Infinite elements are an extension of the concept of finite elements in which the element has an infinite extent in one or more directions to address this limitation. In the special wave infinite element developed in this study multiple waves propagating in different directions are considered, in contrast to conventional infinite elements in which only a single wave propagating in the radial direction is considered. The shape functions of the special wave infinite elements contain trigonometric functions to describe multiple waves, and the amplitude decay factor to satisfy the radiation condition. The special wave infinite elements become a straightforward extension to the special wave finite elements for wave problems in an unbounded domain.

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 2014. / In this thesis we will establish effective numerical schemes appropriate for the solution of a non-linear partial differential equation modelling heat transfer in one dimensional longitudinal fins. We will consider the problem as it stands without any physical simplification. The main methodology is based on balancing the non-linear source term as well as the application of numerical relaxation techniques. In either approach we will incorporate the no-flux condition for singular fins. By doing so, we obtain appropriate numerical schemes which improve results found in literature. To generalize, we will provide a relaxed numerical scheme that is applicable for both integer and fractional order non-linear heat transfer equations for one dimensional longitudinal fins.

by Chung Shun Shing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 158-170). / Abstracts in English and Chinese. / List of Figures --- p.x / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Background and Historical Development --- p.1 / Chapter 1.2 --- Practical Problems --- p.3 / Chapter 1.2.1 --- Statistics --- p.3 / Chapter 1.2.2 --- Aerodynamics --- p.4 / Chapter 1.2.3 --- Factory Allocation Problem --- p.5 / Chapter 1.2.4 --- Parameter Problem --- p.5 / Chapter 1.2.5 --- Chemical Engineering --- p.5 / Chapter 1.2.6 --- Operational Research --- p.6 / Chapter 1.2.7 --- Economics --- p.6 / Chapter 1.3 --- Mathematical Models for Optimization Problems --- p.6 / Chapter 1.4 --- Unconstrained Optimization Techniques --- p.8 / Chapter 1.4.1 --- Direct Method - Differential Calculus --- p.8 / Chapter 1.4.2 --- Iterative Methods --- p.10 / Chapter 1.5 --- Main Objectives of the Thesis --- p.11 / Chapter 2 --- Basic Concepts in Optimizations of Smooth Func- tions --- p.14 / Chapter 2.1 --- Notation --- p.14 / Chapter 2.2 --- Different Types of Minimizer --- p.16 / Chapter 2.3 --- Necessary and Sufficient Conditions for Optimality --- p.18 / Chapter 2.4 --- Quadratic Functions --- p.22 / Chapter 2.5 --- Convex Functions --- p.24 / Chapter 2.6 --- "Existence, Uniqueness and Stability of a Minimum" --- p.29 / Chapter 2.6.1 --- Existence of a Minimum --- p.29 / Chapter 2.6.2 --- Uniqueness of a Minimum --- p.30 / Chapter 2.6.3 --- Stability of a Minimum --- p.31 / Chapter 2.7 --- Types of Convergence --- p.34 / Chapter 2.8 --- Minimization of Functionals --- p.35 / Chapter 3 --- Steepest Descent Method --- p.37 / Chapter 3.1 --- Background --- p.37 / Chapter 3.2 --- Line Search Method and the Armijo Rule --- p.39 / Chapter 3.3 --- Steplength Control with Polynomial Models --- p.43 / Chapter 3.3.1 --- Quadratic Polynomial Model --- p.43 / Chapter 3.3.2 --- Safeguarding --- p.45 / Chapter 3.3.3 --- Cubic Polynomial Model --- p.46 / Chapter 3.3.4 --- General Line Search Strategy --- p.49 / Chapter 3.3.5 --- Algorithm of Steepest Descent Method --- p.51 / Chapter 3.4 --- Advantages of the Armijo Rule --- p.54 / Chapter 3.5 --- Convergence Analysis --- p.56 / Chapter 4 --- Iterative Methods Using Second Derivatives --- p.63 / Chapter 4.1 --- Background --- p.63 / Chapter 4.2 --- Newton's Method --- p.64 / Chapter 4.2.1 --- Basic Concepts --- p.64 / Chapter 4.2.2 --- Convergence Analysis of Newton's Method --- p.65 / Chapter 4.2.3 --- Newton's Method with Steplength --- p.69 / Chapter 4.2.4 --- Convergence Analysis of Newton's Method with Step-length --- p.70 / Chapter 4.3 --- Greenstadt's Method --- p.72 / Chapter 4.4 --- Marquardt-Levenberg Method --- p.74 / Chapter 4.5 --- Fiacco and McComick Method --- p.76 / Chapter 4.6 --- Matthews and Davies Method --- p.79 / Chapter 4.7 --- Numerically Stable Modified Newton's Method --- p.80 / Chapter 4.8 --- The Role of the Second Derivative Methods --- p.89 / Chapter 5 --- Multi-step Methods --- p.92 / Chapter 5.1 --- Background --- p.93 / Chapter 5.2 --- Heavy Ball Method --- p.94 / Chapter 5.3 --- Conjugate Gradient Method --- p.99 / Chapter 5.3.1 --- Some Types of Conjugate Gradient Method --- p.99 / Chapter 5.3.2 --- Convergence Analysis of Conjugate Gradient Method --- p.108 / Chapter 5.4 --- Methods of Variable Metric and Methods of Conju- gate Directions --- p.111 / Chapter 5.5 --- Other Approaches for Constructing the First-order Methods --- p.116 / Chapter 6 --- Quasi-Newton Methods --- p.121 / Chapter 6.1 --- Disadvantages of Newton's Method --- p.122 / Chapter 6.2 --- General Idea of Quasi-Newton Method --- p.124 / Chapter 6.2.1 --- Quasi-Newton Methods --- p.124 / Chapter 6.2.2 --- Convergence of Quasi-Newton Methods --- p.129 / Chapter 6.3 --- Properties of Quasi-Newton Methods --- p.131 / Chapter 6.4 --- Some Particular Algorithms for Quasi-Newton Methods --- p.137 / Chapter 6.4.1 --- Single-Rank Algorithms --- p.137 / Chapter 6.4.2 --- Double-Rank Algorithms --- p.144 / Chapter 6.4.3 --- Other Applications --- p.149 / Chapter 6.5 --- Conclusion --- p.152 / Chapter 7 --- Choice of Methods in Optimization Problems --- p.154 / Chapter 7.1 --- Choice of Methods --- p.154 / Chapter 7.2 --- Conclusion --- p.157 / Bibliography --- p.158