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.

Approved for Public Release; Distribution is Unlimited / The impact of non-conventional observations and increased horizontal resolution on the numerical weather prediction (NWP) system of the National Center for Aeronautic Meteorology and Climatology of the Italian Air Force (CNMCA) has been investigated. The present study is part of ongoing research activities whose goal is the improvement of CNMCA's operational numerical weather prediction capabilities through the assimilation of non-conventional observations. Additional data derived from satellite observations, such as 10 m wind retrieved from Quikscat polar-orbit satellite, atmospheric motion vectors (AMVs) from Meteosat geostationary satellites and manual and automated aircraft observations were used. The NWP system, which is in operational use, is based on an "observation space" version of the 3D-Var method for the objective analysis component (3D-PSAS), while the prognostic component is based on the High Resolution Regional Model (HRM) of the German Meteorological Service (DWD). The analysis and forecast fields derived from the NWP system were objectively evaluated through comparisons with radiosonde and conventional surface observations. Comparisons with parallel runs of the HRM model starting from the 3D-Var operational analysis have showed that each of those observations have a measurable positive impact on forecast skill. / Captain, Italian Air Force

The realization of high-speed numeric computation is a sought-after commodity for real world applications, including high-speed scientific computation, digital signal processing, and embedded computers. An example of this is the generation of elementary functions, such as sin( ) x , x e and log( ) x . Sasao, Butler and Reidel [Ref. 1] developed a high speed numeric function generator using a look-up table (LUT) cascade. Their method used a piecewise linear segmentation algorithm to generate the functions [Ref. 1]. In this thesis, two alternative segmentation algorithms are proposed and compared to the results of Sasao, Butler and Reidel [Ref.1]. The first algorithm is the Constant Approximation. This algorithm uses lines of slope zero to approximate a curve. The second algorithm is the power-of-2-approximation. This method uses 2i x to approximate a curve. The constant approximation eliminates the need for a multiplier and adder, while the power-of-2-approximations eliminates the need for multiplier, thus improving the computation speed. Tradeoffs between the three methods are examined. Specifically, the implementation of the piecewise linear algorithm requires the most amount of hardware and is slower than the other two. The advantage that it has is that it yields the least amount of segments to generate a function. The constant approximation requires the most amount of hardware to realize a function, but is the fastest implementation. The power-of-2 approximation is an intermediate choice that balances speed and hardware requirements.

No description available.

No description available.

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.

When planning for a tunnel, the ground conditions in which the tunnel is going to be excavated through will be investigated to different extent. Lack of relevant pre-investigation data or misinterpretations of the available data can cause both economical and/or unexpected stability problems. Weakness zones that are expected to cross the tunnel could be investigated thoroughly with a variety of methods. Refraction seismicity survey and 2D resistivity survey are two geophysical methods that are common in Norway for obtaining information about the rock quality in weakness zones. In this work, a twin tunnel under construction in Bodø (northern Norway) called the Bodøtunnel is studied. The predictions based on the pre-investigation for crossing of some expected weakness zones are compared to the actual conditions encountered during tunneling. Tunneling observations (Geological mapping and photos), rock samples and measurement while drilling (MWD) were used to describe the weakness zones that were encountered during tunneling. Rock samples were collected from two weakness zones and the general rock mass. These samples were tested in a point bearing machine for determination of their uniaxial compressive strength (UCS). These results indicated that the rock samples gathered from the weakness zones had significantly lower UCS than the samples from the rock mass. This was exceedingly clear for the samples of fault rock gathered in connection with a shear zone. The results from this work demonstrate that refraction seismicity had a high success rate for locating weakness zones, with the exception for the crossed narrow zones that were interpreted lacking a shear component. Empirical formulas relating Q-value and UCS with the seismic wave speed were used for calculating these factors for some interesting locations. The empirically calculated UCS was similar to the obtained UCS from the point bearing tests, while the empirically calculated Q-value showed large deviations from the mapped Q-value. The resistivity measurements had a low success rate so far in this project; the reason for this could be disturbances in the ground and the location of the resistivity profiles, which had to adapted to the nearby railroad. It should be noted that only one full resistivity profile has been crossed and the rest of the profiles are expected to be more accurate. Based on the results from the crossed profile(s), the suitability of resistivity survey 2D in urban areas can be brought to question. This work also stumbled upon problems regarding the definition of weakness zones. Shear/fault zones are one of the more common type of weakness zones encountered in tunneling. These kind of zones often consists of different parts. Depending on which parts are regarded as a weakness zone by the responsible engineers, the Q-value might differ due to the SRF. Different scenarios were also evaluated with numerical modeling for the expected remaining major weakness zones. This analysis highlights the importance of differentiation between more fractured zones and zones containing fault rock, such as breccia. The width of the zone had a major impact on the stability while the dip for wide zones had a minor impact on the stability, as long the zones dip is not so small that both tunnels are intersected at the same time. The rock mechanical parameter of the weakness zones that had the most impact on the overall stability was the cohesion.

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 Yuen Wong. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1990. / Chapter Chapter 1 --- Introduction --- p.1 / Chapter Chapter 2 --- Basic Results --- p.5 / Chapter §2.1 --- General Results --- p.5 / Chapter §2.2 --- Williamson's Method --- p.8 / Chapter Chapter 3 --- Algorithms And Subroutines --- p.15 / Chapter §3.1 --- Introduction --- p.15 / Chapter §3.2 --- Increasing Determinant Method --- p.15 / Chapter §3.3 --- Williamson's Method - Direct Computation --- p.21 / Chapter §3.4 --- Williamson's Method - Increasing Determinant --- p.26 / Chapter Chapter 4 --- Comparisons And Recommendations On Algorithms --- p.32 / Chapter §4.1 --- Introduction --- p.32 / Chapter §4.2 --- Comparisons And Recommendations On IMPROV(N) --- p.32 / Chapter §4.3 --- Comparisons And Recommendations On GENHA(N) --- p.34 / Chapter §4.4 --- Comparisons And Recommendations On VTID(N) --- p.35 / Chapter §4.5 --- Summary --- p.37 / Chapter Chapter 5 --- Applications Of Hadamard Matrices --- p.38 / Chapter §5.1 --- Hadamard Matrices And Balanced Incomplete Block Designs' --- p.38 / Chapter §5.2 --- Hadamard Matrices And Optimal Weighing Designs --- p.43 / Chapter Chapter 6 --- Conclusion --- p.51 / References --- p.52 / Appendices --- p.53

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