Encounters with turbulence generated by complex topography, convection, or mechanical forcing present a significant threat to military aircraft operations. Properly forecasting the initiation, duration, and intensity of such encounters is a tremendous challenge to forecasters often resulting in the over-forecasting of turbulence. Over-forecasting the presence or intensity of turbulence can result in unnecessary mission delays, cancellations, and re-routing. The lack of observations and the fact that turbulence is a microscale phenomenon which Numerical Weather Prediction (NWP) models currently can not resolve are what make forecasting turbulence so difficult. Progress has been made in the last several decades in both the observation of turbulence and the resolution of NWP models. A new turbulence forecast approach has been created based on recent developments in observing turbulence and using automated turbulence diagnostics. The development of an in-situ observation platform, using the Eddy Dissipation Rate (EDR), and the Graphical Turbulence Guidance (GTG) model are discussed. A turbulence forecast approach is derived that includes the synoptic patterns which create or allow the turbulent environment to exist, the use of current tools to observe turbulence, and the use of models to help form the turbulence forecast. A turbulence forecasting manual has been created to give the new forecaster improved guidance to effectively forecast turbulence.

CIVINS / Due to environmental regulations, waste water disposal for US Navy ships has become a requirement which impacts both operations and the US Navy's budget. In 2006, the cost for waste water disposal Navy-wide was 54 million dollars. There are many advanced waste water treatment technologies in the research and development stage at academic institutions, private corporations, and government labs. Additionally, considerable progress has been made in installing and operating unique waste water treatment systems onboard merchant and commercial vessels, showing that waste water treatment technologies are near the maturity level required for installation on US Navy ships. Installation and operations costs can be estimated from data collected from merchant ships, but the accompanying life cycle liquid disposal costs savings can be difficult to estimate. A cost estimator is presented which allows variations in ship's operational schedule and aids in determining the total life cycle savings, and the time for return on investment, when waste destruction technologies are installed in a class of ship. Additionally, the properties of one waste water destruction medium, supercritical water, are reviewed and its use in efficient and environmentally safe chemical processes are discussed. In particular, supercritical water is the medium of choice for the performance of a biomass to synthetic natural gas conversion process. The supercritical water is utilized to aid in a vital salt separation process which allows for efficient 4 hydrothermal gasification. Numerical simulations of the salt separation process are completed which help in understanding the flow properties. The results will aid in yielding an optimized salt separation process, improving the efficiency and viability of the conversion process. / CIVINS

This paper explores the potential applications of existing spectral clustering algorithms to real life problems through experiments on existing road traffic data. The analysis begins with an overview of previous unsupervised machine learning techniques and constructs an effective spectral clustering algorithm that demonstrates the analytical power of the method. The paper focuses on the spectral embedding method’s ability to project non-linearly separable, high dimensional data into a more manageable space that allows for accurate clustering. The key step in this method involves solving a normalized eigenvector problem in order to construct an optimal representation of the original data. While this step greatly enhances our ability to analyze the relationships between data points and identify the natural clusters within the original dataset, it is difficult to comprehend the eigenvalue representation of the data in terms of the original input variables. The later sections of this paper will explore how the careful framing of questions with respect to available data can help researchers extract tangible decision driving results from real world data through spectral clustering analysis.

In this thesis, we develop multiscale models for particle simulations in population dynamics. These models are characterised by prescribing particle motion on two spatial scales: microscopic and macroscopic.At the microscopic level, each particle has its own mass, position and velocity, while at the macroscopic level the particles are interpolated to a continuum quantity whose evolution is governed by a system of transport equations.This way, one can prescribe various types of interactions on a global scale, whilst still maintaining high simulation speed for a large number of particles. In addition, the interplay between particle motion and interaction is well tuned in both regions of low and high densities. We analyse links between models on these two scales and prove that under certain conditions, a system of interacting particles converges to a nonlinear coupled system of transport equations.We use this as a motivation to derive a model defined on both modelling scales and prescribe the intercommunication between them. Simulation takes place in inhomogeneous domains with arbitrary conditions at inflow and outflow boundaries. We realise this by modelling obstacles, sources and sinks.Integrating these aspects into the simulation requires a route planning algorithm for the particles. Several algorithms are considered and evaluated on accuracy, robustness and efficiency. All aspects mentioned above are combined in a novel open source prototyping simulation framework called Mercurial. This computational framework allows the design of geometries and is built for high performance when large numbers of particles are involved. Mercurial supports various types of inhomogeneities and global systems of equations. We apply our framework to simulate scenarios in crowd dynamics.We compare our results with test cases from literature to assess the quality of the simulations. / <p>Master Thesis in Industrial and Applied Mathematics</p>

The purpose of this paper is to investigate the nature and practical uses of Chebyshev polynomials. Chapter I gives recognition to mathematicians responsible for studies in this area. Chapter II enumerates several mathematical situations in which the polynomials naturally arise and suggests reasons for the pursuance of their study. Chapter III includes: Chebyshev polynomials as related to "best" polynomial approximation, Chebyshev series, and methods of producing polynomial approximations to continuous functions. Chapter IV discusses the use of Chebyshev polynomials to solve certain differential equations and Chebyshev-Gauss quadrature.

M.Sc. (Applied Mathematics) / Numerous studies on linear and nonlinear Volterra integral equations (VIEs), have been performed. These studies mainly considered the existence and uniqueness of the solution, and numerical solutions of these equations. In this work, a class of nonlinear (nonstandard) Volterra integral equation that has received very little attention in the literature is considered. The existence and uniqueness of the solution for the nonlinear VIE is proved using the contraction mapping theorem in the space C[0; d]. Collocation methods, repeated trapezoidal rule and repeated Simpson's rule are used to solve the nonlinear (nonstandard) VIE. For the collocation solutions we considered two cases: implicit Euler method and implicit midpoint method. Examples are used to compare the performance of these methods and the results show that the repeated Simpson's rule performs better than the other methods. An analysis of the collocation solution and the solution by the repeated trapezoidal rule is performed. Su cient conditions for existence and uniqueness of the numerical solution are given. The collocation methods and repeated trapezoidal rule yield convergence of order one.

A Dissertation submitted to the Faculty of Science, University of the Witwatersrand, in fulfillment of the requirements for the degree of Master of Science. Johannesburg, 2013 / Many of the problems in mathematics have very elegant solutions. As complex, real–world geometries come into play, however, this elegance is often lost. This is particularly the case with meshes of physical, real–world problems. Domain mapping helps to move problems from some geometrically complex domain to a regular, easy to use domain. Shape transformation, specifically, allows one to do this in 2D domains where mesh construction can be difficult. Numerical methods usually work over some mesh on the target domain. The structure and detail of these meshes affect the overall computation and accuracy immensely. Unfortunately, building a good mesh is not always a straight forward task. Finite Element Analysis, for example, typically requires 4–10 times the number of tetrahedral elements to achieve the same accuracy as the corresponding hexahedral mesh. Constructing this hexahedral mesh, however, is a difficult task; so in practice many people use tetrahedral meshes instead. By mapping the geometrically complex domain to a regular domain, one can easily construct elegant meshes that bear useful properties. Once a domain has been mapped to a regular domain, the mesh can be constructed and calculations can be performed in the new domain. Later, results from these calculations can be transferred back to the original domain. Using harmonic functions, source domains can be parametrised to spaces with many different desired properties. This allows one to perform calculations that would be otherwise expensive or inaccurate. This research implements and extends the methods developed in Voruganti et al. [2006 2008] for domain mapping using harmonic functions. The method was extended to handle cases where there are voids in the source domain, allowing the user to map domains that are not topologically equivalent to the equivalent dimension hypersphere. This is accomplished through the use of various boundary conditions as the void is mapped to the target domains which allow the user to reshape and shrink the void in the target domain. The voids can now be reduced to arcs, radial lines and even shrunk to single points. The algorithms were implemented in two and three dimensions and ultimately parallelised to run on the Centre for High Performance Computing clusters. The parallel code also allows for arbitrary dimension genus-0 source domains. Finally, applications, such as remeshing and robot path planning were investigated and illustrated.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Master of Science. Johannesburg, 2014. / This thesis aims to provide a summary on computational approaches to solving the Compressed Sensing problem. The theoretical problem of solving systems of linear equations has long been investigated in academic literature. A relatively new field, Compressed Sensing is an application of such a problem. Specifically, with the ability to change the way in which we obtain and process signals. Under the assumption of sparse signals, Compressed Sensing is able to recover signals sampled at a rate much lower than that of the current Shannon/Nyquist sampling rate. The primary goal of this thesis, is to describe major algorithms currently used in the Compressed Sensing problem. This is done as a means to provide the reader with sufficient up to date knowledge on current approaches as well as their means of implementation, on central processing units (CPUs) and graphical processing units (GPUs), when considering computational concerns such as computational time, storage requirements and parallelisability.

A dissertation submitted to the Faculty of Engineering and the Built Environment, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Master of Science in Engineering Johannesburg, May 2017 / Flow around flat plates that were uniformly accelerated from rest with acceleration of 13g is analysed with overset mesh from Star CCM+ commercial CFD software. The particular interest is more on the vortices shed from the plate edges. Three 8mm thick plates of the same cross-sectional areas (108mm length equilateral triangular, 71mm length square and 80mm diameter circular) were simulated. The validation of the numerical method was achieved by using laser vapor sheet method to visualize the flow profiles of accelerating circular plate and comparing the CFD and experimental results. The CFD and experimental results were consistent with each other. It was found that when a plate accelerated in air, it displaced air particles out of its way. The shear layers of air separated from the front edges of the plate and rolled around a vortex core forming a primary vortex ring in the plate wake. The size of the primary vortex increased with Reynolds number (Re) that was increasing with time. This was because as Re increased, more fluid particles were displaced from the front face of the plate at a time. More displacement of the fluid particles led to shear layers separating from the plate edges with stronger momentum resulting in larger vortex ring. The shape of the primary vortex depended on the shape of the accelerating plate. For the circular plate, all the points on the front edge being equidistant from the plate centroid, fluid particles were evenly displaced from that separation edge. The result was an axis-symmetric ring of primary vortex around a circular vortex core. The asymmetric plates (triangular and square) did not evenly displace air particles from their edges of separation. The result was an asymmetric vortex ring. More air particles separated from the plate at separation points closest to the plate centroid and led to the largest vortical structure there. That is; the primary vortex ring was largest at the midpoints of the plate edges because they were the closest points of separation from the plate centroid. The size of the primary vortex continuously reduced from the mid-points of the plate edges to the corners. The corners had the smallest primary vortical structure due to being furthest points of separation from the plate centroid. The parts of the vortex ring from the two edges of the plate interacted at the corner connecting those edges. / MT 2017

Este trabalho apresenta um estudo a respeito do comportamento do solo durante a escavação de um túnel, com foco na metodologia de escavação mecanizada. São abordadas diretrizes para a modelagem numérica de túneis com o emprego de diferentes métodos de cálculo, distintos modelos constitutivos para representação do comportamento do solo e duas maneiras de se representar o revestimento da escavação. Alguns conceitos fundamentais relacionados a escavações de túneis de grande diâmetro em solo com o emprego de máquinas tuneladoras são descritos, como os tipos de máquinas existentes (com ênfase nas máquinas do tipo EPB - Earth Pressure Balance), princípios de funcionamento e modos de operação, além de uma comparação deste método com a metodologia sequêncial tradicional. É realizada uma revisão bibliográfica sobre distintos métodos de cálculo para estimar a estabilidade de uma escavação, a previsão da deformação no maciço e os esforços atuantes no revestimento do túnel. O trabalho também apresenta uma revisão de publicações recentes a respeito da modelagem numérica de túneis escavados mecanicamente, destacando algumas diferenças em relação à metodologia sequencial. Uma das diferenças é com relação à simulação do revestimento do túnel, com a possibilidade de se considerar as propriedades das juntas dos anéis. Outro ponto distinto é o método de cálculo empregado, levando em conta na simulação aspectos específicos da escavação mecanizada, como pressão de frente aplicada, vazio anelar e injeção de grout. É feita uma breve introdução e são apresentados os conceitos básicos de dois modelos constitutivos do solo: o modelo elastoplástico perfeito conhecido como Mohr-Coulomb e o modelo elastoplástico com endurecimento Hardening Soil. Utilizando alguns dos conceitos estudados na revisão bibliográfica, são realizadas análises numéricas bidimensionais retroanalisando um caso real de escavação com tuneladora. Foi utilizado o programa de elementos finitos Plaxis para realizar comparações entre os métodos de cálculo do alívio de tensões e da contração, além da simulação do revestimento com ou sem a consideração das juntas. Por fim é feita uma análise crítica dos resultados obtidos nas diferentes modelagens numéricas. / This research presents a study about the soil behavior during the excavation of a tunnel, focusing on the mechanized excavation methodology. Guidelines for the numerical modeling of tunnels with the use of different calculation methods, distinct constitutive models to characterize the soil behavior and two ways of representing the lining of the excavation are addressed. Relevant aspects regarding excavations of large diameter tunnels in soil with the use of TBM (Tunnel Boring Machine) are presented, like types of machines, with the focus on the Earth Pressure Balance (EPB) machines, fundamental concepts of the methodology and operation modes, besides a comparison of this method with the traditional sequential methodology. A literature review on different methods of calculation to estimate the excavation stability, the prediction of ground deformation and the efforts acting in the tunnel lining is made. The research also presents a review of recent publications regarding numerical modeling of mechanically excavated tunnels, highlighting some differences with the sequential method. One of these differences is related to the simulation of the tunnel lining, with the possibility of considering the properties of the ring joints. Another different point is the calculation method employed, considering in the simulation specific aspects of mechanized excavation, as front pressure applied, ring void and grout injection. A brief introduction is made about constitutive models to represent the soil behavior and the basic concepts of two constitutive soil models are presented: Mohr-Coulomb and Hardening Soil. Using some of the concepts studied in the literature review, it is performed two-dimensional numerical analysis with a back-analysis of a real case, using the finite element model program Plaxis, comparing the calculation methods of stress relieving and contraction, as well as the tunnel lining simulation with or without consideration of the joints. Finally, it is made a critical analysis of the results of the numerical simulations.