Dissertations / Theses on the topic 'Second order difference array'

This thesis covers topics such as finite difference schemes, mean-square convergence, modelling, and numerical approximations of second order quasi-linear stochastic partial differential equations (SPDE) driven by white noise in less than three space dimensions. The motivation for discussing and expanding these topics lies in their implications in such physical phenomena as signal and information flow, gravitational and electromagnetic fields, large scale weather systems, and macro-computer networks. Chapter 2 delves into the hyperbolic SPDE in one space and one time dimension. This is an important equation to such fields as signal processing, communications, and information theory where singularities propagate throughout space as a function of time. Chapter 3 discusses some concepts and implications of elliptic SPDE's driven by additive noise. These systems are key for understanding steady state phenomena. Chapter 4 presents some numerical work regarding elliptic SPDE's driven by multiplicative and general noise. These SPDE's are open topics in the theoretical literature, hence numerical work provides significant insight into the nature of the process. Chapter 5 presents some numerical work regarding quasi-geostrophic geophysical fluid dynamics involving stochastic noise and demonstrates how these systems can be represented as a combination of elliptic and hyperbolic components.

Slotted waveguide array (SWGA) antennas find application in systems which require planarity, low profile, high power handling capabilities such as radars. In this thesis, a planar, low sidelobe, phased array antenna, capable of electronically beam scanning in E-plane is designed, manufactured and measured. In the design, slot characterization is done with HFSS and by measurements, and mutual coupling between slots are calculated analytically. A MATLAB code is developed for the synthesis of the SWGA antenna. Grating lobe problem in the scanning array, which is caused by the slot positions, is solved using baffles on the array. A high power feeding section for the planar array, having an amplitude tapering to get low sidelobes is also designed using a corrugated E-plane sectoral horn. The power divider is designed analytically, and simulated and optimized with HFSS.

Placing wave energy devices within close proximity to each other can be beneficial as the costs of deployment, maintenance and infrastructure are reduced significantly compared to if the devices are deployed in isolation. A mathematical model is presented in this thesis which combines linear wave theory with a series of linear driven harmonic oscillators to model an array (group) of floating wave energy devices which move predominantly in heave (vertically) in a train of incident regular waves. Whilst similar mathematical models have been used previously to investigate interactions between fluids and groups of structures, much of the published work does not address array configurations or device constraints that are relevant to designers of structure-supported array devices. The suitability of linear theory for application to closely spaced arrays is assessed in this thesis through comparison to small-scale experimental data and by evaluation of the magnitude of second-order hydrodynamic forces. Values of mechanical damping and mass are determined for each element of an array in order to achieve the maximum power from an array of floats without requiring the knowledge of the motion of every float within the array in order to apply the forces to any one float. Further to this, the analysis of floats of varying geometry is performed in order to assess the possibility of array optimisation through the variation of float geometries within a closely spaced array.It is shown in this thesis that linear theory provides a reasonable prediction of the response of floats that are sufficiently close together to interact for most wave frequencies to which the arrays are likely to be subjected. Under the assumption of easily implementable mechanical damping, it is determined that the power output from an array of floats of equal geometry can be increased by specifying different magnitudes of mechanical damping on each float independently of the radiation damping. Variations in submerged float geometries for the purpose of manipulating array characteristics according to the incident wave frequency are best applied through the variation in draft of a single geometry. Variations in submerged float geometry which occur close to the free surface are found to be of the greatest significance. Where the float is uniform in cross-section, the most appropriate method to select float drafts within an array is found to be based on the evaluation of the total damping on each float.

Este trabalho apresenta uma técnica de diferenças finitas para resolver a equação constitutiva Fluido de Segunda Ordem para escoamentos tridimensionais com superfície livre. As equações governantes são resolvidas pelo método de diferenças finitas em uma malha deslocada 3D. A superfície livre é modelada por células marcadoras (Marker-and-Cell) e as condições de contorno a superfície livre são empregadas. O método numérico apresentado neste trabalho foi validado pela comparação entre as soluções numéricas obtidas para o escoamento em um tubo com a solução analítica correspondente para Fluidos de Segunda Ordem. Ao fazer refinamento de malha, a convergência do método numérico foi verificada. Resultados numéricos da simulação do problema do inchamento do extrudado para números de Deborah De \'< OU =\' 0:3 são apresentados
This work presents a finite difference method to simulate three-dimensional viscoelastic flow with free surfaces governed by the constitutive equation Second Order Fluid. The governing equations are solved by the finite difference method in a three-dimensional shifted mesh. The free surface of fluid is modeled by the Marker-and-Cell method which allows for the visualization and the location of the free surface of fluid. The full free surface stress conditions are employed. The numerical method developed in this work is validated by comparing the numerical and analytic solutions for the steady state flow of a Second Order Fluid in a pipe. By using mesh refinement convergence results are given. Numerical results of the simulation of the transient extrudate swell of a Second Order Fluid of the Deborah number De \'< OR =\' 0:3 are presented

Fundação Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro
Considera-se neste trabalho um modelo matemático para escoamentos com duas e três fases em reservatórios petrolíferos e a modelagem computacional do sistema de equações governantes para a sua solução numérica. Os fluidos são imiscíveis e incompressíveis e as heterogeneidades da rocha reservatório são modeladas estocasticamente. Além disso, é modelado o fenômeno de histerese para a fase óleo via funções de permeabilidades relativas. No caso de escoamentos trifásicos água-óleo-gás a escolha de expressões gerais para as funções de permeabilidades relativas pode levar à perda de hiperbolicidade estrita e, desta maneira, à existência de uma região elíptica ou de pontos umbílicos para o sistema não linear de leis de conservação hiperbólicas que descreve o transporte convectivo das fases fluidas. Como conseqüência, a perda de hiperbolicidade estrita pode levar à existência de choques não clássicos (também chamados de choques transicionais ou choques subcompressivos) nas soluções de escoamentos trifásicos, de difícil simulação numérica. Indica-se um método numérico com passo de tempo fracionário, baseado em uma técnica de decomposição de operadores, para a solução numérica do sistema governante de equações diferenciais parciais que modela o escoamento bifásico água-óleo imiscível em reservatórios de petróleo heterogêneos. Um simulador numérico bifásico água-óleo eficiente desenvolvido pelo grupo de pesquisa no qual o autor está inserido foi modificado com sucesso para incorporar a histerese sob as hipóteses consideradas. Os resultados numéricos obtidos para este caso indicam fortes evidências que o método proposto pode ser estendido para o caso trifásico água-óleo-gás. A técnica de decomposição de operadores em dois níveis permite o uso de passos de tempo distintos para os quatro problemas definidos pelo procedimento de decomposição: convecção, difusão, pressão-velocidade e relaxação para histerese. O problema de transporte convectivo (hiperbólico) das fases fluidas é aproximado por um esquema central de diferenças finitas explícito, conservativo, não oscilatório e de segunda ordem. Este esquema é combinado com elementos finitos mistos, localmente conservativos, para a aproximação dos problemas de transporte difusivo (parabólico) e de pressão-velocidade (elíptico). O operador temporal associado ao problema parabólico de difusão é resolvido fazendo-se uso de uma estratégia implícita de solução (Backward Euler). Uma equação diferencial ordinária é resolvida (analiticamente) para a relaxação relacionada à histerese. Resultados numéricos para o problema bifásico água-óleo em uma dimensão espacial em concordância com resultados semi-analíticos disponíveis na literatura foram reproduzidos e novos resultados em meios heterogêneos, em duas dimensões espaciais, são apresentados e a extensão desta técnica para o caso de problemas trifásicos água-óleo-gás é proposta.
We consider in this work a mathematical model for two- and three-phase flow problems in petroleum reservoirs and the computational modeling of the governing equations for its numerical solution. We consider two- (water-oil) and three-phase (water-gas-oil) incompressible, immiscible flow problems and the reservoir rock is considered to be heterogeneous. In our model, we also take into account the hysteresis effects in the oil relative permeability functions. In the case of three-phase flow, the choice of general expressions for the relative permeability functions may lead to the loss of strict hyperbolicity and, therefore, to the existence of an elliptic region or umbilic points for the system of nonlinear hyperbolic conservation laws describing the convective transport of the fluid phases. As a consequence, the loss of hyperbolicity may lead to the existence of nonclassical shocks (also called transitional shocks or undercompressive shocks) in three-phase flow solutions. We present a new, accurate fractional time-step method based on an operator splitting technique for the numerical solution of a system of partial differential equations modeling two-phase, immiscible water-oil flow problems in heterogeneous petroleum reservoirs. An efficient two-phase water-oil numerical simulator developed by our research group was sucessfuly extended to take into account hysteresis effects under the hypotesis previously annouced. The numerical results obtained by the procedure proposed indicate numerical evidence the method at hand can be extended for the case of related three-phase water-gas-oil flow problems. A two-level operator splitting technique allows for the use of distinct time steps for the four problems defined by the splitting procedure: convection, diffusion, pressure-velocity and relaxation for hysteresis. The convective transport (hyperbolic) of the fluid phases is approximated by a high resolution, nonoscillatory, second-order, conservative central difference scheme in the convection step. This scheme is combined with locally conservative mixed finite elements for the numerical solution of the diffusive transport (parabolic) and the pressure-velocity (elliptic) problems. The time discretization of the parabolic problem is performed by means of the implicit Backward Euler method. An ordinary diferential equation is solved (analytically) for the relaxation related to hysteresis. Two-phase water-oil numerical results in one space dimensional, in which are in a very good agreement with semi-analitycal results available in the literature, were computationaly reproduced and new numerical results in two dimensional heterogeneous media are also presented and the extension of this technique to the case of three-phase water-oil-gas flows problems is proposed.

This thesis considers the problem of angle-of-arrival (AOA) estimation in the context of its application to electronic surveillance systems. Due to the operational requirements of such systems, the AOA estimation algorithm must be computationally fast, accurate and will need to be implemented using sparse, large aperture arrays. Interferometry is proposed as a suitable algorithm that meets the operational requirements of electronic surveillance systems. However, for sparse array geometries, phase wrapping effects introduce ambiguities to the phase measurements and so unambiguous AOA estimation requires the use of computationally intensive ambiguity resolution algorithms using three or more antennas. Beamforming and array processing techniques are another class of AOA estimation algorithms that can unambiguously estimate the AOA using sparse, large aperture arrays. While these techniques generally offer better AOA estimation performance than interferometric techniques, they are also comparatively more computationally intensive algorithms. Furthermore, by virtue of using very sparse arrays, high sidelobes in the array beampattern may cause incorrect AOA estimation. This thesis will introduce the concept of using second-order difference array (SODA) geometries which allow unambiguous AOA estimation to be performed in a computationally effcient manner. In the context of interferometry, the so-called “SODA interferometer" will be shown to synthesise the equivalent output of a smaller virtual aperture to allow unambiguous AOA estimation to be performed at the expense of a coarser estimation performance compared to the physical aperture of the array. It will also be shown that the coarse SODA AOA estimate can be used to cue the conventional ambiguity resolution algorithms to provide higher accuracy in a computationally efficient manner. This thesis will also show that the creation of virtual arrays from SODA geometries can be generalised to a larger number of antennas to allow conventional array processing techniques to perform unambiguous AOA estimation in a computationally fast manner. The AOA estimation performance of each algorithm is compared through simulations and also verified using experimental data. This thesis will show that the SODA interferometer, SODA-cued ambiguity resolution algorithms and so-called “second-order array processors" can be used to obtain high accuracy AOA estimates in a more computationally efficient manner than the conventional algorithms.
Thesis (Ph.D.) -- University of Adelaide, School of Electrical and Electronic Engineering, 2013

碩士
中原大學
資訊工程研究所
82
Kalman filter has been one of the most popular methods applied in the area of modern control, signal processing etc. But it can not process applications in real time , due to its complicated processing of matrix computations. The only way to extensive Kalman filter in real time processing is parallel processing. The goal of this paper is the parallizing processing of second-order Kalman filter. The second-order Kalman filter is to solve the second-order system state value , so must spend more computation. Systolic array can be made of VLSI chip, it has the advantage of processing great amount data. There are a lots of reseaches in first-order Kalman filter using systolic array. For the sake of parallelizing second-order Kalman filter, we analyze the dependency of the Kalman filter equation's data in advance. According to data dependency graphy. The ovelall design of implentention is partitioned into four stages. For reducing cost, we just design two kind of systolic array structures to process our second- order Kalman filter. When we arrange system structure, we use the concept of interarray pipelining, concurrent processing and the proper arrangment of equations to improve 19n^{3} computation time of sequential processing into 17n+5r-2. Using in the first-order Kalman filter, we need only 7n+5r-2. Since the matrix's dimension of the second-order is twice as that in the first-order system. Assume the dimension n=r, the computation time of the first-order and second-order are 12n-2 and 11n-2 respectively. Therefore the second-order Kalman filter is faster than the first-order filter. Our proposed mathod is faster than the first-order Kalman filter processing which has been proposed by Taylor and Liu. We propose this analytic structure which can be designed into VLSI chip. It can provide to various practical applications and will increase the efficiency and accuracy of the systems.

碩士
國立交通大學
應用數學系所
99
In this thesis, we discuss two distinct dynamics of the difference equation ∆[p∆x(t-1)]+qx(t)=f(x(t-1)) or f(x(t)), t∈Z, where ∆x(t-1)=ax(t)-bx(t-1). These two dynamics are the behavior of globally attracting and topological chaos. We have several results. Under some conditions of a, b, p and q, every orbit of the equation asymptotically converges to a global attractor. See theorems 2.2 and 2.3. If there exists a function relating to f which has more than one simple zeros or positive topological entropy at an expected parametric value, then the shift map restricted to the set of solutions of this equation has topological chaos. See theorems 2.6, 2.7, 2.8 and 2.9. Finally, we transform this equation into a parameterized continuous function by changing variables. We can also write it as the form of a discrete Hamiltonian system. For the case f(x(t)), theorem 2.10 says that there exists a function relating to f which has positive topological entropy such that the corresponding function has topological chaos. For the case f(x(t-1)), with an additional assumption that the function relating to f is locally trapping, theorem 2.11 says that the corresponding function has also topological chaos.

The determination of first-arrival seismic traveltimes radiating outward from a point-source in an arbitrary slowness field plays an important role in methods which require knowledge of curved wavefronts in a complex domain. Adaptations of the 1st order upwind finite difference Godunov and 2nd MUSCL schemes are presented which model the two dimensional Eikonal equation. These formulations prove to be a numerically straightforward and computationally efficient alternative to ray tracing and other finite difference methods. Initial results demonstrate convergence of the methods. Other examples are introduced and analyzed to demonstrate the effectiveness of the method in regions of velocity contrast. Specifically, we show that head waves and crossing rays are modeled correctly to produce a true first-arrival traveltime field. These examples demonstrate that the method is well suited for many structural velocity models, though success is not guaranteed if the family of computational fronts can not expand in the direction of increasing time.

Bragg reflection waveguides (BRW) or one-dimensional photonic bandgap structures have been demonstrated for phase-matching chi(2) nonlinearities in AlxGa1-xAs. The method exploits strong modal dispersion of a Bragg mode and total internal reflection modes co-propagating inside the waveguide. It is shown that phase-matching is attained among the lowest order modes of interacting harmonics, which allows maximizing the utilization of harmonics powers for nonlinear interactions. As our first demonstration, we report second-harmonic generation (SHG) of a 2-ps telecommunication pump in a 2.4 mm long slab BRW. The conversion efficiency is estimated as 2.0 %/W.cm^2 with a generated SH power of 729 nW. This efficiency has been considerably improved by introducing lateral confinement of optical modes in ridge structures. Characterizations denote that efficiency of SHG in ridge BRWs can increase by over an order of magnitude in comparison to that of the slab device. Also, we report continuous-wave SHG in BRWs. Using a telecommunication pump with a power of 98 mW, the continuous-wave SH power of 23 nW is measured in a 2.0 mm long device. Significant enhancements of chi(2) interactions is obtained in the modified design of matching-layer enhanced BRW (ML-BRW). For the first time, we report type-II SHG in ML-BRW, where the second-harmonic power of 60 µW is measured for a pump power of 3.3 mW in a 2.2 mm long sample. Also, we demonstrate the existence of type-0 SHG, where both pump and SH signal have an identical TM polarization state. It is shown that the efficiency of the type-0 process is comparable to type-I and type-II processes with the phase-matching wavelengths of all three interactions lying within a spectral window as small as 17 nm. ML-BRW is further reported for sum-frequency and difference-frequency generations. For applications requiring high pump power, a generalized ML-BRW design is proposed and demonstrated. The proposed structure offsets the destructive effects of third-order nonlinearities on chi(2) processes when high power harmonics are involved. This is carried out through incorporation of larger bandgap materials by using high aluminum content AlxGa1-xAs layers without undermining the nonlinear conversion efficiency. Theoretical investigations of BRWs as integrated sources of photon-pairs with frequency correlation properties are discussed. It is shown that the versatile dispersion properties in BRWs enables generation of telecommunication anti-correlated photon-pairs with bandwidth tunablity between 1 nm and 450 nm.

This thesis examines second-order optical nonlinear wave mixing processes in domain-disordered quasi-phase matching waveguides and evaluates their potential use in compact, monolithically integrated wavelength conversion devices. The devices are based on a GaAs/AlGaAs superlattice-core waveguide structure with an improved design over previous generations. Quantum-well intermixing by ion-implantation is used to create the quasi-phase matching gratings in which the nonlinear susceptibility is periodically suppressed. Photoluminescence experiments showed a large band gap energy blue shift around 70 nm after intermixing. Measured two-photon absorption coefficients showed a significant polarization dependence and suppression of up to 80% after intermixing. Similar polarization dependencies and suppression were observed in three-photon absorption and nonlinear refraction. Advanced modeling of second-harmonic generation showed reductions of over 50% in efficiency due to linear losses alone. Self-phase modulation was found to be the dominant parasitic nonlinear effect on the conversion efficiency, with reductions of over 60%. Simulations of group velocity mismatch showed modest reductions in efficiency of less than 10%. Experiments on second-harmonic generation showed improvements in efficiency over previous generations due to low linear loss and improved intermixing. The improvements permitted demonstration of continuous wave second-harmonic generation for the first time in such structures with output power exceeding 1 µW. Also, Type-II phase matching was demonstrated for the first time. Saturation was observed as the power was increased, which, as predicted, was the result of self-phase modulation when using 2 ps pulses. By using 20 ps pulses instead, saturation effects were avoided. Thermo-optically induced bistability was observed in continuous wave experiments. Difference frequency generation was demonstrated with wavelengths from the optical C-band being converted to the L- and U-bands with continuous waves. Conversion for Type-I phase matching was demonstrated over 20 nm with signal and idler wavelengths being separated by over 100 nm. Type-II phase matched conversion was also observed. Using the experimental data for analysis, self-pumped conversion devices were found to require external amplification to reach practical output powers. Threshold pump powers for optical parametric oscillators were calculated to be impractically large. Proposed improvements to the device design are predicted to allow more practical operation of integrated conversion devices based on quasi-phase matching superlattice waveguides.

Locating a radiating source based on range or range measurements obtained from a network of passive sensors has been a subject of research over the past two decades due to the problem’s importance in applications in wireless communications, surveillance, navigation, geosciences, and several other fields. In this thesis, we develop new solution methods for the problem of localizing a single radiating source based on range and range-difference measurements. Iterative re-weighting algorithms are developed for both range-based and range-difference-based least squares localization. Then we propose a penalty convex-concave procedure for finding an approximate solution to nonlinear least squares problems that are related to the range measurements. Finally, the sequential convex relaxation procedures are proposed to obtain the nonlinear least squares estimate of source coordinates. Localization in wireless sensor network, where the RF signals are used to derive the ranging measurements, is the primary application area of this work. However, the solution methods proposed are general and could be applied to range and range-difference measurements derived from other types of signals.
Graduate
0544
ismailds@uvic.ca

The fast-paced growth in microelectromechanical systems (MEMS), microfluidic fabrication, porous media applications, biomedical assemblies, space propulsion, and vacuum technology demands accurate and practical transport equations for rarefied gas flows. It is well-known that in rarefied situations, due to strong deviations from the continuum regime, traditional fluid models such as Navier-Stokes-Fourier (NSF) fail. The shortcoming of continuum models is rooted in nonequilibrium behavior of gas particles in miniaturized and/or low-pressure devices, where the Knudsen number (Kn) is sufficiently large. Since kinetic solutions are computationally very expensive, there has been a great desire to develop macroscopic transport equations for dilute gas flows, and as a result, several sets of extended equations are proposed for gas flow in nonequilibrium states. However, applications of many of these extended equations are limited due to their instabilities and/or the absence of suitable boundary conditions. In this work, we concentrate on regularized 13-moment (R13) equations, which are a set of macroscopic transport equations for flows in the transition regime, i.e., Kn≤1. The R13 system provides a stable set of equations in Super-Burnett order, with a great potential to be a powerful CFD tool for rarefied flow simulations at moderate Knudsen numbers. The goal of this research is to implement the R13 equations for problems of practical interest in arbitrary geometries. This is done by transformation of the R13 equations and boundary conditions into general curvilinear coordinate systems. Next steps include adaptation of the transformed equations in order to solve some of the popular test cases, i.e., shear-driven, force-driven, and temperature-driven flows in both planar and curved flow passages. It is shown that inexpensive analytical solutions of the R13 equations for the considered problems are comparable to expensive numerical solutions of the Boltzmann equation. The new results present a wide range of linear and nonlinear rarefaction effects which alter the classical flow patterns both in the bulk and near boundary regions. Among these, multiple Knudsen boundary layers (mechanocaloric heat flows) and their influence on mass and energy transfer must be highlighted. Furthermore, the phenomenon of temperature dip and Knudsen paradox in Poiseuille flow; Onsager's reciprocity relation, two-way flow pattern, and thermomolecular pressure difference in simultaneous Poiseuille and transpiration flows are described theoretically. Through comparisons it is shown that for Knudsen numbers up to 0.5 the compact R13 solutions exhibit a good agreement with expensive solutions of the Boltzmann equation.