Dissertations / Theses on the topic 'Orthogonal coordinates'

PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO
CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO
O trabalho desenvolveu uma metodologia de solução numérica de escoamentos em geometrias complexas, numa formulação incompressível e bi-dimensional. As equações de conservação são discretizas com o emprego da técnica de volumes finitos em coordenadas não ortogonais. Esta técnica mapeia o espaço real num espaço transformado, no qual as fronteiras do domínio de cálculo coincidem com as fronteiras do domínio físico. Os componentes contravariantes da velocidade foram empregados como variáveis dependentes nas equações de conservação de quantidade de movimento. Estas equações foram obtidas em coordenadas não ortogonais pela manipulação algébrica das equações discretizadas para os componentes cartesianos. Este procedimento, que emprega um sistema de coordenadas auxiliar fixo localmente, evita o surgimento dos diversos termos oriundos da curvutura e da não ortogonalidade da malha, que seriam obtidos caso fosse empregada a análise tensorial para a derivação destas equações. O ocoplamento pressão-velocidade é feito utilizando SIMPLEC. O conjunto de equações algébricas resultante é resolvido por um esquema de solução segregado, no qual é empregado um esquema de solução linha-a-a linha(TDMA), com um processo de correção por blocos para acelerar a convergência. A metodologia desenvolvida foi utilizada para solução de diversos problemas visando analisar o seu desempenho. Foram estudados os seguintes casos-escoamento laminar entre dois cilindros, convecção natural entre dois cilidros excêntricos, escoamento induzido numa cavidade trapezoidal pelo movimento de suas bases, escoamento laminar num canal, escoamento axi-simétrico num duto com estrangulamento.Tendo em vista os bons resultados obtidos para testes, pode-se concluir que as opções realizadas para a confeção do esquema desenvolvido foram corretas, pois geraram um algoritimo efeciente e versátil.
A solution method for bi-dimensional incompressibible fluid flow problems in complex geometrics is developed in this work. The method solves the conservation equations in nonorthogonal coordinate system using the finite volumes technique. The contravariant velocities are kept as dependent variables in the momentum equations. These equations are obtained by an algebric manipulation of the discretization equations written in locally fixed coordinate system. This producedure avoids the treatment of the extra terms if the discretization equations for the curvilinear velocities are obtained in the conventional manner. The coupling of pressure and velocities are performed by the SIMPLEC algorithm. The set of algebric equations are solved using an iterative method in conjunction with coefficient update for linerization. In the computer implementation of the proposed scheme a line-by-line algorithm (TDMA) has been employed with a block corretion procedure to enhance the convergence. The method is tested by solving a variety of problems. The problems include-flow between two concentric rotating cylinders, natural convection in an eccentric annuli, driven flow in a trapezoidal cavity with moving lids, laminar flow in a channel, exismetric flow in duct with reduced cross section and laminar and turbulent flow through a tube with an axisimetric constriction. The objetive of these tests is to establish the validity of the proposed scheme and demonstrate its applicability to a wide variety of problems.

This thesis is concerned with the approximation of Cartesian coordinate data by parametric curves and surfaces, with an emphasis upon a technique known as parametric orthogonal distance regression (parametric ODR). The technique has become increasingly popular in the literature over the past decade and has applications in a wide range of fields, including metrology-the science of measurement, and computer aided design (CAD) modelling. Typically, the data are obtained by recording points measured in the surface of some physical artefact, such as a manufactured part. Parametric ODR involves minimizing the shortest distances from the data to the curve or surface in some norm. Under moderate assumptions, these shortest distances are orthogonal projections from the data onto the approximant, hence the nomenclature ODR. The motivation behind this type of approximation is that, by using a distance-based measure, the resulting best fit curve or surface is independent of the position or orientation of the physical artefact from which the data is obtained. The thesis predominately concerns itself with parametric ODR in a least squares setting, although it is indicated how the techniques described can be extended to other error measures in a fairly straightforward manner. The parametric ODR problem is formulated mathematically, and a detailed survey of the existing algorithms for solving it is given. These algorithms are then used as the basis for developing new techniques, with an emphasis placed upon their efficiency and reliability. The algorithms (old and new) detailed in this thesis are illustrated by problems involving well-known geometric elements such as lines, circles, ellipse and ellipsoids, as well as spline curves and surfaces. Numerical considerations specific to these individual elements, including ones not previously reported in the literature, are addressed. We also consider a sub-problem of parametric ODR known as template matching, which involves mapping in an optimal way a set of data into the same frame of reference as a fixed curve or surface.

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Made available in DSpace on 2018-02-07T13:41:46Z (GMT). No. of bitstreams: 1 VERA SOLANGE DE OLIVEIRA FARIAS – TESE PPGEP 2011.pdf: 7179434 bytes, checksum: 4a30c9a95f4a089e00fa550fbf1b42b8 (MD5) Previous issue date: 2011-04-29
CNPq
Este trabalho apresenta a solução numérica da equação de difusão tridimensional em regime transiente, para um domínio arbitrário. Para atingir os objetivos, a equação de difusão foi discretizada usando coordenadas generalizadas via método dos volumes finitos com uma formulação totalmente implícita, para condições de contorno de equilíbrio e convectiva. Para cada passo no tempo, o sistema de equações obtido para uma dada malha estruturada foi resolvido pelo método de Gauss-Seidel. O código computacional foi desenvolvido em FORTRAN, usando o estúdio CVF 6.6.0, na plataforma Windows Vista. A solução proposta foi validada usando soluções analíticas e numéricas da equação de difusão para várias geometrias, permitindo validar malhas ortogonais e não-ortogonais. A análise e comparação dos resultados mostraram que a solução proposta forneceu resultados coerentes para todos os casos investigados. O código computacional desenvolvido foi aplicado na simulação, a partir de dados experimentais da secagem de telhas cerâmicas para as seguintes condições experimentais: temperaturas de 55,6 °C, 69,7 °C, 82,7 °C e 98,6 °C e teor de umidade inicial variando de 0,2345 até 0,2405 (b.s.). A simulação tornou possível determinar o coeficiente de difusão efetivo em função da razão de umidade e da temperatura do ar de secagem e também o valor do coeficiente de transferência convectivo de massa correspondente para cada temperatura.
This work presents a three-dimensional numerical solution for the diffusion equation in transient state, in an arbitrary domain. The diffusion equation was discretized using the finite volume method with a fully implicit formulation and generalized coordinates, for the equilibrium and convective boundary condition. For each time step, the system of equations obtained for a given structured mesh was solved by the Gauss-Seidel method. A computational code in FORTRAN, using the CFV 6.6.0 Studio, in a Windows Vista platform was developed. The proposed solution was validated by analytical and numerical solutions of the diffusion equation for several geometries. The geometries tested enabled to validate both orthogonal and non-orthogonal meshes. The analysis and comparison of the results showed that the proposed solution provides correct results for all cases investigated. The developed computational code was applied in the simulation, using experimental data of the drying of ceramic roof tiles, for the following experimental conditions: temperature from 55.6; 69.7; 82.7; 72.8 and 98.7 °C, initial moisture content from 0.2345 up to 0.2405 (d.b.). The simulation makes it possible to determine an expression for the diffusion coefficient as a function of the moisture content and temperature of the drying air, and also the value of the convective mass transfer coefficient corresponding to each temperature.

In the first part of this dissertation, a novel medium access protocol for the Internet of Thing (IoT) networks is introduced. The Internet of things (IoT), which is the network of physical devices embedded with sensors, actuators, and connectivity, is being accelerated into the mainstream by the emergence of 5G wireless networking. This work presents an uncoordinated non-orthogonal random-access protocol, which is an enhancement to the recently introduced slotted ALOHA- NOMA (SAN) protocol that provides high throughput, while being matched to the low complexity requirements and the sporadic traffic pattern of IoT devices. Under ideal conditions it has been shown that slotted ALOHA-NOMA (SAN), using power- domain orthogonality, can significantly increase the throughput using SIC (Successive Interference Cancellation) to enable correct reception of multiple simultaneous transmitted signals. For this ideal performance, the enhanced SAN receiver adaptively learns the number of active devices (which is not known a priori) using a form of multi-hypothesis testing. For small numbers of simultaneous transmissions, it is shown that there can be substantial throughput gain of 5.5 dB relative to slotted ALOHA (SA) for 0.07 probability of transmission and up to 3 active transmitters. As a further enhancement to SAN protocol, the SAN with beamforming (BF-SAN) protocol was proposed. The BF-SAN protocol uses beamforming to significantly improve the throughput to 1.31 compared with 0.36 in conventional slotted ALOHA when 6 active IoT devices can be successfully separated using 2×2 MIMO and a SIC (Successive Interference Cancellation) receiver with 3 optimum power levels. The simulation results further show that the proposed protocol achieves higher throughput than SAN with a lower average channel access delay. In the second part of this dissertation a novel Machine Learning (ML) approach was applied for proactive mobility management in 5G Virtual Cell (VC) wireless networks. Providing seamless mobility and a uniform user experience, independent of location, is an important challenge for 5G wireless networks. The combination of Coordinated Multipoint (CoMP) networks and Virtual- Cells (VCs) are expected to play an important role in achieving high throughput independent of the mobile’s location by mitigating inter-cell interference and enhancing the cell-edge user throughput. User- specific VCs will distinguish the physical cell from a broader area where the user can roam without the need for handoff, and may communicate with any Base Station (BS) in the VC area. However, this requires rapid decision making for the formation of VCs. In this work, a novel algorithm based on a form of Recurrent Neural Networks (RNNs) called Gated Recurrent Units (GRUs) is used for predicting the triggering condition for forming VCs via enabling Coordinated Multipoint (CoMP) transmission. Simulation results show that based on the sequences of Received Signal Strength (RSS) values of different mobile nodes used for training the RNN, the future RSS values from the closest three BSs can be accurately predicted using GRU, which is then used for making proactive decisions on enabling CoMP transmission and forming VCs. Finally, the work in the last part of this dissertation was directed towards applying Bayesian games for cell selection / user association in 5G Heterogenous networks to achieve the 5G goal of low latency communication. Expanding the cellular ecosystem to support an immense number of connected devices and creating a platform that accommodates a wide range of emerging services of different traffic types and Quality of Service (QoS) metrics are among the 5G’s headline features. One of the key 5G performance metrics is ultra-low latency to enable new delay-sensitive use cases. Some network architectural amendments are proposed to achieve the 5G ultra-low latency objective. With these paradigm shifts in system architecture, it is of cardinal importance to rethink the cell selection / user association process to achieve substantial improvement in system performance over conventional maximum signal-to- interference plus noise ratio (Max-SINR) and Cell Range Expansion (CRE) algorithms employed in Long Term Evolution- Advanced (LTE- Advanced). In this work, a novel Bayesian cell selection / user association algorithm, incorporating the access nodes capabilities and the user equipment (UE) traffic type, is proposed in order to maximize the probability of proper association and consequently enhance the system performance in terms of achieved latency. Simulation results show that Bayesian game approach attains the 5G low end-to-end latency target with a probability exceeding 80%.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
The Geodesy methods and equipments progress brings with himself the growing need on geodetic surveys. Thereby, it is possible to make compatible any surveys with no extension boundary like from plane models and with no points overlap. These surveys are usually from geodetic satellites for that specific purpose. However, there are cases under precision restriction by nature of these systems, as well as physical conditions lack and equipments deficiency disables such operations. Thus, it becomes necessary to take place surveys from a topographical origin to a geocentric origin. To other cases, it is not enough just to know the geodetic coordinates but it is necessary to know its equivalent ones in the topographical plan, mostly to section and calculus of areas besides point locations. In these cases a solution of great use is the transformation of the geocentric coordinates to the UTM Projection plane cartographic coordinates, usually being ignored the deformations. Therefore it is a mistaken solution. Among the appropriate solutions to transform coordinates between surfaces from a topographical origin to a geocentric origin it is the Orthogonal Transformation. This method is more concise than the traditional Puissant´s methodology and to make possible the conversion of topographical coordinateds to geocentric coordinates and vice-versa. In this work the precision of Orthogonal Transformation method was compared initially with the Puissant´s method. Also were compared the differences among areas calculated from the UTM cartographic plan coordinates and topographical coordinates, besides having established the position discrepancy among boundary points calculated in a certain plan and implanted in other one without the true conversions. The results shows that the Orthogonal Transformation method precision is equivalent to Puissant´s method for observed ranges. It was possible to note that the values to areas calculated from coordinates related to models or different surfaces presents discrepancy. The areas section and estimation from the UTM cartographic projection plane coordinates, even with scale factor and elevation corrections, has been different to the area from the local topographical plan, although it was similar when under corrections. Points from section of geodetic areas can not to have implant in the topographical plane under use of UTM cartographic projection plane coordinates due to the position difference between it and the topographical coordinates. The more appropriate solution, due to good precision and easiness to section and estimate geodetic areas from geocentric coordinates, is the change to topographical ones using the Orthogonal Transformation. The Orthogonal Transformation of Coordinates is a practice and quick solution to make geodetic points from topographical surveys as well as to plot geodetic points in the topographical plan, and it is possible to implement it in the Electronic Total Stations in a easy way.
O avanço da geodésia, em seus métodos e equipamentos traz consigo a crescente necessidade da realização de levantamentos georreferenciados. Assim é possível compatibilizar vários levantamentos sem limites de extensão impostos pelo modelo plano e sem a sobreposição de pontos. Estes levantamentos geralmente são executados pelo rastreio de satélites que operam para este fim. Contudo, em certos casos as limitações de precisão e acurácia impostos pelos princípios destes sistemas, bem como a falta de condições físicas e deficiência de equipamentos impossibilita tal operação. Torna-se necessário então realizar levantamentos com a origem topocêntrica e transforma-la em geocêntrica. Em outros casos, não basta conhecer somente as coordenadas georreferenciadas, sendo necessário conhecer suas equivalentes no plano topográfico, principalmente para calcular e dividir áreas e implantar pontos no plano local. Nestes casos uma solução de grande utilização é transformação das coordenadas elipsoidicas geocêntricas em coordenadas planas associadas ao plano da Projeção Cartográfica UTM, geralmente ignorando-se as deformações, portanto consistindo em uma solução equivocada. Entre as soluções adequadas para a conversão de coordenadas de uma superfície para outra quando a origem dos sistemas difere entre topocêntrica e geocêntrica é a Transformação Ortogonal. Este método, além de ser mais sucinto que a metodologia de Puissant tradicionalmente utilizada permite a conversão tanto de coordenadas topocêntricas em geocêntricas bem como o inverso. Neste trabalho inicialmente se comparou a precisão do método da Transformação Ortogonal com o método de Puissant. Também foram comparadas as diferenças entre áreas calculadas apartir de coordenadas no plano cartográfico UTM e coordenadas topocêntricas, além de estabelecida a diferença de posição entre pontos de divisa calculados em um plano e implantados em outro sem as devidas conversões. O resultados obtidos demonstraram que a precisão do método de Transformação Ortogonal nas distancias observadas equivale com a metodologia de Puissant. Foi possível observar que os valores referentes a áreas quando calculadas em relação a coordenadas associadas a modelos ou superfícies diferentes apresentam variação. O cálculo e divisão de áreas utilizando coordenadas planas no plano da projeção cartográfica UTM, mesmo com correções de fator de escala e elevação diferiu da área no plano topográfico local, embora tenha se aproximado quando efetuadas correções. Pontos de divisão de áreas georreferenciadas não devem ser implantados no plano topográfico local utilizando-se coordenadas planas no plano da projeção cartográfica UTM, pois existe diferença de posição entre estas e as coordenadas topográficas. A solução mais adequada pela precisão e facilidade para cálculo e divisão de áreas georreferenciadas a partir de coordenadas geocêntricas é a conversão destas para topocêntricas pela transformação ortogonal de coordenadas. A transformação ortogonal de coordenadas constitui-se de uma solução prática e rápida tanto para o georreferenciamento de pontos oriundos de levantamentos topográficos, bem como para implantação de pontos georreferenciados no plano topográfico local, podendo ser inclusive facilmente implementado em estações totais topográficas.

This thesis is devoted to creating a systematic way of determining all inequivalent orthogonal coordinate systems which separate the Hamilton-Jacobi equation for a given natural Hamiltonian defined on three-dimensional spaces of constant, non-zero curvature. To achieve this, we represent the problem with Killing tensors and employ the recently developed invariant theory of Killing tensors. Killing tensors on the model spaces of spherical and hyperbolic space enjoy a remarkably simple form; even more striking is the fact that their parameter tensors admit the same symmetries as the Riemann curvature tensor, and thus can be considered algebraic curvature tensors. Using this property to obtain invariants and covariants of Killing tensors, together with the web symmetries of the associated orthogonal coordinate webs, we establish an equivalence criterion for each space. In the case of three-dimensional spherical space, we demonstrate the surprising result that these webs can be distinguished purely by the symmetries of the web. In the case of three-dimensional hyperbolic space, we use a combination of web symmetries, invariants and covariants to achieve an equivalence criterion. To completely solve the equivalence problem in each case, we develop a method for determining the moving frame map for an arbitrary Killing tensor of the space. This is achieved by defining an algebraic Ricci tensor. Solutions to equivalence problems of Killing tensors are particularly useful in the areas of multiseparability and superintegrability. This is evidenced by our analysis of symmetric potentials defined on three-dimensional spherical and hyperbolic space. Using the most general Killing tensor of a symmetry subspace, we derive the most general potential “compatible” with this Killing tensor. As a further example, we introduce the notion of a joint invariant in the vector space of Killing tensors and use them to characterize a well-known superintegrable potential in the plane. xiii

碩士
國立交通大學
電控工程研究所
106
We propose a mixed precoding strategy for CoMP systems, which allows that the orthogonal Block Diagonalization (BD) and the non-orthogonal Maximum Ratio Transmission (MRT) techniques can be simultaneously used for reducing network backhaul loading. We formulated the issues into two optimization problems. Namely, 1) minimizing the backhaul loading under a minimum target rate constraint and 2) maximizing the sum rate under a minimum target rate constraint. We derive closed-form solutions for these two problems and verify them by the Matlab CVX tool box. The de- rived closed-form solutions can signicantly reduce the computational time compared to the CVX. Simulation results show that the proposed precoding schemes can signicantly reduce the backhaul loading and outperform con- ventional schemes in terms of outage probability in achieving the minimum rate constraint.

碩士
國立臺灣大學
電信工程學研究所
106
Non-orthogonal multiple access (NOMA) can be effectively integrated in coordinated multipoint (CoMP) mobile networks to enhance transmission performance and spectral efficiency at cell boundaries. In this paper, a strong-channel-selection NOMA (SC-NOMA) scheme is proposed to achieve better throughput than joint-reception NOMA (JR-NOMA) scheme for uplink CoMP networks and comparable throughput than joint-transmission NOMA (JT-NOMA) scheme for downlink CoMP networks. Hybrid JT+SC-NOMA scheme that combines strong-channel-selection and joint-transmission strategies is also proposed to provide prevailing throughput for downlink CoMP networks. Moreover, the α-fairness JT-NOMA scheme and the product-rate-based NOMA (PR-based NOMA) schemes are demonstrated to provide comparable system throughput to the sum-rate-based NOMA schemes while yielding better user fairness.

碩士
國立清華大學
通訊工程研究所
105
In this work, coordinated beamforming and power allocation schemes are proposed for multicell downlink non-orthogonal multiple access (NOMA) systems. Beamforming is adopted at the base-station to mitigate both intercell and intracell interference whereas power-domain multiplexing between a pair of users is considered in each spatial dimension. Given the beamforming, intra-pair power allocation is first determined by maximizing the weighted sum rate under worst-case interference. Then, two coordinated beamforming schemes that take into consideration the effect of power-domain multiplexing are proposed, namely, the coordinated scheduling with pair-wise zero-forcing (CSPZF) and the maximum pair-wise signal-to-leakage-plus-noise ratio (PSLNR) beamforming schemes. CSPZF beamforming utilizes the available spatial degrees of freedom at each base-station to separate signals intended for intracell user pairs as well as to eliminate strong out-of-cell interference. PSLNR beamforming, on the other hand, enhances the signal intended for each user pair while suppressing its leakage towards other users (including those in other cells). For each scheme, the coordinated user schedule of multicell is proposed to maximize the weighted sum rate. CSPZF is shown to outperform PSLNR beamforming in the high SNR region where the performance is interference-limited and, vice versa. The effectiveness of our proposed schemes are demonstrated via simulations.

碩士
國立成功大學
電腦與通信工程研究所
107
In this study ,we investigates a power allocation problem for maximizing the minimum achievable users’ throughput in downlink coordinated multipoint(CoMP) with non-orthogonal multiple access (NOMA) using successive interference cancellation (SIC).First, non-orthogonal multiple access (NOMA) is one of the promising candidates in 5G system. NOMA offer a set of benefits, including great channel gain and spectrum efficiency. Second, we combine CoMP in able to enhance the quality of service for the Cell edge user in Heterogeneous Network. The original mathematical model of the network is a nonlinear programming problem. Converting the original model into a mixed integer linear programming problem with piecewise-RLT and mathematical approximate skills of the logarithmic term. Finally, using C++ combined with CPLEX to simulate, get the best solution of the whole model.

碩士
國立清華大學
資訊工程學系
103
The downlink non-orthogonal multiple access (NOMA) with successive interference cancellation (SIC) improves both the system throughput and cell edge user equipments (UEs) throughput. However, there are constraints on position and transmit power of UEs in NOMA. The base stations (BSs) transmit signals to the UE nearby and the UE far away with lower power and higher power, respectively. Without the constraint, we propose a probability model to derive the exactly throughput of UEs in NOMA. Moreover, the concept of the NOMA can combine with that of coordinated multipoint (CoMP). The associated BSs transmit signals to the UEs near themselves while also transmit signal to a cell-edge UE between two cells; with that it can improve the system throughput. We also propose a probability model for NOMA+CoMP. Based on the probability model, we propose an efficient resource allocation algorithm whose objective function is to maximize the number of satisfied UEs. The simulation results show that the proposed algorithm outperforms the compared algorithm in number of the satisfied UEs.

碩士
國立清華大學
資訊工程學系
104
Fourth-generation mobile networks such as LTE and LTE-advanced [1], which are based on orthogonal frequency division multiple access (OFDMA), can only serve one user equipment (UE) at a time with a resource block (RB). In this thesis, we study non-orthogonal multiple access (NOMA), a multiple access technique that can serve two or more UEs at a time with a RB in order to increase overall performance. We consider the resource allocation problem in the case with heterogeneous RBs. Moreover, we combine NOMA with Coordinated Multi-Point (CoMP) [2] for edge UEs. We propose two resource allocation algorithms, which are called MaxWeightMatching NOMA (MEM NOMA) and MaxIndependentSet NOMA (MIS NOMA). The objective is to maximize the number of satisfied UEs. The simulation results show that the proposed algorithms have better performance, compared with other resource allocation algorithms.

碩士
國立交通大學
土木工程學系
99
This study develops a semi-3D model based on a vertical-horizontal splitting (VHS) method to analyze the flow in open-channel bends. In horizontal, the surface elevation and depth-averaged velocity components are computed by 2D depth-averaged model. In vertical, assume the 3D velocity profile of Navier-Stokes equations is equal to the depth-averaged velocity plus the deviation of velocity profile, and then the vertical governing equations can be derived by subtracting the 2D depth-averaged equations to the 3D Navier-Stokes equations. In order to fit the complex geometry in both side wall and bed slope of channel, the orthogonal curvilinear coordinate system is used in horizontal gird, and sigma coordinate system is used in vertical grid. As for the numerical solution procedure, the two-step split-operator approach, which includes dispersion process (advection and diffusion terms) and propagation process (bed shear stress and pressure terms), is adopted to solve the 2D depth-averaged flow equations to improve the application flexibility. Implicit difference methods are adopted to relax the time step restriction allowing large time steps. Finally, three sets of experimental data including mildly curved, sharply curved and meandering channel are used to demonstrate the capability and accuracy of the semi-3D model, and the results of 2D depth-averaged model are also compared. The simulation results of semi-3D model show well agreement with experimental data considering different curved channels, bend lengths, secondary current and transverse mixing conditions.