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Numerical Study of Mach Number Effects on Combustion Instability / Etude numérique des effets du nombre de Mach sur les instabilités de combustionWieczorek, Kerstin 08 November 2010 (has links)
L'évolution des turbines à gaz vers des régimes de combustion en mélange pauvre augmente la sensibilité de la flamme aux perturbations de l'écoulement. Plus particulièrement, cela augmente le risque que des instabilités de combustion apparaissent. Comme ces oscillations peuvent affecter le processus de combustion, il est très important d'être capable de prédire ce comportement au niveau de la conception.L'objectif du travail présenté est de développer un solveur numérique qui permet de décrire ces instabilités, et d'évaluer les effets du nombre de Mach de l'écoulement moyen sur ce phénomène. L'approche choisie consiste à résoudre les équations d'Euler linéarisées, qui sont écrites dans le domaine fréquentiel sous la forme d'un problème aux valeurs propres. Ce système d'équations permets de prendre en compte la vitesse moyenne de l'écoulement, et donc d'évaluer les effets causés par la convection et leur impact sur la stabilité des modes. Parmi les mécanismes qui peuvent être étudiés se trouve notamment l'effet des ondes d'entropie convectées, ce qui est particulièrement intéressant dans le contexte des chambres de combustions. Afin de déterminer l'effet des termes liés à la vitesse de l'écoulement moyen sur la stabilité des modes, une analyse de l'énergie contenue dans les perturbations est effectuée. Finalement, l'aspect de la non-orthogonalité des modes propres, qui permet une croissance d'énergie transitoire dans un système linéairement stable, est abordé. / The development of gas turbines towards lean combustion increases the susceptibility of the flame to flow perturbations, and leads more particularly to a higher risk of combustion instability. As these self-sustained oscillations may affect the performance of the combustion device, it is very important to be able to predict them at the design level. At present, several methods are used to describe combustion instabilities, ranging from complex LES and DNS calculations to low-order network models. An intermediate method consists in solving a set of equations describing the acoustic field using a finite volume technique, which is the approach used in the present study.This thesis discusses the impact of a non zero Mach number mean flow field on thermoacoustic instability. The study is based on the linearized Euler equations, which are stated in the frequency domain in the form of an eigenvalue problem. Using the linearized Euler equations rather than the Helmholtz equation avoids making the commonly used assumption of the mean flow being at rest, and allows to take into account convection effects and their impact on the stability of the system. Among the mechanisms that can be studied using the present approach is namely the impact of convected entropy waves, which is especially interesting in combustion applications.For this study, a 1D and a 2D numerical solver have been developed and are presented in this thesis. In order to asses the effect of the mean flow terms on the modes' stability, an analysis of the disturbance energy budget is performed. Finally, the aspect of the eigenmodes being non-orthogonal and thus allowing for transient growth in linearly stable systems is adressed.
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Stability Analysis of Systems of Difference EquationsClinger, Richard A. 01 January 2007 (has links)
Difference equations are the discrete analogs to differential equations. While the independent variable of differential equations normally is a continuous time variable, t, that of a difference equation is a discrete time variable, n, which measures time in intervals. A feature of difference equations not shared by differential equations is that they can be characterized as recursive functions. Examples of their use include modeling population changes from one season to another, modeling the spread of disease, modeling various business phenomena, discrete simulations applications, or giving rise to the phenomena chaos. The key is that they are discrete, recursive relations. Systems of difference equations are similar in structure to systems of differential equations. Systems of first-order linear difference equations are of the form x(n + 1) = Ax(n) , and systems of first-order linear differential equations are of the form x(t) = Ax(t). In each case A is a 2x2 matrix and x(n +1), x(n), x(t), and x(t) are all vectors of length 2. The methods used in analyzing systems of difference equations are similar to those used in differential equations.Solutions of scalar, second-order linear difference equations are similar to those of scalar, second-order differential equations, but with one major difference: the composition of their general solutions. When the eigenvalues of A, λ1 and λ2, are real and distinct, general solutions of differential equations are of the form x(t) = c1eλ1t +c2eλ2t, while general solutions of difference equations are of form x(n) = 1λn1 + c2λn2. So, on the one hand, while the methods used in examining systems of difference equations are similar to those used for systems of differential equations; on the other hand, their general solutions can exhibit significantly different behavior.Chapter 1 will cover systems of first-order and second-order linear difference equations that are autonomous (all coefficients are constant). Chapter 2 will apply that theory to the local stability analysis of systems of nonlinear difference equations. Finally, Chapter 3 will give some example of the types of models to which systems of difference equations can be applied.
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Identification of Stiffness Reductions Using Partial Natural Frequency DataSokheang Thea (6620237) 15 May 2019 (has links)
In vibration-based damage detection in structures, often changes in the
dynamic properties such as natural frequencies, modeshapes, and derivatives of
modeshapes are used to identify the damaged elements. If only a partial list of
natural frequencies is known, optimization methods may need to be used to
identify the damage. In this research, the algorithm proposed by Podlevskyi & Yaroshko (2013) is used to determine
the stiffness distribution in shear building models. The lateral load resisting
elements are presented as a single equivalent spring, and masses are lumped at
floor levels. The proposed method calculates stiffness values directly, i.e., without optimization, from the known partial list of natural frequency data and mass
distribution. It is shown that if the number of stories with reduced
stiffness is smaller than the number of known natural frequencies, the stories
with reduced stiffnesses can be identified. Numerical studies on building
models with two stories and four stories are used to illustrate the solution
method. Effect of error or noise in given natural frequencies on stiffness
estimates and, conversely, sensitivity of natural frequencies to changes in
stiffness are studied using 7-, 15-, 30-, and 50-story numerical models. From
the studies, it is learnt that as the number of stories increases, the natural
frequencies become less sensitive to stiffness changes. Additionally, eight
laboratory experiments were conducted on a five-story aluminum structural
model. Ten slender columns were used in each story of the specimen. Damage was
simulated by removing columns in one, two, or three stories. The method can
locate and quantify the damage in cases presented in the experimental studies. It
is also applied to a 1/3 scaled 18-story steel moment frame building tested on
an earthquake simulator (Suita et al., 2015) to identify the reduction in the
stiffness due to fractures of beam flanges. Only the first two natural
frequencies are used to determine the reductions in the stiffness since the
third mode of the tower is torsional and no reasonable planar spring-mass model
can be developed to present all of the translational modes. The method produced possible cases of the
softening when the damage was assumed to occur at a single story.
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Faber-Krahn Type Inequalities for TreesBiyikoglu, Türker, Leydold, Josef January 2003 (has links) (PDF)
The Faber-Krahn theorem states that among all bounded domains with the same volume in Rn (with the standard Euclidean metric), a ball that has lowest first Dirichlet eigenvalue. Recently it has been shown that a similar result holds for (semi-)regular trees. In this article we show that such a theorem also hold for other classes of (not necessarily non-regular) trees. However, for these new results no couterparts in the world of the Laplace-Beltrami-operator on manifolds are known. / Series: Preprint Series / Department of Applied Statistics and Data Processing
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Inner-outer iterative methods for eigenvalue problems : convergence and preconditioningFreitag, Melina January 2007 (has links)
Many methods for computing eigenvalues of a large sparse matrix involve shift-invert transformations which require the solution of a shifted linear system at each step. This thesis deals with shift-invert iterative techniques for solving eigenvalue problems where the arising linear systems are solved inexactly using a second iterative technique. This approach leads to an inner-outer type algorithm. We provide convergence results for the outer iterative eigenvalue computation as well as techniques for efficient inner solves. In particular eigenvalue computations using inexact inverse iteration, the Jacobi-Davidson method without subspace expansion and the shift-invert Arnoldi method as a subspace method are investigated in detail. A general convergence result for inexact inverse iteration for the non-Hermitian generalised eigenvalue problem is given, using only minimal assumptions. This convergence result is obtained in two different ways; on the one hand, we use an equivalence result between inexact inverse iteration applied to the generalised eigenproblem and modified Newton's method; on the other hand, a splitting method is used which generalises the idea of orthogonal decomposition. Both approaches also include an analysis for the convergence theory of a version of inexact Jacobi-Davidson method, where equivalences between Newton's method, inverse iteration and the Jacobi-Davidson method are exploited. To improve the efficiency of the inner iterative solves we introduce a new tuning strategy which can be applied to any standard preconditioner. We give a detailed analysis on this new preconditioning idea and show how the number of iterations for the inner iterative method and hence the total number of iterations can be reduced significantly by the application of this tuning strategy. The analysis of the tuned preconditioner is carried out for both Hermitian and non-Hermitian eigenproblems. We show how the preconditioner can be implemented efficiently and illustrate its performance using various numerical examples. An equivalence result between the preconditioned simplified Jacobi-Davidson method and inexact inverse iteration with the tuned preconditioner is given. Finally, we discuss the shift-invert Arnoldi method both in the standard and restarted fashion. First, existing relaxation strategies for the outer iterative solves are extended to implicitly restarted Arnoldi's method. Second, we apply the idea of tuning the preconditioner to the inner iterative solve. As for inexact inverse iteration the tuned preconditioner for inexact Arnoldi's method is shown to provide significant savings in the number of inner solves. The theory in this thesis is supported by many numerical examples.
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Fault Tolerance in Linear Algebraic Methods using Erasure Coded ComputationsXuejiao Kang (5929862) 16 January 2019 (has links)
<p>As
parallel and distributed systems scale to hundreds of thousands of
cores and beyond, fault tolerance becomes increasingly important --
particularly on systems with limited I/O capacity and bandwidth.
Error correcting codes (ECCs) are used in communication systems where
errors arise when bits are corrupted silently in a message. Error
correcting codes can detect and correct erroneous bits. Erasure
codes, an instance of error correcting codes that deal with data
erasures, are widely used in storage systems. An erasure code
addsredundancy to the data to tolerate erasures.
</p>
<p><br>
</p>
<p>In
this thesis, erasure coded computations are proposed as a novel
approach to dealing with processor faults in parallel and distributed
systems. We first give a brief review of traditional fault tolerance
methods, error correcting codes, and erasure coded storage. The
benefits and challenges of erasure coded computations with respect to
coding scheme, fault models and system support are also presented.</p>
<p><br>
</p>
<p>In
the first part of my thesis, I demonstrate the novel concept of
erasure coded computations for linear system solvers. Erasure coding
augments a given problem instance with redundant data. This augmented
problem is executed in a fault oblivious manner in a faulty parallel
environment. In the event of faults, we show how we can compute the
true solution from potentially fault-prone solutions using a
computationally inexpensive procedure. The results on diverse linear
systems show that our technique has several important advantages: (i)
as the hardware platform scales in size and in number of faults, our
scheme yields increasing improvement in resource utilization,
compared to traditional schemes; (ii) the proposed scheme is easy to
code as the core algorithm remains the same; (iii) the general scheme
is flexible to accommodate a range of computation and communication
trade-offs.
</p>
<p><br>
</p>
<p>We
propose a new coding scheme for augmenting the input matrix that
satisfies the recovery equations of erasure coding with high
probability in the event of random failures. This coding scheme also
minimizes fill (non-zero elements introduced by the coding block),
while being amenable to efficient partitioning across processing
nodes. Our experimental results show that the scheme adds minimal
overhead for fault tolerance, yields excellent parallel efficiency
and scalability, and is robust to different fault arrival models and
fault rates.</p>
<p><br>
</p>
<p>Building
on these results, we show how we can minimize, to optimality, the
overhead associated with our problem augmentation techniques for
linear system solvers. Specifically, we present a technique that
adaptively augments the problem only when faults are detected. At any
point during execution, we only solve a system with the same size as
the original input system. This has several advantages in terms of
maintaining the size and conditioning of the system, as well as in
only adding the minimal amount of computation needed to tolerate the
observed faults. We present, in details, the augmentation process,
the parallel formulation, and the performance of our method.
Specifically, we show that the proposed adaptive fault tolerance
mechanism has minimal overhead in terms of FLOP counts with respect
to the original solver executing in a non-faulty environment, has
good convergence properties, and yields excellent parallel
performance.</p>
<p><br>
</p>
<p>Based
on the promising results for linear system solvers, we apply the
concept of erasure coded computation to eigenvalue problems, which
arise in many applications including machine learning and scientific
simulations. Erasure coded computation is used to design a fault
tolerant eigenvalue solver. The original eigenvalue problem is
reformulated into a generalized eigenvalue problem defined on
appropriate augmented matrices. We present the augmentation scheme,
the necessary conditions for augmentation blocks, and the proofs of
equivalence of the original eigenvalue problem and the reformulated
generalized eigenvalue problem. Finally, we show how the eigenvalues
can be derived from the augmented system in the event of faults.
</p>
<p><br>
</p>
<p>We
present detailed experiments, which demonstrate the excellent
convergence properties of our fault tolerant TraceMin eigensolver in
the average case. In the worst case where the row-column pairs that
have the most impact on eigenvalues are erased, we present a novel
scheme that computes the augmentation blocks as the computation
proceeds, using the estimates of leverage scores of row-column pairs
as they are computed by the iterative process. We demonstrate low
overhead, excellent scalability in terms of the number of faults, and
the robustness to different fault arrival models and fault rates for
our method.</p>
<p><br>
</p>
<p>In
summary, this thesis presents a novel approach to fault tolerance
based on erasure coded computations, demonstrates it in the
context of important linear algebra kernels, and validates its
performance on a diverse set of problems on scalable parallel
computing platforms. As parallel systems scale to hundreds of
thousands of processing cores and beyond, these techniques present
the most scalable fault tolerant mechanisms currently available.</p><br>
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Grafos e suas aplicações / Graphs and their applicationsSantos Júnior, Jânio Alves dos 14 December 2016 (has links)
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Previous issue date: 2016-12-14 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This work aims to study some topics of graph theory in order to solve some problems. In order to complement, we approached a light study of matrices, eigenvalues and eigenvectors. The first problem is known as Königsberg Bridge Problem, where this was considered the problem that gave rise to the study on graphs. The House Problem is a joke, which shows us several propositions about planar and bipartite graphs. Some models we can relate graphs, such as we can observe in the problem of cannibals and in the game of chess. Finally, we will work with applications in the adjacency matrix as in the Problem of the Condominium of Farms and in the Number of Possible Paths in a graph, where we will work with geometric figures, apparently resolving a counting problem using eigenvalues and graph. As a methodological support will be approached Linear Algebra. / O objetivo deste trabalho é estudar alguns tópicos da teoria de grafos com o intuito de resolver alguns problemas. Para complementar, abordamos um leve estudo de matrizes, autovalores e autovetores. O primeiro problema é conhecido como o Problema da Ponte de Königsberg, onde tal, foi considerado o problema que deu origem ao estudo sobre grafos. O Problema das Casas que é uma brincadeira, que nos mostra várias proposições sobre grafos planares e bipartidos. Alguns modelos que podemos relacionar grafos, tais como veremos no problema dos canibais e no jogo de xadrez. Por fim, trabalharemos com aplicações na matriz de adjacência como no problema do Condomínio de Chácaras e no Número de Caminhos Possíveis em um Grafo, onde trabalharemos com figuras geométricas, resolvendo aparentemente um problema de contagem, utilizando autovalores e grafos. Como suporte metodológico será abordado Álgebra Linear.
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Using factor analysis to determine why students select UWC as higher education institute.Osman, Abuelgasim Ahemd Atta-Almanan. January 2009 (has links)
<p>This study investigates the most important reasons behind the rst-year students' decision to select University of the Western Cape (UWC) as higher education institution.<br />
These reasons were organized into a few factors for easy interpretation. The data to be analyzed for this project is a subsection of the data collected during the orientation period of 2008. During the orientation week of 2008, the questionnaires were completed on a voluntary basis by new rst-year students. All questionnaires were anonymously completed and therefore the data does not contain any information that could be linked to any individual. For the purpose of this study, only the black African and coloured students were considered. The other racial groups were not analyzed due to too small sample sizes. Questionnaires with missing information on the reasons for selecting UWC were not  / nalyzed. We ended up with a sample of size 600. The data were statistically analyzed, using descriptive statistics, bivariate analyses, factor analysis, coefficient of congruence and bootstrap factor analysis. The results indicated that the most important reasons aecting students to choose UWC were identied as good academic reputation, family member's advice, UWC graduates are successful and UWC graduates get good jobs. The least important reasons were found to be not accepted anywhere, parents / family members graduated from UWC, recruited by UWC and wanted to study near to home. The results also indicated that there were significant differences among students according to population groups, parent's monthly income and grade 12 average. Factor analysis of 12 variables yielded three extracted factors upon which student decisions were based. Similarities of these three factors were tested, and a high similarity among demographic characteristics and grade 12 average were found. Additional analyses were conducted to measure the accuracy of factor analyses models constructed using Spearman and Polychoric correlation matrices. The results indicated that both correlation matrices were  / nbiased, with higher variance and higher loadings when the Polychoric correlation matrix was used to construct a factor analysis model for categorical data.</p>
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Development of New Monte Carlo Methods in Reactor Physics : Criticality, Non-Linear Steady-State and Burnup ProblemsDufek, Jan January 2009 (has links)
The Monte Carlo method is, practically, the only approach capable of giving detail insight into complex neutron transport problems. In reactor physics, the method has been used mainly for determining the keff in criticality calculations. In the last decade, the continuously growing computer performance has allowed to apply the Monte Carlo method also on simple burnup simulations of nuclear systems. Nevertheless, due to its extensive computational demands the Monte Carlo method is still not used as commonly as deterministic methods. One of the reasons for the large computational demands of Monte Carlo criticality calculations is the necessity to carry out a number of inactive cycles to converge the fission source. This thesis presents a new concept of fission matrix based Monte Carlo criticality calculations where inactive cycles are not required. It is shown that the fission matrix is not sensitive to the errors in the fission source, and can be thus calculated by a Monte Carlo calculation without inactive cycles. All required results, including keff, are then derived via the final fission matrix. The confidence interval for the estimated keff can be conservatively derived from the variance in the fission matrix. This was confirmed by numerical test calculations of Whitesides's ``keff of the world problem'' model where other Monte Carlo methods fail to estimate the confidence interval correctly unless a large number of inactive cycles is simulated. Another problem is that the existing Monte Carlo criticality codes are not well shaped for parallel computations; they cannot fully utilise the processing power of modern multi-processor computers and computer clusters. This thesis presents a new parallel computing scheme for Monte Carlo criticality calculations based on the fission matrix. The fission matrix is combined over a number of independent parallel simulations, and the final results are derived by means of the fission matrix. This scheme allows for a practically ideal parallel scaling since no communication among the parallel simulations is required, and no inactive cycles need to be simulated. When the Monte Carlo criticality calculations are sufficiently fast, they will be more commonly applied on complex reactor physics problems, like non-linear steady-state calculations and fuel cycle calculations. This thesis develops an efficient method that introduces thermal-hydraulic and other feedbacks into the numerical model of a power reactor, allowing to carry out a non-linear Monte Carlo analysis of the reactor with steady-state core conditions. The thesis also shows that the major existing Monte Carlo burnup codes use unstable algorithms for coupling the neutronic and burnup calculations; therefore, they cannot be used for fuel cycle calculations. Nevertheless, stable coupling algorithms are known and can be implemented into the future Monte Carlo burnup codes. / QC 20100709
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Step by step eigenvalue analysis with EMTP discrete time solutionsHollman, Jorge 11 1900 (has links)
The present work introduces a methodology to obtain a discrete time state space representation of an electrical network using the nodal [G] matrix of the Electromagnetic Transients Program (EMTP) solution. This is the first time the connection between the EMTP nodal analysis solution and a corresponding state-space formulation is presented. Compared to conventional state space solutions, the nodal EMTP solution is computationally much more efficient. Compared to the phasor solutions used in transient stability analysis, the proposed approach captures a much wider range of eigenvalues and system operating states. A fundamental advantage of extracting the system eigenvalues directly from the EMTP solution is the ability of the EMTP to follow the characteristics of nonlinearities. The system's trajectory can be accurately traced and the calculated eigenvalues and eigenvectors correctly represent the system's instantaneous dynamics. In addition, the algorithm can be used as a tool to identify network partitioning subsystems suitable for real-time hybrid power system simulator environments, including the implementation of multi-time scale solutions. The proposed technique can be implemented as an extension to any EMTP-based simulator. Within our UBC research group, it is aimed at extending the capabilities of our real-time PC-cluster
Object Virtual Network Integrator (OVNI) simulator.
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