Kirchhoff's laws are well-studied for electrical networks with voltage and current sources, and edges marked by resistors. Kirchhoff's voltage law states that the sum of voltages around any circuit of the network graph is zero, while Kirchhoff's current law states that the sum of the currents along any cutset of the network graph is zero. Given a network, these requirements may be encoded by the circuit matrix and cutset matrix of the network graph. The columns of these matrices are indexed by the edges of the network graph, and their row spaces are orthogonal complements. For (chemical or electrochemical) reaction networks, one must naturally study the opposite problem, beginning with the stoichiometric matrix rather than the network graph. This leads to the following question: given such a matrix, what is a suitable graphic rendering of a network that properly visualizes the underlying chemical reactions? Although we can not expect uniqueness, the goal is to prove existence of such a graph for any matrix. Specifically, we study Kirchhoff graphs, originally introduced by Fehribach. Mathematically, Kirchhoff graphs represent the orthocomplementarity of the row space and null space of integer-valued matrices. After introducing the definition of Kirchhoff graphs, we will survey Kirchhoff graphs in the context of several diverse branches of mathematics. Beginning with combinatorial group theory, we consider Cayley graphs of the additive group of vector spaces, and resolve the existence problem for matrices over finite fields. Moving to linear algebra, we draw a number of conclusions based on a purely matrix-theoretic definition of Kirchhoff graphs, specifically regarding the number of vector edges. Next we observe a geometric approach, reviewing James Clerk Maxwell's theory of reciprocal figures, and presenting a number of results on Kirchhoff duality. We then turn to algebraic combinatorics, where we study equitable partitions, quotients, and graph automorphisms. In addition to classifying the matrices that are the quotient of an equitable partition, we demonstrate that many Kirchhoff graphs arise from equitable edge-partitions of directed graphs. Finally we study matroids, where we review Tutte's algorithm for determining when a binary matroid is graphic, and extend this method to show that every binary matroid is Kirchhoff. The underlying theme throughout each of these investigations is determining new ways to both recognize and construct Kirchhoff graphs.

Underwater acoustic modelling is an important aspect of Synthetic Aperture Sonar (SAS) system design and algorithm development. Sea-trials are an expensive and time-consuming exercise and simulations provide an efficient and economic alternative. However, there are few simulators (in the open literature) that can efficiently provide realistic SAS data for large, complicated scenes. Conventional side-scan sonar simulators are not suitable for SAS data simulation. These simulators utilise narrow-beam and narrow-band approximations; typical SAS systems are wide-beam and wide-band and these approximations are invalid. Moreover, conventional side-scan sonar is a non-coherent imaging technique and SAS processing relies on the phase. Existing SAS simulators are capable of modelling very simple scenes only. They utilise a decomposition of the scene into point or smooth facet primitives, which is very inefficient. Many primitives are required and this imposes a severe restriction on scene complexity and size. This thesis presents a rigorous mathematical framework for the modelling of SAS imagery. A novel acoustic scattering model is developed and its implementation in a wide-beam and wide-band, multiple-receiver Interferometric SAS (InSAS) simulator is detailed. The scattering model utilises a decomposition of the scene into rough (rather than smooth) facet primitives. The use of rough facet primitives provides a significant increase in computational efficiency since scenes are decomposed into fewer primitives. This facilitates the simulation of larger and more complicated scenes. Each rough facet is characterised by its far-field beampattern. The statistics of the beampattern are related to the facet shape and roughness statistics using the Kirchhoff approximation. The beampattern is realised from its first and second-order statistics. The SAS imagery is obtained using a coherent sum of the facet responses and occlusions and multiple-scattering are resolved by ray-tracing. The simulator is implemented for use on a parallel computing cluster. The simulator is shown to provide realistic SAS data that is qualitatively and quantitatively similar to real data. The simulated results are considered, in many ways, superior to the simulated results in the literature.

Made available in DSpace on 2016-08-29T15:32:56Z (GMT). No. of bitstreams: 1 tese_568_Edmilson Bermudes Rocha Junior.pdf: 525051 bytes, checksum: 0b490fd7042c272e01d671bea06adcd6 (MD5) Previous issue date: 2004-12-22 / A proposta deste trabalho é analisar a dinâmica de um sistema formado por um motor elétrico de corrente contínua desbalanceada, sustentando por uma estrutura elástica. A importância do estudo de problemas envolvendo o acoplamento da dinâmica de diversos sistemas tem aumentado recentemente pelas características construtivas das máquinas e estruturas. Crê-se que a tendência é a de que as máquinas rotativas sejam mais flexíveis e devem operar em rotações mais altas. Assim, fenômenos que não eram observados em gerações de máquinas anteriores se fazem presentes e sua explicação exige a adoção de modelos mais completos. Pelo lado operacional, exige-se mais dos sistemas de controle. O conjunto de equações que governam o sistema em estudo motor+estrutura é composto pelas equações mecânicas obtidas a partir das equações de Lagrange e pela equação do motor obtida através da lei de tensão de Kirchhoff. Aplica-se o método de Runge-Kutta de quinta ordem, com passo variável, para a simulação do sistema. Por fim, é feita a análise de rendimento do motor elétrico de excitação independente e série, considerando a influência da estrutura sobre a sua rotação. Palavras-chave: Efeito Sommerfeld, Lagrange, Kirchhoff, rendimento, Sistema não ideal.

No description available.

This investigation analyzes the effect of vortex wakes on the Lagrangian displacement of particles induced by the passage of an obstacle in a two-dimensional incompressible, inviscid fluid such that the flow is potential and time-independent in a suitable frame of reference. In addition to the trajectories of individual particles, we also study their drift and the corresponding total drift areas in the Föppl and Kirchhoff potential flow models. Our findings, which are obtained numerically and in some regimes are also supported by asymptotic analysis, are compared to the wakeless potential flow which serves as a reference. We show that in the presence of the Föppl vortex wake some of the particles follow more complicated trajectories featuring a second loop. The appearance of an additional stagnation point in the Föppl flow is identified as a source of this effect. It is also demonstrated that, while the total drift area increases with the size of the wake for large vortex strengths, it is actually decreased for small circulation values. On the other hand, the Kirchhoff flow model is shown to have an unbounded total drift area. By providing a systematic account of the wake effects on the drift, the results of this study will allow for more accurate modeling of hydrodynamic stirring. / Thesis / Master of Science (MSc)

Este trabalho contém resultados sobre a existência, unicidade e comportamento assintótico de soluções para uma equação de viga não linear do tipo Kirchhoff, \'u IND. tt\' \'+ \'DELTA\' POT. 2\' u - M(\'INT.IND. OMEGA\' | \'NABLA\' u| 2 dx) \'DELTA\' u+ f (\'u IND. t\' ) +g(u) = h em × R +, onde \'R POT. N\' é um domínio limitado com fronteira regular \\GAMA. Essa equação é um modelo matemático para pequenas vibrações transversais de vigas ou placas extensíveis. O termo não local M(\'INT.IND. OMEGA\' | \\NABLA u |2 dx) u está relacionado à variação de tensão na viga devida à sua extensibilidade. O termo f (\'u IND. t\' ) representa uma dissipação para o sistema e g(u) representa a força exercida pelo meio. A função h representa uma força externa adicional. Consideramos o problema com as condições de fronteira u|×R + = \'INT. u SUP. \'INT. v\' | \\\'GAMA\' ×\'R +\' = 0, que corresponde ao modelo de vigas fixadas pelo bordo \\\'GAMA\'. Discutiremos o caso em que a dissipação é linear e o caso em que é não linear. Mostraremos que em ambos os casos o sistema dinâmico associado ao problema possui um atrator global. Entretanto, para o caso em que a dissipação é linear, obtemos num espaço de fase mais regular, a existência de um conjunto inércia de dimensão finita, que atrai exponencialmente todos os limitados deste espaço / This work contains some results on the existence, uniqueness and asymptotic behavior of solutions for a nonlinear beam equation of Kirchhoff type, \'u IND. tt\' + \' DELTA POT. 2\' u+ M(\'INT. IND.\' |u| 2 dx) u + g(\'u IND. t\') + f (u) = h; where \'R POT. N\' is a bounded domain with smooth boundary . This equation is a model for small vibrations of extensible beams. The nonlocal term M(\' INT. IND.\' |u| 2 dx) u is related to the variation of tensions in the beam due to its extensibility. The term f (\'u IND. t\') represents a damping mechanism for the system and g(u) represents the force exerted by the foundation. The function h represents an additional external force. We consider the problem with boundary condition u|×R+ = \' u SUP. \' |×R+ = 0, which corresponds to the model of clamped beams. We discuss the cases where the dissipation is linear and the case nonlinear. We show that in both cases, the dynamical system associated to the problem has a global attractor. However, when the dissipation is linear, we obtain, in a more regular space, the existence of an inertial set of finite dimension, which attracts exponentially all bounded sets of this space

Este trabalho contém resultados sobre a existência, unicidade e comportamento assintótico de soluções para uma equação de viga não linear do tipo Kirchhoff, \'u IND. tt\' \'+ \'DELTA\' POT. 2\' u - M(\'INT.IND. OMEGA\' | \'NABLA\' u| 2 dx) \'DELTA\' u+ f (\'u IND. t\' ) +g(u) = h em × R +, onde \'R POT. N\' é um domínio limitado com fronteira regular \\GAMA. Essa equação é um modelo matemático para pequenas vibrações transversais de vigas ou placas extensíveis. O termo não local M(\'INT.IND. OMEGA\' | \\NABLA u |2 dx) u está relacionado à variação de tensão na viga devida à sua extensibilidade. O termo f (\'u IND. t\' ) representa uma dissipação para o sistema e g(u) representa a força exercida pelo meio. A função h representa uma força externa adicional. Consideramos o problema com as condições de fronteira u|×R + = \'INT. u SUP. \'INT. v\' | \\\'GAMA\' ×\'R +\' = 0, que corresponde ao modelo de vigas fixadas pelo bordo \\\'GAMA\'. Discutiremos o caso em que a dissipação é linear e o caso em que é não linear. Mostraremos que em ambos os casos o sistema dinâmico associado ao problema possui um atrator global. Entretanto, para o caso em que a dissipação é linear, obtemos num espaço de fase mais regular, a existência de um conjunto inércia de dimensão finita, que atrai exponencialmente todos os limitados deste espaço / This work contains some results on the existence, uniqueness and asymptotic behavior of solutions for a nonlinear beam equation of Kirchhoff type, \'u IND. tt\' + \' DELTA POT. 2\' u+ M(\'INT. IND.\' |u| 2 dx) u + g(\'u IND. t\') + f (u) = h; where \'R POT. N\' is a bounded domain with smooth boundary . This equation is a model for small vibrations of extensible beams. The nonlocal term M(\' INT. IND.\' |u| 2 dx) u is related to the variation of tensions in the beam due to its extensibility. The term f (\'u IND. t\') represents a damping mechanism for the system and g(u) represents the force exerted by the foundation. The function h represents an additional external force. We consider the problem with boundary condition u|×R+ = \' u SUP. \' |×R+ = 0, which corresponds to the model of clamped beams. We discuss the cases where the dissipation is linear and the case nonlinear. We show that in both cases, the dynamical system associated to the problem has a global attractor. However, when the dissipation is linear, we obtain, in a more regular space, the existence of an inertial set of finite dimension, which attracts exponentially all bounded sets of this space

Modern oil and gas exploration depends on a variety of geophysical prospect tools. One of them is reflection seismology that allows to obtain interwell information of sufficient resolution economically. This exploration method collects reflection seismic data on the surface of an area of prospect interest and then uses them to build seismic images of the subsurface. All imaging approaches can be divided into two groups: wave equation-based methods and integral schemes. Kirchhoff migration, which belongs to the second group, is an indispensable tool in seismic imaging due to its flexibility and relatively low computational cost. Unfortunately, the classic formulation of this method images only a part of the surface data, if so-called multipathing is present in it. That phenomenon occurs in complex geologic settings, such as subsalt areas, when seismic waves travel between a subsurface point and a surface location through more than one path. The quality of imaging with Kirchhoff migration in complex geological areas can be improved if multiple paths of ray propagation are included in the integral. Multiple arrivals can be naturally incorporated into the imaging operator if it is expressed as an integral over subsurface take-off angles. In this form, the migration operator involves escape functions that connect subsurface locations with surface seismic data values through escape traveltime and escape positions. These escape quantities are functions of phase space coordinates that are simply related to the subsurface reflection system. The angle-domain integral operator produces output scattering- and dip-angle image gathers, which represent a convenient domain for subsurface analysis. Escape functions for angle-domain imaging can be simply computed with initial-value ray tracing, a Lagrangian computational technique. However, the computational cost of such a bottom-up approach can be prohibitive in practice. The goal of this work was to construct a computationally efficient phase space imaging framework. I designed several approaches to computing escape functions directly in phase space for mapping surface seismic reflection data to the subsurface angle domain. Escape equations have been introduced previously to describe distribution of escape functions in the phase space. Initially, I employed these equations as a basis for building an Eulerian numerical scheme using finite-difference method in the 2-D case. I show its accuracy constraints and suggest a modification of the algorithm to overcome them. Next, I formulate a semi-Lagrangian approach to computing escape functions in 3-D. The second method relies on the fundamental property of continuity of these functions in the phase space. I define locally constrained escape functions and show that a global escape solution can be reconstructed from local solutions iteratively. I validate the accuracy of the proposed methods by imaging synthetic seismic data in several complex 2-D and 3-D models. I draw conclusions about efficiency by comparing the compute time of the imaging tests with the compute time of a well-optimized conventional initial-value ray tracing. / text

No description available.

Ingeniero Civil Mecánico / Los fenómenos físicos, presentes en las ciencias y en las diferentes áreas de la ingeniería, a menudo son modelados por Ecuaciones Diferenciales Parciales (EDP). Los problemas de valor de frontera resultantes en muchos casos carecen de soluciones analíticas. Para resolver tales problemas, uno puede hacer suposiciones que simplifiquen el problema, o usar métodos numéricos para aproximar la solución. Dentro de los métodos numéricos actualmente existentes, el más popular es el Método de Elementos Finitos (FEM), que es la base de diferentes programas comerciales, como ADINA o ANSYS, entre muchos otros. La desventaja de este método es la gran cantidad de recursos computacionales y los tiempos de iteración requeridos para obtener una solución precisa del problema. Dada esta desventaja, Hughes desarrolló el Análisis IsoGeométrico (IGA). Este método permite integrar el modelo CAD con el Análisis de Elementos Finitos (FEA), por lo tanto, reduce los tiempos y los recursos necesarios para obtener una solución precisa. Pero a su vez, el IGA no tiene flexibilidad para obtener soluciones de ciertos problemas, ya que usa las mismas funciones bases para parametrizar tanto la geometría como el campo de solución. Debido a esto último, surge el Análisis IsoGeométrico Generalizado (GIFT) como una generalización del IGA, este método utiliza diferentes funciones bases para parametrizar la geometría del objeto y el campo de solución, permitiendo la selección de funciones que se adapten mejor al problema estudiado. En trabajos anteriores, el GIFT ha sido aplicado a problemas de la Ecuación de Laplace y de Elasticidad Lineal. El objetivo principal de este trabajo es estudiar el rendimiento del GIFT para problemas de flexión y de vibraciones de placas delgadas. El estudio consiste en implementar el GIFT para 3 placas diferentes y comparar los resultados numéricos con lo predicho por la Teoría de Placas de Kirchhoff-Love (KLPT). Se consideran una placa de geometría circular simple, una placa de geometría circular de dos parches y una placa cuadrada con un agujero de forma compleja, modelada por 8 parches. Las placas están parametrizadas por NURBS, mientras que las soluciones se aproximan por un parche usando NURBS o B-Splines. Los resultados se muestran en términos de curvas de convergencia, modos de vibración y frecuencias naturales. Los resultados numéricos se comparan con las soluciones analíticas para problemas con geometría simple y con la solución FEM para el problema de una placa más compleja. El análisis realizado indica que, para la misma parametrización de geometría (uniforme), (a) la solución se puede aproximar mediante un parche NURBS o B-Splines, manteniendo inalterada la geometría original, (b) los resultados obtenidos con las aproximaciones de campo NURBS y B-Splines son idénticas, (c) la tasa de convergencia depende del grado de aproximación de la solución. Para parametrizaciones geométricas no uniformes, el método no produce una tasa de convergencia óptima o resultados suficientemente precisos, al igual que el IGA tradicional.