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Thesis (Ph. D.)--University of Texas at Austin, 2005. / Supervisor: Thomas F. Edgar. Vita. Includes bibliographical references.

No description available.

As the second task of this thesis, we shall propose some mathematical model to simulate the movement of a floating bridge under some moving loadings. The floating bridge system consists of three parts, i.e. river (fluid), floating bridge (multibody system) and vehicles (load) which pass the bridge. Our objective is to find the motion and dynamical responses of the floating bridge with a truck or tracklayer passing on it. The floating bridge is a system of steel rectangular boxes which can be seen as rigid bodies connected by some kinematic joints. In fact, such system is a fluid-structure coupled system and one must treat the governing equations for the floating bridge and fluid, i.e. Euler-Lagrange equations and Navier-Stokes equations, simultaneously. In our work, we apply the one-leg method and operator splitting arbitrary Lagrangian-Eulerian method to solve the coupled system. / When performing dynamical analysis of a constrained mechanical system, a set of index-3 differential algebraic equations, i.e. Euler-Lagrange equations, are often needed to describe the time evolution of the mechanical system. In this thesis, we apply one-leg multi-step methods to integrate the DAEs directly. To overcome some difficulties leading to certain numerical instabilities, a velocity elimination technique is applied to generate a framework that the position and velocity profiles can be obtained in two separate stages: only the position variables and Lagrange multipliers take part in the convergent nonlinear iterations at each time step while the velocity is calculated by the multi-step formula directly without any iteration. The framework is constructed in a manner such that it satisfies all the constraints at the position level and involves variables as few as possible during the iteration. Some convergence analysis are presented and good stability and high efficiency can be seen through the experiments of some benchmark problems. / Zhao Yubo. / "July 2005." / Adviser: Zou Jun. / Source: Dissertation Abstracts International, Volume: 67-01, Section: B, page: 0310. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (p. 244-276). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

<p>Den kvasilinjära formen av differential-algebraiska ekvationer är både en mycket allmängiltig generalisering av den linjära tidsinvarianta formen, och en form som visar sig lämpa sig väl för indexreduktionsmetoder som vi hoppas ska komma att bli både praktiskt tillämpbara och väl förstådda i framtiden.</p><p>Kuperingsalgoritmen (engelska: the shuffle algorithm) användes ursprungligen för att bestämma konsistenta initialvillkor för linjära tidsinvarianta differential-algebraiska ekvationer, men har även andra tillämpningar, till exempel det grundläggande problemet numerisk integration. I syfte att förstå hur kuperingsalgoritmen kan tillämpas på kvasilinjära differential-algebraiska ekvationer som inte låter sig analyseras utifrån mönstret av nollor, har problemet att förstå singulära perturbationer i differential-algebraiska ekvationer uppstått. Den här avhandlingen presenterar en indexreduktionsmetod där behovet framgår tydligt, och visar att algoritmen inte bara generaliserar kuperingsalgoritmen, utan även är ett specialfall av den mer allmänna strukturalgoritmen (engelska: the structure algorithm) för att invertera system av Li och Feng.</p><p>Ett kapitel av den här avhandlingen söker av en klass av ekvations-former efter former som är mindre generella än den kvasilinjära, men som en algoritm lik vår kan anpassas till. Det visar sig att indexreduktionen ofta förstör strukturella egenskaper hos ekvationerna, och att det därför är naturligt att arbeta med den mest allmänna kvasilinjära formen.</p><p>Avhandlingen innehåller också några tidiga resultat gällande hur perturbationerna kan hanteras. Huvudresultaten är inspirerade av den modellering i skilda tidskalor som görs i teorin om singulära perturbationer (engelska: singular perturbation theory). Medan teorin om singulära perturbationer betraktar inverkan av en försvinnande skalär i ekvationerna, betraktar analysen häri en okänd matris vars norm begränsas av en liten skalär. Resultaten är begränsade till linjära tidsinvarianta ekvationer av index inte högre än 1, men det är värt att notera att index 0-fallet självt innebär en intressant generalisering av teorin för singulära perturbationer för ordinära differentialekvationer.</p> / <p>The quasilinear form of differential-algebraic equations is at the same time both a very versatile generalization of the linear time-invariant form, and a form which turns out to suit methods for index reduction which we hope will be practically applicable and well understood in the future.</p><p>The shuffle algorithm was originally a method for computing consistent initial conditions for linear time-invariant differential algebraic equations, but has other applications as well, such as the fundamental task of numerical integration. In the prospect of understanding how the shuffle algorithm can be applied to quasilinear differential-algebraic equations that cannot be analyzed by zero-patterns, the question of understanding singular perturbation in differential-algebraic equations has arose. This thesis details an algorithm for index reduction where this need is evident, and shows that the algorithm not only generalizes the shuffle algorithm, but also specializes the more general structure algorithm for system inversion by Li and Feng.</p><p>One chapter of this thesis surveys a class of forms of equations, searching less general forms than the quasilinear, to which an algorithm like ours can be tailored. It is found that the index reduction process often destroys structural properties of the equations, and hence that it is natural to work with the quasilinear form in its full generality.</p><p>The thesis also contains some early results on how the perturbations can be handled. The main results are inspired by the separate timescale modeling found in singular perturbation theory. While the singular perturbation theory considers the influence of a vanishing scalar in the equations, the analysis herein considers an unknown matrix bounded in norm by a small scalar. Results are limited to linear time-invariant equations of index at most 1, but it is worth noting that the index 0 case in itself holds an interesting generalization of the singular perturbation theory for ordinary differential equations.</p> / Report code: LiU-TEK-LIC-2007:27.

Den kvasilinjära formen av differential-algebraiska ekvationer är både en mycket allmängiltig generalisering av den linjära tidsinvarianta formen, och en form som visar sig lämpa sig väl för indexreduktionsmetoder som vi hoppas ska komma att bli både praktiskt tillämpbara och väl förstådda i framtiden. Kuperingsalgoritmen (engelska: the shuffle algorithm) användes ursprungligen för att bestämma konsistenta initialvillkor för linjära tidsinvarianta differential-algebraiska ekvationer, men har även andra tillämpningar, till exempel det grundläggande problemet numerisk integration. I syfte att förstå hur kuperingsalgoritmen kan tillämpas på kvasilinjära differential-algebraiska ekvationer som inte låter sig analyseras utifrån mönstret av nollor, har problemet att förstå singulära perturbationer i differential-algebraiska ekvationer uppstått. Den här avhandlingen presenterar en indexreduktionsmetod där behovet framgår tydligt, och visar att algoritmen inte bara generaliserar kuperingsalgoritmen, utan även är ett specialfall av den mer allmänna strukturalgoritmen (engelska: the structure algorithm) för att invertera system av Li och Feng. Ett kapitel av den här avhandlingen söker av en klass av ekvations-former efter former som är mindre generella än den kvasilinjära, men som en algoritm lik vår kan anpassas till. Det visar sig att indexreduktionen ofta förstör strukturella egenskaper hos ekvationerna, och att det därför är naturligt att arbeta med den mest allmänna kvasilinjära formen. Avhandlingen innehåller också några tidiga resultat gällande hur perturbationerna kan hanteras. Huvudresultaten är inspirerade av den modellering i skilda tidskalor som görs i teorin om singulära perturbationer (engelska: singular perturbation theory). Medan teorin om singulära perturbationer betraktar inverkan av en försvinnande skalär i ekvationerna, betraktar analysen häri en okänd matris vars norm begränsas av en liten skalär. Resultaten är begränsade till linjära tidsinvarianta ekvationer av index inte högre än 1, men det är värt att notera att index 0-fallet självt innebär en intressant generalisering av teorin för singulära perturbationer för ordinära differentialekvationer. / The quasilinear form of differential-algebraic equations is at the same time both a very versatile generalization of the linear time-invariant form, and a form which turns out to suit methods for index reduction which we hope will be practically applicable and well understood in the future. The shuffle algorithm was originally a method for computing consistent initial conditions for linear time-invariant differential algebraic equations, but has other applications as well, such as the fundamental task of numerical integration. In the prospect of understanding how the shuffle algorithm can be applied to quasilinear differential-algebraic equations that cannot be analyzed by zero-patterns, the question of understanding singular perturbation in differential-algebraic equations has arose. This thesis details an algorithm for index reduction where this need is evident, and shows that the algorithm not only generalizes the shuffle algorithm, but also specializes the more general structure algorithm for system inversion by Li and Feng. One chapter of this thesis surveys a class of forms of equations, searching less general forms than the quasilinear, to which an algorithm like ours can be tailored. It is found that the index reduction process often destroys structural properties of the equations, and hence that it is natural to work with the quasilinear form in its full generality. The thesis also contains some early results on how the perturbations can be handled. The main results are inspired by the separate timescale modeling found in singular perturbation theory. While the singular perturbation theory considers the influence of a vanishing scalar in the equations, the analysis herein considers an unknown matrix bounded in norm by a small scalar. Results are limited to linear time-invariant equations of index at most 1, but it is worth noting that the index 0 case in itself holds an interesting generalization of the singular perturbation theory for ordinary differential equations. / Report code: LiU-TEK-LIC-2007:27.

No description available.

<p> Systems of differential-algebraic equations (DAEs) arise in numerious applications, and there has been considerable research on solving DAE initial value problems (IVPs). Existing methods and software for solving DAEs usually handle at most index-three problems. However, DAE problems of index three and higher do arise, for example, in actuator dynamics, multi-stage processes, and optimization.</p> <p> We present the method of J. Pryce and N. Nedialkov for solving DAEs, which can be of high index, fully implicit, and contain derivatives of order higher than one. We solve such DAEs by expanding their solution in Taylor series (TS). To compute Taylor coefficients, we employ J. Pryce's structural analysis and automatic differentiation. Then we compute an approximate TS solution with appropriate stepsize and project this solution to satisfy the constraints (explicit and hidden) of the problem.</p> <p> This thesis discusses the algorithms involved in this method, including the algorithms for Taylor coefficients computation, consistent point projection, error estimation, stepsize control, and the overall integration process. The author has implemented a software package named HIDAETS (High-Index DAE by Taylor Series). In this thesis, we present the specification, design, implementation, and usage of HIDAETS. Numerical results on several high-index DAEs are reported. These results demonstrate that HIDAETS is efficient and accurate for solving IVP in DAEs.</p> / Thesis / Master of Science (MSc)

<p> Systems of differential-algebraic equations (DAEs) arise in applications such as circuit simulation, models of chemical processes, optimal control, and multi-body dynamics. Informally, the index of a DAE is the number of differentiations needed to convert it to an ordinary differential equation. The index generally indicates the difficulty of solving a DAE problem. The higher the index of a DAE, the more difficult it is to solve it numerically.</p> <p> Structural index analysis plays a crucial role in solving DAE problems. In this thesis, we present two methods for index analysis, namely, Pryce's structural analysis (SA) and Linninger's symbolic-numeric (SN) analysis. We provide a Matlab tool implementing these two approaches: an Automatic Structural Index Analyzer (ASIA). We discuss the underlying algorithms, which include generating a signature matrix and computing SA index, computing a system Jacobian, and generating a symbolic-numeric matrix and computing SN index. We also present implementation issues and illustrate how ASIA is used.</p> <p> Numerical experiments show that ASIA can produce reliable structural information. We also show examples on which structural analysis fails, and how ASIA detects such situations.<p> / Thesis / Master of Science (MSc)

We propose two efficient automatic differentiation (AD) schemes to compute incrementally Taylor series and System Jacobian for solving differential-algebraic equations (DAEs) by Taylor series. Our schemes are based on topological ordering of a DAE's computational graph and then partitioning the topologically sorted nodes using structural information obtained from the DAE. Solving a DAE by Taylor series is carried out in stages. From one stage to another, partitions of the computational graph are incrementally activated so that we can reuse Taylor coefficients and gradients computed in previous stages. As a result, the computational complexity of evaluating a System Jacobian is independent of the number of stages. We also develop a common subexpression elimination (CSE) method to build a compact computational graph through operator overloading. The CSE method is of linear time complexity, which makes it suitable as a preprocessing step for general operator overloaded computing. By applying CSE, all successive overloaded computation can save time and memory. Furthermore, the computational graph of a DAE reveals its internal sparsity structure. Based on it, we devise an algorithm to propagate gradients in the forward mode of AD using compressed vectors. This algorithm can save both time and memory when computing the System Jacobian for sparse DAEs. We have integrated our approaches into the \daets solver. Computational results show multiple-fold speedups against two popular AD tools, \FAD~and ADOL-C, when solving various sparse and dense DAEs. / Thesis / Master of Science (MSc)