Thesis (Ph. D.)--University of Tennessee, Knoxville, 2005. / Title from title page screen (viewed on Feb. 14, 2006). Vita. Includes bibliographical references.

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<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)

We first provide a detailed background of a geometric projection methodology developed by Professor Roswitha Marz at Humboldt University in Berlin for showing uniqueness and existence of solutions for ordinary differential-algebraic equations (DAEs). Because of the geometric and operator-theoretic aspects of this particular method, it can be extended to the case of infinite-dimensional abstract DAEs. For example, partial differential equations (PDEs) are often formulated as abstract Cauchy or evolution problems which we label abstract ordinary differential equations or AODE. Using this abstract formulation, existence and uniqueness of the Cauchy problem has been studied. Similarly, we look at an AODE system with operator constraint equations to formulate an abstract differential-algebraic equation or ADAE problem. Existence and uniqueness of solutions is shown under certain conditions on the operators for both index-1 and index-2 abstract DAEs. These existence and uniqueness results are then applied to some index-1 DAEs in the area of thermodynamic modeling of a chemical vapor deposition reactor and to a structural dynamics problem. The application for the structural dynamics problem, in particular, provides a detailed construction of the model and development of the DAE framework. Existence and uniqueness are primarily demonstrated using a semigroup approach. Finally, an exploration of some issues which arise from discretizing the abstract DAE are discussed. / Ph. D.

Transformers are very important parts of any electrical network, and their size increase so does their price. Protecting these important devices is a daunting task due to the wide variety of operating conditions. This thesis develops a new protection scheme based on state estimation.The foundation upon which our protection scheme is built is the modeling of the single phase transformer system of equations. The transformer equations are composed of polynomial and differential equations and this system of equations involving the transformer's electrical quantities are modeled into a system of equations such that highest degree of each of the system's equations is quadratic―in a process named Quadratization and then integrated using a technique called Quadratic integration to give a set of algebraic companion equations that can be solved numerically to determine the health of the transformer.

We consider the study of symmetry analysis of difference equations. The original work done by Lie about a century ago is known to be one of the best methods of solving differential equations. Lie's theory of difference equations on the contrary, was only first explored about twenty years ago. In 1984, Maeda [42] constructed the similarity methods for difference equations. Some work has been done in the field of symmetries of difference equations for the past years. Given an ordinary or partial differential equation (PDE), one can apply Lie algebra techniques to analyze the problem. It is commonly known that the number of independent variables can be reduced after the symmetries of the equation are obtained. One can determine the optimal system of the equation in order to get a reduction of the independent variables. In addition, using the method, one can obtain new solutions from known ones. This feature is interesting because some differential equations have apparently useless trivial solutions, but applying Lie symmetries to them, more interesting solutions are obtained. The question arises when it happens that our equation contains a discrete quantity. In other words, we aim at investigating steps to be performed when we have a difference equation. Doing so, we find symmetries of difference equations and use them to linearize and reduce the order of difference equations. In this work, we analyze the work done by some researchers in the field and apply their results to some examples. This work will focus on the topical review of symmetries of difference equations and going through that will enable us to make some contribution to the field in the near future. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2011.

This thesis addressed non-smooth dynamics of lumped parameter systems, and was restricted to Filippov-type systems. The main objective of this thesis was twofold. Firstly, modeling aspects of Filippov-type non-smooth dynamical systems were addressed with an emphasis on the constitutive assumptions and mathematical structure behind these models. Secondly, robust algorithms were presented to obtain numerical solutions for various Filippov-type lumped parameter systems. Governing equations were written using two different mathematical approaches. The first approach was based on differential inclusions and the second approach was based on differential-algebraic equations. The differential inclusions approach is more amenable to mathematical analysis using existing mathematical tools. On the other hand, the approach based on differential-algebraic equations gives more insight into the constitutive assumptions of a chosen model and easier to obtain numerical solutions. Bingham-type models in which the force cannot be expressed in terms of kinematic variables but the kinematic variables can be expressed in terms of force were considered. Further, Coulomb friction was considered in which neither the force can be expressed in terms of kinematic variables nor the kinematic variables in terms of force. However, one can write implicit constitutive equations in terms of kinematic quantities and force. A numerical framework was set up to study such systems and the algorithm was devised. Towards the end, representative dynamical systems of practical significance were considered. The devised algorithm was implemented on these systems and the results were obtained. The results show that the setting offered by differential-algebraic equations is appropriate for studying dynamics of lumped parameter systems under implicit constitutive models.

Thesis (Ph. D.)--Aerospace Engineering, Georgia Institute of Technology, 2009. / Committee Chair: Olivier A. Bauchau; Committee Member: Andrew Makeev; Committee Member: Carlo L. Bottasso; Committee Member: Dewey H. Hodges; Committee Member: Massimo Ruzzene. Part of the SMARTech Electronic Thesis and Dissertation Collection.