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Title:On the application of numerical continuation to large-scale dynamical systems
Author(s):Saghafi, Mehdi
Director of Research:Dankowicz, Harry
Doctoral Committee Chair(s):Dankowicz, Harry
Doctoral Committee Member(s):Hiskens, Ian; Mehta, Prashant; West, Matthew
Department / Program:Mechanical Sci & Engineering
Discipline:Mechanical Engineering
Degree Granting Institution:University of Illinois at Urbana-Champaign
Subject(s):Large-scale systems
Numerical Continuations
Power Systems
Aerosol Particles
Nonlinear Structures
Error Analysis
Abstract:The system approach in dealing with large groups of components with nonlinear interactions has received constantly growing attention in recent years in wide variety of applications from system biology to the size evolution of aerosol particles. These systems exhibit complex behaviors that are not predictable unless the entire system as well as their interactions are taken into account. The complexity of these high-dimensional problems is exacerbated when parameter space of interest is also sizable. The focus of this dissertation is addressing several physical and engineering problems within the context of large-scale dynamical systems through the development as well as implementation of novel computational algorithms based on numerical path-following techniques. For all systems investigated in this study, a suitable boundary value problem is formulated and the numerical continuation method is used for covering the parameter space using the software-package coco. In this dissertation, we first briefly consider the reachability problem in power system networks in which the uncertainties in the loads require robust design of system parameters to ensure safe operation of the network under undesired perturbation in the system. This analysis is based on an existing matlab-code which utilizes a so-called numerical shooting method for the parametric analysis of power systems. A wrapper is developed to make the custom-designed systems in this code compatible with the asynchronous collocation toolbox in coco. The asynchronous collocation toolbox developed in this dissertation extends the capability of common implementation of collocation methods for differential equations by reducing the number of mesh points required specifically for problems with slow-fast dynamics. This class of dynamical systems appears in many large-scale physical systems in which changes of states are governed by different time scales. The transient growth of aerosol particles in a humid environment is then investigated in order to explore the effect of some system parameters, such as the initial size distribution and the rate of temperature decay, on the formation of cloud droplets. For a fixed fraction of droplet forming particles in a rising parcel, a suitably formulated boundary value problem is used along with numerical continuation to obtain the solutions in the parameter space of interest. Results obtained from the proposed numerical scheme are compared with the estimated fractions from available criteria in the literature to show the estimation error due to a phenomenon known as kinetic limitation. In the second part of this dissertation, our objective is to identify and, where possible, resolve singularities that may arise in the discretization of spatiotemporal boundary-value problems governing the steady-state behavior of nonlinear beam structures. Of particular interest is the formulation of nondegenerate continuation problems of a geometrically-nonlinear model of a slender beam, subject to a uniform harmonic excitation, that may be analyzed numerically in order to explore the parameter-dependence of the excitation response. Several methods for breaking both the spatial and temporal equivariance are proposed. We then use these findings to suitably discretize the mixed formulation of the governing equations of a beam in the longitudinal and transverse directions where inertia effects are taken into account. The numerical results corresponding to the free vibrations are compared to a perturbation analysis obtained using the multiple-scales method to show the validity of the numerical scheme. The developed numerical technique is then employed to investigate the geometry as well as parameter dependence of the range of the linear regime in the beam's forced response. Finally, the convergence associated with the collocation scheme is investigated in order to rigorously analyze the capabilities of this method. The dissertation ends with a discussion of further development of asynchronous discretization scheme as well as several physical questions to be addressed.
Issue Date:2015-01-21
Rights Information:Copyright 2014 Mehdi Saghafi
Date Available in IDEALS:2015-01-21
Date Deposited:2014-12

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