In this thesis, a reduced order model (ROM) based on the proper orthogonal decomposition (POD) for the transport phenomena in fluidized beds has been developed. The reduced order model is tested first on a gas-only flow. Two different strategies and implementations are described for this case. Next, a ROM for a two-dimensional gas-solids fluidized bed is presented. A ROM is developed for a range of diameters of the solids particles. The reconstructed solution is calculated and compared against the full order solution. The differences between the ROM and the full order solution are smaller than 3.2% if the diameters of the solids particles are in the range of diameters used for POD database generation. Otherwise, the errors increase up to 10% for the cases presented herein. The computational time of the ROM varied between 25% and 33% of the computational time of the full order solution. The computational speed-up depended on the complexity of the transport phenomena, ROM methodology and reconstruction error. In this thesis, we also investigated the accuracy of the reduced order model based on the POD. When analyzing the accuracy, we used two simple sets of governing partial differential equations: a non-homogeneous Burgers' equation and a system of two coupled Burgers' equations.

In this thesis, we adapt and apply methods from linear control theory to shear flows. The challenge of this task is to build a linear dynamic system that models the evolution of the flow, using the Navier--Stokes equations, then to define sensors and actuators, that can sense the flow state and affect its evolution. We consider flows exposed to stochastic excitations. This framework allows to account for complex sources of excitations, often present in engineering applications. Once the system is built, including dynamic model, sensors, actuators, and sources of excitations, we can use standard optimization techniques to derive a feedback law. We have used feedback control to stabilize unstable flows, and to reduce the energy level of sensitive flows subject to external excitations. / QC 20100830

Modern nonlinear programming solvers can efficiently handle very large scale optimization problems when accurate derivative information is available. However, black box or derivative free modeling components are often unavoidable in practice when the modeled phenomena may cross length and time scales. This work is motivated by examples in chemical process optimization where most unit operations have well-known equation oriented representations, but some portion of the model (e.g. a complex reactor model) may only be available with an external function call. The concept of a surrogate model is frequently used to solve this type of problem. A surrogate model is an equation oriented approximation of the black box that allows traditional derivative based optimization to be applied directly. However, optimization tends to exploit approximation errors in the surrogate model leading to inaccurate solutions and repeated rebuilding of the surrogate model. Even if the surrogate model is perfectly accurate at the solution, this only guarantees that the original problem is feasible. Since optimality conditions require gradient information, a higher degree of accuracy is required. In this work, we consider the general problem of hybrid glass box/black box optimization, or gray box optimization, with focus on guaranteeing that a surrogate-based optimization strategy converges to optimal points of the original detailed model. We first propose an algorithm that combines ideas from SQP filter methods and derivative free trust region methods to solve this class of problems. The black box portion of the model is replaced by a sequence of surrogate models (i.e. surrogate models) in trust region subproblems. By carefully managing surrogate model construction, the algorithm is guaranteed to converge to true optimal solutions. Then, we discuss how this algorithm can be modified for effective application to practical problems. Performance is demonstrated on a test set of benchmarks as well as a set of case studies relating to chemical process optimization. In particular, application to the oxycombustion carbon capture power generation process leads to significant efficiency improvements. Finally, extensions of surrogate-based optimization to other contexts is explored through a case study with physical properties.

Simulating the complex physics and dynamics associated with unsteady aeroelastic systems is often attempted with high-fidelity numerical models. While these high-fidelity approaches are powerful in terms of capturing the main physical features, they may not discern the role of underlying phenomena that are interrelated in a complex manner. This often makes it difficult to characterize the relevant causal mechanisms of the observed features. Besides, the extensive computational resources and time associated with the use these tools could limit the capability of assessing different configurations for design purposes. These shortcomings present the need for the development of simplified and reduced-order models that embody relevant physical aspects and elucidate the underlying phenomena that help in characterizing these aspects. In this work, different fluid and aeroelastic systems are considered and reduced-order models governing their behavior are developed. In the first part of the dissertation, a methodology, based on the method of multiple scales, is implemented to show its usefulness and effectiveness in the characterization of the physics underlying the system, the implementation of control strategies, and the identification of high-impact system parameters. In the second part, the unsteady aerodynamic aspects of flapping micro air vehicles (MAVs) are modeled. This modeling is required for evaluation of performance requirements associated with flapping flight. The extensive computational resources and time associated with the implementation of high-fidelity simulations limit the ability to perform optimization and sensitivity analyses in the early stages of MAV design. To overcome this and enable rapid and reasonably accurate exploration of a large design space, a medium-fidelity aerodynamic tool (the unsteady vortex lattice method) is implemented to simulate flapping wing flight. This model is then combined with uncertainty quantification and optimization tools to test and analyze the performance of flapping wing MAVs under varying conditions. This analysis can be used to provide guidance and baseline for assessment of MAVs performance in the early stages of decision making on flapping kinematics, flight mechanics, and control strategies. / Ph. D.

The characterization of damping is important in making accurate predictions of both the true response and the frequency response of any device or structure dominated by energy dissipation. The process of modeling damping matrices and experimental verification of those is challenging because damping can not be determined via static tests as can mass and stiffness. Furthermore, damping is more difficult to determine from dynamic measurements than natural frequency. However, damping is extremely important in formulating predictive models of structures. In addition, damping matrix identification may be useful in diagnostics or health monitoring of structures. The objective of this work is to find a robust, practical procedure to identify damping matrices. All aspects of the damping identification procedure are investigated. The procedures for damping identification presented herein are based on prior knowledge of the finite element or analytical mass matrices and measured eigendata. Alternately, a procedure is based on knowledge of the mass and stiffness matrices and the eigendata. With this in mind, an exploration into model reduction and updating is needed to make the problem more complete for practical applications. Additionally, high performance computing is used as a tool to deal with large problems. High Performance Fortran is exploited for this purpose. Finally, several examples, including one experimental example are used to illustrate the use of these new damping matrix identification algorithms and to explore their robustness. / Ph. D.

Due to the complexity, multi-scale, and multi-physics nature of the mathematical models for fires, current numerical models require too much computational effort to be useful in design and real-time decision making, especially when dealing with fires over large domains. To reduce the computational time while retaining the complexity of the domain and physics, our research has focused on several reduced-order modeling techniques. Our contributions are improving wildland fire reduced-order models (ROMs), creating new ROM techniques for nonlinear systems, and preserving optimality when discretizing a continuous-time ROM. Currently, proper orthogonal decomposition (POD) is being used to reduce wildland fire-spread models with limited success. We use a technique known as the discrete empirical interpolation method (DEIM) to address the slowness due to the nonlinearity. We create new methods to reduce nonlinear models, such as the Burgers' equation, that perform better than POD over a wider range of input conditions. Further, these ROMs can often be constructed without needing to capture full-order solutions a priori. This significantly reduces the off-line costs associated with creating the ROM. Finally, we investigate methods of time-discretization that preserve the optimality conditions in a certain norm associated with the input to output mapping of a dynamical system. In particular, we are able to show that the Crank-Nicholson method preserves the optimality conditions, but other single-step methods do not. We further clarify the need for these discrete-time ROMs to match at infinity in order to ensure local optimality. / Ph. D.

In this thesis, we apply nonlinear Koopman mode analysis to decompose the swing dynamics of a power system into modes of oscillation, which are identified by analyzing the Koopman operator, a linear infinite-dimensional operator that may be defined for any nonlinear dynamical system. Specifically, power system modes of oscillation are identified through spectral analysis of the Koopman operator associated with a particular observable. This means that they can be determined directly from measurements. These modes, referred to as Koopman modes, are single-frequency oscillations, which may be extracted from nonlinear swing dynamics under small and large disturbances. They have an associated temporal frequency and growth rate. Consequently, they may be viewed as a nonlinear generalization of eigen-modes of a linearized system. Koopman mode analysis has been also applied to identify coherent swings and coherent groups of machines of a power system. This will allow us to carry out a model reduction of a large-scale system and to derive a precursor to monitor the loss of transient stability. / Master of Science

In this thesis a numerical study of the one dimensional viscous Burgers equation is conducted. The discretization techniques Finite Differences, Finite Element Method and Group Finite Elements are applied and their impact on model reduction techniques, namely Proper Orthogonal Decomposition (POD), Group POD and the Discrete Empirical Interpolation Method (DEIM), is studied. This study is facilitated by examination of several common ODE solvers. Embedded in this process, some results on the structure of the POD basis and an alternative algorithm to compute the POD subspace are presented. Various numerical studies are conducted to compare the different methods and the to study the interaction of the spatial discretization on the ROM through the basis functions. Moreover, the results are used to investigate the impact of Reduced Order Models (ROM) on Optimal Control Problems. To this end, the ROM is embedded in a Trust Region Framework and the convergence results of Arian et al. (2000) is extended to POD-DEIM. Based on the convergence theorem and the results of the numerical studies, the emphasis is on implementation strategies for numerical speedup. / Master of Science

To add damping to structures, viscoelastic materials (VEM) are added to structures. In order to enhance the damping effect of the VEM, a constraining layer is attached, creating a passive constrained layer damping treatment (PCLD). When this constraining layer is an active element, the treatment is called active constrained layer damping (ACLD). Recently, the investigation of ACLD treatments has shown it to be an effective method of vibration suppression. In this work, two new hybrid configurations are introduced by separating the passive and active elements. In the first variation, the active and passive element are constrained to the same side of the beam. The other variation allows one of the treatments to be placed on the opposite side of the beam. A comparison will be made with pure active, PCLD, ACLD and a variation which places the active element underneath PCLD. Energy methods and Lagrange's equation are used to obtain equations of motion, which are discretized using assumed modes method. The frequency dependent damping is modeled using the Golla-Hughes-McTavish (GHM) method and the system is analyzed in the time domain. GHM increases the size of the original system by adding fictitious dissipation coordinates that account for the frequency dependent damping. An internally balanced model reduction method is used to reduce the equations of motion to their original size. A linear quadratic regulator and output feedback are used to actively control vibration. The length and placement of treatment is optimized using different criteria. It is shown that placing the active element on the opposite side of the passive element is capable of vibration suppression with lower control effort and more inherent damping. If the opposite surface is not available for treatment, a suitable alternative places the PZT underneath the PCLD. LQR provides the best control, since it assumes all states are available for feedback. Usually only select states are available and output feedback is used. It is shown that output feedback, while not as effective as full state feedback, is still able to damp vibration. / Ph. D.

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