In this thesis, we develop a data-driven correction reduced order model (DDC-ROM) for numerical simulation of fluid flows. The general DDC-ROM involves two stages: (1) we apply ROM filtering (such as ROM projection) to the full order model (FOM) and construct the filtered ROM (F-ROM). (2) We use data-driven modeling to model the nonlinear interactions between resolved and unresolved modes, which solves the F-ROM's closure problem. In the DDC-ROM, a linear or quadratic ansatz is used in the data-driven modeling step. In this thesis, we propose a new cubic ansatz. To get the unknown coefficients in our ansatz, we solve an optimization problem that minimizes the difference between the FOM data and the ansatz. We test the new DDC-ROM in the numerical simulation of the one-dimensional Burgers equation with a small diffusion coefficient. Furthermore, we perform a cross-validation of the DDC-ROM to investigate whether it can be successful in computational settings that are different from the training regime. / M.S. / Practical engineering and scientific problems often require the repeated simulation of unsteady fluid flows. In these applications, the computational cost of high-fidelity full-order models can be prohibitively high. Reduced order models (ROMs) represent efficient alternatives to brute force computational approaches. In this thesis, we propose a data-driven correction ROM (DDC-ROM) in which available data and an optimization problem are used to model the nonlinear interactions between resolved and unresolved modes. In order to test the new DDC-ROM's predictability, we perform its cross-validation for the one-dimensional viscous Burgers equation and different training regimes.

Hot section components of land-based gas turbines are subject to extremely harsh, high temperature environments and require the use of advanced materials. Directionally solidified Ni-base superalloys are often chosen as materials for these hot section components due to their excellent creep resistance and fatigue properties at high temperatures. These blades undergo complex thermomechanical loading conditions throughout their service life, and the influences of blade geometry and variable operation can make life prediction difficult. Accurate predictions of material response under thermomechanical loading conditions is essential for life prediction of these components. Complex crystal viscoplasticity models are often used to capture the behavior of Ni-base superalloys. While accurate, these models are computationally expensive and are not suitable for all phases of design. This work involves the calibration of a previously developed reduced-order, macroscale transversely isotropic viscoplasticity model to a directionally solidified Ni-base superalloy. The unified model is capable of capturing isothermal and thermomechanical responses in addition to secondary creep behavior. An extreme reduced order microstructure-sensitive constitutive model is also developed using an artificial neural network to provide a rapid first-order approximation of material response under various temperatures, rates of loading, and material orientation from the axis of solidification.

<p>Asymptotic Modal Analysis (AMA) is a computationally efficient and accurate method for studying the response of dynamical systems experiencing banded, random harmonic excitation at high frequencies when the number of responding modes is large. In this work, AMA has been extended to systems of coupled continuous components as well as nonlinear systems. Several prototypical cases are considered to advance the technique from the current state-of-the-art. The nonlinear problem is considered in two steps. First, a method for solving problems involving nonlinear continuous multi-mode components, called Iterative Modal Analysis (IMA), is outlined. Secondly, the behavior of a plate carrying a nonlinear spring-mass system is studied, showing how nonlinear effects on system natural frequencies may be accounted for in AMA. The final chapters of this work consider the coupling of continuous systems. For example, two parallel plates coupled at a point are studied. The principal novel element of the two-plate investigation reduces transfer function sums of the coupled system to an analytic form in the AMA approximation. Secondly, a stack of three parallel plates where adjacent plates are coupled at a point are examined. The three-plate investigation refines the reduction of transfer function sums, studies spatial intensification in greater detail, and offers insight into the diminishing response amplitudes in networks of continuous components excited at one location. These chapters open the door for future work in networks of vibrating components responding to banded, high-frequency, random harmonic excitation in the linear and nonlinear regimes.</p> / Dissertation

This work presents a number of the practical aspects of developing reduced- order models (ROMs) based on proper orthogonal decomposition (POD). ROMS are derived and implemented for multiphase flow, quasi-2D nozzle flow and 2D inviscid channel flow. Results are presented verifying the ROMs against existing full-order models (FOM). POD is a method for separating snapshots of a flow field that varies in both time and space into spatial basis functions and time coefficients. The partial differential equations that govern fluid flow can then be pro jected onto these basis functions, generating a system of ordinary differential equations where the unknowns are the time coefficients. This results in the reduction of the number of equations to be solved from hundreds of thousands or more to hundreds or less. A ROM is implemented for three-dimensional and non-isothermal multiphase flows. The derivation of the ROM is presented. Results are compared against the FOM and show that the ROM agrees with the FOM. While implementing the ROM for multiphase flow, moving discontinuities were found to be a ma jor challenge when they appeared in the void fraction around gas bubbles. A point-mode POD approach is proposed and shown to have promise. A simple test case for moving discontinuities, the first order wave equation, is used to test an augmentation method for capturing the discontinuity exactly. This approach is shown to remove the unphysical oscillations that appear around the discontinuityin traditional approaches. A ROM for quasi-2D inviscid nozzle flow is constructed and the results are com- pared to a FOM. This ROM is used to test two approaches, POD-Analytical and POD-Discretized. The stability of each approach is assessed and the results are used in the implementation of a ROM for the Navier-Stokes equations. A ROM for a Navier-Stokes solver is derived and implemented using the results of the nozzle flow case. Results are compared to the FOM for channel flow with a bump. The computational speed-up of the ROM is discussed. Two studies are presented with practical aspects of the implementation of POD- based ROMs. The first shows the effect of the snapshot sampling on the accuracy of the POD basis functions. The second shows that for multiphase flow, the cross- coupling between field variables should not be included when computing the POD basis functions.

<p>A new method of stabilizing low-order, proper orthogonal decomposition based reduced-order models of the Navier Stokes equations is proposed. Unlike traditional approaches, this method does not rely on empirical turbulence modeling or modification of the Navier-Stokes equations. It provides spatial basis functions different from the usual proper orthogonal decomposition basis function in that, in addition to optimally representing the solution, the new proposed basis functions also provide stable reduced-order models. The proposed approach is illustrated with two test cases: two-dimensional flow inside a square lid-driven cavity and a two-dimensional mixing layer.</p> / Dissertation

Interactions between the aerodynamics and dynamics of bluff bodies are important in many engineering applications, including suspension bridges, tall buildings, oil platforms, wind turbine towers, air drops, and construction with cranes. In the rotorcraft field, bluff bodies are commonly suspended underneath the vehicle by tethers. This approach is often the only practical way to deliver a payload in a reasonable amount of time in disaster relief efforts, search-and-rescue operations, and military operations. However, currently a fundamental understanding of the aerodynamics of these bluff bodies is lacking, and accurate dynamic simulation models for predicting the safe flying speed are not available. In order to address these shortcomings, two main advancements are presented in this thesis. The aerodynamics of several three-dimensional canonical bluff bodies are examined over a range of Reynolds numbers representative of wind-tunnel-scale to full-scale models. Numerical experiments are utilized, with a focus on uncertainty analysis and validation of the computations. Mean and unsteady forces and moments for these bluff bodies have been evaluated, and empirical models of the shear layer characteristics have been extracted to quantify the behaviors and provide predictive capability. In addition, a physics-based reduced-order simulation model has been developed for bluff bodies. The physics-based approach is necessary to ensure that the predicted behavior of new configurations is accurate, and it is made possible by the breakthroughs in three-dimensional bluff body aerodynamics presented in this thesis. The integrated aerodynamic forces and moments and dynamic behavior predicted by model are extensively validated with data from wind tunnels, flight tests, and high-fidelity computations. Furthermore, successful stability predictions for tethered loads are demonstrated. The model is applicable to the simulation of any generic bluff body configuration, is readily extensible, and has low computational cost.

Thermal diffusivity in an important thermophysical property that quantifies the ratio of the rate at which heat is conducted through a material to the amount of energy stored in a material. The pulsed laser diffusion (PLD) method is a widely used technique for measuring thermal diffusivities of materials. This technique is based on the fact that the diffusivity of a sample may be inferred from measurement of the time-dependent temperature profile at a point on the surface of a sample that has been exposed to a pulse of radiant energy from a laser or flash lamp. An accepted standard approach for the PLD method is based on a simple model of a PLD measurement system. However, the standard approach is based on idealizations that are difficult to achieve in practice. Therefore, models that treat a PLD measurement system with greater fidelity are desired. The objective of this research is to develop and test a higher fidelity model that more accurately represents the spatial and temporal variations in the input power. This higher fidelity model is referred to as Distributed Source Finite Absorption (DSFA) model. The cost of the increased fidelity associated with the DSFA model is an increase in the complexity of inferring values of the thermal diffusivity. A new method of extracting values from time dependent temperature measurements based on a genetic algorithm and on reduced order modeling was developed. The primary contribution of this thesis is a detailed discussion of the development and numerical verification of this proposed new method for measuring the thermal diffusivity of various materials. Verification of the proposed new method was conducted using numerical experiments. A detailed model of a PLD system was created using advanced engineering software, and detailed simulations, including conjugate heat transfer and solution of the full Navier-Stokes equations, were used to generate multiple numerical data sets. These numerical data sets were then used to infer the thermal diffusivity and other properties of the sample using the proposed new method. These numerical data sets were also used as inputs to the standard approach. The results of this verification study show that the proposed new method is able to infer the thermal diffusivity of samples to within 4.93%, the absorption coefficient to within 10.57 % and the heat capacity of the samples to within 5.37 %. Application of the standard approach to these same data sets gave much poorer estimates of the thermal diffusivity, particularly when the absorption coefficient of the material was relatively low.

<div>This dissertation is devoted to developing a first-of-a-kind uncertainty characterization framework (UCF) providing comprehensive, efficient and scientifically defendable methodologies for uncertainty characterization (UC) in best-estimate (BE) reactor physics simulations. The UCF is designed with primary application to CANDU neutronics calculations, but could also be applied to other thermal-spectrum reactor systems. The overarching goal of the UCF is to propagate and prioritize all sources of uncertainties, including those originating from nuclear data uncertainties, modeling assumptions, and other approximations, in order to reliably use the results of BE simulations in the various aspects of reactor design, operation, and safety. The scope of this UCF is to propagate nuclear data uncertainties from the multi-group format, representing the input to lattice physics calculations, to the few-group format, representing the input to nodal diffusion-based core simulators and quantify the uncertainties in reactor core attributes.</div><div>The main contribution of this dissertation addresses two major challenges in current uncertainty analysis approaches. The first is the feasibility of the UCF due to the complex nature of nuclear reactor simulation and computational burden of conventional uncertainty quantification (UQ) methods. The second goal is to assess the impact of other sources of uncertainties that are typically ignored in the course of propagating nuclear data uncertainties, such as various modeling assumptions and approximations.</div>To deal with the first challenge, this thesis work proposes an integrated UC process employing a number of approaches and algorithms, including the physics-guided coverage mapping (PCM) method in support of model validation, and the reduced order modeling (ROM) techniques as well as the sensitivity analysis (SA) on uncertainty sources, to reduce the dimensionality of uncertainty space at each interface of neutronics calculations. In addition to the efficient techniques to reduce the computational cost, the UCF aims to accomplish four primary functions in uncertainty analysis of neutronics simulations. The first function is to identify all sources of uncertainties, including nuclear data uncertainties, modeling assumptions, numerical approximations and technological parameter uncertainties. Second, the proposed UC process will be able to propagate the identified uncertainties to the responses of interest in core simulation and provide uncertainty quantifications (UQ) analysis for these core attributes. Third, the propagated uncertainties will be mapped to a wide range of reactor core operation conditions. Finally, the fourth function is to prioritize the identified uncertainty sources, i.e., to generate a priority identification and ranking table (PIRT) which sorts the major sources of uncertainties according to the impact on the core attributes’ uncertainties. In the proposed implementation, the nuclear data uncertainties are first propagated from multi-group level through lattice physics calculation to generate few-group parameters uncertainties, described using a vector of mean values and a covariance matrix. Employing an ROM-based compression of the covariance matrix, the few-group uncertainties are then propagated through downstream core simulation in a computationally efficient manner.<div>To explore on the impact of uncertainty sources except for nuclear data uncertainties on the UC process, a number of approximations and assumptions are investigated in this thesis, e.g., modeling assumptions such as resonance treatment, energy group structure, etc., and assumptions associated with the uncertainty analysis itself, e.g., linearity assumption, level of ROM reduction and associated number of degrees of freedom employed. These approximations and assumptions have been employed in the literature of neutronic uncertainty analysis yet without formal verifications. The major argument here is that these assumptions may introduce another source of uncertainty whose magnitude needs to be quantified in tandem with nuclear data uncertainties. In order to assess whether modeling uncertainties have an impact on parameter uncertainties, this dissertation proposes a process to evaluate the influence of various modeling assumptions and approximations and to investigate the interactions between the two major uncertainty sources. To explore this endeavor, the impact of a number of modeling assumptions on core attributes uncertainties is quantified.</div><div>The proposed UC process has first applied to a BWR application, in order to test the uncertainty propagation and prioritization process with the ROM implementation in a wide range of core conditions. Finally, a comprehensive uncertainty library for CANDU uncertainty analysis with NESTLE-C as core simulator is generated compressed uncertainty sources from the proposed UCF. The modeling uncertainties as well as their impact on the parameter uncertainty propagation process are investigated on the CANDU application with the uncertainty library.</div>

An efficient solution method for evaluating radiative transport in a foam layer is a valuable tool for predicting the properties of the layer. Two different solution methods have been investigated. First, a reverse Monte Carlo (RMC) simulation has been developed. In the RMC simulation photon bundles are traced backwards from a detector to the source where they were emitted. The RMC method takes advantage of time reflection symmetry, allowing the photons to be traced backwards in the same manner they are tracked in a standard forward Monte Carlo scheme. Second, a reduced order model based on the singular value decomposition (ROM) has been developed. ROM uses solutions of the reflectance-time profiles found for specific values of the governing parameters to form a solution basis that can be used to generate the profile for any arbitrary values of the parameter set. The governing parameters that were used in this study include the foam layer thickness, the asymmetry parameter, and the scattering coefficient. Layer thicknesses between 4 cm and 20 cm were considered. Values of the asymmetry parameter varied between 0.2 and .08, while the scattering coefficient ranged from 2800 m-1 to 14000 m-1. Ten blind test cases with parameters chosen randomly from these ranges were run and compared to an established forward Monte Carlo (FMC) solution to determine the accuracy and efficiency of both methods. For both RMC and ROM methods the agreement with FMC is good. The average difference in areas under the curves relative to the FMC curve for the ten cases of RMC is 7.1% and for ROM is 7.6%. One of the ten cases causes ROM to extrapolate outside of its data set. If this case is excluded the average error for the remaining nine cases is 5.3%. While the efficiency of RMC for this case is not much greater than that of FMC, it is advantageous in that a solution over a specified time range can be found, as opposed to the FMC where the entire profile must be found. ROM is a very efficient solution method. After a library of solutions is developed, a separated solution with different parameters can be found essentially in real-time. Because of the efficiency of this ROM it is a very promising solution technique for property analysis using inverse methods.

The objective of this dissertation is to present a modeling and simulation methodology for MEMS devices and identify and understand the associated nonlinearities due to large deflections, electric actuation, impacts, and friction. In the first part of the dissertation, we introduce a reduced-order model of flexible microplates under electric excitation. The model utilizes the von Karman plate equations to account for geometric nonlinearities due to large plate deflections. The Galerkin approach is employed to reduce the partial-differential equations of motion and associated boundary conditions into a finite dimensional system of nonlinearly coupled ordinary-differential equations. We use the reduced-order model to analyze the mechanical behavior of a simply supported microplate and a fully clamped microplate. Effect of various design parameters on both the static and dynamic characteristics of microplates is studied. The second part of the dissertation presents comprehensive modeling and simulation tools for impact microactuators. Nonsmooth dynamics due to impacts and friction are studied, combining various approaches, including direct numerical integration, root-finding technique for periodic motions, continuation of grazing periodic orbits, and local analysis of the near grazing dynamics. The transition between nonimpacting and impacting long term motions, referred to as grazing bifurcations, indicates the transition between on and off states of an impact microactuator. Three different on-off switching mechanisms are identified for the Mita microactuator. These mechanisms also generalize to arbitrary impacting systems with a similar nonlinearity. A local map based on the concept of discontinuity mapping provides an effcient and accurate tool for the grazing bifurcation analysis. Nonlinear impacting dynamics of the microactuator are studied in detail to identify various bifurcations and parameter ranges corresponding to chaotic motions. We find that the frequency-response curves of the impacting dynamics are significantly different from those of the nonimpacting dynamics. / Ph. D.