The need for low order models capable of performing damage identification has become apparent in many structural dynamics applications where structural health monitoring and damage prognosis programs are implemented. These programs require that damage identification routines have low computational requirements and be reliable with some quantifiable degree of accuracy. Response surface metamodels (RSMs) are proposed to fill this need. Popular in the fields of chemical and industrial engineering, RSMs have only recently been applied in the field of structural dynamics and to date there have been no studies which fully demonstrate the potential of these methods. In this thesis, several RSMs are developed in order to demonstrate the potential of the methodology. They are shown to be robust to noise (experimental variability) and have success in solving the damage identification problem, both locating and quantifying damage with some degree of accuracy, for both linear and nonlinear systems. A very important characteristic of the RSMs developed in this thesis is that they require very little information about the system in order to generate relationships between damage indicators and measureable system responses for both linear and nonlinear structures. As such, the potential of these methods for damage identification has been demonstrated and it is recommended that these methods be developed further. / Master of Science

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An investigation into the responses of microbeams to electric actuations is presented. Attention is focused mainly on the use of microbeams in two important MEMS-based devices: capacitive microswitches and resonant microsensors. Nonlinear models are developed to simulate the behavior of the microbeams in each device. The models account for mid-plane stretching, an applied axial load, a DC electrostatic force, and, for the case of resonant sensors, an AC harmonic force. Further, a novel method that uses a reduced-order model is introduced for simulating the behavior of microbeams under a DC electrostatic force. The presented method shows attractive features, like for example, a high stability near the pull-in and a low computational cost. Thus, it can be of significant benefit to the development of MEMS design software. The static behavior of microbeams under electrostatic forces is studied using two methods. One method employs a shooting technique for solving the boundary-value problem that governs the static behavior. The second method is based on solving an algebraic system of equations obtained from the reduced-order model. Further, the eigenvalue problem describing the vibrations of a microbeam around its statically deflected position is solved using a shooting method to obtain the microbeam mode shapes and natural frequencies. The dynamic behavior of resonant microbeams is also investigated. A perturbation method, the method of multiple scales, is used to obtain two first-order nonlinear ordinary-differential equations that describe the amplitude and phase of the response and its stability. The results show that an inaccurate representation of the system nonlinearities may lead to an erroneous prediction of the nonlinear resonance frequency of a microbeam. The case of three-to-one internal resonance between the lowest two modes is treated. Finally, the reduced-order model is used to study the dynamic behavior of the electrostatically actuated microbeams. The proposed models are validated by comparing their results with experimental results available in the literature. / Master of Science

Lean-premixed combustion has the advantage of low emissions for modern gas turbines, but it is susceptible to thermoacoustic instabilities, which can result in large amplitude pressure oscillations in the combustion chamber. The thermoacoustic limit cycle is generated by the unsteady heat release dynamics coupled to the combustor acoustics. In this dissertation, we focused on reduced-order modeling of the dynamics of a laminar premixed flame. From first principles of combustion dynamics, a physically-based, reduced-order, nonlinear model was developed based on the proper orthogonal decomposition technique and generalized Galerkin method. In addition, the describing function for the flame was measured experimentally and used to identify an empirical nonlinear flame model. Furthermore, a linear acoustic model was developed and identified for the Rijke tube experiment. Closed-loop thermoacoustic modeling using the first principles flame model coupled to the linear acoustics successfully reproduced the linear instability and predicted the thermoacoustic limit cycle amplitude. With the measured experimental flame data and the modeled linear acoustics, the describing function technique was applied for limit cycle analysis. The thermoacoustic limit cycle amplitude was predicted with reasonable accuracy, and the closed-loop model also predicted the performance for a phase shift controller. Some problems found in the predictions for high heat release cases were documented. / Ph. D.

Composite materials hold numerous advantages over conventional aircraft grade metals. These include high stiffness/strength-to-weight ratios and beneficial stiffness coupling typically used for aeroelastic tailoring. Due to the complexity of modeling composites, designers often select safe, simple geometry and layup schedules for their wing/blade cross-sections. An example of this might be a box-beam made up of 4 laminates, all of which are quasi-isotropic. This results in neglecting more complex designs that might yield a more effective solution, but require a greater analysis effort. The present work aims to show that the incorporation of complex cross-sections are feasible in the early design process through the use of cross-sectional analysis in conjunction with Timoshenko beam theory. It is important to note that in general, these cross-sections can be inhomogeneous: made up of any number of various materials systems. In addition, these materials could all be anisotropic in nature. The geometry of the cross-sections can take on any shape. Through this reduced order modeling scheme, complex structures can be reduced to 1 dimensional beams. With this approach, the elastic behavior of the structure can be captured, while also allowing for accurate 3D stress and strain recovery. This efficient structural modeling would be ideal in the preliminary design optimization of a wing structure. Furthermore, in conjunction with an efficient unsteady aerodynamic model such as the doublet lattice method, the dynamic aeroelastic stability can also be efficiently captured. This work introduces a comprehensively verified, open source python API called AeroComBAT (Aeroelastic Composite Beam Analysis Tool). By leveraging cross-sectional analysis, Timoshenko beam theory, and unsteady doublet-lattice method, this package is capable of efficiently conducting linear static structural analysis, normal mode analysis, and dynamic aeroelastic analysis. AeroComBAT can have a significant impact on the design process of a composite structure, and would be ideally implemented as part of a design optimization. / Master of Science

Mathematical models describing real world processes are becoming increasingly complex to better match the dynamics of the true system. While this is a positive step towards more complete knowledge of our world, numerical evaluations of these models become increasingly computationally inefficient, requiring increased resources or time to evaluate. This has led to the need for simplified surrogates to these complex mathematical models. A growing surrogate modeling solution is with the usage of neural networks. Neural networks (NN) are known to generalize an approximation across a diverse dataset and minimize the solution along complex nonlinear boundaries. Additionally, these surrogate models can be found using only incomplete knowledge of the true dynamics. However, NN surrogates often suffer from a lack of interpretability, where the decisions made in the training process are not fully understood, and the roles of individual neurons are not well defined. We present two solutions towards this lack of interpretability. The first focuses on mimicking polynomial chaos (PC) modeling techniques, modifying the structure of a NN to produce polynomial approximations of the underlying dynamics. This methodology allows for an extractable meaning from the network and results in improvement in accuracy over traditional PC methods. Secondly, we examine the construction of a reduced order modeling scheme using NN autoencoders, guiding the decisions of the training process to better match the real dynamics. This guiding process is performed via a physics-informed (PI) penalty, resulting in a speed-up in training convergence, but still results in poor performance compared to traditional schemes. / Master of Science / The world is an elaborate system of relationships between diverse processes. To accurately represent these relationships, increasingly complex models are defined to better match what is physically seen. These complex models can lead to issues when trying to use them to predict a realistic outcome, either requiring immensely powerful computers to run the simulations or long amounts of time to present a solution. To fix this, surrogates or approximations to these complex models are used. These surrogate models aim to reduce the resources needed to calculate a solution while remaining as accurate to the more complex model as possible. One way to make these surrogate models is through neural networks. Neural networks try to simulate a brain, making connections between some input and output given to the network. In the case of surrogate modeling, the input is some current state of the true process, and the output is what is seen later from the same system. But much like the human brain, the reasoning behind why choices are made when connecting the input and outputs is often largely unknown. Within this paper, we seek to add meaning to neural network surrogate models in two different ways. In the first, we change what each piece in a neural network represents to build large polynomials (e.g., $x^5 + 4x^2 + 2$) to approximate the larger complex system. We show that the building of these polynomials via neural networks performs much better than traditional ways to construct them. For the second, we guide the choices made by the neural network by enforcing restrictions in what connections it can make. We do this by using additional information from the larger system to ensure the connections made focus on the most important information first before trying to match the less important patterns. This guiding process leads to more information being captured when the surrogate model is compressed into only a few dimensions compared to traditional methods. Additionally, it allows for a faster learning time compared to similar surrogate models without the information.

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.

In this thesis we propose, analyze, and investigate numerically a novel two-level Galerkin reduced order model (2L-ROM) for the efficient and accurate numerical simulation of the steady Navier-Stokes equations. In the first step of the 2L-ROM, a relatively low-dimensional nonlinear system is solved. In the second step, the Navier-Stokes equations are linearized around the solution found in the first step, and a higher-dimensional system for the linearized problem is solved. We prove an error bound for the new 2L-ROM and compare it to the standard Galerkin ROM, or one-level ROM (1L-ROM), in the numerical simulation of the steady Burgers equation. The 2L-ROM significantly decreases (by a factor of 2 and even 3) the 1L-ROM computational cost, without compromising its numerical accuracy. / Master of Science / In this thesis we introduce a new method for efficiently and accurately simulating fluid flow, the Navier-Stokes equations, called the two-level Galerkin reduced order model (2L-ROM). The 2L-ROM involves solving a relatively low-dimensional nonlinear system in the first step, followed by a higher-dimensional linearized system in the second step. We show that this method produces highly accurate results while significantly reducing computational costs compared to previous methods. We provide a comparison between the 2L-ROM and the standard Galerkin ROM, or one-level ROM (1L-ROM), by modeling the steady Burgers equation, as an example. Our results demonstrate that the 2L-ROM reduces the computational cost of the 1L-ROM by a factor of 2 to 3 without sacrificing accuracy.

Dynamical systems are useful tools for modeling many complex physical phenomena. In many situations, we do not have access to the governing equations to create these models. Instead, we have access to data in the form of input-output measurements. Data-driven approaches use these measurements to construct reduced order models (ROMs), a small scale model that well approximates the true system, directly from input/output data. Frequency domain data-driven methods, which require access to values (and in some cases to derivatives) of the transfer function, have been very successful in constructing high-fidelity ROMs from data. However, at times this frequency domain data can be difficult to obtain or one might have only access to time-domain data. Recently, Burohman et al. [2020] introduced a framework to approximate transfer function values using only time-domain data. We first discuss improvements to this method to allow a more efficient and more robust numerical implementation. Then, we develop an algorithm that performs optimal-H2 approximation using purely time-domain data; thus significantly extending the applicability of H2-optimal approximation without a need for frequency domain sampling. We also investigate how well other established frequency-based ROM techniques (such as the Loewner Framework, Adaptive Anderson-Antoulas Algorithm, and Vector Fitting) perform on this identified data, and compare them to the optimal-H2 model. / Master of Science / Dynamical systems are useful tools for modeling many phenomena found in physics, chemistry, biology, and other fields of science. A dynamical system is a system of ordinary differential equations (ODEs), together with a state to output mapping. These typically result from a spatial discretization of a partial differential equation (PDE). For every dynamical system, there is a corresponding transfer function in the frequency domain that directly links an input to the system with its corresponding output. For some phenomena where the underlying system does not have a known governing PDE, we are forced to use observations of system input-output behavior to construct models of the system. Such models are called data-driven models. If in addition, we seek a model that can well approximate the true system while keeping the number of degrees of freedom low (e.g., for fast simulation of the system or lightweight memory requirements), we refer to the resulting model as a reduced order model (ROM). There are well established ROM methods that assume access to transfer function input-output data, but such data may be costly or impossible to obtain. This thesis expands upon a method introduced by Burohman et al. [2020] to infer values and derivatives of the transfer function using time domain input-output data. The first contribution of this thesis is to provide a robust and efficient implementation for the data informativity framework. We then provide an algorithm for constructing a ROM that is optimal in a frequency domain sense from time domain data. Finally, we investigate how other established frequency domain ROM techniques perform on the learned frequency domain data.

This dissertation uses spatial filtering to develop a large eddy simulation reduced order model (LES-ROM) framework for fluid flows. Proper orthogonal decomposition is utilized to extract the dominant spatial structures of the system. Within the general LES-ROM framework, two approaches are proposed to address the celebrated ROM closure problem. No phenomenological arguments (e.g., of eddy viscosity type) are used to develop these new ROM closure models. The first novel model is the approximate deconvolution ROM (AD-ROM), which uses methods from image processing and inverse problems to solve the ROM closure problem. The AD-ROM is investigated in the numerical simulation of a 3D flow past a circular cylinder at a Reynolds number $Re=1000$. The AD-ROM generates accurate results without any numerical dissipation mechanism. It also decreases the CPU time of the standard ROM by orders of magnitude. The second new model is the calibrated-filtered ROM (CF-ROM), which is a data-driven ROM. The available full order model results are used offline in an optimization problem to calibrate the ROM subfilter-scale stress tensor. The resulting CF-ROM is tested numerically in the simulation of the 1D Burgers equation with a small diffusion parameter. The numerical results show that the CF-ROM is more efficient than and as accurate as state-of-the-art ROM closure models. / Ph. D.