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A two-dimensional, nonlinear dynamic analysis is conducted on a moored breakwater configuration to investigate snap loads in mooring lines. Breakwaters are structures used to attenuate or eliminate waves and protect shorelines, harbors, and other natural and man-made marine structures from wave damage. The breakwater in this investigation is modeled both as a point mass and as a rigid body. Both models are subjected to free undamped motions and forced undamped wave motion. Energy is dissipated through the use of a coefficient of restitution applied when a mooring line becomes taut (i.e., reaches its natural length). The mooring line is modeled as an inextensible cable with no axial or bending resistance when slack. Snap loading arises when a mooring line transitions suddenly from a slack condition to a taut condition. The analysis was conducted on a breakwater configured upside down and hanging by two mooring lines. The length of the mooring lines, coefficient of restitution, size and shape of the breakwater, initial position of the breakwater, amplitude of wave forcing, ratio of vertical to horizontal forcing, and frequency of forcing were all varied in the analysis. The results show that the rotations of the rigid body and the wave forcing have a significant role in the analysis, indicating that a rigid-body model for a moored breakwater under wave forcing is the more accurate model. / Master of Science

In recent decades, computational models have become critical to how engineers and mathematicians understand nature; as a result they have become an integral part of the design process in most engineering disciplines. Moore's law anticipates computing power doubling every two years; a prediction that has historically been realized. As modern computing power increases, problems that were previously too complex to solve by hand or by previous computing abilities become tractable. This has resulted in the development of increasingly complex computational models simulating increasingly complex dynamics. Unfortunately, this has also resulted in increased challenges in fields related to model development, such as model validation and model based control, which are needed to make models useful in the real world. Much of the validation literature to date has focused on spatial and spatiotemporal simulations; validation approaches are well defined for such models. For most time series simulations, simulated and experimental trajectories can be directly compared negating the need for specialized validation tools. In the study of some ship motion behavior, chaos exists, which results in chaotic time series simulations. This presents novel challenges for validation; direct comparison may not be the most apt approach. For these applications, there is a need to develop appropriate metrics for model validation. A major thrust of the current work seeks to develop a set of validation metrics for such chaotic time series data. A complementary but separate portion of work investigates Non-Intrusive Polynomial Chaos as an approach to reduce the computational costs associated with uncertainty analysis and other stochastic investigations into the behavior of nonlinear, chaotic models. A final major thrust of this work focuses on contributing to the control of nonlinear marine systems, specifically the autonomous recovery of an unmanned surface vehicle utilizing motion prediction information. The same complexity and chaotic nature that makes the validation of ship motion models difficult can also make the development of reliable, robust controllers difficult as well. This body of work seeks to address several facets of this broad need that has developed due to our increased computational abilities by providing validation metrics and robust control laws. / Ph. D.

This dissertation presents useful nonlinear models for fluid forcing on a moving body in two distinct contexts, and methods for analyzing the geometric structure within those and other mathematical models. This manuscript style dissertation presents three works within the theme of understanding fluid forcing and geometric structure. When a bluff body is free to move in the presence of an incoming bluff body wake, the average forcing on the body is dependent on its position relative to the upstream bluff body. This position-dependent forcing can be conceptualized as a stiffness, much like a spring. This work presents an updated model for the quasi-steady fluid forcing of a wake and extends the notion of wake stiffness to consider a nonlinear spring. These results are compared with kinematic experimental results to provide an example of the application of this framework. Fluid force models also play a role in understanding the behavior of passive aerodynamic gliders, such as gliding animals or plant material. The forces a glider experiences depend on the angle that its body makes with respect to its direction of motion. Modeling the glider as capable of pitch control, this work considers a glider with a fixed angle with respect to the ground. Within this model, all trajectories in velocity space collapse to a 1-dimensional invariant manifold known as the terminal velocity manifold. This work presents methods to identify the terminal velocity manifold, investigates its properties, and extends it to a 2-dimensional invariant manifold in a 3-dimensional space. Finally, in the search for manifolds such as the terminal velocity manifold, this dissertation introduces a new diagnostic for identifying the low dimensional geometric structure of models. The trajectory divergence rate uses instantaneous vector field information to identify regions of large normal stretching and strong normal convergence between nearby invariant manifolds. This work lays out the mathematical basis of the trajectory divergence rate and shows its application to approximate a variety of structures including slow manifolds and Lagrangian coherent structures. This dissertation applies nonlinear theoretical and numerical techniques to analyze models of fluid forcing and their geometric structure. The tools developed in this dissertation lay the groundwork for future research in the fields of flow-induced vibration, plant and animal biomechanics, and dynamical systems. / Ph. D. / When an object moves through a fluid such as air or water, the motion of the surrounding fluid generates forces on the moving object, affecting its motion. The moving object, in turn, affects the motion of the surrounding fluid. This interaction is complicated, nonlinear, and hard to even simulate numerically. This dissertation aims to analyze simplified models for these interactions in a way that gives a deeper understanding of the physics of the interaction between an object and a surrounding fluid. In order to understand these interactions, this dissertation looks at the geometric structure of the models. Very often, there are low-dimensional points, curves, or surfaces which have a very strong effect on the behavior of the system. The search for these geometric structures is another key theme of this dissertation. This dissertation presents three independent studies, with an introduction and conclusion to discuss the overall themes. The first work focuses on the forces acting on a cylinder in the wake of another cylinder. These forces are important to understand, because the vibrations that arise from wake forcing are important to consider when designing bridges, power cables, or pipes to carry oil from the ocean floor to offshore oil platforms. Previous studies have shown that the wake of a circular cylinder acts like a spring, pulling harder on the downstream cylinder the more it is moved from the center of the wake. In this work, I extend this idea of the wake as a spring to consider a nonlinear spring, which keeps the same idea, but provides a more accurate representation of the forces involved. The second work considers a simple model of gliding flight, relevant to understanding the behavior of gliding animals, falling leaves, or passive engineered gliders. Within this model, a key geometric feature exists on which the majority of the motion of the glider occurs, representing a 2-dimensional analogy to terminal velocity. In this work, I study the properties of this influential curve, show several ways to identify it, and extend the idea to a surface in a 3-dimensional model. The third study of this dissertation introduces a new mathematical quantity for studying models of systems, for fluid-body interaction problems, ocean flows, chemical reactions, or any other system that can be modeled as a vector field. This quantity, the trajectory divergence rate, provides an easily computed measurement of highly attracting or repelling regions of the states of a model, which can be used to identify influential geometric structures. This work introduces the quantity, discusses its properties, and shows its application to a variety of systems.

Thesis advisor: Pablo Guerron-Quintana / The dissertation consists of three independent chapters that study nonlinear effects in international finance and macroeconomics. The implications of presence of nonlinear effects are examined both in the context of a puzzle in international financial markets, a constrained policy within a closed economy, and are also ap- proached as a general problem in macro and macroeconometric modeling. I quantify the role of nonlinear effects in these contexts, and make a case for the application of nonlinear modeling techniques.The first chapter of the dissertation titled “Sovereign Default Risk and Currency Returns” is solo-authored. Many currencies exhibit non-zero average returns with respect to US dollar, in an apparent violation of textbook uncovered and covered interest parities. I first show that in the cross-section of countries foreign currency returns are positively related to the sovereign default risk, and then reconcile this finding with the standard theory via the “peso problem”. Market players collect premium for bearing the risk of sharp devaluation in case of default. Since defaults are rare in the data, default premium manifests itself in higher currency returns. To formalize the link between default risk and currency returns, I discipline quantitatively a model “with default” based on Arellano (2008) for a set of developing countries. I then use the implications of this model to construct an econometric model for cross-section of currency returns that I estimate using extended Fama and MacBeth (1973) method. I find strong evidence supporting the “peso problem” explanation: credit default swaps’ spreads serving as proxy for the risk of default explain around 25% of the cross-country variation of average currency returns. I also estimate that the market participants expect a 50% depreciation of national currency upon default. The second chapter is titled “Nonlinear Dynamic Factor Model in Application to Financial and Macroeconomic data”, and is joint work with Pablo Guerron- Quintana and Molin Zhong. Through the lens of a nonlinear dynamic factor model, we study the role of exogenous shocks and internal propagation forces in driving the fluctuations of macroeconomic and financial data. The proposed model 1) allows for nonlinear dynamics in the state and measurement equations; 2) can generate asymmetric, state-dependent, and size-dependent responses of observables to shocks; 3) and can produce time-varying volatility and asymmetric tail risks in predictive distributions. We find evidence in favor of the nonlinear factor model over its linear counterpart in applications that include interest rates with zero lower bounds, credit default swap spreads for European countries, and nonfinancial cor- porate credit default swap spreads in the U.S. We extract a shadow interest rate comparable to those in the literature. The results hint to an important role for a nonlinear internal propagation element to exogenous shocks during periods of tur- bulence such as the European debt crisis and the Great Recession. This nonlinear term allows the model to forecast better during the early stages of the Covid-19 crisis. The third chapter is titled “Local Government spending and business cycle” and is based on a solo-authored paper. Local government revenues and spending in the United States are procyclical due to constitutional constraints of states and municipalities. As a result, the local government policies can act as amplifiers of the business cycle. This paper introduces fiscal policy conducted by local governments to an otherwise standard New Keynesian closed economy model to assess quantitatively the contribution of spending policies into the business cycle. The procyclical nature of local government spending generates an amplification mechanism that accounts for around 15% of fluctuations in output and hours worked. / Thesis (PhD) — Boston College, 2022. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Economics.

A three-dimensional, nonlinear dynamic analysis is conducted on a fully submerged, rigid, solid cylinder to be used as a breakwater. The breakwater could potentially be used as a single cylinder to protect small structures. Alternatively, multiple cylinders could be positioned in series to protect shorelines, harbors, or moored vessels from destructive incident water waves. The cylinder is positioned with its axis horizontal and is moored to the seafloor with four symmetrically placed massless mooring lines connected at the ends of the cylinder. The mooring lines are modeled as both linearly elastic ("regular") springs and compressionless springs. All six degrees of freedom of the structure are considered. The breakwater is modeled in air with a net buoyant force acting through the cylinder's center of gravity. The six "dry" natural frequencies of the structure are computed. Both linear and nonlinear free vibrations of the structure are considered. Linear damping is used to model the fluid and mooring damping effects. Normal and oblique harmonic wave forces at various frequencies and amplitudes are applied to the cylinder. The effects of the forcing amplitude and frequency, and the coefficient of damping, on the motion of the breakwater are studied. The results show that more erratic behavior occurs for the breakwater with compressionless springs, mainly due to the development of snap loads in the mooring lines. / Master of Science

Motion capture (mocap) systems integrated with force plates and electromyography (EMG) collect detailed kinematic and kinetic data on subjects, including stride length, width, cadence, speed, and other spatiotemporal parameters. These systems allow clinicians and researchers to analyze movements, both cyclic (e.g., walking, running) and non-cyclic (e.g., jumping, falling), which is crucial for understanding movement patterns and identifying abnormalities. Clinical gait analysis, a key application, focuses on detecting musculoskeletal issues and walking impairments. While essential for diagnosing gait disorders and planning interventions, clinical gait analysis faces challenges such as noise, outliers, and marker occlusion in optical motion tracking data, requiring complex post-processing. Additionally, the measurement of ground reaction forces (GRFs) and moments (GRMs) is limited due to the restricted number of force plates. There are also challenges in EMG data collection, such as finding optimal MVC positions and developing nonlinear normalization techniques to replace traditional methods.To address these challenges, this research aims to develop an AI-driven gait analysis system that is cost-effective, user-independent, and relies solely on kinematic and EMG data for real-time analysis. The system is specifically designed to assess musculoskeletal characteristics in individuals with special needs, walking disabilities, or injuries, where measuring MVC levels is impractical or unsafe. The research has four main objectives: (1) standardize MVC positions for four lower limb muscles, (2) develop an alternative EMG normalization technique using nonlinear data analysis, (3) create an unsupervised framework using transformers for missing marker recovery without perfect ground-truth data, and (4) generate GRFs, GRMs, and JMs from lower limb kinematics using a 1D-CNN, improving accuracy and efficiency with transfer learning, without requiring force plates. While addressing these challenges, the proposed system aims to minimize user interaction, reduce pre- and post-processing, and lower costs for researchers and clinicians. The designed tool will integrate with existing optical marker-based mocap systems, providing greater flexibility and usability. In educational settings, it will offer students hands-on experience in advanced gait analysis techniques. Economically, widespread adoption of the tool in research and clinical settings will reduce data collection and analysis costs, making advanced gait analysis more accessible. Additionally, this tool can be applied to other fields, such as precision manufacturing, security, and predictive maintenance, where analyzing data can predict failures. Consequently, this research will significantly advance the field of human movement, increasing the volume and quality of research using optical marker-based mocap systems.

The term spatiotemporal chaos refers to physical phenomena that exhibit irregular oscillations in both space and time. Examples of such phenomena range from cardiac dynamics to fluid turbulence, where the motion is described by nonlinear partial differential equations. It is well known from the studies of low dimensional chaotic systems that the state space, the space of solutions to the governing dynamical equations, is shaped by the invariant sets such as equilibria, periodic orbits, and invariant tori. State space of partial differential equations is infinite dimensional, nevertheless, recent computational advancements allow us to find their invariant solutions (exact coherent structures) numerically. In this thesis, we try to elucidate the chaotic dynamics of nonlinear partial differential equations by studying their exact coherent structures and invariant manifolds. Specifically, we investigate the Kuramoto-Sivashinsky equation, which describes the velocity of a flame front, and the Navier-Stokes equation for an incompressible fluid in a circular pipe. We argue with examples that this approach can lead to a theory of turbulence with predictive power.