<p dir="ltr">This thesis presents results in Optimal Transport theory and applications to unsuper- vised statistical learning and robust asset pricing. In unsupervised learning applications, we assume that we observe the distribution of some data of interest which might be too big in size, have a high-dimensional structure or be polluted with noise. We investigate the construction of an optimal distribution that precedes the given data distribution in convex order, which means that the given distribution is a dispersion of it. The intention is to use this construction to estimate a concise, lower-dimensional or unpolluted version of the given data. We provide existence and convergence results and show that popular methods including k-means and principal curves can be unified under this model. We further investi- gate a relaxation of the order relation that leads to similar results in terms of existence and convergence and broadens the range of applications to include e.g. the Principal Compo- nent Analysis and the Factor model. We relate the two versions and show that the relaxed problem can be described as a bilinear optimization with a tractable computational method. As examples, we apply our method to generate fixed-weight k-means, principal curves with bounded curvature that are actual generalizations of PCA, and a latent factor structure in a classical Gaussian setting. In robust finance applications, we investigate the Vectorial Martingale Optimal Transport problem, the geometry of its solutions, and its application to model-free asset pricing. We consider a multi-asset, two-period contract pricing model and show that the solution to this problem with a sub or supermodular payoff function reduces to a single factor in the first period in the case of two underlying assets (d = 2), but not in general for a greater number of assets. This result for d = 2 enables the construction of a joint distribution of prices at the first period from market data, which adds information to the model-free pricing method and reduces the computational dimensionality. We provide an improved version of an existing pricing method and show numerical evidence of increased accuracy.</p>

<p> The question of how the plastic strength of alloys depends on composition is critical to alloy design. Numerous classical works have tackled this question in the past. Yet, the models available to date primarily focus on the strength of alloys at the onset of yielding and seldom address the role of alloy composition in the hardening and dislocation microstructure evolution regime. The above question becomes even more important in situations in which the alloys are compositionally nonuniform at the mesoscale, as in spinodally decomposed alloys, irradiated alloys, high entropy alloys, and additive-manufactured alloys. In this work, the interaction between alloy plasticity and compositional inhomogeneity is addressed from a discrete dislocation dynamics (DDD) perspective. A framework comprising of three components: (1) analysis of the 3D composition morphology in inhomogeneous alloys with tendency to undergo spinodal/spinodal-like instability, (2) atomistic simulations of the dislocation mobility as a function of the local composition, and (3) dislocation dynamics simulations, has been utilized to understand the collective dynamics of dislocations and mesoscale plasticity in inhomogeneous alloys. Irradiated FeCrAl has been used as a model alloy for the implementation of the current framework and subsequent investigations. The investigation reveals that the composition inhomogeneity plays a crucial role in influencing microplasticity and macroscopic plasticity in inhomogeneous alloys. This happens due to the motion of dislocations taking place in a wavy fashion due to coherency stresses and locally varying dislocation velocities. </p> <p>To further understand alloy microplasticity from a single dislocation perspective, Cahn’s theory of hardening in compositionally modulated alloys based on coherency stresses has been modified to account for superposition of solid solution strengthening on spinodal strengthening due to the composition modulation. A new definition for the CRSS in compositionally modulated alloys is provided. Subsequently, CRSS is determined as a function of dislocation line direction, amplitude, and wavelength of the composition fluctuations.</p> <p>Lastly, an application of the developed framework is demonstrated where plasticity in irradiated FeCrAl nanopillars is investigated using DDD, with a comparison to transmission electron microscopic in situ tensile tests of ion- and neutron-irradiated commercial FeCrAl C35M alloy.</p>

<div>A leading cause of premature failure and excessive wear and tear in mechanical components that rely on compression springs for their operation is the development of unwanted side forces when the spring is compressed.</div><div>These side forces are usually around 10% - 20% of the magnitude of the axial load and point in different directions in the plane perpendicular to the axis of the spring.</div><div>The magnitude and direction of the resultant of side forces varies very non-linearly and unpredictably even though the axial force behavior of the spring is very consistent and predictable.</div><div>Since these side forces have to be resisted by the housing components that hold the spring in place, it is difficult to design these components for optimal operation.</div><div><br></div><div>The hypothesis of this study is that side forces are highly sensitive to small changes in spring geometry and its placement configuration in the housing. <br></div><div><div>Several experiments are conducted to measure the axial and side forces in barrel springs and two different types of finite element models are developed and calibrated to model the spring behavior. </div><div>Spring geometry and placement are parameterized using several control variables and an approach based on design of experiments is used to identify the critical parameters that control the behavior of side-forces. </div><div>The models resulted in deeper insight into the development of side forces as the spring is progressively loaded and how its contact interactions with the housing lead to changes in the side force.</div><div>It was found that side-forces are indeed sensitive to variations in spring geometry and placement.</div><div>These sensitivities are quantified to enable designers to and manufacturers of such springs to gain more control of side force variations between different spring specimens.</div></div>

<p>Fractional calculus is the integration and differentiation to an arbitrary or fractional order. The techniques of fractional calculus are not commonly taught in engineering curricula since physical laws are expressed in integer order notation. Dr. Richard Magin (2006) notes how engineers occasionally encounter dynamic systems in which the integer order methods do not properly model the physical characteristics and lead to numerous mathematical operations. In the following study, the application of fractional order calculus to approximate the angular position of the disk oscillating in a Newtonian fluid was experimentally validated. The proposed experimental study was conducted to model the nonlinear response of an oscillating system using fractional order calculus. The integer and fractional order mathematical models solved the differential equation of motion specific to the experiment. The experimental results were compared to the integer order and the fractional order analytical solutions. The fractional order mathematical model in this study approximated the nonlinear response of the designed system by using the Bagley and Torvik fractional derivative. The analytical results of the experiment indicate that either the integer or fractional order methods can be used to approximate the angular position of the disk oscillating in the homogeneous solution. The following research was in collaboration with Dr. Richard Mark French, Dr. Garcia Bravo, and Rajarshi Choudhuri, and the experimental design was derived from the previous experiments conducted in 2018.</p>

Microelectromechanical systems (MEMS) depend on mechanical deformation to sense their environment, enhance electrical circuitry, or store data. Nonlinear forces arising from multiphysics phenomena at the micro- and nanoscale -- van der Waals forces, electrostatic fields, dielectric charging, capillary forces, surface roughness, asperity interactions -- lead to challenging problems for analysis, simulation, and measurement of the deforming device elements. Herein, a foundation for the study of mechanical deformation is provided through computational and experimental studies of MEMS microcantilever capacitive switches. Numerical techniques are built to capture deformation equilibria expediently. A compact analytical model is developed from principle multiphysics governing operation. Experimental measurements support the phenomena predicted by the analytical model, and finite element method (FEM) simulations confirm device-specific performance. Altogether, the static multistability and quasistatic performance of the electrostatically-actuated switches are confirmed across analysis, simulation, and experimentation. <p><br></p> <p>The nonlinear multiphysics forces present in the devices are critical to the switching behavior exploited for novel applications, but are also a culprit in a common failure mode when the attractive forces overcome the restorative and repulsive forces to result in two elements sticking together. Quasistatic operation is functional for switching between multistable states during normal conditions, but is insufficient under such stiction-failure. Exploration of dynamic methods for stiction release is often the only option for many system configurations. But how and when is release achieved? To investigate the fundamental mechanism of dynamic release, an atomic force microscopy (AFM) system -- a microcantilever with a motion-controlled base and a single-asperity probe tip, measured and actuated via lasers -- is configured to replicate elements of a stiction-failed MEMS device. Through this surrogate, observable dynamic signatures of microcantilever deflection indicate the onset of detachment between the probe and a sample.</p>

<p dir="ltr">Autonomous robots can perform dangerous and tedious tasks, eliminating the need for human involvement. To deploy an autonomous robot in the field, a typical planning and control hierarchy is used, consisting of a high-level planner, a mid-level motion planner, and a low-level tracking controller. In applications such as simultaneous localization and mapping, package delivery, logistics, and surveillance, a group of autonomous robots can be more efficient and resilient than a single robot. However, deploying a multi-robot team by directly aggregating each robot's planning hierarchy into a larger, centralized hierarchy faces challenges related to scalability, resilience, and real-time computation. Distributed algorithms offer a promising solution for introducing effective coordination within a network of robots, addressing these issues. This thesis explores the application of distributed algorithms in multi-robot systems, focusing on several essential components required to enable distributed multi-robot coordination, both in general terms and for specific applications.</p>