Magnetic Resonance Elastography (MRE) is a medical imaging technique used to generate a map of tissue elasticity. The resulting image is known as an elastogram, and gives a quantitative measure of stiffness in the examined tissue. The method is indirect; the elasticity, itself, is not measured. Instead, the physical response to a known stress is captured using magnetic resonance imaging, and is related to an elasticity parameter through a mathematical model of the tissue. In dynamic elastography, a harmonic stress is externally applied by a mechanical actuator, which is oriented to induce shear waves through the tissue. Once the system reaches a quasi-steady state, the displacement field is measured at a sequence of points in time. This data is the input to elasticity reconstruction algorithms. In this dissertation, the tissue is modelled as a linearly viscoelastic, isotropic continuum, undergoing harmonic motion with a known fundamental frequency. With this model, viscoelasticity is described by the complex versions of Lamé's first and second parameters. The second parameter, known as the complex shear modulus, is the one of interest. The term involving the first parameter is usually deemed negligible, so is ignored. The task is to invert the tissue model, a system of linear differential equations, to find the desired parameter. Direct inversion methods use the measured data directly in the model. Most current direct methods assume the shear modulus can be approximated locally by a constant, so ignore all derivative terms. This is known as the local homogeneity assumption, and allows for a simple, algebraic solution. The accuracy, however, is limited by the validity of the assumption. One of the purposes of MRE is to find pathological tissue marked by a higher than normal stiffness. At the boundaries of such diseased tissue, the stiffness is expected to change, invalidating the local homogeneity assumption, and hence, the shear modulus estimate. In order to capture the true shape of any stiff regions, a method must allow for local variations. Two new inversion methods are derived. In the first, a Green's function is introduced in an attempt to solve the differential equations. To simplify the system, the tissue is taken to be incompressible, another common assumption in direct inversion methods. Unfortunately, without designing an iterative procedure, the method still requires a homogeneity assumption, limiting potential accuracy. However, it is very fast and robust. In the second new inversion method, neither of the local homogeneity or incompressibility assumptions are made. Instead, the problem is re-posed in a quadratic optimization form. The system of linear differential equations is set as a constraint, and any free parameters are steered through quadratic programming techniques. It is found that, in most cases, there are no degrees of freedom in the optimization problem. This suggests that the system of differential equations has a fully determined solution, even without initial, boundary, or regularization conditions. The result is that estimates of the shear modulus and its derivatives can be obtained, locally, without requiring any assumptions that might invalidate the solution. The new inversion algorithms are compared to a few prominent, existing ones, testing accuracy and robustness. The Green's function method is found to have a comparable accuracy and noise performance to existing techniques. The second inversion method, employing quadratic optimization, is shown to be significantly more accurate, but not as robust. It seems the two goals of increasing accuracy and robustness are somewhat conflicting. One possible way to improve performance is to gather and use more data. If a second displacement field is generated using a different actuator location, further differential equations are obtained, resulting in a larger system. This enlarged system is better determined, and has improved signal-to-noise properties. It is shown that using data from a second field can increase accuracy for all methods.

Magnetic Resonance Elastography (MRE) is a medical imaging technique used to generate a map of tissue elasticity. The resulting image is known as an elastogram, and gives a quantitative measure of stiffness in the examined tissue. The method is indirect; the elasticity, itself, is not measured. Instead, the physical response to a known stress is captured using magnetic resonance imaging, and is related to an elasticity parameter through a mathematical model of the tissue. In dynamic elastography, a harmonic stress is externally applied by a mechanical actuator, which is oriented to induce shear waves through the tissue. Once the system reaches a quasi-steady state, the displacement field is measured at a sequence of points in time. This data is the input to elasticity reconstruction algorithms. In this dissertation, the tissue is modelled as a linearly viscoelastic, isotropic continuum, undergoing harmonic motion with a known fundamental frequency. With this model, viscoelasticity is described by the complex versions of Lamé's first and second parameters. The second parameter, known as the complex shear modulus, is the one of interest. The term involving the first parameter is usually deemed negligible, so is ignored. The task is to invert the tissue model, a system of linear differential equations, to find the desired parameter. Direct inversion methods use the measured data directly in the model. Most current direct methods assume the shear modulus can be approximated locally by a constant, so ignore all derivative terms. This is known as the local homogeneity assumption, and allows for a simple, algebraic solution. The accuracy, however, is limited by the validity of the assumption. One of the purposes of MRE is to find pathological tissue marked by a higher than normal stiffness. At the boundaries of such diseased tissue, the stiffness is expected to change, invalidating the local homogeneity assumption, and hence, the shear modulus estimate. In order to capture the true shape of any stiff regions, a method must allow for local variations. Two new inversion methods are derived. In the first, a Green's function is introduced in an attempt to solve the differential equations. To simplify the system, the tissue is taken to be incompressible, another common assumption in direct inversion methods. Unfortunately, without designing an iterative procedure, the method still requires a homogeneity assumption, limiting potential accuracy. However, it is very fast and robust. In the second new inversion method, neither of the local homogeneity or incompressibility assumptions are made. Instead, the problem is re-posed in a quadratic optimization form. The system of linear differential equations is set as a constraint, and any free parameters are steered through quadratic programming techniques. It is found that, in most cases, there are no degrees of freedom in the optimization problem. This suggests that the system of differential equations has a fully determined solution, even without initial, boundary, or regularization conditions. The result is that estimates of the shear modulus and its derivatives can be obtained, locally, without requiring any assumptions that might invalidate the solution. The new inversion algorithms are compared to a few prominent, existing ones, testing accuracy and robustness. The Green's function method is found to have a comparable accuracy and noise performance to existing techniques. The second inversion method, employing quadratic optimization, is shown to be significantly more accurate, but not as robust. It seems the two goals of increasing accuracy and robustness are somewhat conflicting. One possible way to improve performance is to gather and use more data. If a second displacement field is generated using a different actuator location, further differential equations are obtained, resulting in a larger system. This enlarged system is better determined, and has improved signal-to-noise properties. It is shown that using data from a second field can increase accuracy for all methods.

<p>Within the past decade, advances in small-scale electronics have reduced power consumption requirements such that mechanisms for harnessing ambient kinetic energy for self-sustenance are a viable technology. Such devices, known as energy harvesters, may enable self-sustaining wireless sensor networks for applications ranging from Tsunami warning detection to environmental monitoring to cost-effective structural health diagnostics in bridges and buildings. In particular, flexible electroelastic materials such as lead-zirconate-titanate (PZT) are sought after in designing such devices due to their superior efficiency in transforming mechanical energy into the electrical domain in comparison to induction methods. To date, however, material and dynamic nonlinearities within the most popular type of energy harvester, an electroelastically laminated cantilever beam, has received minimal attention in the literature despite being readily observed in laboratory experiments. </p><p>In the first part of this dissertation, an experimentally validated first-principles based modeling framework for quantitatively characterizing the intrinsic nonlinearities and moderately large amplitude response of a cantilevered electroelastic generator is developed. Nonlinear parameter identification is facilitated by an analytic solution for the generator's dynamic response alongside experimental data. The model is shown to accurately describe amplitude dependent frequency responses in both the mechanical and electrical domains and implications concerning the conventional approach to resonant generator design are discussed. Higher order elasticity and nonlinear damping are found to be critical for correctly modeling the harvester response while inclusion of a proof mass is shown to invigorate nonlinearities a much lower driving amplitudes in comparison to electroelastic harvesters without a tuning mass.</p><p>The second part of the dissertation concerns dynamical systems design to purposefully engage nonlinear phenomena in the mechanical domain. In particular, two devices, one exploiting hysteretic nonlinearities and the second featuring homoclinic bifurcation are investigated. Both devices exploit nonlinear magnet interactions with piezoelectric cantilever beams and a first principles modeling approach is applied throughout. The first device is designed such that both softening and hardening nonlinear resonance curves produces a broader response in comparison to the linear equivalent oscillator. The second device makes use of a supercritical pitchfork bifurcation wrought by nonlinear magnetic repelling forces to achieve a bistable electroelastic dynamical system. This system is also analytically modeled, numerically simulated, and experimentally realized to demonstrate enhanced capabilities and new challenges. In addition, a bifurcation parameter within the design is examined as a either a fixed or adaptable tuning mechanism for enhanced sensitivity to ambient excitation. Analytical methodologies to include the method of Harmonic Balance and Melnikov Theory are shown to provide superior insight into the complex dynamics of the bistable system in response to deterministic and stochastic excitation.</p> / Dissertation

A fascinating feature of bucket brigade assembly lines is that work load on workers is balanced spontaneously as workers follow some simple rules in the assembly process. This self-organizing property significantly reduces the management effort on an assembly line. We generalize this idea in several directions. These include an adapted bucket brigade protocol for complex assembly networks, a generalized model that permits chaotic behavior, and a more detailed model for a flow line in which jobs arrive arbitrarily in time and are introduced into the system at several points on the line.

In this study, the behavior of pressure in the Rapid Distortion Limit, along with its concomitant modeling, are addressed. In the first part of the work, the role of pressure in the initiation, propagation and suppression of flow instabilities for quadratic flows is analyzed. The paradigm of analysis considers the Reynolds stress transport equations to govern the evolution of a dynamical system, in a state space composed of the Reynolds stress tensor components. This dynamical system is scrutinized via the identification of the invariant sets and the bifurcation analysis. The changing role of pressure in quadratic flows, viz. hyperbolic, shear and elliptic, is established mathematically and the underlying physics is explained. Along the maxim of "understanding before prediction", this allows for a deeper insight into the behavior of pressure, thus aiding in its modeling. The second part of this work deals with Rapid Pressure Strain Correlation modeling in earnest. Based on the comprehension developed in the preceding section, the classical pressure strain correlation modeling approaches are revisited. Their shortcomings, along with their successes, are articulated and explained, mathematically and from the viewpoint of the governing physics. Some of the salient issues addressed include, but are not limited to, the requisite nature of the model, viz. a linear or a nonlinear structure, the success of the extant models for hyperbolic flows, their inability to capture elliptic flows and the use of RDT simulations to validate models. Through this analysis, the schism between mathematical and physical guidelines and the engineering approach, at present, is substantiated. Subsequently, a model is developed that adheres to the classical modeling framework and shows excellent agreement with the RDT simulations. The performance of this model is compared to that of other nominations prevalent in engineering simulations. The work concludes with a summary, pertinent observations and recommendations for future research in the germane field.

A stochastic differential equation with vanishing martingale term is studied. Specifically, given a domain D, the asymptotic scaling properties of both the exit time from the domain and the exit distribution are considered under the additional (non-standard) hypothesis that the initial condition also has a scaling limit. Methods from dynamical systems are applied to get more complete estimates than the ones obtained by the probabilistic large deviation theory. Two situations are completely analyzed. When there is a unique critical saddle point of the deterministic system (the system without random effects), and when the unperturbed system escapes the domain D in finite time. Applications to these results are in order. In particular, the study of 2-dimensional heteroclinic networks is closed with these results and shows the existence of possible asymmetries. Also, 1-dimensional diffusions conditioned to rare events are further studied using these results as building blocks. The approach tries to mimic the well known linear situation. The original equation is smoothly transformed into a very specific non-linear equation that is treated as a singular perturbation of the original equation. The transformation provides a classification to all 2-dimensional systems with initial conditions close to a saddle point of the flow generated by the drift vector field. The proof then proceeds by estimates that propagate the small noise nature of the system through the non-linearity. Some proofs are based on geometrical arguments and stochastic pathwise expansions in noise intensity series.

In this dissertation, we demonstrate the importance of invariant objects in many areas of applied research. The areas of application we consider are chemistry, celestial mechanics and aerospace engineering, plasma physics, and coupled map lattices. In the context of chemical reactions, stable and unstable manifolds of fixed points separate regions of phase space that lead to a certain outcome of the reaction. We study how these regions change under the influence of exposing the molecules to a laser. In celestial mechanics and aerospace engineering, we compute periodic orbits and their stable and unstable manifolds for a object of negligible mass (e.g. a satellite or spacecraft) under the presence of Jupiter and two of its moons, Europa and Ganymede. The periodic orbits serve as convenient spot to place a satellite for observation purposes, and computing their stable and unstable manifolds have been used in constructing low-energy transfers between the two moons. In plasma physics, an important and practical problem is to study barriers for heat transport in magnetically confined plasma undergoing fusion. We compute barriers for which heat cannot pass through. However, such barriers break down and lead to robust partial barriers. In this latter case, heat can flow across the barrier, but at a very slow rate. Finally, infinite dimensional coupled map lattice systems are considered in a wide variety of areas, most notably in statistical mechanics, neuroscience, and in the discretization of PDEs. We assume that the interaction amont the lattice sites decays with the distance of the sites, and assume the existence of an invariant whiskered torus that is localized near a collection of lattice sites. We prove that the torus has invariant stable and unstable manifolds that are also localized near the torus. This is an important step in understanding the global dynamics of such systems and opens the door to new possible results, most notably studying the problem of energy transfer between the sites. / text

Dynamical Systems Theory (DST) (De Bot, Lowie, & Vespoor 2007; Larsen-Freeman 1997, 2007; Larsen-Freeman & Cameron 2008; Dörnyei 2009; and van Lier 2000) represents a scientific paradigm shift derived from the fields of physics, engineering and theoretical mathematics that attempts to solve real-world scenarios that do not respond to scientific reductionism, otherwise known as ‘analysis’. The purpose of this dissertation is to (re)frame foreign language learning/use as a dynamical process that that involves interplay among what Dörnyei (2009) terms the language, the agent and the environment. More specifically, this dissertation presents a quasi-experimental, psycholinguistic study that looks at the interface between language (in this case the talk that resulted from NS-NNS interactions) and agent (as defined by a set of personal traits, or Individual Differences [IDs], including motivation, attitudes, personality and aptitude) in order to answer the research question: Do IDs vary in conjunction with language learning/use, and if so, how? Eight tutored Spanish learners were followed over the course of 16 weeks during which time they participated in 8 chat sessions with a native Spanish-speaker. Their ID profiles were measured immediately before and after each session and sessions with significant pre- to post-session ID shifts were analyzed to determine to what extent such shifts correlated with certain types of talk and/or think-aloud sequences. Results indicated that all participants’ pre- and post-interactional ID profiles fluctuated measurably and significantly, even within the span of a single interaction. Moreover, those sessions with significantly positive ID shifts were qualitatively different in terms of language-related episodes (LREs), conversation management/pragmatic markers, and metacognition from those with significantly negative ID shifts. Other unexpected findings revealed, for example, that LREs (especially NS-initiated LREs) negatively impacted motivations and attitudes and, therefore, the language-learning process itself. Taken together, the results of this study indicate that the agent’s IDs and their (inter)language co-emerge; that is to say, they evolve simultaneously and in response to one another. Moreover, this study suggests that DST can indeed be quasi-experimentally applied to the study of SLA, thus necessitating further development in DST-oriented methodologies and research questions. / text

Efficient and robust numerical simulation of multiscale problems encountered in science and engineering is a formidable challenge. Full resolution of multiscale problems using direct numerical simulations requires enormous amounts of computational time and resources. This thesis develops seamless multiscale methods for ordinary and partial differential equations under the framework of the heterogeneous multiscale method (HMM). The first part of the thesis is devoted to the development of seamless multiscale integrators for ordinary differential equations. The first method, which we call backward-forward HMM (BFHMM), uses splitting and on-the-fly filtering techniques to capture slow variables of highly oscillatory problems without any a priori information. The second method, denoted by variable step size HMM (VSHMM), as the name implies, uses variable mesoscopic step sizes for the unperturbed equation, which gives computational efficiency and higher accuracy. VSHMM can be applied to dissipative problems as well as highly oscillatory problems, while BFHMM has some difficulties when applied to the dissipative case. The effect of variable time stepping is analyzed and the two methods are tested numerically. Multi-spatial problems and numerical methods are discussed in the second part. Seamless heterogeneous multiscale methods (SHMM) for partial differential equations, especially the parabolic case without scale separation are proposed. SHMM is developed first for the multiscale heat equation with a continuum of scales in the diffusion coefficient. This seamless method uses a hierarchy of local grids to capture effects from each scale and uses filtering in Fourier space to impose an artificial scale gap. SHMM is then applied to advection enhanced diffusion problems under incompressible turbulent velocity fields. / text

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