The processes studied by nuclear engineers generally include coupled physics phenomena (Thermal-Hydraulics, Neutronics, Material Mechanics, etc.) and modeling such multiphysics processes numerically can be computationally intensive. A way to reduce the computational burden is to use spatial meshes that are optimally suited for a specific solution; such meshes are obtained through a process known as Adaptive Mesh Refinement (AMR). AMR can be especially useful for modeling multiphysics phenomena by allowing each solution component to be computed on an independent mesh (Multimesh AMR). Using AMR on time dependent problems requires the spatial mesh to change in time as the solution changes in time. Current algorithms presented in the literature address this concern by adapting the spatial mesh at every time step, which can be inefficient. This Thesis proposes an algorithm for saving computational resources by using a spatially adapted mesh for multiple time steps, and only adapting the spatial mesh when the solution has changed significantly. This Thesis explores the mechanisms used to determine when and where to spatially adapt for time dependent, coupled physics problems. The algorithm is implemented using the Deal.ii fiinite element library [1, 2], in 2D and 3D, and is tested on a coupled neutronics and heat conduction problem in 2D. The algorithm is shown to perform better than a uniformly refined static mesh and, in some cases, a mesh that is spatially adapted at every time step.

Spatiotemporal properties of classical coupled nonlinear oscillators are investigated in this thesis. Chapter 1 gives an introduction to nonlinear lattices and to the concept of breathers, that are spatially localized and temporally periodic excitation in nonlinear lattices. The concept of anti-continuous limit that provides the basic methodology in probing spatiotemporal breather properties is discussed. In Chapter 2, the general approach for finding exact breather solutions from the anti-continuous limit is examined, and the rotating wave approximation(RWA) is applied to probe the spatial structure of static breathers. Numerical evidence reveals that the RWA relates the spatial structure of stable multi-breathers to a single breather of the same frequency. Chapter 3 presents linear stability analysis of static breathers and gives a systematic way to construct mobile breathers. Formation and collision properties of this moving breathers are also studied. Chapter 4 discusses dynamics of kinks and anti-kinks in hydrogen-bonded chains in the context of two-component soliton model. From molecular dynamics simulations with finite temperature, it is observed that, in a real system (eg. ice), a pair of kink and anti-kink can evolve into a moving-breather-like excitation. Chapter 5 is devoted to the understand of the effects of disorder in the Holstein model. The summary is given in Chapter 6.

Coupled nonlinear Schrodinger equations (CNLS) govern many physical phenomena, such as nonlinear optics and Bose-Einstein condensates. For their wide applications, many studies have been carried out by physicists, mathematicians and engineers from different respects. In this dissertation, we focused on standing wave solutions, which are of particular interests for their relatively simple form and the important roles they play in studying other wave solutions. We studied the multiplicity of this type of solutions of CNLS via variational methods and bifurcation methods. Variational methods are useful tools for studying differential equations and systems of differential equations that possess the so-called variational structure. For such an equation or system, a weak solution can be found through finding the critical point of a corresponding energy functional. If this equation or system is also invariant under a certain symmetric group, multiple solutions are often expected. In this work, an integer-valued function that measures symmetries of CNLS was used to determine critical values. Besides variational methods, bifurcation methods may also be used to find solutions of a differential equation or system, if some trivial solution branch exists and the system is degenerate somewhere on this branch. If local bifurcations exist, then new solutions can be found in a neighborhood of each bifurcation point. If global bifurcation branches exist, then there is a continuous solution branch emanating from each bifurcation point. We consider two types of CNLS. First, for a fully symmetric system, we introduce a new index and use it to construct a sequence of critical energy levels. Using variational methods and the symmetric structure, we prove that there is at least one solution on each one of these critical energy levels. Second, we study the bifurcation phenomena of a two-equation asymmetric system. All these bifurcations take place with respect to a positive solution branch that is already known. The locations of the bifurcation points are determined through an equation of a coupling parameter. A few nonexistence results of positive solutions are also given

We consider a type of forced "Van Der Pol" oscillator where the forced function is periodic and oscillatory around the t-axis. This problem derived from an electrical model. The important issues here is that this circuits presents the spiking phenomena over a one time period and it has important applications in signal processing and digital communication. The three most important problems that we addressed here in this thesis are to compute the number of spikes a solution completes in one time period (it can be used to transform the analog signal into digital information), how the dynamics of the number of spikes change with respect to the parameters amplitude (k) and frequency (w), and when the coupled circuits synchronize (i.e., the driver and the respond are on synchronous). Sophisticated mathematical and numerical analysis has been developed that enable us to give a complete study of the problems above described.

The thesis is devoted to the study of spontaneous synchronization of coupled nonlinear oscillators. It consists of two major parts. The first describes synchronization of Josephson junctions embedded in a transmission line. I consider in detail a new phenomenon ??eation of inert oscillator pairs ??ich was observed in analytical studies. The second part of the thesis describes synchronization of an array of single mode fiber lasers, with special interest in the phenomenon of synchronization of subsets of fiber lasers in a two dimensional array through a specific arrangement of the under-pumped lasers.

<p>This study focuses on developing a new online recursive numerical algorithm for a coupled nonlinear inverse heat conduction-microstructure problem. This algorithm is essential in identifying, designing and controlling many industrial applications such as the quenching process for heat treating of materials, chemical vapor deposition and industrial baking. In order to develop the above algorithm, a systematic four stage research plan has been conducted. </P> <p> The first and second stages were devoted to thoroughly reviewing the existing inverse heat conduction techniques. Unlike most inverse heat conduction solution methods that are batch form techniques, the online input estimation algorithm can be used for controlling the process in real time. Therefore, in the first stage, the effect of different parameters of the online input estimation algorithm on the estimate bias has been investigated. These parameters are the stabilizing parameter, the measurement errors standard deviation, the temporal step size, the spatial step size, the location of the thermocouple as well as the initial assumption of the state error covariance and error covariance of the input estimate. Furthermore, three different discretization schemes; namely: explicit, implicit and Crank-Nicholson have been employed in the input estimation algorithm to evaluate their effect on the algorithm performance. </p> <p> The effect of changing the stabilizing parameter has been investigated using three different forms of boundary conditions covering most practical boundary heat flux conditions. These cases are: square, triangular and mixed function heat fluxes. The most important finding of this investigation is that a robust range of the stabilizing parameter has been found which achieves the desired trade-off between the filter tracking ability and its sensitivity to measurement errors. For the three considered cases, it has been found that there is a common optimal value of the stabilizing parameter at which the estimate bias is minimal. This finding is important for practical applications since this parameter is usually unknown. Therefore, this study provides a needed guidance for assuming this parameter. </p> <p> In stage three of this study, a new, more efficient direct numerical algorithm has been developed to predict the thermal and microstructure fields during quenching of steel rods. The present algorithm solves the full nonlinear heat conduction equation using a central finite-difference scheme coupled with a fourth-order Runge-Kutta nonlinear solver. Numerical results obtained using the present algorithm have been validated using experimental data and numerical results available in the literature. In addition to its accurate predictions, the present algorithm does not require iterations; hence, it is computationally more efficient than previous numerical algorithms. </p> <p> The work performed in stage four of this research focused on developing and applying an inverse algorithm to estimate the surface temperatures and surface heat flux of a steel cylinder during the quenching process. The conventional online input estimation algorithm has been modified and used for the first time to handle this coupled nonlinear problem. The nonlinearity of the problem has been treated explicitly which resulted in a non-iterative algorithm suitable for real-time control of the quenching process. The obtained results have been validated using experimental data and numerical results obtained by solving the direct problem using the direct solver developed in stage three of this work. These results showed that the algorithm is efficiently reconstructing the shape of the convective surface heat flux. </P> / Thesis / Doctor of Philosophy (PhD)

Le contexte du sujet est la gestion des systèmes aquifères, en particulier le contrôle de leur exploitation et de leur éventuelle pollution. Comme exemple d'application, nous nous focalisons sur le problème d'eau salée dans les aquifères côtiers. Plus généralement, le travail s'applique à tout écoulement miscible et stratifié dans un milieu poreux faiblement déformable. Le but est d'obtenir un modèle robuste pour modéliser le déplacement des fronts de l'eau salée et de la surface supérieure de l'aquifère. Nous avons proposé une approche mixte entre interface diffuse et interface abrupte ce qui a l'avantage de respecter la réalité physique du problème tout en conservant l'efficacité numérique. De plus, nous réussissons à modéliser ce problème 3D par un modèle dynamique 2D où la 3ème dimension est traitée via l'évolution des fronts d'eau salée et de la surface libre supérieure de l'aquifère en prenant en compte l'épaisseur des zones de transition (transition entre eau salée et eau claire, transition entre zone saturée et zone insaturée). Le modèle est basé sur les lois de conservation dans le domaine de l'eau salée et dans celui de l'eau douce, les deux domaines (à frontière libre) étant couplés par un modèle intermédiaire de changement de phase. De plus, nous avons effectué des simulations numériques pour comparer notre modèle 2D issu de l'approche mixte avec un modèle 3D d'écoulement de deux fluides miscibles en milieu compressible saturé. Puis, des simulations sont faites sur notre modèle 2D pour illustrer son efficacité (cette fois dans le cas insaturé). / The context of the subject is the management of aquifers, in especially the control of their operations and their possible pollution. A critical case is the saltwater intrusion problem in costal aquifers. The goal is to obtain efficient and accurate models to simulate the displacement of fresh and salt water fronts in coastal aquifer for the optimal exploitation of groundwater. More generally, the work applies for miscible and stratified displacements in slightly deformable porous media. In this work we propose an original model mixing abrupt interfaces/diffuse interfaces approaches. The advantage is to adopt the (numerical) simplicity of a sharp interface approach, and to take into account the existence of diffuse interfaces. The model is based on the conservation laws written in the saltwater zone and in the freshwater zone, these two free boundary problems being coupled through an intermediate phase field model. An upscaling procedure let us reduce the problem to a two-dimensional setting. The theoretical analysis of the new model is performed. We also present numerical simulations comparing our 2D model with the classical 3D model for miscible displacement in a confined aquifer. Physical predictions from our new model are also given for an unconfined setting.

We clear up two issues regarding the eigenvalue problem for the Manakov system; these problems relate directly to the existence of the soliton [sic] effect in fiber optic cables. The first issue is a bound on the eigenvalues of the Manakov system: if the parameter ξ is an eigenvalue, then it must lie in a certain region in the complex plane. The second issue has to do with a chirped Manakov system. We show that if a system is chirped too much, the soliton effect disappears. While this has been known for some time experimentally, there has not yet been a theoretical result along these lines for the Manakov system. / Ph. D.

The human sense of hearing is characterized by its exquisite sensitivity, sharp frequency selectivity, and wide dynamic range. These features depend on an active process that in the inner ear boosts vibrations evoked by auditory stimuli. Spontaneous otoacoustic emissions constitute a demonstrative manifestation of this physiologically vulnerable mechanism. In the cochlea, sensory hair bundles transduce sound-induced vibrations into neural signals. Hair bundles can power mechanical movements of their tip, oscillate spontaneously, and operate as tuned nonlinear amplifiers of weak periodic stimuli. Active hair-bundle motility constitutes a promising candidate with respect to the biophysical implementation of the active process underlying human hearing. The responsiveness of isolated hair bundles, however, is seriously hampered by intrinsic fluctuations. In this thesis, we present theoretical and experimental results concerning the noise-imposed limitations of nonlinear amplification by active sensory hair bundles. We analyze the effect of noise within the framework of a stochastic description of hair-bundle dynamics and relate our findings to generic aspects of the stochastic dynamics of oscillatory systems. Hair bundles in vivo are often elastically coupled by overlying gelatinous membranes. In addition to theoretical results concerning the dynamics of elastically coupled hair bundles, we report on an experimental study. We have interfaced dynamic force clamp performed on a hair bundle from the sacculus of the bullfrog with real-time stochastic simulations of hair-bundle dynamics. By means of this setup, we could couple a hair bundle to two virtual neighbors, called cyber clones. Our theoretical and experimental work shows that elastic coupling leads to an effective noise reduction. Coupled hair bundles exhibit an increased coherence of spontaneous oscillations and an enhanced amplification gain. We therefore argue that elastic coupling by overlying membranes constitutes a morphological specialization for reducing the detrimental effect of intrinsic fluctuations.

The human sense of hearing is characterized by its exquisite sensitivity, sharp frequency selectivity, and wide dynamic range. These features depend on an active process that in the inner ear boosts vibrations evoked by auditory stimuli. Spontaneous otoacoustic emissions constitute a demonstrative manifestation of this physiologically vulnerable mechanism. In the cochlea, sensory hair bundles transduce sound-induced vibrations into neural signals. Hair bundles can power mechanical movements of their tip, oscillate spontaneously, and operate as tuned nonlinear amplifiers of weak periodic stimuli. Active hair-bundle motility constitutes a promising candidate with respect to the biophysical implementation of the active process underlying human hearing. The responsiveness of isolated hair bundles, however, is seriously hampered by intrinsic fluctuations. In this thesis, we present theoretical and experimental results concerning the noise-imposed limitations of nonlinear amplification by active sensory hair bundles. We analyze the effect of noise within the framework of a stochastic description of hair-bundle dynamics and relate our findings to generic aspects of the stochastic dynamics of oscillatory systems. Hair bundles in vivo are often elastically coupled by overlying gelatinous membranes. In addition to theoretical results concerning the dynamics of elastically coupled hair bundles, we report on an experimental study. We have interfaced dynamic force clamp performed on a hair bundle from the sacculus of the bullfrog with real-time stochastic simulations of hair-bundle dynamics. By means of this setup, we could couple a hair bundle to two virtual neighbors, called cyber clones. Our theoretical and experimental work shows that elastic coupling leads to an effective noise reduction. Coupled hair bundles exhibit an increased coherence of spontaneous oscillations and an enhanced amplification gain. We therefore argue that elastic coupling by overlying membranes constitutes a morphological specialization for reducing the detrimental effect of intrinsic fluctuations.