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Cheung, Yat Ming. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 75-78). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Symplectic Geometry and Special Lagrangian Graphs in Cn --- p.10 / Chapter 2.1 --- Symplectic and Lagrangian Geometry of Cn --- p.10 / Chapter 2.2 --- Calibrated and Spccial Lagrangian Geometries in Cn --- p.13 / Chapter 2.3 --- Special Lagrangian Differential Equation --- p.16 / Chapter 3 --- Contact Geometry in S2n-1 --- p.20 / Chapter 3.1 --- Contact and Legendrian Geometries in S2n-1 --- p.20 / Chapter 3.2 --- Special Lagrangian Cone in R2n --- p.24 / Chapter 3.3 --- The Second Fundamental Form of Lagrangian Cone in E2n --- p.26 / Chapter 4 --- Geometry of Grassmannians --- p.29 / Chapter 4.1 --- Locally Symmetric Space --- p.29 / Chapter 4.2 --- "The Grassmann manifold G(n, m)" --- p.33 / Chapter 4.3 --- "Leichtweiss' Formula for Curvature Tensor in G(n, m)" --- p.36 / Chapter 4.4 --- "Normal Neighbourhoods of a Point in G(n, m)" --- p.39 / Chapter 4.5 --- Some Remarks on Lagrangian Grassmannians --- p.49 / Chapter 5 --- Harmonic Maps between Riemannian Manifolds --- p.51 / Chapter 5.1 --- Energy Functional and Tension Field --- p.52 / Chapter 5.2 --- Harmonic Map and Euler-Lagrange Equation --- p.56 / Chapter 5.3 --- The Gauss Map and its Tension Field --- p.59 / Chapter 5.4 --- Simple Riemannian Manifolds and A Liouville-Type Result of Har- monic Maps --- p.63 / Chapter 6 --- Bernstein-Type Results for Special Lagrangian Graphs --- p.65 / Chapter 6.1 --- Convexity and Bounded Slope Assumption --- p.65 / Chapter 6.2 --- Spherical Bernstein-Type Result --- p.68 / Chapter 6.3 --- Bernstein-Type Result with only Bounded Slope --- p.72 / Bibliography --- p.75

In this work, I seek to model swarms of fruit flies, drosophila melanogaster, whose flights are characterized by straight flight segments interrupted by rapid turns called saccades. These flights are reminiscent of Levy-distributed random walks which are known to lead to efficient search behavior. I build two types of model for swarms of foraging fruit flies, whose behavior depends on swarm density and chemoattractant concentration, using rules inspired by experimentally observed flight patterns. First I will present a Lagrangian model where the path of each individual fly is tracked. I will also consider an Eulerian model where the fruit fly density evolves as a function of time and position in space. I will discuss the advantages and disadvantages of the two models and the relationship between them.

Bohm's model for quantum mechanics is examined and a well-known drawback of the model is considered, namely the fact that the model does not conserve energy and momentum.It is shown that the Lagrangian formalism and the use of energy-momentum tensors provide a way of addressing this non-conservation aspect once the model is considered from the point of view of an interacting particle-field system. The full mathematical formulation that is then presented demonstrates that conservation can be reintroduced without disrupting the present agreement of Bohm's model with experiment. / Doctor of Philosphy (PhD)

Thesis (Ph. D.)--West Virginia University, 2006. / Title from document title page. Document formatted into pages; contains xxiv, 163 p. : ill. (some col.). + QuickTime and media files. Includes video files in the mpeg and wmv formats. Vita. Includes abstract. Includes bibliographical references (p. 126-136).

To differ from the usually applied Eulerian method for describing the motion of fluid, the governing equations complete in the Lagrangian form for describing three-dimensional progressive and short-crested waves system are derived in this paper. A systematical ordering expansion by an appropriate perturbation approximation is developed, and the exactly satisfactory solutions in a form of functional, up to third-order progressive waves and up to second-order short-crested waves, are obtained. The kinematic properties of the waves, including the surface profile, pressure, the paths of fluid particles, and the mass transport velocity, are then described directly. The obtained solution for the short-crested waves system is successfully verified by reducing to two special cases, one is the two-dimensional simple progressive waves, and the other is the two-dimensional standing waves. Also, the analytical results are compared with experimental data including the surface profiles, the pressures and the paths of fluid particles for validation.

Controlled cooling is an important part of steel production industry that affects the properties of the outcome steel. Many of the researches done in controlled cooling are experimental. Due to progress in the numerical techniques and high cost of experimental works in this field the numerical work seems more feasible. Heat transfer analysis is the necessary element of successful controlled cooling and ultimately achievement of novel properties in steel. Heat transfer on the surface of the plate normally contains different regimes such as film boiling, nucleate boiling, transition boiling and radiation heat transfer. This makes the analysis more complicated. In order to perform the heat transfer analysis often empirical correlations are being used. In these correlations the velocity and pressure within the fluid domain is involved. Therefore in order to obtain a better understanding of heat transfer process, study of hydrodynamics of the fluid becomes necessary. Circular jet due to its high efficiency has been used vastly in the industry. Although some experimental studies of round jet arrays have been done, yet the characteristics of a single jet with industrial geometric and flow parameters on the surface of a flat plate is not fully understood. Study of hydrodynamics of the jet impingement is the first step to achieve better understanding of heat transfer process. Finite element method as a popular numerical method has been used vastly to simulate different domains. Traditional approaches of finite element method, Lagrangian and Eulerian, each has its own benefits and drawbacks. Lagrangian approach has been used widely in solid domains and Eulerian approach has been widely used in fluid fields. Jet impingement problem, due to its unknown free surface and the change in the boundary, falls in the category of special problems and none of the traditional approaches is suitable for this application. The Arbitrary Lagrangian Eulerian (ALE) formulation has emerged as a technique that can alleviate many of the shortcomings of the traditional Lagrangian and Eulerian formulations in handling these types of problems. Using the ALE formulation the computational grid need not adhere to the material (Lagrangian) nor be fixed in space (Eulerian) but can be moved arbitrarily. Two distinct techniques are being used to implement the ALE formulation, namely the operator split approach and the fully coupled approach. This thesis presents a fully coupled ALE formulation for the simulation of flow field. ALE form of Navier-Stokes equations are derived from the basic principles of continuum mechanics and conservation laws in the fluid. These formulations are then converted in to ALE finite element equations for the fluid flow. The axi-symmetric form of these equations are then derived in order to be used for jet impingement application. In the ALE Formulation as the mesh or the computational grid can move independent of the material and space, an additional set of unknowns representing mesh movement appears in the equations. Prescribing a mesh motion scheme in order to define these unknowns is problem-dependent and has not been yet generalized for all applications. After investigating different methods, the Winslow method is chosen for jet impingement application. This method is based on adding a specific set of partial differential Equations(Laplace equations) to the existing equations in order to obtain enough equations for the unknowns. Then these set of PDEs are converted to finite element equations and derived in axi-symmetric form to be used in jet impingement application. These equations together with the field equations are then applied to jet impingement problem. Due to the number of equations and nonlinearity of the field equations the solution of the problem faces some challenges in terms of convergence characteristics and modeling strategies. Some suggestions are made to deal with these challenges and convergence problems. Finally the numerical treatment and results of analyzing hydrodynamics of the Jet Impingement is presented. The work in this thesis is confined to the numerical simulation of the jet impingement and the specifications of an industrial test setup only have been used in order to obtain the parameters of the numerical model.

This thesis explores three consolidation problems derived from the daily operations of major international airfreight forwarders. First, we study the freight forwarder's unsplittable shipment planning problem in an airfreight forwarding network where a set of cargo shipments have to be transported to given destinations. We provide mixed integer programming formulations that use piecewise-linear cargo rates and account for volume and weight constraints, flight departure/arrival times, as well as shipment-ready times. After exploring the solution of such models using CPLEX, we devise two solution methodologies to handle large problem sizes. The first is based on Lagrangian relaxation, where the problems decompose into a set of knapsack problems and a set of network flow problems. The second is a local branching heuristic that combines branching ideas and local search. The two approaches show promising results in providing good quality heuristic solutions within reasonable computational times, for difficult and large shipment consolidation problems. Second, we further explore the freight forwarder's shipment planning problem with a different type of discount structure - the system-wide discount. The forwarder's cost associated with one flight depends not only on the quantity of freight assigned to that flight, but also on the total freight assigned to other flights operated by the same carrier. We propose a multi-commodity flow formulation that takes shipment volume and over-declaration into account, and solve it through a Lagrangian relaxation approach. We also model the "double-discount" scheme that incorporates both the common flight-leg discount (the one used in the unsplittable shipment problem) and the system-wide discount offered by cargo airlines. Finally, we focus on palletized loading using unit loading devices (ULDs) with pivots, which is different from what we assumed in the previous two research problems. In the international air cargo business, shipments are usually consolidated into containers; those are the ULDs. A ULD is charged depending on whether the total weight exceeds a certain threshold, called the pivot weight. Shipments are charged the under-pivot rate up to the pivot weight. Additional weight is charged at the over-pivot rate. This scheme is adopted for safety reasons to avoid the ULD overloading. We propose three solution methodologies for the air-cargo consolidation problem under the pivot-weight (ACPW), namely: an exact solution approach based on branch-and-price, a best fit decreasing loading heuristic, and an extended local branching. We found superior computational performance with a combination of the multi-level variables and a relaxation-induced neighborhood search for local branching.

In many engineering applications involving mixing of highly viscous fluids or mixing at micro-scales, efficient mixing must be accomplished in the absence of turbulence. Similarly in geophysical flows large-scale, deterministic flow structures can account for a considerable portion of global transport and mixing. For these types of problems, concepts from non-linear dynamical systems and the theory of chaotic advection provide the tools for understanding, quantifying, and optimizing transport and mixing processes. In this thesis chaotic advection is studied numerically in three, steady, experimentally realizable, three-dimensional flows: 1) steady vortex breakdown flow in a cylindrical container with bottom rotating lid, 2) flow in a cylindrical container with exactly counter rotating lids, and 3) flow in a new model stirred-tank with counter-rotating disks. For all cases the three-dimensional Navier-Stokes equations are solved numerically and the Lagrangian properties of the computed velocity fields are analyzed using a variety of computational and theoretical tools. For the flow in the interior of vortex breakdown bubbles it is shown that even though from the Eulerian viewpoint the simulated flow fields are steady and nearly axisymmetric the Lagrangian dynamics could be chaotic. Silnikovs mechanism is shown to play a critical role in breaking up the invariance of the bubble and giving rise to chaotic dynamics. The computations for the steady flow in a cylindrical container with two exactly counter-rotating lids confirm for the first time the findings of recent linear stability studies. Above a threshold Reynolds number the equatorial shear layer becomes unstable to azimuthal modes and an intricate web of radial (cats eyes) and axial, azimuthally-inclined vortices emerge in the flow paving the way for extremely complex chaotic dynamics. Using these fundamental insights, a new stirring tank device with exactly counter-rotating disks is proposed. Results show for the first time that counter rotation of the middle disk in a three-disk stirred tank can create a flow with large chaotic regions. The results of this thesis serve to demonstrate that fundamental studies of chaotic mixing are both important from a theoretical standpoint and can potentially lead to valuable technological breakthroughs.

Three-dimensional short-crested waves in Lagrangian form was already solved by Wang(2007). By employing the technique of perturbation analysis, the solution for the entire wave filed was obtained and the results are verified to be correct to second-order. The period of the trajectory of fluid particle in short-crested wave field was manifested in Lagrangian form. Consequently, all the characteristics of the flow field can be vividly described including the moving trajectory of fluid particle. To distinguish two different ways that short-crested waves might take place, Wang(2007)¡¦s results were extended to perturbation¡¦s third-order. The mechanism of resonance phenomenon is then clearly explained. In this study, the analytical results for the three-dimensional short-crested wave field correct to third-order were explicitly derived. The fluid particle with different initial positions or different phases has different moving trajectories. Besides, the period of the trajectory of fluid particle varies with different water depths. These are obviously revealed in our perturbation solutions. The three-dimensional short-crested wave system is successfully verified by reducing to two special cases, two-dimensional progressive waves and standing waves. Also, the analytical results were compared with experimental data including the surface profiles, the pressures, and the paths of fluid particles for validation. Furthermore, the mechanism of resonance phenomenon and the property of angular frequency were explained. Thus, the exactness and generality of the results are firm certified.