Thesis (Ph. D.)--University of Missouri-Columbia, 1997. / Typescript. Vita. Includes bibliographical references (leaves 119-128). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 1997. / Typescript. Vita. Includes bibliographical references (leaves 119-128). Also available on the Internet.

Thesis (Ph. D.)--University of Chicago, Dept. of Geophysical Sciences, August 1999. / Includes bibliographical references. Also available on the Internet.

Thesis (Ph. D.)--University of Hong Kong, 2006. / Title proper from title frame. Also available in printed format.

Thesis: Ph. D., Massachusetts Institute of Technology, Sloan School of Management, 1981 / Vita. / Bibliography: leaves 97-98. / by Dorothy Elliott Sempolinski. / Ph. D. / Ph. D. Massachusetts Institute of Technology, Sloan School of Management

published_or_final_version / Civil Engineering / Doctoral / Doctor of Philosophy

Classically, fluid dynamics have been dealt with analytically because of the lack of numerical resources (Yeung, 1982). With the development of computational ability, many formulations have been developed which typically use the traditional Navier-Stokes equations along with an Eulerian grid. Today, there exists the possibility of using a moving grid (Lagrangian) along with a meshless discretization. The first issue in meshless fluid dynamics is the equations of motion. There are currently two types of Lagrangian formulations. Spherical Particle Hydrodynamics (SPH) is a method which calculates all equations of motion explicitly. The Moving Particle Semi-implicit (MPS) method uses a mathematical foundation based on SPH. However, instead of calculating all laws of motion explicitly, a fractional time step is performed to calculate pressure. A proposed method, Lagrange Implicit Fraction Step (LIFS), has been created which improves the mathematical formulations on the fluid domain. The LIFS method returns to Continuum mechanics to construct the laws of motion based on decomposing all forces of a volume. It is assumed that all forces on this volume can be linearly superposed to calculate the accelerations of each mass. The LIFS method calculates pressure from a boundary value problem with the inclusion of proper flux boundary conditions. The second issue in meshless Lagrangian dynamics is the calculation of derivatives across a domain. The Monte Carlo Integration (MCI) method uses weighted averages to calculate operators. However, the MCI method can be very inaccurate, and is not suitable for sparse grids. The Radial Basis Function (RBF) method is introduced and studied as a possibility to calculate meshless operators. The RBF method involves a solution of a system of equations to calculate interpolants. Machine expenses are shown to limit the viability of the RBF method for large domains. A new method of calculation has been created called Multi-dimensional Lagrange Interpolating Polynomials (MLIP). While Lagrange Interpolating Polynomials are essentially a one-dimensional interpolation, the use of "dimensional-cuts" and Gaussian quadratures can provide multi-dimensional interpolation. This paper is divided into three sections. The first section specifies the equations of motion. The second section provides the mathematical basis for meshless calculations. The third section evaluates the effectiveness of the meshless calculations and compares two fluiddynamic codes. / Fund number: N62271-97-G-0041. / US Navy (USN) author.

A variety of techniques are available to enable the fabrication of photonic crystal structures. Multi-beam-interference lithography (MBIL) is a relatively new technique which offers many advantages over more traditional means of fabrication. Unlike the more common fabrication methods such as optical and electron-beam lithography, MBIL is a method that can produce both two- and three-dimensional large-area photonic crystal structures for use in the infrared and visible light regimes. While multi-beam-interference lithography represents a promising methodology for the fabrication of PC structures, there has been an incomplete understanding of MBIL itself. The research in this thesis focuses on providing a more complete, systematic description of MBIL in order to demonstrate its full capabilities. Analysis of both three- and four-beam interference is investigated and described in terms of contrast and crystallography. The concept of a condition for primitive-lattice-vector-direction equal contrasts} is introduced in this thesis. These conditions are developed as nonlinear constraints when optimizing absolute contrast for producing lithographically useful interference patterns (meaning high contrast and localized intensity extrema). By understanding the richness of possibilities within MBIL, a number of useful interference patterns are found that can be created in a straightforward manner. These patterns can be both lithographically useful and structurally useful (providing interference contours that can define wide-bandgap photonic crystals). Included within this investigation are theoretical calculations of band structures for photonic crystals that are fabricatable through MBIL. The resulting calculations show that not only do most MBIL-defined structures exhibit similar performance characteristics compared to conventionally designed photonic crystal structures, but in some cases MBIL-defined structures show a significant increase in bandgap size. Using the results from this analysis, a number of hexagonal photonic crystals are fabricated using a variety of process conditions. It is shown that both rod- and hole-type photonic crystal structures can be fabricated using processes based on both positive and negative photoresist. The "light-field" and "dark-field" interference patterns used to define the hexagonal photonic crystal structures are quickly interchanged by the proper adjustment of each beam's intensity and polarization. The resulting structures, including a large area (~1 cm², 1 x 10⁹ lattice points) photonic crystal are imaged using a scanning electron microscope. Multi-beam-interference lithography provides an enabling initial step for the wafer-scale, cost-effective integration of the impressive PC-based devices into manufacturable DIPCS. While multi-beam-interference lithography represents a promising methodology for the fabrication of PC structures, it lacks in the ability to produce PC-based integrated photonic circuits. Future research will target the lack of a large-scale, cost-effective fabrication methodology for photonic crystal devices. By utilizing diffractive elements, a photo-mask will be able to combine both MBIL and conventional lithography techniques into a single fabrication technology while taking advantage of the inherent positive attributes of both.

published_or_final_version / abstract / Mechanical Engineering / Doctoral / Doctor of Philosophy

In this thesis, we extend De Giorgi's interpolation method to a class of parabolic equations which are not gradient flows but possess an entropy functional and an underlying Lagrangian. The new fact in the study is that not only the Lagrangian may depend on spatial variables, but also it does not induce a metric. Assuming the initial condition is a density function, not necessarily smooth, but solely of bounded first moments and finite "entropy", we use a variational scheme to discretize the equation in time and construct approximate solutions. Moreover, De Giorgi's interpolation method is revealed to be a powerful tool for proving convergence of our algorithm. Finally, we analyze uniqueness and stability of our solution in L¹.