In this thesis, we examine the equation describing fluid flow through saturated porous medium in order to develop a new method for approximating hydraulic head values in the subsurface. In particular, we show that under reasonable assumptions, the local explicit equation (LEE) method, an accurate, finite-difference based method that is highly sensitive to changes in the assumed location of hydraulic flow parameters, can be used to approximate hydraulic head values throughout a subsurface domain of interest. This forward solution of the fluid flow equation is solved using an altered finite difference scheme, designed to account for discontinuous jumps often encountered between subsurface material types. While the method is able to handle complicated discontinuities arising from the intermingling of various underground materials, the method determines values at nodes on an easy-to-use uniform Cartesian grid and only requires information from immediately adjacent points. The results of this research directly support the development of more accurate subsurface fluid flow models for use in a wide variety of real-world situations in areas such as water management, contaminant remediation and waste storage. Furthermore, the general development of the LEE method allows it to be used as an approximation technique for any equation where the media of interest encounters a jump.

Until now, few rigorous studies of the precursors in structures exhibiting superluminal group velocities have been performed. One dimensional photonic crystals(1DPC) and active Lorentzian media are among the ones which are able to exhibit superluminal propagation. In the first part of the thesis we have studied the evolution of the precursors in active Lorentzian media and 1DPC. The problem of the propagation of the precursors in active Lorentzian media is addressed, by employing the steepest descent method to provide a detailed description of the propagation of the pulse inside the dispersive medium in the time domain. The problem of the time and frequency evolution of the precursors in 1DPC is studied, using the finite-difference time-domain (FDTD) techniques in conjunction with joint time-frequency analysis (JTFA). Our study clearly shows that the precursor fields associated with superluminal pulse propagation travel at subluminal speeds. It is also shown that FDTD analysis and JTFA can be combined to study the dynamic evolution of the transient and steady state pulse propagation in dispersive media. The second part of the thesis concentrates on the applications of the precursors. An interesting property of the precursors is their lower than exponential attenuation rate inside a lossy dielectric, such as water. This property of the precursors has made them an interesting candidate for applications such as ground penetrating radar and underwater communication. It was recently pointed out that a pulse which is generated inside of water and assumes the shape of the Brillouin precursor would be optimally suited for long range propagation in water (described by the single-pole Debye model). Here, we have considered the optimal pulse propagation problem, accounting for the interaction of the pulse with the air/water interface at oblique incidence. In addition, we argue that pulse excitations which are rough approximation of the Brillouin precursor will eventually evolve into the Brillouin precursor itself shortly after they enter water. Therefore, the excitation of a long-propagating pulse is not sensitive to its shape. Finally, we studied the performance of the optimized pulse in terms of the energy of the scattered field from an object inside water. Based on the simulation results the optimized pulse scattered field has higher energy compared to pulses with the same energy and different temporal distribution. The FDTD technique is employed in all the simulations.

Until now, few rigorous studies of the precursors in structures exhibiting superluminal group velocities have been performed. One dimensional photonic crystals(1DPC) and active Lorentzian media are among the ones which are able to exhibit superluminal propagation. In the first part of the thesis we have studied the evolution of the precursors in active Lorentzian media and 1DPC. The problem of the propagation of the precursors in active Lorentzian media is addressed, by employing the steepest descent method to provide a detailed description of the propagation of the pulse inside the dispersive medium in the time domain. The problem of the time and frequency evolution of the precursors in 1DPC is studied, using the finite-difference time-domain (FDTD) techniques in conjunction with joint time-frequency analysis (JTFA). Our study clearly shows that the precursor fields associated with superluminal pulse propagation travel at subluminal speeds. It is also shown that FDTD analysis and JTFA can be combined to study the dynamic evolution of the transient and steady state pulse propagation in dispersive media. The second part of the thesis concentrates on the applications of the precursors. An interesting property of the precursors is their lower than exponential attenuation rate inside a lossy dielectric, such as water. This property of the precursors has made them an interesting candidate for applications such as ground penetrating radar and underwater communication. It was recently pointed out that a pulse which is generated inside of water and assumes the shape of the Brillouin precursor would be optimally suited for long range propagation in water (described by the single-pole Debye model). Here, we have considered the optimal pulse propagation problem, accounting for the interaction of the pulse with the air/water interface at oblique incidence. In addition, we argue that pulse excitations which are rough approximation of the Brillouin precursor will eventually evolve into the Brillouin precursor itself shortly after they enter water. Therefore, the excitation of a long-propagating pulse is not sensitive to its shape. Finally, we studied the performance of the optimized pulse in terms of the energy of the scattered field from an object inside water. Based on the simulation results the optimized pulse scattered field has higher energy compared to pulses with the same energy and different temporal distribution. The FDTD technique is employed in all the simulations.

Seismic modeling is a technique for simulating wave propagation through the subsurface. For a given geological model, seismic modeling allows us to generate snapshots of wave propagation and synthetic data. In my dissertation, for real seismic events I have chosen to implement the finite-difference modeling technique. When adequate discretization in space and time is possible, the finite-difference technique is by far one of the most accurate tools for simulating elastic-wave propagation through complex geological models. In recent years, a significant amount of work has been done in our group using 2D finite-difference modeling. For complex salt structures which exploration and pro- duction industries meet today, 2D finite-difference modeling is not sufficient to study subsalt imaging or the demultiple of subsalt models. That is why I have developed a 3D finite-difference modeling code. One of the key challenges that I have met in developing the 3D finite-difference code is to adapt the absorbing boundary conditions. Absorbing boundary conditions are needed to describe the infinite geological models by limited computing domain. I have validated the 3D finite-difference code by comparing its results with analytic solutions. I have used 3D finite-difference program to generate data corresponding to 3D complex model which describes salt and subsalt structures of Gulf of Mexico. The resulting data include reflections, diffractions and other scattering phenomena. I have also used finite-difference program in anisotropic context to show that we can effectively predict shear-wave splitting and triplication in the data. There are new sets of events that are not directly recorded in seismic data, they have been called virtual events. These events are turning to be as important as real events in modern data processing. Therefore we also have to learn how to model them. Unfortunately, they cannot yet be modeled directly from finite-difference. Here I will describe how to model these events by using cross correlation type representation theorem. As illustration of how important of virtual events for seismic data process- ing, I also described an internal multiple attenuation technique which utilized virtual events.

It is well known that the finite difference time domain (FDTD) method is a powerful numerical analysis tool for solving electromagnetic problems. In a simulated area, in order to discretize an object which is much smaller than the others, a very small space increment is needed and hence the time step should be decreased too for stability consideration in traditional FDTD. The small space and time increments will respectively increase the memory requirement and calculation time. To overcome these problems, some numerical methods were developed, such as the subcell and nonuniform grid method, to handle the small feature size. This thesis describes an efficient method for generating FDTD subcell equations. We construct a second order macromodel system instead of the subcell region in conventional FDTD. The macromodel system can be reduced with model order reduction techniques (MOR) and then translated into new FDTD update equations. When the problem contains several objects of the same size and material properties, the MOR subcell has the advantage of reusability. This means that the reduce-order model of the object needs to be generated only once nonetheless can be applied to every position where the objects originally occupied.

¡@In this thesis, a novel subgridding scheme is proposed based on the hybridization of the FDTD and CNDG-FDTD algorithms. The FDTD method is applied to the coarse grid region, while the CNDG-FDTD method is used in the fine grid region. Because of the unconditional stability of the CNDG scheme, the temporal step size can be set equal to that in the coarse grid region to speed up the computation in the fine grid region. Furthermore, the temporal interpolation at the fine and coarse grids interface is no longer necessary and thus the complexity of spatial interpolation is largely reduced. ¡@As the CNDG-FDTD method is free from the CFL condition restraint, it saves a large amount of CPU time. Numerical results agree very well with that of the FDTD scheme. But it requires a larger amount of computer memory, at least 20% more than the FDTD method. A modified version of the CNDG-FDTD scheme with increased memory efficiency is also presented. It has not only eliminated the restraint of the CFL condition, but also achieved a more efficient saving of CPU time and computer memory requirements.

A mathematical model considering ellipsoidal surface for the forming tube is proposed in this work to examine the plastic deformation behavior of a thin-walled tube during tube bulge hydroforming process in an open die. In the formulation of this mathematical model, nonuniform thinning in the free bulged region and sticking and sliding friction modes between the tube and die are considered. In the sticking friction mode, the elements in contact with the die do not move or slide after contact with the die. Whereas, in the sliding friction mode, the elements in contact with the die will continue to deform plastically in the subsequent forming process. The relationship between the internal pressure and the bulge height of the tube is examined. The effects of various forming parameters such as the die entry radius, the initial thickness, the length/diameter ratio, material property, etc., upon the forming pressure and the thickness distribution of products were discussed systematically.

When we use the FDTD method to analyze electromagnetic problems, it has to properly discretize the space and time. Automeshing can non-uniformly discretize the simulated structure and generate gradual grids. To improve the efficiency of automeshing, we optimize the parameter of automeshing using the genetic algorithm. Without sacrificing accuracy, it searches a suitable ratio to reduce the generated grids and to save simulation time. At last, we optimize the PIFA using genetic algorithm and search automatically the height of the substrate and the feed position in order to obtain optimal performance. When we use the genetic algorithm, it is the key point to define an objective function evaluating the fitness of the optimized problem. It is important that the function has to appropriately describe the performance at that time.

In this work, based on both experimental and theoretical approach, the contact ESD behavior on a PCB circuit is investigated. The discharge mechanisms of ESD (Electrostatic Discharge) phenomena are discussed by both practical measurement and mathematic analysis. Simplified mathematic models include CR-R¡BCR-C and CR-L are proposed to explain the low frequency phenomena of ESD discharge events. Moreover, some experimental setups with good repeatability are demonstrated for measuring the ESD-induced noise on high-speed PCB and some countermeasures are suggested to reduce ESD damage.

In part I, a numerical study of the mixed states in a mesoscopic type-II superconducting cylinder is described. Steady-state configurations and transient behavior of the magnetic vortices for various values of the applied magnetic field H are presented. Transitions between different multi-vortex states as H is changed is demonstrated by abrupt changes in vortex configurations and jumps in the B vs H plot. An efficient scheme to determine the equilibrium vortex configuration in a mesoscopic system at any given applied field, not limited to the symmetry of the system, is devised and demonstrated. In part II, a superconducting thin film is subject to a non-uniform magnetic field from a vertical magnetic dipole, consisting of two magnetic monopoles of opposite charges. For a film with constant thickness and with no pins, it has been found that the film carries two pairs of vortex-antivortex in the steady state in the imposed flux range of 2.15 < (Phi)+ < 2.90 (in units of flux quantum) and no vortex at all for (Phi)+ <= 2.15. Transitions from a superconducting state with 3 pairs of vortex-antivortex to one with 2 pairs, where a pair of vortex-antivortex annihilates, have been observed in the pseudo-time sequence. With a perturbation with antidots (holes), vortexantivortex pair has been created for lower magnetic fluxes down to (Phi)+ = 1.3. In the sample of size 16(Xi) x 16(Xi), the attraction force between the vortex and antivortex always dominates over the pinning force, so that they eventually come out of pins, move toward each other, and annihilate each other. The annihilation rate, measured with time taken for the annihilation, is reduced noticeably by the increase of the distance between pins, or the increase in the pin size. A simulation of the magnetic vortex pinning in the sample of size 32(Xi) x 32(Xi) suggests we are likely to achieve pinning of the vortex-antivortex pair with the sample size around this and vortex-antivortex separation of 22(Xi). Using this sample as a template, the maximum density of pinned vortices achievable is calculates to be about 7.6 x 10^14 vortices/m2 for (Xi) =~ 1.6A°.