Thesis (M.S.)--Ohio State University, 2005. / Includes bibliographical references (leaves 65-66). Available online via OhioLINK's ETD Center

Thesis (Ph.D.) -- McMaster University, 2002. / Includes bibliographical references (p. 159-170).absrobing boundary Also available via World Wide Web.

The Finite-Difference Time-Domain (FDTD) method has been used extensively in electromagnetic field modeling because of its ability to robustly handle interactions of fields with complex heterogeneous structures. In particular, the total/scattered field formulation has allowed for efficient implementation of arbitrarily directed uniform plane waves, consequently facilitating efficient modeling of far-field scattering problems. The total/scattered approach is not restricted to plane waves and can be expanded to any waveforms that can be described in analytical or semi-analytical form. While existing formulations of FDTD have been immensely successful, they are not well suited to problems that involve near field scattering/interaction problems, where both the source and object are in the same domain but at a substantial distance from each other. This is due to the exceedingly high demands for computational resources that may result from the domain size, and/or dramatically different requirements for the mesh density in the source and object areas. One solution to this problem is to separate the domain into source and scatterer regions coupled by surface boundary radiation conditions. However, this method can incur large storage requirements for calculation of the radiation conditions. A specific near-field situation of interest to the utility industry is the case of workers near high voltage powerlines. In this instance, the field pattern takes on a cylindrical, transverse electromagnetic character. More general radiating sources can be accurately represented in the near-field by using spherical wave expansions, which are often used to represent antennas measured on test ranges. Successful implementation of these analytic solutions is feasible within the FDTD framework, and would allow for the illumination of the scatterer modeled at a considerably lower cost than in the standard approach. This thesis presents a method where these non-uniform, near-field, sources can be implemented implicitly as source conditions in an existing FDTD method. The specific case of powerline fields is described first, followed by the more general case of spherical waves. The analytic solution for powerline fields is implemented to show that near-field source configuration can be successfully modeled implicitly with accurate and efficient results. The method is validated by comparing with known analytic solutions, with very good accuracy being achieved. Then, a specific example of a human under a powerline close by is modeled to examine predictions made earlier under the assumption of a plane wave source condition. For a similar powerline source configuration, results of organ dosimetry predict that induced fields are from ten to sixty percent greater than predicted with the plane wave source. This same approach is applied to model a more general and difficult problem, namely spherical waves as sources in the total/scattered FDTD, called the SW-FDTD. Since transverse properties of spherical modes are known, the behavior of a mode can be represented on a one-dimensional radial grid. Thus, much like the plane wave sources in the FDTD method, the spherical wave modes are time-stepped on one-dimensional staggered electric/magnetic field source grids in the radial direction, representing mode propagation in free space. Spherical wave modes can then be interpolated and summed on the Huygens’ surface to represent the total field of the source, thus providing the coupling between the complex source and a scatterer using one-dimensional grids. It is assumed that the object of interest is beyond the reactive near-field of the source, and therefore there is no significant coupling between source and object. The SW-FDTD method is validated by comparing simulations with several analytic solutions that increase in complexity, demonstrating very good accuracy. Issues relating to the numerical implementation are discussed, including the effects of numerical dispersion, stability, and simple Mur first order boundary conditions. Incorporation of the method as a source condition in an existing FDTD program, and validation of this synthesis, show that the SW-FDTD method can implictly model sources as accurately as explicit models do. The efficiency, and the reduction of errors remain issues for further research to improve the overall utility of the SW-FDTD method. / Graduate

We consider the finite difference method applied to a class of financial problems. Specifically, we investigate the properties of the Du Fort and Frankel finite difference scheme and experiment with adaptations of the scheme to improve on its consistency properties. The Du Fort and Frankel finite difference scheme is applied to a number of problems that frequently occur in finance. We specifically investigate problems associated with jumps, discontinuous behavior, free boundary problems and multi dimensionality. In each case we consider adaptations to the Du Fort and Frankel scheme in order to produce reliable results. Copyright / Dissertation (MSc)--University of Pretoria, 2009. / Mathematics and Applied Mathematics / unrestricted

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This thesis provides insight into methods for estimating blood perfusion, emphasizing the need for accurate modeling in dynamic physiological environments. The thesis critically examines conventional error function solutions used in steady state or gradually changing blood flow scenarios, revealing their shortcomings in accurately reflecting more rapid changes in blood perfusion. To address this limitation, this study introduces a novel prediction model based on the finite-difference method (FDM) specifically designed to produce accurate results under different blood flow perfusion conditions. A comparative analysis concludes that the FDM-based model is consistent with traditional error function methods under constant blood perfusion conditions, thus establishing its validity under dynamic and steady blood flow conditions. In addition, the study attempts to determine whether analytical solutions exist that are suitable for changing perfusion conditions. Three alternative analytical estimation methods were explored, each exposing the common thread of inadequate responsiveness to sudden changes in blood perfusion. Based on the advantages and disadvantages of the error function and FDM estimation, a combination of these two methods was developed. Utilizing the simplicity and efficiency of the error function, the prediction of contact resistance and core temperature along with the initial blood perfusion was first made at the beginning of the data. Then the subsequent blood perfusion values were predicted using the FDM, as the FDM can effectively respond to changing blood perfusion values. / Master of Science / Blood perfusion, the process of blood flowing through our body's tissues, is crucial for our health. It's like monitoring traffic flow on roads, which is especially important during rapid changes, such as during exercise or medical treatments. Traditional methods for estimating blood perfusion, akin to older traffic monitoring techniques, struggle to keep up with these rapid changes. This research introduces a new approach, using a method often found in engineering and physics, called the finite-difference method (FDM), to create more accurate models of blood flow in various conditions. This study puts this new method to the test against the old standards. We discover that while both are effective under steady conditions, the FDM shines when blood flow changes quickly. We also examined three other methods, but they, too, fell short in these fast-changing scenarios. This work is more than just numbers and models; it's about potentially transforming how we understand and manage health. By combining the simplicity of traditional methods for initial blood flow estimates with the dynamic capabilities of the FDM, we're paving the way for more precise medical diagnostics and treatments.

Nowadays, subsalt oil and gas exploration is drawing more and more attention from the hydrocarbon industry. Hydrocarbon exploitation requires detailed geological information beneath the surface. Seismic imaging is a powerful tool employed by the hydrocarbon industry to provide subsurface characterization and monitoring information. Traditional wave-equation migration algorithms are based on the one- way-in-depth propagation using the scalar wave equation. These algorithms focus on downward continuing the upcoming waves. However, it is still really difficult for conventional seismic imaging methods, which have dip limitations, to get a correct image for the edge and shape of the salt body and the corresponding subsalt structure. The dip limitation problem in seismic imaging can be solved completely by switching to Reverse-Time Migration (RTM). Unlike old methods, which deal with the one-way wave equation, RTM propagator is two-way and, as a result, it no longer imposes dip limitations on the image. It can also handle complex waveforms, including prismatic waves. Therefore it is a powerful tool for subsalt imaging. RTM involves wave extrapolation forward and backward in time. In order to accurately and efficiently extrapolate the wavefield in heterogeneous media, I develop three novel methods for seismic wave modeling in both isotropic and tilted transversely isotropic (TTI) media. These methods overcome the space-wavenumber mixed-domain problem when solving the acoustic two-way wave equation. The first method involves cascading a Fourier Transform operator and a finite difference (FD) operator to form a chain operator: Fourier Finite Differences (FFD). The second method is lowrank finite differences (LFD), whose FD schemes are derived from the lowrank approximation of the mixed-domain operator and are represented using adapted coefficients. The third method is lowrank Fourier finite differences (LFFD), which use LFD to improve the accuracy of TTI FFD mothod. The first method, FFD, may have an advantage in efficiency, because it uses only one pair of multidimensional forward and inverse FFTs (fast Fourier transforms) per time step. The second method, LFD, as an accurate FD method, is free of FFTs and in return more suitable for massively parallel computing. It can also be applied to the FFD method to reduce the dispersion in TTI case, which results in the third method, LFFD. LFD and LFFD are based on lowrank approx- imation which is a general method to handle mixed-domain operators and can be easily applied to more complicated mixed-domain operators. I show pseudo-acoustic modeling in orthorhombic media by lowrank approximation as an example. / text

published_or_final_version / abstract / toc / Civil Engineering / Master / Master of Philosophy

Synthetic seismograms are a computer-generated aid in the search for hydrocarbons. Heretofore the solution has been done by z-transforms. This thesis presents a solution based on the method of finite differences. The resulting algorithm is fast and compact. The method is applied to three variations of the problem, all three are reduced to the same approximating equation, which is shown to be optimal, in that grid refinement does not change it. Two types of algorithms are derived from the equation. The number of obvious multiplications, additions and subtractions of each is analyzed. Critical section of each requires one multiplication, two additions and two subtractions. Four sample synthetic seismograms are shown. Implementation of the new algorithm runs twice as fast as previous computer program.