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Title: On superconvergent discontinuous Galerkin methods for Schrödinger equations and sparse grid central discontinuous Galerkin method
Creator: Chen, Anqi
Date: 2019
Collection: Electronic Theses & Dissertations
Description: "In this thesis, we design and analyze a discontinuous Galerkin (DG) method for one-dimensional Schrodinger equations under a general class of numerical fluxes, and another efficient DG method for high-dimensional hyperbolic equations.In the first DG method, we develop an ultra-weak discontinuous Galerkin (UWDG) method to solve the one-dimensional nonlinear Schrödinger equation. Stability conditions and error estimates are derived for the scheme with a general class of numerical fluxes. The...
Show more"In this thesis, we design and analyze a discontinuous Galerkin (DG) method for one-dimensional Schrodinger equations under a general class of numerical fluxes, and another efficient DG method for high-dimensional hyperbolic equations.In the first DG method, we develop an ultra-weak discontinuous Galerkin (UWDG) method to solve the one-dimensional nonlinear Schrödinger equation. Stability conditions and error estimates are derived for the scheme with a general class of numerical fluxes. The error estimates are based on detailed analysis of the projection operator associated with each individual flux choice. Depending on the parameters, we find out that in some cases, the projection can be defined element-wise, facilitating analysis. In most cases, the projection is global, and its analysis depends on the resulting 2 x 2 block-circulant matrix structures. For a large class of parameter choices, optimal a priori L2 error estimates can be obtained. Numerical examples are provided verifying theoretical results.In addition to the stability and error analysis, we analyze the superconvergence properties of the UWDG method for one-dimensional linear Schrodinger equation with various choices of flux parameters. Depending on the flux choices and if the polynomial degree k is even or odd, we prove 2k or (2k - 1)-th order superconvergence rate for cell averages and numerical flux of the function, as well as (2k - 1) or (2k - 2)-th order for numerical flux of the derivative. In addition, we prove superconvergence of (k + 2) or (k + 3)-th order of the UWDG solution towards a special projection. At a class of special points, the function values and the first and second order derivatives of the UWDG solution are superconvergent with order k + 2, k + 1, k, respectively. The proof relies on the correction function techniques initiated in Cao, et. al. (2014), and applied to Cao, et. al. (2017) for direct DG (DDG) methods for diffusion problems. By negative norm estimates, we apply the post processing technique and show that the accuracy of our scheme can be enhanced to order 2k. Theoretical results are verified by numerical experiments.In the second DG method, we develop sparse grid central discontinuous Galerkin (CDG) scheme for linear hyperbolic systems with variable coefficients in high dimensions. The scheme combines the CDG framework with the sparse grid approach, with the aim of breaking the curse of dimensionality. A new hierarchical representation of piecewise polynomials on the dual mesh is introduced and analyzed, resulting in a sparse finite element space that can be used for non-periodic problems. Theoretical results, such as L2 stability and error estimates are obtained for scalar problems. CFL conditions are studied numerically comparing discontinuous Galerkin (DG), CDG, sparse grid DG and sparse grid CDG methods. Numerical results including scalar linear equations, acoustic and elastic waves are provided."--Pages ii-iii.
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Title: Applications of evolutionary algorithms to electromagnetic materials characterization and design problems
Creator: Frasch, Jonathan Lemoine
Date: 2017
Collection: Electronic Theses & Dissertations
Description: Determining the electrical permittivity and magnetic permeability of materials is an important task in electromagnetics research. The method using reflection and transmission scattering parameters to determine these constants has been widely employed for many years, ever since the work of Nicolson, Ross, and Weir in the 1970's. For general materials that are homogeneous, linear, and isotropic, the method they developed (the NRW method) works very well and provides an analytical solution.For...
Show moreDetermining the electrical permittivity and magnetic permeability of materials is an important task in electromagnetics research. The method using reflection and transmission scattering parameters to determine these constants has been widely employed for many years, ever since the work of Nicolson, Ross, and Weir in the 1970's. For general materials that are homogeneous, linear, and isotropic, the method they developed (the NRW method) works very well and provides an analytical solution.For materials which possess a metal backing or are applied as a coating to a metal surface, it can be difficult or even impossible to obtain a transmission measurement, especially when the coating is thin. In such a circumstance, it is common to resort to a method which uses two reflection type measurements. There are several such methods for free-space measurements, using multiple angles or polarizations for example. For waveguide measurements, obtaining two independent sources of information from which to extract two complex parameters can be a challenge.This dissertation covers three different topics. Two of these involve different techniques to characterize conductor-backed materials, and the third proposes a method for designing synthetic validation standards for use with standard NRW measurements. All three of these topics utilize modal expansions of electric and magnetic fields to analyze propagation in stepped rectangular waveguides. Two of the projects utilize evolutionary algorithms (EA) to design waveguide structures. These algorithms were developed specifically for these projects and utilize fairly recent innovations within the optimization community.The first characterization technique uses two different versions of a single vertical step in the waveguide. Samples to be tested lie inside the steps with the conductor reflection plane behind them. If the two reflection measurements are truly independent it should be possible to recover the values of two complex parameters, but success of the technique ultimately depends upon how independent the measurements actually are.Next, a method is demonstrated for developing synthetic verification standards. These standards are created from combinations of vertical steps formed from a single piece of metal or metal coated plastic. These fully insertable structures mimic some of the measurement characteristics of typical lab specimens and thus provide a useful tool for verifying the proper calibration and function of the experimental setup used for NRW characterization. These standards are designed with the use an EA, which compares possible designs based on the quality of the match with target parameter values. Several examples have been fabricated and tested, and the design specifications and results are presented.Finally, a second characterization technique is considered. This method uses multiple vertical steps to construct an error reducing structure within the waveguide, which allows parameters to be reliably extracted using both reflection and transmission measurements. These structures are designed with an EA, measuring fitness by the reduction of error in the extracted parameters. An additional EA is used to assist in the extraction of the material parameters supplying better initial guesses to a secant method solver. This hybrid approach greatly increases the stability of the solver and increases the speed of parameter extractions. Several designs have been identified and are analyzed.
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Title: Machine learning method for authorship attribution
Creator: Hu, Xianfeng
Date: 2015
Collection: Electronic Theses & Dissertations
Description: MACHINE LEARNING METHOD FOR AUTHORSHIP ATTRIBUTIONBy Xianfeng HuBroadly speaking, the authorship identification or authorship attribution problem is to determine the authorship of a given sample such as text, painting and so on. Our main work is to develop an effective and mathe-sound approach for the analysis of authorship of doubted books.Inspired by various authorship attribution problems in the history of literature and the application of machine learning in the study of literary...
Show moreMACHINE LEARNING METHOD FOR AUTHORSHIP ATTRIBUTIONBy Xianfeng HuBroadly speaking, the authorship identification or authorship attribution problem is to determine the authorship of a given sample such as text, painting and so on. Our main work is to develop an effective and mathe-sound approach for the analysis of authorship of doubted books.Inspired by various authorship attribution problems in the history of literature and the application of machine learning in the study of literary stylometry, we develop a rigorous new method for the mathematical analysis of authorship by testing for a so-called chrono-divide in writing styles. Our method incorporates some of the latest advances in the study of au- thorship attribution, particularly techniques from support vector machines. By introducing the notion of relative frequency of word and phrases as feature ranking metrics our method proves to be highly effective and robust.Applying our method to the classical Chinese novel Dream of the Red Chamber has led to convincing if not irrefutable evidence that the first 80 chapters and the last 40 chapters of the book were written by two different authors.Also applying our method to the English novel Micro, we are able to confirm the existence of the chrono-divide and identify its location so that we can differentiate the contribution of Michael Crichton and Richard Preston, the authors of the novel.We have also tested our method to the other three Great Classical Novels in Chinese. As expected no chrono-divides have been found in these novels. This provides further evidenceof the robustness of our method. We also proposed a new approach to authorship identification to solve the open classproblem where the candidate group is nonexistent or very large, which is reliably scaled from a new method we have developed for the closed class problem in which the candidates author pool is small. This is attained by using support vector machines and by analyzing the relative frequencies of common words in the function words dictionary and most frequently used words. This method scales very nicely to the open class problem through a novel author randomization technique, where an author in question is compared repeatedly to randomly selected authors. The author randomization technique proves to be highly robust and effective. Using our approaches we have found answers to three well known authorship controversies: (1) Did Robert Galbraith write Cuckoo’s Calling? (2) Did Harper Lee write To Kill a Mockingbird or did her friend Truman Capote write it? (3) Did Bill Ayers write Obama’s autobiography Dreams From My Father?
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Title: A novel approach to blind source separation and extraction in audio
Creator: Wang, Xun
Date: 2015
Collection: Electronic Theses & Dissertations
Description: In this thesis, the blind source separation (BSS) on digitalaudio signals via background learning by several differentmethodologies is carefully studied. In the daily auditory, acoustic audio signals are usually mixtures of different sources, including foreground andbackground noises. Most of the time, we only want to receive the foregroundsources and get rid of the background ones. Because of the randomness ofvarious situations, it is very difficult to perform this separation without knowing...
Show moreIn this thesis, the blind source separation (BSS) on digitalaudio signals via background learning by several differentmethodologies is carefully studied. In the daily auditory, acoustic audio signals are usually mixtures of different sources, including foreground andbackground noises. Most of the time, we only want to receive the foregroundsources and get rid of the background ones. Because of the randomness ofvarious situations, it is very difficult to perform this separation without knowing the detailed information. Even if the background noises are not dominating the foreground sources, or even much weaker, it is still a difficult problem, especially for the case that there are more than three sources including the noise. This also makes it even more difficult to separate different sources from mixed signals. In this thesis, a novel approach to solve cancellation kernels is provided by using a modified sigular value decomposition method. The main focus is to use this new technique to estimate the cacellation kernels with good results in short computational time.In this work, some background information for blind source separation of audio will be first introduced. Next, the knowledge of four different methods for solving this type of problems is mentioned. Split Bregman has been studied by others in solving cancellation kernels for the separation of speech signals. We apply proximity operator method to solve the cancellation kernels for BSS of audio signal processing. It has been applied to study image processing by other researchers. Quadratic programming method has been applied to solve cancellation kernels by Wang and Zhou. We provide a new approach to bring sparseness to cancellation kernels by using quadratic programming. We developed a modified singular value decomposition (SVD) algorithm based on the numerical experiments. It is a new technique to estimate cancellation kernels for BSS of audio signals. The detailed information and schemes are presented in Chapter 3. Then, in the fouth chapter, there are different numerical simulation examples according to different scenarios. We compare the results of our modified SVD method with others methods, and conclude that our modified SVD is the best approach.
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Title: A global modeling framework for plasma kinetics : development and applications
Creator: Parsey, Guy Morland
Date: 2017
Collection: Electronic Theses & Dissertations
Description: The modern study of plasmas, and applications thereof, has developed synchronously with com-puter capabilities since the mid-1950s. Complexities inherent to these charged-particle, many-body, systems have resulted in the development of multiple simulation methods (particle-in-cell,fluid, global modeling, etc.) in order to both explain observed phenomena and predict outcomesof plasma applications. Recognizing that different algorithms are chosen to best address specifictopics of interest, this...
Show moreThe modern study of plasmas, and applications thereof, has developed synchronously with com-puter capabilities since the mid-1950s. Complexities inherent to these charged-particle, many-body, systems have resulted in the development of multiple simulation methods (particle-in-cell,fluid, global modeling, etc.) in order to both explain observed phenomena and predict outcomesof plasma applications. Recognizing that different algorithms are chosen to best address specifictopics of interest, this thesis centers around the development of an open-source global model frame-work for the focused study of non-equilibrium plasma kinetics. After verification and validationof the framework, it was used to study two physical phenomena: plasma-assisted combustion andthe recently proposed optically-pumped rare gas metastable laser.Global models permeate chemistry and plasma science, relying on spatial averaging to focusattention on the dynamics of reaction networks. Defined by a set of species continuity and energyconservation equations, the required data and constructed systems are conceptually similar acrossmost applications, providing a light platform for exploratory and result-search parameter scan-ning. Unfortunately, it is common practice for custom code to be developed for each application-an enormous duplication of effort which negatively affects the quality of the software produced.Presented herein, the Python-based Kinetic Global Modeling framework (KGMf) was designed tosupport all modeling phases: collection and analysis of reaction data, construction of an exportablesystem of model ODEs, and a platform for interactive evaluation and post-processing analysis. Asymbolic ODE system is constructed for interactive manipulation and generation of a Jacobian,both of which are compiled as operation-optimized C-code.Plasma-assisted combustion and ignition (PAC/PAI) embody the modernization of burning fuelby opening up new avenues of control and optimization. With applications ranging from engineefficiency and pollution control to stabilized operation of scramjet technology in hypersonic flows,developing an understanding of the underlying plasma chemistry is of the utmost importance.While the use of equilibrium (thermal) plasmas in the combustion process extends back to the ad-vent of the spark-ignition engine, works from the last few decades have demonstrated fundamentaldifferences between PAC and classical combustion theory. The KGMf is applied to nanosecond-discharge systems in order to analyze the effects of electron energy distribution assumptions onreaction kinetics and highlight the usefulness of 0D modeling in systems defined by coupled andcomplex physics.With fundamentally different principles involved, the concept of optically-pumped rare gasmetastable lasing (RGL) presents a novel opportunity for scalable high-powered lasers by takingadvantage of similarities in the electronic structure of elements while traversing the periodic ta-ble. Building from the proven concept of diode-pumped alkali vapor lasers (DPAL), RGL systemsdemonstrate remarkably similar spectral characteristics without problems associated with heatedcaustic vapors. First introduced in 2012, numerical studies on the latent kinetics remain immature.This work couples an analytic model developed for DPAL with KGMf plasma chemistry to bet-ter understand the interaction of a non-equilibrium plasma with the induced laser processes anddetermine if optical pumping could be avoided through careful discharge selection.
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Title: Geometric evolution of single-layer interfaces in the functionalized Cahn-Hilliard equation
Creator: Hayrapetyan, Gurgen Ruben
Date: 2011
Collection: Electronic Theses & Dissertations
Description: We study the Functionalized Cahn-Hilliard Energy (FCH), which is a higher-order reformulation of the Cahn-Hilliard energy, as a model for network formation in polymer-solvent mixtures. The model affords a finite interfacial width, accommodates merging and other topological reorganization, and couples naturally to momentum balance and other macroscopic mass transport equations.The corresponding constrained L2 gradient flow has a rich family of approximately steady-state...
Show moreWe study the Functionalized Cahn-Hilliard Energy (FCH), which is a higher-order reformulation of the Cahn-Hilliard energy, as a model for network formation in polymer-solvent mixtures. The model affords a finite interfacial width, accommodates merging and other topological reorganization, and couples naturally to momentum balance and other macroscopic mass transport equations.The corresponding constrained L2 gradient flow has a rich family of approximately steady-state solutions that include not only the single-layer heteroclinic front profile seen in gradient flows of the Cahn-Hilliard energy, but also a novel one parameter family of homoclinic bi-layer solutions. In this thesis we rigorously derive the geometric evolution of the single-layer polymer-solvent interface.We form a manifold of quasi-equilbria by "dressing" a large family of co-dimension one interfaces immersed in Rd with heteroclinic solutions of a one-dimensional equilibrium equation derived from the first variation of the FCH energy. We show that solutions of the gradient flow that start sufficiently close to the manifold remain close, and moreover the flow can be decomposed, at leading order, as a normal velocity for the underlying co-dimension one interface. Assuming the smoothness of the interface under this flow, we develop rigorous estimates on the proximity of the true solution to the manifold, in an appropriate norm, for long time.
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Title: Segmented nano-force sensor
Creator: Dharuman, Gautham
Date: 2013
Collection: Electronic Theses & Dissertations
Description: Nanoscale force sensors are finding widespread applications in atomic and biological force sensing where forces involved range from zeptonewtons to several nanonewtons. Different methods of nanoscale force sensing based on optical, electrical or purely mechanical schemes have been reported. However, each technique is limited by factors such as large size, low resolution, slow response, force range and alignment issues. In this research, a new device structure which could overcome the above...
Show moreNanoscale force sensors are finding widespread applications in atomic and biological force sensing where forces involved range from zeptonewtons to several nanonewtons. Different methods of nanoscale force sensing based on optical, electrical or purely mechanical schemes have been reported. However, each technique is limited by factors such as large size, low resolution, slow response, force range and alignment issues. In this research, a new device structure which could overcome the above mentioned constraints is studied theoretically and experimentally for the possibility of its application in nano-scale force sensing.
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Title: Modeling of charge injection and transport in organic semiconductors with applications to conducting atomic force microscopy
Creator: Pimcharoen, Kanokkorn
Date: 2018
Collection: Electronic Theses & Dissertations
Description: Charge injection and transport in organic semiconductors are key factors controlling the device performance, and have been intensively investigated by conductive atomic force microscope (c-AFM) experiments in the space-charge-limited current (SCLC) regime. The simplified SCLC theory, despite being widely used to describe the unipolar SCLC, has limitations in explaining the current-voltage responses of c-AFM measurements due to two major reasons. First, the conventional planar model does not...
Show moreCharge injection and transport in organic semiconductors are key factors controlling the device performance, and have been intensively investigated by conductive atomic force microscope (c-AFM) experiments in the space-charge-limited current (SCLC) regime. The simplified SCLC theory, despite being widely used to describe the unipolar SCLC, has limitations in explaining the current-voltage responses of c-AFM measurements due to two major reasons. First, the conventional planar model does not include the effect of current spreading commonly found beneath the conducting tip. Secondly, the theory only considers drift transport, and assumes that charge diffusion can be neglected, causing discrepancies in its predictions of transport behaviors that will be discussed thoroughly here. The focus of this thesis is on developing numerical models for hole-only devices with the full description of drift and diffusion transport mechanisms, which is called the drift-diffusion (DD-) SCLC model. The applications of the models in the analysis of c-AFM experimental data are presented. We generalize the theory which takes both drift and diffusion currents into account, leading to more realistic DD-SCLC models for several applications. We then develop numerical approaches that efficiently simulate the hole-only SCLCs for one-, two-, and three- dimensional systems. In the case of fully 3-D calculations, the DD-SCLC model is able to treat inhomogeneous systems including spatially varying trap distributions, nanoscale morphologies, and the tip-plane (c-AFM) geometry. In the theoretical studies, the device simulations elucidate a number of crucial factors that affect the charge transport in the SCLC regime, including charge diffusion, traps, as well as, nanoscale morphology. We introduce the methodology of characterizing the current-voltage responses from c-AFM measurements, which has been used in elucidating the experiments on semiconductor poly(3-hexylthiophene) (P3HT) thin films that develop fibrous morphologies after thermal annealing. We generalize the theory which takes both drift and diffusion currents into account, leading to more realistic DD-SCLC models for several applications. We then develop numerical approaches that efficiently simulate the hole-only SCLCs for one-, two-, and three- dimensional systems. In the case of fully 3-D calculations, the DD-SCLC model is able to treat inhomogeneous systems including spatially varying trap distributions, nanoscale morphologies, and the tip-plane (c-AFM) geometry. In the theoretical studies, the device simulations elucidate a number of crucial factors that affect the charge transport in the SCLC regime, including charge diffusion, traps, as well as, nanoscale morphology. We introduce the methodology of characterizing the current-voltage responses from c-AFM measurements, which has been used in elucidating the experiments on semiconductor poly(3-hexylthiophene) (P3HT) thin films that develop fibrous morphologies after thermal annealing.
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Title: Discontinuous Galerkin methods for Hamilton-Jacobi equations and high-dimensional elliptic equations
Creator: Wang, Zixuan
Date: 2015
Collection: Electronic Theses & Dissertations
Description: This thesis focuses on two related topics, which are to design efficient discontinuous Galerkin (DG) schemes for Hamilton-Jacobi (HJ) equations and high-dimensional elliptic equations.In the first part, we propose a new DG method that solves for the viscosity solution of the general HJ equations. The new method is compact and easy to implement. We avoid the reconstruction of the solution across elements by utilizing the interfacial terms involving the Roe speed. A penalty term proportional to...
Show moreThis thesis focuses on two related topics, which are to design efficient discontinuous Galerkin (DG) schemes for Hamilton-Jacobi (HJ) equations and high-dimensional elliptic equations.In the first part, we propose a new DG method that solves for the viscosity solution of the general HJ equations. The new method is compact and easy to implement. We avoid the reconstruction of the solution across elements by utilizing the interfacial terms involving the Roe speed. A penalty term proportional to the jump of the normal derivative of the numerical solution is added to fix the entropy violation, which was inspired by the Harten and Hymans entropy fix [53] for Roe scheme for the conservation laws. Numerical experiments demonstrate good performance for general Hamiltonians, including nonconvex Hamiltonians.In the second part, we develop an interior penalty DG method on sparse grids for efficient computations of high-dimensional second-order elliptic problems. Using a hierarchical basis representation, we construct a sparse finite element approximation space, reducing the degree of freedom from the standard O(h−d) to O(h−1|log2h|d−1) for d-dimensional problems, where h is the uniform mesh size in each dimension. Compared to the traditional full grid approaches, the accuracy of the numerical approximation of this method is only slightly deteriorated by a factor of | log2 h|d−1 in the energy norm. Error estimates are provided and confirmed by numerical tests in multi-dimensions.
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Title: Stability estimates for electromagnetic scattering from open cavity
Creator: Zheng, Qiong
Date: 2013
Collection: Electronic Theses & Dissertations
Description: The electromagnetic scattering from an open large cavity embedded in an infinite plane is of practical importance due to its significant industrial and military applications. Examples of cavities include jet engine inlet ducts, exhaust nozzles and cavity-backed antennas [3, 4]. In many practical applications, one is interested in the cavity problem with either a large wave number $k$ or a large diameter $a$, in which case the solution has a highly oscillatory nature [7]. While the original...
Show moreThe electromagnetic scattering from an open large cavity embedded in an infinite plane is of practical importance due to its significant industrial and military applications. Examples of cavities include jet engine inlet ducts, exhaust nozzles and cavity-backed antennas [3, 4]. In many practical applications, one is interested in the cavity problem with either a large wave number $k$ or a large diameter $a$, in which case the solution has a highly oscillatory nature [7]. While the original time-harmonic problem is modeled by Helmholtz equation in the unbounded domain, the reformulated model through Fourier transform is essentially a Helmholtz equation in the bounded domain with mixed nonlocal boundary condition. Deriving an explicit dependency between the wave energy and the wave number is mathematically interesting and challenging. The stability estimate is also important as it defines relations between the wave number and the discretization parameters in the error analysis [16]. For the open cavity problem, while the stability analysis for the rectangular cavity was derived recently [8] as described , the stability results for more general shapes of cavities are to be explored. The objective of this thesis work is to partially answer this question by imposing some geometric assumptions.We first start from considering a class of cavity with a strong geometric constraint. The energy stability is established by careful choices of the parameters, and test functions, which take full advantage of geometric properties. The arguments are based on the appropriate usage of the real and imaginary part of the weak formulation of the problem, the separation of lower frequency and higher frequency part, and connections between frequency components and spatial components. The energy in cavity is bounded by the energy of incoming field with coefficient in terms of powers of wave number. Next, we investigate the case where a weaker geometric constraint is imposed. A new auxiliary function with compact support near the boundary of the cavity is carefully constructed to reformulate the problem. However, the original homogeneous Helmholtz equation is changed to a non-homogeneous one, all previous work in homogeneous equation must be suitably modified, and the estimate in terms of wave number $k$ is obtained from detailed analysis of this auxiliary function. The energy norm is proved to be at most in the order of $k^{frac{7}{10}}$, which is the same in terms of the power of wave number $k$ as the case with strong geometric conditions but with other additional terms. Furthermore, we studied the case where the cavity domain is of rectangular-like shape, where new test function is introduced and new inequalities are established to derive the energy estimate.
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Title: Modeling and simulation of strongly coupled plasmas
Creator: Chowdhury, Rahnuma Rifat
Date: 2016
Collection: Electronic Theses & Dissertations
Description: The objective of this work is to develop new modeling and simulation tools for studying strongly coupled plasmas (SCP). Strongly coupled plasmas are different from traditional plasmas as potential energy is larger than the kinetic energy. The standard plasma model does not account for some major effects in SCP: 1) the change in the permittivity 2) the impact on relaxation of the charged particles undergoing Coulomb collisions in a system with weakly shielded long range interactions3) the...
Show moreThe objective of this work is to develop new modeling and simulation tools for studying strongly coupled plasmas (SCP). Strongly coupled plasmas are different from traditional plasmas as potential energy is larger than the kinetic energy. The standard plasma model does not account for some major effects in SCP: 1) the change in the permittivity 2) the impact on relaxation of the charged particles undergoing Coulomb collisions in a system with weakly shielded long range interactions3) the impact of statistical fluctuations in strongly coupled plasmas that leads to non-Markovian effects. Proper modeling of such systems through consideration of Lévy flight processes gives rise to fractional derivatives in time that result in an incorporation of time history in the model. A Lévy flight is a random walk in which the steps are defined in terms of the step-lengths, which have a certain probability distribution, with the directions of the steps being isotropic and random. Lévy processes in the plasma give rise to fluctuations in medium through which the electromagnetic waves are propagating. Averaging over the Lévy processes will allow us to relate to other important parameters in the plasma.
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Title: Rectangular waveguide material characterization : anisotropic property extraction and measurement validation
Creator: Crowgey, Benjamin Reid
Date: 2013
Collection: Electronic Theses & Dissertations
Description: Rectangular waveguide methods are appealing for measuring isotropic and anisotropic materials because of high signal strength due to field confinement, and the ability to control the polarization of the applied electric field. As a stepping stone to developing methods for characterizing materials with fully-populated anisotropic tensor characteristics, techniques are presented in this dissertation to characterize isotropic, biaxially anisotropic, and gyromagnetic materials. Two...
Show moreRectangular waveguide methods are appealing for measuring isotropic and anisotropic materials because of high signal strength due to field confinement, and the ability to control the polarization of the applied electric field. As a stepping stone to developing methods for characterizing materials with fully-populated anisotropic tensor characteristics, techniques are presented in this dissertation to characterize isotropic, biaxially anisotropic, and gyromagnetic materials. Two characterization techniques are investigated for each material, and thus six different techniques are described. Additionally, a waveguide standard is introduced which may be used to validate the measurement of the permittivity and permeability of materials at microwave frequencies. The first characterization method examined is the Nicolson-Ross-Weir (NRW) technique for the extraction of isotropic parameters of a sample completely filling the cross-section of a rectangular waveguide. A second technique is proposed for the characterization of an isotropic conductor-backed sample filling the cross-section of a waveguide. If the sample is conductor-backed, and occupies the entire cross-section, a transmission measurement is not available, and thus a method must be found for providing two sufficiently different reflection measurements.The technique proposed here is to place a waveguide iris in front of the sample, exposing the sample to a spectrum of evanescent modes. By measuring the reflection coefficient with and without an iris, the necessary two data may be obtained to determine the material parameters. A mode-matching approach is used to determine the theoretical response of a sample placed behind the waveguide iris. This response is used in a root-searching algorithm to determine permittivity and permeability by comparing to measurements of the reflection coefficient.For the characterization of biaxially anisotropic materials, the first method considers an extension of the NRW technique for characterization of a sample filling the cross-section of a waveguide. Due to the rectangular nature of the waveguide, typically three different samples are manufactured from the same material in order to characterize the six complex material parameters. The second technique for measuring the electromagnetic properties of a biaxially anisotropic material sample uses a reduced-aperture waveguide sample holder designed to accommodate a cubical sample. All the tensor material parameters can then be determined by measuring the reflection and transmission coefficients of a single sample placed into several orientations. The parameters are obtained using a root-searching algorithm by comparing theoretically computed and measured reflection and transmission coefficients. The theoretical coefficients are determined using a mode matching technique. The first technique for characterizing the electromagnetic properties of gyromagnetic materials considers requires filling the cross-section of a waveguide. The material parameters are extracted from the measured reflection and transmission coefficients. Since the cross-sectional dimensions of waveguides become prohibitively large at low frequencies, and it is at these frequencies that the gyromagnetic properties are most pronounced, sufficiently large samples may not be available. Therefore, the second technique uses a reduced-aperture sample holder that does not require the sample to fill the entire cross section of the guide. The theoretical reflection and transmission coefficients for both methods are determined using a mode matching technique. A nonlinear least squares method is employed to extract the gyromagnetic material parameters. Finally, this dissertation introduces a waveguide standard that acts as a surrogate material with both electric and magnetic properties and is useful for verifying systems designed to characterize engineered materials using the NRW technique. A genetic algorithm is used to optimize the all-metallic structure to produce a surrogate with both relative permittivity and permeability near six across S-band, and with low sensitivity to changes in geometry to reduce the effects of fabrication errors.
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Title: Modeling of hydrogen-based plasmas in microwave plasma-assisted chemical vapor deposition reactors at moderate pressures
Creator: Meierbachtol, Collin Stephen
Date: 2013
Collection: Electronic Theses & Dissertations
Description: Microwave Plasma-Assisted Chemical Vapor Deposition (MPACVD) systems are used in the deposition of high quality diamond films. These systems have traditionally been operated at less than 20% atmospheric pressure (atm), resulting in growth rates up to 5 μm/hr. Under such conditions, the system operation and plasma behavior are well-understood and have been successfully modeled. Recent experiments at pressures approaching 40% atm have demonstrated faster growth rates and better quality...
Show moreMicrowave Plasma-Assisted Chemical Vapor Deposition (MPACVD) systems are used in the deposition of high quality diamond films. These systems have traditionally been operated at less than 20% atmospheric pressure (atm), resulting in growth rates up to 5 μm/hr. Under such conditions, the system operation and plasma behavior are well-understood and have been successfully modeled. Recent experiments at pressures approaching 40% atm have demonstrated faster growth rates and better quality samples. At these increased pressures, the system operation and plasma behavior are not completely understood, with unusual plasma behavior sometimes observed. Experimental measurements within these systems can be difficult, making numerical models attractive for aiding in understanding this behavior. This thesis presents a self-consistent multiphysics numerical model of MPACVD systems, which is accurate under these operating conditions. Electromagnetic field propagation, chemical reactions, species diffusion, thermal processes, energy transfer, and convective flows are all included in the multiphysics model. The model is verified against canonical problems and validated against experimental data. Extensive numerical results are provided for different operating conditions and system configurations.
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Title: Linear response density functional theory for metal surfaces with application to second harmonic generation
Creator: Droba, Justin
Date: 2014
Collection: Electronic Theses & Dissertations
Description: This thesis is a study of electronic excitations at metal surfaces, as described within the context of density functional theory (DFT). Before presenting any physics, the document develops an adaptive spline collocation method that is the workhorse for most of the numerical computations presented afterwards. The background and mathematical foundations of DFT are briefly explored. From there, two different techniques for computing ground state densities---orbital-free density functional theory...
Show moreThis thesis is a study of electronic excitations at metal surfaces, as described within the context of density functional theory (DFT). Before presenting any physics, the document develops an adaptive spline collocation method that is the workhorse for most of the numerical computations presented afterwards. The background and mathematical foundations of DFT are briefly explored. From there, two different techniques for computing ground state densities---orbital-free density functional theory and Kohn-Sham density functional theory---are fully realized. Development of algorithms for numerical computation is the primary component of the presentation, but both techniques are also given rigorous theoretical treatment in full proofs of asymptotic results. What follows next is a rigorous derivation of linear response theory from first principles. Great effort is then spent to develop a scheme for numerical computation of excited responses via linear response theory. Finally, the thesis concludes by demonstrating the application of the techniques developed in the previous chapters to a nonlinear optical phenomenon called second harmonic generation.
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Title: High frequency computation in wave equations and optimal design for a cavity
Creator: Lai, Jun
Date: 2013
Collection: Electronic Theses & Dissertations
Description: Two types of problems are studied in this thesis.One part of the thesis is devoted to high frequency computation. Motivated by fast multiscale Gaussian wavepacket transforms and multiscale Gaussian beam methods which were originally designed for initial value problems of wave equations in the high frequency regime, we develop fast multiscale Gaussian beam methods for wave equations in bounded convex domains in the high frequency regime. To compute the wave propagation in bounded convex...
Show moreTwo types of problems are studied in this thesis.One part of the thesis is devoted to high frequency computation. Motivated by fast multiscale Gaussian wavepacket transforms and multiscale Gaussian beam methods which were originally designed for initial value problems of wave equations in the high frequency regime, we develop fast multiscale Gaussian beam methods for wave equations in bounded convex domains in the high frequency regime. To compute the wave propagation in bounded convex domains, we have to take into account reflecting multiscale Gaussian beams, which are accomplished by enforcing reflecting boundary conditions during beam propagation and carrying out suitable reflecting beam summation. To propagate multiscale beams efficiently, we prove that the ratio of the squared magnitude of beam amplitude and the beam width is roughly conserved, and accordingly we propose an effective indicator to identify significant beams. We also prove that the resulting multiscale Gaussian beam methods converge asymptotically. Numerical examples demonstrate the accuracy and efficiency of the method.The second part of the thesis studies the reduction of backscatter radar cross section (RCS) for a cavity embedded in the ground plane. One approach for RCS reduction is through the coating material. Assume the bottom of the cavity is coated by a thin, multilayered radar absorbing material (RAM) with possibly different permittivities. The objective is to minimize the backscatter RCS by the incidence of a plane wave over a single or a set of incident angles and frequencies. By formulating the scattering problem as a Helmholtz equation with artificial boundary condition, the gradient with respect to the material permittivities is determined efficiently by the adjoint state method, which is integrated into a nonlinear optimization scheme. Numerical example shows the RCS may be significantly reduced.Another approach is through shape optimization. By introducing a transparent boundary condition, the unbounded scattering problem from a cavity is reduced to a bounded domain problem. RCS reduction for the cavity is formulated as a shape optimization problem involving the Helmholtz equation. The existence of the minimizer is proved under an appropriate constraint. Descent directions of the objective function with respect to the boundary may be found via the domain derivative. It is used in a gradient-based optimization scheme to find the optimal shape of the cavity. Numerical examples show that the RCS is effectively reduced at different incident frequencies.
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Title: Near-field imaging via inverse scattering
Creator: Lin, Junshan
Date: 2011
Collection: Electronic Theses & Dissertations
Description: Near-field optics is an emerging research topic in the past fewdecades, mostly motivated by applications in the near-fieldmicroscopy in an effort to break the diffraction limit. In thefar-field imaging, only the propagating wave components with spatialfrequency below the wavenumber are available, and it is well-knownthat the resolution of the image is approximately half of the wavelength(diffraction limit). In the near field, however, the bandwidth ofthe spatial frequency may be expanded by...
Show moreNear-field optics is an emerging research topic in the past fewdecades, mostly motivated by applications in the near-fieldmicroscopy in an effort to break the diffraction limit. In thefar-field imaging, only the propagating wave components with spatialfrequency below the wavenumber are available, and it is well-knownthat the resolution of the image is approximately half of the wavelength(diffraction limit). In the near field, however, the bandwidth ofthe spatial frequency may be expanded by taking account of theevanescent (exponentially decayed) waves. Nowadays there existvarious configurations for the near-field microscopies. However, itis recognized that the images obtained from the near-fieldmicroscopies are problematic by visualizing the object in ananalogical way. Therefore, the inverse scattering theory is appliedto understand how the structure of the scattering object is encodedin the measured scattered field. When single scattering (or Bornapproximation) is assumed, the studies are complete for thenear-field scanning optical microscopy (NSOM) and the total internalreflection microscopy (TIRT) within the framework of the inversescattering theory.In this thesis, we focus on one specific problem where the imagingtarget is a ground plane with some local disturbance. The data iscollected in the near-field regime with a distance above the surfacedisplacement that is smaller than the wavelength. In the recentpaper by G. Derveaux, G. Papanicolaou, and C. Tsogka, a linearizedmodel has been introduced for the nonlinear inverse scatteringproblem by the single scattering assumption. The authors alsoproposed a broadband imaging strategy for denoising and improvingthe resolution of the image. In the thesis, we investigate the moregeneral case by considering the full scattering model, for which thelinearized model is no longer valid. By the analysis of thescattered field, it is confirmed that the evanescent wave modeswhich are not accessible in the far-field regime become significantin the near field. Evanescent wave modes make it possible to breakthe diffraction limit. It is shown that such exponentially decayedmodes of the scattered wave contain the high spatial frequencyinformation (fine features) of the profile. We formulate explicitlythe connection between the evanescent wave modes and the highfrequency components of the surface displacement, and present a newnumerical scheme to reconstruct the surface displacement from theboundary measurements. By extracting the information carried by theevanescent modes effectively, it is shown that the resolution of thereconstructed image is significantly improved in the near field.Numerical examples show that images with a resolution oftenth of wavelength are obtained.To overcome the ill-posedness and the presence of local minimaassociated with this nonlinear imaging problem, we propose to usemultiple frequency data to image the profile of the surfacedisplacement in the second part of the thesis. The main idea is tomarch from the lowest wavenumber to the highest wavenumber. At thefixed wavenumber, by an analysis of the domain derivative for theforward scattering map, a vector field is chosen such that thedefined cost functional decreases. The reconstructed profile evolveswith the chosen vector field at the fixed wavenumber and theevolution process continues until it reaches the highest wavenumber.The proposed reconstructed scheme is able to capture the mainfeature of the profile at low frequency and recover the fine detailsat higher frequency. In particular, for a multiple scale profile, itresolves the fine scale with sufficiently high frequency.
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Title: Electrostatic particle based modeling and simulation of ultra cold plasma
Creator: Jain, Mayur
Date: 2015
Collection: Electronic Theses & Dissertations
Description: We model moderately coupled ultra cold plasma based on experimental setups and investigate the influence of external electric and magnetic fields by simulating the interaction of this plasma with constant magnetic field and radio frequency fields in the form of continuous application and short pulses. A density dependent resonant response is observed through these simulations and we infer the cause to be rapid energy transfer to individual electrons from electric fields through the collective...
Show moreWe model moderately coupled ultra cold plasma based on experimental setups and investigate the influence of external electric and magnetic fields by simulating the interaction of this plasma with constant magnetic field and radio frequency fields in the form of continuous application and short pulses. A density dependent resonant response is observed through these simulations and we infer the cause to be rapid energy transfer to individual electrons from electric fields through the collective motion of the electron cloud rather than a collision based mechanism since collisional time scales are found to be larger than the response period. It is also observed that electron evaporation influences the UCP expansion by reducing theelectron temperature signicantly. These arguments are corroborated by experimental results. We report diagnostics such as temperature, potential and density evolution, electron and ion pair correlation functions, and estimate the size of the UCP with varying initial ionization energies for the ultra cold plasma throughout complete simulation.
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Title: Computational study of strongly coupled charged particle systems
Creator: Dharuman, Gautham
Date: 2018
Collection: Electronic Theses & Dissertations
Description: Exciting experiments in ultracold neutral plasmas, laser-matter interaction, charged particle stopping, mixing under extreme conditions etc., at academic facilities or at even larger facilities such as the National Ignition Facility, Z machine or the Linac Coherent Light Source, have necessitated the need for models that can simulate these systems at large length- and time-scales. This thesis summarizes my research work, falling within the category of computational plasma physics, aimed at...
Show moreExciting experiments in ultracold neutral plasmas, laser-matter interaction, charged particle stopping, mixing under extreme conditions etc., at academic facilities or at even larger facilities such as the National Ignition Facility, Z machine or the Linac Coherent Light Source, have necessitated the need for models that can simulate these systems at large length- and time-scales. This thesis summarizes my research work, falling within the category of computational plasma physics, aimed at three aspects: effective quantum potentials based method for non-equilibrium quantum electron dynamics at scale, efficient force calculation method for molecular dynamics simulation with screened Coulomb interactions, and an avenue based on compressed gases for creation of laboratory-scale tunable strongly coupled plasmas as a platform for understanding large-scale experiments. Effective classical dynamics provide a potentially powerful avenue for modeling large-scale dynamical quantum systems. We have examined the accuracy of a Hamiltonian-based approach that employs effective momentum-dependent potentials (MDPs) within a molecular-dynamics framework through studies of atomic ground states, excited states, ionization energies and scattering properties of continuum states. Working exclusively with the Kirschbaum-Wilets (KW) formulation with empirical MDPs [C. L. Kirschbaum and L. Wilets, PRA 21, 834 (1980)], leads to very accurate ground-state energies for several elements (e.g., N, F, Ne, Al, S, Ar and Ca) relative to Hartree-Fock values. The KW MDP parameters obtained are found to be correlated, thereby revealing some degree of transferability in the empirically determined parameters. We have studied excited-state orbits of electron-ion pair to analyze the consequences of the MDP on the classical Coulomb catastrophe. From the ground-state energies, we find that the experimental first- and second-ionization energies are fairly well predicted. Finally, electron-ion scattering was examined by comparing the predicted momentum transfer cross section to a semi-classical phase-shift calculation; optimizing the MDP parameters for the scattering process yielded rather poor results, suggesting a limitation of the use of the KW MDPs for plasmas. Efficient force calculation methods are needed for molecular dynamics simulation with medium-range interactions. Such interactions occur in a wide range of systems, including charged-particle systems with varying screening lengths. We generalize the Ewald method to charged systems described by interactions involving an arbitrary dielectric response function. We provide an error estimate and optimize the generalization to find the break-even parameters that separate a neighbor list-only algorithm from the particle-particle particle-mesh (PPPM) algorithm. We examine the implications of different choices of the screening length for the computational cost of computing the dynamic structure factor. We then use our new method in molecular dynamics simulations to compute the dynamic structure factor for a model plasma system and examine the wave-dispersion properties of this system. Laboratory-scale non-ideal plasmas with controllable properties over a wide range of densities below solid density are needed for understanding large-scale plasma experiments. Based on a suite of molecular dynamics simulations, we propose a general paradigm for producing such controllable non-ideal plasmas. We simulated the formation of non-equilibrium plasmas from photoionized, cool gases that are spatially precorrelated through neutral-neutral interactions that are important at moderate to high pressures. We examined the plasma-formation process over orders-of-magnitude variations in the initial gas pressure to characterize variations in several physical properties, including Coulomb collisional rates, partial pressures, screening strengths, continuum lowering, interspecies Coulomb coupling, electron degeneracy and ionization states. We find that variations in the initial gas pressure lead to controllable variations in a wide range of plasma properties, including the equation of state, collisional processes, atomic processes and basic plasma properties (coupling, screening and degeneracy). This paradigm has significant advantages over solid-density experiments because the collisional, collective and recombination timescales are reduced by a factor of 3 to 10, potentially broadening the efficacy of diagnostics. The paradigm also has advantages over ultracold plasma experiments because the trapping and cooling phases are avoided.
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Title: System performance and performance enhancement relative to element position location errors for distributed linear antenna arrays
Creator: Adrian, Andrew
Date: 2014
Collection: Electronic Theses & Dissertations
Description: ABSTRACTSYSTEM PERFORMANCE AND PERFORMANCE ENHANCEMENTRELATIVE TO ELEMENT POSITION LOCATION ERRORS FOR DISTRIBUTED LINEAR ANTENNA ARRAYSByAndrew AdrianFor the most part, antenna phased arrays have traditionally been comprised of antenna elements that are very carefully and precisely placed in very periodic grid structures. Additionally, the relative positions of the elements to each other are typically mechanically fixed as best as possible. There is never an assumption the relative positions...
Show moreABSTRACTSYSTEM PERFORMANCE AND PERFORMANCE ENHANCEMENTRELATIVE TO ELEMENT POSITION LOCATION ERRORS FOR DISTRIBUTED LINEAR ANTENNA ARRAYSByAndrew AdrianFor the most part, antenna phased arrays have traditionally been comprised of antenna elements that are very carefully and precisely placed in very periodic grid structures. Additionally, the relative positions of the elements to each other are typically mechanically fixed as best as possible. There is never an assumption the relative positions of the elements are a function of time or some random behavior. In fact, every array design is typically analyzed for necessary element position tolerances in order to meet necessary performance requirements such as directivity, beamwidth, sidelobe level, and beam scanning capability.Consider an antenna array that is composed of several radiating elements, but the position of each of the elements is not rigidly, mechanically fixed like a traditional array. This is not to say that the element placement structure is ignored or irrelevant, but each element is not always in its relative, desired location. Relative element positioning would be analogous to a flock of birds in flight or a swarm of insects. They tend to maintain a near fixed position with the group, but not always. In the antenna array analog, it would be desirable to maintain a fixed formation, but due to other random processes, it is not always possible to maintain perfect formation. This type of antenna array is referred to as a distributed antenna array.A distributed antenna array's inability to maintain perfect formation causes degradations in the antenna factor pattern of the array. Directivity, beamwidth, sidelobe level and beam pointing error are all adversely affected by element relative position error. This impact is studied as a function of element relative position error for linear antenna arrays. The study is performed over several nominal array element spacings, from λ to λ, several sidelobe levels (20 to 50 dB) and across multiple array illumination tapers.Knowing the variation in performance, work is also performed to utilize a minimum variance array processing method to minimize the effects of the distributed array element mis-positioning. The extent of array factor performance enhancement is demonstrated for several linear distributed array designs where the input to the enhancement algorithm is only the element position information.
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Title: A robust stabilization methodology for time domain integral equations in electromagnetics
Creator: Pray, Andrew J.
Date: 2014
Collection: Electronic Theses & Dissertations
Description: Time domain integral equations (TDIEs) are an attractive framework from which to analyze electromagnetic scattering problems. Casting problems in the time domain enables study of systems with nonlinearities, characterization of transient behavior both at the early and late time, and broadband analysis within a single simulation. Integral equation frameworks have the advantages of restricting the computational domain to the scatterer surface (boundary integral equations) or volume (volume...
Show moreTime domain integral equations (TDIEs) are an attractive framework from which to analyze electromagnetic scattering problems. Casting problems in the time domain enables study of systems with nonlinearities, characterization of transient behavior both at the early and late time, and broadband analysis within a single simulation. Integral equation frameworks have the advantages of restricting the computational domain to the scatterer surface (boundary integral equations) or volume (volume integral equations), implicitly satisfying the radiation boundary condition, and being free of numerical dispersion error. Despite these advantages, TDIE solvers are not widely used by computational practitioners; principally because TDIE solutions are susceptible to late-time instability. While a plethora of stabilization schemes have been developed, particularly since the early 1980s, most of these schemes either do not guarantee stability, are difficult to implement, or are impractical for certain problems. The most promising methods seem to be the {\em space-time Galerkin} schemes. These are very challenging to implement as they require the accurate evaluation of 4-dimensional spatial integrals. The most successful recent approach to implementing these schemes has been to approximate a subset of these integrals, and evaluate the remaining integrals analytically. This approach describes the {\em quasi-exact integration} methods [Shanker et al. IEEE TAP 2009, Shi et al. IEEE TAP 2011]. The method of [Shanker et al. IEEE TAP 2009] approximates 2 of the 4 dimensions using numerical quadrature. The remaining integrals are evaluated analytically by determining shadow boundaries on the domain of integration. In [Shi et al. IEEE TAP 2011], only 1 dimension is approximated, but the procedure also relies on analytical integration between shadow boundaries. These two characteristics-the need to find shadow boundaries and develop analytical integration rules-prevent these methods from being extended to higher order tessellations of scattering surfaces. This is an important restriction as the use of curvilinear elements can greatly improve the accuracy of the geometric representation. The need for a method to accurately evaluate the spatial integrals involved in these formulations on higher order surface tessellations motivates this thesis.The major novelty of this thesis is a space-time separated expansion of the convolution with the retarded potential Green's function. This separation leads to integrands that are smooth over the entire domain of integration. This implies that integration can be accurately carried out via numerical quadrature (not analytically) and shadow boundaries do not need to be found, unlike in the quasi-exact integration methods. The numerical nature of the method allows it to trivially be implemented on higher order surface descriptions. In this thesis, we will detail the procedure of the separable expansion and investigate, both numerically and analytically, the error incurred in truncating the expansion to a given upper limit. We will validate the stability of the resulting scheme by (1) observing the late time behavior of solutions to scattering from a variety of objects and (2) deriving and implementing an eigenvalue analysis to demonstrate the absence of growing terms. Additionally, this thesis will detail the use of the separable expansion in tandem with the plane wave time domain algorithm to accelerate the solution. Also, we will present extension of the space-time separation to analysis of penetrable materials using the PMCHWT formulation. A prescription for integrating singular and near-singular kernels over curved elements will also be given. The final contribution of this thesis is the application of the space-time separation to the generalized method of moments (GMM) with smooth surface parameterization.
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