Browsing by Author "Dr. Zhilin Li, Committee Member"

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Characteristic Methods for Solving the Particle Transport Equation in 1-D Spherical Geometry
(2009-04-15) Fleming, John Thomas III; Dr. Paul J. Turinsky, Committee Member; Dr. Yousry Y. Azmy , Committee Member; Dr. Zhilin Li, Committee Member; Dr. Dmitriy Y. Anistratov, Committee Chair
A family of numerical methods for solving the particle transport equation in 1-D spherical geometry are developed using the method of characteristics. The development of these methods is driven by a desire to: (i) provide solutions to transport problems which cannot otherwise be determined using analytic techniques and (ii) provide comparative solutions to test methods developed for other curvilinear geometries. Problems that are of increasing importance to the transport community are those that contain subdomains which are considered optically thick and diffusive. These problems result in high computational costs due to the grid refinement necessary to generate acceptable solutions. As a result, we look to develop vertex-based characteristic methods that can reproduce these diffusive solutions without resorting to significant spatial grid refinement. This research will allow for continued development of advanced conservative characteristic methods with better properties for R-Z geometries. The transport methods derived here are based on a change of coordinates that removes the angular derivative term in the differential operator resulting in a first order differential equation which can be discretized using methods similar to those found in 1-D slab geometry. In this study, we present a family of characteristic methods; Vladimirov's method of characteristics, a conservative long characteristic method, two locally conservative short characteristic methods, a linear long characteristic method, and an explicit slope long characteristic method. The numerical results presented in this thesis demonstrate the performance of each method. We found that the linear and explicit slope long characteristic methods generated numerical solutions which are well behaved in some diffusive problems. Also, we analyzed several of these methods using asymptotic diffusion limit analysis and found that the linear long characteristic method limits to a discretized version of the diffusion equation.
Finite-Difference Time-Domain Methods for Electromagnetic Problems Involving Biological Bodies
(2006-03-13) Schmidt, Stefan; Dr. Zhilin Li, Committee Member; Dr. Gregory T. Byrd, Committee Member; Dr. Gianluca Lazzi, Committee Chair; Dr. Brian L. Hughes, Committee Member
As more applications of wireless devices in the personal space are emerging, the analysis of interactions between electromagnetic energy and the human body will become increasingly important. Due to the risk of adverse health effects caused by the use of wireless devices adjacent to or implanted into the human body, it is important to minimize their electromagnetic interaction with biological objects. Efficient numerical methods may play an integral role in the design and analysis of wireless telemetry in implanted biomedical devices, as well as computation and minimization of the specific absorption rate (SAR) associated with wireless devices as an alternative to repetitive design prototyping and measurements. Research presented in this dissertation addresses the need to develop efficient numerical methods for the solution of such bio-electromagnetic problems. Bio-electromagnetic problems involving inhomogeneous dispersive media are traditionally solved using the Finite-Difference Time-Domain (FDTD) method. In this class of problems, the spatial discretization is often dominated by very fine geometric details rather than the smallest wavelength of interest. For an explicit FDTD scheme, these fine details dictate a small time-step due to the Courant-Friedrichs-Lewy (CFL) stability bound, which in turn leads to a large number of computational steps. In this dissertation, numerical methods are considered that overcome the CFL stability bound for particular bio-electromagnetic problems. One such method is to incorporate a thin wire sub-cell model into the explicit FDTD method for the computation of inductive coupling. The sub-cell model allows the use of larger FDTD cells, hence relaxing the CFL stability bound. A novel stability bound for the method is derived. Furthermore, an extension to the Thin-Strut FDTD method is proposed for the modeling of thin wire elements in lossy dielectric materials. Numerical results obtained by the Thin-Strut FTDT method were compared with measurements. Furthermore, the Partial Inductance Method (PIM) was implemented using arbitrarily oriented cylindrical wire elements to obtain an analytical approximation of inductive coupling and to verify the Thin-Strut FDTD method. The PIM method was also shown to be a very efficient tool for the approximation of free-space or low frequency inductive coupling problems for biomedical applications. The Alternating-Direction-Implicit (ADI) FDTD method is another method considered in this dissertation. Due to its unconditional stability, the ADI FDTD method alleviates the CFL stability bound. The objective is to apply the ADI method to the simulation of bio-electromagnetic problems and the computation of the SAR. For large time-steps, the ADI method has larger dispersion and phase errors than the explicit FDTD method, but it is still useful for the computation of SAR where those errors are tolerable. An improved anisotropic-material Perfectly-Matched-Layer (PML) Absorbing-Boundary-Condition(ABC) is presented for the ADI FDTD method. The material independent D-H-field formulation of the PML ABC leads to an efficient and simple implementation and allows the truncation of dispersive material models. Furthermore, this formulation is easily extended to n[superscript th]-order dispersive materials. Numerical results for reflection errors associated with the PML and their dependence on parameters like PML conductivity and time-step size are investigated. Furthermore, uniform and expanding grid implementations of the ADI FDTD method are used to compute the Specific Absorption Rate (SAR) distribution inside spherical objects representative of bio-electromagnetic problem. Different grid implementation sizes are considered, and errors associated with the ADI FDTD method are investigated by comparing numerical results to those obtained using the explicit FDTD method.
Higher-Order Cartesian Grid Based Finite Difference Methods for Elliptic Equations on Irregular Domains and Interface Problems and their Applications
(2004-05-13) Kyei, Yaw; Dr. Kazufumi Ito, Committee Chair; Dr. Zhilin Li, Committee Member; Dr. Ralph Smith, Committee Member; Dr Hien T. Tran, Committee Member
This thesis describes higher-order finite difference methods for solving elliptic equations on irregular domains with general boundary conditions and the corresponding elliptic interface problems. We develop second and fourth order methods for two and three dimensions using uniform Cartesian grids. However, with an irregular domain we cannot apply the standard finite difference schemes directly at the grid points near the boundary and therefore some treatment is required in order to use the uniform Cartesian grids. Our approach involves modifying the standard finite difference schemes. In particular, we use the standard five-point and standard compact nine-point stencil schemes for the second and fourth order methods, respectively. That is, on the standard stencils that contain the boundary, we carry out the modification by applying the continuation of solution from the inside of domain to the outside. The method of continuation of solution uses Taylor series expansion of the solution about selected boundary points, the equation and the boundary values of the local and their nearby boundary points. Naturally, second and fourth order Taylor series expansions about the boundary points are used for the second and fourth order methods respectively. Our methods have a unified and an effective approach to deal with general boundary conditions and capture the boundary and its local geometrical properties by the level set function and the local coordinate system at the boundary points. The resulting finite difference system matrices of our methods remain symmetric positive definite and maintain the sparsity of the standard finite difference schemes. As part of our main objective, we apply our fourth order method to semi-discretize the corresponding parabolic equation in space on the irregular domain and obtain an ODE system. The validity and effectiveness of the proposed method is clearly demonstrated through the computation of eigenvalues of the associated eigenvalue problem for the circular domain and other irregular domains. As one of the essential applications of our method, we design a state feedback controller for the Dirichlet boundary control problem of the heat equation. By the standard $LQR$ theory of the optimal state-feedback design, we solve the associated Riccati equation where the ODE system resulting from our fourth order method of semi-discretization serves as the state equation. Conventional second order methods have system matrices of higher dimensions which makes the Riccatti equation almost impractical to solve numerically. But with our nine-point compact fourth order methods, we capture the essential properties of the equation through our low order system matrices and demonstrate the capability of our approach through our computations. Another important contribution of the thesis is that we extend our fourth order method to develop a nine-point compact finite difference method for the variable elliptic equation and the corresponding interface problem. For the interface problem, the method is based on the continuation of solution procedure and in this case, across the material interface from one side to the other. Specifically we develop the procedure to define the scheme near the interface through the optimization technique to preserve the M-property of finite difference schemes.
Implantable Devices for a Retinal Prosthesis: Design and Electromagnetic and Thermal Effects
(2009-12-16) Singh, Vinit; Dr. Gianluca Lazzi, Committee Chair; Dr. Zhilin Li, Committee Member; Dr. Griff Bilbro, Committee Member; Dr. Mehmet Ozturk, Committee Member
A retinal prosthesis, wherein electrical stimulation is provided to the retina of a person inflicted with outer retinal degenerative diseases such as Retinitis Pigmentosa and Age-related Macular degeneration, has been clinically tested and has succeeded in providing limited vision; such as shape recognition. It is hoped that an increased electrode count will improve visual acuity. The retinal prosthesis considered in this work is a dual-unit system with an external (outside the human body) unit and an internal (inside the body) unit with a wireless link for power and data transfer between them. Such a system poses possible health risks due to the incident electromagnetic energy of the wireless link and the power dissipated by the internal components, particularly the processing chip which drives the electrodes responsible for eliciting a neural response from the retina. Tissue damage via heating is one the primary concerns for such a system making it necessary to obtain via simulation and in-vivo and in-vitro experiments, accurate estimates of thermal elevation due to the operation of the such devices. In this work, numerical methods have been developed to compute temperature increases and electromagnetic effects due to the prosthesis components in anatomically correct human head models. The explicit and the Alternating-Direction Implicit (ADI) Finite-Difference Time-Domain (FDTD) have been used. Further, a hybrid explicit-ADI method was developed for the heat equation which provided simulation speedup of more than 10x over conventional methods for the models considered. FDTD methods were employed to compute the induced current densities and Specific Absorption Rate (SAR) in the human head due the inductive link comprising the primary coil (external) and a secondary coil (internal). Different orientations of the primary coil were considered in a frequency range of 1 MHz-20 MHz to provide guidelines for choosing eventual frequency and power parameters to conform to international safety standards. A novel displacement field excitation method was used for the spiral primary coil and verified with analytical results. In an effort to reduce the size of the internal unit and to allow integration of a patch antenna (for a separate data link), and the active devices on a single substrate, a 3-D trench inductor geometry was investigated. To enable patterning of structured surface, a custom experimental setup was designed and maintained to process a positive tone PEPR2400 electro-depositable photoresist.
Integration of Interconnect Models in a Circuit Simulator
(2003-01-22) Mohan, Ramya; Dr. Gianluca Lazzi, Committee Member; Dr. Michael B. Steer, Committee Chair; Dr. Zhilin Li, Committee Member
A novel approach is implemented for synthesizing equivalent circuits. The approach called the Foster's approach finds its application in the transient simulation of distributed structures. The implementation is analogous to that of a Voltage Controlled Current Source, as it is a natural way to handle Admittance Matrix. The two main features of this method are its guaranteed causality and good numerical stability. The method is tested by simulating a six port power/ground plane and comparing the results with measurements. Also, different analyses types are compared and conclusions are made.
Inverse Problems and Post Analysis Techniques for a Stenosis-Driven Acoustic Wave Propagation Model
(2008-06-18) Samuels, John Richard Jr; Dr. Hien Tran, Committee Member; Dr. Zhilin Li, Committee Member; Dr. Marie Davidian, Committee Member; Dr. H.T. Banks, Committee Chair
Liquid-encapsulated Czochralski Growth of Compound Semiconductor Crystals with Steady and Rotating Magnetic Fields
(2006-07-13) Yang, Mei; Dr. Nancy Ma, Committee Chair; Dr. Kevin Lyons, Committee Member; Dr. Zhilin Li, Committee Member; Dr. Tarek Echekki, Committee Member
Integrated circuits and optoelectronic devices are produced on surfaces of thin wafers sliced from a photonic or compound semiconductor crystal. The growth of compound semiconductor crystals is critically important because viable substrates which are compositionally uniform both within a wafer and from wafer to wafer are needed. A dopant is an element that is added to the melt during growth to give the semiconductor crystal specific electrical and/or optical properties. More and better compound semiconductor crystals are needed for advanced optoelectronic devices. This investigation is focused on developing mathematical and numerical models to understand transport phenomena during bulk growth of compound semiconductor crystals. Since molten semiconductors are electrical conductors, magnetic fields can be used to control the melt motion in order to control the crystal's dopant distribution. Compound semiconductor crystals can be grown from the melt by the liquid-encapsulated Czochralski (LEC) process with a steady magnetic field. During this process, the molten semiconductor (melt) is covered with a layer of liquid encapsulant in order to prevent the escape of the volatile component. In this dissertation, we treat several different problems. We investigate the coupling of free convections in the melt and liquid encapsulant in a rectangular enclosure with steady vertical and horizontal magnetic fields, and find that these flows are coupled and the competition between these flows determines the direction of the horizontal velocity of the encapsulant-melt interface. We also investigate the dopant transport during the LEC process with a steady axial magnetic field, and find that both the radial and axial homogeneity of the crystal improves as the magnetic field strength decreases. With magnetic stabilization alone, however, the radially-inward flow below the crystal-melt interface does not become large enough to produce acceptable levels of segregation. A transverse magnetic field which rotates around the centerline of the melt can provide an electromagnetic stirring of the melt, and may represent a promising means to produce a crystal with good homogeneity. We investigate LEC growth with a combination of steady and rotating magnetic fields, and find that a rotating field can increase the magnitude of the radially-inward flow below the crystal-melt interface.
A Model-Based Closure Approach for Turbulent Combustion using the One-Dimensional Turbulence Model
(2007-03-21) Ranganath, Bhargav Bindiganavile; Dr. Tarek Echekki, Committee Chair; Dr. William Roberts, Committee Member; Dr. Kevin Lyons, Committee Member; Dr. Zhilin Li, Committee Member
A new model-based closure approach for turbulent combustion using the One-Dimensional turbulence model (ODT) is developed and validated in context to a turbulent jet diffusion flame. The interaction of turbulence and chemistry provides interesting finite rate chemistry effects including the phenomena of extinction and re-ignition. The ODT model resolves both spatially and temporally all the scales in a turbulent reaction flow problem, thus, combining the accuracy of a DNS like solver with efficiency by reduction in the number of dimensions. The closure approach is based on identifying the mechanisms responsible for the above mentioned effects and parameterizing the ODT results with a minimum set of scalars transported in the coarse grained solvers like the Reynolds-Averaged Navier-Stokes (RANS) or Large Eddy Simulation (LES). Thus, the closure from ODT is based on a "one-way" coupling between the coarse grained solvers and ODT. Two approaches for closure are developed in the present work with respect to a RANS solver; however, they can be easily extended to LES. The first approach relies on ODT to provide the history effects associated with the geometry, which represent the interactions of turbulence and chemistry, by tabulating scalar statistics (first and second moments) on two parameters measuring, the extent of mixing, the radial mean mixture fraction, and the extent of entrainment, the centerline mean mixture fraction. However, based on the above parameterization, the approach is limited to jet diffusion flame geometry. Furthermore, the closure requires a one to one correspondence between the flames simulated in the coarse grained solver and ODT. As a second approach, the results from ODT are parameterized based on general representative scalars; mixture fraction, which specifies the mixedness of the mixture and temperature, which specifies the reactedness of the mixture. The history effects associated with the flow geometry are provided by the RANS solver in the form of probability distribution functions (PDFs). Two classes of turbulent jet diffusion flames; hydrogen⁄air (Flame H3) and piloted methane/air (Sandia flames D and F), are considered for validation of the above ODT-based closure approaches. The piloted methane air flames, owing to higher turbulence, exhibit severe extinction in the near field followed by re-ignition around the flame height. Good comparisons of the conditional statistics for temperature and reactive scalars with the experiments are obtained for both the flames. Good predictions of entrainment as well as mixing for both the flames, as seen in the comparisons of Favre averaged axial and radial profiles, are obtained. Furthermore, the correct trends of extinction and re-ignition are predicted successfully for the piloted methane/air flames. Thus, the results show the capability of ODT to address the closure needs for a turbulent combustion problem both at molecular length scales (conditional profiles) and integral length scales (Favre averaged axial and radial profile) successfully. Refinements in terms correct representation of the PDFs for the second closure approach can be recognized, whereas, a robust "two-way" coupling of RANS and ODT may yield good results.
Novel Compact Antennas for Biomedical Implants and Wireless Applications
(2005-08-09) Gosalia, Keyoor Chetan; Dr. Robert J. Trew, Committee Member; Dr. Zhilin Li, Committee Member; Dr. Gianluca Lazzi, Committee Chair; Dr. Brian Hughes, Committee Member
Novel design methodologies and implementation techniques for antennas with an extremely small form factor (kr < 1; k is the wavenumber and r is the radius of enclosing spherical volume) are presented. These size reduction techniques are applied to design antennas for two emerging fields: Short (or long) range wireless connectivity and human body implants (prosthetic devices). The first test bed describes compact microstrip patch antennas employing polarization diversity for optimizing the available channel bandwidth in conventional wireless communications. Extremely small antennas (for implantation in an eye ball) operating at microwave frequencies for a visual prosthesis are designed and implemented for the second test bed. The visual prosthesis under consideration is an implantable prosthetic device which attempts to restore partial vision in the blind (patients suffering from retinal degeneration) by artificial stimulation of the retinal cells. Mutually exclusive power and data transfer via a wireless link with the implanted device is proposed where inductive coil coupling transfers power at low frequencies while data communication is performed using extraocular and intraocular antennas at microwave frequencies. The microwave data telemetry link is characterized computationally (using Finite Difference Time Domain-FDTD) and experimentally with appropriately sized external and implanted antennas. It is observed that the head and eye tissues act as a form of dielectric lens and improve the coupling performance between the two antennas (with intraocular antenna embedded in the eye ball) as compared to coupling in free space. The data telemetry link is characterized with novel small microstrip patch and planar wire dipole as intraocular antennas. An electromagnetic and thermal analysis of the operation of such a visual prosthesis is performed. Electromagnetic power deposition in the head is evaluated in terms of Specific Absorption Rate (SAR). Temperature rise in the tissues is characterized by computationally discretizing and implementing the bio-heat equation in three dimensions in an anatomically accurate head model.
Skin Friction versus Fire Propagation
(2006-08-13) Gibson, LaTosha; Dr. William Roberts, Committee Member; Dr. Kevin Lyons, Committee Chair; Dr. Zhilin Li, Committee Member
In examination of skin friction versus fire propagation, two methods of solution were of interest: (1) the viscous solution of the incompressible stagnation point velocity flow and (2) the Amplification Theory. For stagnation point velocity flow, the velocity is assumed to be zero at the stagnation point for the viscous solution. The Amplification Theory, however, deduces that the velocity is characterized by vortexes at the stagnation point. Therefore it was hypothesized that turbulence intensity through the Amplification Theory would render higher values for skin friction. The accounting of flame stretch was believed to have a small effect on the value of skin friction since the stretched laminar burning velocity is a product of the laminar burning velocity, and the pressure and temperature risen by a small power. Because of the direct correlation between the wall heat flux at the stagnation point and shear stress, the associated analytical heat flux equation utilizing the stagnation velocity gradient as a function of turbulent intensity was believed to be in a closer approximation to empirical values than the heat flux associated with the viscous solution for the incompressible stagnation point flow. Overall, the values from the viscous solution of the stagnation point velocity reported lower values than values of the K-epsilon solution involving premixed combustion. However, factoring stretch decreased the skin friction within the stagnation region. The empirical heat flux formula was shown to be in closer proximity to experimental values than the semi-analytical and theoretical heat flux solution.