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Hall, Brenton TaylorUsing the Non-Uniform Dynamic Mode Decomposition to Reduce the Storage Required for PDE Simulations
Master of Mathematical Sciences, The Ohio State University, 2017, Mathematical Sciences
Partial Differential Equation simulations can produce large amounts of data that are very slow to transfer. There have been many model reduction techniques that have been proposed and utilized over the past three decades. Two popular techniques Proper Orthogonal Decomposition and Dynamic Mode Decomposition have some hindrances. Non-Uniform Dynamic Mode Decomposition (NU-DMD), which was introduced in 2015 by Gueniat et al., that overcomes some of these hindrances. In this thesis, the NU-DMD's mathematics are explained in detail, and three versions of the NU-DMD's algorithm are outlined. Furthermore, different numerical experiments were performed on the NU-DMD to ascertain its behavior with repect to errors, memory usage, and computational efficiency. It was shown that the NU-DMD could reduce an advection-diffusion simulation to 6.0075% of its original memory storage size. The NU-DMD was also applied to a computational fluid dynamics simulation of a NASA single-stage compressor rotor, which resulted in a reduced model of the simulation (using only three of the five simulation variables) that used only about 4.67% of the full simulation's storage with an overall average percent error of 8.90%. It was concluded that the NU-DMD, if used appropriately, could be used to possibly reduce a model that uses 400GB of memory to a model that uses as little as 18.67GB with less than 9% error. Further conclusions were made about how to best implement the NU-DMD.

Committee:

Ching-Shan Chou (Advisor); Jen-Ping Chen (Committee Member)

Subjects:

Aerospace Engineering; Applied Mathematics; Computer Science; Mathematics; Mechanical Engineering

Keywords:

Fluid Dynamics; Fluid Flow; Model Reduction; Partial Differential Equations; reducing memory; Dynamic Mode Decomposition; Decomposition; memory; Non-Uniform Dynamic Mode Decomposition

Ignatyev, OleksiyThe Compact Support Property for Hyperbolic SPDEs: Two Contrasting Equations
PHD, Kent State University, 2008, College of Arts and Sciences / Department of Mathematical Science

In this Dissertation we investigate the compact support property of the solutions of two hyperbolic stochastic partial differential equations (SPDEs) whose initial condition function is deterministic and compactly supported. First, we consider the hyperbolic semi-SPDE treated by Allouba and Goodman and others in Financial mathematics modelling. This is an SPDE in both time and space in which all the derivatives (including in the noise) are only with respect to the time parameter, and hence the name semi-SPDE. It turns out that, under appropriate conditions on the diffusion coefficient, the semi-SPDE preserves the compact support property. I.e., starting from a compactly supported initial solution u0(x), the solution u(t,x) is compactly supported in x for all times t>0.

Second, we consider a rotated wave SPDE in time-space considered by Allouba. Our approach here is to use the Allouba stochastic differential-difference equations (SDDE) approach. In this approach, we start by discretizing space, leaving time continuous, thereby obtaining a simpler version of the SPDE under question. We then resolve the question for the SDDE (or SPDE on the spatial lattice) and then use a limiting argument – as the mesh size of the spatial lattice goes to zero – to transfer regularity results to the associated SPDE. We also prove a noncompact support result for the SPDE. It turns out that in the rotated wave SPDE case, the compact support property is not preserved with positive probability. The contrast between the two SPDEs’ behaviors is due to the extra differentiation in space in the second SPDE which plays a crucial role in the behavior of solutions.

Committee:

Prof. Hassan Allouba (Committee Chair); Prof. Brett Ellman (Committee Member); Prof. Andrew Tonge (Committee Member); Prof. Deng-Ke Yang (Committee Member); Prof. Volodymyr Andriyevskyy (Committee Member)

Subjects:

Mathematics

Keywords:

Stochastic Partial Differential Equations; Compact Support Property

Manukian, Vahagn EmilExistence and stability of multi-pulses with applicatons to nonlinear optics
Doctor of Philosophy, The Ohio State University, 2005, Mathematics

In the present work we study the existence and stability of multi-pulses in dynamical systems that arise as traveling-wave equations for a partial differential equation (PDE) with symmetries. The motivation comes from two different models that describe the propagation of pulses in optical fibers.

In the first part of the thesis we consider reversible, ℤ2 symmetric dynamical systems with heteroclinic orbits related via symmetries. The heteroclinic orbits are assumed to undergo an orbit flip bifurcation upon changing appropriate parameters. We construct multi-bump solutions close to the heteroclinic orbits and investigate their PDE stability by using Lin's method and Lyapunov-Schmidt reduction. We apply this abstract theory to a model equation that describes the propagation of pulses in optical fibers with phase sensitive amplifiers. Our results show that stable multi-pulses exist. In the second part we consider parameter-dependent dynamical systems with reflection and SO(2) symmetry, which possess a homoclinic solution to a saddle focus. The reflection symmetry is broken by the second parameter which plays the role of the wave speed. We derive the bifurcation equations for the existence of N-pulse solutions and solve them for N=3. As a result we obtain standing and traveling 3-pulse solutions which we describe through the phase differences and the distances between consecutive bumps. We also investigate stability of these 3-pulses. We derive the stability matrix for multi-bump solutions and compute to the leading order the location of the eigenvalues for the 3-pulses.

Committee:

Bjorn Sandstede (Advisor)

Subjects:

Mathematics

Keywords:

partial differential equations; heteroclinic and homoclinic bifurcations; multi-pulses; stability

Kramer, EugeneNonhomogeneous Boundary Value Problems for the Korteweg-de Vries Equation on a Bounded Domain
PhD, University of Cincinnati, 2009, Arts and Sciences : Mathematical Sciences
The Korteweg-de Vries equation models unidirectional propagation of small finite amplitude long waves in a non-dispersive medium. The well-posedness, that is the existence, uniqueness of the solution, and continuous dependence on data, has been studied on unbounded,periodic, and bounded domains.

This research focuses on an initial and boundary value problem (IBVP) for the Korteweg-de Vries (KdV) equation posed on a bounded interval with general nonhomogeneous boundary conditions. Using Kato smoothing properties of an associated linear problem and the contraction mapping principle, the IBVP is shown to be locally well-posed given several conditions on the parameters for the boundary conditions, in the L²-based Sobolev space Hs(0, 1) for any s≥0.

Committee:

Bingyu Zhang, PhD (Committee Chair); H Dumas, PhD (Committee Member); Anthony Leung, PhD (Committee Member); Philip Korman, PhD (Committee Member)

Subjects:

Mathematics

Keywords:

Partial Differential Equations;Korteweg-de Vries;KdV equation;well-posedness

Seadler, Bradley T.Signed-Measure Valued Stochastic Partial Differential Equations with Applications in 2D Fluid Dynamics
Doctor of Philosophy, Case Western Reserve University, 2012, Mathematics
We note the interesting phenomenon that the Kantorovich-Rubinstein metric is not complete on the space of signed measures. Consequently, we introduce a new metric with a useful partial completeness property. With this metric, a general result about the Hahn-Jordan decomposition of solutions of stochastic partial differential equations is shown. These general results are applied to the smoothed Stochastic Navier-Stokes equations. As an application, we derive that the vorticity of the fluid is conserved for a solution of the Stochastic Navier-Stokes equations.

Committee:

Dr. Peter Kotelenez, PhD (Committee Chair); Dr. Elizabeth Meckes, PhD (Committee Member); Dr. Manfred Denker, PhD (Committee Member); Dr. Marshall Leitman, PhD (Committee Member)

Subjects:

Aerospace Engineering; Mathematics; Physics

Keywords:

Stochastic; Partial; Differential; Equations; Hahn; Jordan; Fluid; Dynamics; 2D

Brubaker, Lauren P.Completely Residual Based Code Verification
Master of Science, University of Akron, 2006, Applied Mathematics
Mathematical models of physical processes often include partial differential equations (PDEs). Oftentimes solving PDEs analytically is not feasible and a numerical method is implemented to obtain an approximate solution. Too often the assumption is made that the solution should be trusted when codes are prone to implementation errors. Code verification is a field of mathematics that shows the algorithm has been implemented without mistakes and has correctly solved the problem. Currently no one method of code verification is universally accepted. The Method of Manufactured Exact Solutions (MMES) is the most commonly used, but it has a considerable disadvantage of altering the code after verification. We have developed a new method, Completely Residual Based Code Verification (CRBCV), which does not require any modification. By using several solution methods, we have shown that CRBCV is dependable when verifying the heat equation with linear and nonlinear source terms and a frontal polymerization model.

Committee:

Laura Gross (Advisor)

Subjects:

Mathematics

Keywords:

Code verification; Partial Differential Equations; Numerical Methods; Method of Manufactured Exact Solutions; Frontal Polymerization; Heat Equation; Residual

Johnston, Joshua D.Analytically and Numerically Modeling Reservoir-Extended Porous Slider and Journal Bearings Incorporating Cavitation Effects
Doctor of Philosophy, University of Akron, 2011, Applied Mathematics

The technology of porous bearings is well-known in industry. In classical cases, the porous medium acts as an external reservoir making their use ideal for applications where an external lubricant supply is undesirable or impractical or when fluid has to be delivered on a continual basis. The work considered here looks to extend the benefits of typical porous bearings to allow for the bearing to be sealed, containing, from the onset of operation, all necessary lubricant.

The goal of this work is to demonstrate a bearing that circulates the fluid between a fluid film and an eccentric reservoir, using a porous medium as an intermediary; a system that is capable of supporting a realistic load, while simultaneously pumping the fluid back and forth between the lubricating region and the reservoir. The method used to investigate such a bearing is a mixture of analytical and numerical techniques. For the analysis, a non-dimensionalization scheme is used to analyze both the momentum and thermal governing equations at their differing orders of magnitude. Upon doing so, the governing momentum equations are reduced considerably which allows for a straight-forward numerical solution procedure. The governing thermal equations are solved using an asymptotic expansion approach, keeping the first and second order terms and equations. This is done so to more accurately model the effects the circulating fluid has on the thermal performance of the bearing.

The phenomenon of cavitation is also discussed, utilizing a method that integrates cavitation into the governing equations and numerical solution procedure. Unlike other cavitation models that decouple cavitation from the governing momentum equations, this model accounts for mass flow continuity which leads to more realistic results.

Practical design considerations, including how to determine the effective permeability and the effective heat transfer coefficient at the exterior wall of the bearing, are discussed. These parameters, used extensively in the analytical and numerical modeling of the bearing, are essentially functions of other physical parameters. Once these relationships are established, their values can be utilized by someone looking to design a bearing considered in this work with a set of performance criteria in mind.

The combination of analytical work and numerical computations produces a comprehensive look at this new type of bearing. A long slider bearing and both long and short journal bearings are discussed in a parametric fashion, whereby the effects of varying the operational and geometric parameters are investigated by examining the accompanying pressure and temperature fields. The model presented here demonstrates the feasibility of a bearing that is capable of supporting a load while eliminating the need for an external lubricant supply and the necessary infrastructure that is required to actively feed a bearing with lubricant. It is shown that the temperatures stay within operating limits utilizing a realistic heat transfer coefficient and a realistic thermal conductivity for the lubricant while generating pressures inside the film that can support a load and simultaneously pump fluid between the film and reservoir regions.

Committee:

Gerald Young, Dr. (Advisor); Minel Braun, Dr. (Advisor); Kevin Kreider, Dr. (Committee Member); Joseph Wilder, Dr. (Committee Member); Scott Sawyer, Dr. (Committee Member); Alex Povitsky, Dr. (Committee Member); S.I. Hariharan, Dr. (Committee Member)

Subjects:

Mathematics; Mechanical Engineering

Keywords:

bearing; cavitation; numerical analysis; partial differential equations; slider; journal; porous medium; heat transfer

Ingraham, DanielExternal Verification Analysis: A Code-Independent Approach to Verifying Unsteady Partial Differential Equation Solvers
Doctor of Philosophy, University of Toledo, 2015, Mechanical Engineering
External Verification Analysis (EVA), a new approach to verifying unsteady partial differential equation codes, is presented. After a review of the relevant code verification literature, the mathematical foundation and solution method of the EVA tool is discussed in detail. The implementation of the EVA tool itself is verified through an independent Python program. A procedure for code verification with the EVA tool is described and then applied to the three-dimensional form of a high-order non-linear computational aeroacoustics code.

Committee:

Ray Hixon (Advisor); Sorin Cioc (Committee Member); James DeBonis (Committee Member); Mehdi Pourazady (Committee Member); Chunhua Sheng (Committee Member)

Subjects:

Aerospace Engineering; Fluid Dynamics; Mechanical Engineering

Keywords:

code verification; computational aeroacoustics; computational fluid dynamics; numerical partial differential equations

Li, YinyunComputational Modeling of Slow Axonal Transport of Neurofilaments
Doctor of Philosophy (PhD), Ohio University, 2013, Physics and Astronomy (Arts and Sciences)
Neurons communicate with each other through dendrites and axons. Typically, dendrites are responsible for receiving signals from other neurons, while axons are the pathways to send out signals. Signal propagation through axons is closely correlated with their morphology. It is well known that the rate of signal propagation is proportional to the caliber of axons[2]. The intrinsic determinant of axonal caliber is the abundance of cytoskeletal protein, neurofilament (NF)[6]. NFs are not static but undergo "slow axonal transport", which is characterized by rapidly intermittent, asynchronous and bidirectional motion[21-23]. Many neurodegenerative diseases are related to the malfunction of neurofilament transport, either by accumulation of neurofilaments leading to swelling of the axon or by deficiency in neurofilaments resulting in axonal atrophy[9-12]. The mechanism of neurofilament transport can be explained by the "stop-and-go"; hypothesis[21, 24, 28], according to which neurofilaments spend long periods of time pausing interrupted by bouts of rapid movements. By the "stop-and-go" hypothesis, a compact and powerful mathematical model was proposed in [27], which connects the group behavior of neurofilaments as a wave to the individual neurofilament kinetics, which are observed directly from time-lapse imaging. Our main hypothesis is that axonal morphology is determined by the kinetics of NFs. According to this hypothesis, an increase in axonal caliber must go along with a decrease in speed of NFs and accordingly a modified kinetics. Two main examples, the distally increasing accumulation of NFs in the mouse optic nerve and the constrictions of myelinated axons at the nodes of Ranvier, demonstrate this hypothesis and support it with detailed experimental data. In the mouse optic nerve, sufficient data about the abundance of NFs proximal to distal as well as kinetic data are available to extract differential kinetics using our computational model. The most remarkable result is that NF accumulation is extremely sensitive to changes in kinetic rates. Using the same data, it was possible to exclude a conceptual model for slow axonal transport which was based on the hypothesis that NFs can deposit into a stationary cytoskeleton, settling a 20-year controversy in neurobiology. One of the biggest challenges is to extract the rate constants intrinsic to our model from experiments. Key for this analysis is the pulse-escape experiment. The kinetics of photo activatable Green Fluorescence Proteins (GFP)-labeled NFs is measured and simulated using our model. Tracking the departure of neurofilaments from an activation window, experimentally and computationally allows us to determine all rate constants except the reversal rates. With those tools available, the differential kinetics of the NF transport in myelinated axons was studied. The axon of a myelinated nerve exhibits segments where it is myelinated (internode) and not myelinated (node of Ranvier). At the node of Ranvier, the axon is significantly constricted in many cases and displays a sausage-like morphology. The goal of this study is to correlate NF content with differential kinetics and to determine how the rate of movement is differentially regulated at the nodes and internodes.

Committee:

Peter Jung (Advisor); David F. J. Tees (Committee Member); Markus Böttcher (Committee Member); Ralph DiCaprio (Committee Member)

Subjects:

Biophysics; Physics

Keywords:

slow axonal transport of neurofilaments; stop and go; 6-state model; partial differential equations; computational modeling; pulse-escape method; laplace transformation