Browsing by Subject "Numerical analysis"
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Item A numerical analysis of jet impingement cooling of a rotating disk(Texas Tech University, 1982-08) Hung, Ying TsengThe problem of jet impingement cooling of a rotating disk was investigated by numerical techniques. The integral approach was applied to the hydrodynamic and energy equations to obtain ordinary differential equations. These equations were solved numerically for the hydrodynamic and thermal boundary layer thicknesses. The thermal boundary layer thickness was used to calculate the average Nusselt number. The results were compared with experimental data to assess the validity of the method of solution. For the limiting case of a non-rotating disk this study predicted average Nusselt numbers that were good approximations of experimental results. For the case of a rotating disk the average Nusselt numbers predicted by this study were on the order of one half that of experimental results and followed the same trends as experimental results.Item An analysis of an antiplane shear crack in a nonhomogeneous elastic medium(Texas Tech University, 1988-08) Timmons, William ToddIn this thesis, a rigorous derivation of the energy release rate based on the change in potential energy of a body is given for a nonhomogeneous linear elastic medium. The energy release rate is calculated for an antiplane shear crack whose shear modulus corresponds to a reduced rigidity about the crack tip. Plastic zones about the crack tip are calculated based upon the yielding condition of von Mises, and the effect of decreasing rigidity upon these zones is displayed. In addition, the crack problem is analyzed within the framework of the strain energy density theory and the maximum cleavage stress theory.Item An extension of the Newton-Raphson method for non-linear simultaneous equations(Texas Tech University, 1972-08) McCoy, J. RobertNot availableItem Analysis of a novel thermoelectric generator in the built environment(2011-08) Lozano, Adolfo; Webber, Michael E., 1971-; Schmidt, Philip S.This study centered on a novel thermoelectric generator (TEG) integrated into the built environment. Designed by Watts Thermoelectric LLC, the TEG is essentially a novel assembly of thermoelectric modules whose required temperature differential is supplied by hot and cold streams of water flowing through the TEG. Per its recommended operating conditions, the TEG nominally generates 83 Watts of electrical power. In its default configuration in the built environment, solar-thermal energy serves as the TEG’s hot stream source and geothermal energy serves as its cold stream source. Two systems-level, thermodynamic analyses were performed, which were based on the TEG’s upcoming characterization testing, scheduled to occur later in 2011 in Detroit, Michigan. The first analysis considered the TEG coupled with a solar collector system. A numerical model of the coupled system was constructed in order to estimate the system’s annual energetic performance. It was determined numerically that over the course of a sample year, the solar collector system could deliver 39.73 megawatt-hours (MWh) of thermal energy to the TEG. The TEG converted that thermal energy into a net of 266.5 kilowatt-hours of electricity in that year. The second analysis focused on the TEG itself during operation with the purpose of providing a preliminary thermodynamic characterization of the TEG. Using experimental data, this analysis found the TEG’s operating efficiency to be 1.72%. Next, the annual emissions that would be avoided by implementing the zero-emission TEG were considered. The emission factor of Michigan’s electric grid, RFCM, was calculated to be 0.830 tons of carbon dioxide-equivalent (CO2e) per MWh, and with the TEG’s annual energy output, it was concluded that 0.221 tons CO2e would be avoided each year with the TEG. It is important to note that the TEG can be linearly scaled up by including additional modules. Thus, these benefits can be multiplied through the incorporation of more TEG units. Finally, the levelized cost of electricity (LCOE) of the TEG integrated into the built environment with the solar-thermal hot source and passive ground-based cold source was considered. The LCOE of the system was estimated to be approximately $8,404/MWh, which is substantially greater than current generation technologies. Note that this calculation was based on one particular configuration with a particular and narrow set of assumptions, and is not intended to be a general conclusion about TEG systems overall. It was concluded that while solar-thermal energy systems can sustain the TEG, they are capital-intensive and therefore not economically suitable for the TEG given the assumptions of this analysis. In the end, because of the large costs associated with the solar-thermal system, waste heat recovery is proposed as a potentially more cost-effective provider of the TEG’s hot stream source.Item Coupled flow systems, adjoint techniques and uncertainty quantification(2012-08) Garg, Vikram Vinod, 1985-; Carey, Graham F.; Prudhomme, Serge M.; Dawson, Clint N.; Gamba, Irene; Ghattas, Omar; Oden, J. Tinsley; Carey, VarisCoupled systems are ubiquitous in modern engineering and science. Such systems can encompass fluid dynamics, structural mechanics, chemical species transport and electrostatic effects among other components, all of which can be coupled in many different ways. In addition, such models are usually multiscale, making their numerical simulation challenging, and necessitating the use of adaptive modeling techniques. The multiscale, multiphysics models of electrosomotic flow (EOF) constitute a particularly challenging coupled flow system. A special feature of such models is that the coupling between the electric physics and hydrodynamics is via the boundary. Numerical simulations of coupled systems are typically targeted towards specific Quantities of Interest (QoIs). Adjoint-based approaches offer the possibility of QoI targeted adaptive mesh refinement and efficient parameter sensitivity analysis. The formulation of appropriate adjoint problems for EOF models is particularly challenging, due to the coupling of physics via the boundary as opposed to the interior of the domain. The well-posedness of the adjoint problem for such models is also non-trivial. One contribution of this dissertation is the derivation of an appropriate adjoint problem for slip EOF models, and the development of penalty-based, adjoint-consistent variational formulations of these models. We demonstrate the use of these formulations in the simulation of EOF flows in straight and T-shaped microchannels, in conjunction with goal-oriented mesh refinement and adjoint sensitivity analysis. Complex computational models may exhibit uncertain behavior due to various reasons, ranging from uncertainty in experimentally measured model parameters to imperfections in device geometry. The last decade has seen a growing interest in the field of Uncertainty Quantification (UQ), which seeks to determine the effect of input uncertainties on the system QoIs. Monte Carlo methods remain a popular computational approach for UQ due to their ease of use and "embarassingly parallel" nature. However, a major drawback of such methods is their slow convergence rate. The second contribution of this work is the introduction of a new Monte Carlo method which utilizes local sensitivity information to build accurate surrogate models. This new method, called the Local Sensitivity Derivative Enhanced Monte Carlo (LSDEMC) method can converge at a faster rate than plain Monte Carlo, especially for problems with a low to moderate number of uncertain parameters. Adjoint-based sensitivity analysis methods enable the computation of sensitivity derivatives at virtually no extra cost after the forward solve. Thus, the LSDEMC method, in conjuction with adjoint sensitivity derivative techniques can offer a robust and efficient alternative for UQ of complex systems. The efficiency of Monte Carlo methods can be further enhanced by using stratified sampling schemes such as Latin Hypercube Sampling (LHS). However, the non-incremental nature of LHS has been identified as one of the main obstacles in its application to certain classes of complex physical systems. Current incremental LHS strategies restrict the user to at least doubling the size of an existing LHS set to retain the convergence properties of LHS. The third contribution of this research is the development of a new Hierachical LHS algorithm, that creates designs which can be used to perform LHS studies in a more flexibly incremental setting, taking a step towards adaptive LHS methods.Item A direct numerical simulation of fully developed turbulent channel flow with spanwise wall oscillation(2005) Zhou, Dongmei; Ball, K. S.; Bogard, David G.Low-Reynolds-number, fully developed turbulent channel flow with wall motion has been simulated by direct numerical simulation to examine the effectiveness and the near-wall mechanics using spanwise wall oscillation to reduce friction drag. The three-dimensional unsteady Navier-Stokes (and energy) equations are solved using Fourier-Chebyshev-Tau spectral methods combined with a second-order semi-implicit time-advancement scheme. The effects of spatial resolution and computational box size on the computed turbulence and the drag reduction percentage were investigated. Finer spanwise resolution has a greater effect on achieving a better solution and the turbulent flow is well resolved for a spanwise grid spacing of Δ 3 <10 + x . It was also confirmed that the dynamics of turbulence in a natural full channel could be reproduced by a minimal channel. Parameter studies have been performed to examine the variation of drag reduction value with wall oscillation frequency, velocity amplitude, peak-to-peak amplitude, and oscillation orientation, and drag reduction data were discovered to correlate better with peak-to-peak amplitude for frequencies 01 > 0. + f in contrast to the previous finding of its correlation with peak-wall-speed. At the optimal wall oscillation conditions, net power savings of about 5% are obtained after the power input to move the wall is accounted for, even though more than 40% friction drag reduction has been achieved in the turbulence flow. Significant drag reduction is accompanied by the suppression of the turbulent bursting process and production of turbulence, and by a reduction in the intensity of streaks and streamwise vortices. A thickened viscous sublayer is indicated through the observed outward shift of statistical quantities such as velocity fluctuations and Reynolds shear stress in the moving-wall channel flow. Drag reduction by spanwise wall oscillation is mainly due to the suppression of ejection-sweep motions and the disruption in the cycle of the turbulence selfsustaining process, starting with the wall streaks that are distorted and reduced in number and extent. The intensity and the number of vortical structures are also reduced by the wall motion. The suppression of the regeneration of new streamwise vortices above the wall in turn further suppresses the ejection-sweep motions, thus leading to the reduced skin-friction levels at the wall.Item Dynamics of boundary-controlled convective reaction-diffusion equations(Texas Tech University, 1995-05) Okasha, Nahed AThis research is concerned with an initial value boundary problem for a class of convective reaction-diffusion equations for which a feedback control law is implemented through the boundary conditions. This class contains, as a special case the well-known Burgers' equation which has been studied rather extensively. Using methods based on Functional Analysis, in particular, the energy method and Galerkin Approximations, solvability for the above class is established. In addition, we prove the global in time existence and the regularity of solutions of the controlled problem for sufficiently small L^2-initial data. To do this, additional explicit restrictions on the nonlinear terms are imposed. Then we prove the local Lyapunov stability of the system, the existence of an absorbing ball, and the existence of a compact local attractor in this ball. Similar results for the same equation with Dirichlet boundary conditions are obtained for arbitrary L^2-initial data. The solutions of the boundary-controlled problem are shown to depend continuously on the boundary control parameters. As these parameters tend to infinity, we prove that the trajectories of the boundary-controlled problem converge, uniformly on any finite interval, to the trajectories of the corresponding problem with Dirichlet boundary conditions.Item Isogeometric analysis and numerical modeling of the fine scales within the variational multiscale method(2007-08) Cottrell, John Austin, 1980-; Hughes, Thomas J. R.This work discusses isogeometric analysis as a promising alternative to standard finite element analysis. Isogeometric analysis has emerged from the idea that the act of modeling a geometry exactly at the coarsest levels of discretization greatly simplifies the refinement process by obviating the need for a link to an external representation of that geometry. The NURBS based implementation of the method is described in detail with particular emphasis given to the numerous refinement possibilities, including the use of functions of higher-continuity and a new technique for local refinement. Examples are shown that highlight each of the major features of the technology: geometric flexibility, functions of high continuity, and local refinement. New numerical approaches are introduced for modeling the fine scales within the variational multiscale method. First, a general framework is presented for seeking solutions to differential equations in a way that approximates optimality in certain norms. More importantly, it makes possible for the first time the approximation of the fine-scale Green's functions arising in the formulation, leading to a better understanding of machinery of the variational multiscale method and opening new avenues for research in the field. Second, a simplified version of the approach, dubbed the "parameter-free variational multiscale method," is proposed that constitutes an efficient stabilized method, grounded in the variational multiscale framework, that is free of the ad hoc stabilization parameter selection that has plagued classical stabilized methods. Examples demonstrate the efficacy of the method for both linear and nonlinear equations.Item Majorization in the Bergman space(Texas Tech University, 1991-05) Richards, Kendall C.; Bernard, RogerWe will discuss some generalizations of the zero condition as well as investigate the extent to which we are able to extend Korenblum's method to the general Ap space. Next, we will approach the problem of improving c using the series representation of the Bergman norm. We obtain a domination result under an additional condition on the Taylor coefficients of the functions under consideration. Finally, some sharp values of the inner radius of the annulus of majorization are determined in the case that one of the functions in question is a monomial.Item Numerical solutions of an observability problem for the heat equation(Texas Tech University, 1988-12) Anglin, Quanna LeahIn this thesis we will consider numerical solutions to the discrete observability problem of the heat equation with periodic boundary conditions. The problem to be discussed is that of one-dimensional circular geometry, modeled by an insulated ring of wire. It is known that the discrete observability of the heat equation is preserved by two appropriately chosen spatial samples and an infinite set of discrete temporal samples. The main result of this thesis is a numerical examination of this result.Item On the growth of condition numbers in finite element calculations(Texas Tech University, 2003-08) Kulish, KandleIn this paper, we study the finite element method in one dimension using the classical basis functions and our new nonconforming basis functions. Comparison between the two methods is made via condition numbers. We show that our nonconforming basis functions yield slower growth when compared to the classical approach, giving us more confidence in our approximated solution.Item Roots of univariate polynomials via the euclidean algorithm(2012-05) Gonzalez, Elias; Weinberg, David A.; Lee, Jeffrey M.In this paper, we will consider univariate polynomials with real coefficients in degrees three, four, and five in great detail. Polynomial conditions will be found in terms of the coefficients of the univariate polynomial that will be able to describe the configuration of the roots. These conditions will be able to tell what the multiplicities of all the roots of the univariate polynomial are, as well as how many of these roots are real and how many are complex. In the scenario where the polynomial has real roots of different multiplicities, there will be conditions found that will describe the relative position of these real roots with respect to their multiplicities.Item Stability of dual discretization methods for partial differential equations(2011-05) Gillette, Andrew Kruse; Bajaj, Chandrajit; Demkowicz, Leszek; Gonzalez, Oscar; Luecke, John; Reid, Alan; Vick, JamesThis thesis studies the approximation of solutions to partial differential equations (PDEs) over domains discretized by the dual of a simplicial mesh. While `primal' methods associate degrees of freedom (DoFs) of the solution with specific geometrical entities of a simplicial mesh (simplex vertices, edges, faces, etc.), a `dual discretization method' associates DoFs with the geometric duals of these objects. In a tetrahedral mesh, for instance, a primal method might assign DoFs to edges of tetrahedra while a dual method for the same problem would assign DoFs to edges connecting circumcenters of adjacent tetrahedra. Dual discretization methods have been proposed for various specific PDE problems, especially in the context of electromagnetics, but have not been analyzed using the full toolkit of modern numerical analysis as is considered here. The recent and still-developing theories of finite element exterior calculus (FEEC) and discrete exterior calculus (DEC) are shown to be essential in understanding the feasibility of dual methods. These theories treat the solutions of continuous PDEs as differential forms which are then discretized as cochains (vectors of DoFs) over a mesh. While the language of DEC is ideal for describing dual methods in a straightforward fashion, the results of FEEC are required for proving convergence results. Our results about dual methods are focused on two types of stability associated with PDE solvers: discretization and numerical. Discretization stability analyzes the convergence of the approximate solution from the discrete method to the continuous solution of the PDE as the maximum size of a mesh element goes to zero. Numerical stability analyzes the potential roundoff errors accrued when computing an approximate solution. We show that dual methods can attain the same approximation power with regard to discretization stability as primal methods and may, in some circumstances, offer improved numerical stability properties. A lengthier exposition of the approach and a detailed description of our results is given in the first chapter of the thesis.Item Stability of the diamond difference approximation in energy to the Spencer-Lewis equation of electron transport(Texas Tech University, 1987-12) Seth, Daniel L.Consider the Spencer-Lewis equation (S-L) of electron transport in an azimuthally symmetric slab geometry setting with energy restricted to a finite interval. Further, S-L is subject to boundary conditions in the form of known incident particle fluxes at the slab faces. The one-dimensional diamond difference approximation is applied in the energy variable to the continuous slowing down term (i.e., the energy derivative) of S-L, This results in a semi-discrete system of integro-differential equations in the spatial and angular variables (D.E.S-L). The numerical stability of D,E.S-L as an approximation to S-L is demonstrated for solutions of D.E,S-L that belong to an L2 function space. The system of integro-differential equations may be rewritten as a system of integral equations. Under certain reasonable conditions on the data, the existence-uniqueness of solutions of the integral equations in a Banach space of square integrable functions with weighted norms is established. This implies the existence of L2 solutions of the integral equations. Under further assumptions on the data, the solutions of the integral equations are shown to be the required solutions of the integro-differential equations as well. Moreover, if the data are all bounded, the solutions are bounded.Item The inverse power method for multiplication factors in the neutron transport equation(Texas Tech University, 2001-05) Berry, Robb MNot availableItem Toward seamless multiscale computations(2013-05) Lee, Yoonsang, active 2013; Engquist, Björn, 1945-Efficient and robust numerical simulation of multiscale problems encountered in science and engineering is a formidable challenge. Full resolution of multiscale problems using direct numerical simulations requires enormous amounts of computational time and resources. This thesis develops seamless multiscale methods for ordinary and partial differential equations under the framework of the heterogeneous multiscale method (HMM). The first part of the thesis is devoted to the development of seamless multiscale integrators for ordinary differential equations. The first method, which we call backward-forward HMM (BFHMM), uses splitting and on-the-fly filtering techniques to capture slow variables of highly oscillatory problems without any a priori information. The second method, denoted by variable step size HMM (VSHMM), as the name implies, uses variable mesoscopic step sizes for the unperturbed equation, which gives computational efficiency and higher accuracy. VSHMM can be applied to dissipative problems as well as highly oscillatory problems, while BFHMM has some difficulties when applied to the dissipative case. The effect of variable time stepping is analyzed and the two methods are tested numerically. Multi-spatial problems and numerical methods are discussed in the second part. Seamless heterogeneous multiscale methods (SHMM) for partial differential equations, especially the parabolic case without scale separation are proposed. SHMM is developed first for the multiscale heat equation with a continuum of scales in the diffusion coefficient. This seamless method uses a hierarchy of local grids to capture effects from each scale and uses filtering in Fourier space to impose an artificial scale gap. SHMM is then applied to advection enhanced diffusion problems under incompressible turbulent velocity fields.Item Weakly non-local arbitrarily-shaped absorbing boundary conditions for acoustics and elastodynamics theory and numerical experiments(2004) Lee, Sanghoon; Kallivokas, Loukas F.