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Experimental computation with oscillatory integrals

Description: A previous study by one of the present authors, together with D. Borwein and I. Leonard [8], studied the asymptotic behavior of the p-norm of the sinc function: sinc(x) = (sin x)/x and along the way looked at closed forms for integer values of p. In this study we address these integrals with the tools of experimental mathematics, namely by computing their numerical values to high precision, both as a challenge in itself, and also in an attempt to recognize the numerical values as closed-form constants. With this approach, we are able to reproduce several of the results of [8] and to find new results, both numeric and analytic, that go beyond the previous study.
Date: June 26, 2009
Creator: Bailey, David H. & Borwein, Jonathan M.
Partner: UNT Libraries Government Documents Department

Magnetosonic Eigenmodes Near the Magnetic Field Well in a Spherical Torus

Description: The structure and spectrum of magnetosonic Alfven eigenmodes in spherical torus in the presence of magnetic field well are studied. Analytical solution for eigenmodes localized in the well is obtained and compared with the numerical one. The possibility of using the eigenmode spectrum measurements for reconstructing the magnetic field well, and, thus, central magnetic safety factor profile is discussed.
Date: July 10, 1998
Creator: Gorelenkova, M. V. & Gorelenkov, N. N.
Partner: UNT Libraries Government Documents Department

Tracer Mixing at Fracture Intersections

Description: Discrete network models are one of the approaches used to simulate a dissolved contaminant, which is usually represented as a tracer in modeling studies, in fractured rocks. The discrete models include large numbers of individual fractures within the network structure, with flow and transport described on the scale of an individual fracture. Numerical simulations for the mixing characteristics and transfer probabilities of a tracer through a fracture intersection are performed for this study. A random-walk, particle-tracking model is applied to simulate tracer transport in fracture intersections by moving particles through space using individual advective and diffusive steps. The simulation results are compared with existing numerical and analytical solutions for a continuous intersection over a wide range of Peclet numbers. This study attempts to characterize the relative concentration at the outflow branches for a continuous intersection with different flow fields. The simulation results demonstrate that the mixing characteristics at the fracture intersections are a function not only of the Peclet number but also of the flow field pattern.
Date: February 10, 2001
Creator: Li, Guomin
Partner: UNT Libraries Government Documents Department


Description: In the mid 80’s, a four-group/two-region, entirely analytical 1D nodal benchmark appeared. It was readily acknowledged that this special case was as far as one could go in terms of group number and still achieve an analytical solution. In this work, we show that by decomposing the solution to the multigroup diffusion equation into homogeneous and particular solutions, extension to any number of groups is a relatively straightforward exercise using the mathematics of linear algebra.
Date: September 1, 2008
Creator: Ganapol, B. D. & Nigg, D. W.
Partner: UNT Libraries Government Documents Department

Developments in realistic design for aperiodic Mo/Si multilayermirrors

Description: Aperiodic multilayers have been designed for various applications, using numeric algorithms and analytical solutions, for many years with varying levels of success. This work developed a more realistic model for simulating aperiodic Mo/Si multilayers to be used in these algorithms by including the formation of MoSi{sub 2}. Using a genetic computer code we were able to optimize a 45{sup o} multilayer for a large bandpass reflection multilayer that gave good agreement with the model.
Date: April 5, 2006
Creator: Aquila, A.L.; Salmassi, F.; Dollar, F.; Liu, Y. & Gullikson, E.M.
Partner: UNT Libraries Government Documents Department

A Semi-Analytical Solution for Large-Scale Injection-Induced PressurePerturbation and Leakage in a Laterally Bounded Aquifer-AquitardSystem

Description: A number of (semi-)analytical solutions are available to drawdown analysis and leakage estimation of shallow aquifer-aquitard systems. These solutions assume that the systems are laterally infinite. When a large-scale pumping from (or injection into) an aquifer-aquitard system of lower specific storativity occurs, induced pressure perturbation (or hydraulic head drawdown/rise) may reach the lateral boundary of the aquifer. We developed semi-analytical solutions to address the induced pressure perturbation and vertical leakage in a 'laterally bounded' system consisting of an aquifer and an overlying/underlying aquitard. A one-dimensional radial flow equation for the aquifer was coupled with a one-dimensional vertical flow equation for the aquitard, with a no-flow condition imposed on the outer radial boundary. Analytical solutions were obtained for (1) the Laplace-transform hydraulic head drawdown/rise in the aquifer and in the aquitard, (2) the Laplace-transform rate and volume of leakage through the aquifer-aquitard interface integrated up to an arbitrary radial distance, (3) the transformed total leakage rate and volume for the entire interface, and (4) the transformed horizontal flux at any radius. The total leakage rate and volume depend only on the hydrogeologic properties and thicknesses of the aquifer and aquitard, as well as the duration of pumping or injection. It was proven that the total leakage rate and volume are independent of the aquifer's radial extent and wellbore radius. The derived analytical solutions for bounded systems are the generalized solutions of infinite systems. Laplace-transform solutions were numerically inverted to obtain the hydraulic head drawdown/rise, leakage rate, leakage volume, and horizontal flux for given hydrogeologic and geometric conditions of the aquifer-aquitard system, as well as injection/pumping scenarios. Application to a large-scale injection-and-storage problem in a bounded system was demonstrated.
Date: July 15, 2008
Creator: Zhou, Quanlin; Birkholzer, Jens T. & Tsang, Chin-Fu
Partner: UNT Libraries Government Documents Department

An analytical solution for transient radial flow through unsaturated fractured porous media

Description: This paper presents analytical solutions for one-dimensional radial transient flow through horizontal, unsaturated fractured rock formation. In these solutions, unsaturated flow through fractured media is described by a linearized Richards' equation, while fracture-matrix interaction is handled using the dual-continuum concept. Although linearizing Richards' equation requires a specially correlated relationship between relative permeability and capillary pressure functions for both fractures and matrix, these specially formed relative permeability and capillary pressure functions are still physically meaningful. These analytical solutions can thus be used to describe the transient behavior of unsaturated flow in fractured media under the described model conditions. They can also be useful in verifying numerical simulation results, which, as demonstrated in this paper, are otherwise difficult to validate.
Date: February 13, 2004
Creator: Wu, Yu-Shu & Pan, Lehua
Partner: UNT Libraries Government Documents Department

Correspondence of the Gardner and van Genuchten/Mualem relativepermeability function parameters

Description: The Gardner and van Genuchten models of relativepermeability are widely used in analytical and numerical solutions toflow problems. However, the applicab ility of the Gardner model to realproblems is usually limited, because empirical relative permeability datato calibrate the model are not routinely available. In contrast, vanGenuchten parameters can be estimated using more routinely availablematric potential and saturation data. However, the van Genuchten model isnot amenable to analytical solutions. In this paper, we introducegeneralized conversion formulae that reconcile these two models. Ingeneral, we find that the Gardner parameter alpha G is related to the vanGenuchten parameters alpha vG and n by alpha G=alpha vG ~; 1:3 n. Thisconversion rule will allow direct recasting of Gardner-based analyticalsolutions in the van Genuchten parameter space. The validity of theproposed formulae was tested by comparing the predicted relativepermeability of various porous media with measured values.
Date: January 3, 2007
Creator: Ghezzehei, Teamrat A.; Kneafsey, Timothy J. & Su, Grace W.
Partner: UNT Libraries Government Documents Department

Fraction of space debris collisions that are catastrophic

Description: Analytic calculations estimate the fraction of catalog collisions that are catastrophic by a modification of collision rates. Most catalog collisions are catastrophic. Impactors of 60 kg or larger participate in about half of the catastrophic collisions. Analytic estimates give accurate values for catastrophic collisions, which are complicated numerically.
Date: August 1, 1996
Creator: Canavan, G. H.
Partner: UNT Libraries Government Documents Department

An Analytical Solution for Slug-Tracer Tests in FracturedReservoirs

Description: The transport of chemicals or heat in fractured reservoirs is strongly affected by the fracture-matrix interfacial area. In a vapor-dominated geothermal reservoir, this area can be estimated by inert gas tracer tests, where gas diffusion between the fracture and matrix causes the tracer breakthrough curve (BTC) to have a long tail determined by the interfacial area. For water-saturated conditions, recent studies suggest that sorbing solute tracers can also generate strong tails in BTCs that may allow a determination of the fracture-matrix interfacial area. To theoretically explore such a useful phenomenon, this paper develops an analytical solution for BTCs in slug-tracer tests in a water-saturated fractured reservoir. The solution shows that increased sorption should have the same effect on BTCs as an increase of the diffusion coefficient. The solution is useful for understanding transport mechanisms, verifying numerical codes, and for identifying appropriate chemicals as tracers for the characterization of fractured reservoirs.
Date: March 2, 2005
Creator: Shan, Chao & Pruess, Karsten
Partner: UNT Libraries Government Documents Department

An Analytical Model for Solute Transport in Unsaturated Flowthrough a Single Fracture and Porous Rock Matrix

Description: Exact analytical solutions are presented for solute transport in an unsaturated fracture and porous rock matrix. The problem includes advective transport in the fracture and rock matrix as well as advective and diffusive fracture-matrix exchange. Linear sorption in the fracture and matrix and radioactive decay are also treated. The solution is for steady, uniform transport velocities within the fracture and matrix, but allows for independent specification of each of the velocities. The problem is first solved in terms of the solute concentrations that result from an instantaneous point source. Superposition integrals are then used to derive the solute mass flux at a fixed downstream position from an instantaneous point source and for the solute concentrations that result from a continuous point source. Solutions are derived for cases with the solute source in the fracture and the solute source in the matrix. The analytical solutions are closed-form and are expressed in terms of algebraic functions, exponentials, and error functions. Comparisons between the analytical solutions and numerical simulations, as well as sensitivity studies, are presented. Increased sensitivity to cross-flow and solute source location is found for increasing Peclet number. The numerical solutions are found to compare well with the analytical solutions at lower Peclet numbers ,but show greater deviation at higher Peclet numbers.
Date: September 16, 2004
Creator: Houseworth, J.E.
Partner: UNT Libraries Government Documents Department

Domain evolution and polarization of continuously graded ferroelectric films

Description: A thermodynamic analysis of graded ferroelectric films demonstrates that in the equilibrium state the films are subdivided into a single-domain band and a polydomain band which consists of wedge-shape domains. Polarization under an external electrostatic field proceeds through an inter-band boundary movement due to growth or shrinkage of the wedge domains. It is shown how the domain structure and evolution are determined by the principal characteristics of the film: the distribution of the spontaneous polarization and dielectric constant. Graded films exhibit a sharp increase of polarization with the field for weak fields, with a drop of the dielectric constant when the field is increasing. A general approach to finding the dependence of the displacement and the wedge-domain shape on the field as well as analytical solutions for the p{sup 4} Landau-Devonshire and parabolic potentials are presented.
Date: January 1, 2008
Creator: Roytburd, A. & Roytburd, V.
Partner: UNT Libraries Government Documents Department

Nonlinear Accelerator with Transverse Motion Integrable in Normalized Polar Coordinates

Description: Several families of nonlinear accelerator lattices with integrable transverse motion were suggested recently. One of the requirements for the existence of two analytic invariants is a special longitudinal coordinate dependence of fields. This paper presents the particle motion analysis when a problem becomes integrable in the normalized polar coordinates. This case is distinguished from the others: it yields an exact analytical solution and has a uniform longitudinal coordinate dependence of the fields (since the corresponding nonlinear potential is invariant under the transformation from the Cartesian to the normalized coordinates). A number of interesting features are revealed: while the frequency of radial oscillations is independent of the amplitude, the spread of angular frequencies in a beam is absolute. A corresponding spread of frequencies of oscillations in the Cartesian coordinates is evaluated via the simulation of transverse Schottky noise.
Date: May 1, 2012
Creator: Nagaitsev, S.; /Fermilab; Kharkov, Y.; Morozov, I.A.; /Novosibirsk, IYF; Zolkin, T.V. et al.
Partner: UNT Libraries Government Documents Department

Fracture-Flow-Enhanced Solute Diffusion into Fractured Rock

Description: We propose a new conceptual model of fracture-flow-enhanced matrix diffusion, which correlates with fracture-flow velocity, i.e., matrix diffusion enhancement induced by rapid fluid flow within fractures. According to the boundary-layer or film theory, fracture flow enhanced matrix diffusion may dominate mass-transfer processes at fracture-matrix interfaces, because rapid flow along fractures results in large velocity and concentration gradients at and near fracture-matrix interfaces, enhancing matrix diffusion at matrix surfaces. In this paper, we present a new formulation of the conceptual model for enhanced fracture-matrix diffusion, and its implementation is discussed using existing analytical solutions and numerical models. In addition, we use the enhanced matrix diffusion concept to analyze laboratory experimental results from nonreactive and reactive tracer breakthrough tests, in an effort to validate the new conceptual model.
Date: December 15, 2007
Creator: Wu, Yu-Shu; Ye, Ming & Sudicky, E.A.
Partner: UNT Libraries Government Documents Department

Analytic solutions for seismic travel time and ray path geometry through simple velocity models.

Description: The geometry of ray paths through realistic Earth models can be extremely complex due to the vertical and lateral heterogeneity of the velocity distribution within the models. Calculation of high fidelity ray paths and travel times through these models generally involves sophisticated algorithms that require significant assumptions and approximations. To test such algorithms it is desirable to have available analytic solutions for the geometry and travel time of rays through simpler velocity distributions against which the more complex algorithms can be compared. Also, in situations where computational performance requirements prohibit implementation of full 3D algorithms, it may be necessary to accept the accuracy limitations of analytic solutions in order to compute solutions that satisfy those requirements. Analytic solutions are described for the geometry and travel time of infinite frequency rays through radially symmetric 1D Earth models characterized by an inner sphere where the velocity distribution is given by the function V (r) = A-Br{sup 2}, optionally surrounded by some number of spherical shells of constant velocity. The mathematical basis of the calculations is described, sample calculations are presented, and results are compared to the Taup Toolkit of Crotwell et al. (1999). These solutions are useful for evaluating the fidelity of sophisticated 3D travel time calculators and in situations where performance requirements preclude the use of more computationally intensive calculators. It should be noted that most of the solutions presented are only quasi-analytic. Exact, closed form equations are derived but computation of solutions to specific problems generally require application of numerical integration or root finding techniques, which, while approximations, can be calculated to very high accuracy. Tolerances are set in the numerical algorithms such that computed travel time accuracies are better than 1 microsecond.
Date: December 1, 2007
Creator: Ballard, Sanford
Partner: UNT Libraries Government Documents Department

Numerical Tests and Properties of Waves in Radiating Fluids

Description: We discuss the properties of an analytical solution for waves in radiating fluids, with a view towards its implementation as a quantitative test of radiation hydrodynamics codes. A homogeneous radiating fluid in local thermodynamic equilibrium is periodically driven at the boundary of a one-dimensional domain, and the solution describes the propagation of the waves thus excited. Two modes are excited for a given driving frequency, generally referred to as a radiative acoustic wave and a radiative diffusion wave. While the analytical solution is well known, several features are highlighted here that require care during its numerical implementation. We compare the solution in a wide range of parameter space to a numerical integration with a Lagrangian radiation hydrodynamics code. Our most significant observation is that flux-limited diffusion does not preserve causality for waves on a homogeneous background.
Date: September 3, 2009
Creator: Johnson, B M & Klein, R I
Partner: UNT Libraries Government Documents Department

A dual-porosity reservoir model with an improved coupling term

Description: A new dual-porosity model is developed for single-phase flow in fractured/porous media. As in the commonly-used approach, flow is assumed to take place through the fracture network, and between the fractures and matrix blocks. The matrix blocks are treated in a lumped-parameter manner, with a single average pressure used for each matrix block. However, instead of assuming that fracture/matrix flux is proportional to the difference between the fracture pressure and matrix pressure at each point, as in the Warren-Root model, a nonlinear equation is used which accurately models the flux at both early and late times. This flux equation is verified against analytical solutions for spherical blocks with prescribed pressure variations on their boundaries. This equation is then used as a source/sink term in the numerical simulator TOUGH. The modified code allows more accurate simulations than the conventional Warren-Root method, and with a large savings in computational time compared to methods which explicitly discretize the matrix blocks.
Date: January 1, 1992
Creator: Zimmerman, Robert W.; Chen, Gang & Bodvarsson, Gudmundur S.
Partner: UNT Libraries Government Documents Department

A hybrid method for computing forces on curved dislocations threading to free surfaces

Description: Dislocations threading to free surfaces present a challenge for numerical implementation of traction-free boundary conditions. The difficulty arises when canonical (singular) solutions of dislocation mechanics are used in combination with the Finite Element or Boundary Element (Green's function) methods. A new hybrid method is developed here in which the singular part and the non-singular (regular) part of the image stress are dealt with separately. A special analytical solution for a semi-infinite straight dislocation intersecting the surface of a half-space is used to account for the singular part of the image stress, while the remaining regular part of the image stress field is treated using the standard Finite Element Method. The numerical advantages of such regularization are demonstrated with examples.
Date: June 6, 2005
Creator: Tang, M; Cai, W; Xu, G & Bulatov, V V
Partner: UNT Libraries Government Documents Department

Esimation of field-scale thermal conductivities of unsaturatedrocks from in-situ temperature data

Description: A general approach is presented here which allows estimationof field-scale thermal properties of unsaturated rock using temperaturedata collected from in situ heater tests. The approach developed here isused to determine the thermal conductivities of the unsaturated host rockof the Drift Scale Test (DST) at Yucca Mountain, Nevada. The DST wasdesigned to obtain thermal, hydrological, mechanical, and chemical (THMC)data in the unsaturated fractured rock of Yucca Mountain. Sophisticatednumerical models have been developed to analyze these THMC data. However,though the objective of those models was to analyze "field-scale" (of theorder of tens-of-meters) THMC data, thermal conductivities measured from"laboratory-scale" core samples have been used as input parameters.While, in the absence of a better alternative, using laboratory-scalethermal conductivity values in field-scale models can be justified, suchapplications introduce uncertainties in the outcome of the models. Thetemperature data collected from the DST provides a unique opportunity toresolve some of these uncertainties. These temperature data can be usedto estimate the thermal conductivity of the DST host rock and, given thelarge volume of rock affected by heating at the DST, such an estimatewill be a more reliable effective thermal conductivity value for fieldscale application. In this paper, thus, temperature data from the DST areused to develop an estimate of the field-scale thermal conductivityvalues of the unsaturated host rock of the DST. An analytical solution isdeveloped for the temperature rise in the host rock of the DST; and usinga nonlinear fitting routine, a best-fit estimate of field-scale thermalconductivity for the DST host rock is obtained. Temperature data from theDST show evidence of two distinct thermal regimes: a zone below boiling(wet) and a zone above boiling (dry). Estimates of thermal conductivityfor both the wet and dry zones are obtained in this paper. Sensitivity ofthese estimates to the input heating power of the DST is alsoinvestigated in this paper. These estimated ...
Date: June 26, 2006
Creator: Mukhopadhyay, Sumit; Tsang, Yvonne W. & Birkholzer, Jens T.
Partner: UNT Libraries Government Documents Department

River meander modeling and confronting uncertainty.

Description: This study examines the meandering phenomenon as it occurs in media throughout terrestrial, glacial, atmospheric, and aquatic environments. Analysis of the minimum energy principle, along with theories of Coriolis forces (and random walks to explain the meandering phenomenon) found that these theories apply at different temporal and spatial scales. Coriolis forces might induce topological changes resulting in meandering planforms. The minimum energy principle might explain how these forces combine to limit the sinuosity to depth and width ratios that are common throughout various media. The study then compares the first order analytical solutions for flow field by Ikeda, et al. (1981) and Johannesson and Parker (1989b). Ikeda's et al. linear bank erosion model was implemented to predict the rate of bank erosion in which the bank erosion coefficient is treated as a stochastic variable that varies with physical properties of the bank (e.g., cohesiveness, stratigraphy, or vegetation density). The developed model was used to predict the evolution of meandering planforms. Then, the modeling results were analyzed and compared to the observed data. Since the migration of a meandering channel consists of downstream translation, lateral expansion, and downstream or upstream rotations several measures are formulated in order to determine which of the resulting planforms is closest to the experimental measured one. Results from the deterministic model highly depend on the calibrated erosion coefficient. Since field measurements are always limited, the stochastic model yielded more realistic predictions of meandering planform evolutions. Due to the random nature of bank erosion coefficient, the meandering planform evolution is a stochastic process that can only be accurately predicted by a stochastic model.
Date: May 1, 2011
Creator: Posner, Ari J. (University of Arizona Tucson, AZ)
Partner: UNT Libraries Government Documents Department

On Sub-linear Convergence for Linearly Degenerate Waves in Capturing Schemes

Description: A common attribute of capturing schemes used to find approximate solutions to the Euler equations is a sub-linear rate of convergence with respect to mesh resolution. Purely nonlinear jumps, such as shock waves produce a first-order convergence rate, but linearly degenerate discontinuous waves, where present, produce sub-linear convergence rates which eventually dominate the global rate of convergence. The classical explanation for this phenomenon investigates the behavior of the exact solution to the numerical method in combination with the finite error terms, often referred to as the modified equation. For a first-order method, the modified equation produces the hyperbolic evolution equation with second-order diffusive terms. In the frame of reference of the traveling wave, the solution of a discontinuous wave consists of a diffusive layer that grows with a rate of t{sup 1/2}, yielding a convergence rate of 1/2. Self-similar heuristics for higher order discretizations produce a growth rate for the layer thickness of {Delta}t{sup 1/(p+1)} which yields an estimate for the convergence rate as p/(p+1) where p is the order of the discretization. In this paper we show that this estimated convergence rate can be derived with greater rigor for both dissipative and dispersive forms of the discrete error. In particular, the form of the analytical solution for linear modified equations can be solved exactly. These estimates and forms for the error are confirmed in a variety of demonstrations ranging from simple linear waves to multidimensional solutions of the Euler equations.
Date: March 17, 2008
Creator: Banks, J W; Aslam, T & Rider, W J
Partner: UNT Libraries Government Documents Department