Results for “Number theory” — Page 26

NASA NTRS2007

Incorporation of Half-Cycle Theory Into Ko Aging Theory for Aerostructural Flight-Life Predictions

The half-cycle crack growth theory was incorporated into the Ko closed-form aging theory to improve accuracy in the predictions of operational flight life of failure-critical aerostructural components. A new crack growth computer program was written for reading the maximum and minimum loads of each half-cycle from the random loading spectra for crack growth calculations and generation of in-flight crack growth curves. The unified theories were then applied to calculate the number of flights (operational life) permitted for B-52B pylon hooks and Pegasus adapter pylon hooks to carry the Hyper-X launching vehicle that air launches the X-43 Hyper-X research vehicle. A crack growth curve for each hook was generated for visual observation of the crack growth behavior during the entire air-launching or captive flight. It was found that taxiing and the takeoff run induced a major portion of the total crack growth per flight. The operational life theory presented can be applied to estimate the service life of any failure-critical structural components.

Ko, William L.↗

DOE OSTI2025

Toward Establishing Uniqueness of Experimentally Determined Transference Numbers

The passage of current through a battery results in the development of concentration gradients in the electrolytic phase. For a fully characterized binary electrolyte, where the conductivity, salt diffusion coefficient, cation transference number, and the thermodynamic factor are known, concentration and potential gradients in the electrolytic phase can be modeled using Newman’s concentrated solution theory. We report two methods for measuring the transference number: the standard method based on electrochemical measurements ( t + , echem 0 ) and electrophoretic NMR ( t + , eNMR 0 ). The electrochemical approach requires combining measurements from multiple experiments; the equations used to determine the cation transference number and the thermodynamic factor are coupled, nonlinear algebraic equations. In the electrophoretic-NMR-based approach, however, the equations used to determine the cation transference number and the thermodynamic factor are decoupled. We find for a liquid electrolyte comprised of a lithium salt dissolved in tetraglyme, the values of the transference numbers obtained by these two methods are distinct. For example, at 30 °C, t + , echem 0 = −1.02 ± 1.11 and t + , eNMR 0 = 0.25 ± 0.04. The corresponding thermodynamic factors are also different. While the magnitude of the predicted concentration gradients based on the two sets of parameters are different, the predicted current-voltage relationships are similar.

Hickson, Darby T. (ORCID:0000000251339755)↗

NASA NTRS2001

Studies of Fundamental Particle Dynamics in Microgravity

This work summarizes theoretical and experimental concepts used to design the flight experiment mission for SHIVA - Spaceflight Holography Investigation in a Virtual Apparatus. SHIVA is a NASA project that exploits a unique, holography-based, diagnostics tool to understand the behavior of small particles subjected to transient accelerations. The flight experiments are designed for testing model equations, measuring g, g-jitter, and other microgravity phenomena. Data collection will also include experiments lying outside of the realm of existing theory. The regime under scrutiny is the low Reynolds number, Stokes regime or creeping flow, which covers particles and bubbles moving at very low velocity. The equations describing this important regime have been under development and investigation for over 100 years and yet a complete analytical solution of the general equation had remained elusive yielding only approximations and numerical solutions. In the course of the ongoing NASA NRA, the first analytical solution of the general equation was produced by members of the investigator team using the mathematics of fractional derivatives. This opened the way to an even more insightful and important investigation of the phenomena in microgravity. Recent results include interacting particles, particle-wall interactions, bubbles, and Reynolds numbers larger than unity. The Space Station provides an ideal environment for SHIVA. Limited ground experiments have already confirmed some aspects of the theory. In general the space environment is required for the overall experiment, especially for cases containing very heavy particles, very light particles, bubbles, collections of particles and for characterization of the space environment and its effect on particle experiments. Lightweight particles and bubbles typically rise too fast in a gravitational field and heavy particles sink too fast. In a microgravity environment, heavy and light particles can be studied side-by-side for long periods of time.

Rangel, Roger↗

DOE OSTI2021

Strong coupling expansion of free energy and BPS Wilson loop in $$ \mathcal{N} $$ = 2 superconformal models with fundamental hypermultiplets

As a continuation of the study (in arXiv:2102.07696 and arXiv:2104.12625 ) of strong-coupling expansion of non-planar corrections in $ \mathcal{N} $ = 2 4d superconformal models we consider two special theories with gauge groups SU( N ) and Sp(2 N ). They contain N -independent numbers of hypermultiplets in rank 2 antisymmetric and fundamental representations and are planar-equivalent to the corresponding $ \mathcal{N} $ = 4 SYM theories. These $ \mathcal{N} $ = 2 theories can be realised on a system of N D3-branes with a finite number of D7-branes and O7-plane; the dual string theories should be particular orientifolds of AdS 5 × S 5 superstring. Starting with the localization matrix model representation for the $ \mathcal{N} $ = 2 partition function on S 4 we find exact differential relations between the 1 /N terms in the corresponding free energy F and the $ \frac{1}{2} $ -BPS Wilson loop expectation value $ \left\langle \mathcal{W}\right\rangle $ and also compute their large ’t Hooft coupling ( λ » 1) expansions. The structure of these expansions is different from the previously studied models without fundamental hypermultiplets. In the more tractable Sp(2 N ) case we find an exact resummed expression for the leading strong coupling terms at each order in the 1 /N expansion. We also determine the exponentially suppressed at large λ contributions to the non-planar corrections to F and $ \left\langle \mathcal{W}\right\rangle $ and comment on their resurgence properties. We discuss dual string theory interpretation of these strong coupling expansions.

1/N expansion↗

NASA NTRS1977

Assessment of existing analytic methods for prediction of high angle-of-attack loads on delta wings at supersonic speeds

An assessment of the applicability of four loading prediction methods to high angle-of-attack conditions for simplified wing-body configurations is provided. The methods are: The tangent wedge approximation, the linear theory methods of Middleton and Woodward, and a shock-fitting finite-difference technique. Estimates obtained by these methods were compared with experimental pressure data on delta wings to examine the effects of Mach number, camber, sweep angle, and angle of attack. Results indicate that all of the methods provided reasonable estimates at moderate angles of attack. At these moderate angles of attack, the methods of Middleton and Woodward provided good estimates at Mach numbers higher than those usually associated with linear theory. Only the finite-difference method provided reasonable load estimates at high angles of attack.

Landrum, E. J.↗

NASA NTRS1996

Generation of Squeezed Light Using Photorefractive Degenerate Two-Wave Mixing

We present a quantum nonlinear model of two-wave mixing in a lossless photorefractive medium. A set of equations describing the quantum nonlinear coupling for the field operators is obtained. It is found that, to the second power term, the commutation relationship is maintained. The expectation values for the photon number concur with those of the classical electromagnetic theory when the initial intensities of the two beams are strong. We also calculate the quantum fluctuations of the two beams initially in the coherent state. With an appropriate choice of phase, quadrature squeezing or number state squeezing can be produced.

Lu, Yajun↗

NASA NTRS1988

Thermal properties of heterogeneous grains

Cometary dust is not spherical nor homogeneous, yet these are the assumptions used to model its thermal, optical, and dynamical properties. To better understand the effects of heterogeneity on the thermal and optical properties of dust grains, the effective dielectric constant for an admixture of magnetite and a silicate were calculated using two different effective medium theories: the Maxwell-Garnett theory and the Bruggeman theory. In concept, the MG theory describes the effective dielectric constant of a matrix material into which is embedded a large number of very small inclusions of a second material. The Bruggeman theory describes the dielectric constant of a well mixed aggregate of two or more types of materials. Both theories assume that the individual particles are much smaller than the wavelength of the incident radiation. The refractivity for a heterogeneous grain using the MG theory is very similar to the refractivity of the matrix material, even for large volume fractions of the inclusion. The equilibrium grain temperature for spherical particles sized from .001 to 100 microns in radius at 1 astronomical unit from the sun was calculated. Further explanation is given.

Lien, David J.↗

DOE OSTI2026

Extended JT supergravity and random matrix models: The power of the string equation

A number of supersymmetric Jackiw-Teitelboim (JT) gravity theories are known to be described (in the Euclidean path integral formulation) by double-scaled random matrix models. Such matrix models can be characterized using a certain “string equation”. It was shown recently that in extended supergravity, when the number of BPS states scales as e$^{S_0}$, where $S_0$ is the extremal entropy, a special ansatz for the leading order solution of the string equation yields the supergravity spectrum. Somewhat miraculously, the construction showed that the functional form of the non-BPS (continuum) sector predicts the precise form of the BPS sector, showing the robustness of the supergravity/matrix-model correspondence. In this paper, we refine the analysis and show that the string equation, combined with some simple requirements on solutions, are powerful tools for constraining the spectrum of extended JT supergravity theories. We re-explore the cases of $\mathcal{N} = 2$ and (small) $\mathcal{N} = 4$ JT supergravity, and then explore the new cases of spectra from $\mathcal{N} = 3$ and $\mathcal{N} = 4$ large JT supergravity (recently derived by Heydeman, Shi, and Turiaci) showing that our approach also works naturally for (nearly) all the models. Based on this success, we conjecture that these new supergravity models also have matrix model descriptions.

Extended Supersymmetry↗

DOE OSTI2020

Dakota A Multilevel Parallel Object-Oriented Framework for Design Optimization Parameter Estimation Uncertainty Quantification and Sensitivity Analysis: Version 6.12 Theory Manual

The Dakota toolkit provides a flexible and extensible interface between simulation codes and iterative analysis methods. Dakota contains algorithms for optimization with gradient and nongradient-based methods; uncertainty quantification with sampling, reliability, and stochastic expansion methods; parameter estimation with nonlinear least squares methods; and sensitivity/variance analysis with design of experiments and parameter study methods. These capabilities may be used on their own or as components within advanced strategies such as surrogate-based optimization, mixed integer nonlinear programming, or optimization under uncertainty. By employing object-oriented design to implement abstractions of the key components required for iterative systems analyses, the Dakota toolkit provides a flexible and extensible problem-solving environment for design and performance analysis of computational models on high performance computers. This report serves as a theoretical manual for selected algorithms implemented within the Dakota software. It is not intended as a comprehensive theoretical treatment, since a number of existing texts cover general optimization theory, statistical analysis, and other introductory topics. Rather, this manual is intended to summarize a set of Dakota-related research publications in the areas of surrogate-based optimization, uncertainty quantification, and optimization under uncertainty that provide the foundation for many of Dakota's iterative analysis capabilities.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Dakota A Multilevel Parallel Object-Oriented Framework for Design Optimization Parameter Estimation Uncertainty Quantification and Sensitivity Analysis (V.6.14) (Theory Manual)

The Dakota toolkit provides a flexible and extensible interface between simulation codes and iterative analysis methods. Dakota contains algorithms for optimization with gradient and nongradient-based methods; uncertainty quantification with sampling, reliability, and stochastic expansion methods; parameter estimation with nonlinear least squares methods; and sensitivity/variance analysis with design of experiments and parameter study methods. These capabilities may be used on their own or as components within advanced strategies such as surrogate-based optimization, mixed integer nonlinear programming, or optimization under uncertainty. By employing object-oriented design to implement abstractions of the key components required for iterative systems analyses, the Dakota toolkit provides a flexible and extensible problem-solving environment for design and performance analysis of computational models on high performance computers. This report serves as a theoretical manual for selected algorithms implemented within the Dakota software. It is not intended as a comprehensive theoretical treatment, since a number of existing texts cover general optimization theory, statistical analysis, and other introductory topics. Rather, this manual is intended to summarize a set of Dakota-related research publications in the areas of surrogate-based optimization, uncertainty quantification, and optimization under uncertainty that provide the foundation for many of Dakota's iterative analysis capabilities.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Dakota, A Multilevel Parallel Object-Oriented Framework for Design Optimization, Parameter Estimation, Uncertainty Quantification, and Sensitivity Analysis: Version 6.13 Theory Manual

The Dakota toolkit provides a flexible and extensible interface between simulation codes and iterative analysis methods. Dakota contains algorithms for optimization with gradient and nongradient-based methods; uncertainty quantification with sampling, reliability, and stochastic expansion methods; parameter estimation with nonlinear least squares methods; and sensitivity/variance analysis with design of experiments and parameter study methods. These capabilities may be used on their own or as components within advanced strategies such as surrogate-based optimization, mixed integer nonlinear programming, or optimization under uncertainty. By employing object-oriented design to implement abstractions of the key components required for iterative systems analyses, the Dakota toolkit provides a flexible and extensible problem-solving environment for design and performance analysis of computational models on high performance computers. This report serves as a theoretical manual for selected algorithms implemented within the Dakota software. It is not intended as a comprehensive theoretical treatment, since a number of existing texts cover general optimization theory, statistical analysis, and other introductory topics. Rather, this manual is intended to summarize a set of Dakota-related research publications in the areas of surrogate-based optimization, uncertainty quantification, and optimization under uncertainty that provide the foundation for many of Dakota's iterative analysis capabilities.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Dakota, A Multilevel Parallel Object-Oriented Framework for Design Optimization, Parameter Estimation, Uncertainty Quantification, and Sensitivity Analysis: Version 6.16 Theory Manual

The Dakota toolkit provides a flexible and extensible interface between simulation codes and iterative analysis methods. Dakota contains algorithms for optimization with gradient and nongradient-based methods; uncertainty quantification with sampling, reliability, and stochastic expansion methods; parameter estimation with nonlinear least squares methods; and sensitivity/variance analysis with design of experiments and parameter study methods. These capabilities may be used on their own or as components within advanced strategies such as surrogate-based optimization, mixed integer nonlinear programming, or optimization under uncertainty. By employing object-oriented design to implement abstractions of the key components required for iterative systems analyses, the Dakota toolkit provides a flexible and extensible problem-solving environment for design and performance analysis of computational models on high performance computers. This report serves as a theoretical manual for selected algorithms implemented within the Dakota software. It is not intended as a comprehensive theoretical treatment, since a number of existing texts cover general optimization theory, statistical analysis, and other introductory topics. Rather, this manual is intended to summarize a set of Dakota-related research publications in the areas of surrogate-based optimization, uncertainty quantification, and optimization under uncertainty that provide the foundation for many of Dakota’s iterative analysis capabilities.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Dakota, A Multilevel Parallel Object-Oriented Framework for Design Optimization, Parameter Estimation, Uncertainty Quantification, and Sensitivity Analysis: Theory Manual (V.6.15)

The Dakota toolkit provides a flexible and extensible interface between simulation codes and iterative analysis methods. Dakota contains algorithms for optimization with gradient and nongradient-based methods; uncertainty quantification with sampling, reliability, and stochastic expansion methods; parameter estimation with nonlinear least squares methods; and sensitivity/variance analysis with design of experiments and parameter study methods. These capabilities may be used on their own or as components within advanced strategies such as surrogate-based optimization, mixed integer nonlinear programming, or optimization under uncertainty. By employing object-oriented design to implement abstractions of the key components required for iterative systems analyses, the Dakota toolkit provides a flexible and extensible problem-solving environment for design and performance analysis of computational models on high performance computers. This report serves as a theoretical manual for selected algorithms implemented within the Dakota software. It is not intended as a comprehensive theoretical treatment, since a number of existing texts cover general optimization theory, statistical analysis, and other introductory topics. Rather, this manual is intended to summarize a set of Dakota-related research publications in the areas of surrogate-based optimization, uncertainty quantification, and optimization under uncertainty that provide the foundation for many of Dakota's iterative analysis capabilities.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1949

Aerodynamic Study of a Wing-fuselage Combination Employing a Wing Swept Back 63 Degrees : Characteristics at a Mach Number of 1.53 Including Effect of Small Variations of Sweep

Measured values of lift, drag, and pitching moment at a Mach number of 1.53 and Reynolds numbers of 0.31, 0.62, and 0.84 million are presented for a wing-fuselage combination having a wing leading-edge sweep angle of 63 degrees, an aspect ratio of 3.42, a taper ratio of 0.25, and an NACA 64A006 section in the stream direction. Data are also presented for sweep angles of 57.0 degrees, 60.4 degrees, 67.0 degrees, and 69.9 degrees. The experimentally determined characteristics were less favorable than indicated by the linear theory but the experimental and theoretical trends with sweep were in good agreement. Boundary-layer-flow tests showed that laminar boundary-layer separation was the primary cause of the differences between experiment and theory.

REYNOLDS NUMBER EFFECTS - COMPLETE WING↗

NASA NTRS1973

Valence-bond study of the /H2, D2/ exchange reaction mechanism.

The exchange reaction of H2 with D2 to form 2 HD is important in that it is fundamentally the simplest four-body exchange reaction and should therefore represent a model system on which various theories of reactions dynamics might be tested. A number of theoretical and experimental investigations carried out on this system are reviewed. It is concluded that a Y yields T yields Y mechanism for the (H2, D2) exchange is not a low energy pathway that would make theory compatible with the shock-tube experiments of Bauer and Ossa (1966) and of Burcat and Lifshits (1967).

Freihaut, B.↗

NASA NTRS1986

The structure of separated flow regions occurring near the leading edge of airfoils including transition

A semi-empirical method for predicting separation bubble characteristics was evaluated using low Reynolds number test data. On the basis of this data, several observations were made. First, a sizable growth in the momentum thickness can occur in the laminar portion of a separation bubble. This is in direct contrast to the theory and is apparently due to low Reynolds number effects. Secondly, the transition Reynolds number (R sub l sub 1) which governs the extent of a bubble's laminar region, was found to be much lower than that used in the method. At present, there does not seem to be any evidence supporting a single value for R sub l sub 1. Apparently, R sub l sub 1 is affected by the freestream disturbance environment, and airfoil's pressure distribution, and possibly the chord Reynolds number as well. Thirdly, the growth in momentum thickness over a bubble's turbulent region was predicted reasonably well by the method, provided that Roberts' suggested value for the mean dissipation coefficient was used. Finally, the present data does not substantiate the universality of the velocity profile at reattachment. However, measurement error may be responsible for this result.

Mueller, T. J.↗

NASA NTRS1983

Development of a distributed-parameter mathematical model for simulation of cryogenic wind tunnels

A one-dimensional distributed-parameter dynamic model of a cryogenic wind tunnel was developed which accounts for internal and external heat transfer, viscous momentum losses, and slotted-test-section dynamics. Boundary conditions imposed by liquid-nitrogen injection, gas venting, and the tunnel fan were included. A time-dependent numerical solution to the resultant set of partial differential equations was obtained on a CDC CYBER 203 vector-processing digital computer at a usable computational rate. Preliminary computational studies were performed by using parameters of the Langley 0.3-Meter Transonic Cryogenic Tunnel. Studies were performed by using parameters from the National Transonic Facility (NTF). The NTF wind-tunnel model was used in the design of control loops for Mach number, total temperature, and total pressure and for determining interactions between the control loops. It was employed in the application of optimal linear-regulator theory and eigenvalue-placement techniques to develop Mach number control laws.

Tripp, J. S.↗

NASA NTRS1986

Convection without eddy viscosity: An attempt to model the interiors of giant planets

In the theory of hydrostatic quasi-geostrophic flow in the Earth's atmosphere the principal results do not depend on the eddy viscosity. This contrasts with published theories of convection in deep rotating fluid spheres, where the wavelength of the fastest growing disturbance varies as E sup 1/3, where E, the Ekman number, is proportional to the eddy viscosity. A new theory of quasi-columnar motions in stably stratified fluid spheres attempts to capture the luck of the meteorologists. The theory allows one to investigate the stability of barotropic and baroclinic zonal flows that extend into the planetary interior. It is hypothesized that the internal heat Jupiter and Saturn comes out not radially but on sloping surfaces defined by the internal entropy distribution. To test the hypothesis one searches for basic states in which the wavelength of the fastest-growing disturbance remains finite as E tends to zero, and is which the heat flux vector is radially outward and poleward.

Ingersoll, A. P.↗