Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “EXPONENTIAL FUNCTION”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 379 records · Page 21
All sourcesNASA NTRSDOE OSTI
NASA NTRS1992

ROSAT all-sky survey on the Einstein EMSS sample

The cosmological evolution and the luminosity function (XLF) of X ray selected Active Galactic Nuclei (AGN's) are discussed. The sample used is extracted from the Einstein Observatory Extended Medium Sensitivity Surveys (EMSS) and consists of more than 420 objects. Preliminary results from the ROSAT All-Sky Survey data confirm the correctness of the optical identification of the EMSS sources, thus giving confidence to the results obtained from the analysis of the AGN's sample. The XLF observed at different redshifts (up to z approx. 2) gives direct evidence of cosmological evolution. Data have been analyzed within the framework of luminosity evolution models and the two most common evolutionary forms, L sub x(Z) = L sub x(0) x e(sup Cr) and L sub x(Z) = L sub x(0) x (1 + z)(exp C), have been considered. Luminosity dependent evolution is required if the evolution function has the exponential form, whereas the simpler pure luminosity evolution model is still acceptable if the evolution function has the power law form. Using the whole sample of objects the number-counts and the de-evolved (z = 0) XLF have been derived. A comparison of the EMSS data with preliminary ROSAT results presented at this meeting indicates an overall agreement.

Maccacaro, Tomasso↗
NASA NTRS1992

The evolution of AGN's at x ray wavelengths

The cosmological evolution and the Luminosity Function (LF) of x-ray selected Active Galactic Nuclei (AGN) are discussed. The sample used is extracted from the Einstein Observatory Extended Medium Sensitivity Survey (EMSS) and consists of more than 420 objects. Preliminary results from the Rosat all sky survey data confirm the correctness of the optical identification of the EMSS sources, thus giving confidence to the results obtained from the analysis of the AGN sample. The XLF observed at different redshifts gives direct evidence of cosmological evolution. Data were analyzed within the framework of luminosity evolution models and the two most common evolutionary forms were considered. Luminosity dependent evolution is required if the evolution function has the exponential form, whereas the simpler pure luminosity evolution model is still acceptable if the evolution function has the power law form. Using the whole sample of objects the number counts and the devolved XLF were derived. A comparison of the EMSS Data with preliminary Rosat results indicates an overall agreement.

Maccacaro, Tommaso↗
DOE OSTI2011

Atom-atom correlations in time-of-flight imaging of ultracold bosons in optical lattices

We study the spatial correlations of strongly interacting bosons in a ground state, confined in a two-dimensional square and a three-dimensional cubic lattice. Using the combined Bogoliubov method and the quantum rotor approach, we map the Hamiltonian of strongly interacting bosons onto U(1) phase action in order to calculate the atom-atom correlations' decay along the principal axis and a diagonal of the lattice-plane direction as a function of distance. Lower tunneling rates lead to quicker decays of the correlations, whose character becomes exponential. Finally, correlation functions allow us to calculate quantities that are directly bound to experimental outcomes, namely time-of-flight absorption images and resulting visibility. Our results contain all the characteristic features present in experimental data (transition from Mott insulating blob to superfluid peaks, etc.), emphasizing the usability of the proposed approach.

74 ATOMIC AND MOLECULAR PHYSICS↗
NASA NTRS1993

The analytical representation of viscoelastic material properties using optimization techniques

This report presents a technique to model viscoelastic material properties with a function of the form of the Prony series. Generally, the method employed to determine the function constants requires assuming values for the exponential constants of the function and then resolving the remaining constants through linear least-squares techniques. The technique presented here allows all the constants to be analytically determined through optimization techniques. This technique is employed in a computer program named PRONY and makes use of commercially available optimization tool developed by VMA Engineering, Inc. The PRONY program was utilized to compare the technique against previously determined models for solid rocket motor TP-H1148 propellant and V747-75 Viton fluoroelastomer. In both cases, the optimization technique generated functions that modeled the test data with at least an order of magnitude better correlation. This technique has demonstrated the capability to use small or large data sets and to use data sets that have uniformly or nonuniformly spaced data pairs. The reduction of experimental data to accurate mathematical models is a vital part of most scientific and engineering research. This technique of regression through optimization can be applied to other mathematical models that are difficult to fit to experimental data through traditional regression techniques.

Hill, S. A.↗
DOE OSTI2023

Exploiting Kronecker structure in exponential integrators: Fast approximation of the action of φ $-$functions of matrices via quadrature

Here, in this article, we propose an algorithm for approximating the action of φ $-$ functions of matrices against vectors, which is a key operation in exponential time integrators. In particular, we consider matrices with Kronecker sum structure, which arise from problems admitting a tensor product representation. The method is based on quadrature approximations of the integral form of the φ $-$ functions combined with a scaling and modified squaring method. Owing to the Kronecker sum representation, only actions of 1D matrix exponentials are needed at each quadrature node and assembly of the full matrix can be avoided. Additionally, we derive a priori bounds for the quadrature error, which show that, as expected by classical theory, the rate of convergence of our method is supergeometric. Guided by our analysis, we construct a fast and robust method for estimating the optimal scaling factor and number of quadrature nodes that minimizes the total cost for a prescribed error tolerance. We investigate the performance of our algorithm by solving several linear and semilinear time-dependent problems in 2D and 3D. The results show that our method is accurate and orders of magnitude faster than the current state-of-the-art.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1992

The properties of X-ray selected active galactic nuclei. II - A deeper look at the cosmological evolution

A detailed study of the cosmological properties of X-ray selected AGN is presented. The data are analyzed within the framework of a pure luminosity evolution (PLE) model and the two most population evolutionary forms. Evidence is found for luminosity-dependent luminosity evolution if the evolution function has the exponential form. The simpler PLE model is more acceptable if the data are fitted with a power-law evolution function. Similar results are obtained in the optical domain from an analysis of a sample of optically selected QSOs with z less than 2.2 and B less than 20.

Della Ceca, Roberto↗
DOE OSTI2014

Universal scaling of potential energy functions describing intermolecular interactions. I. Foundations and scalable forms of new generalized Mie, Lennard-Jones, Morse, and Buckingham exponential-6 potentials

Based on the formulation of the analytical expression of the potential V(r) describing intermolecular interactions in terms of the dimensionless variables r*=r/rm and !*=V/!, where rm is the separation at the minimum and ! the well depth, we propose more generalized scalable forms for the commonly used Lennard-Jones, Mie, Morse and Buckingham exponential-6 potential energy functions (PEFs). These new generalized forms have an additional parameter from and revert to the original ones for some choice of that parameter. In this respect, the original forms can be considered as special cases of the more general forms that are introduced. We also propose a scalable, but nonrevertible to the original one, 4-parameter extended Morse potential.

Xantheas, Sotiris S.↗
NASA NTRS2010

A Study of Cirrus Ice Particle Size Distribution Using TC4 Observations

An analysis of two days of in situ observations of ice particle size spectra, in convectively generated cirrus, obtained during NASA s Tropical Composition, Cloud, and Climate Coupling (TC4) mission is presented. The observed spectra are examined for their fit to the exponential, gamma, and lognormal function distributions. Characteristic particle size and concentration density scales are determined using two (for the exponential) or three (for the gamma and lognormal functions) moments of the spectra. It is shown that transformed exponential, gamma, and lognormal distributions should collapse onto standard curves. An examination of the transformed spectra, and of deviations of the transformed spectra from the standard curves, shows that the lognormal function provides a better fit to the observed spectra.

Tian, Lin↗
DOE OSTI2014

Molecular simulation of thermodynamic and transport properties for the H{sub 2}O+NaCl system

Molecular dynamics and Monte Carlo simulations have been carried out to obtain thermodynamic and transport properties of the binary mixture H{sub 2}O+NaCl at temperatures from T = 298 to 473 K. In particular, vapor pressures, liquid densities, viscosities, and vapor-liquid interfacial tensions have been obtained as functions of pressure and salt concentration. Several previously proposed fixed-point-charge models that include either Lennard-Jones (LJ) 12-6 or exponential-6 (Exp6) functional forms to describe non-Coulombic interactions were studied. In particular, for water we used the SPC and SPC/E (LJ) models in their rigid forms, a semiflexible version of the SPC/E (LJ) model, and the Errington-Panagiotopoulos Exp6 model; for NaCl, we used the Smith-Dang and Joung-Cheatham (LJ) parameterizations as well as the Tosi-Fumi (Exp6) model. While none of the model combinations are able to reproduce simultaneously all target properties, vapor pressures are well represented using the SPC plus Joung-Cheathem model combination, and all LJ models do well for the liquid density, with the semiflexible SPC/E plus Joung-Cheatham combination being the most accurate. For viscosities, the combination of rigid SPC/E plus Smith-Dang is the best alternative. For interfacial tensions, the combination of the semiflexible SPC/E plus Smith-Dang or Joung-Cheatham gives the best results. Inclusion of water flexibility improves the mixture densities and interfacial tensions, at the cost of larger deviations for the vapor pressures and viscosities. The Exp6 water plus Tosi-Fumi salt model combination was found to perform poorly for most of the properties of interest, in particular being unable to describe the experimental trend for the vapor pressure as a function of salt concentration.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2015

Thresholded power law size distributions of instabilities in astrophysics

Power-law-like size distributions are ubiquitous in astrophysical instabilities. There are at least four natural effects that cause deviations from ideal power law size distributions, which we model here in a generalized way: (1) a physical threshold of an instability; (2) incomplete sampling of the smallest events below a threshold x{sub 0}; (3) contamination by an event-unrelated background x{sub b}; and (4) truncation effects at the largest events due to a finite system size. These effects can be modeled in the simplest terms with a “thresholded power law” distribution function (also called generalized Pareto [type II] or Lomax distribution), N(x)dx∝(x+x{sub 0}){sup −a}dx, where x{sub 0} > 0 is positive for a threshold effect, while x{sub 0} < 0 is negative for background contamination. We analytically derive the functional shape of this thresholded power law distribution function from an exponential growth evolution model, which produces avalanches only when a disturbance exceeds a critical threshold x{sub 0}. We apply the thresholded power law distribution function to terrestrial, solar (HXRBS, BATSE, RHESSI), and stellar flare (Kepler) data sets. We find that the thresholded power law model provides an adequate fit to most of the observed data. Major advantages of this model are the automated choice of the power law fitting range, diagnostics of background contamination, physical instability thresholds, instrumental detection thresholds, and finite system size limits. When testing self-organized criticality models that predict ideal power laws, we suggest including these natural truncation effects.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS1984

Function algorithms for MPP scientific subroutines, volume 1

Design documentation and user documentation for function algorithms for the Massively Parallel Processor (MPP) are presented. The contract specifies development of MPP assembler instructions to perform the following functions: natural logarithm; exponential (e to the x power); square root; sine; cosine; and arctangent. To fulfill the requirements of the contract, parallel array and solar implementations for these functions were developed on the PDP11/34 Program Development and Management Unit (PDMU) that is resident at the MPP testbed installation located at the NASA Goddard facility.

Gouch, J. G.↗
NASA NTRS2018

A Simple, Velocity Dependent, Collision Probability Algorithm for Small Combined Hard Body Radii Close Approach Events

Over the last few years progress has been made in the velocity dependent collision probability problem. Recent progress in this problem has been approached via integration of the probability flux into the surface of the combined hard body object. The algorithm presented uses the surface flux approach and is designed to compute the collision probability rate between two objects given their time dependent states and state error covariance matrices. With regards to the computation of the collision probability (Pc) rate, two major differences exist between the development of this algorithm and the usual Pc algorithm. First, the shape of the at-risk volume is assumed to be a cube rather than a sphere. The size of the cube is chosen so that it circumscribes the usual hard body sphere chosen for the spherical Pc problem. This will result in half the length of a side of the cube being equal to the hard body radius (HBR) of the sphere. Second, it is assumed that the HBR of the cube is much smaller than the smallest combined position uncertainty (σ(sub min) > 5*HBR) at each time point of evaluation. This is necessary as the actual collision probability calculation is based on a first order, small variable expansion in the position components of the Gaussian probability density function. This leads to the collision probability rate being to second order in the combined HBR. The collision probability rate is in a concise, closed form containing exponential and error functions.

Frisbee, Joe↗
DOE OSTI2021

Spin dynamics of the antiferromagnetic Heisenberg model on a kagome bilayer

We research the spin dynamics of a classical Heisenberg antiferromagnet with nearest-neighbor interactions on a quasi-two-dimensional kagome bilayer. This geometrically frustrated lattice consists of two kagome layers connected by a triangular-lattice linking layer. By combining Monte Carlo with precessional spin dynamics simulations, we compute the dynamical structure factor of the classical spin liquid in the kagome bilayer and investigate the thermal and dilution effects. While the low-frequency and long-wavelength dynamics of the cooperative paramagnetic phase is dominated by spin diffusion, weak magnon excitations persist at higher energies, giving rise to the half-moon pattern in the dynamical structure factor. In the presence of spin vacancies, the dynamical properties of the diluted system can be understood within the two-population picture. The spin diffusion of the “correlated” spin clusters is mainly driven by the zero-energy weather-vane modes, giving rise to an autocorrelation function that decays exponentially with time. On the other hand, the diffusive dynamics of the quasifree “orphan” spins leads to a distinctive longer-time power-law tail in the autocorrelation function. We discuss the implications of our work for the glassy behaviors observed in the archetypal frustrated magnet SrCr 9p Ga 12-9 pO 19 (SCGO).

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2022

Multirate Exponential Rosenbrock Methods

In this paper we propose a novel class of methods for high-order accurate integration of multirate systems of ordinary differential equation initial-value problems. The proposed methods construct multirate schemes by approximating the action of matrix φ functions within explicit exponential Rosenbrock (ExpRB) methods, thereby called multirate ExpRB (MERB) methods. They consist of the solution to a sequence of modified “fast” initial-value problems, which may themselves be approximated through subcycling any desired initial-value problem solver. In addition to proving how to construct MERB methods from certain classes of ExpRB methods, we provide rigorous convergence analysis of these methods and derive efficient MERB schemes of orders 2 through 6 (the highest-order infinitesimal multirate methods to date). Lastly, we then present numerical simulations to confirm these theoretical convergence rates and to compare the efficiency of MERB methods against other recently introduced high-order multirate methods.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Exact Fluctuations of Nonequilibrium Steady States from Approximate Auxiliary Dynamics

We describe a framework to reduce the computational effort to evaluate large deviation functions of time integrated observables within nonequilibrium steady states. We do this by incorporating an auxiliary dynamics into trajectory based Monte Carlo calculations, through a transformation of the system's propagator using an approximate guiding function. This procedure importance samples the trajectories that most contribute to the large deviation function, mitigating the exponential complexity of such calculations. Here, we illustrate the method by studying driven diffusion and interacting lattice models in one and two spatial dimensions. Finally, our work offers an avenue to calculate large deviation functions for high dimensional systems driven far from equilibrium.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1982

The stability of quasi-geostrophic fields induced by potential vorticity sources

A determination of the conditions leading to instabilities which grow in place in quasi-geostrophic fields induced by localized potential vorticity sources is presented. The analysis considers a quasi-geostrophic barotropic flow in a horizontally open domain, and a nondimensional quasi-geostrophic vorticity equation is defined which governs the deviation stream function. Weak forcing is considered, and it is shown that the field induced by the topography is stable while the field induced by the potential vorticity source can be unstable. The growth rate is exponential and a function of nonlinearity and friction, which, if absent, indicates an unstable flowfield. The instability is shown to be capable of changing the flowfield from one quasi-steady state to another, with energy extraction occurring at the source

Merkine, L.-O.↗