Search NASA⌕ Search

SEARCH · Search NASA

Results for “Analytic linearization”

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 1,189 records · Page 66
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Beam focusing and consequences for Doppler backscattering measurements

The phenomenon of focusing of microwave beams in a plasma near a turning-point caustic is discussed by exploiting the analytical solution to the Gaussian beam-tracing equations in the two-dimensional (2-D) linear-layer problem. The location of maximum beam focusing and the beam width at that location are studied in terms of the beam initial conditions. This focusing must be taken into account to interpret Doppler backscattering (DBS) measurements. We find that the filter function that characterises the scattering intensity contribution along the beam path through the plasma is inversely proportional to the beam width, predicting enhanced scattering from the beam focusing region. We show that the DBS signal enhancement for decreasing incident angles between the beam path and the density gradient is due to beam focusing and not due to forward scattering, as was originally proposed by (Gusakov et al., (Plasma Phys. Contr. Fusion, vol. 56, 2014, p. 0250092014, 2017); Plasma Phys. Rep. vol. 43(6), 2017, pp. 605–613). The analytic beam model is used to predict the measurement of the k y density-fluctuation wavenumber power spectrum via DBS, showing that, in an NSTX-inspired example, the spectral exponent of the turbulent, intermediate-to-high k y density-fluctuation spectrum might be quantitatively measurable via DBS, but not the spectral peak corresponding to the driving scale of the turbulent cascade.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
NASA NTRS1989

Acoustic emission from a growing crack

An analytical method is being developed to determine the signature of an acoustic emission waveform from a growing crack and the results of this analysis are compared to experimentally obtained values. Within the assumptions of linear elastic fracture mechanics, a two dimensional model is developed to examine a semi-infinite crack that, after propagating with a constant velocity, suddenly stops. The analytical model employs an integral equation method for the analysis of problems of dynamic fracture mechanics. The experimental procedure uses an interferometric apparatus that makes very localized absolute measurements with very high fidelity and without acoustically loading the specimen.

Jacobs, Laurence J.↗
DOE OSTI2014

ANALYTICAL MODELS OF EXOPLANETARY ATMOSPHERES. I. ATMOSPHERIC DYNAMICS VIA THE SHALLOW WATER SYSTEM

Within the context of exoplanetary atmospheres, we present a comprehensive linear analysis of forced, damped, magnetized shallow water systems, exploring the effects of dimensionality, geometry (Cartesian, pseudo-spherical, and spherical), rotation, magnetic tension, and hydrodynamic and magnetic sources of friction. Across a broad range of conditions, we find that the key governing equation for atmospheres and quantum harmonic oscillators are identical, even when forcing (stellar irradiation), sources of friction (molecular viscosity, Rayleigh drag, and magnetic drag), and magnetic tension are included. The global atmospheric structure is largely controlled by a single key parameter that involves the Rossby and Prandtl numbers. This near-universality breaks down when either molecular viscosity or magnetic drag acts non-uniformly across latitude or a poloidal magnetic field is present, suggesting that these effects will introduce qualitative changes to the familiar chevron-shaped feature witnessed in simulations of atmospheric circulation. We also find that hydrodynamic and magnetic sources of friction have dissimilar phase signatures and affect the flow in fundamentally different ways, implying that using Rayleigh drag to mimic magnetic drag is inaccurate. We exhaustively lay down the theoretical formalism (dispersion relations, governing equations, and time-dependent wave solutions) for a broad suite of models. In all situations, we derive the steady state of an atmosphere, which is relevant to interpreting infrared phase and eclipse maps of exoplanetary atmospheres. We elucidate a pinching effect that confines the atmospheric structure to be near the equator. Our suite of analytical models may be used to develop decisively physical intuition and as a reference point for three-dimensional magnetohydrodynamic simulations of atmospheric circulation.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2015

SU-E-T-422: Fast Analytical Beamlet Optimization for Volumetric Intensity-Modulated Arc Therapy

Purpose: To implement a fast optimization algorithm on CPU/GPU heterogeneous computing platform and to obtain an optimal fluence for a given target dose distribution from the pre-calculated beamlets in an analytical approach. Methods: The 2D target dose distribution was modeled as an n-dimensional vector and estimated by a linear combination of independent basis vectors. The basis set was composed of the pre-calculated beamlet dose distributions at every 6 degrees of gantry angle and the cost function was set as the magnitude square of the vector difference between the target and the estimated dose distribution. The optimal weighting of the basis, which corresponds to the optimal fluence, was obtained analytically by the least square method. Those basis vectors with a positive weighting were selected for entering into the next level of optimization. Totally, 7 levels of optimization were implemented in the study.Ten head-and-neck and ten prostate carcinoma cases were selected for the study and mapped to a round water phantom with a diameter of 20cm. The Matlab computation was performed in a heterogeneous programming environment with Intel i7 CPU and NVIDIA Geforce 840M GPU. Results: In all selected cases, the estimated dose distribution was in a good agreement with the given target dose distribution and their correlation coefficients were found to be in the range of 0.9992 to 0.9997. Their root-mean-square error was monotonically decreasing and converging after 7 cycles of optimization. The computation took only about 10 seconds and the optimal fluence maps at each gantry angle throughout an arc were quickly obtained. Conclusion: An analytical approach is derived for finding the optimal fluence for a given target dose distribution and a fast optimization algorithm implemented on the CPU/GPU heterogeneous computing environment greatly reduces the optimization time.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2016

Exact relativistic expressions for polarization of incoherent Thomson scattering

We present a derivation of the degree of polarization for incoherent Thomson scattering (TS) using Mueller matrix formalism. An exact analytic solution is obtained for spectrum-integrated matrix elements. The solution is valid for the full range of incident polarizations, scattering angles, and electron thermal motion from non-relativistic to ultra-relativistic. It is based on a newly developed theoretical model, a finite transit time (FTT) correction to previous theoretical work on TS polarization. The Mueller matrix elements are substantially different from previous calculations without the FTT correction, even to the lowest linear order in T e =m e c 2 <<1. Mathematically, the derivation is a unique example of fully analytical integration of the 3D scattering operator over a relativistic Maxwellian distribution function; experimentally, the results have application to the use of the polarization properties of Thomson scattered light as a method of electron temperature measurement. The results can also be used as a reliable tool for benchmarking and verification of numerical codes for frequency resolved properties of TS polarization.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2014

A semi-analytical solver for the Grad-Shafranov equation

In toroidally confined plasmas, the Grad-Shafranov equation, in general, a non-linear partial differential equation, describes the hydromagnetic equilibrium of the system. This equation becomes linear when the kinetic pressure is proportional to the poloidal magnetic flux and the squared poloidal current is a quadratic function of it. In this work, the eigenvalue of the associated homogeneous equation is related with the safety factor on the magnetic axis, the plasma beta, and the Shafranov shift; then, the adjustable parameters of the particular solution are bounded through physical constrains. The poloidal magnetic flux becomes a linear superposition of independent solutions and its parameters are adjusted with a non-linear fitting algorithm. This method is used to find hydromagnetic equilibria with normal and reversed magnetic shear and defined values of the elongation, triangularity, aspect-ratio, and X-point(s). The resultant toroidal and poloidal beta, the safety factor at the 95% flux surface, and the plasma current are in agreement with usual experimental values for high beta discharges and the model can be used locally to describe reversed magnetic shear equilibria.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2016

Non-linear theory of a cavitated plasma wake in a plasma channel for special applications and control

We introduce a complete semi-analytical model for a cavitated electron wake driven by an electron beam in a radially inhomogeneous plasma. The electron response to the driver, dynamics of electrons in a thin sheath surrounding the cavity, as well as accelerating and focusing fields inside the cavity are calculated in the quasistatic approximation. Our theory holds for arbitrary radial density profiles and reduces to known models in the limit of a homogeneous plasma. A free-propagating blow-out in an evacuated channel experiences longitudinal squeezing, qualitatively the same as observed in particle-in-cell simulations for the laser pulse-driven case [Pukhov et al., Phys. Rev. Lett. 113, 245003 (2014)]. Our model also permits qualitative interpretation of the earlier observed cancellation of the focusing gradient in the cavity [Pukhov et al., Phys. Rev. Lett. 113, 245003 (2014)]. In this work, we show the underlying mechanism that causes the radial fields in the vacuum part of a channel to become defocussing.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2016

Reply to Levermann et al.: Linear scaling for monsoons based on well-verified balance between adiabatic cooling and latent heat release

We demonstrated (1) that an analytical model of monsoons reflects no abrupt response to forcing changes when one assumes, consistent with observations (2), that monsoon circulations export energy away from their ascending branch. We claimed that a previous model (3) incorrectly produced an abrupt response because it neglected the adiabatic cooling of rising air in a stably stratified troposphere. Levermann et al. argue (4) that the abrupt response is preserved in the ref. 3 model as long as adiabatic cooling does not balance the latent heat released during precipitation formation, and that neither theory nor observations support the idea that “most of the energy from the rain’s latent heat release is consumed by adiabatic cooling.”

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2021

Dynamic Catalysis Fundamentals: I. Fast calculation of limit cycles in dynamic catalysis

Dynamic catalysis—the forced oscillation of catalytic reaction coordinate potential energy surfaces (PES)—has recently emerged as a promising method for the acceleration of heterogeneously-catalyzed reactions. Theoretical study of enhancement of rates and supra-equilibrium product yield via dynamic catalysis has, to-date, been severely limited by onerous computational demands of forward integration of stiff, coupled ordinary differential equations (ODEs) that are necessary to quantitatively describe periodic cycling between PESs. Here, we establish a new approach that reduces, by ≳108×, the computational cost of finding the time-averaged rate at dynamic steady state (i.e. the limit cycle for linear and nonlinear systems of kinetic equations). Our developments are motivated by and conceived from physical and mathematical insight drawn from examination of a simple, didactic case study for which closed-form solutions of rate enhancement are derived in explicit terms of periods of oscillation and elementary step rate constants. Generalization of such closed-form solutions to more complex catalytic systems is achieved by introducing a periodic boundary condition requiring the dynamic steady state solution to have the same periodicity as the kinetic oscillations and solving the corresponding differential equations by linear algebra or Newton-Raphson-based approaches. The methodology is well-suited to extension to non-linear systems for which we detail the potential for multiple solutions or solutions with different periodicities. For linear and non-linear systems alike, the acute decrement in computational expense enables rapid optimization of oscillation waveforms and, consequently, accelerates understanding of the key catalyst properties that enable maximization of reaction rates, conversions, and selectivities during dynamic catalysis.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Analytic and numerical vibronic spectra from quasi-classical trajectory ensembles

The truncated Wigner approximation to quantum dynamics in phase space is explored in the context of computing vibronic line shapes for monomer linear optical spectra. We consider multiple model potential forms including a shifted harmonic oscillator with both equal and unequal frequencies on the ground and excited state potentials as well as a shifted Morse potential model. For the equal-frequency shifted harmonic oscillator model, we derive an analytic expression for the exact vibronic line shape that emphasizes the importance of using a quantum mechanical distribution of phase space initial conditions. For the unequal-frequency shifted harmonic oscillator model, we are no longer able to obtain an exact expression for the vibronic line shape in terms of independent deterministic classical trajectories. We show how one can rigorously account for corrections to the truncated Wigner approximation through nonlinear responses of the line shape function to momentum fluctuations along a classical trajectory and demonstrate the qualitative improvement in the resulting spectrum when the leading-order quantum correction is included. Finally, we numerically simulate absorption spectra of a highly anharmonic shifted Morse potential model. We find that, while finite quantization and the dissociation limit are captured with reasonable accuracy, there is a qualitative breakdown of the quasi-classical trajectory ensemble’s ability to describe the vibronic line shape when the relative shift in Morse potentials becomes large. The work presented here provides clarity on the origin of unphysical negative features known to contaminate absorption spectra computed with quasi-classical trajectory ensembles.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
NASA NTRS1985

On the adequacy of meridional resolution of linear and quasi-linear barotropic models

The minimum meridional resolution needed for an adequate numerical simulation of the linear and 'quasi-linear' barotropic vorticity dynamics in the vicinity of a critical latitude (CL) is determined by using a semispectral nondivergent barotropic model on a sphere. The behavior of the analytic as well as the meridionally finite differenced form of the refractive index in the vicinity of a CL is discussed. An estimation of an upper bound on the latitudinal gridsize to be used in linear calculations is given, and constraint on the latitudinal grid size are considered. The dynamical model utilized is briefly described.

Nigam, S.↗
NASA NTRS1975

Dynamic analysis of a flexible spacecraft with rotating components. Volume 1: Analytical developments

Analytical procedures and digital computer code are presented for the dynamic analysis of a flexible spacecraft with rotating components. Topics, considered include: (1) nonlinear response in the time domain, and (2) linear response in the frequency domain. The spacecraft is assumed to consist of an assembly of connected rigid or flexible subassemblies. The total system is not restricted to a topological connection arrangement and may be acting under the influence of passive or active control systems and external environments. The analytics and associated digital code provide the user with the capability to establish spacecraft system nonlinear total response for specified initial conditions, linear perturbation response about a calculated or specified nominal motion, general frequency response and graphical display, and spacecraft system stability analysis.

Bodley, C. S.↗
DOE OSTI2014

Weakly nonlinear Rayleigh-Taylor instability of a finite-thickness fluid layer

A weakly nonlinear (WN) model has been developed for the Rayleigh-Taylor instability of a finite-thickness incompressible fluid layer (slab). We derive the coupling evolution equations for perturbations on the (upper) “linearly stable” and (lower) “linearly unstable” interfaces of the slab. Expressions of temporal evolutions of the amplitudes of the perturbation first three harmonics on the upper and lower interfaces are obtained. The classical feedthrough (interface coupling) solution obtained by Taylor [Proc. R. Soc. London A 201, 192 (1950)] is readily recovered by the first-order results. Our third-order model can depict the WN perturbation growth and the saturation of linear (exponential) growth of the perturbation fundamental mode on both interfaces. The dependence of the WN perturbation growth and the slab distortion on the normalized layer thickness (kd) is analytically investigated via the third-order solutions. Comparison is made with Jacobs-Catton's formula [J. W. Jacobs and I. Catton, J. Fluid Mech. 187, 329 (1988)] of the position of the “linearly unstable” interface. Using a reduced formula, the saturation amplitude of linear growth of the perturbation fundamental mode is studied. It is found that the finite-thickness effects play a dominant role in the WN evolution of the slab, especially when kd < 1. Thus, it should be included in applications where the interface coupling effects are important, such as inertial confinement fusion implosions and supernova explosions.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
NASA NTRS1973

Experimental and analytical determination of stability parameters for a balloon tethered in a wind

Experimental and analytical techniques for determining stability parameters for a balloon tethered in a steady wind are described. These techniques are applied to a particular 7.64-meter-long balloon, and the results are presented. The stability parameters of interest appear as coefficients in linearized stability equations and are derived from the various forces and moments acting on the balloon. In several cases the results from the experimental and analytical techniques are compared and suggestions are given as to which techniques are the most practical means of determining values for the stability parameters.

Redd, L. T.↗
DOE OSTI2021

Analytic Binary Alloy Volume–Concentration Relations and the Deviation from Zen’s Law

Alloys expand or contract as concentrations change, and the resulting relationship between atomic volume and alloy content is an important property of the solid. While a well-known approximation posits that the atomic volume varies linearly with concentration (Zen’s law), the actual variation is more complicated. Here we use the apparent size of the solute (solvent) atom and the elasticity to derive explicit analytical expressions for the atomic volume of binary solid alloys. Two approximations, continuum and terminal, are proposed. Deviations from Zen’s law are studied for 22 binary alloy systems.

36 MATERIALS SCIENCE↗
DOE OSTI2015

Matrix effect of sodium compounds on the determination of metal ions in aqueous solutions by underwater laser-induced breakdown spectroscopy

A significant portion of the carbon sequestration research being performed in the United States involves the risk assessment of injecting large quantities of carbon dioxide into deep saline aquifers. Leakage of CO 2 has the potential to affect the quality of groundwater supplies in case contaminants migrate through underlying conduits. New remote sensing and near-surface monitoring technologies are needed to ensure that injection, abandoned, and monitoring wells are structurally sound, and that CO 2 remains within the geologic storage reservoir. In this paper, we propose underwater laser-induced breakdown spectroscopy (underwater LIBS) as an analytical method for monitoring naturally occurring elements that can act as tracers to detect a CO 2 leak from storage sites. Laboratory-scale experiments were conducted to measure Sr 2+ , Ca 2+ , K + , and Li + in bulk solutions to ascertain the analytical performance of underwater LIBS. We compared the effect of NaCl, Na 2 CO 3 , and Na 2 SO 4 on the analytes calibration curves to determine underwater LIBS’ ability to analyze samples of sodium compounds. In all cases, the calibration curves showed a good linearity within 2 orders of magnitude. The limit of detections (LODs) obtained for K + (30±1 ppb) and Li + (60±2 ppb) were in ppb range, while higher LODs were observed for Ca 2+ (0.94±0.14 ppm) and Sr 2+ (2.89±0.11 ppm). Evaluation of the calibration curves for the analytes in mixed solutions showed dependence of the lines’ intensity with the sodium compounds. The intensities increased respectively in the presence of dissolved NaCl and Na 2 SO 4 , whereas the intensities slightly decreased in the presence of Na 2 CO 3 . Lastly, the capabilities of underwater LIBS to detect certain elements in the ppb or in the low ppm range make it particularly appealing for in situ monitoring of a CO 2 leak.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Surface science insight note: A linear algebraic approach to elucidate native films on Fe 3 O 4 surface

Standard materials are often used to obtain spectra that can be compared to those from unknown samples. Spectra measured from these known substances are also used as a means of computing sensitivity factors to allow quantification by X-ray photoelectron spectroscopy (XPS) of less well-defined materials. Spectra from known materials also provide line shapes suitable for inclusion in spectral models which, when fitted to spectra, permit the chemical state for a sample to be assessed. Both types of information depend on isolating photoemission signals from the inelastically scattered signal. In this Insight note, technical issues associated with the use of XPS of as received Fe 3 O 4 powder sample surface are discussed. The Insight note is designed to show how linear algebraic techniques applied to data collected from a sample marketed as pure Fe 3 O 4 powder are used to verify that XPS has been performed on chemistry representative of the sample. The methods described in this Insight note can further be utilized in elucidating complex XPS data obtained from thin films formed or evolved during cyclic/non-steady use of complex (electro)catalyst surfaces, especially in the presence of contaminants.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

Elastic Solutions to 2D Plane Strain Problems: Nonlinear Contact and Settlement Analysis for Shallow Foundations

The classical Neumann boundary value problem of an isotropic, homogeneous elastic half-plane under plane strain conditions is readdressed as the limiting case of the fully three-dimensional problem. Analytical solutions of the stress and strain tensors are obtained by taking the limit from known three-dimensional solutions. It is shown that the displacement fields for the plane strain problem are not well defined. A small number of simple expressions are developed, which provide a general solution for linearly-varying traction over arbitrary regions on the boundary. A simple, efficient, and rapidly convergent algorithm is developed which uses these solutions as analytic elements and provides a solution approach to the general boundary value problem. The method is verified against known solutions for Hertzian contact between parallel cylinders. Two numerical examples are presented for the analysis of shallow foundation systems. In the first, the boundary conditions are informed by analytical elastoplastic calculations and a strain influence analysis is performed and compared with the Schmertmann method. Subsequently, empirical laboratory contact traction distributions measured by Bauer et al., in both the normal and tangential directions are employed as boundary conditions for an analysis of the underlying stress field.

42 ENGINEERING↗