SEARCH · Engineering Papers

Results for “Differential Geometry”

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.

SourceResearch connectionFrom yearThrough year

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

At least 307 records · Page 17

All sources NASA NTRS DOE OSTI

NASA NTRS1983

Angular distribution for electron excitation of the 4(2)S yields 4(2)P transition in Zn II - Comparison of experiment and theory

Differential electron-scattering cross sections for inelastic excitation of an ion have been measured for the first time. Experiments were carried out in a cross electron-ion beam geometry for the 4(2)S yields 4(2)P transition in Zn II at 75 eV. In addition, differential cross sections were calculated at energies between 15 and 100 eV in a five-state close-coupling approximation in which 4s, 4p, 3d(9)4s(2), 5s, and 4 d states were included. Agreement in shape between theory and experiment at 75 eV is excellent.

Chutjian, A.↗

NASA NTRS1990

Time-optimal Aircraft Pursuit-evasion with a Weapon Envelope Constraint

The optimal pursuit-evasion problem between two aircraft including a realistic weapon envelope is analyzed using differential game theory. Six order nonlinear point mass vehicle models are employed and the inclusion of an arbitrary weapon envelope geometry is allowed. The performance index is a linear combination of flight time and the square of the vehicle acceleration. Closed form solution to this high-order differential game is then obtained using feedback linearization. The solution is in the form of a feedback guidance law together with a quartic polynomial for time-to-go. Due to its modest computational requirements, this nonlinear guidance law is useful for on-board real-time implementation.

Menon, P. K. A.↗

DOE OSTI2024

Improving River Routing Using a Differentiable Muskingum‐Cunge Model and Physics‐Informed Machine Learning

Recently, rainfall-runoff simulations in small headwater basins have been improved by methodological advances such as deep neural networks (NNs) and hybrid physics-NN models—particularly, a genre called differentiable modeling that intermingles NNs with physics to learn relationships between variables. However, hydrologic routing simulations, necessary for simulating floods in stem rivers downstream of large heterogeneous basins, had not yet benefited from these advances and it was unclear if the routing process could be improved via coupled NNs. We present a novel differentiable routing method (δMC-Juniata-hydroDL2) that mimics the classical Muskingum-Cunge routing model over a river network but embeds an NN to infer parameterizations for Manning's roughness (n) and channel geometries from raw reach-scale attributes like catchment areas and sinuosity. The NN was trained solely on downstream hydrographs. Synthetic experiments show that while the channel geometry parameter was unidentifiable, n can be identified with moderate precision. With real-world data, the trained differentiable routing model produced more accurate long-term routing results for both the training gage and untrained inner gages for larger subbasins (>2,000 km2) than either a machine learning model assuming homogeneity, or simply using the sum of runoff from subbasins. The n parameterization trained on short periods gave high performance in other periods, despite significant errors in runoff inputs. The learned n pattern was consistent with literature expectations, demonstrating the framework's potential for knowledge discovery, but the absolute values can vary depending on training periods. The trained n parameterization can be coupled with traditional models to improve national-scale hydrologic flood simulations.

54 ENVIRONMENTAL SCIENCES↗

NASA NTRS1993

A simulation of GPS and differential GPS sensors

The Global Positioning System (GPS) is a revolutionary advance in navigation. Users can determine latitude, longitude, and altitude by receiving range information from at least four satellites. The statistical accuracy of the user's position is directly proportional to the statistical accuracy of the range measurement. Range errors are caused by clock errors, ephemeris errors, atmospheric delays, multipath errors, and receiver noise. Selective Availability, which the military uses to intentionally degrade accuracy for non-authorized users, is a major error source. The proportionality constant relating position errors to range errors is the Dilution of Precision (DOP) which is a function of the satellite geometry. Receivers separated by relatively short distances have the same satellite and atmospheric errors. Differential GPS (DGPS) removes these errors by transmitting pseudorange corrections from a fixed receiver to a mobile receiver. The corrected pseudorange at the moving receiver is now corrupted only by errors from the receiver clock, multipath, and measurement noise. This paper describes a software package that models position errors for various GPS and DGPS systems. The error model is used in the Real-Time Simulator and Cockpit Technology workstation simulations at NASA-LaRC. The GPS/DGPS sensor can simulate enroute navigation, instrument approaches, or on-airport navigation.

Rankin, James M.↗

DOE OSTI2018

Differentiating the shape of stellarator coils with respect to the plasma boundary

The task of designing the geometry of a set of current-carrying coils that produce the magnetic field required to confine a given plasma equilibrium in stellarators is expressed as a minimization principle, namely that the coils minimize a suitably defined error expressed as a surface integral, which is recognized as the quadratic-flux. A penalty on the coil length is included to avoid pathological solutions. A simple expression for how the quadratic-flux and coil length vary as the coil geometry varies is derived, and an expression describing how this varies with variations in the surface geometry is derived. In conclusion, these expressions allow efficient coil-design algorithms to be implemented, and also enable efficient algorithms for varying the shape of the plasma surface in order to simplify the coil geometry, and a numerical illustration of this is given.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2015

THE ANALYTICAL STRUCTURE OF THE PRIMARY INTERSTELLAR HELIUM DISTRIBUTION FUNCTION IN THE HELIOSPHERE

A new analytical model based on the previous work of Lee et al. is presented for the distribution of interstellar helium in the heliosphere. The model is tailored for comparison with the IBEX-Lo observations in order to determine the bulk velocity and temperature of helium in the local interstellar cloud. The model includes solar gravity, spherically symmetric stationary ionization rates, transformation to the Earth/IBEX frame of reference, the IBEX viewing geometry with small spin-axis tilt, and integration of the atom differential intensity over energy and the instrument collimator solid angle. The analysis employs an expansion of the count rate about the peak of the velocity distribution to second order in the magnitudes of several small quantities: the ratio of the helium thermal speed to its bulk speed, the angle between the bulk velocity and the ecliptic, the two angles describing the tilt of the IBEX spin-axis away from Sun-pointing, the collimator angular width, and the angular difference between the observing longitude and the longitude where the projection of the bulk velocity onto the ecliptic is tangential to Earth's orbit. The model reveals the evolving ellipsoidal shape of the helium distribution as it moves along its average hyperbolic orbit. For specified interstellar parameters, the model predicts the latitudinal and longitudinal structure of the helium distribution. The model is in reasonable agreement with IBEX observations and the predictions of the other available models.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2019

Magnetic Eddy Viscosity of Mean Sheared Flows in Two-Dimensional Magnetohydrodynamics

Magnetic induction in magnetohydrodynamic fluids at magnetic Reynolds number (Rm) less than 1 has long been known to cause magnetic drag. Here we show that when Rm $\gg 1$ and the fluid is in a hydrodynamic-dominated regime in which the magnetic energy is much smaller than the kinetic energy, induction due to a mean shear flow leads to a magnetic eddy viscosity. The magnetic viscosity is derived from simple physical arguments, where a coherent response due to shear flow builds up in the magnetic field until decorrelated by turbulent motion. The dynamic viscosity coefficient is approximately $(B^2_p/2_{μ_0}) τ_{corr}$, the poloidal magnetic energy density multiplied by the correlation time. We confirm the magnetic eddy viscosity through numerical simulations of two-dimensional incompressible magnetohydrodynamics. We also consider the three-dimensional case, and in cylindrical or spherical geometry, theoretical considerations similarly point to a nonzero viscosity whenever there is differential rotation. Hence, these results serve as a dynamical generalization of Ferraro's law of isorotation. The magnetic eddy viscosity leads to transport of angular momentum and may be of importance to zonal flows in astrophysical domains such as the interior of some gas giants.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2023

Multislice forward modeling of coherent surface scattering imaging on surface and interfacial structures

To study nanostructures on substrates, surface-sensitive reflection-geometry scattering techniques such as grazing incident small angle X-ray scattering are commonly used to yield an averaged statistical structural information of the surface sample. Grazing incidence geometry can probe the absolute three-dimensional structural morphology of the sample if a highly coherent beam is used. Coherent surface scattering imaging (CSSI) is a powerful yet non-invasive technique similar to coherent X-ray diffractive imaging (CDI) but performed at small angles and grazing-incidence reflection geometry. A challenge with CSSI is that conventional CDI reconstruction techniques cannot be directly applied to CSSI because the Fourier-transform-based forward models cannot reproduce the dynamical scattering phenomenon near the critical angle of total external reflection of the substrate-supported samples. To overcome this challenge, we have developed a multislice forward model which can successfully simulate the dynamical or multi-beam scattering generated from surface structures and the underlying substrate. The forward model is also demonstrated to be able to reconstruct an elongated 3D pattern from a single shot scattering image in the CSSI geometry through fast-performing CUDA-assisted PyTorch optimization with automatic differentiation.

77 NANOSCIENCE AND NANOTECHNOLOGY↗

DOE OSTI2021

Deformation and strength of mantle relevant garnets: Implications for the subduction of basaltic-rich crust

Garnet is an important mineral phase in the upper mantle as it is both a key component in bulk mantle rocks, and a primary phase at high pressure within subducted basalt. In this work, we focus on the strength of garnet and the texture that develops within garnet during accommodation of differential deformational strain. We use X-ray diffraction in a radial geometry to analyze texture development in situ in three garnet compositions under pressure at 300 K: a natural garnet (Prp 60 Alm 37 ) to 30 GPa, and two synthetic majorite-bearing compositions (Prp 59 Maj 41 and Prp 42 Maj 58 ) to 44 GPa. All three garnets develop a modest (100) texture at elevated pressure under axial compression. Elasto-viscoplastic self-consistent (EVPSC) modeling suggests that two slip systems are active in the three garnet compositions at all pressures studied: {110}<$1\bar{1}1$> and {001}<110>. We determine a flow strength of ~5 GPa at pressures between 10 to 15 GPa for all three garnets; these values are higher than previously reported yield strengths measured on natural and majoritic garnets. Strengths calculated using the experimental lattice strain differ from the strength generated from those calculated using EVPSC. Prp 67 Alm 33 , Prp 59 Maj 41 , and Prp 42 Maj 58 are of comparable strength to each other at room temperature, which indicates that majorite substitution does not greatly affect the strength of garnets. Additionally, all three garnets are of similar strength as lower mantle phases such as bridgmanite and ferropericlase, suggesting that garnet may not be notably stronger than the surrounding lower mantle/deep upper mantle phases at the base of the upper mantle.

58 GEOSCIENCES↗

DOE OSTI2022

Taylor wave solution for a general equation of state

This document describes a solution procedure for calculating the Taylor wave behind an unsupported Chapman–Jouguet (CJ) detonation in planar, cylindrical, and spherical geometries given a general equation of state. The resulting semi-analytic solution can be utilized to examine new equation of state models for detonation products and during the verification of hydrodynamic codes. The governing partial differential equations are reduced to ordinary differential equations in both characteristic and self-similar forms. The first-order systems corresponding to each geometry are amenable to solution numerically using commonly available methods. A difficulty arises at the CJ point in radial coordinates where the similarity equations become singular. Two separate strategies are proposed to integrate the first-order system. The first one uses an asymptotic approximation near the CJ point that can be used to perturb the boundary conditions. The second one applies a change of variables which removes the singularity at the expense of an additional equation to be integrated. A test problem is provided for the Davis products equation of state to illustrate the qualitative features of the Taylor wave in each geometric configuration and compared with a Lagrangian hydrodynamics research code. A Python code listing gives an implementation using the SciPy library to assists users in generating the results.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1999

The Magnetospheric Multiscale Mission...Resolving Fundamental Processes in Space Plasmas

The Magnetospheric Multiscale (MMS) mission is a multiple-spacecraft Solar-Terrestrial Probe designed to study the microphysics of magnetic reconnection, charged particle acceleration, and turbulence in key boundary regions of Earth's magnetosphere. These three processes, which control the flow of energy, mass, and momentum within and across plasma boundaries, occur throughout the universe and are fundamental to our understanding of astrophysical and solar system plasmas. Only in Earth's magnetosphere, however, are they readily accessible for sustained study through in-situ measurement. MMS will employ five co-orbiting spacecraft identically instrumented to measure electric and magnetic fields, plasmas, and energetic particles. The initial parameters of the individual spacecraft orbits will be designed so that the spacecraft formation will evolve into a three-dimensional configuration near apogee, allowing MMS to differentiate between spatial and temporal effects and to determine the three dimensional geometry of plasma, field, and current structures. In order to sample all of the magnetospheric boundary regions, MMS will employ a unique four-phase orbital strategy involving carefully sequenced changes in the local time and radial distance of apogee and, in the third phase, a change in orbit inclination from 10 degrees to 90 degrees. The nominal mission operational lifetime is two years. Launch is currently scheduled for 2006.

Curtis, S.↗

NASA NTRSDate not supplied

Formal Verification of the Interaction Between Semi-Algebraic Sets and Real Analytic Functions

Semi-algebraic sets and real analytic functions are fundamental concepts in Real Algebraic Geometry and Real Analysis, respectively. These concepts interact in the study of Differential Equations, where the real analytic solution to a differential equation is known to enter or exit a semi-algebraic set in a predicable way. Motivated to enhance the capability to reason about differential equations in the Prototype Verification System (PVS), a formalization of multivariate polynomials, semi-algebraic sets, and real analytic functions is developed. The favorable way that a real analytic function enters and exits a semi-algebraic set is proven. It is further shown that if the function is assumed to be smooth, a slightly weaker assumption than real analytic, these favorable interactions with semi-algebraic sets may fail.

Real analytic functions↗

DOE OSTI2013

Cartan gravity, matter fields, and the gauge principle

Gravity is commonly thought of as one of the four force fields in nature. However, in standard formulations its mathematical structure is rather different from the Yang–Mills fields of particle physics that govern the electromagnetic, weak, and strong interactions. This paper explores this dissonance with particular focus on how gravity couples to matter from the perspective of the Cartan-geometric formulation of gravity. There the gravitational field is represented by a pair of variables: (1) a ‘contact vector’ V{sup A} which is geometrically visualized as the contact point between the spacetime manifold and a model spacetime being ‘rolled’ on top of it, and (2) a gauge connection A{sub μ}{sup AB}, here taken to be valued in the Lie algebra of SO(2,3) or SO(1,4), which mathematically determines how much the model spacetime is rotated when rolled. By insisting on two principles, the gauge principle and polynomial simplicity, we shall show how one can reformulate matter field actions in a way that is harmonious with Cartan’s geometric construction. This yields a formulation of all matter fields in terms of first order partial differential equations. We show in detail how the standard second order formulation can be recovered. In particular, the Hodge dual, which characterizes the structure of bosonic field equations, pops up automatically. Furthermore, the energy–momentum and spin-density three-forms are naturally combined into a single object here denoted the spin-energy–momentum three-form. Finally, we highlight a peculiarity in the mathematical structure of our first-order formulation of Yang–Mills fields. This suggests a way to unify a U(1) gauge field with gravity into a SO(1,5)-valued gauge field using a natural generalization of Cartan geometry in which the larger symmetry group is spontaneously broken down to SO(1,3)×U(1). The coupling of this unified theory to matter fields and possible extensions to non-Abelian gauge fields are left as open questions. -- Highlights: •Develops Cartan gravity to include matter fields. •Coupling to gravity is done using the standard gauge prescription. •Matter actions are manifestly polynomial in all field variables. •Standard equations recovered on-shell for scalar, spinor and Yang–Mills fields. •Unification of a U(1) field with gravity based on the orthogonal group SO(1,5)

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

NASA NTRS1984

Development of MCAERO wing design panel method with interactive graphics module

A reliable and efficient iterative method has been developed for designing wing section contours corresponding to a prescribed subcritical pressure distribution. The design process is initialized by using MCAERO (MCAIR 3-D Subsonic Potential Flow Analysis Code) to analyze a baseline configuration. A second program DMCAERO is then used to calculate a matrix containing the partial derivative of potential at each control point with respect to each unknown geometry parameter by applying a first-order expansion to the baseline equations in MCAERO. This matrix is calculated only once but is used in each iteration cycle to calculate the geometry perturbation and to analyze the perturbed geometry. The potential on the new geometry is calculated by linear extrapolation from the baseline solution. This extrapolated potential is converted to velocity by numerical differentiation, and velocity is converted to pressure by using Bernoulli's equation. There is an interactive graphics option which allows the user to graphically display the results of the design process and to interactively change either the geometry or the prescribed pressure distribution.

Hawk, J. D.↗

NASA NTRS1974

Three-dimensional elastic stress and displacement analysis of finite geometry solids containing cracks

The line method of analysis is applied to the Navier-Cauchy equations of elastic equilibrium to calculate the displacement distributions in various bodies containing cracks. The application of this method to these equations leads to coupled sets of simultaneous ordinary differential equations whose solutions are obtained along sets of lines in a discretized region. When decoupling the equations and their boundary conditions is not possible, the use of a successive approximation procedure permits the analytical solution of the resulting ordinary differential equations. The results obtained show a considerable potential for using this method in the three-dimensional analysis of finite geometry solids and suggest a possible extension of this technique to nonlinear material behavior.

Gyekenyesi, J. P.↗

NASA NTRS1975

Three-dimensional elastic stress and displacement analysis of finite geometry solids containing cracks

Gyekenyesi, J. P.↗

NASA NTRS1990

The method of lines in analyzing solids containing cracks

A semi-numerical method is reviewed for solving a set of coupled partial differential equations subject to mixed and possibly coupled boundary conditions. The line method of analysis is applied to the Navier-Cauchy equations of elastic and elastoplastic equilibrium to calculate the displacement distributions in various, simple geometry bodies containing cracks. The application of this method to the appropriate field equations leads to coupled sets of simultaneous ordinary differential equations whose solutions are obtained along sets of lines in a discretized region. When decoupling of the equations and their boundary conditions is not possible, the use of a successive approximation procedure permits the analytical solution of the resulting ordinary differential equations. The use of this method is illustrated by reviewing and presenting selected solutions of mixed boundary value problems in three dimensional fracture mechanics. These solutions are of great importance in fracture toughness testing, where accurate stress and displacement distributions are required for the calculation of certain fracture parameters. Computations obtained for typical flawed specimens include that for elastic as well as elastoplastic response. Problems in both Cartesian and cylindrical coordinate systems are included. Results are summarized for a finite geometry rectangular bar with a central through-the-thickness or rectangular surface crack under remote uniaxial tension. In addition, stress and displacement distributions are reviewed for finite circular bars with embedded penny-shaped cracks, and rods with external annular or ring cracks under opening mode tension. The results obtained show that the method of lines presents a systematic approach to the solution of some three-dimensional mechanics problems with arbitrary boundary conditions. The advantage of this method over other numerical solutions is that good results are obtained even from the use of a relatively coarse grid.

Gyekenyesi, John P.↗

DOE OSTI2014

Ultrahigh-Resolution Differential Ion Mobility Separations of Conformers for Proteins above 10 kDa: Onset of Dipole Alignment?

Biomacromolecules tend to assume numerous structures in solution or the gas phase. It has been possible to resolve disparate conformational families but not unique geometries within each, and drastic peak broadening has been the bane of protein analyses by chromatography, electrophoresis, and ion mobility spectrometry (IMS). The new differential IMS (FAIMS) approach using hydrogen-rich gases was recently found to separate conformers of a small protein ubiquitin with same peak width and resolving power up to ~400 as for peptides. Present work explores the reach of this approach for larger proteins, exemplified by cytochrome c and myoglobin. Resolution similar to that for ubiquitin was largely achieved with longer separations, while the onset of peak broadening and coalescence with shorter separations suggests the limitation of present technique to proteins under ~20 kDa. This capability may enable distinguishing whole proteins with differing residue sequences or localizations of posttranslational modifications. Small features at negative compensation voltages that markedly grow from cytochrome c to myoglobin indicate the dipole alignment of rare conformers in accord with theory, further supporting the concept of pendular macroions in FAIMS.

Environmental Molecular Sciences Laboratory↗