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Results for “higher-order derivatives”

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.

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At least 145 records · Page 8
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Achieving Higher Order Accuracy in Space in Hydrodynamic Simulations of Self-Gravitating Gas

Modern astrophysical simulation codes employ a variety of numerical algorithms capable of achieving higher-order accuracy in both space and time. Albeit they succeed in achieving an effective higher spatial resolution and in suppressing the numerical damping of waves, to our knowledge, all current astrophysical simulations invoking self-gravity are limited to second-order accuracy in space. If we can devise an algorithm to evaluate self-gravity with a higher-order spatial accuracy, we can better the evaluation of the gravitational acceleration and gravitational energy release which dictate the evolution of many astrophysical systems. Herein, we present a numerical algorithm for self-gravitating hydrodynamics capable of achieving fourth order accuracy for a given density distribution on a Cartesian uniform grid. First, we derive the cell-averaged gravitational potential at fourth-order accuracy from the cell-averaged density by solving the Poisson equation. Next, we obtain the cell average of the product of the density and gravitational acceleration, which differs from the cell-averaged density multiplied by the cell-averaged gravitational acceleration. We then show the verification of the algorithm by applying it to critical test problems: (1) maintaining equilibria of self-gravitating slabs, even upon advection, (2) evolving a polytropic sphere with a massive power-law envelope, and (3) conservation of specific entropy during the propagation of a sound wave.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

The Principle of Maximum Conformality Correctly Resolves the Renormalization-Scheme-Dependence Problem

In this paper, we clarify a serious misinterpretation and consequent misuse of the Principle of Maximum Conformality (PMC), which also can serve as a mini-review of PMC. In a recently published article, P. M. Stevenson has claimed that “the PMC is ineffective and does nothing to resolve the renormalization-scheme-dependence problem”, concluding incorrectly that the success of PMC predictions is due to the PMC being a “laborious, ad hoc, and back-door” version of the Principle of Minimal Sensitivity (PMS). We show that such conclusions are incorrect, deriving from a misinterpretation of the PMC and an overestimation of the applicability of the PMS. The purpose of the PMC is to achieve precise fixed-order pQCD predictions, free from conventional renormalization schemes and scale ambiguities. We demonstrate that the PMC predictions satisfy all the self-consistency conditions of the renormalization group and standard renormalization-group invariance; the PMC predictions are thus independent of any initial choice of renormalization scheme and scale. The scheme independence of the PMC is also ensured by commensurate scale relations, which relate different observables to each other. Moreover, in the Abelian limit, the PMC dovetails into the well-known Gell-Mann–Low framework, a method universally revered for its precision in QED calculations. Due to the elimination of factorially divergent renormalon terms, the PMC series not only attains a convergence behavior far superior to that of its conventional counterparts but also deftly curtails any residual scale dependence caused by the unknown higher-order terms. This refined convergence, coupled with its robust suppression of residual uncertainties, furnishes a sound and reliable foundation for estimating the contributions from unknown higher-order terms. Anchored in the bedrock of standard renormalization-group invariance, the PMC simultaneously eradicates the factorial divergences and eliminates superfluous systematic errors, which inversely provides a good foundation for achieving high-precision pQCD predictions. Consequently, owing to its rigorous theoretical underpinnings, the PMC is eminently applicable to virtually all high-energy hadronic processes.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1975

High-order numerical solutions using cubic splines

The cubic spline collocation procedure for the numerical solution of partial differential equations was reformulated so that the accuracy of the second-derivative approximation is improved and parallels that previously obtained for lower derivative terms. The final result is a numerical procedure having overall third-order accuracy for a nonuniform mesh and overall fourth-order accuracy for a uniform mesh. Application of the technique was made to the Burger's equation, to the flow around a linear corner, to the potential flow over a circular cylinder, and to boundary layer problems. The results confirmed the higher-order accuracy of the spline method and suggest that accurate solutions for more practical flow problems can be obtained with relatively coarse nonuniform meshes.

Rubin, S. G.↗
NASA NTRS1987

A higher-order theory for geometrically nonlinear analysis of composite laminates

A third-order shear deformation theory of laminated composite plates and shells is developed, the Navier solutions are derived, and its finite element models are developed. The theory allows parabolic description of the transverse shear stresses, and therefore the shear correction factors of the usual shear deformation theory are not required in the present theory. The theory also accounts for the von Karman nonlinear strains. Closed-form solutions of the theory for rectangular cross-ply and angle-ply plates and cross-ply shells are developed. The finite element model is based on independent approximations of the displacements and bending moments (i.e., mixed finite element model), and therefore, only C sup o -approximation is required. The finite element model is used to analyze cross-ply and angle-ply laminated plates and shells for bending and natural vibration. Many of the numerical results presented here should serve as references for future investigations. Three major conclusions resulted from the research: First, for thick laminates, shear deformation theories predict deflections, stresses and vibration frequencies significantly different from those predicted by classical theories. Second, even for thin laminates, shear deformation effects are significant in dynamic and geometrically nonlinear analyses. Third, the present third-order theory is more accurate compared to the classical and firt-order theories in predicting static and dynamic response of laminated plates and shells made of high-modulus composite materials.

Reddy, J. N.↗
NASA NTRS1987

A study of some approximations in the modeling of a coupled rotor/fuselage aeromechanical system

Equations of motion for a three-bladed hingeless rotor coupled to a gimbal-supported fuselage are developed. A brief derivation of the equations is presented, including the assumptions made, the ordering scheme, and a discussion of both the source of the aerodynamic axial load term and the reason for differences due to hinge sequence. The solution of the equations is discussed, including linearization, the use of multiblade coordinates to eliminate periodic coefficients, and specialization to a root-sprung model. The treatment of the blade root offset by using an ordering scheme to neglect higher-order terms is found to affect results in pitch and roll. A small aerodynamic loading term in the blade axial direction has a significant effect at large collective pitch settings, as does the choice of a flap-lag or a lag-flap transformation sequence for blade motion. The results for a hingeless rotor also differ significantly from those for the root-sprung model when the nonrotating natural frequencies are matched.

Blelloch, P. A.↗
NASA NTRS1987

Covariance and the hierarchy of frame bundles

This is an essay on the general concept of covariance, and its connection with the structure of the nested set of higher frame bundles over a differentiable manifold. Examples of covariant geometric objects include not only linear tensor fields, densities and forms, but affinity fields, sectors and sector forms, higher order frame fields, etc., often having nonlinear transformation rules and Lie derivatives. The intrinsic, or invariant, sets of forms that arise on frame bundles satisfy the graded Cartan-Maurer structure equations of an infinite Lie algebra. Reduction of these gives invariant structure equations for Lie pseudogroups, and for G-structures of various orders. Some new results are introduced for prolongation of structure equations, and for treatment of Riemannian geometry with higher-order moving frames. The use of invariant form equations for nonlinear field physics is implicitly advocated.

Estabrook, Frank B.↗
NASA NTRS1989

Derivation of a Tappered p-Version Beam Finite Element

A tapered p-version beam finite element suitable for dynamic applications is derived. The taper in the element is represented by allowing the area moments of inertia to vary as quartic polynomials along the length of the beam, and the cross-sectional area to vary as a quadratic polynomial. The p-version finite-element characteristics are implemented through a set of polynomial shape functions. The lower-order shape functions are identical to the classical cubic and linear shape functions normally associated with a beam element. The higher-order shape functions are a hierarchical set of polynomials that are integrals of orthogonal polynomials. Explicit expressions for the mass and stiffness matrices are presented for an arbitrary value of p. The element has been verified to be numerically stable using shape functions through 22nd order.

Hinnant, Howard E.↗
NASA NTRS1990

TE scattering by a two-dimensional groove in a ground plane using higher order boundary conditions

Higher order boundary conditions involve derivatives of the fields beyond the first and were recently shown to be more effective than traditional first order conditions in modeling dielectric coatings and layers. In this report an application of a third order generalized boundary condition to scattering by a filled rectangular groove is presented. Deficiencies of such higher-order boundary conditions are addressed and a correction is proposed for the present case. As part of the process of examining and correcting the accuracy of the proposed generalized boundary conditions, an exact solution is developed and a comparison is provided with a solution based on the standard impedance boundary condition.

Barkeshli, Kasra↗
NASA NTRS1992

Oligomeric baroeffect and gas aggregation states

The baroeffect is analyzed to include a gas that aggregates into higher-order polymers or oligomers. The resulting pressure change is found to vary independently of the molecular weight of the gas components and to depend only on the aggregation or oligomeric order of the gas. With increasing aggregation, diffusive slip velocities are found to increase. The calculations are extended to include general counterdiffusion of two distinct aggregation states (k-, j-mer) for the gas, and the pressure change is derived as a function that is independent of both molecular weight and the absolute aggregation. The only parameter that determines the baroeffect is the ratio of aggregated states, beta = k/j. For gases that reversibly aggregate, possible oscillatory behavior and complex dynamics for pressure are discussed. Gas aggregation may play a role for low-temperature crystal-growth conditions in which vapor concentrations of one (or more) species are high.

Noever, David A.↗
NASA NTRS1992

Near-wall reconstruction of higher order moments and length scales using the POD

An analysis of the near-wall behavior of the proper orthogonal decomposition (POD) eigenfunctions derived from direct numerical simulation (DNS) of channel flow is performed. Consistent with previous studies, a low order multi-mode reconstruction of the kinetic energy and Reynolds shear stress suffices. A similar reconstruction of the isotropic dissipation rate is shown to be insufficient, however. An analysis is performed of the multi-mode composition of the dissipation rate in the near-wall region, and it is shown that a significant number of higher-order modes are required to achieve the correct asymptotic consistency in the near-wall region. In an attempt to avoid this problem, a length scale definition is proposed in terms of an integration of the correlation tensor which factors in the presence of the wall. The wall is accounted for by only integrating out to 2y(+) and not over the entire domain. Viscous and inviscid estimates for the dissipation were used in the near-wall and core regions respectively, in conjunction with this length scale representation to obtain an estimate of the dissipation throughout the domain. The resulting dissipation exhibits the proper behavior near the wall and in the inertial layer. A 1 POD mode estimate of the length scale is computed and found to agree quite well with the length scale obtained when the entire correlation tensor is used.

Glauser, Mark N.↗
NASA NTRS1993

High-frequency techniques for RCS prediction of plate geometries and a physical optics/equivalent currents model for the RCS of trihedral corner reflectors

Part 1 of this report continues the investigation, initiated in previous reports, of scattering from rectangular plates coated with lossy dielectrics. The hard polarization coefficients given in the last report are incorporated into a model, which includes second- and third-order diffractions, for the coated plate. Computed results from this model are examined and compared to measured data. A breakdown of the contribution of each of the higher-order terms to the total radar cross section (RCS) is given. The effectiveness of the uniform theory of diffraction (UTD) model in accounting for the coating effect is investigated by examining a Physical Optics (PO) model which incorporates the equivalent surface impedance approximation used in the UTD model. The PO, UTD, and experimental results are compared. Part 2 of this report presents a RCS model, based on PO and the Method of Equivalent Currents (MEC), for a trihedral corner reflector. PO is used to account for the reflected fields, while MEC is used for the diffracted fields. Single, double, and triple reflections and first-order diffractions are included in the model. A detailed derivation of the E(sub theta)-polarization, monostatic RCS is included. Computed results are compared with finite-difference time-domain (FDTD) results for validation. The PO/MEC model of this report compares very well with the FDTD model, and it is a much faster model in terms of computational speed.

Balanis, Constantine A.↗
NASA NTRS1992

Variable frequency pulsed phase-locked loop method for measuring material nonlinearity

The design of a pressure vessel with an isothermal volume to measure changes in the natural velocities of material continuously over a wide temperature range is reported. Highly sensitive pulsed phase-locked loop technology is used to measure the natural velocity of fatigued aluminum 2024-T4 as a function of pressure at different temperatures. Second derivative variations with respect to pressure and temperature are found to exhibit order-of-magnitude changes at a gauge pressure near 950 kPa at room temperature. This indicates a significant change in certain combinations of second-order and higher-order elastic constants at relatively low pressures. Fractional frequency change as a function of applied pressure for aluminum 2024-T4 with fatigue damage is illustrated.

Yost, William T.↗
NASA NTRS1993

Collisionless parallel shocks

Consideration is given to a collisionless parallel shock based on solitary-type solutions of the modified derivative nonlinear Schroedinger equation (MDNLS) for parallel Alfven waves. The standard derivative nonlinear Schroedinger equation is generalized in order to include the possible anisotropy of the plasma distribution and higher-order Korteweg-de Vies-type dispersion. Stationary solutions of MDNLS are discussed. The anisotropic nature of 'adiabatic' reflections leads to the asymmetric particle distribution in the upstream as well as in the downstream regions of the shock. As a result, nonzero heat flux appears near the front of the shock. It is shown that this causes the stochastic behavior of the nonlinear waves, which can significantly contribute to the shock thermalization.

Khabibrakhmanov, I. KH.↗
NASA NTRS1994

A comparison of Lorentz, planetary gravitational, and satellite gravitational resonances

We consider a charged dust grain whose orbital motion is dominated by a planet's point-source gravity, but perturbed by higher-order terms in the planet's gravity field as well as by the Lorentz force arising from an asymmetric planetary magnetic field. Perturbations to Keplerian orbits due to a nonspherical gravity field are expressed in the traditional way: in terms of a disturbing function which can be expanded in a series of spherical harmonics (W. M. Kaula, 1966). In order to calculate the electromagnetic perturbation, we first write the Lorentz force in terms of the orbital elements and then substitute it into Gauss' perturbation equations. We use our result to derive strengths of Lorentz resonances and elucidate their properties. In particular, we compare Lorentz resonances to two types of gravitational resonances: those arising from periodic tugs of a satellite and those due to the attraction of an arbitrarily shaped planet. We find that Lorentz resonances share numerous properties with their gravitational counterparts and show, using simple physical arguments, that several of these patterns are fundamental, applying not only to our expansions, but to all quantities expressed in terms of orbital elements. Some of these patterns have been previously called 'd'Alembert rules' for satellite resonances. Other similarities arise because, to first-order in the perturbing force, the three problems share an integral of the motion. Yet there are also differences; for example, first-order inclination resonances exist for perturbations arising from planetary gravity and from the Lorentz force, but not for those due to an orbiting satellite. Finally, we provide a heuristic treatment of a particle's orbital evolution under the influence of drag and resonant forces. Particles brought into mean-motion resonances experience either trapping or resonant 'jumps,' depending on the direction from which the resonance is approached. We show that this behavior does not depend on the details of the perturbing force but rather is fundamental to all mean-motion resonances.

Hamilton, Douglas P.↗
NASA NTRS1993

Higher-order ionospheric effects on the GPS observables and means of modeling them

Based on relaistic modeling of the electron density of the ionosphere and using a dipole moment approximation for the earth magnetic field, we are able to estimate the effect of the ionosphere on the Global Positioning Systems (GPS) signal for a ground user. The lowest-order (1/f(exp 2)) effect, which is of the order of .1 - 30 meters of zenith group delay, is subtracted out by forming a linear combination of the dual frequencies of the GPS signal. One is left with second- (1/f(exp 3)) and third-order (1/f(exp 4)) effects which are estimated typically to be approximately 0 - 2 cm, and approximately 0 - 2 mm at zenith respectively, depending on the time of day, time of year, the solar cycle and the relative geometry of the magnetic field and the line of sight. Given the total electron content along a line of sight, we derive an approximation to the second-order term which is accurate to approximately 90% within the magnetic dipole moment model; this approximation can be used to reduce the second-order term to the millimeter level, thus potentially improving precise positioning in space and on the ground. The induced group delay, or phase advance, due to second- and third-order effects are examined for two ground receivers located at equatorial and mid-latitude regions tracking several GPS satellites.

Bassiri, Sassan↗
NASA NTRS1997

Viscous Effects in the Elastodynamics of Thick Beams

A viscoelastic higher-order thick beam finite element formulation is extended to include elastodynamic deformations. The material constitutive law is a special differential form of the Maxwell solid. In the constitutive model, the elastic strains and the conjugate viscous strains are coupled through a system of first- order ordinary differential equations. The total time-dependent stress is the superposition of its elastic and viscous components. The elastodynamic equations of motion are derived from the virtual work principle. Computational examples are carried out for a thick orthotropic cantilevered beam. A quasi-static relaxation problem is employed as a validation test for the elastodynamic algorithm. The elastodynamic code is demonstrated by analyzing the damped vibrations of the beam which is deformed and then released to freely vibrate.

Johnson, A. R.↗
NASA NTRS2018

Representativeness of CO and O3 Along ATom Transects Derived from GEOS-5 and GMI-CTM Simulations

One major goal for the NASA Atmospheric Tomography Mission (ATom) is producing an observation-based chemical climatology to represent the atmospheric heterogeneity. In this study, we use CO and O3 observations and global atmospheric model simulations to examine the spatial representativeness of the ATom-1 and -2 transects within a 4D framework provided by the NASA GEOS-5 and GMI-CTM models. Based on the probability density functions, we find that the variability of CO and O3 along the flight tracks is well hindcast by the model when sampled per ATom flights. The CO variations along the ATom-transect are likely representative of the typical CO variations over the whole Pacific basin during both the ATom-1 and -2 periods, the northern Atlantic during the ATom-1 period, and the tropical Atlantic in the ATom-2 period. Over southern Atlantic, CO along the ATom-1 transects is likely less well mixed than that of the broader region, but is still representative of the median CO concentration. CO along the ATom-2 transect is likely higher than the median CO concentration over this region. For O3, the agreements between PDFs of O3 sampled along the ATom transects and over the broader regions are fair to good over all six regions (Scores > 0.65) with notable discrepancies over some regions. For example, in ATom-1 over the northern Pacific and Atlantic, the transect samples air masses with higher O3 levels. During ATom-2, the transect over-represents the occurrence of O3 plumes over tropical Pacific. Over the southern Pacific and Atlantic for both ATom-1 and -2, the transects have a less uniform distribution compared to the surrounding basins, but still represent the median O3 abundance. Overall, we conclude in most cases that ATom measurements represent the statistical variations of these two species over the ocean basins at the time of measurement. Higher-order statistics, including covariance of species, has not been tested in this study.

ozone↗
NASA NTRS2021

Evaluation of Higher-order Quadrature Schemes in Improving Computational Efficiency for Orientation-averaged Single-Scattering Properties of Nonspherical Ice Particles

We evaluate several high-order quadrature schemes for accuracy and efficacy in obtaining orientation-averaged single-scattering properties (SSPs). We use the highly efficient MIDAS to perform electromagnetic scattering calculations to evaluate the gain in efficiency from these schemes. MIDAS is shown to be superior to DDSCAT, a popular discrete dipole approximation (DDA) method. This study is motivated by the fact that quality physical precipitation retrievals rely on using accurate orientation-averaged SSPs derived from realistic hydrometeors as input to radiative transfer Models (RTMs). The DDA has been a popular choice for single-scattering calculations, due to its versatility with respect to target geometry. However, being iterative-solver-based (ISB), the most used DDA codes, e.g. DDSCAT and ADDA, must solve the scattering problem for each orientation of the target separately. As the size parameter and geometric anisotropy of the hydrometeor increase, the number of orientations needed to obtain accurate orientation-averages can increase drastically and so does the computation cost incurred by the ISB-DDA methods. MIDAS is a Direct-Solver-Based (DSB) code, its decomposition of the original large matrix with a high rank into multiple more manageable smaller matrices of lower ranks makes it much more computationally efficient while maintaining excellent accuracy. In addition, direct solvers consider all requested orientations at once, giving MIDAS further advantage over popular ISB-DDA methods. MIDAS, when combined with high-order quadrature for orientation averaging, can be greater than three orders of magnitude more efficient in obtaining RTM-ready SSPs of complex-shaped hydrometeors than existing ISB-DDA methods, with the native quadrature schemes they offer.

Ines Fenni↗