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NASA NTRS1988

Total Variation Diminishing (TVD) schemes of uniform accuracy

Explicit second-order accurate finite-difference schemes for the approximation of hyperbolic conservation laws are presented. These schemes are nonlinear even for the constant coefficient case. They are based on first-order upwind schemes. Their accuracy is enhanced by locally replacing the first-order one-sided differences with either second-order one-sided differences or central differences or a blend thereof. The appropriate local difference stencils are selected such that they give TVD schemes of uniform second-order accuracy in the scalar, or linear systems, case. Like conventional TVD schemes, the new schemes avoid a Gibbs phenomenon at discontinuities of the solution, but they do not switch back to first-order accuracy, in the sense of truncation error, at extrema of the solution. The performance of the new schemes is demonstrated in several numerical tests.

Hartwich, PETER-M.↗

NASA NTRS1987

A finite difference scheme for three-dimensional steady laminar incompressible flow

A finite difference scheme for three-dimensional steady laminar incompressible flows is presented. The Navier-Stokes equations are expressed conservatively in terms of velocity and pressure increments (delta form). First order upwind differences are used for first order partial derivatives of velocity increments resulting in a diagonally dominant matrix system. Central differences are applied to all other terms for second order accuracy. The SIMPLE pressure correction algorithm is used to satisfy the continuity equation. Numerical results are presented for cubic cavity flow problems for Reynolds numbers up to 2000 and are in good agreement with other numerical results.

Hwang, Danny P.↗

NASA NTRS1987

A finite difference scheme for three-dimensional steady laminar incompressible flow

Hwang, Danny P.↗

NASA NTRS2016

Transit Timing Variations for Planets Near Eccentricity-Type Mean Motion Resonances

We derive the transit timing variations (TTVs) of two planets near a second-order mean motion resonance (MMR) on nearly circular orbits. We show that the TTVs of each planet are given by sinusoids with a frequency of jn(sub 2) - ( j - 2)n(sub 1), where j ≽ 3 is an integer characterizing the resonance and n(sub 2) and n(sub 1) are the mean motions of the outer and inner planets, respectively. The amplitude of the TTV depends on the mass of the perturbing planet, relative to the mass of the star, and on both the eccentricities and longitudes of pericenter of each planet. The TTVs of the two planets are approximated anti-correlated, with phases of ϕ and ≈ϕ + π, where the phase ϕ also depends on the eccentricities and longitudes of pericenter. Therefore, the TTVs caused by proximity to a second-order MMR do not in general uniquely determine both planet masses, eccentricities, and pericenters. This is completely analogous to the case of TTVs induced by two planets near a first-order MMR. We explore how other TTV signals, such as the short-period synodic TTV or a first-order resonant TTV, in combination with the second-order resonant TTV, can break degeneracies. Finally, we derive approximate formulae for the TTVs of planets near any order eccentricity-type MMR; this shows that the same basic sinusoidal TTV structure holds for all eccentricity-type resonances. Our general formula reduces to previously derived results near first-order MMRs.

Celestial mechanics↗

NASA NTRS1993

Relativistic effects on the bonding and properties of the hydrides of platinum

The ground state of PtH2 and several low-lying states of PtH(+) and PtH have been studied at the all-electron self-consistent-field level of theory to examine the importance of relativistic effects. The results of calculations based on Dirac-Hartree-Fock theory, nonrelativistic theory, and the spin-free no-pair relativistic approximation of Hess are compared to separate the effects of the spin-free terms and the spin-orbit terms of the Hamiltonian on the relativistic corrections to the molecular properties. Comparison is also made between first-order perturbation theory including the one-electron spin-free terms and the method of Hess to determine the size of effects beyond first order. It is found that the spin-orbit interaction significantly affects the properties and energetics of these molecules because of the participation of the Pt 5d orbitals in the bonding, and that effects beyond first order in perturbation theory are large. Any treatment of Pt compounds will have to include both the spin-free and spin-orbit interactions for an accurate description.

Dyall, Kenneth G.↗

NASA NTRS1995

The origin of spurious solutions in computational electromagnetics

The origin of spurious solutions in computational electromagnetics, which violate the divergence equations, is deeply rooted in a misconception about the first-order Maxwell's equations and in an incorrect derivation and use of the curl-curl equations. The divergence equations must be always included in the first-order Maxwell's equations to maintain the ellipticity of the system in the space domain and to guarantee the uniqueness of the solution and/or the accuracy of the numerical solutions. The div-curl method and the least-squares method provide rigorous derivation of the equivalent second-order Maxwell's equations and their boundary conditions. The node-based least-squares finite element method (LSFEM) is recommended for solving the first-order full Maxwell equations directly. Examples of the numerical solutions by LSFEM for time-harmonic problems are given to demonstrate that the LSFEM is free of spurious solutions.

Jiang, Bo-Nan↗

NASA NTRS1974

A parametric analysis of transport aircraft system weights and costs

In determining unit and operating costs for advanced aircraft, it has been found that by having first-order weight and performance approximations for the aircraft systems and structural components, a step increase in cost prediction accuracy results. This paper presents first-order approximation equations for these systems and components. These equations were developed from data for most current jet transports, and they have been ordered to use a minimum number of performance parameters such as aircraft style, number of passengers, empty and gross weight, cargo load, and operating range. A NASA Ames Research Center aircraft cost program has been used to compare calculated and actual weights for the same aircraft. Good aircraft cost correlation is shown to exist between calculated first-order and actual aircraft weight data.

Anderson, J. L.↗

NASA NTRS1990

Optimal guidance law development for an advanced launch system

A regular perturbation analysis is presented. Closed-loop simulations were performed with a first order correction including all of the atmospheric terms. In addition, a method was developed for independently checking the accuracy of the analysis and the rather extensive programming required to implement the complete first order correction with all of the aerodynamic effects included. This amounted to developing an equivalent Hamiltonian computed from the first order analysis. A second order correction was also completed for the neglected spherical Earth and back-pressure effects. Finally, an analysis was begun on a method for dealing with control inequality constraints. The results on including higher order corrections do show some improvement for this application; however, it is not known at this stage if significant improvement will result when the aerodynamic forces are included. The weak formulation for solving optimal problems was extended in order to account for state inequality constraints. The formulation was tested on three example problems and numerical results were compared to the exact solutions. Development of a general purpose computational environment for the solution of a large class of optimal control problems is under way. An example, along with the necessary input and the output, is given.

Calise, Anthony J.↗

NASA NTRS1984

A refined nonlinear theory of plates with transverse shear deformation

A higher-order shear deformation theory of plates accounting for the von Karman strains is presented. The theory contains the same dependent unknowns as in the Hencky-Mindlin type first-order shear deformation theory and accounts for parabolic distribution of the transverse shear strains through the thickness of the plate. Exact solutions of simply supported plates are obtained using the linear theory and the results are compared with the exact solutions of three-dimensional elasticity theory, the first order shear deformation theory, and the classical plate theory. The present theory predicts the deflections, stresses, and frequencies more accurately when compared to the first-order theory and the classical plate theory.

Reddy, J. N.↗

NASA NTRS2011

Effective Tree Scattering and Opacity at L-Band

This paper investigates vegetation effects at L-band by using a first-order radiative transfer (RT) model and truck-based microwave measurements over natural conifer stands to assess the applicability of the tau-omega) model over trees. The tau-omega model is a zero-order RT solution that accounts for vegetation effects with effective vegetation parameters (vegetation opacity and single-scattering albedo), which represent the canopy as a whole. This approach inherently ignores multiple-scattering effects and, therefore, has a limited validity depending on the level of scattering within the canopy. The fact that the scattering from large forest components such as branches and trunks is significant at L-band requires that zero-order vegetation parameters be evaluated (compared) along with their theoretical definitions to provide a better understanding of these parameters in the retrieval algorithms as applied to trees. This paper compares the effective vegetation opacities, computed from multi-angular pine tree brightness temperature data, against the results of two independent approaches that provide theoretical and measured optical depths. These two techniques are based on forward scattering theory and radar corner reflector measurements, respectively. The results indicate that the effective vegetation opacity values are smaller than but of similar magnitude to both radar and theoretical estimates. The effective opacity of the zero-order model is thus set equal to the theoretical opacity and an explicit expression for the effective albedo is then obtained from the zero- and first- order RT model comparison. The resultant albedo is found to have a similar magnitude as the effective albedo value obtained from brightness temperature measurements. However, it is less than half of that estimated using the theoretical calculations (0.5 - 0.6 for tree canopies at L-band). This lower observed albedo balances the scattering darkening effect of the large theoretical albedo with a first-order multiple-scattering contribution. The retrieved effective albedo is different from theoretical definitions and not the albedo of single forest elements anymore, but it becomes a global parameter, which depends on all the processes taking place within the canopy, including multiple-scattering.

Kurum, Mehmet↗

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 NTRS1989

Rotordynamic coefficients for labyrinth seals calculated by means of a finite difference technique

The compressible, turbulent, time dependent and three dimensional flow in a labyrinth seal can be described by the Navier-Stokes equations in conjunction with a turbulence model. Additionally, equations for mass and energy conservation and an equation of state are required. To solve these equations, a perturbation analysis is performed yielding zeroth order equations for centric shaft position and first order equations describing the flow field for small motions around the seal center. For numerical solution a finite difference method is applied to the zeroth and first order equations resulting in leakage and dynamic seal coefficients respectively.

Nordmann, R.↗

NASA NTRS1992

Angle-dependent Green's functions for relativistic Compton reflection

Using analytic methods, we derived the first-order Green's functions, in terms of the observing angle, for the reflected spectra after a single Compton scattering of relativistic photons by a plane-parallel semiinfinite medium of cold electrons. We also calculated the total Green's functions for incident photons of 511 keV, using Monte Carlo methods. We show that both the first-order and total Green's functions have strong angular dependences and that the first-order Green's functions play a dominant role in determining the properties of the total reflected spectra.

Hua, Xin-Min↗

NASA NTRS1993

Microdensity in space using microelectronic circuits

Single event phenomena in biological organisms are compared with similar phenomena in microeletronic circuits and the similarities and differences are discussed. Both types of phenomena appear to obey the same model, at least to first order. The general applicability of this first-order model appears to reflect the need to operate despite the noise inherent in the storage and processing of information within microscopic volume elements. Arrays of p-n junctions, each having dimensions of a few microns, are proposed as solid state microdosimeters.

Mcnulty, P. J.↗

NASA NTRS2002

Process Cost Modeling for Multi-Disciplinary Design Optimization

For early design concepts, the conventional approach to cost is normally some kind of parametric weight-based cost model. There is now ample evidence that this approach can be misleading and inaccurate. By the nature of its development, a parametric cost model requires historical data and is valid only if the new design is analogous to those for which the model was derived. Advanced aerospace vehicles have no historical production data and are nowhere near the vehicles of the past. Using an existing weight-based cost model would only lead to errors and distortions of the true production cost. This report outlines the development of a process-based cost model in which the physical elements of the vehicle are costed according to a first-order dynamics model. This theoretical cost model, first advocated by early work at MIT, has been expanded to cover the basic structures of an advanced aerospace vehicle. Elemental costs based on the geometry of the design can be summed up to provide an overall estimation of the total production cost for a design configuration. This capability to directly link any design configuration to realistic cost estimation is a key requirement for high payoff MDO problems. Another important consideration in this report is the handling of part or product complexity. Here the concept of cost modulus is introduced to take into account variability due to different materials, sizes, shapes, precision of fabrication, and equipment requirements. The most important implication of the development of the proposed process-based cost model is that different design configurations can now be quickly related to their cost estimates in a seamless calculation process easily implemented on any spreadsheet tool. In successive sections, the report addresses the issues of cost modeling as follows. First, an introduction is presented to provide the background for the research work. Next, a quick review of cost estimation techniques is made with the intention to highlight their inappropriateness for what is really needed at the conceptual phase of the design process. The First-Order Process Velocity Cost Model (FOPV) is discussed at length in the next section. This is followed by an application of the FOPV cost model to a generic wing. For designs that have no precedence as far as acquisition costs are concerned, cost data derived from the FOPV cost model may not be accurate enough because of new requirements for shape complexity, material, equipment and precision/tolerance. The concept of Cost Modulus is introduced at this point to compensate for these new burdens on the basic processes. This is treated in section 5. The cost of a design must be conveniently linked to its CAD representation. The interfacing of CAD models and spreadsheets containing the cost equations is the subject of the next section, section 6. The last section of the report is a summary of the progress made so far, and the anticipated research work to be achieved in the future.

Bao, Han P.↗

NASA NTRS1975

Remarks on the theory of aerodynamic noise

The accuracy of approximations employed in the Lighthill theory in current aerodynamic noise research is critically evaluated. Based on the method of matched asymptotic expansions, the full Navier-Stokes equations are expanded for small Mach numbers. To the first order, the near-field is generally nonisentropic. The pressure field (pseudo-sound) is generated by the incompressible Reynolds stresses in the turbulent flow and the velocity, pressure perturbation, and their derivatives on the boundaries. In the far-field, the first-order pressure (acoustic) field satisfying a linear wave equation is obtained by matching with the pseudo-sound field. A uniformly valid solution for the pressure field in a stationary or uniformly moving medium is obtained. The solution shows that the generation of the first-order aerodynamic noise does not depend on the viscous, thermal, or entropy effects in the adiabatic flow nor on the shear stress on a smooth rigid boundary.

Pan, Y. S.↗

NASA NTRS1988

A laboratory verification sensor

The use of a variant of the Hartmann test is described to sense the coalignment of the 36 primary mirror segments of the Keck 10-meter Telescope. The Shack-Hartmann alignment camera is a surface-tilt-error-sensing device, operable with high sensitivity over a wide range of tilt errors. An interferometer, on the other hand, is a surface-height-error-sensing device. In general, if the surface height error exceeds a few wavelengths of the incident illumination, an interferogram is difficult to interpret and loses utility. The Shack-Hartmann aligment camera is, therefore, likely to be attractive as a development tool for segmented mirror telescopes, particularly at early stages of development in which the surface quality of developmental segments may be too poor to justify interferometric testing. The constraints are examined which would define the first-order properties of a Shack-Hartmann alignment camera and the precision and range of measurement one could expect to achieve with it are investigated. Fundamental constraints do arise, however, from consideration of geometrical imaging, diffraction, and the density of sampling of images at the detector array. Geometrical imagining determines the linear size of the image, and depends on the primary mirror diameter and the f-number of a lenslet. Diffraction is another constraint; it depends on the lenslet aperture. Finally, the sampling density at the detector array is important since the number of pixels in the image determines how accurately the centroid of the image can be measured. When these factors are considered under realistic assumptions it is apparent that the first order design of a Shack-Hartmann alignment camera is completely determined by the first-order constraints considered, and that in the case of a 20-meter telescope with seeing-limited imaging, such a camera, used with a suitable detector array, will achieve useful precision.

Vaughan, Arthur H.↗

NASA NTRS2016

High-Order Residual-Distribution Schemes for Discontinuous Problems on Irregular Triangular Grids

In this paper, we develop second- and third-order non-oscillatory shock-capturing hyperbolic residual distribution schemes for irregular triangular grids, extending our second- and third-order schemes to discontinuous problems. We present extended first-order N- and Rusanov-scheme formulations for hyperbolic advection-diffusion system, and demonstrate that the hyperbolic diffusion term does not affect the solution of inviscid problems for vanishingly small viscous coefficient. We then propose second- and third-order blended hyperbolic residual-distribution schemes with the extended first-order Rusanov-scheme. We show that these proposed schemes are extremely accurate in predicting non-oscillatory solutions for discontinuous problems. We also propose a characteristics-based nonlinear wave sensor for accurately detecting shocks, compression, and expansion regions. Using this proposed sensor, we demonstrate that the developed hyperbolic blended schemes do not produce entropy-violating solutions (unphysical stocks). We then verify the design order of accuracy of these blended schemes on irregular triangular grids.

Mazaheri, Alireza↗