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NASA NTRS1991

Weak Hamiltonian finite element method for optimal control problems

A temporal finite element method based on a mixed form of the Hamiltonian weak principle is developed for dynamics and optimal control problems. The mixed form of Hamilton's weak principle contains both displacements and momenta as primary variables that are expanded in terms of nodal values and simple polynomial shape functions. Unlike other forms of Hamilton's principle, however, time derivatives of the momenta and displacements do not appear therein; instead, only the virtual momenta and virtual displacements are differentiated with respect to time. Based on the duality that is observed to exist between the mixed form of Hamilton's weak principle and variational principles governing classical optimal control problems, a temporal finite element formulation of the latter can be developed in a rather straightforward manner. Several well-known problems in dynamics and optimal control are illustrated. The example dynamics problem involves a time-marching problem. As optimal control examples, elementary trajectory optimization problems are treated.

Hodges, Dewey H.↗
NASA NTRS1990

A weak Hamiltonian finite element method for optimal control problems

A temporal finite element method based on a mixed form of the Hamiltonian weak principle is developed for dynamics and optimal control problems. The mixed form of Hamilton's weak principle contains both displacements and momenta as primary variables that are expanded in terms of nodal values and simple polynomial shape functions. Unlike other forms of Hamilton's principle, however, time derivatives of the momenta and displacements do not appear therein; instead, only the virtual momenta and virtual displacements are differentiated with respect to time. Based on the duality that is observed to exist between the mixed form of Hamilton's weak principle and variational principles governing classical optimal control problems, a temporal finite element formulation of the latter can be developed in a rather straightforward manner. Several well-known problems in dynamics and optimal control are illustrated. The example dynamics problem involves a time-marching problem. As optimal control examples, elementary trajectory optimization problems are treated.

Hodges, Dewey H.↗
NASA NTRS1987

A Taylor weak-statement algorithm for hyperbolic conservation laws

Finite element analysis, applied to computational fluid dynamics (CFD) problem classes, presents a formal procedure for establishing the ingredients of a discrete approximation numerical solution algorithm. A classical Galerkin weak-statement formulation, formed on a Taylor series extension of the conservation law system, is developed herein that embeds a set of parameters eligible for constraint according to specification of suitable norms. The derived family of Taylor weak statements is shown to contain, as special cases, over one dozen independently derived CFD algorithms published over the past several decades for the high speed flow problem class. A theoretical analysis is completed that facilitates direct qualitative comparisons. Numerical results for definitive linear and nonlinear test problems permit direct quantitative performance comparisons.

Baker, A. J.↗
NASA NTRS1989

A finite element based method for solution of optimal control problems

A temporal finite element based on a mixed form of the Hamiltonian weak principle is presented for optimal control problems. The mixed form of this principle contains both states and costates as primary variables that are expanded in terms of elemental values and simple shape functions. Unlike other variational approaches to optimal control problems, however, time derivatives of the states and costates do not appear in the governing variational equation. Instead, the only quantities whose time derivatives appear therein are virtual states and virtual costates. Also noteworthy among characteristics of the finite element formulation is the fact that in the algebraic equations which contain costates, they appear linearly. Thus, the remaining equations can be solved iteratively without initial guesses for the costates; this reduces the size of the problem by about a factor of two. Numerical results are presented herein for an elementary trajectory optimization problem which show very good agreement with the exact solution along with excellent computational efficiency and self-starting capability. The goal is to evaluate the feasibility of this approach for real-time guidance applications. To this end, a simplified two-stage, four-state model for an advanced launch vehicle application is presented which is suitable for finite element solution.

Bless, Robert R.↗
NASA NTRS2002

Fundamental Mixing and Combustion Experiments for Propelled Hypersonic Flight

The first experiment is a study of a coaxial jet discharging into stagnant laboratory air, with center jet of a mixture of 5% oxygen and 95% helium by volume and coflow jet of air. The exit flow pressure of both center-jet and coflow nozzles is 1 atmosphere. The presence of oxygen in the center jet is to allow the use of an oxygen flow-tagging technique (RELIEF4) to obtain non-intrusive velocity measurements. Both jets are nominally Mach 1.8, but, because of the greater speed of sound, the center jet velocity is more than twice that of the coflow. The mixing layer which forms between the center jet and the coflow near the nozzle exit is compressible, with a calculated convective Mach number of approximately 0.7. This geometry has several advantages: The streamwise development of the flow is generally dominated by turbulent stresses (rather than pressure forces), and thus calculations are sensitive to turbulence modeling. It includes features present in supersonic combustors, including a compressible mixing layer near the nozzle exit and a light-gas/air plume downstream. Since it is a free jet, it provides easy access for both optical instrumentation and probes. Since it is axisymmetric, it requires fewer experimental measurements to fully characterize, and calculations can be performed with more modest computer resources. However, weak shock waves formed at the nozzle exit strengthen and turn normal as they approach the axis, complicating the flow. Care is thus taken in the design of the facility to provide as near as possible to 1-D flow at the exit of both center and coflow nozzles, and to minimize the strength of waves generated at the nozzle exit. Results from this experiment are compared to CFD solutions obtained by VULCAN, a previously developed code used in engine analysis. The second experiment is a study of a supersonic combustor consisting of a diverging duct with single downstream-angled wall injector. Thus, the geometry is relatively simple and large regions of subsonic recirculating flow are avoided. The nominal entrance Mach number is 2 and the enthalpy of the test gas (hot air "simulant") is nominally that of Mach 7 flight. It was believed, on the basis of calculations performed that this would produce mixing-limited flow, that is to say, one for which chemical reaction to equilibrium proceeds at a much greater rate than mixing. It later proved that this was not the case. The primary experimental technique employed is coherent anti-Stokes Raman spectroscopy, known by its acronym CARS. The species probed is molecular nitrogen and the quantity measured is temperature. Intrusive probes, such as Pitot, total temperature, hot-wire, etc., are not used due to access difficulty and high heat flux in the combustor, and because they may alter the flow. CARS has several advantages over other optical methods. It is a relatively mature and well-understood technique. Signal levels are relatively high and the signal is in the form of a coherent (laser) beam that can be collected through small windows. Incoherent (non-CARS) interferences are rejected by spatial filtering.

Diskin, G. S.↗
NASA NTRS1989

A weak Hamiltonian finite element method for optimal guidance of an advanced launch vehicle

A temporal finite-element method based on a mixed form of the Hamiltonian weak principle is presented for optimal control problems. The mixed form of this principle contains both states and costates as primary variables, which are expanded in terms of nodal values and simple shape functions. Time derivatives of the states and costates do not appear in the governing variational equation; the only quantities whose time derivatives appear therein are virtual states and virtual costates. Numerical results are presented for an elementary trajectory optimization problem; they show very good agreement with the exact solution along with excellent computational efficiency and self-starting capability. The feasibility of this approach for real-time guidance applications is evaluated. A simplified model for an advanced launch vehicle application that is suitable for finite-element solution is presented.

Hodges, Dewey H.↗
NASA NTRS1972

Solar large-scale positive polarity magnetic fields and geomagnetic disturbances

Unlike the negative polarity solar magnetic field large-scale regular features that correlate with enhanced solar activity regions, the positive polarity regular formations formed in the weak and old background magnetic fields seem to correlate well with geomagnetically enhanced periods of time (shifted for 4 days), which means that they seem to be the source of the quiet solar wind. This behavior of the large intervals of heliographic longitude with prevailing positive polarity fields may be followed to the end of the 18th cycle, during the declining part of the 19th cycle, and during the first half of the present 20th cycle of solar activity.

Bumba, V.↗
NASA NTRS1984

Track damage and erosion of insulators by ion-induced electronic processes

Track damage and the associated ejection of atoms and molecules from insulators, which occur as a result of ion-induced electronic excitation, are of interest both in their own right and because of the mechanisms through which the energy in the excited electrons is transformed into atomic motion. In this paper an overview is given of the phenomena that are observed. We show that there is a remarkable similarity between the damage profile along the ion's track in the solid and the yield of ejected atoms at the energy that corresponds to each point on the track. It is also seen that the density of extended defects (or, correspondingly, the ejected particle yield) appears to have a 'universal' form that is weakly dependent on the type of material. In the model presented this is a consequence of the inner-shell ionization of light elements in the solid by the incident ion; the resulting Auger decay produces an intense ionization spike that locally triggers the track formation/erosion process. This model allows the estimation of erosion yields/damage profiles for different ions and materials.

Tombrello, T. A.↗
NASA NTRS1985

A Numerical Method of Calculating Propeller Noise Including Acoustic Nonlinear Effects

Using the transonic flow fields(s) generated by the NASPROP-E computer code for an eight blade SR3-series propeller, a theoretical method is investigated to calculate the total noise values and frequency content in the acoustic near and far field without using the Ffowcs Williams - Hawkings equation. The flow field is numerically generated using an implicit three dimensional Euler equation solver in weak conservation law form. Numerical damping is required by the differencing method for stability in three dimensions, and the influence of the damping on the calculated acoustic values is investigated. The acoustic near field is solved by integrating with respect to time the pressure oscillations induced at a stationary observer location. The acoustic far field is calculated from the near field primitive variables as generated by NASPROP-E computer code using a method involving a perturbation velocity potential as suggested by Hawkings in the calculation of the acoustic pressure time-history at a specified far field observed location. the methodologies described are valid for calculating total noise levels and are applicable to any propeller geometry for which a flow field solution is available.

Korkan, K. D.↗
NASA NTRS1988

Far-infrared observations of a luminous dust-shrouded source in the nucleus of NGC 4945

High-resolution far-infrared observations of the galaxy NGC 4945 have been obtained from the Kuiper Airborne Observatory. Using new observational techniques and nonlinear deconvolution routines, it is found that virtually all of the far-infrared luminosity originates from a nuclear source no larger than 12 arcsec x 9 arcsec (225 pc x 170 pc) in extent. This size constraint, coupled with the far-infrared dust temperature, indicates that the source is deeply embedded in dust: the lower limit for the 100 micron optical depth is 0.35, which is by far the largest yet measured in an external galaxy. Published optical spectra of NGC 4945 reveal a heavily obscured nonthermal source which exhibits broad line profiles typical of a Seyfert 2 active nucleus; it is concluded that the far-infrared emission is probably due to thermal radiation from dust grains surrounding the nonthermal nuclear source. A compact cluster of massive young stars may also contribute to the infrared luminosity, but the evidence for such star-forming activity is weak.

Brock, David↗
NASA NTRS1999

Frequency-Dependent Viscosity of Xenon Near the Critical Point

We used a novel, overdamped oscillator aboard the Space Shuttle to measure the viscosity eta of xenon near its critical density rho(sub c), and temperature T(sub c). In microgravity, useful data were obtained within 0.1 mK of T(sub c), corresponding to a reduced temperature t = (T -T(sub c))/T(sub c) = 3 x 10(exp -7). The data extend two decades closer to T(sub c) than the best ground measurements, and they directly reveal the expected power-law behavior eta proportional to t(sup -(nu)z(sub eta)). Here nu is the correlation length exponent, and our result for the small viscosity exponent is z(sub eta) = 0.0690 +/- 0.0006. (All uncertainties are one standard uncertainty.) Our value for z(sub eta) depends only weakly on the form of the viscosity crossover function, and it agrees with the value 0.067 +/- 0.002 obtained from a recent two-loop perturbation expansion. The measurements spanned the frequency range 2 Hz less than or equal to f less than or equal to 12 Hz and revealed viscoelasticity when t less than or equal to 10(exp -1), further from T(sub c) than predicted. The viscoelasticity scales as Af(tau), where tau is the fluctuation-decay time. The fitted value of the viscoelastic time-scale parameter A is 2.0 +/- 0.3 times the result of a one-loop perturbation calculation. Near T(sub c), the xenon's calculated time constant for thermal diffusion exceeded days. Nevertheless, the viscosity results were independent of the xenon's temperature history, indicating that the density was kept near rho(sub c), by judicious choices of the temperature vs. time program. Deliberately bad choices led to large density inhomogeneities. At t greater than 10(exp -5), the xenon approached equilibrium much faster than expected, suggesting that convection driven by microgravity and by electric fields slowly stirred the sample.

Berg, Robert F.↗
NASA NTRS2007

Observed Enhancement of Reflectivity and Electric Field in Long-Lived Florida Anvils

A study of two long-lived Florida anvils showed that reflectivity greater than 20 dBZ increased in area, thickness and sometimes magnitude at mid-level well downstream of the convective cores. In these same regions electric fields maintained strengths greater than 10 kV m(sup -1) for many tens of minutes and became quite uniform over tens of kilometers. Millimetric aggregates persisted at 9 to 10 km for extended times and distances. Aggregation of ice particles enhanced by strong electric fields might have contributed to reflectivity growth in the early anvil, but is unlikely to explain observations further out in the anvil. The enhanced reflectivity and existence of small, medium and large ice particles far out into the anvil suggest that an updraft was acting, perhaps in weak convective cells formed by instability generated from the evaporation and melting of falling ice particles. We conclude that charge separation must have occurred in these anvils, perhaps at the melting level but also at higher altitudes, in order to maintain fields greater than 10 kV m(sup -1) at 9 to 10 km for extended periods of time over large distances. We speculate that charge separation occurred as a result of ice-ice particle collisions (without supercooled water being present) via either a noninductive or perhaps even an inductive mechanism, given the observed broad ice particle spectra, the strong pre-existing electric fields and the many tens of minutes available for particle interactions. The observations, particularly in the early anvil, show that the charge structure in these anvils was quite complex.

Dye, James E.↗
NASA NTRS2007

Observed Enhancement of Reflectivity and Electric Field in Long-Lived Florida Anvils

A study of two long-lived Florida anvils showed that reflectivity >20 dBZ increased in area, thickness and sometimes magnitude at mid-level well downstream of the convective cores. In these same regions electric fields maintained strengths >10 kV m1 for many tens of minutes and became quite uniform over tens of kilometers. Millimetric aggregates persisted at 9 to 10 km for extended times and distances. Aggregation of ice particles enhanced by strong electric fields might have contributed to reflectivity growth in the early anvil, but is unlikely to explain observations further out in the anvil. The enhanced reflectivity and existence of small, medium and large ice particles far out into the anvil suggest that an updraft was acting, perhaps in weak convective cells formed by instability generated from the evaporation and melting of falling ice particles. We conclude that charge separation must have occurred in these anvils, perhaps at the melting level but also at higher altitudes, in order to maintain fields >10 kV m 1 at 9 to 10 km for extended periods of time over large distances. We speculate that charge separation occurred as a result of ice-ice particle collisions (without supercooled water being present) via either a non-inductive or perhaps even an inductive mechanism, given the observed broad ice particle spectra, the strong pre-existing electric fields and the many tens of minutes available for particle interactions. The observations, particularly in the early anvil, show that the charge structure in these anvils was quite complex.

Dye, James E.↗
NASA NTRS1986

Comet-solar wind interaction - Dynamical length scales and models

ICE magnetometer measurements at Comet Giacobini-Zinner and model simulations of comet-solar wind interactions are analyzed. The magnetometer data reveal the existence of intense hydromagnetic turbulence, a draping of the magnetic field lines to form a magnetotail, a weak shock, and a magnetic barrier region in the magnetosphere. The global models of the comet-solar wind interaction are described. The observed data and models are compared and good correlation is displayed.

Mendis, D. A.↗
NASA NTRS1986

An analysis of finite-difference and finite-volume formulations of conservation laws

Finite-difference and finite-volume formulations are analyzed in order to clear up the confusion concerning their application to the numerical solution of conservation laws. A new coordinate-free formulation of systems of conservation laws is developed, which clearly distinguishes the role of physical vectors from that of algebraic vectors which characterize the system. The analysis considers general types of equations--potential, Euler, and Navier-Stokes. Three-dimensional unsteady flows with time-varying grids are described using a single, consistent nomeclature for both formulations. Grid motion due to a non-inertial reference frame as well as flow adaptation is covered. In comparing the two formulations, it is found useful to distinguish between differences in numerical methods and differences in grid definition. The former plays a role for non-Cartesian grids, and results in only cosmetic differences in the manner in which geometric terms are handled. The differences in grid definition for the two formulations is found to be more important, since it affects the manner in which boundary conditions, zonal procedures, and grid singularities are handled at computational boundaries. The proper interpretation of strong and weak conservation-law forms for quasi-one-dimensional and axisymmetric flows is brought out.

Vinokur, Marcel↗
NASA NTRS1988

Energy dependent pitch angle distributions of auroral primary electrons

Double-layer/parallel-electric field accelerations and the subsequent electron-beam plasma interactions involving Cerenkov and anomalous cyclotron resonances are considered. It is found that these phenomena yield pitch angle distributions as noted from rocket and satellite studies. Although the electron acceleration by weak parallel electric fields forming a runaway electron tail is limited to a critical parallel energy determined by the anomalous cyclotron resonance, such a limitation does not occur with acceleration by a localized parallel electric field such as that in a double layer.

Singh, N.↗
NASA NTRS1988

Systematic properties of CO emission from galaxies. II - Weighted correlations

The dependence of molecular gas emission on various global galaxy properties is tested by examining a series of correlations. CO luminosities of entire galaxies are correlated with their H I content, blue luminosity, disk area, far-infrared luminosity, Hubble type, Elmegreen arm type, DDO luminosity class, total B-V color, disk B-V color, and H-alpha emission. A weighted correlation test is derived and applied to correct for the strong Malmquist bias in the available CO observations of galaxies. The relative strengths of the correlations indicate that molecular emission has a strong dependence on galaxy scale, a weak dependence on galaxy form, and is closely linked to tracers of star forming regions.

Verter, Frances↗
NASA NTRS1989

An analysis of finite-difference and finite-volume formulations of conservation laws

Finite-difference and finite-volume formulations are analyzed in order to clear up the confusion concerning their application to the numerical solution of conservation laws. A new coordinate-free formulation of systems of conservation laws is developed, which clearly distinguishes the role of physical vectors from that of algebraic vectors which characterize the system. The analysis considers general types of equations: potential, Euler, and Navier-Stokes. Three-dimensional unsteady flows with time-varying grids are described using a single, consistent nomenclature for both formulations. Grid motion due to a non-inertial reference frame as well as flow adaptation is covered. In comparing the two formulations, it is found useful to distinguish between differences in numerical methods and differences in grid definition. The former plays a role for non-Cartesian grids, and results in only cosmetic differences in the manner in which geometric terms are handled. The differences in grid definition for the two formulations is found to be more important, since it affects the manner in which boundary conditions, zonal procedures, and grid singularities are handled at computational boundaries. The proper interpretation of strong and weak conservation-law forms for quasi-one-dimensional and axisymmetric flows is brought out.

Vinokur, Marcel↗