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NASA NTRS1999

Computation of Steady and Unsteady Laminar Flames: Theory

In this paper we describe the numerical analysis underlying our efforts to develop an accurate and reliable code for simulating flame propagation using complex physical and chemical models. We discuss our spatial and temporal discretization schemes, which in our current implementations range in order from two to six. In space we use staggered meshes to define discrete divergence and gradient operators, allowing us to approximate complex diffusion operators while maintaining ellipticity. Our temporal discretization is based on the use of preconditioning to produce a highly efficient linearly implicit method with good stability properties. High order for time accurate simulations is obtained through the use of extrapolation or deferred correction procedures. We also discuss our techniques for computing stationary flames. The primary issue here is the automatic generation of initial approximations for the application of Newton's method. We use a novel time-stepping procedure, which allows the dynamic updating of the flame speed and forces the flame front towards a specified location. Numerical experiments are presented, primarily for the stationary flame problem. These illustrate the reliability of our techniques, and the dependence of the results on various code parameters.

Hagstrom, Thomas↗

NASA NTRS1988

The dynamics and control of large-flexible space structures, part 10

A mathematical model is developed to predict the dynamics of the proposed orbiting Spacecraft Control Laboratory Experiment (SCOLE) during the station keeping phase. The equations of motion are derived using a Newton-Euler formulation. The model includes the effects of gravity, flexibility, and orbital dynamics. The control is assumed to be provided to the system through the Shuttle's three torquers, and through six actuators located by pairs at two points on the mast and at the mass center of the reflector. The modal shape functions are derived using the fourth order beam equation. The generic mode equations are derived to account for the effects of the control forces on the modal shape and frequencies. The equations are linearized about a nominal equilibrium position. The linear regulator theory is used to derive control laws for both the linear model of the rigidized SCOLE as well as that of the actual SCOLE including the first four flexible modes. The control strategy previously derived for the linear model of the rigidized SCOLE is applied to the nonlinear model of the same configuration of the system and preliminary single axis slewing maneuvers conducted. The results obtained confirm the applicability of the intuitive and appealing two-stage control strategy which would slew the SCOLE system, as if rigid to its desired position and then concentrate on damping out the residual flexible motions.

Bainum, Peter M.↗

NASA NTRS1988

The dynamics and control of the orbiting spacecraft control laboratory experiment (SCOLE) during station keeping

A mathematical model is developed to predict the dynamics of the proposed orbiting Spacecraft Control Laboratory Experiment during the station keeping phase. The Shuttle as well as the reflector are assumed to be rigid, the mast is flexible and is assumed to undergo elastic displacements very small as compared with its length. The equations of motion are derived using a Newton-Euler formulation. The model includes the effects of gravity, flexibility, and orbital dynamics. The control is assumed to be provided to the system through the Shuttle's three torquers, and through six actuators located by pairs at two points on the mast and at the mass center of the reflector. At each of the locations, an actuator acts parallel to the roll axis while the other one acts parallel to the pitch axis. It is seen that, in the presence of gravity-gradient torques in the system dynamics, the system assumes a new equilibrium position about which the equations must be linearized, primarily due to the offset in the mast attachment point to the reflector. The linear regulator theory is used to derive control laws for the linear model of the SCOLE including the first four flexible modes. Numerical results confirm the robustness of this control strategy for station keeping with maximum control efforts significantly below saturation levels.

Bainum, Peter M.↗

NASA NTRS2012

The Outer Limits of Galaxy Clusters: Observations to the Virial Radius with Suzaku, XMM,and Chandra

The outskirts of galaxy clusters, near the virial radius, remain relatively unexplored territory and yet are vital to our understanding of cluster growth, structure, and mass. In this presentation, we show the first results from a program to constrain the sate of the outer intra-cluster medium (ICM) in a large sample of galaxy clusters, exploiting the strengths of three complementary X-ray observatories: Suzaku (low, stable background), XMM-Newton (high sensitivity),and Chandra (good spatial resolution). By carefully combining observations from the cluster core to beyond r200, we are able to identify and reduce systematic uncertainties that would impede our spatial and spectral analysis using a single telescope. Our sample comprises nine clusters at z is approximately 0.1-0.2 fully covered in azimuth to beyond r200, and our analysis indicates that the ICM is not in hydrostatic equilibrium in the cluster outskirts, where we see clear azimuthal variations in temperature and surface brightness. In one of the clusters, we are able to measure the diffuse X-ray emission well beyond r200, and we find that the entropy profile and the gas fraction are consistent with expectations from theory and numerical simulations. These results stand in contrast to recent studies which point to gas clumping in the outskirts; the extent to which differences of cluster environment or instrumental effects factor in this difference remains unclear. From a broader perspective, this project will produce a sizeable fiducial data set for detailed comparison with high-resolution numerical simulations.

Miller, Eric D.↗

NASA NTRS1987

Analysis of a parallelized nonlinear elliptic boundary value problem solver with application to reacting flows

A parallelized finite difference code based on the Newton method for systems of nonlinear elliptic boundary value problems in two dimensions is analyzed in terms of computational complexity and parallel efficiency. An approximate cost function depending on 15 dimensionless parameters is derived for algorithms based on stripwise and boxwise decompositions of the domain and a one-to-one assignment of the strip or box subdomains to processors. The sensitivity of the cost functions to the parameters is explored in regions of parameter space corresponding to model small-order systems with inexpensive function evaluations and also a coupled system of nineteen equations with very expensive function evaluations. The algorithm was implemented on the Intel Hypercube, and some experimental results for the model problems with stripwise decompositions are presented and compared with the theory. In the context of computational combustion problems, multiprocessors of either message-passing or shared-memory type may be employed with stripwise decompositions to realize speedup of O(n), where n is mesh resolution in one direction, for reasonable n.

Keyes, David E.↗

NASA NTRS2004

The Sombrero: An Edge-On Spiral: Halo Dynamics, A Massive Dark Object, and XRBS

We present an analysis of the XMM-Newton and Chandra ACIS-S observations of the LINER nucleus of the Sombrero galaxy, and we discuss possible explanations for its very Sub-Eddington luminosity by complementing the X-ray results with high angular resolution observations in other bands. The X-ray investigation shows a hard (gamma=1.89) and moderately absorbed (NH=1.8x1021 per square centimeter) nuclear source of 1.5x1040 ergs per second in the 2-10 keV band, surrounded by hot gas at a temperature of approximately 0.6 keV. The bolometric nuclear luminosity is at least approximately 200 times lower than expected if mass accreted on the supermassive black hole, which Hubble Space Telescope shows to reside at the center of this galaxy, at the rate predicted by the spherical and adiabatic Bondi accretion theory and with the high radiative efficiency of a standard accretion disk. The low luminosity, coupled to the observed absence of Fe K emission in the nuclear spectrum, indicates that such a disk is not present. This nucleus also differs from bright unobscured active galactic nuclei in the lack of high flux variability and prominent broad H emission. However, it is also too faint for the predictions of simple radiatively inefficient accretion taking place at the Bondi rate; it could instead be too radio bright for radiatively inefficient accretion that includes strong mass outflows or convection. This discrepancy could be solved by the possible presence of nuclear radio jets. An alternative explanation of the low luminosity, in place of radiative inefficiency, could be unsteady accretion.

Fabbiano, G.↗

NASA NTRS2003

The Sombrero: An Edge-On Spiral: Halo Dynamics, A Massive Dark Object and XRBS

We present an analysis of the XMM-Newton and Chandra ACIS-S observations of the LINER nucleus of the Sombrero galaxy and we discuss possible explanations for its very sub-Eddington luminosity by complementing the X-ray results with high angular resolution observations in other bands. The X-ray investigation shows a hard (Gamma=1.89) and moderately absorbed ( N(sub H) = 1.8 x l0(exp 21)/sq cm) nuclear source of 1.5 x l0(exp 40) erg/s in the 2-10 keV band, surrounded by hot gas at a temperature of approx. 0.6 keV. The bolometric nuclear luminosity is at least approx. 200 times lower than expected if mass accreted on the supermassive black hole, that HST shows to reside at the center of this galaxy, at the rate predicted by the spherical and adiabatic Bondi accretion theory and with the high radiative efficiency of a standard accretion disc. The low luminosity, coupled to the observed absence of Fe-K emission in the nuclear spectrum, indicates that such a disc is not present. This nucleus differs from bright unobscured AGNs also for the lack of high flux variability and of prominent broad Halpha emission. However, it is also too faint for the predictions of simple radiatively inefficient accretion taking place at the Bondi rate; it could be too radio bright, instead, for radiatively inefficient accretion that includes strong mass outflows or convection. This discrepancy could be solved by the possible presence of nuclear radio jets. An alternative explanation of the low luminosity, in place of radiative inefficiency, could be unsteady accretion.

Mushotzky, Richard F.↗

NASA NTRS1993

Adjoint methods for aerodynamic wing design

A model inverse design problem is used to investigate the effect of flow discontinuities on the optimization process. The optimization involves finding the cross-sectional area distribution of a duct that produces velocities that closely match a targeted velocity distribution. Quasi-one-dimensional flow theory is used, and the target is chosen to have a shock wave in its distribution. The objective function which quantifies the difference between the targeted and calculated velocity distributions may become non-smooth due to the interaction between the shock and the discretization of the flowfield. This paper offers two techniques to resolve the resulting problems for the optimization algorithms. The first, shock-fitting, involves careful integration of the objective function through the shock wave. The second, coordinate straining with shock penalty, uses a coordinate transformation to align the calculated shock with the target and then adds a penalty proportional to the square of the distance between the shocks. The techniques are tested using several popular sensitivity and optimization methods, including finite-differences, and direct and adjoint discrete sensitivity methods. Two optimization strategies, Gauss-Newton and sequential quadratic programming (SQP), are used to drive the objective function to a minimum.

Grossman, Bernard↗

NASA NTRS1972

Numerical method for the solution of large systems of differential equations of the boundary layer type

A numerical method for the solution of large systems of nonlinear differential equations of the boundary-layer type is described. The method is a modification of the technique for satisfying asymptotic boundary conditions. The present method employs inverse interpolation instead of the Newton method to adjust the initial conditions of the related initial-value problem. This eliminates the so-called perturbation equations. The elimination of the perturbation equations not only reduces the user's preliminary work in the application of the method, but also reduces the number of time-consuming initial-value problems to be numerically solved at each iteration. For further ease of application, the solution of the overdetermined system for the unknown initial conditions is obtained automatically by applying Golub's linear least-squares algorithm. The relative ease of application of the proposed numerical method increases directly as the order of the differential-equation system increases. Hence, the method is especially attractive for the solution of large-order systems. After the method is described, it is applied to a fifth-order problem from boundary-layer theory.

Green, M. J.↗

NASA NTRS1989

Theory of wide-angle photometry from standard stars

Wide angle celestial structures, such as bright comet tails and nearby galaxies and clusters of galaxies, rely on photographic methods for quantified morphology and photometry, primarily because electronic devices with comparable resolution and sky coverage are beyond current technological capability. The problem of the photometry of extended structures and of how this problem may be overcome through calibration by photometric standard stars is examined. The perfect properties of the ideal field of view are stated in the guise of a radiometric paraxial approximation, in the hope that fields of view of actual telescopes will conform. Fundamental radiometric concepts are worked through before the issue of atmospheric attenuation is addressed. The independence of observed atmospheric extinction and surface brightness leads off the quest for formal solutions to the problem of surface photometry. Methods and problems of solution are discussed. The spectre is confronted in the spirit of standard stars and shown to be chimerical in that light, provided certain rituals are adopted. After a brief discussion of Baker-Sampson polynomials and the vexing issue of saturation, a pursuit is made of actual numbers to be expected in real cases. While the numbers crunched are gathered ex nihilo, they demonstrate the feasibility of Newton's method in the solution of this overdetermined, nonlinear, least square, multiparametric, photometric problem.

Usher, Peter D.↗

NASA NTRS2007

A Stellar-mass Black Hole in the Ultra-luminous X-ray Source M82 X-1

We have analyzed the archival XMM-Newton data of the archetypal Ultra-Luminous X-ray Source (ULX) M82 X-1 with an LO5 ksec exposure when the source was in the steady state. Thanks to the high photon statistics from the large effective area and long exposure, we were able to discriminate different X-ray continuum spectral models. Neither the standard accretion disk model (where the radial dependency of the disk effective temperature is T(r) proportional to r(sup -3/4)) nor a power-law model gives a satisfactory fit. In fact, observed curvature of the M82 X-1 spectrum was just between those of the two models. When the exponent of the radial dependence (p in T(r) proportional to r(sup -P)) of the disk temperature is allowed to be free, we obtained p = 0.61 (sup +0.03)(sub -0.02). Such a reduction of p from the standard value 3/4 under extremely high mass accretion rates is predicted from the accretion disk theory as a consequence of the radial energy advection. Thus, the accretion disk in M82 X-1 is considered to be in the Slim disk state, where an optically thick Advection Dominant Accretion Flow (ADAF) is taking place. We have applied a theoretical slim disk spectral model to M82 X-1, and estimated the black hole mass approximately equal to 19 - 32 solar mass. We conclude that M82 X-1 is a stellar black hole which has been produced through evolution of an extremely massive star, shining at a several times the super-Eddington luminosity.

Okajima, Takashi↗

NASA NTRSDate not supplied

Off-Nominal Event Analysis in Autonomous Flights Based on Explainable Artificial Intelligence

A key objective in the Urban Air Mobility program at NASA is to intelligently perform an autonomous flight in a complex urban environment under all weather conditions with guaranteed levels of safety. To accomplish this, the mission manager (central decision-making module) of the vehicle needs to make informed decisions between various Courses of Action (CoA) based on its' interpretation of the inputs it receives. If an off-nominal event is detected either based on the amalgamation of sensor data or the use of machine learning models, the mission manager may greatly benefit from identification of the input features that most likely contributed to that specific event. Such an understanding is usually not possible to obtain from the classical machine learning models (deep learning) due to the inherent black box like structure. However, this understanding is achieved using eXplainable Artificial Intelligence (XAI) models that provide a human interpretable rationale for the predictions made. This work presents a game theory inspired XAI model for the off-nominal assessment of autonomous flights. The proposed approach based on Shapley values is model agnostic, provides local as well as global explanation and satisfies the four axioms (efficiency, symmetry, dummy, additivity) to achieve fair contribution. The versatility of the approach is first demonstrated on a simulated dataset in which the significance of each input to flight phase prediction is clearly identified. Subsequently, data from simulated flight trajectories are fed into the model which reveal the input features that most likely contributed to a rotor failure event thereby empowering the mission manager to take the appropriate CoA.

autonomy↗

NASA NTRSDate not supplied

Off-Nominal Event Analysis in Autonomous Flights Based on Explainable Artificial Intelligence

autonomy↗

NASA NTRS1995

Modified Interior Distance Functions (Theory and Methods)

In this paper we introduced and developed the theory of Modified Interior Distance Functions (MIDF's). The MIDF is a Classical Lagrangian (CL) for a constrained optimization problem which is equivalent to the initial one and can be obtained from the latter by monotone transformation both the objective function and constraints. In contrast to the Interior Distance Functions (IDF's), which played a fundamental role in Interior Point Methods (IPM's), the MIDF's are defined on an extended feasible set and along with center, have two extra tools, which control the computational process: the barrier parameter and the vector of Lagrange multipliers. The extra tools allow to attach to the MEDF's very important properties of Augmented Lagrangeans. One can consider the MIDFs as Interior Augmented Lagrangeans. It makes MIDF's similar in spirit to Modified Barrier Functions (MBF's), although there is a fundamental difference between them both in theory and methods. Based on MIDF's theory, Modified Center Methods (MCM's) have been developed and analyzed. The MCM's find an unconstrained minimizer in primal space and update the Lagrange multipliers, while both the center and the barrier parameter can be fixed or updated at each step. The MCM's convergence was investigated, and their rate of convergence was estimated. The extension of the feasible set and the special role of the Lagrange multipliers allow to develop MCM's, which produce, in case of nondegenerate constrained optimization, a primal and dual sequences that converge to the primal-dual solutions with linear rate, even when both the center and the barrier parameter are fixed. Moreover, every Lagrange multipliers update shrinks the distance to the primal dual solution by a factor 0 less than gamma less than 1 which can be made as small as one wants by choosing a fixed interior point as a 'center' and a fixed but large enough barrier parameter. The numericai realization of MCM leads to the Newton MCM (NMCM). The approximation for the primal minimizer one finds by Newton Method followed by the Lagrange multipliers update. Due to the MCM convergence, when both the center and the barrier parameter are fixed, the condition of the MDF Hessism and the neighborhood of the primal ninimizer where Newton method is 'well' defined remains stable. It contributes to both the complexity and the numerical stability of the NMCM.

Polyak, Roman A.↗

NASA NTRS2002

Measurement of Absolute Excitation Cross Sections in Highly-Charged Ions Using Electron Energy Loss and Merged Beams

There is increasing emphasis during this decade on understanding energy balance and phenomena observed in high electron temperature plasmas. The UV spectral return from FUSE, the X-ray spectral return from the HETG on Chandra and the LETGS 011 XMM-Newton are just beginning. Line emissions are almost entirely from highly-charged ions (HCIs) of C, N, 0, Ne, Mg, S, Si, Ca, and Fe. The Constellation-X mission will provide X-ray spectroscopy up to photon energies of 0.12 nm (10 keV) where primary line emitters will be HCIs. A variety of atomic parameters are required to model the stellar and solar plasma. These include cross sections for excitation, ionization, charge-exchange, X-ray emission, direct and indirect recombination, lifetimes and branching ratios, and dependences on l, m mixing by external E and B fields. In almost all cases the atomic quantities are calculated, and few comparisons to experiment have been carried out. Collision strengths and Einstein A-values are required to convert the observed spectral intensities to electron temperatures and densities in the stellar plasma. The JPL electron energy-loss and merged beam approach has been used to measure absolute collision strengths in a number of ions, with critical comparison made to the best available theories.

Chutjian, A.↗

NASA NTRS1998

An Investigation into the Elementary Temporal Structure of Solar Flare Hard X-Ray Bursts Using BATSE

The research performed under this contract is part of an on-going investigation to explore the finest time-resolution hard X-ray data available on solar flares. Since 1991, the Burst and Transient Source Experiment (BATSE) aboard the Compton Gamma Ray Observatory has provided almost continual monitoring of the Sun in the hard X-ray and gamma-ray region of the spectrum. BATSE provides for the first time a temporal resolution in the data comparable to the timescales on which flare particle energization occurs. Under this contract, we have employed an important but under-utilized BATSE data type, the Time-To-Spill (TTS) data, to address the question of how fine a temporal structure exists in flare hard X-ray emission. By establishing the extent to which "energy release fragments," or characteristic (recurrent) time structures, are building blocks of flare emission, it is possible to place constraints on particle acceleration theories. We have utilized a spectral estimation technique, known as Lomb's normalized periodogram, to overcome the challenge of computing the power spectra of the unevenly sampled TTS data. By comparing the flare's power spectra to the expected power arising from Poisson noise, we obtain measurements of the smallest, statistically significant timescales present in the data. We have found, in an initial sample of 100 flares, that the smallest statistically significant timescales detected in a single flare are: 89 ms (30 May 1991 11:26:06) in channel 0, 117 ms (17 May 1991 09:03:20) in channel 1, 167 ms (31 May 1991 16:53:12) in channel 2, and 1.55 s (06 June 1991 01:02:08) in channel 3. We have also found some evidence for the existence of preferred timescales, however, the significance of this finding awaits a larger sample of flares.

Newton , Elizabeth↗

NASA NTRS2005

Discovery of a 105-ms X-ray Pulsar in Kesteven-79: On the Nature of Compact Central Objects in Supernova Remnants

We report the discovery of 105-ms X-ray pulsations from the compact central object (CCO) in the supernova remnant \snr\ using data acquired with the {\it Newton X-Ray Multi-Mirror Mission). Using two observations of the pulsar taken 6-days apart we derive an upper limit on its spin-down rate of $\dot P < 9 \times 10"{-14}$-s-~$~{-l)$,a nd find no evidence for binary orbital motion. The implied energy loss rate is $\dot E < 3 \times 10A{36)$-ergs-s$A{-1)$, polar magnetic field strength is $B-{\rm p) < 3 \times 10A{12)$-G, and spin-down age is $\tau > 18.5$-kyr. The latter exceeds the remnant's estimated age, suggesting that the pulsar was born spinning near its current period. The X-ray spectrum of \psr\ is best characterized as a blackbody of temperature $kT {BB) =, 0.43\pm0.02$ keV, radius $R-{BB) \approx 1.3$-km, and $~I{\rm bol) = 5.2 \times 10A{33)$ ergs-sSA{-1)$ at $d = 7.1$-kpc. The sinusoidal light curve is modulated with a pulsed fraction of $>45\%$, suggestive of a small hot spot on the surface of the rotating neutron star. The lack of a discernible pulsar wind nebula is consistent with an interpretation of \psr\ as a rotation-powered pulsar whose spin-down luminosity falls below the empirical threshold for generating bright wind nebulae, $\dot E-{\rm c) = 4 \times 10A{36)$-ergs-sSA{-I)$. The age discrepancy suggests that its $\dot E$ has always been below $\dot E c$, perhaps a distinguishing property of the CCOs. Alternatively, the X-ray spectrum of \psr\ suggests a low-luminosity AXP, but the weak inferred $B-{\rm p)$ field is incompatible with a magnetar theory of its X-ray luminosity. The ordinary spin parameters discovered from \psr\ highlight the inability of existing theories to explain the high luminosities and temperatures of CCO thermal X-ray spectra.

Gotthelf, E. V.↗

NASA NTRS1995

Bicycle ergometer instrumentation to determine muscle and bone forces during exercise

It is hypothesized that bone loss experienced by astronauts in zero gravity conditions may be curtailed by appropriate exercise. According to Wolf's law, bone regenerates when muscles produce stresses by pulling on the bone during daily activity and/or exercise on Earth. to use this theory to prevent or decrease bone loss, one needs to quantify musculoskeletal loads and relate them to bone density changes. In the context of the space program, it is desirable to determine musculoskeletal loads during exercise (using the bicycle ergometer in this case) so that one may make similar measurements on Earth and in space. In this manner, load measurements on Earth may be used as reference to generate similar loads during exercise in space. The work reported in this document entails a musculoskeletal load measurement system that, when complete, will provide forces at muscle insertion points and other contact points, on bone. This data will be used by Dr. Beth A. Todd, who is also a SSF working with Dr. Shackelford, as input to a finite element model of bone sections to determine stress distributions. A bicycle ergometer has been instrumented to measure parameters needed to determine musculoskeletal forces during exercise. A primary feature of the system is its compactness. It uses small/light sensors without line-of-sight requirements. The system developed includes sensors, signal processing, a data acquisition system, and software to collect the data. The sensors used include optical encoders to measure position and orientation of the pedal (foot), accelerometers to determine kinematic parameters of the shank and thigh, load cells to measure pedal forces on the sagittal plane, and EMG probes to measure muscle activity. The signals are processed using anti-aliasing filters and amplifiers. The sensors' output is digitized using 30 channels of a board mounted inside a 486 class PC. A program sets the data acquisition parameters and collects data during a time period specified by the user. The data is put directly into a file on the hard disk in binary form. The 30 channels are sampled at 200 KHz, and each 30 channel scan is done at a rate of 1000 Hz. The instrumented ergometer has been flown in the KC-135 zero-gravity (zero-g) flight to collect information needed to determine musculoskeletal forces under these conditions. Similar information has been collected in 1-g conditions for comparision with the results from the zero-g case. At this time, the sets of data from both experiments are being processed. An existing methodology will be used to determine the kinematic parameters of the shank and thigh using accelerometer and encoder data. This methodology was developed during the fellow's previous NASA/ASEE fellowship and thanks to a Director's Grant. In the future, a methodology to determine the musculoskeletal forces using Newton's Law of Motion and optimization techniques will be developed to determine forces exerted by particular muscles.

Figueroa, Fernando↗