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NASA NTRS1987

Adaptive finite element methods for high-speed compressible flows

An adaptive finite element algorithm for solving the unsteady Euler equations is described. The finite element algorithm is based on a Taylor/Galerkin formulation and uses a very fast and efficient data structure to refine and unrefine the grid in order to optimize the approximation. A general version of the method which can be applied to moving grids with sliding interfaces is given, and results for a transient supersonic calculation of rotor-stator interaction are presented.

Oden, J. T.↗

NASA NTRS1994

The classification of the Arctic Sea ice types and the determination of surface temperature using advanced very high resolution radiometer data

The accurate quantification of new ice and open water areas and surface temperatures within the sea ice packs is a key to the realistic parameterization of heat, moisture, and turbulence fluxes between ocean and atmosphere in the polar regions. Multispectral NOAA advanced very high resolution radiometer/2 (AVHRR/2) satellite images are analyzed to evaluate how effectively the data can be used to characterize sea ice in the Bering and Greenland seas, both in terms of surface type and physical temperature. The basis of the classification algorithm, which is developed using a late wintertime Bering Sea ice cover data, is that frequency distributions of 10.8- micrometers radiances provide four distinct peaks, represeting open water, new ice, young ice, and thick ice with a snow cover. The results are found to be spatially and temporally consistent. Possible sources of ambiguity, especially associated with wider temporal and spatial application of the technique, are discussed. An ice surface temperature algorithm is developed for the same study area by regressing thermal infrared data from 10.8- and 12.0- micrometers channels against station air temperatures, which are assumed to approximate the skin temperatures of adjacent snow and ice. The standard deviations of the results when compared with in situ data are about 0.5 K over leads and polynyas to about 0.5-1.5 K over thick ice. This study is based upon a set of in situ data limited in scope and coverage. Cloud masks are applied using a thresholding technique that utilizes 3.74- and 10.8- micrometers channel data. The temperature maps produced show coherence with surface features like new ice and leads, and consistency with corresponding surface type maps. Further studies are needed to better understand the effects of both the spatial and temporal variability in emissivity, aerosol and precipitable atmospheric ice particle distribution, and atmospheric temperature inversions.

Massom, Robert↗

NASA NTRS1997

Convergence Estimates for Multidisciplinary Analysis and Optimization

A quantitative analysis of coupling between systems of equations is introduced. This analysis is then applied to problems in multidisciplinary analysis, sensitivity, and optimization. For the sensitivity and optimization problems both multidisciplinary and single discipline feasibility schemes are considered. In all these cases a "convergence factor" is estimated in terms of the Jacobians and Hessians of the system, thus it can also be approximated by existing disciplinary analysis and optimization codes. The convergence factor is identified with the measure for the "coupling" between the disciplines in the system. Applications to algorithm development are discussed. Demonstration of the convergence estimates and numerical results are given for a system composed of two non-linear algebraic equations, and for a system composed of two PDEs modeling aeroelasticity.

Arian, Eyal↗

NASA NTRS1994

A Numerical Comparison of Barrier and Modified Barrier Methods for Large-Scale Bound-Constrained Optimization

When a classical barrier method is applied to the solution of a nonlinear programming problem with inequality constraints, the Hessian matrix of the barrier function becomes increasingly ill-conditioned as the solution is approached. As a result, it may be desirable to consider alternative numerical algorithms. We compare the performance of two methods motivated by barrier functions. The first is a stabilized form of the classical barrier method, where a numerically stable approximation to the Newton direction is used when the barrier parameter is small. The second is a modified barrier method where a barrier function is applied to a shifted form of the problem, and the resulting barrier terms are scaled by estimates of the optimal Lagrange multipliers. The condition number of the Hessian matrix of the resulting modified barrier function remains bounded as the solution to the constrained optimization problem is approached. Both of these techniques can be used in the context of a truncated-Newton method, and hence can be applied to large problems, as well as on parallel computers. In this paper, both techniques are applied to problems with bound constraints and we compare their practical behavior.

Nash, Stephen G.↗

NASA NTRS2000

Analysis of Phase-Type Stochastic Petri Nets With Discrete and Continuous Timing

The Petri net formalism is useful in studying many discrete-state, discrete-event systems exhibiting concurrency, synchronization, and other complex behavior. As a bipartite graph, the net can conveniently capture salient aspects of the system. As a mathematical tool, the net can specify an analyzable state space. Indeed, one can reason about certain qualitative properties (from state occupancies) and how they arise (the sequence of events leading there). By introducing deterministic or random delays, the model is forced to sojourn in states some amount of time, giving rise to an underlying stochastic process, one that can be specified in a compact way and capable of providing quantitative, probabilistic measures. We formalize a new non-Markovian extension to the Petri net that captures both discrete and continuous timing in the same model. The approach affords efficient, stationary analysis in most cases and efficient transient analysis under certain restrictions. Moreover, this new formalism has the added benefit in modeling fidelity stemming from the simultaneous capture of discrete- and continuous-time events (as opposed to capturing only one and approximating the other). We show how the underlying stochastic process, which is non-Markovian, can be resolved into simpler Markovian problems that enjoy efficient solutions. Solution algorithms are provided that can be easily programmed.

Jones, Robert L.↗

NASA NTRSDate not supplied

Uncertainty Estimation for SMAP Level-1 Brightness Temperature Assimilation at Different Timescales

In this study, we assimilate Soil Moisture Active Passive (SMAP) mission brightness temperature (Tb) observations into NASA’s Catchment Land Surface Model using an Ensemble Kalman filter to update surface and root-zone soil moisture simulations. Different time series components of the Tb observations are assimilated including anomalies, inter-annual variations, and high-frequency variations. To optimize the weights that the data assimilation (DA) puts on the observations, the ratio between the un-certainties of modeled and observed Tb is approximated using modeled and observed soil moisture uncertainties estimated using triple collocation analysis. Results are compared to a benchmark experiment that mimics the operational SMAP Level-4 algorithm, which assimilates Tb observations using a spatially-constant 4 Kelvin (K) observation uncertainty. All DA experiments exhibit notable skill improvements in most regions. Improvements are greatest for the inter-annual variations in the simulations of both surface and root-zone soil moisture (mean improvements in terms of Pearson correlation (-) are 0.08 and 0.06, respectively). Anomaly simulations improve similarly (0.07), and improvements in the high-frequency variations are only observed for sur-face soil moisture simulations (0.06). Strikingly, however, no notable difference in skill—neither improvement nor deterioration—is observed between the experiments that use optimized observation uncertainty parameters and the 4 K benchmark experiment. We show, analytically, that this may be explained by the presence of large observation operator errors, which have the potential to render post-update uncertainty insensitive to inaccuracies in the Kalman gain.

Alexander Gruber↗

NASA NTRS2005

Radiative Studies of Planetary Atmospheres

Retrieval algorithms and associated software for application to CIRS infrared spectral data have been developed and coded. A general forward radiative transfer code has been written that runs efficiently on a Macintosh, even at high spectral resolution (0.5 per centimeter). It makes use of the correlated-k approach for representation of the gaseous absorption and can include those gases listed in the HITRAN and GEISA atlases, along with collision-induced absorption. Cloud effects are included as spectrally dependent absorbers. Provision has been made for future extension to include particle scattering in an n-stream approximation. The primary purpose of the code is to produce synthetic data and to serve as the forward calculating element in gas and cloud retrieval programs developed for the Mac as well as other platforms. Initial development of algorithms and production software suitable for application to CIRS data to be obtained from Jupiter, Saturn and Titan has been completed, and production versions of the software for application to the spectral data are in place. This includes temperature, gaseous constituent, and cloud opacity retrieval, algorithms that can be applied to both nadir and limb data. This work has been done as a cooperative effort between Conrath and Matcheva (Cornell), Achterberg (GSFWSSAI), and Flasar (GSFC).

Conrath, Barney J.↗

NASA NTRS2010

Analyzing Non Stationary Processes in Radiometers

The lack of well-developed techniques for modeling changing statistical moments in our observations has stymied the application of stochastic process theory for many scientific and engineering applications. Non linear effects of the observation methodology is one of the most perplexing aspects to modeling non stationary processes. This perplexing problem was encountered when modeling the effect of non stationary receiver fluctuations on the performance of radiometer calibration architectures. Existing modeling approaches were found not applicable; particularly problematic is modeling processes across scales over which they begin to exhibit non stationary behavior within the time interval of the calibration algorithm. Alternatively, the radiometer output is modeled as samples from a sequence random variables; the random variables are treated using a conditional probability distribution function conditioned on the use of the variable in the calibration algorithm. This approach of treating a process as a sequence of random variables with non stationary stochastic moments produce sensible predictions of temporal effects of calibration algorithms. To test these model predictions, an experiment using the Millimeter wave Imaging Radiometer (MIR) was conducted. The MIR with its two black body calibration references was configured in a laboratory setting to observe a third ultra-stable reference (CryoTarget). The MIR was programmed to sequentially sample each of the three references in approximately a 1 second cycle. Data were collected over a six-hour interval. The sequence of reference measurements form an ensemble sample set comprised of a series of three reference measurements. Two references are required to estimate the receiver response. A third reference is used to estimate the uncertainty in the estimate. Typically, calibration algorithms are designed to suppress the non stationary effects of receiver fluctuations. By treating the data sequence as an ensemble collection, it is possible to apply temporal algorithms which exacerbate the non stationary effects. By varying the algorithm, information about the properties of the non stationary receiver fluctuations is obtained. Comparisons of analytical calculations and statistical analysis of data demonstrate impressive agreement.

Racette, Paul↗

NASA NTRS1989

Flux splitting algorithms for two-dimensional viscous flows with finite-rate chemistry

The Roe flux-difference splitting method has been extended to treat two-dimensional viscous flows with nonequilibrium chemistry. The derivations have avoided unnecessary assumptions or approximations. For spatial discretization, the second-order Roe upwind differencing is used for the convective terms and central differencing for the viscous terms. An upwind-based TVD scheme is applied to eliminate oscillations and obtain a sharp representation of discontinuities. A two-stage Runge-Kutta method is used to time integrate the discretized Navier-Stokes and species transport equations for the asymptotic steady solutions. The present method is then applied to two types of flows: the shock wave/boundary layer interaction problems and the jet in cross flows.

Shuen, Jian-Shun↗

NASA NTRS2003

Relative Navigation Algorithms for Phase 1 of the MMS Formation

This paper evaluates several navigation approaches for the first phase of the Magnetospheric Multiscale (MMS) mission, which consists of a tetrahedral formation of four satellites in highly eccentric Earth orbits of approximately 1.2 by 12 Earth radii at an inclination of 10 degrees. The inter-satellite separation is approximately 10 kilometers near apogees. Navigation approaches were studied using ground station m g e =d two-way Doppler measurements, Global Positioning System (GPS) pseudorange measurements, crosslink range measurements among the members flying in formation, and various combinations of these measurement types. An absolute position accuracy of 10 kilometers or better can be achieved with most of the approaches studied and a relative position accuracy of 100 meters or better can be achieved at apogee in some cases. Among the various approaches studied, the approaches that use a combination of GPS and crosslink measurements were found to be more reliable in terms of absolute and relative navigation accuracies and operational flexibility.

Kelbel, David↗

NASA NTRS1991

Monte-Carlo methods make Dempster-Shafer formalism feasible

One of the main obstacles to the applications of Dempster-Shafer formalism is its computational complexity. If we combine m different pieces of knowledge, then in general case we have to perform up to 2(sup m) computational steps, which for large m is infeasible. For several important cases algorithms with smaller running time were proposed. We prove, however, that if we want to compute the belief bel(Q) in any given query Q, then exponential time is inevitable. It is still inevitable, if we want to compute bel(Q) with given precision epsilon. This restriction corresponds to the natural idea that since initial masses are known only approximately, there is no sense in trying to compute bel(Q) precisely. A further idea is that there is always some doubt in the whole knowledge, so there is always a probability p(sub o) that the expert's knowledge is wrong. In view of that it is sufficient to have an algorithm that gives a correct answer a probability greater than 1-p(sub o). If we use the original Dempster's combination rule, this possibility diminishes the running time, but still leaves the problem infeasible in the general case. We show that for the alternative combination rules proposed by Smets and Yager feasible methods exist. We also show how these methods can be parallelized, and what parallelization model fits this problem best.

Kreinovich, Vladik YA.↗

NASA NTRS2002

Geolocation Assessment Algorithm for CALIPSO Using Coastline Detection

Cloud-Aerosol Lidar Infrared Pathfinder Satellite Observations (CALIPSO) is a joint satellite mission between NASA and the French space agency CNES. The investigation will gather long-term, global cloud and aerosol optical and physical properties to improve climate models. The CALIPSO spacecraft is scheduled to launch in 2004 into a 98.2 inclination, 705 km circular orbit approximately 3 minutes behind the Aqua spacecraft. The payload consists of a two-wavelength polarization-sensitive lidar, and two passive imagers operating in the visible (0.645 mm) and infrared (8.7 - 12.0 mm) spectral regions. The imagers are nadir viewing and co-aligned with the lidar. Earth viewing measurements are geolocated to the Earth fixed coordinate system using satellite ephemeris, Earth rotation and geoid, and instrument pointing data. The coastline detection algorithm will assess the accuracy of the CALIPSO geolocation process by analyzing Wide Field Camera (WFC) visible ocean land boundaries. Processing space-time coincident MODIS and WFC scenes with the coastline algorithm will help verify the co-registration requirement with Moderate Resolution Imaging Spectrometer (MODIS) data. This paper quantifies the accuracy of the coastline geolocation assessment algorithm.

Currey, J. Chris↗

NASA NTRS1994

Accelerated panel methods using the fast multipole method

Panel methods are commonly used in computational fluid dynamics for the solution of potential flow problems. The methods are a numerical technique based on the surface distribution of singularity elements. The solution is the process of finding the strength of the singularity elements distributed over the body's surface. This process involves the solution of the matrix problem Pq = p' for a set of unknowns q. The Fast Multipole Method is used to directly compute q without using matrix solvers. The algorithm works in O(N) time for N points, a great improvement over standard matrix solvers. In panel methods, the surface of a body is divided into a series of quadrilateral panels. The methods involve the computation of the influence of all other panels on each individual panel. The influence is based on the surface distribution, though this can be approximated by the area for distant panels. An alternative approximation, though with arbitrary accuracy, is to develop a multipole expansion about the center of the panel to describe the effect of a given panel on distant points in space. The expansion is based on the moments of the panel, thus allow the use of various surface distributions without changing the basic algorithm, just the computation of the various moments. The expansions are then manipulated in a tree walk to develop Taylor series expansions about a point in space which describe the effect of all distant panels on any point within a volume of convergence. The effect of near panels then needs to be computed directly, but the effect of all distant panels can be computed by simply evaluating the resulting expansion. The Fast Multipole Method has been applied to panel methods for the solution of source and doublet distributions. A major feature of the algorithm is that the algorithm does not change to derive the potential and velocity for sources and doublets. The same expansions can be used for both sources and doublets. Since the velocity is related to the potential, and the doublet potential is related to the z-component of the source velocity, all values can be derived from the same expansion by taking a series of partial derivatives. This requires more expansion terms to be kept since terms are lost in the process of taking partial derivatives. Thus to maintain accuracy for the doublet computation, more terms are required than if just evaluating for sources. The resulting Fast Multipole code should then parallelize better than classical panel methods due to the locality of data dependencies found in the Fast Multipole Method. Theoretically the parallelized code should execute in O(log N) time with O(N) processors, though this is not practical. Ongoing work includes implementing the parallel accelerated panel method, including methods to improve the load balancing of the problem by taking advantage of the known geometry of panels, and to encorporate sensitivity analysis into the algorithm.

Leathrum, James F., Jr.↗

NASA NTRS2023

A Framework for Evaluating Distributed Electric Propulsion on the SUSAN Electrofan Aircraft

This work presents a framework for evaluating models and algorithms for Distributed Electric Propulsion (DEP) on the SUSAN Electrofan Aircraft. Throughout the development of the SUSAN aircraft, the performance of various configurations of the aircraft will need to be analyzed. However, the static behavior alone is not sufficient to describe the performance of these configurations. Therefore, simulation with fully integrated subsystem models is required. The proposed framework considers the vehicle aerodynamic, propulsion, and control subsystems. The presented framework automatically generates control laws for any vehicle configuration in response to changes in these subsystems. To compare these different vehicle configurations, various time and frequency domain performance metrics are compared. Three different system modifications are used as cases to evaluate this framework. The first modification integrates the propulsion control system with the flight controller to enable differential thrust without stalling the main engine. This evaluation case is used to validate the framework for aircraft configurations with coupled subsystems. The second modification compares the effect of the vertical tail size on open and closed loop performance. This evaluation case is used to validate the framework for controlling different configurations and tuning towards comparable closed loop performance despite changes to the aircraft's aerodynamic model. The third modification implements two different control allocation schemes. This evaluation case demonstrates the framework's ability to evaluate allocation modifications needed to take advantage of DEP. The first evaluation case is used to show that controller integration enables differential thrust, improving realized wingfan bandwidth by up to 40\% in simulation. The second evaluation case demonstrates that the framework can stabilize the reduced tail size aircraft with closed loop control. The third evaluation case demonstrates that a pseudoinverse control allocation scheme improves lateral velocity settling time by approximately 17~seconds over a symmetric-thrust allocation. These cases show that the framework is useful for evaluating the performance of integrated system designs, enabling analyses of new models and algorithms for the SUSAN distributed electric propulsion vehicle.

Nicholas C Ogden↗

NASA NTRS1972

Control system estimation and design for aerospace vehicles with time delay

The problems of estimation and control of discrete, linear, time-varying systems are considered. Previous solutions to these problems involved either approximate techniques, open-loop control solutions, or results which required excessive computation. The estimation problem is solved by two different methods, both of which yield the identical algorithm for determining the optimal filter. The partitioned results achieve a substantial reduction in computation time and storage requirements over the expanded solution, however. The results reduce to the Kalman filter when no delays are present in the system. The control problem is also solved by two different methods, both of which yield identical algorithms for determining the optimal control gains. The stochastic control is shown to be identical to the deterministic control, thus extending the separation principle to time delay systems. The results obtained reduce to the familiar optimal control solution when no time delays are present in the system.

Allgaier, G. R.↗

NASA NTRS1976

On the computational cost of approximating and recognizing noise-perturbed straight lines and quadratic arcs in the plane

Recognition of underlying straight lines and quadratic arcs in line drawings, and approximation of very noisy data by such line/curve segments, is addressed as a subproblem of the more general problem of optimum recognition of complicated line/curve drawings. Some specific algorithms are presented with extensions and interpretations for more complicated applications, and a data generation model is developed for the problem. Data are generated as a perturbation of a single underlying straight line or an elliptic or hyperbolic arc. Recursive estimation techniques, minimization of central processing unit time, decision making with controlled error probabilities, and modeling and recognition of pictures consisting of noisy curves are dealt with. Applications envisaged include: picture data compression, contour line representation in maps, intelligent data searches, and ballistic missile decoy tracking.

Cooper, D. B.↗

NASA NTRS1982

Approximate factorization with an elliptic pressure solver for incompressible flow

Two-dimensional curvilinear coordinates are used to solve the incompressible Navier-Stokes equations, in conjunction with approximate factorization for the solution of the momentum equation and the successive overrelaxation by lines method for the solution of a Poisson equation for the pressure. The combined algorithm, although not fully explicit, is marginally stable at Reynolds numbers lower than 10,000 and time increments of 0.01. Pressure distributions calculated for attack angles of zero and 6 deg are of the same shape as the experimental curves, but are shifted to one side.

Bernard, R. S.↗

NASA NTRS1985

Solution algorithms for nonlinear transient heat conduction analysis employing element-by-element iterative strategies

The particular problems investigated in the present study arise from nonlinear transient heat conduction. One of two types of nonlinearities considered is related to a material temperature dependence which is frequently needed to accurately model behavior over the range of temperature of engineering interest. The second nonlinearity is introduced by radiation boundary conditions. The finite element equations arising from the solution of nonlinear transient heat conduction problems are formulated. The finite element matrix equations are temporally discretized, and a nonlinear iterative solution algorithm is proposed. Algorithms for solving the linear problem are discussed, taking into account the form of the matrix equations, Gaussian elimination, cost, and iterative techniques. Attention is also given to approximate factorization, implementational aspects, and numerical results.

Winget, J. M.↗