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NASA NTRS2010

Broken Ergodicity in MHD Turbulence

Ideal magnetohydrodynamic (MHD) turbulence may be represented by finite Fourier series, where the inherent periodic box serves as a surrogate for a bounded astrophysical plasma. Independent Fourier coefficients form a canonical ensemble described by a Gaussian probability density function containing a Hermitian covariance matrix with positive eigenvalues. The eigenvalues at lowest wave number can be very small, resulting in a large-scale coherent structure: a turbulent dynamo. This is seen in computations and a theoretical explanation in terms of 'broken ergodicity' contains Taylor s theory of force-free states. An important problem for future work is the case of real, i.e., dissipative flows. In real flows, broken ergodicity and coherent structure are still expected to occur in MHD turbulence at the largest scale, as suggested by low resolution simulations. One challenge is to incorporate coherent structure at the largest scale into the theory of turbulent fluctuations at smaller scales.

Shebalin, John V.↗

NASA NTRS1995

An aspect ration bound for triangulating a d-grid cut by a hyperplane

We consider the problem of triangulating a d-dimensional uniform grid of d-cubes that is cut by a k-dimensional affine subspace. The goal is to obtain a triangulation with bounded aspect ratio. To achieve this goal, we allow some of the box faces near the affine subspace to be displaced. This problem has applications to finite element mesh generation. For general d and k, the bound on aspect ratio that we attain is double-exponential in d. For the important special case of d = 3, the aspect ratio bound is small enough that the technique is useful in practice.

Mitchell, Scott A.↗

NASA NTRS1987

Theoretical constraints on oxygen and carbon dixoide concentrations in the Precambrian atmosphere

Theoretical arguments which bear on the time histories of atmospheric oxygen and carbon dioxide during the Precambrian are reviewed and extended. It is shown that reasonably tight constraints can be placed on atmospheric pCO2 during the early and late Proterozoic, based on the observation that parts of the earth were glaciated at those times. It is demonstrated that an upper bound on early Proterozoic pO2 can be derived from a simple box model of the atmosphere-ocean system.

Kasting, James F.↗

NASA NTRS1983

A new algorithm for constrained nonlinear least-squares problems, part 1

A Gauss-Newton algorithm is presented for solving nonlinear least squares problems. The problem statement may include simple bounds or more general constraints on the unknowns. The algorithm uses a trust region that allows the objective function to increase with logic for retreating to best values. The computations for the linear problem are done using a least squares system solver that allows for simple bounds and linear constraints. The trust region limits are defined by a box around the current point. In its current form the algorithm is effective only for problems with small residuals, linear constraints and dense Jacobian matrices. Results on a set of test problems are encouraging.

Hanson, R. J.↗

NASA NTRS2010

Orion Entry Performance-Based Center-of-Gravity Box

The Orion capsule is designed both for Low Earth Orbit missions to the ISS and for missions to the moon. For ISS class missions, the capsule will use an Apollo-style direct entry. For lunar return missions, depending on the timing of the mission, the capsule could perform a direct entry or a skip entry of up to 4800 n.mi. in order to land in the coastal waters of California. The physics of atmospheric re-entry determine the capability of the Orion vehicle. For a given vehicle mass and shape, physics tells us that the driving parameters for an entry vehicle are the hypersonic lift-to-drag ratio (L/D) and the flight path angle at entry interface (gamma(sub EI)). The design of the Orion atmospheric re-entry must meet constraints during both nominal and dispersed flight conditions on landing accuracy, heating rate, total heat load, sensed acceleration, and proper disposal of the Service Module. These constraints define an entry corridor in the space of L/D-gamma(sub EI); if the vehicle falls within this corridor, then all constraints are met. The gamma(sub EI) dimension of the corridor can be further constrained by the gloads experienced during emergency entries. Thus, the entry performance for the Orion vehicle can be described completely by the L/D. Bounds on the hypersonic L/D necessary to achieve all the mission requirements can be defined for the given entry corridor. Landing accuracy performance drives the lower limit on L/D. In order to achieve the desired landing accuracy, a minimum L/D must be ensured. The design of the Thermal Protection System (TPS) drives the upper limit on L/D. A higher L/D can drive mass into the design of the TPS. Conversely, once the TPS is designed, the L/D must be ensured to stay below a certain limit in order for the TPS to stay within its design envelop. The L/D must stay within its upper and lower bounds during dispersed flight conditions. L/D is a function of both the aerodynamics and the center-of-gravity (CG) of the vehicle. The aerodynamics of the vehicle are determined by Computational Fluid Mechanics (CFD) and wind tunnel tests. However, the aerodynamics are not known precisely. Instead, an aerodynamic database has been developed where the aerodynamic coefficients are known to fall within a probabilistic band defined by upper and lower bounds. It is expected that the probabilistic band will shrink after the first missions are flown and real-world data is collected. Until that time, the Orion must be designed to the current aerodynamic database. Thus, for a given aerodynamic database with given uncertainties, the allowable range in L/D can be mapped to an allowable box for the CG location. The CG box is used to set requirements on the dispersions allowed for vehicle packaging and cargo storage. As the aerodynamic uncertainties decrease, the size of the CG box can increase. This paper discusses the technique used to map the minimum and maximum L/D bounds set by the entry performance requirements to the allowable dispersions in CG while accounting for aerodynamic uncertainties. The L/D is defined as the ratio of the lift force to the drag force. It is equivalent to the ratio of lift coefficient (C(sub L)) over drag coefficient (C(sub D)). C(sub L) and C(sub D) are functions of Mach number (M) and angle of attack (alpha). A Mach number of 25 is used as a measuring point of the hypersonic L/D. Variations in C(sub L), C(sub D) and alpha cause variations in L/D. Equation (1) shows the three contributions to the variation in L/D.

Rea, Jeremy R.↗

NASA NTRS1994

The Accuracy of Parameter Estimation in System Identification of Noisy Aircraft Load Measurement

This thesis focuses on the subject of the accuracy of parameter estimation and system identification techniques. Motivated by a complicated load measurement from NASA Dryden Flight Research Center, advanced system identification techniques are needed. The objective of this problem is to accurately predict the load experienced by the aircraft wing structure during flight determined from a set of calibrated load and gage response relationship. We can then model the problem as a black box input-output system identification from which the system parameter has to be estimated. Traditional LS (Least Square) techniques and the issues of noisy data and model accuracy are addressed. A statistical bound reflecting the change in residual is derived in order to understand the effects of the perturbations on the data. Due to the intrinsic nature of the LS problem, LS solution faces the dilemma of the trade off between model accuracy and noise sensitivity. A method of conflicting performance indices is presented, thus allowing us to improve the noise sensitivity while at the same time configuring the degredation of the model accuracy. SVD techniques for data reduction are studied and the equivalence of the Correspondence Analysis (CA) and Total Least Squares Criteria are proved. We also looked at nonlinear LS problems with NASA F-111 data set as an example. Conventional methods are neither easily applicable nor suitable for the specific load problem since the exact model of the system is unknown. Neural Network (NN) does not require prior information on the model of the system. This robustness motivated us to apply the NN techniques on our load problem. Simulation results for the NN methods used in both the single load and the 'warning signal' problems are both useful and encouraging. The performance of the NN (for single load estimate) is better than the LS approach, whereas no conventional approach was tried for the 'warning signals' problems. The NN design methodology is also presented. The use of SVD, CA and Collinearity Index methods are used to reduce the number of neurons in a layer.

Kong, Jeffrey↗

NASA NTRS1993

Exact and quasi-classical density matrix and Wigner functions for a particle in the box and half space

The exact expressions for density matrix and Wigner functions of quantum systems are known only in special cases. Corresponding Hamiltonians are quadratic forms of Euclidean coordinates and momenta. In this paper we consider the problem of one-dimensional free particle movement in the bounded region 0 is less than x is less than a (including the case a = infinity).

Akhundova, E. A.↗

NASA NTRS2012

Mars Science Laboratory Rover System Thermal Test

On November 26, 2011, NASA launched a large (900 kg) rover as part of the Mars Science Laboratory (MSL) mission to Mars. The MSL rover is scheduled to land on Mars on August 5, 2012. Prior to launch, the Rover was successfully operated in simulated mission extreme environments during a 16-day long Rover System Thermal Test (STT). This paper describes the MSL Rover STT, test planning, test execution, test results, thermal model correlation and flight predictions. The rover was tested in the JPL 25-Foot Diameter Space Simulator Facility at the Jet Propulsion Laboratory (JPL). The Rover operated in simulated Cruise (vacuum) and Mars Surface environments (8 Torr nitrogen gas) with mission extreme hot and cold boundary conditions. A Xenon lamp solar simulator was used to impose simulated solar loads on the rover during a bounding hot case and during a simulated Mars diurnal test case. All thermal hardware was exercised and performed nominally. The Rover Heat Rejection System, a liquid-phase fluid loop used to transport heat in and out of the electronics boxes inside the rover chassis, performed better than predicted. Steady state and transient data were collected to allow correlation of analytical thermal models. These thermal models were subsequently used to predict rover thermal performance for the MSL Gale Crater landing site. Models predict that critical hardware temperatures will be maintained within allowable flight limits over the entire 669 Sol surface mission.

Novak, Keith S.↗

NASA NTRS2012

Mars Science Laboratory Rover System Thermal Test

Novak, Keith S.↗

NASA NTRS1973

Non-classical adhesive-bonded joints in practical aerospace construction

Solutions are derived for adhesive-bonded joints of non-classical geometries. Particular attention is given to bonded doublers and to selective reinforcement by unidirectional composites. Non-dimensionalized charts are presented for the efficiency limit imposed on the skin as the result of the eccentricity in the load path through the doubler. It is desirable to employ a relativly large doubler to minimize the effective eccentricity in the load path. The transfer stresses associated with selective reinforcement of metal structures by advanced composites are analyzed. Reinforcement of bolt holes in composites by bonded metal doublers is covered quantitatively. Also included is the adhesive joint analysis for shear flow in a multi-cell torque box, in which the bond on one angle becomes more critical sooner than those on the others, thereby restricting the strength to less than the total of each maximum strength when acting alone. Adhesive plasticity and adherend stiffness and thermal imbalances are included. A simple analysis/design technique of solution in terms of upper and lower bounds on an all-plastic adhesive analysis is introduced.

Hart-Smith, L. J.↗

NASA NTRS2020

Squash-Box Feasibility Driven Differential Dynamic Programming

Recently, Differential Dynamic Programming (DDP) and other similar algorithms have become the solvers of choice when performing non-linear Model Predictive Control (nMPC) with modern robotic devices. The reason is that they have a lower computational cost per iteration when compared with off-the-shelf Non-Linear Programming (NLP) solvers, which enables its online operation. However, they cannot handle constraints, and are known to have poor convergence capabilities. In this paper, we propose a method to solve the optimal control problem with control bounds through a squashing function (i.e., a sigmoid, which is bounded by construction). It has been shown that a naive use of squashing functions damage the convergence rate. To tackle this, we first propose to add a quadratic barrier that avoids the difficulty of the plateau produced by the sigmoid. Second, we add an outer loop that adapts both the sigmoid and the barrier; it makes the optimal control problem with the squashing function converge to the original control-bounded problem. To validate our method, we present simulation results for different types of platforms including a multi-rotor, a biped, a quadruped and a humanoid robot.

Navarro, Angel Santamaria↗

NASA NTRS1991

Direct simulations of compressible wall-bounded turbulence

Several direct numerical simulations of high-speed turbulent Couette flow were performed with a new spectral code. Mach numbers up to three and a Reynolds number of 3000 were used. A new time-integration scheme was developed to handle Mach numbers above 1.5, which require greater accuracy and stability than lower Mach numbers. At low Mach number, the large streamwise eddies found by M. J. Lee in high incompressible Couette flow simulations were reproduced. At higher Mach numbers these structures still exist, but they become considerably less organized (although the disorganization may be a function of the spanwise box size). While the same types of vortical structures seen in the incompressible flow are observed at higher Mach numbers, a new structure involving the divergence of the velocity is also observed. This structure is generally associated with low shear areas next to the walls, but it has not been determined whether it is a cause or an effect of the low shear. A 'nonphysical' simulation was performed to determine by what mechanism the Mach number affects the flow. It appears that pressure gradient (acoustic) effects are more important than variable viscosity effects in determining the wall shear, but the size of vortical structures is determined more by the local kinematic viscosity. Low-order mean statistics are provided to help quantify these effects.

Buell, Jeffrey C.↗

NASA NTRS2006

Convective Weather Forecast Quality Metrics for Air Traffic Management Decision-Making

Since numerical weather prediction models are unable to accurately forecast the severity and the location of the storm cells several hours into the future when compared with observation data, there has been a growing interest in probabilistic description of convective weather. The classical approach for generating uncertainty bounds consists of integrating the state equations and covariance propagation equations forward in time. This step is readily recognized as the process update step of the Kalman Filter algorithm. The second well known method, known as the Monte Carlo method, consists of generating output samples by driving the forecast algorithm with input samples selected from distributions. The statistical properties of the distributions of the output samples are then used for defining the uncertainty bounds of the output variables. This method is computationally expensive for a complex model compared to the covariance propagation method. The main advantage of the Monte Carlo method is that a complex non-linear model can be easily handled. Recently, a few different methods for probabilistic forecasting have appeared in the literature. A method for computing probability of convection in a region using forecast data is described in Ref. 5. Probability at a grid location is computed as the fraction of grid points, within a box of specified dimensions around the grid location, with forecast convection precipitation exceeding a specified threshold. The main limitation of this method is that the results are dependent on the chosen dimensions of the box. The examples presented Ref. 5 show that this process is equivalent to low-pass filtering of the forecast data with a finite support spatial filter. References 6 and 7 describe the technique for computing percentage coverage within a 92 x 92 square-kilometer box and assigning the value to the center 4 x 4 square-kilometer box. This technique is same as that described in Ref. 5. Characterizing the forecast, following the process described in Refs. 5 through 7, in terms of percentage coverage or confidence level is notionally sound compared to characterizing in terms of probabilities because the probability of the forecast being correct can only be determined using actual observations. References 5 through 7 only use the forecast data and not the observations. The method for computing the probability of detection, false alarm ratio and several forecast quality metrics (Skill Scores) using both the forecast and observation data are given in Ref. 2. This paper extends the statistical verification method in Ref. 2 to determine co-occurrence probabilities. The method consists of computing the probability that a severe weather cell (grid location) is detected in the observation data in the neighborhood of the severe weather cell in the forecast data. Probabilities of occurrence at the grid location and in its neighborhood with higher severity, and with lower severity in the observation data compared to that in the forecast data are examined. The method proposed in Refs. 5 through 7 is used for computing the probability that a certain number of cells in the neighborhood of severe weather cells in the forecast data are seen as severe weather cells in the observation data. Finally, the probability of existence of gaps in the observation data in the neighborhood of severe weather cells in forecast data is computed. Gaps are defined as openings between severe weather cells through which an aircraft can safely fly to its intended destination. The rest of the paper is organized as follows. Section II summarizes the statistical verification method described in Ref. 2. The extension of this method for computing the co-occurrence probabilities in discussed in Section HI. Numerical examples using NCWF forecast data and NCWD observation data are presented in Section III to elucidate the characteristics of the co-occurrence probabilities. This section also discusses the procedure for computing throbabilities that the severity of convection in the observation data will be higher or lower in the neighborhood of grid locations compared to that indicated at the grid locations in the forecast data. The probability of coverage of neighborhood grid cells is also described via examples in this section. Section IV discusses the gap detection algorithm and presents a numerical example to illustrate the method. The locations of the detected gaps in the observation data are used along with the locations of convective weather cells in the forecast data to determine the probability of existence of gaps in the neighborhood of these cells. Finally, the paper is concluded in Section V.

Chatterji, Gano B.↗

NASA NTRS1972

Bounded state space

This investigation is divided functionally into three different areas: (1) study of bounded state space, (2) nonlinear smoothing theory, and (3) system identification. (1) Study of bounded state space: necessary and sufficient conditions for an optimal control are obtained for a bounded state space optimal control problem. The difficulty of determining the so-called jump conditions is eliminated; however, the problem of determining the points where the response either enters or leaves the boundary still remains unsolved. (2) Nonlinear smoothing theory: nonlinear fixed-interval, fixed-point and fixed-lag smoothing of a random signal generated by a stochastic differential equation are investigated. Results on the asymptotic stability of a linear constant-parameter fixed-interval smoothing filter are obtained. (3) System identification: a particular stochastic modelling problem is solved. An Ito stochastic integral equation is used to mathematically model a black box having multiple inputs and multiple outputs. A new method for identifying system parameters is presented.

Eyman, E. D.↗

NASA NTRS2012

A Geometric Analysis to Protect Manned Assets from Newly Launched Objects - COLA Gap Analysis

A safety risk was identified for the International Space Station (ISS) by The Aerospace Corporation following the launch of GPS IIR-20 (March 24, 2009), when the spent upper stage of the launch vehicle unexpectedly crossed inside the ISS notification box shortly after launch. This event highlighted a 56-hour vulnerability period following the end of the launch Collision Avoidance (COLA) process where the ISS would be unable to react to a conjunction with a newly launched object. Current launch COLA processes screen each launched object across the launch window to determine if an object's nominal trajectory is predicted to pass within 200 km of the ISS (or any other manned/mannable object), resulting in a launch time closure. These launch COLA screens are performed from launch through separation plus I 00 minutes. Once the objects are in orbit, they are cataloged and evaluated as part of routine on-orbit conjunction assessment processes. However, as the GPS IIR-20 scenario illustrated, there is a vulnerability period in the time line between the end of launch COLA coverage and the beginning of standard on-orbit COLA assessment activities. The gap between existing launch and on-orbit COLA processes is driven by the time it takes to track and catalog a launched object, identify a conjunction, and plan and execute a collision avoidance maneuver. For the ISS, the total time required to accomplish an of these steps is 56 hours. To protect human lives, NASA/JSC has requested that an US launches take additional steps to protect the ISS during this "COLA gap" period. The uncertainty in the state of a spent upper stage can be quite large after all bums are complete and all remaining propellants are expelled to safe the stage. Simply extending the launch COLA process an additional 56 hours is not a viable option as the 3-sigma position uncertainty will far exceed the 200 km miss-distance criterion. Additionally, performing a probability of collision (Pc) analysis over this period is also not practical due to the limiting effects of these large orbit state uncertainties. An estimated upper bound for Pc for a typical spent upper stage if nominally aligned for a direct broadside collision with the ISS is only on the order of 10-6. For a smaller manned object such as a Soyuz capsule, the risk level decreases to an order of 10'8 . In comparison, the Air Force Range policy (AFI 91-217) for launch COLAs would only eliminate launch opportunities when conjunctions with objects exceed a Pc of 10'5 This paper demonstrates a conservative geometry-based methodology that may be used to determine if launch opportunities pose a threat to the ISS during the COLA gap period. The NASA Launch Services Program at Kennedy Space Center has developed this COLA gap analysis method and employed it fQr three NASA missions to identify potential ISS conjunctions and corresponding launch window closures during the 56-hour at-risk period. In the analysis, for each launch opportunity, the nominal trajectory of the spent upper stage and the orbit state of the ISS are propagated over the 56 hour period. Each time the upper stage crosses the orbit plane of the ISS, the relative radial and argument of latitude separations are calculated. A window cutout is identified if these separation differences fall within a mission-specific violation box, which is determined from the evaluation of a Monte Carlo dispersions analysis that quantifies the potential variation in the upper stage radial and argument of latitude differences. This paper details the results of these analyses and their impacts to each mission.

Hametz, Mark E.↗

NASA NTRS2002

Digital Distortion Caused by Traveling- Wave-Tube Amplifiers Simulated

Future NASA missions demand increased data rates in satellite communications for near real-time transmission of large volumes of remote data. Increased data rates necessitate higher order digital modulation schemes and larger system bandwidth, which place stricter requirements on the allowable distortion caused by the high-power amplifier, or the traveling-wave-tube amplifier (TWTA). In particular, intersymbol interference caused by the TWTA becomes a major consideration for accurate data detection at the receiver. Experimentally investigating the effects of the physical TWTA on intersymbol interference would be prohibitively expensive, as it would require manufacturing numerous amplifiers in addition to acquiring the required digital hardware. Thus, an accurate computational model is essential to predict the effects of the TWTA on system-level performance when a communication system is being designed with adequate digital integrity for high data rates. A fully three-dimensional, time-dependent, TWT interaction model has been developed using the electromagnetic particle-in-cell code MAFIA (Solution of Maxwell's equations by the Finite-Integration-Algorithm). It comprehensively takes into account the effects of frequency-dependent AM (amplitude modulation)/AM and AM/PM (phase modulation) conversion, gain and phase ripple due to reflections, drive-induced oscillations, harmonic generation, intermodulation products, and backward waves. This physics-based TWT model can be used to give a direct description of the effects of the nonlinear TWT on the operational signal as a function of the physical device. Users can define arbitrary excitation functions so that higher order modulated digital signals can be used as input and that computations can directly correlate intersymbol interference with TWT parameters. Standard practice involves using communication-system-level software packages, such as SPW, to predict if adequate signal detection will be achieved. These models use a nonlinear, black-box model to represent the TWTA. The models vary in complexity, but most make several assumptions regarding the operation of the high-power amplifier. When the MAFIA TWT interaction model was used, these assumptions were found to be in significant error. In addition, digital signal performance, including intersymbol interference, was compared using direct data input into the MAFIA model and using the system-level analysis tool SPW for several higher order modulation schemes. Results show significant differences in predicted degradation between SPW and MAFIA simulations, demonstrating the significance of the TWTA approximations made in the SPW model on digital signal performance. For example, a comparison of the SPW and MAFIA output constellation diagrams for a 16-ary quadrature amplitude modulation (16-QAM) signal (data shown only for second and fourth quadrants) is shown. The upper-bound degradation was calculated from the corresponding eye diagrams. In comparison to SPW simulations, the MAFIA data resulted in a 3.6-dB larger degradation.

Kory, Carol L.↗

NASA NTRSDate not supplied

A Simplified Model of VIPER Thermal Management System. Part II: Integrated Vehicle

NASA’s Volatiles Investigating Polar Exploration Rover (VIPER) thermal management system (TMS) relies on four loop heat pipes (LHPs) to transport electronic waste heat to the vehicle cooling radiative surface and avoid overheating. The TMS has also ten constant conductance heat pipes (CCHPs) dedicated to balance the thermal load within the internal environment where the avionics boxes are mounted, also called warm electronic box (WEB), and to transport the heat from two of the science payload instruments. The TMS also uses two thermal straps to thermally link the batteries to the WEB. These thermal components, in addition to heaters, thermostat, multi-layer insulation (MLIs), and isolators forms the core of the VIPER TMS. The complex heat transport balance managed by the TMS is challenging to characterize and model. The more fidelity and granularity of a model, the more costly the computational resources needed and the longer the simulation and modeling time. When the priority is to provide quick but reliable assessments of the thermal performance or real time thermal feedback for training of console operators, simplified modeling tools are needed. To satisfy that need, this paper describes the effort to develop and correlate a model of VIPER TMS based on control volume approach. The correlation effort in particular focuses on hibernation, cold thermal balance, and hot thermal balance data from the integrated vehicle thermal vacuum (TVAC) test. Thus, the correlated model captures the heat leaks during hibernations and the performance at two extremes, bounding, operating scenarios.

Loop Heat Pipe↗

NASA NTRSDate not supplied

A Simplified Model of VIPER Thermal Management System. Part II: Integrated Vehicle

Thermal Modeling↗