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NASA NTRS1992

Stochastic stability properties of jump linear systems

Jump linear systems are defined as a family of linear systems with randomly jumping parameters (usually governed by a Markov jump process) and are used to model systems subject to failures or changes in structure. The authors study stochastic stability properties in jump linear systems and the relationship among various moment and sample path stability properties. It is shown that all second moment stability properties are equivalent and are sufficient for almost sure sample path stability, and a testable necessary and sufficient condition for second moment stability is derived. The Lyapunov exponent method for the study of almost sure sample stability is discussed, and a theorem which characterizes the Lyapunov exponents of jump linear systems is presented.

Feng, Xiangbo↗

NASA NTRS2014

Combined Uncertainty and A-Posteriori Error Bound Estimates for General CFD Calculations: Theory and Software Implementation

This workshop presentation discusses the design and implementation of numerical methods for the quantification of statistical uncertainty, including a-posteriori error bounds, for output quantities computed using CFD methods. Hydrodynamic realizations often contain numerical error arising from finite-dimensional approximation (e.g. numerical methods using grids, basis functions, particles) and statistical uncertainty arising from incomplete information and/or statistical characterization of model parameters and random fields. The first task at hand is to derive formal error bounds for statistics given realizations containing finite-dimensional numerical error [1]. The error in computed output statistics contains contributions from both realization error and the error resulting from the calculation of statistics integrals using a numerical method. A second task is to devise computable a-posteriori error bounds by numerically approximating all terms arising in the error bound estimates. For the same reason that CFD calculations including error bounds but omitting uncertainty modeling are only of limited value, CFD calculations including uncertainty modeling but omitting error bounds are only of limited value. To gain maximum value from CFD calculations, a general software package for uncertainty quantification with quantified error bounds has been developed at NASA. The package provides implementations for a suite of numerical methods used in uncertainty quantification: Dense tensorization basis methods [3] and a subscale recovery variant [1] for non-smooth data, Sparse tensorization methods[2] utilizing node-nested hierarchies, Sampling methods[4] for high-dimensional random variable spaces.

CFD↗

NASA NTRS1987

A system identification technique based on the random decrement signatures. Part 2: Experimental results

Identification of the system parameters of a randomly excited structure may be treated using a variety of statistical techniques. Of all these techniques, the Random Decrement is unique in that it provides the homogeneous component of the system response. Using this quality, a system identification technique was developed based on a least-squares fit of the signatures to estimate the mass, damping, and stiffness matrices of a linear randomly excited system. The results of an experiment conducted on an offshore platform scale model to verify the validity of the technique and to demonstrate its application in damage detection are presented.

Bedewi, Nabih E.↗

NASA NTRS2012

Recursive Branching Simulated Annealing Algorithm

This innovation is a variation of a simulated-annealing optimization algorithm that uses a recursive-branching structure to parallelize the search of a parameter space for the globally optimal solution to an objective. The algorithm has been demonstrated to be more effective at searching a parameter space than traditional simulated-annealing methods for a particular problem of interest, and it can readily be applied to a wide variety of optimization problems, including those with a parameter space having both discrete-value parameters (combinatorial) and continuous-variable parameters. It can take the place of a conventional simulated- annealing, Monte-Carlo, or random- walk algorithm. In a conventional simulated-annealing (SA) algorithm, a starting configuration is randomly selected within the parameter space. The algorithm randomly selects another configuration from the parameter space and evaluates the objective function for that configuration. If the objective function value is better than the previous value, the new configuration is adopted as the new point of interest in the parameter space. If the objective function value is worse than the previous value, the new configuration may be adopted, with a probability determined by a temperature parameter, used in analogy to annealing in metals. As the optimization continues, the region of the parameter space from which new configurations can be selected shrinks, and in conjunction with lowering the annealing temperature (and thus lowering the probability for adopting configurations in parameter space with worse objective functions), the algorithm can converge on the globally optimal configuration. The Recursive Branching Simulated Annealing (RBSA) algorithm shares some features with the SA algorithm, notably including the basic principles that a starting configuration is randomly selected from within the parameter space, the algorithm tests other configurations with the goal of finding the globally optimal solution, and the region from which new configurations can be selected shrinks as the search continues. The key difference between these algorithms is that in the SA algorithm, a single path, or trajectory, is taken in parameter space, from the starting point to the globally optimal solution, while in the RBSA algorithm, many trajectories are taken; by exploring multiple regions of the parameter space simultaneously, the algorithm has been shown to converge on the globally optimal solution about an order of magnitude faster than when using conventional algorithms. Novel features of the RBSA algorithm include: 1. More efficient searching of the parameter space due to the branching structure, in which multiple random configurations are generated and multiple promising regions of the parameter space are explored; 2. The implementation of a trust region for each parameter in the parameter space, which provides a natural way of enforcing upper- and lower-bound constraints on the parameters; and 3. The optional use of a constrained gradient- search optimization, performed on the continuous variables around each branch s configuration in parameter space to improve search efficiency by allowing for fast fine-tuning of the continuous variables within the trust region at that configuration point.

Bolcar, Matthew↗

NASA NTRS1974

A design method for minimizing sensitivity to plant parameter variations

A method is described for minimizing the sensitivity of multivariable systems to parameter variations. The variable parameters are considered as random variables and their effect is included in a quadratic performance index. The performance index is a weighted sum of the state and control covariances that stem from both the random system disturbances and the parameter uncertainties. The numerical solution of the problem is described and application of the method to several initially sensitive tracking systems is discussed. The sensitivity factor of reduction was typically 2 or 3 over a system based on random system noise only, and yet resulted in state RMS increases of only about a factor of two.

Hadass, Z.↗

NASA NTRS1987

A system identification technique based on the random decrement signatures. Part 1: Theory and simulation

Identification of the system parameters of a randomly excited structure may be treated using a variety of statistical techniques. Of all these techniques, the Random Decrement is unique in that it provides the homogeneous component of the system response. Using this quality, a system identification technique was developed based on a least-squares fit of the signatures to estimate the mass, damping, and stiffness matrices of a linear randomly excited system. The mathematics of the technique is presented in addition to the results of computer simulations conducted to demonstrate the prediction of the response of the system and the random forcing function initially introduced to excite the system.

Bedewi, Nabih E.↗

NASA NTRS1976

Response of linear dynamic systems with random coefficients

Numerous models of physical systems contain parameters whose values are not known exactly. The physical and mathematical complexities arising in the prediction of the statistical behavior of such systems are discussed. Although the discussions are far from providing a satisfactory solution to such problems, they perhaps, by utilization of simple examples, will create a greater awareness of the statistical effect of random parameters.

Dickerson, J.↗

NASA NTRS2014

Deducing Electronic Unit Internal Response During a Vibration Test Using a Lumped Parameter Modeling Approach

During random vibration testing of electronic boxes there is often a desire to know the dynamic response of certain internal printed wiring boards (PWBs) for the purpose of monitoring the response of sensitive hardware or for post-test forensic analysis in support of anomaly investigation. Due to restrictions on internally mounted accelerometers for most flight hardware there is usually no means to empirically observe the internal dynamics of the unit, so one must resort to crude and highly uncertain approximations. One common practice is to apply Miles Equation, which does not account for the coupled response of the board in the chassis, resulting in significant over- or under-prediction. This paper explores the application of simple multiple-degree-of-freedom lumped parameter modeling to predict the coupled random vibration response of the PWBs in their fundamental modes of vibration. A simple tool using this approach could be used during or following a random vibration test to interpret vibration test data from a single external chassis measurement to deduce internal board dynamics by means of a rapid correlation analysis. Such a tool might also be useful in early design stages as a supplemental analysis to a more detailed finite element analysis to quickly prototype and analyze the dynamics of various design iterations. After developing the theoretical basis, a lumped parameter modeling approach is applied to an electronic unit for which both external and internal test vibration response measurements are available for direct comparison. Reasonable correlation of the results demonstrates the potential viability of such an approach. Further development of the preliminary approach presented in this paper will involve correlation with detailed finite element models and additional relevant test data.

random vibration response↗

NASA NTRS1994

Temperature dependence of Lorentz air-broadening and pressure-shift coefficients of (12)CH4 lines in the 2.3-micron spectral region

High-resolution (0.01/cm) absorption spectra of lean mixtures of CH4 in dry air were recorded with the McMath-Pierce Fourier transform spectrometer (FTS) of the National Solar Observatory on Kitt Peak at various temperatures between 24 and -61 C. The spectra have been analyzed to determine the values at room temperature of pressure-broadened widths and pressure-induced shifts of more than 740 transitions. The temperature dependence of air-broadened widths and pressure-induced shifts was deduced for approx. 370 transitions in the nu(sub 1) + nu(sub 4), nu(sub 3) + nu(sub 4), and nu(sub 2) + nu(sub 3) bands of (12)CH4 located between 4118 and 4615/cm. These results were obtained by analyzing a total of 29 spectra simultaneously using a multi-spectral non-linear least-squares fitting technique. This new technique allowed the determination of correlated spectral line parameters (e.g. intensity and broadening coefficient) better than the procedure of averaging values obtained by fitting the spectra individually. This method also provided a direct determination of the uncertainties in the retrieved parameters due to random errors. For each band analysed in this study the dependence of the various spectral line parameters upon the tetrahedral symmetry species and the rotational quantum numbers of the transitions is also presented.

Devi, V. Malathy↗

NASA NTRS1988

Predictions of silicon avalanche photodiode detector performance in water vapor differential absorption lidar

Performance analyses are presented which establish that over most of the range of signals expected for a down-looking differential absorption lidar (DIAL) operated at 16 km the silicon avalanche photodiode (APD) is the preferred detector for DIAL measurements of atmospheric water vapor in the 730 nm spectral region. The higher quantum efficiency of the APD's, (0.8-0.9) compared to a photomultiplier's (0.04-0.18) more than offsets the higher noise of an APD receiver. In addition to offering lower noise and hence lower random error the APD's excellent linearity and impulse recovery minimize DIAL systematic errors attributable to the detector. Estimates of the effect of detector system parameters on overall random and systematic DIAL errors are presented, and performance predictions are supported by laboratory characterization data for an APD receiver system.

Kenimer, R. L.↗

NASA NTRS1992

A stochastic damage model for the rupture prediction of a multi-phase solid. I - Parametric studies. II - Statistical approach

A stochastic damage model for predicting the rupture of a brittle multiphase material is developed, based on the microcrack-macrocrack interaction. The model, which incorporates uncertainties in locations, orientations, and numbers of microcracks, characterizes damage by microcracking and fracture by macrocracking. A parametric study is carried out to investigate the change of the stress intensity at the macrocrack tip by the configuration of microcracks. The inherent statistical distribution of the fracture toughness arising from the intrinsic random nature of microcracks is explored using a statistical approach. For this purpose, a computer simulation model is introduced, which incorporates a statistical characterization of geometrical parameters of a random microcrack array.

Lua, Yuan J.↗

NASA NTRS1990

First-passage problems: A probabilistic dynamic analysis for degraded structures

Structures subjected to random excitations with uncertain system parameters degraded by surrounding environments (a random time history) are studied. Methods are developed to determine the statistics of dynamic responses, such as the time-varying mean, the standard deviation, the autocorrelation functions, and the joint probability density function of any response and its derivative. Moreover, the first-passage problems with deterministic and stationary/evolutionary random barriers are evaluated. The time-varying (joint) mean crossing rate and the probability density function of the first-passage time for various random barriers are derived.

Shiao, Michael C.↗

NASA NTRS1997

Particle Scattering in the Resonance Regime: Full-Wave Solution for Axisymmetric Particles with Large Aspect Ratios

Reliable descriptions of the optical properties of clouds and aerosols are essential for studies of radiative transfer in planetary atmospheres. The scattering algorithms provide accurate estimates of these properties for spherical particles with a wide range of sizes and refractive indices, but these methods are not valid for non-spherical particles (e.g., ice crystals, mineral dust, and smoke). Even though a host of methods exist for deriving the optical properties of nonspherical particles that are very small or very large compared with the wavelength, only a few methods are valid in the resonance regime, where the particle dimensions are comparable with the wavelength. Most such methods are not ideal for particles with sharp edges or large axial ratios. We explore the utility of an integral equation approach for deriving the single-scattering optical properties of axisymmetric particles with large axial ratios. The accuracy of this technique is shown for spheres of increasing size parameters and an ensemble of randomly oriented prolate spheroids of size parameter equal to 10.079368. In this last case our results are compared with published results obtained with the T-matrix approach. Next we derive cross sections, single-scattering albedos, and phase functions for cylinders, disks, and spheroids of ice with dimensions extending from the Rayleigh to the geometric optics regime. Compared with those for a standard surface integral equation method, the storage requirement and the computer time needed by this method are reduced, thus making it attractive for generating databases to be used in multiple-scattering calculations. Our results show that water ice disks and cylinders are more strongly absorbing than equivalent volume spheres at most infrared wavelengths. The geometry of these particles also affects the angular dependence of the scattering. Disks and columns with maximum linear dimensions larger than the wavelength scatter much more radiation in the forward and backward directions and much less radiation at intermediate phase angles than equivalent volume spheres.

Zuffada, Cinzia↗

NASA NTRS1987

Probabilistic SSME blades structural response under random pulse loading

The purpose is to develop models of random impacts on a Space Shuttle Main Engine (SSME) turbopump blade and to predict the probabilistic structural response of the blade to these impacts. The random loading is caused by the impact of debris. The probabilistic structural response is characterized by distribution functions for stress and displacements as functions of the loading parameters which determine the random pulse model. These parameters include pulse arrival, amplitude, and location. The analysis can be extended to predict level crossing rates. This requires knowledge of the joint distribution of the response and its derivative. The model of random impacts chosen allows the pulse arrivals, pulse amplitudes, and pulse locations to be random. Specifically, the pulse arrivals are assumed to be governed by a Poisson process, which is characterized by a mean arrival rate. The pulse intensity is modelled as a normally distributed random variable with a zero mean chosen independently at each arrival. The standard deviation of the distribution is a measure of pulse intensity. Several different models were used for the pulse locations. For example, three points near the blade tip were chosen at which pulses were allowed to arrive with equal probability. Again, the locations were chosen independently at each arrival. The structural response was analyzed both by direct Monte Carlo simulation and by a semi-analytical method.

Shiao, Michael↗

NASA NTRS1989

Analysis of random drop for gateway congestion control

Lately, the growing demand on the Internet has prompted the need for more effective congestion control policies. Currently No Gateway Policy is used to relieve and signal congestion, which leads to unfair service to the individual users and a degradation of overall network performance. Network simulation was used to illustrate the character of Internet congestion and its causes. A newly proposed gateway congestion control policy, called Random Drop, was considered as a promising solution to the pressing problem. Random Drop relieves resource congestion upon buffer overflow by choosing a random packet from the service queue to be dropped. The random choice should result in a drop distribution proportional to the bandwidth distribution among all contending TCP connections, thus applying the necessary fairness. Nonetheless, the simulation experiments demonstrate several shortcomings with this policy. Because Random Drop is a congestion control policy, which is not applied until congestion has already occurred, it usually results in a high drop rate that hurts too many connections including well-behaved ones. Even though the number of packets dropped is different from one connection to another depending on the buffer utilization upon overflow, the TCP recovery overhead is high enough to neutralize these differences, causing unfair congestion penalties. Besides, the drop distribution itself is an inaccurate representation of the average bandwidth distribution, missing much important information about the bandwidth utilization between buffer overflow events. A modification of Random Drop to do congestion avoidance by applying the policy early was also proposed. Early Random Drop has the advantage of avoiding the high drop rate of buffer overflow. The early application of the policy removes the pressure of congestion relief and allows more accurate signaling of congestion. To be used effectively, algorithms for the dynamic adjustment of the parameters of Early Random Drop to suite the current network load must still be developed.

Hashem, Emam Salaheddin↗

NASA NTRS1993

The effect of dissipative inhomogeneous medium on the statistics of the wave intensity

One of the main theoretical points in the theory of wave propagation in random medium is the derivation of closed form equations to describe the statistics of the propagating waves. In particular, in one dimensional problems, the closed form representation of the multiple scattering effects is important since it contributes in understanding such problems like wave localization, backscattering enhancement, and intensity fluctuations. In this the propagation of plane waves in a layer of one-dimensional dissipative random medium is considered. The medium is modeled by a complex permittivity whose real part is a constant representing the absorption. The one dimensional problem is mathematically equivalent to the analysis of a transmission line with randomly perturbed distributed parameters and a single mode lossy waveguide and the results can be used to study the propagation of radio waves through atmosphere and the remote sensing of geophysical media. It is assumed the scattering medium consists of an ensemble of one-dimensional point scatterers randomly positioned in a layer of thickness L with diffuse boundaries. A Poisson impulse process with density lambda is used to model the position of scatterers in the medium. By employing the Markov properties of this process an exact closed form equation of Kolmogorov-Feller type was obtained for the probability density of the reflection coefficient. This equation was solved by combining two limiting cases: (1) when the density of scatterers is small; and (2) when the medium is weakly dissipative. A two variable perturbation method for small lambda was used to obtain solutions valid for thick layers. These solutions are then asymptotically evaluated for small dissipation. To show the effect of dissipation, the mean and fluctuations of the reflected power are obtained. The results were compared with a lossy homogeneous medium and with a lossless inhomogeneous medium and the regions where the effect of absorption is not essential were discussed.

Saatchi, Sasan S.↗

NASA NTRS2000

Statistical 3D 'Atomistic' Simulation of Decanano MOSFETs

A 3D statistical 'atomistic' simulation technique has been developed to study the effect of the random dopant induced parameter fluctuations in aggressively scaled MOSFETs. Efficient implementation of the 'atomistic' simulation approach has been used to investigate the threshold voltage standard deviation and lowering in the case of uniformly doped MOSFETs, and in fluctuation-resistant architectures utilising epitaxial-layers and delta-doping. The effect of the random doping in the polysilicon gate on the threshold voltage fluctuations has also been thoroughly investigated. The influence of a single-charge trapping on the channel conductivity in decanano MOSFETs is studied in the 'atomistic' framework as well. Quantum effects are taken into consideration in our 'atomistic' simulations using the density gradient formalism.

Asenov, Asen↗

NASA NTRS2011

Calculating Launch Vehicle Flight Performance Reserve

This paper addresses different methods for determining the amount of extra propellant (flight performance reserve or FPR) that is necessary to reach orbit with a high probability of success. One approach involves assuming that the various influential parameters are independent and that the result behaves as a Gaussian. Alternatively, probabilistic models may be used to determine the vehicle and environmental models that will be available (estimated) for a launch day go/no go decision. High-fidelity closed-loop Monte Carlo simulation determines the amount of propellant used with each random combination of parameters that are still unknown at the time of launch. Using the results of the Monte Carlo simulation, several methods were used to calculate the FPR. The final chosen solution involves determining distributions for the pertinent outputs and running a separate Monte Carlo simulation to obtain a best estimate of the required FPR. This result differs from the result obtained using the other methods sufficiently that the higher fidelity is warranted.

Hanson, John M.↗