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NASA NTRS2000

Analytical and Computational Properties of Distributed Approaches to MDO

Historical evolution of engineering disciplines and the complexity of the MDO problem suggest that disciplinary autonomy is a desirable goal in formulating and solving MDO problems. We examine the notion of disciplinary autonomy and discuss the analytical properties of three approaches to formulating and solving MDO problems that achieve varying degrees of autonomy by distributing the problem along disciplinary lines. Two of the approaches-Optimization by Linear Decomposition and Collaborative Optimization-are based on bi-level optimization and reflect what we call a structural perspective. The third approach, Distributed Analysis Optimization, is a single-level approach that arises from what we call an algorithmic perspective. The main conclusion of the paper is that disciplinary autonomy may come at a price: in the bi-level approaches, the system-level constraints introduced to relax the interdisciplinary coupling and enable disciplinary autonomy can cause analytical and computational difficulties for optimization algorithms. The single-level alternative we discuss affords a more limited degree of autonomy than that of the bi-level approaches, but without the computational difficulties of the bi-level methods. Key Words: Autonomy, bi-level optimization, distributed optimization, multidisciplinary optimization, multilevel optimization, nonlinear programming, problem integration, system synthesis

Alexandrov, Natalia M.↗
NASA NTRS1995

Profiles of Gamma-Ray Bursts and Their Component Pulses

One physically informative regularity of their otherwise heterogeneous ensemble, is that many Gamma-Ray Bursts consist of well defined pulses. To objectively quantify the temporal structure of BATSE bursts, we have developed an automatic modeling procedure that separates overlapping pulses and determines the energy-dependence of the pulse-shape parameters. No binning of photon arrival times is needed, so when applied to time-tagged events (TTE) the procedure captures variability information down to the shortest time scales present in the raw data. Maximizing the Bayesian likelihood function Pr(data/model) yields estimates of the model parameters, including the number of pulses present, and allows intercomparison of models of different forms. As with any nonlinear optimization, good initial guesses are crucial to avoid convergence to undesirable local minima. We find excellent initial pulse decompositions by wavelet-denoising a cumulative distribution of the raw photon arrival data; differentiation then gives a time profile mostly free of the systematic effects of degraded resolution (as in ordinary Fourier smoothing) and binning. We present statistical information on pulse rise-time, decay-time, peakedness, and amplitudes, plus their energy dependences - both within a single burst and for a large ensemble of bursts.

Scargle, Jeff D.↗
NASA NTRS2003

Binary-Phase Fourier Gratings for Nonuniform Array Generation

We describe a design method for a binary-phase Fourier grating that generates an array of spots with nonuniform, user-defined intensities symmetric about the zeroth order. Like the Dammann fanout grating approach, the binary-phase Fourier grating uses only two phase levels in its grating surface profile to generate the final spot array. Unlike the Dammann fanout grating approach, this method allows for the generation of nonuniform, user-defined intensities within the final fanout pattern. Restrictions governing the specification and realization of the array's individual spot intensities are discussed. Design methods used to realize the grating employ both simulated annealing and nonlinear optimization approaches to locate optimal solutions to the grating design problem. The end-use application driving this development operates in the near- to mid-infrared spectrum - allowing for higher resolution in grating specification and fabrication with respect to wavelength than may be available in visible spectrum applications. Fabrication of a grating generating a user-defined nine spot pattern is accomplished in GaAs for the near-infrared. Characterization of the grating is provided through the measurement of individual spot intensities, array uniformity, and overall efficiency. Final measurements are compared to calculated values with a discussion of the results.

Keys, Andrew S.↗
NASA NTRS2007

Iterative-Transform Phase Retrieval Using Adaptive Diversity

A phase-diverse iterative-transform phase-retrieval algorithm enables high spatial-frequency, high-dynamic-range, image-based wavefront sensing. [The terms phase-diverse, phase retrieval, image-based, and wavefront sensing are defined in the first of the two immediately preceding articles, Broadband Phase Retrieval for Image-Based Wavefront Sensing (GSC-14899-1).] As described below, no prior phase-retrieval algorithm has offered both high dynamic range and the capability to recover high spatial-frequency components. Each of the previously developed image-based phase-retrieval techniques can be classified into one of two categories: iterative transform or parametric. Among the modifications of the original iterative-transform approach has been the introduction of a defocus diversity function (also defined in the cited companion article). Modifications of the original parametric approach have included minimizing alternative objective functions as well as implementing a variety of nonlinear optimization methods. The iterative-transform approach offers the advantage of ability to recover low, middle, and high spatial frequencies, but has disadvantage of having a limited dynamic range to one wavelength or less. In contrast, parametric phase retrieval offers the advantage of high dynamic range, but is poorly suited for recovering higher spatial frequency aberrations. The present phase-diverse iterative transform phase-retrieval algorithm offers both the high-spatial-frequency capability of the iterative-transform approach and the high dynamic range of parametric phase-recovery techniques. In implementation, this is a focus-diverse iterative-transform phaseretrieval algorithm that incorporates an adaptive diversity function, which makes it possible to avoid phase unwrapping while preserving high-spatial-frequency recovery. The algorithm includes an inner and an outer loop (see figure). An initial estimate of phase is used to start the algorithm on the inner loop, wherein multiple intensity images are processed, each using a different defocus value. The processing is done by an iterative-transform method, yielding individual phase estimates corresponding to each image of the defocus-diversity data set. These individual phase estimates are combined in a weighted average to form a new phase estimate, which serves as the initial phase estimate for either the next iteration of the iterative-transform method or, if the maximum number of iterations has been reached, for the next several steps, which constitute the outerloop portion of the algorithm. The details of the next several steps must be omitted here for the sake of brevity. The overall effect of these steps is to adaptively update the diversity defocus values according to recovery of global defocus in the phase estimate. Aberration recovery varies with differing amounts as the amount of diversity defocus is updated in each image; thus, feedback is incorporated into the recovery process. This process is iterated until the global defocus error is driven to zero during the recovery process. The amplitude of aberration may far exceed one wavelength after completion of the inner-loop portion of the algorithm, and the classical iterative transform method does not, by itself, enable recovery of multi-wavelength aberrations. Hence, in the absence of a means of off-loading the multi-wavelength portion of the aberration, the algorithm would produce a wrapped phase map. However, a special aberration-fitting procedure can be applied to the wrapped phase data to transfer at least some portion of the multi-wavelength aberration to the diversity function, wherein the data are treated as known phase values. In this way, a multiwavelength aberration can be recovered incrementally by successively applying the aberration-fitting procedure to intermediate wrapped phase maps. During recovery, as more of the aberration is transferred to the diversity function following successive iterations around the ter loop, the estimated phase ceases to wrap in places where the aberration values become incorporated as part of the diversity function. As a result, as the aberration content is transferred to the diversity function, the phase estimate resembles that of a reference flat.

Dean, Bruce H.↗
NASA NTRS2011

Broadband Achromatic Phase Shifter for a Nulling Interferometer

Nulling interferometry is a technique for imaging exoplanets in which light from the parent star is suppressed using destructive interference. Light from the star is divided into two beams and a phase shift of radians is introduced into one of the beams. When the beams are recombined, they destructively interfere to produce a deep null. For monochromatic light, this is implemented by introducing an optical path difference (OPD) between the two beams equal to lambda/2, where lambda is the wavelength of the light. For broadband light, however, a different phase shift will be introduced at each wavelength and the two beams will not effectively null when recombined. Various techniques have been devised to introduce an achromatic phase shift a phase shift that is uniform across a particular bandwidth. One popular technique is to use a series of dispersive elements to introduce a wavelength-dependent optical path in one or both of the arms of the interferometer. By intelligently choosing the number, material and thickness of a series of glass plates, a nearly uniform, arbitrary phase shift can be introduced between two arms of an interferometer. There are several constraints that make choosing the number, type, and thickness of materials a difficult problem, such as the size of the bandwidth to be nulled. Several solutions have been found for bandwidths on the order of 20 to 30 percent (Delta(lambda)/lambda(sub c)) in the mid-infrared region. However, uniform phase shifts over a larger bandwidth in the visible regime between 480 to 960 nm (67 percent) remain difficult to obtain at the tolerances necessary for exoplanet detection. A configuration of 10 dispersive glass plates was developed to be used as an achromatic phase shifter in nulling interferometry. Five glass plates were placed in each arm of the interferometer and an additional vacuum distance was also included in the second arm of the interferometer. This configuration creates a phase shift of pi radians with an average error of 5.97 x 10(exp -8) radians and standard deviation of 3.07 x 10(exp -4) radians. To reduce ghost reflections and interference effects from neighboring elements, the glass plates are tilted such that the beam does not strike each plate at normal incidence. Reflections will therefore walk out of the system and not contribute to the intensity when the beams are recombined. Tilting the glass plates, however, introduces several other problems that must be mitigated: (1) the polarization of a beam changes when refracted at an interface at non-normal incidence; (2) the beam experiences lateral chromatic spread as it traverses multiple glass plates; (3) at each surface, wavelength- dependent intensity losses will occur due to reflection. For a fixed angle of incidence, each of these effects must be balanced between each arm of the interferometer in order to ensure a deep null. The solution was found using a nonlinear optimization routine that minimized an objective function relating phase shift, intensity difference, chromatic beam spread, and polarization difference to the desired parameters: glass plate material and thickness. In addition to providing a uniform, broadband phase shift, the configuration achieves an average difference in intensity transmission between the two arms of the interferometer of 0.016 percent with a standard deviation of 3.64 x 10(exp -4) percent, an average difference in polarization between the two arms of the interferometer of 5.47 x 10(exp -5) percent with a standard deviation of 1.57 x 10(exp -6) percent, and an average chromatic beam shift between the two arms of the interferometer of -47.53 microns with a wavelength-by-wavelength spread of 0.389 microns.

Bolcar, Matthew R.↗
NASA NTRS2012

On Heatshield Shapes for Mars Entry Capsules

The 70deg sphere-cone - the standard geometry for all US Mars entry missions - is thoroughly examined via flow field simulations at a select few peak heating points along candidate flight trajectories. Emphasis is placed on turbulent heating based on the Baldwin- Lomax turbulence model. It is shown that increased leeward turbulent heating for a 70 sphere-cone flying at angle of attack is primarily due to the discontinuity in curvature between the spherical nose cap and the conical frustum - the attachment of the sonic line at this sphere-cone junction leads to a supersonic edge Mach number over the leeward acreage. In an attempt to mitigate this problem of elevated turbulent heating, alternate geometries, without any curvature discontinuities in the acreage, are developed. Two approaches, one based on nonlinear optimization with constraints, and one based on the use of non-uniform rational B-splines, are considered. All configurations examined remain axisymmetric. The aerothermal performance of alternate geometries is shown to be superior to that of the 70 sphere-cone.

Prabhu, DInesh K.↗
NASA NTRS2012

Uncertainty Quantification for Polynomial Systems via Bernstein Expansions

This paper presents a unifying framework to uncertainty quantification for systems having polynomial response metrics that depend on both aleatory and epistemic uncertainties. The approach proposed, which is based on the Bernstein expansions of polynomials, enables bounding the range of moments and failure probabilities of response metrics as well as finding supersets of the extreme epistemic realizations where the limits of such ranges occur. These bounds and supersets, whose analytical structure renders them free of approximation error, can be made arbitrarily tight with additional computational effort. Furthermore, this framework enables determining the importance of particular uncertain parameters according to the extent to which they affect the first two moments of response metrics and failure probabilities. This analysis enables determining the parameters that should be considered uncertain as well as those that can be assumed to be constants without incurring significant error. The analytical nature of the approach eliminates the numerical error that characterizes the sampling-based techniques commonly used to propagate aleatory uncertainties as well as the possibility of under predicting the range of the statistic of interest that may result from searching for the best- and worstcase epistemic values via nonlinear optimization or sampling.

Crespo, Luis G.↗
NASA NTRS2012

Cascade Storage and Delivery System for a Multi Mission Space Exploration Vehicle (MMSEV)

NASA is developing a Multi Mission Space Exploration Vehicle (MMSEV) for missions beyond Low Earth Orbit (LEO). The MMSEV is a pressurized vehicle used to extend the human exploration envelope for Lunar, Near Earth Object (NEO), and Deep Space missions. The Johnson Space Center is developing the Environmental Control and Life Support System (ECLSS) for the MMSEV. The MMSEV s intended use is to support longer sortie lengths with multiple Extra Vehicular Activities (EVAs) on a higher magnitude than any previous vehicle. This paper presents an analysis of a high pressure oxygen cascade storage and delivery system that will accommodate the crew during long duration Intra Vehicular Activity (IVA) and capable of multiple high pressure oxygen fills to the Portable Life Support System (PLSS) worn by the crew during EVAs. A cascade is a high pressure gas cylinder system used for the refilling of smaller compressed gas cylinders. Each of the large cylinders are filled by a compressor, but the cascade system allows small cylinders to be filled without the need of a compressor. In addition, the cascade system is useful as a "reservoir" to accommodate low pressure needs. A regression model was developed to provide the mechanism to size the cascade systems subject to constraints such as number of crew, extravehicular activity duration and frequency, and ullage gas requirements under contingency scenarios. The sizing routine employed a numerical integration scheme to determine gas compressibility changes during depressurization and compressibility effects were captured using the Soave-Redlich-Kwong (SRK) equation of state. A multi-dimensional nonlinear optimization routine was used to find the minimum cascade tank system mass that meets the mission requirements. The sizing algorithms developed in this analysis provide a powerful framework to assess cascade filling, compressor, and hybrid systems to design long duration vehicle ECLSS architecture. 1

Yagoda, Evan↗
NASA NTRS2016

Phase-Retrieval Uncertainty Estimation and Algorithm Comparison for the JWST-ISIM Test Campaign

Phase retrieval, the process of determining the exitpupil wavefront of an optical instrument from image-plane intensity measurements, is the baseline methodology for characterizing the wavefront for the suite of science instruments (SIs) in the Integrated Science Instrument Module (ISIM) for the James Webb Space Telescope (JWST). JWST is a large, infrared space telescope with a 6.5-meter diameter primary mirror. JWST is currently NASA's flagship mission and will be the premier space observatory of the next decade. ISIM contains four optical benches with nine unique instruments, including redundancies. ISIM was characterized at the Goddard Space Flight Center (GSFC) in Greenbelt, MD in a series of cryogenic vacuum tests using a telescope simulator. During these tests, phase-retrieval algorithms were used to characterize the instruments. The objective of this paper is to describe the Monte-Carlo simulations that were used to establish uncertainties (i.e., error bars) for the wavefronts of the various instruments in ISIM. Multiple retrieval algorithms were used in the analysis of ISIM phase-retrieval focus-sweep data, including an iterativetransform algorithm and a nonlinear optimization algorithm. These algorithms emphasize the recovery of numerous optical parameters, including low-order wavefront composition described by Zernike polynomial terms and high-order wavefront described by a point-by-point map, location of instrument best focus, focal ratio, exit-pupil amplitude, the morphology of any extended object, and optical jitter. The secondary objective of this paper is to report on the relative accuracies of these algorithms for the ISIM instrument tests, and a comparison of their computational complexity and their performance on central and graphical processing unit clusters. From a phase-retrieval perspective, the ISIM test campaign includes a variety of source illumination bandwidths, various image-plane sampling criteria above and below the Nyquist- Shannon critical sampling value, various extended object sizes, and several other impactful effects.

Design Analysis↗
NASA NTRS2019

Retrieval of Vegetation Water Content Using Brightness Temperatures from the Soil Moisture Active Passive (SMAP) Mission

In this paper, we explore a time series approach to using the tau-omega (τ-ω) model to retrieve vegetation water content (kg/m2) with minimal use of ancillary data. Analytically, this approach calls for nonlinear optimization in two steps. First, multiple days of co-located brightness temperature observations are used to retrieve the effective vegetation opacity, which incorporates the combined radiometric and polarization effects of surface roughness and vegetation opacity. The resulting effective vegetation opacity is then used to retrieve vegetation water content to within a gain factor α and an offset factor β. By using a climatological vegetation water content ancillary database as the one adopted in the development of the SMAP standard and enhanced soil moisture products, α and β can be determined globally using the annual minimum and annual maximum of vegetation water content. The resulting values of α and β can then be used to reconstruct the retrieved vegetation water content. Formulation, assumptions, and limitations of this approach are presented alongside the preliminary global retrieval of vegetation water content using one year (2016) ofSMAP brightness temperature observations.

radiative transfer↗
NASA NTRSDate not supplied

An Interactive MATLAB Program for Fitting Transfer Functions to Frequency Responses

A computer program called FRFit (Frequency Response Fitting) is described for fitting single-input single-output transfer function models to empirical frequency response data. The program is interactive in that the user specifies ``elementary factors'' (gain, delay, pure differentiators and integrators, and first- and second-order zeros and poles) by entering numerical values or moving sliders in a graphical user interface. A nonlinear optimization can then be performed to obtain maximum likelihood estimates of transfer function parameters and uncertainties to provide feedback on the modeling and refine estimates. Several examples are discussed, including the identification of aircraft pitch dynamics from simulation data and data reported in the literature, approximating Theodorsen's function of unsteady aerodynamics, and obtaining a reduced-order model of a computational fluid dynamics code describing the unsteady aerodynamics around an aeroelastic wing. The program has some usefulness as a teaching aid, and can be applied to model structure determination, reduced-order modeling, preliminary analysis, and simple system identification problems. The program was written in MATLAB and is planned for public release through the NASA Software Catalog.

System identification↗
NASA NTRSDate not supplied

An Interactive MATLAB Program for Fitting Transfer Functions to Frequency Responses

A computer program called FRFit (Frequency Response Fitting) for matching single-input single-output (SISO) transfer function models to empirical frequency response data is described. The program was written in MATLAB and has a graphical user interface (GUI). It is interactive in that the user manually builds the transfer function model using ``elementary factors'' (gain, delay, differentiators and integrators, and first- and second-order poles and zeros) and adjusts their values with sliders or entry fields. A nonlinear optimization can also be used to determine maximum-likelihood estimates of the transfer function parameters and their associated uncertainties. The program has some usefulness as a teaching aid, and can be applied to model structure determination, reduced-order modeling, preliminary analysis, and other system identification problems. FRFit is demonstrated using example problems, including the identification of aircraft transfer functions and rational function approximations of Theodorsen's function.

Frequency response↗
NASA NTRS2022

New Constraints on Titan's Stratospheric n-Butane Abundance

Curiously, n-butane has yet to be detected at Titan, though it is predicted to be present in a wide range of abundances that span over 2.5 orders of magnitude. We have searched infrared spectroscopic observations of Titan for signals from n-butane (n-C4H10) in Titan's stratosphere. Three sets of Cassini Composite Infrared Spectrometer Focal Plane 4 (1050–1500 cm−1) observations were selected for modeling, having been collected from different flybys and pointing latitudes. We modeled the observations with the Nonlinear Optimal Estimator for MultivariatE Spectral AnalySIS radiative transfer tool. Temperature profiles were retrieved for each of the data sets by modeling the ν4 emission from methane near 1305 cm−1. Then, incorporating the temperature profiles, we retrieved abundances of all of Titan's known trace gases that are active in this spectral region, reliably reproducing the observations. We then systematically tested a set of models with varying abundances of n-butane, investigating how the addition of this gas affected the fits. We did this for several different photochemically predicted abundance profiles from the literature, as well as for a constant-with-altitude profile. Ultimately, though we did not produce any firm detection of n-butane, we derived new upper limits on its abundance specific to the use of each profile and to multiple different ranges of stratospheric altitudes. These results will tightly constrain the C4 chemistry of future photochemical modeling of Titan's atmosphere and also motivate the continued search for n-butane and its isomer, isobutane.

Brendan L. Steffens↗
NASA NTRS2022

A Rideshare Tensegrity Rover Concept to Explore Titan’s Lands and Oceans

The Tension Adjustable Network for Deploying Entry Membrane (TANDEM) concept is a phase II NASA Innovative Advanced Concepts (NIAC) project. One of the goals for this project was to extend the application of tensegrity-based rovers to a rideshare concept for a Titan mission. This report provides preliminary details of the TANDEM concept for Titan. System analyses of the entry vehicle were performed and compared to data from the Huygens mission to Titan. The entry aeroshell was based on Huygens and Stardust aeroshells. Modeling, simulation and design of the tensegrity deployment were performed using a nonlinear optimization form-finding algorithm. Lastly, a neuroevolution-based machine learning control strategy was applied, which produced efficient rolling locomotion gaits.

tensegrity↗
NASA NTRS1972

Development and optimization of a nonlinear multiparameter model for the human operator

A systematic method is proposed for the development, optimization, and comparison of controller-models for the human operator. This is suitable for any designed model, even multiparameter systems. A random search technique is chosen for the parameter optimization. As valuation criteria for the quality of the model development the criterion function - the comparison between the input and output functions of the human operator and those of the model - and the most important characteristic values and functions of the statistical signal theory are used. A nonlinear multiparameter model for the human operator is being designed which considers the complex input information rate per time in a single display. The nonlinear features of the model are effected by a modified threshold element and a decision algorithm. Different display-configurations as well as various transfer functions of the controlled element are explained by different optimized parameter-combinations.

Johannsen, G.↗
NASA NTRS1981

Nonlinear program based optimization of boost and buck-boost converter designs

The facility of an Augmented Lagrangian (ALAG) multiplier based nonlinear programming technique is demonstrated for minimum-weight design optimizations of boost and buck-boost power converters. Certain important features of ALAG are presented in the framework of a comprehensive design example for buck-boost power converter design optimization. The study provides refreshing design insight of power converters and presents such information as weight and loss profiles of various semiconductor components and magnetics as a function of the switching frequency.

Rahman, S.↗