Search NASASearch

SEARCH · Search NASA

Results for “ERROR FUNCTION”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 37 records · Page 2

Error analysis of multi-conic techniques

A general error analysis of three recently developed multi-conic methods of three-body trajectory integration has been carried out. Single-step error functions for position and velocity have been derived as Taylor series in powers of the time step and also in integral form. These error functions are used to investigate the relative accuracy of the three methods in various regions of the earth-moon space and to provide a method of variable step size control for the trajectory integration procedure. Numerical results are used to compare the multi-step performance of the methods for both large and small step sizes.

D'Amario, L. A.

Application of physical parameter identification to finite element models

A time domain technique for matching response predictions of a structural dynamic model to test measurements is developed. Significance is attached to prior estimates of physical model parameters and to experimental data. The Bayesian estimation procedure allows confidence levels in predicted physical and modal parameters to be obtained. Structural optimization procedures are employed to minimize an error functional with physical model parameters describing the finite element model as design variables. The number of complete FEM analyses are reduced using approximation concepts, including the recently developed convoluted Taylor series approach. The error function is represented in closed form by converting free decay test data to a time series model using Prony' method. The technique is demonstrated on simulated response of a simple truss structure.

Bronowicki, Allen J.

Tolerance analysis of optical telescopes using coherent addition of wavefront errors

A near diffraction-limited telescope requires that tolerance analysis be done on the basis of system wavefront error. One method of analyzing the wavefront error is to represent the wavefront error function in terms of its Zernike polynomial expansion. A Ramsey-Korsch ray trace package, a computer program that simulates the tracing of rays through an optical telescope system, was expanded to include the Zernike polynomial expansion up through the fifth-order spherical term. An option to determine a 3 dimensional plot of the wavefront error function was also included in the Ramsey-Korsch package. Several assimulation runs were analyzed to determine the particular set of coefficients in the Zernike expansion that are effected by various errors such as tilt, decenter and despace. A 3 dimensional plot of each error up through the fifth-order spherical term was also included in the study. Tolerance analysis data are presented.

Davenport, J. W.

Real-Time Adaptive Color Segmentation by Neural Networks

Artificial neural networks that would utilize the cascade error projection (CEP) algorithm have been proposed as means of autonomous, real-time, adaptive color segmentation of images that change with time. In the original intended application, such a neural network would be used to analyze digitized color video images of terrain on a remote planet as viewed from an uninhabited spacecraft approaching the planet. During descent toward the surface of the planet, information on the segmentation of the images into differently colored areas would be updated adaptively in real time to capture changes in contrast, brightness, and resolution, all in an effort to identify a safe and scientifically productive landing site and provide control feedback to steer the spacecraft toward that site. Potential terrestrial applications include monitoring images of crops to detect insect invasions and monitoring of buildings and other facilities to detect intruders. The CEP algorithm is reliable and is well suited to implementation in very-large-scale integrated (VLSI) circuitry. It was chosen over other neural-network learning algorithms because it is better suited to realtime learning: It provides a self-evolving neural-network structure, requires fewer iterations to converge and is more tolerant to low resolution (that is, fewer bits) in the quantization of neural-network synaptic weights. Consequently, a CEP neural network learns relatively quickly, and the circuitry needed to implement it is relatively simple. Like other neural networks, a CEP neural network includes an input layer, hidden units, and output units (see figure). As in other neural networks, a CEP network is presented with a succession of input training patterns, giving rise to a set of outputs that are compared with the desired outputs. Also as in other neural networks, the synaptic weights are updated iteratively in an effort to bring the outputs closer to target values. A distinctive feature of the CEP neural network and algorithm is that each update of synaptic weights takes place in conjunction with the addition of another hidden unit, which then remains in place as still other hidden units are added on subsequent iterations. For a given training pattern, the synaptic weight between (1) the inputs and the previously added hidden units and (2) the newly added hidden unit is updated by an amount proportional to the partial derivative of a quadratic error function with respect to the synaptic weight. The synaptic weight between the newly added hidden unit and each output unit is given by a more complex function that involves the errors between the outputs and their target values, the transfer functions (hyperbolic tangents) of the neural units, and the derivatives of the transfer functions.

Duong, Tuan A.

Compression of ephemerides

An algorithm is proposed for generating sequences of Chebyshev series which are the best approximations of an astronomical ephemeris in the sense of Chebyshev over large intervals of time. The criterion for a polynomial approximation of a function to be the best polynomial approximation of the function is that the error function present certain rippling characteristics as described by Remez (1957). General features of the program in PL/1 are described.

Deprit, A.

Optimization of spray breakup model parameters for predicting fuel spray and film characteristics in gasoline direct injection engines

This study investigated the behavior of gasoline direct injection (GDI) sprays using computational fluid dynamics (CFD). The authors developed an approach to identify optimal spray breakup model parameters by evaluating an error function across numerous simulations, with the goal of minimizing discrepancies from experimental data. Using the optimal setup, the simulated spray matched well with projected liquid volume distributions, liquid penetration, and spray width measured in a constant-pressure continuous-flow chamber. To further validate the approach, the same setup was tested across various fuels, injectors, and operating conditions. Subsequently, the optimal setup, along with a recently developed spray-wall interaction model, were applied to a direct-injection spark-ignited engine under late-injection conditions to predict and evaluate fuel film formation and evolution at varying engine coolant temperatures. Here, with the centrally mounted injector directing the spray toward the piston, simulations indicated that the spray tends to impinge on the piston surface. The proposed simulation framework also accurately captured the aggregate film area on the piston surface, aligning with previously published experimental results. Moreover, simulations showed that increasing the coolant temperature from cold start conditions (333 K) to warm conditions (363 K) reduced the fuel mass deposited on the piston by roughly 50%. Furthermore, for the spray-guided engine configuration studied in this work, the CFD model predicted minimal film deposition on the spark plug electrodes regardless of the coolant temperatures due to a relatively weak in-cylinder flow during the compression phase.

Computational fluid dynamics (CFD)

Thermodynamic and Diffusion Model Estimates on Metamorphic Temperatures and Timescales for Basaltic Eucrite GRA 98098

Introduction: HED meteorites are thought to rep-resent igneous rocks from Vesta’s basaltic crust and preserve evidence of early crustal metamorphism. Determining the temperatures and timescales of thermal metamorphism is important for reconstructing crustal evolution in the early solar system. Here, we study basaltic eucrite Graves Nunataks (GRA) 98098, which has been identified as a highly metamorphosed eucrite [1]. We present new estimates on metamorphic temperatures determined via thermodynamic modeling as well as the initial results from diffusion models constraining timescales of thermal metamorphism. Sample Description: GRA 98098 is an unbrecciated eucrite with a granoblastic plagioclase and pyroxene mineralogy. Millimeter to cm-long lathes of tridymite cross-cut and poikilitically enclose plagioclase and pyroxene [1,this work]. Pyroxene grains have exsolved into Ca-rich (~Wo38En29Fs33) and Ca-poor (~Wo4.5En36Fs59.5) lamellae. Both unzoned and zoned plagioclase grains are observed. Unzoned plagioclase grains are found solely with tridymite laths. These grains have ~An92 compositions. The cores of the zoned plagioclase grains have the same composition and thin, relatively sodic rims (~An67), (Fig. 1). The bulk sample is unusually enriched in highly in-compatible elements and has one of the most fractionated REE patterns reported [1]. Maximum metamorphic temperatures of 985±78°C have been estimated using two-pyroxene thermometry [2]. Methods: Thermodynamic modeling. Thermodynamic models were constructed using the software Perple_X, which employs a Gibbs free energy minimization in order to determine the most stable phase assemblage for a given bulk rock composition [3]. Bulk composition was calculated using mineral com-positions acquired via EMPA (this study) and the observed abundancies present in the thin section. Two bulk compositions were estimated; 1) includes all phases present in the thin section, (assumes that all phases are present during metamorphism), 2) excludes tridymite from the bulk calculation (assumes that tridymite was not present during metamorphism). In order to determine whether metamorphic equilibria was achieved and estimate temperatures of metamorphism, we compared measured pyroxene compositions with thermodynamically predicted compositions [4, Fig. 2]. Diffusion Modeling. Several time-temperature de-pendent diffusion profiles were calculated in order to determine the best match for XAn chemical profiles observed at the edges of the zoned plagioclase (Fig. 3). We assumed that the start condition was a stepwise gradient at the plagioclase/pyroxene interface. We also assumed an average diffusion coefficient (D) and a constant temperature using the equation in [5]. D was determined for two temperatures (T = 1060ºC; near eucrite solidus [6] and T = 985ºC; metamorphism reported in [4]) and then XAn was calculated as a function of distance from plagioclase core to rim using an error function solution to Fick’s second law. Results: Thermodynamic model results are summarized in Fig. 2. For a bulk composition that includes all phases in the thin section, pyroxene endmember compositions plot in the following temperature ranges: Fs ~660-860ºC, En~1000ºC & 1150ºC, and Wo~760-900ºC (Fig. 2a). For a bulk composition that excludes tridymite from the peak metamorphic assemblage (i.e., the bulk composition minus the contribution from tridymite), a temperature range could not be determined for the Fs component of pyroxene. For Wo, T~760-900ºC and En, T~1000ºC & 1150ºC (Fig. 2b). Fig. 3 summarizes the diffusion model results. For T = 1060°C & 985°C, the most appropriate time interval was estimated based on which diffusion curve most matched (solid lines, Fig. 3) the EMPA data. For T = 1060°C, the best looking match was t = 500 ka. For T = 985°C, the best match was t = 7 Ma. Discussion and future work: Temperature estimates from thermodynamic models are not conclusive because the temperature ranges determined for pyroxene endmember stability do not overlap (colored fields in Fig. 2), thus implying that there is disequilibrium between pyroxene crystals and the bulk composition considered [4]. Thus, additional exploration is needed to define a metamorphically equilibrated do-main that accurately records peak temperature. The utility of defining metamorphically equilibrated do-mains to improve the accuracy and level of detail elucidated regarding the petrogenetic history of metamorphose samples has been demonstrated previously [4,7]. We suggest that in the case of Fig. 2a, the thin section composition is not representative of the length scales over which metamorphic equilibrium was achieve and in the case of Fig. 2b, the assumption that tridymite was not present during metamorphism was incorrect. However, results from thermodynamic models can provide insight into the relative timing of mineral and compositional textures. For example from texture alone, it is unclear whether tridymite was igneous in origin and represents the last bits of melt in a crystallizing magma chamber, or if it formed during (and possibly initiated) open system thermal metamorphism. The latter could be consistent with a partial melt hypothesis [8,9] while the former implies that simple fractional crystallization can yield the textures present in GRA 98098. The lack of coincidence be-tween pyroxene endmember compositions in Fig. 2b suggest that the bulk composition minus tridymite was not the assemblage in equilibrium with the pyroxene, suggesting that tridymite was present during metamorphism and formed during igneous crystallization. We conclude that the development of the Na-rich plagioclase rims likely occurred during or immediately after peak thermal metamorphism, because eucrites of similar metamorphic grade and texture have unzoned plagioclase (~An92) [2,4,8], and Na zoning is only observed in the plagioclase not included in the tridymite. This suggests that the zoning formed after tridymite formation, and therefore after igneous crystallization. Thus, the timescales calculated via diffusion modeling possibly represent the time interval over which thermal metamorphism occurred. Cooling rates approximated for the Vestan crust predict that the crust cooled below 300°C around 35-40 Ma after formation[10]. This is consistent with our modeling results that predict formation of the Na rich plagioclase rims occurring at higher temperatures over a period of 0.5 to 7 Ma years. Future work. Additional thermodynamic modeling work will focus on selecting an equilibrated bulk rock domain in which to elucidate metamorphic conditions. Diffusion models currently provide a minimum time-scale, since diffusion slows down as the system cools. Future work with will focus on integrating cooling into the diffusion models and constraining the depth at which thermal metamorphism occurs because it could be used to determine whether the range of time-scales calculated for thermal metamorphism are consistent with the geologic environment.

J S Gorce

Further Development of a Function Generator Technique

Further features and properties of functions generated by generalization of the process originally uncovered to describe the evolution of pressure in a rigid volume due to outgassing or desorption are explored. Properties presented include development of a general Maclaurin series associated with these functions, a limitation in using integration by parts to produce asymptotic expansions, and a general description of an implicit Adams-Moulton method for numerical quadrature of transformed functions. These developments are then applied to develop and explore functions that solve the Sievert integral and the modified Bessel function of the first kind, order zero. Features of the error function and the incomplete lower gamma function are also considered.

Michael S. Woronowicz

Some Insights of Spectral Optimization in Ocean Color Inversion

In the past decades various algorithms have been developed for the retrieval of water constituents from the measurement of ocean color radiometry, and one of the approaches is spectral optimization. This approach defines an error target (or error function) between the input remote sensing reflectance and the output remote sensing reflectance, with the latter modeled with a few variables that represent the optically active properties (such as the absorption coefficient of phytoplankton and the backscattering coefficient of particles). The values of the variables when the error reach a minimum (optimization is achieved) are considered the properties that form the input remote sensing reflectance; or in other words, the equations are solved numerically. The applications of this approach implicitly assume that the error is a monotonic function of the various variables. Here, with data from numerical simulation and field measurements, we show the shape of the error surface, in a way to justify the possibility of finding a solution of the various variables. In addition, because the spectral properties could be modeled differently, impacts of such differences on the error surface as well as on the retrievals are also presented.

Lee, Zhongping

Multilabel proportion prediction and out-of-distribution detection on gamma spectra of short-lived fission products

In the machine learning problem of multilabel classification, the objective is to determine for each test instance which classes the instance belongs to. In this work, we consider an extension of multilabel classification, called multilabel proportion prediction, in the context of radioisotope identification (RIID) using gamma spectra data. We aim to not only predict radioisotope proportions, but also identify out-of-distribution (OOD) spectra. We achieve this goal by viewing gamma spectra as discrete probability distributions, and based on this perspective, we develop a custom semi-supervised loss function that combines a traditional supervised loss with an unsupervised reconstruction error function. Our approach was motivated by its application to the analysis of short-lived fission products from spent nuclear fuel. In particular, we demonstrate that a neural network model trained with our loss function can successfully predict the relative proportions of 37 radioisotopes simultaneously. The model trained with synthetic data was then applied to measurements taken by Pacific Northwest National Laboratory (PNNL) to conduct analysis typically done by subject-matter experts. Here, we also extend our approach to successfully identify when measurements are OOD, and thus should not be trusted, whether due to the presence of a novel source or novel proportions.

Anomaly detection

Reflector Surface Error Compensation in Dual-Reflector Antennas

By probing the field on a small subreflector at a minimal number of points, the main reflector surface errors can be obtained and subsequently used to design a phase-correction subreflector that can compensate for main reflector errors. The compensating phase-error profile across the subreflector can be achieved either by a surface deformation or by the use of an array of elements such as patch antennas that can cause a phase shift between the incoming and outgoing fields. The second option is of primary interest here, but the methodology can be applied to either case. The patch array is most easily implemented on a planar surface. Therefore, the example of a flat subreflector and a parabolic main reflector (a Newtonian dual reflector system) is considered in this work. The subreflector is assumed to be a reflector array covered with patch elements. The phase variation on a subreflector can be detected by a small number of receiving patch elements (probes). By probing the phase change at these few selected positions on the subreflector, the phase error over the entire surface can be recovered and used to change the phase of all the patch elements covering the subreflector plane to compensate for main reflector errors. This is accomplished by using a version of sampling theorem on the circular aperture. The sampling is performed on the phase-error function on the circular aperture of the main reflector by a method developed using Zernike polynomials. This method is based upon and extended from a theory previously proposed and applied to reflector aperture integration. This sampling method provides for an exact retrieval of the coefficients of up to certain orders in the expansion of the phase function, from values on a specifically calculated set of points in radial and azimuthal directions in the polar coordinate system, on the circular reflector aperture. The corresponding points on the subreflector are then obtained and, by probing the fields at these points, a set of phase values is determined that is then transferred back to the main reflector aperture for recovering the phase function. Once this function is recovered, the corresponding phase function on the subreflector is calculated and used to compensate for main reflector surface errors. In going back and forth between sub and main reflectors, geometrical (ray) optics is employed, which even though it ignores edge diffraction and other effects, is shown to be accurate for phase recovery.

Jamnejad, Vahraz

Generation and tooth contact analysis of spiral bevel gears with predesigned parabolic functions of transmission errors

A new approach for determination of machine-tool settings for spiral bevel gears is proposed. The proposed settings provide a predesigned parabolic function of transmission errors and the desired location and orientation of the bearing contact. The predesigned parabolic function of transmission errors is able to absorb piece-wise linear functions of transmission errors that are caused by the gear misalignment and reduce gear noise. The gears are face-milled by head cutters with conical surfaces or surfaces of revolution. A computer program for simulation of meshing, bearing contact and determination of transmission errors for misaligned gear has been developed.

Litvin, Faydor L.

A monitor for the laboratory evaluation of control integrity in digital control systems operating in harsh electromagnetic environments

This paper presents a strategy for dynamically monitoring digital controllers in the laboratory for susceptibility to electromagnetic disturbances that compromise control integrity. The integrity of digital control systems operating in harsh electromagnetic environments can be compromised by upsets caused by induced transient electrical signals. Digital system upset is a functional error mode that involves no component damage, can occur simultaneously in all channels of a redundant control computer, and is software dependent. The motivation for this work is the need to develop tools and techniques that can be used in the laboratory to validate and/or certify critical aircraft controllers operating in electromagnetically adverse environments that result from lightning, high-intensity radiated fields (HIRF), and nuclear electromagnetic pulses (NEMP). The detection strategy presented in this paper provides dynamic monitoring of a given control computer for degraded functional integrity resulting from redundancy management errors, control calculation errors, and control correctness/effectiveness errors. In particular, this paper discusses the use of Kalman filtering, data fusion, and statistical decision theory in monitoring a given digital controller for control calculation errors.

Belcastro, Celeste M.

Adjoint Algorithm for CAD-Based Shape Optimization Using a Cartesian Method

Adjoint solutions of the governing flow equations are becoming increasingly important for the development of efficient analysis and optimization algorithms. A well-known use of the adjoint method is gradient-based shape optimization. Given an objective function that defines some measure of performance, such as the lift and drag functionals, its gradient is computed at a cost that is essentially independent of the number of design variables (geometric parameters that control the shape). More recently, emerging adjoint applications focus on the analysis problem, where the adjoint solution is used to drive mesh adaptation, as well as to provide estimates of functional error bounds and corrections. The attractive feature of this approach is that the mesh-adaptation procedure targets a specific functional, thereby localizing the mesh refinement and reducing computational cost. Our focus is on the development of adjoint-based optimization techniques for a Cartesian method with embedded boundaries.12 In contrast t o implementations on structured and unstructured grids, Cartesian methods decouple the surface discretization from the volume mesh. This feature makes Cartesian methods well suited for the automated analysis of complex geometry problems, and consequently a promising approach to aerodynamic optimization. Melvin et developed an adjoint formulation for the TRANAIR code, which is based on the full-potential equation with viscous corrections. More recently, Dadone and Grossman presented an adjoint formulation for the Euler equations. In both approaches, a boundary condition is introduced to approximate the effects of the evolving surface shape that results in accurate gradient computation. Central to automated shape optimization algorithms is the issue of geometry modeling and control. The need to optimize complex, "real-life" geometry provides a strong incentive for the use of parametric-CAD systems within the optimization procedure. In previous work, we presented an effective optimization framework that incorporates a direct-CAD interface. In this work, we enhance the capabilities of this framework with efficient gradient computations using the discrete adjoint method. We present details of the adjoint numerical implementation, which reuses the domain decomposition, multigrid, and time-marching schemes of the flow solver. Furthermore, we explain and demonstrate the use of CAD in conjunction with the Cartesian adjoint approach. The final paper will contain a number of complex geometry, industrially relevant examples with many design variables to demonstrate the effectiveness of the adjoint method on Cartesian meshes.

Nemec, Marian

Antenna pattern control using impedence surfaces

During this research period, September 16, 1990 to March 15, 1991, a design method for selecting a low-loss impedance material coating for a horn antenna pattern control has been developed. This method and the stepped waveguide technique can be employed to accurately compute the electromagnetic wave phenomenon inside the transition region of the horn antenna, with or without the impedance surfaces, from the feed to the radiating aperture. For moment method solutions of the electric and magnetic current distributions on the radiating aperture and the outer surface of the horn antenna, triangular surface-patch modes are introduced to replace the sinusoidal surface-patch modes as expansion and testing functions to provide a more physical expansion of the current distributions. In the synthesis problem, a numerical optimization process is formulated to minimize the error function between the desired waveguide modes and the modes provided by the horn transition with impedance surfaces. Since the modes generated by the horn transition with impedance surface are computed by analytical techniques, the computational error involved in the synthesis of the antenna pattern is minimum. Therefore, the instability problem can be avoided. A preliminary implementation of the techniques has demonstrated that the developed theory of the horn antenna pattern control using the impedance surfaces is realizable.

Balanis, Constantine A.

Formulation of a strategy for monitoring control integrity in critical digital control systems

Advanced aircraft will require flight critical computer systems for stability augmentation as well as guidance and control that must perform reliably in adverse, as well as nominal, operating environments. Digital system upset is a functional error mode that can occur in electromagnetically harsh environments, involves no component damage, can occur simultaneously in all channels of a redundant control computer, and is software dependent. A strategy is presented for dynamic upset detection to be used in the evaluation of critical digital controllers during the design and/or validation phases of development. Critical controllers must be able to be used in adverse environments that result from disturbances caused by an electromagnetic source such as lightning, high intensity radiated field (HIRF), and nuclear electromagnetic pulses (NEMP). The upset detection strategy presented provides dynamic monitoring of a given control computer for degraded functional integrity that can result from redundancy management errors and control command calculation error that could occur in an electromagnetically harsh operating environment. The use is discussed of Kalman filtering, data fusion, and decision theory in monitoring a given digital controller for control calculation errors, redundancy management errors, and control effectiveness.

Belcastro, Celeste M.

Learning a trajectory using adjoint functions and teacher forcing

A new methodology for faster supervised temporal learning in nonlinear neural networks is presented which builds upon the concept of adjoint operators to allow fast computation of the gradients of an error functional with respect to all parameters of the neural architecture, and exploits the concept of teacher forcing to incorporate information on the desired output into the activation dynamics. The importance of the initial or final time conditions for the adjoint equations is discussed. A new algorithm is presented in which the adjoint equations are solved simultaneously (i.e., forward in time) with the activation dynamics of the neural network. We also indicate how teacher forcing can be modulated in time as learning proceeds. The results obtained show that the learning time is reduced by one to two orders of magnitude with respect to previously published results, while trajectory tracking is significantly improved. The proposed methodology makes hardware implementation of temporal learning attractive for real-time applications.

Toomarian, Nikzad B.

Semi-empirical calculation of depletion region width in n+-p silicon solar cells

An extension of the width calculation for impurity distributions of complementary error function and the Gaussian function, in addition to the exponential function is used in combination with experimental width values deduced from capacitance measurements at zero bias to numerically determine the width occurring in both the n+ and p junctions. The total possible error o the semi-empirical calculation is less than plus or minus 3%. Results show considerable variation in the depletion widths occurring in the n+ and p regions for different impurity profiles.

Wang, E. Y.