Extrapolation techniques applied to matrix methods in neutron diffusion problems
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Mccready, Robert R↗
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An evaluation of rise time characterization and prediction methods
One common method of extrapolating sonic boom waveforms from aircraft to ground is to calculate the nonlinear distortion, and then add a rise time to each shock by a simple empirical rule. One common rule is the '3 over P' rule which calculates the rise time in milliseconds as three divided by the shock amplitude in psf. This rule was compared with the results of ZEPHYRUS, a comprehensive algorithm which calculates sonic boom propagation and extrapolation with the combined effects of nonlinearity, attenuation, dispersion, geometric spreading, and refraction in a stratified atmosphere. It is shown there that the simple empirical rule considerably overestimates the rise time estimate. In addition, the empirical rule does not account for variations in the rise time due to humidity variation or propagation history. It is also demonstrated that the rise time is only an approximate indicator of perceived loudness. Three waveforms with identical characteristics (shock placement, amplitude, and rise time), but with different shock shapes, are shown to give different calculated loudness. This paper is based in part on work performed at the Applied Research Laboratories, the University of Texas at Austin, and supported by NASA Langley.
Robinson, Leick D.↗
Influence of Geometry and Flow Variations on NO Formation in the Quick Mixer of a Staged Combustor
Staged combustion, such as Rich-Burn/Quick-Mix/Lean-Burn (RQL), is a viable strategy to meet nitric oxide (NO) emission goals for both stationary and propulsion gas turbine engines. A critical element of the design is the quick mixer section where the potential for NO production is high. While numerical calculations of the quick mixer under reacting conditions have been conducted, the hostile environment and lack of appropriate diagnostics have, to date, precluded experimental probing of the reacting case. As an alternative to understanding the effect of geometry and flow variations on the production of NO in the quick mixer, the present paper presents (1) a series of non-reacting parametric studies, and (2) a computational method to extrapolate the results of the non-reacting experiments to reacting conditions. The results show that the rate of NO production is highest in the immediate vicinity of the injection plane. For a given momentum flux ratio between the jets and mainstream, the most effective mixing geometry is that which mixes effectively in both (1) the plane of injection, and (2) the wall regions downstream of the plan of injection. The tailoring of the mixing is key to minimize the NO formed. As a result, the best overall mixer with respect to the minimization of NO production may depend on the system specific characteristics of the particular application.
Hatch, M. S.↗
Low Sonic Boom Design Activities at Boeing
Low sonic boom studies have continued during the last year with the goal of exploring the ability of practical airplane designs to achieve significantly reduced sonic boom-loudness with reasonable performance penalties. At the 1993 Sonic Boom Workshop, improvements to the low-boom design methods were described and early results of two low-boom configurations -935 and -936) were presented. Now that the low boom design methods are reasonably mature, recent design activities have broadened somewhat to explore refinements to the -935 and -936 designs. In this paper the results are reported of a detailed systems study and performance sizing of the -935 (Hybrid sonic boom waveform) and the -936 (Flat-top waveform). This analysis included a second design cycle for reduced cruise drag and balance considerations. Another design study was of a small-wing version of the -935. Finally, some preliminary results of the recent LARC UPWT test of the -935 configuration are given, along with a proposed alternative method for extrapolating wind tunnel pressure signatures to the ground. The various configurations studied is also summarized. The topics covered by this paper are as follows: Systems study results of the Baseline -939 and low boom configurations -935 and -936, Small wing derivative of the -935, Wind tunnel test results of the -935, Test-derived F-function and propagation to the ground, and Future considerations (boom-softened baseline, overwater issues, and operations).
Haglund, George T.↗
The 1998-2000 SHADOZ (Southern Hemisphere ADditional OZonesondes) Tropical Ozone Climatology: Ozonesonde Precision, Accuracy and Station-to-Station Variability
As part of the SAFARI-2000 campaign, additional launches of ozonesondes were made at Irene, South Africa and at Lusaka, Zambia. These represent campaign augmentations to the SHADOZ database described in this paper. This network of 10 southern hemisphere tropical and subtropical stations, designated the Southern Hemisphere ADditional OZonesondes (SHADOZ) project and established from operational sites, provided over 1000 profiles from ozonesondes and radiosondes during the period 1998-2000. (Since that time, two more stations, one in southern Africa, have joined SHADOZ). Archived data are available at: http://code9l6.gsfc.nasa.gov/Data-services/shadoz>. Uncertainties and accuracies within the SHADOZ ozone data set are evaluated by analyzing: (1) imprecisions in stratospheric ozone profiles and in methods of extrapolating ozone above balloon burst; (2) comparisons of column-integrated total ozone from sondes with total ozone from the Earth-Probe/TOMS (Total Ozone Mapping Spectrometer) satellite and ground-based instruments; (3) possible biases from station-to-station due to variations in ozonesonde characteristics. The key results are: (1) Ozonesonde precision is 5%; (2) Integrated total ozone column amounts from the sondes are in good agreement (2-10%) with independent measurements from ground-based instruments at five SHADOZ sites and with overpass measurements from the TOMS satellite (version 7 data). (3) Systematic variations in TOMS-sonde offsets and in groundbased-sonde offsets from station to station reflect biases in sonde technique as well as in satellite retrieval. Discrepancies are present in both stratospheric and tropospheric ozone. (4) There is evidence for a zonal wave-one pattern in total and tropospheric ozone, but not in stratospheric ozone.
Witte, J. C.↗
21st Century Extravehicular Activities: Synergizing Past and Present Training Methods for Future Spacewalking Success
Neil Armstrong's understated words, "That's one small step for man, one giant leap for mankind." were spoken from Tranquility Base forty years ago. Even today, those words resonate in the ears of millions, including many who had yet to be born when man first landed on the surface of the moon. By their very nature, and in the the spirit of exploration, extravehicular activities (EVAs) have generated much excitement throughout the history of manned spaceflight. From Ed White's first space walk in June of 1965, to the first steps on the moon in 1969, to the expected completion of the International Space Station (ISS), the ability to exist, live and work in the vacuum of space has stood as a beacon of what is possible. It was NASA's first spacewalk that taught engineers on the ground the valuable lesson that successful spacewalking requires a unique set of learned skills. That lesson sparked extensive efforts to develop and define the training requirements necessary to ensure success. As focus shifted from orbital activities to lunar surface activities, the required skill-set and subsequently the training methods, changed. The requirements duly changed again when NASA left the moon for the last time in 1972 and have continued to evolve through the Skylab, Space Shuttle; and ISS eras. Yet because the visits to the moon were so long ago, NASA's expertise in the realm of extra-terrestrial EVAs has diminished. As manned spaceflight again shifts its focus beyond low earth orbit, EVA success will depend on the ability to synergize the knowledge gained over 40+ years of spacewalking to create a training method that allows a single crewmember to perform equally well, whether performing an EVA on the surface of the Moon, while in the vacuum of space, or heading for a rendezvous with Mars. This paper reviews NASA's past and present EVA training methods and extrapolates techniques from both to construct the basis for future EVA astronaut training.
Moore, Sandra K.↗
The OPALS Plan for Operations: Use of ISS Trajectory and Attitude Models in the OPALS Pointing Strategy
This paper will discuss the OPALS pointing strategy, focusing on incorporation of ISS trajectory and attitude models to build pointing predictions. Methods to extrapolate an ISS prediction based on past data will be discussed and will be compared to periodically published ISS predictions and Two-Line Element (TLE) predictions. The prediction performance will also be measured against GPS states available in telemetry. The performance of the pointing products will be compared to the allocated values in the OPALS pointing budget to assess compliance with requirements.
Optical PAyload for Lasercomm Science (OPALS)↗
Evaluation of Preliminary Buffet Forcing Function Development based on Wind-Tunnel Tests of Geometrically-similar Models
The current state-of-the-art for launch vehicle buffet load estimation involves wind tunnel testing of highly instrumented rigid models and data processing of experimental unsteady pressure measurements to estimate the fluctuating aerodynamic loads. The model and instrumentation requirements to accomplish this type of testing are significant. The rigid model, in particular, must be built with relatively high geometric fidelity to approximate similar aerodynamic phenomena that will occur during the full-scale vehicle flight. Due to this requirement, several iterative wind tunnel tests are often required during the design maturation of the vehicle. For preliminary buffet load estimation, however, data from previous launch vehicle wind tunnel tests of geometrically similar models can be used to provide initial buffet environment estimates before a high geometric fidelity wind tunnel test has been conducted. In this paper, two components of this preliminary buffet estimation process are described: (1) assembly of buffet forcing functions from multiple data sources and (2) scaling of these buffet forcing functions to geometrically-similar components of a target vehicle configuration. The impacts of these estimation methods on the full-vehicle buffet forcing functions, including their longitudinal distribution along the vehicle, frequency-content at individual stations, and station-to-station coherence, are investigated. Considerations for using these methods to extrapolate wind tunnel data to geometrically similar models are also discussed.
Patrick S Heaney↗
Energy Landscape, and Structural and Spectroscopic Characterization of Diazirine and Its Cyclic Isomers
Identifying new nitrogenated hydrocarbon molecules in the interstellar medium (ISM) is challenging due to the lack of comprehensive spectroscopic data from experiments. In this computational work, we focus on investigating the structures, relative energies, spectroscopic constants, and energy landscape of the cyclic isomers of diazirine (c-CH2N2) using ab initio quantum chemical methods. Density functional theory (DFT) methods and coupled cluster theory with singles and doubles including perturbative triples [CCSD(T)] and CCSD(T) with the explicitly correlated F12 b correction [CCSD(T)-F12b] were employed for this purpose along with large correlation consistent cc-pVTZ, cc-pVQZ, and cc-pV5Z basis sets. Harmonic vibrational frequencies, infrared vibrational intensities, rotational constants, and dipole moments are reported. An harmonic vibrational fundamentals along with centrifugal distortion constants, and vibration-rotation interaction constants are also reported for all the cyclic isomers. The energies computed with the CCSD(T) and CCSD(T)-F12b methods were extrapolated to the one-particle complete basis set (CBS) limit following a three-point formula. At the CCSD(T)-F12b/CBS level of theory the 3,3H-diazirine (c-CH2N2) is the lowest energy cyclic isomer followed by 1,3H-diazirine, (E)-1,2H-Diazirine, and (Z)-1,2H-diazirine which are 20.1, 47.8, and 51.3kcal mol-1above the 3,3H-Diazirine, respectively. Accurate structures and spectroscopic constants that are reported here could be useful for future identification of these cyclic nitrogenated organic molecules in the interstellar medium or circumstellar disks.
Energy Landscape↗
Quantum error mitigation by layerwise Richardson extrapolation
A widely used method for mitigating errors in noisy quantum computers is Richardson extrapolation, a technique in which the overall effect of noise on the estimation of quantum expectation values is captured by a single parameter that, after being scaled to larger values, is eventually extrapolated to the zero-noise limit. We generalize this approach by introducing layerwise Richardson extrapolation (LRE), an error mitigation protocol in which the noise of different individual layers (or larger chunks of the circuit) is amplified and the associated expectation values are linearly combined to estimate the zero-noise limit. The coefficients of the linear combination are analytically obtained from the theory of multivariate Lagrange interpolation. LRE leverages the flexible configurational space of layerwise unitary folding, allowing for a more nuanced mitigation of errors by treating the noise level of each layer of the quantum circuit as an independent variable. Furthermore, we provide numerical simulations demonstrating scenarios where LRE achieves superior performance compared to traditional (single-variable) Richardson extrapolation.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
How Should Machine Learning Be Successfully Used for Wind Speed Vertical Extrapolation?
An accurate characterization of the wind resource available at hub-height is required for an efficient and bankable wind farm project. However, direct measurement of wind speed at the constantly increasing height of the hub of commercial wind turbines is oftentimes challenging and expensive, so that it is common practice to vertically extrapolate the wind resource from lower and more easily accessible levels. Conventional techniques for wind speed vertical extrapolation include the use of a power law and a logarithmic profile. While simple, the limits in accuracy of these methods have been shown in various studies. Recently, machine learning has been proposed as a new method to vertically extrapolate winds. All the published studies on the topic assess the performance of machine learning techniques in vertically extrapolating the wind resource at the same location where the algorithm has been trained. However, in real-world applications, the wind resource is measured at the instrument location, but it then needs to be extrapolated at hub height at the location of the wind turbines within the find farm. To be able to fully recommend the use of machine learning techniques over the simple power law and logarithmic law, the spatial variability of the performance improvements of the machine learning approaches needs to be assessed. Here, we propose a round-robin validation of a machine learning-based method for wind speed extrapolation. We use 20 months of observations at four locations spanning a 100 km wide region at the Southern Great Plains (SGP) atmospheric observatory, in north-central Oklahoma. At each location, we train a random forest to predict 30-min average wind speed at 143 m AGL. We use as input features lidar wind speed at 65 m AGL, time of day, sonic anemometer wind speed at 4 m AGL, turbulent kinetic energy, and Obukhov length. First, we perform a same-site comparison of the performance of the proposed random forest against the conventional techniques for wind speed extrapolation (namely power law and logarithmic profile, with widely accepted stability corrections). We find that the random forest outperforms the power law in vertically extrapolating wind speed in all the considered stability regimes, with a 33% reduction in MAE for stable conditions, and a 31% reduction in unstable conditions. Similar results are found when comparing predictions of extrapolated winds from the logarithmic profile and the random forest with the observed values. Next, we propose a round-robin validation, to use the random forest trained at each site to extrapolate wind speed at the remaining three sites. We find that the performance of the random forest approach degrades when the algorithm is tested at a site different than the training one. However, even under those circumstances, the machine learning-based approach still outperforms the conventional techniques for wind speed extrapolation, with, on average, a reduction in mean absolute error between 15 and 20% over the conventional methods, with the largest benefits obtained under stable conditions.
Monte Carlo↗
Scientific study of data analysis
We present a comparison between two numerical methods for the extrapolation of nonlinear force-free magnetic fields, the Iterative Method (IM) and the Progressive Extension Method (PEM). The advantages and disadvantages of these two methods are summarized and the accuracy and numerical instability are discussed. On the basis of this investigation, we claim that the two methods do resemble each other qualitatively.
Wu, S. T.↗
Improved naive Bayesian probability classifier in predictions of nuclear mass
Recently, novel statistical methods such as neural networks and Bayesian learning methods are implemented to describe the nuclear masses. Based on previous studies, an improved naive Bayesian probability (iNBP) classifier is proposed to study the nuclear masses by refining the results of sophisticated nuclear models. In the iNBP method, the prediction for nuclear masses is treated as a classification problem. The residuals are classified into several groups to generate prior and conditional probabilities, and the posterior probabilities are further determined by the Bayesian formula. We choose the expectation with maximum probability as the final prediction. Reliability of the iNBP method is assessed by analyzing the global optimizations and the extrapolating capabilities. Here, the iNBP method exhibits impressive improvements on global descriptions for different mass models. Moreover, the method shows robust extrapolating capabilities. Results demonstrate the iNBP method can be applied to predict the nuclear masses of unknown regions. Considering the local mass relations, the iNBP method can offer considerable fine-tuning of the mass descriptions from nuclear models. The methodology proposed in this paper can also be applied to other model-based extrapolations of nuclear observables.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Airfoil shape optimization using sensitivity analysis on viscous flow equations
An aerodynamic shape optimization method has previously been developed by the authors using the Euler equations and has been applied to supersonic-hypersonic nozzle designs. This method has also included a flowfield extrapolation (or flow prediction) method based on the Taylor series expansion of an existing CFD solution. The present paper reports on the extension of this method to the thin-layer Navier-Stokes equations in order to account for the viscous effects. Also, to test the method under highly nonlinear conditions, it has been applied to the transonic flows. Initially, the success of the flow prediction method is tested. Then, the overall method is demonstrated by optimizing the shapes of two supercritical transonic airfoils at zero angle of attack. The first one is shape optimized to achieve a minimum drag while obtaining a lift above a specified value. Whereas, the second one is shape optimized for a maximum lift while attaining a drag below a specified value. The results of these two cases indicate that the present method can produce successfully optimized aerodynamic shapes.
Eleshaky, Mohamed E.↗
Compact representation and long-time extrapolation of real-time data for quantum systems using the ESPRIT algorithm
Representing real-time data as a sum of complex exponentials provides a compact form that enables both denoising and extrapolation. As a fully data-driven method, the Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) algorithm is agnostic to the underlying physical equations, making it broadly applicable to various observables and experimental or numerical setups. In this work, we consider applications of the ESPRIT algorithm primarily to extend real-time dynamical data from simulations of quantum systems. We evaluate ESPRIT's performance in the presence of noise and compare it to other extrapolation methods. We demonstrate its ability to extract information from short-time dynamics to reliably predict long-time behavior and determine the minimum time interval required for accurate results. We discuss how this insight can be leveraged in numerical methods that propagate quantum systems in time, and we show how ESPRIT can predict infinite-time values of dynamical observables, offering a purely data-driven approach to characterizing quantum phases.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Semiempirical Theories of the Affinities of Negative Atomic Ions
The determination of the electron affinities of negative atomic ions by means of direct experimental investigation is limited. To supplement the meager experimental results, several semiempirical theories have been advanced. One commonly used technique involves extrapolating the electron affinities along the isoelectronic sequences, The most recent of these extrapolations Is studied by extending the method to Include one more member of the isoelectronic sequence, When the results show that this extension does not increase the accuracy of the calculations, several possible explanations for this situation are explored. A different approach to the problem is suggested by the regularities appearing in the electron affinities. Noting that the regular linear pattern that exists for the ionization potentials of the p electrons as a function of Z, repeats itself for different degrees of ionization q, the slopes and intercepts of these curves are extrapolated to the case of the negative Ion. The method is placed on a theoretical basis by calculating the Slater parameters as functions of q and n, the number of equivalent p-electrons. These functions are no more than quadratic in q and n. The electron affinities are calculated by extending the linear relations that exist for the neutral atoms and positive ions to the negative ions. The extrapolated. slopes are apparently correct, but the intercepts must be slightly altered to agree with experiment. For this purpose one or two experimental affinities (depending on the extrapolation method) are used in each of the two short periods. The two extrapolation methods used are: (A) an isoelectronic sequence extrapolation of the linear pattern as such; (B) the same extrapolation of a linearization of this pattern (configuration centers) combined with an extrapolation of the other terms of the ground configurations. The latter method Is preferable, since it requires only experimental point for each period. The results agree within experimental error with all data, except with the most recent value of C, which lies 10% lower.
Edie, John W.↗
Reliable extrapolation of deep neural operators informed by physics or sparse observations
Deep neural operators can learn nonlinear mappings between infinite-dimensional function spaces via deep neural networks. As promising surrogate solvers of partial differential equations (PDEs) for real-time prediction, deep neural operators such as deep operator networks (DeepONets) provide a new simulation paradigm in science and engineering. Pure data-driven neural operators and deep learning models, in general, are usually limited to interpolation scenarios, where new predictions utilize inputs within the support of the training set. However, in the inference stage of real-world applications, the input may lie outside the support, i.e., extrapolation is required, which may result to large errors and unavoidable failure of deep learning models. Here, we address this challenge of extrapolation for deep neural operators. First, we systematically investigate the extrapolation behavior of DeepONets by quantifying the extrapolation complexity, via the 2-Wasserstein distance between two function spaces and propose a new strategy of bias–variance trade-off for extrapolation with respect to model capacity. Subsequently, we develop a complete workflow, including extrapolation determination, and we propose five reliable learning methods that guarantee a safe prediction under extrapolation by requiring additional information—the governing PDEs of the system or sparse new observations. The proposed methods are based on either fine-tuning a pre-trained DeepONet or multifidelity learning. We demonstrate the effectiveness of the proposed framework for various types of parametric PDEs. Furthermore, our systematic comparisons provide practical guidelines for selecting a proper extrapolation method depending on the available information, desired accuracy, and required inference speed.
42 ENGINEERING↗
Heat flux measurements on ceramics with thin film thermocouples
Two methods were devised to measure heat flux through a thick ceramic using thin film thermocouples. The thermocouples were deposited on the front and back face of a flat ceramic substrate. The heat flux was applied to the front surface of the ceramic using an arc lamp Heat Flux Calibration Facility. Silicon nitride and mullite ceramics were used; two thicknesses of each material was tested, with ceramic temperatures to 1500 C. Heat flux ranged from 0.05-2.5 MW/m2(sup 2). One method for heat flux determination used an approximation technique to calculate instantaneous values of heat flux vs time; the other method used an extrapolation technique to determine the steady state heat flux from a record of transient data. Neither method measures heat flux in real time but the techniques may easily be adapted for quasi-real time measurement. In cases where a significant portion of the transient heat flux data is available, the calculated transient heat flux is seen to approach the extrapolated steady state heat flux value as expected.
Holanda, Raymond↗