Results for “Variance stabilization” — Page 11

DOE OSTI2018

The path from litter to soil: Insights into soil C cycling from long-term input manipulation and high-resolution mass spectrometry

The pathways carbon (C) takes from plant litter to soil organic matter (SOM) underlies our understanding of how soil C stocks and microbial decomposition will respond to climate change, and determining whether natural soil C sinks can be enhanced to offset potential losses. Long-term ecosystem-scale litter manipulations and innovations in the molecular characterization of SOM provide a unique opportunity to explore these dynamics. We incubated soils from a 20-year litter-input experiment for 525 days and asked how litter quantity and source (i.e., roots versus aboveground litter) affected soil C cycling, including microbial function and the size and molecular composition of C pools defined by degree of mineral association. Input exclusion led to a 30% loss of soil C, attributable largely to the non-mineral-associated C fraction, and was accompanied by declines in soil C decomposition. The absence of roots led to a shift in the microbial catabolic profile, though there was no evidence that root litter was preferentially stabilized. Carbon did not accumulate under litter addition. However, direct examination of the finest mineral fraction with Fourier transform ion cyclotron resonance mass spectrometry (FTICR-MS) revealed dramatic changes to the chemical composition of carbon. In particular, lipid content increased proportionally with input addition, and was subsequently mineralized during incubation, indicating that this apparently stable fraction is metabolically active. Moreover, non-metric dimensional scaling (NMS) showed that both litter treatments and incubation caused the molecular composition of SOM to diverge, and explained 79% of the variance in soil respiration. We conclude that the path from litter to soil is more complex than previously thought and mineral-associated SOM thought to be stable may be microbially available.

Reynolds, L L.↗

DOE OSTI2022

Parametric model-order-reduction development for unsteady convection

A time-averaged error indicator with POD- h Greedy is developed to drive parametric model order reduction (pMOR) for 2D unsteady natural convection in a high-aspect ratio slot parameterized with the Prandtl number, Rayleigh number, and slot angle with respect to the gravity. The error indicator is extended to accommodate the energy equation and Leray regularization. Despite being two-dimensional and laminar, the target flow regime presents several challenges: 1) there is a bifurcation in the angle parameter space; 2) the solution can be multivalued, even at steady state; and 3) the solution exhibits spatio-temporal chaos at several points in the parameter space. The authors explore several reduced-order models (ROMs) and demonstrate that Leray-regularized Galerkin ROMs provide a robust solution approach for this class of flows. They further demonstrate that error-indicated pMOR can efficiently predict several QOIs, such as mean flow, mean Nusselt number and mean turbulent kinetic energy, even in the presence of a bifurcation. Finally, they show that spatio-temporal chaos can lead to lack of reproducibility in both the full-order model and the reduced-order model and that the variance in the full-order model provides a lower bound on the pMOR error in these cases.

leray regularization↗

DOE OSTI2022

Discovering equations that govern experimental materials stability under environmental stress using scientific machine learning

Abstract While machine learning (ML) in experimental research has demonstrated impressive predictive capabilities, extracting fungible knowledge representations from experimental data remains an elusive task. In this manuscript, we use ML to infer the underlying differential equation (DE) from experimental data of degrading organic-inorganic methylammonium lead iodide (MAPI) perovskite thin films under environmental stressors (elevated temperature, humidity, and light). Using a sparse regression algorithm, we find that the underlying DE governing MAPI degradation across a broad temperature range of 35 to 85 °C is described minimally by a second-order polynomial. This DE corresponds to the Verhulst logistic function, which describes reaction kinetics analogous to self-propagating reactions. We examine the robustness of our conclusions to experimental variance and Gaussian noise and describe the experimental limits within which this methodology can be applied. Our study highlights the promise and challenges associated with ML-aided scientific discovery by demonstrating its application in experimental chemical and materials systems.

36 MATERIALS SCIENCE↗

NASA NTRS2001

Structure and Variability of Water Vapor in the Upper Troposphere and Lower Stratosphere

Upper-tropospheric humidity (UTH) has been synoptically mapped via an algorithm that rejects small-scale undersampled variance, which is intrinsic to asymptotic measurements of water vapor, cloud, and other convective properties. Mapped distributions of UTH have been used, jointly with high-resolution Global Cloud Imagery (GCI), to study how the upper troposphere is humidified. The time-mean distribution of UTH is spatially correlated to the time-mean distribution of cold cloud fraction (eta)(sub c) (T < than 230 K). Regions of large UTH coincide with regions of large eta(sub c), which mark deep convection. They also coincide with regions of reduced vertical stability, in which the vertical gradient of theta is weakened by convective mixing. Coldest cloud cover is attended convective overshoots above the local tropopause, which is simultaneously coldest and highest. Together, these features reflect the upper-troposphere being ventilated by convection, which mixes in moist air from lower levels. Histograms of UTH and eta(sub c) have been applied to construct the joint probability density function, which quantifies the relationship between these properties. The expected value of UTH in convective regions is strongly correlated to the expected value of eta(sub c). In ensembles of asymptotic samples, the correlation between epsilon[UTH] and epsilon[eta(sub c)] exceeds 0.80. As these expectations reflect the most likely values, the strong correlation between epsilon[UTH] and epsilon[eta(sub c)] indicates that the large-scale organization of UTH is strongly shaped by convective pumping of moisture from lower levels. The same relationship holds for unsteady fields - even though, instantaneously, those fields are comprised almost entirely of small-scale convective structure. The spatial autocorrelation of UTH, constructed at high resolution from overpass data along ascending and descending tracks of the orbit, is limited to only a couple of degrees in the horizontal. This mirrors the spatial autocorrelation of eta(sub c), which likewise operates coherently on short scales. The short correlation scale of UTH, which reflects the scale of individual convective systems, is comparable to the spacing of retrievals from MLS. These scales are undersampled in the asynoptic measurements. Despite their prevalence, the mapping algorithm described above successfully recovers synoptic behavior operating coherently on large scales. It reveals eastward migration of anomalous UTH from the Indian ocean to the central Pacific, in association with the modulation of convection by the Madden-Julian oscillation. Additional information is contained in the original extended abstract.

Salby, Murry L.↗

NASA NTRS1989

Daytime turbulent exchange between the Amazon forest and the atmosphere

Detailed observations of turbulence just above and below the crown of the Amazon rain forest during the wet season are presented. The forest canopy is shown to remove high frequency turbulent fluctuations while passing lower frequencies. Filter characteristics of turbulent transfer into the Amazon rain forest canopy are quantified. Simple empirical relations that relate observed turbulent heat fluxes to horizontal wind variance are presented. Changes in the amount of turbulent coupling between the forest and the boundary layer associated with deep convective clouds are presented both as statistical averages and as a series of case studies. These convective processes during the rainy season are shown to alter the diurnal course of turbulent fluxes. In wake of giant coastal systems, no significant heat or moisture fluxes occur for up to a day after the event. Radar data is used to demonstrate that even small raining clouds are capable of evacuating the canopy of substances normally trapped by persistent static stability near the forest floor. Recovery from these events can take more than an hour, even during mid-day. In spite of the ubiquitous presence of clouds and frequent rain during this season, the average horizontal wind speed spectrum is well described by dry CBL similarity hypotheses originally found to apply in flat terrain.

Fitzjarrald, David R.↗

NASA NTRS1990

Daytime turbulent exchange between the Amazon forest and the atmosphere

Detailed observations of turbulence just above and below the crown of the Amazon rain forest during the wet season are presented. The forest canopy is shown to remove high frequency turbulent fluctuations while passing lower frequencies. Filter characteristics of turbulent transfer into the Amazon rain forest canopy are quantified. Simple empirical relations that relate observed turbulent heat fluxes to horizontal wind variance are presented. Changes in the amount of turbulent coupling between the forest and the boundary layer associated with deep convective clouds are presented both as statistical averages and as a series of case studies. These convective processes during the rainy season are shown to alter the diurnal course of turbulent fluxes. In wake of giant coastal systems, no significant heat or moisture fluxes occur for up to a day after the event. Radar data is used to demonstrate that even small raining clouds are capable of evacuating the canopy of substances normally trapped by persistent static stability near the forest floor. Recovery from these events can take more than an hour, even during mid-day. In spite of the ubiquitous presence of clouds and frequent rain during this season, the average horizontal wind speed spectrum is well described by dry CBL similarity hypotheses originally found to apply in flat terrain.

Fitzjarrald, David R.↗

NASA NTRS2023

The Statistical Emulators of GGCMI Phase 2: Responses of Year-to-Year Variation of Crop Yield to CO 2 , Temperature, Water and Nitrogen Perturbations

Understanding the impact of climate change on year-to-year variation of crop yield is critical to global food stability and security. While crop model emulators are believed to be lightweight tools to replace the models, few emulators have been developed to capture such interannual variation of crop yield in response to climate variability. In this study, we developed a statistical emulator with a machine learning algorithm to reproduce the response of year-to-year variation of four crop yields to CO 2 (C), temperature (T), water (W), and nitrogen (N) perturbations defined in the Global Gridded Crop Model Intercomparison Project (GGCMI) phase 2. The emulators were able to explain more than 52 % of the variance of simulated yield and performed well in capturing the year-to-year variation of global average and gridded crop yield over current croplands in the baseline. With the changes in CO 2 –temperature–water–nitrogen (CTWN) perturbations, the emulators could reproduce the year-to-year variation of crop yield well over most current cropland. The variation of R and the mean absolute error was small under the single CTWN perturbations and dual-factor perturbations. These emulators thus provide statistical response surfaces of yield, including both its mean and interannual variability, to climate factors. They could facilitate spatiotemporal downscaling of crop model simulation, projecting the changes in crop yield variability in the future and serving as a lightweight tool for multi-model ensemble simulation. The emulators enhanced the flexibility of crop yield estimates and expanded the application of large-ensemble simulations of crop yield under climate change.

Climate change↗

DOE OSTI2017

An error reduction algorithm to improve lidar turbulence estimates for wind energy

Abstract. Remote-sensing devices such as lidars are currently being investigated as alternatives to cup anemometers on meteorological towers for the measurement of wind speed and direction. Although lidars can measure mean wind speeds at heights spanning an entire turbine rotor disk and can be easily moved from one location to another, they measure different values of turbulence than an instrument on a tower. Current methods for improving lidar turbulence estimates include the use of analytical turbulence models and expensive scanning lidars. While these methods provide accurate results in a research setting, they cannot be easily applied to smaller, vertically profiling lidars in locations where high-resolution sonic anemometer data are not available. Thus, there is clearly a need for a turbulence error reduction model that is simpler and more easily applicable to lidars that are used in the wind energy industry. In this work, a new turbulence error reduction algorithm for lidars is described. The Lidar Turbulence Error Reduction Algorithm, L-TERRA, can be applied using only data from a stand-alone vertically profiling lidar and requires minimal training with meteorological tower data. The basis of L-TERRA is a series of physics-based corrections that are applied to the lidar data to mitigate errors from instrument noise, volume averaging, and variance contamination. These corrections are applied in conjunction with a trained machine-learning model to improve turbulence estimates from a vertically profiling WINDCUBE v2 lidar. The lessons learned from creating the L-TERRA model for a WINDCUBE v2 lidar can also be applied to other lidar devices. L-TERRA was tested on data from two sites in the Southern Plains region of the United States. The physics-based corrections in L-TERRA brought regression line slopes much closer to 1 at both sites and significantly reduced the sensitivity of lidar turbulence errors to atmospheric stability. The accuracy of machine-learning methods in L-TERRA was highly dependent on the input variables and training dataset used, suggesting that machine learning may not be the best technique for reducing lidar turbulence intensity (TI) error. Future work will include the use of a lidar simulator to better understand how different factors affect lidar turbulence error and to determine how these errors can be reduced using information from a stand-alone lidar.

17 WIND ENERGY↗

DOE OSTI2016

A general solution strategy of modified power method for higher mode solutions

A general solution strategy of the modified power iteration method for calculating higher eigenmodes has been developed and applied in continuous energy Monte Carlo simulation. The new approach adopts four features: 1) the eigen decomposition of transfer matrix, 2) weight cancellation for higher modes, 3) population control with higher mode weights, and 4) stabilization technique of statistical fluctuations using multi-cycle accumulations. The numerical tests of neutron transport eigenvalue problems successfully demonstrate that the new strategy can significantly accelerate the fission source convergence with stable convergence behavior while obtaining multiple higher eigenmodes at the same time. The advantages of the new strategy can be summarized as 1) the replacement of the cumbersome solution step of high order polynomial equations required by Booth's original method with the simple matrix eigen decomposition, 2) faster fission source convergence in inactive cycles, 3) more stable behaviors in both inactive and active cycles, and 4) smaller variances in active cycles. Advantages 3 and 4 can be attributed to the lower sensitivity of the new strategy to statistical fluctuations due to the multi-cycle accumulations. The application of the modified power method to continuous energy Monte Carlo simulation and the higher eigenmodes up to 4th order are reported for the first time in this paper. -- Graphical abstract: -- Highlights: •Modified power method is applied to continuous energy Monte Carlo simulation. •Transfer matrix is introduced to generalize the modified power method. •All mode based population control is applied to get the higher eigenmodes. •Statistic fluctuation can be greatly reduced using accumulated tally results. •Fission source convergence is accelerated with higher mode solutions.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2017

Reduced African Easterly Wave Activity with Quadrupled CO 2 in the Superparameterized CESM

African easterly wave (AEW) activity is examined in quadrupled CO 2 experiments with the superparameterized CESM (SP-CESM). The variance of 2–10-day filtered precipitation increases with warming over the West African monsoon region, suggesting increased AEW activity. The perturbation enstrophy budget is used to investigate the dynamic signature of AEW activity. The northern wave track becomes more active associated with enhanced baroclinicity, consistent with previous studies. The southern track exhibits a surprising reduction of wave activity associated with less frequent occurrence of weak waves and a slight increase in the occurrence of strong waves. These changes are connected to changes in the profile of vortex stretching and tilting that can be understood as interconnected consequences of increased static stability from the lapse rate response, weak temperature gradient balance, and the fixed anvil temperature hypothesis.

58 GEOSCIENCES↗

NASA NTRS1982

Synthesis of a modern/classical controller for large space structures

The paper is concerned with wide-sense stationary random vibrations of a representative large space structure subject to nonwhite excitations, and with synthesis of its controller amid modeling uncertainties due to truncated modes. Dynamics is analyzed by residue calculus in frequency domain. Steady state variance of a modal coordinate is shown to depend on, among other things, power spectral density of excitation at the modal frequency. Modal cost analysis is performed for an order reduction with a performance objective of attitude, attitude rate and energy control. Optimal and suboptimal output feedback controllers are compared. By treating deleted modes as an additive and a multiplicative perturbation in an idealized transfer function, the low frequency modes omitted according to modal cost analysis are found to cause instability. Finally, the size of the idealized model is shown to be governed by light damping and stringency in performance and stability specifications.

Hablani, H. B.↗

NASA NTRS1985

A class of optimum digital phase locked loops for the DSN advanced receiver

A class of optimum digital filters for digital phase locked loop of the deep space network advanced receiver is discussed. The filter minimizes a weighted combination of the variance of the random component of the phase error and the sum square of the deterministic dynamic component of phase error at the output of the numerically controlled oscillator (NCO). By varying the weighting coefficient over a suitable range of values, a wide set of filters are obtained such that, for any specified value of the equivalent loop-noise bandwidth, there corresponds a unique filter in this class. This filter thus has the property of having the best transient response over all possible filters of the same bandwidth and type. The optimum filters are also evaluated in terms of their gain margin for stability and their steady-state error performance.

Hurd, W. J.↗

DOE OSTI2017

An Approximate Entropy Based Approach for Quantifying Stability in Spatio-Temporal Data with Limited Temporal Observations

Identifying erratic or unstable time-series is an area of interest to many fields. Recently, there have been successful developments towards this goal. These newly developed methodologies however come from domains where it is typical to have several thousand or more temporal observations. This creates a challenge when attempting to apply these methodologies to time-series with much fewer temporal observations such as for spatio-temporal socio-cultural understanding, a domain where a typical time series of interest might only consist of 20-30 annual observations. Identifying instability in spatio-temporal trends is critical for understanding global dynamics and finding areas of potential concern or intervention. In the case of spatio-temporal socio-cultural understanding, instability is marked by two characteristics, 1) how widely varying the values are and 2) how predictable that variance is from one observation to the next. Approaches for simultaneously addressing both of these concerns have been limited. In this paper, we introduce an approximate entropy based method for characterizing the behaviour of a time series with limited temporal observations. For Geocomputation, this methodology represents a novel additional tool for researchers to use for exploring and understanding spatio-temporal data, specifically with limited temporal observations. As a case study, we look at national youth male unemployment across the world from 1991-2014.

Piburn, Jesse↗

DOE OSTI2021

Impact of cyberattacks on safety and stability of connected and automated vehicle platoons under lane changes

Connected and automated vehicles (CAVs) offer a huge potential to improve the operations and safety of transportation systems. However, the use of smart devices and communications in CAVs introduce new risks. CAVs would leverage vehicle to vehicle (V2V) and vehicle to infrastructure (V2I) communication, thus providing additional system access points compared to traditional systems. Automation makes these systems more vulnerable and increases the consequences of cyberattacks. This study utilizes an infrastructure-based communication platform consisting of cooperative adaptive cruise control and lane control advisories developed by the authors to perform cyber risk assessment of CAVs. The study emulates three types of cyberattacks (message falsification, dedicated denial of service, and spoofing attacks) in a representative traffic environment consisting of multiple CAV platoons and lane change events to analyze the safety and stability impacts of the cyberattacks. Simulation experiments using VISSIM reveals that traffic stream and CAV string is unstable under all three types of cyberattacks. The worst case is represented by the message falsification attack. Increases in volatility are observed over a no attack case, with variations increasing by an average of 43%–51% along with an increase of over 3000 crash conflicts. Similarly, lane change crash conflicts are observed to be more severe compared to rear end crash conflicts, showing a higher probability of severe injuries. Further, the case of slight cyberattack on a single CAV also creates significant disruption in the traffic stream. Analysis of variance (ANOVA) reveals the statistical significance of the results. Furthermore, these results pave the way for future design of secure systems from a monitoring perspective.

97 MATHEMATICS AND COMPUTING↗

NASA NTRSDate not supplied

Effects of Mild Hypobaric Hypoxia on Visual Field Impairment

INTRODUCTION: The primary objective of the Exploration Atmosphere (EA) study is to validate a new prebreathe protocol necessary before exposure to a suited hypobaric environment during extravehicular activity (EVA) from a habitat ‘exploration atmosphere’ of 56.5kPa (8.2 psia), 34% O2, 66% N2. Prebreathe protocols must be time and resource efficient while also controlling Decompression Sickness (DCS) risk to within acceptable limits. The habitat hypobaric exploration atmosphere also results in a mildly hypoxic environment (piO2 = 128mmHg). As a secondary objective of the EA study, we characterized the effects of 11-day exposure to a mild hypobaric hypoxic environment on visual performance. METHODS: Two, 11-day hypobaric chamber tests were performed (EA-1and EA-2, n=8 each) in NASA’s 20-foot chamber at Johnson Space Center where subjects lived in the exploration atmosphere. Subjects also underwent simulated 6-hour EVAs at 85% O2 and 29.6 kPa during EVA on days 3, 5, 7, 9, and 11. Visual acuity (VA), a measure of spatial resolution, and contrast sensitivity (CS), a measure of ability to distinguish ever finer increments of brightness, were assessed on non-EVA days by using gapped Landolt C testing. The luminance was controlled by using a booth, a light monitor, and a lighting rheostat. EA-1 data had revealed problems in lighting control impacting consistency of the data. Procedures were subsequently updated for the EA-2 test. RESULTS: EA-1 data revealed large variance and data recording errors and was removed from the analysis. One participant left the study at Test Day 3 during EA-2. EA-2 ANOVA results showed CS (mean change=0.010 logCSWeber) and VA (mean change, -0.016 logMAR) between pre-test and 11-day test phases; however, neither VA nor CS changes were statistically significant. There were non-statistically significant declines in VA across test phases. DISCUSSION: Overall, visual field performance did not exhibit clinically significant changes (3 lines or greater change in LogMar chart) during exposure to the mild hypoxic exploration atmosphere environment compared to pre-test baseline. The consistency and stability of VA data during EA-2 suggests that the mild hypobaric hypoxic environment did not cause clinically significant negative impacts to participants’ visual field performance.

Visual Acuity↗

NASA NTRSDate not supplied

Effects of Mild Hypobaric Hypoxia on Visual Field Impairment

INTRODUCTION: The primary objective of the Exploration Atmosphere (EA) study is to validate a new prebreathe protocol necessary before exposure to a suited hypobaric environment during extravehicular activity (EVA) from a habitat ‘exploration atmosphere’ of 56.5kPa (8.2 psia), 34% O2, 66% N2. Prebreathe protocols must be time and resource efficient while also controlling Decompression Sickness (DCS) risk to within acceptable limits. The habitat hypobaric exploration atmosphere also results in a mildly hypoxic environment (piO2 = 128mmHg). As a secondary objective of the EA study, we characterized the effects of 11-day exposure to a mild hypobaric hypoxic environment on visual performance. METHODS: Two, 11-day hypobaric chamber tests were performed (EA-1and EA-2, n=8 each) in NASA’s 20-foot chamber at Johnson Space Center where subjects lived in the exploration atmosphere. Subjects also underwent simulated 6-hour EVAs at 85% O2 and 29.6 kPa during EVA on days 3, 5, 7, 9, and 11. Visual acuity (VA), a measure of spatial resolution, and contrast sensitivity (CS), a measure of ability to distinguish ever finer increments of brightness, were assessed on non-EVA days by using gapped Landolt C testing. The luminance was controlled by using a booth, a light monitor, and a lighting rheostat. EA-1 data had revealed problems in lighting control impacting consistency of the data. Procedures were subsequently updated for the EA-2 test. RESULTS: EA-1 data revealed large variance and data recording errors and was removed from the analysis. One participant left the study at Test Day 3 during EA-2. EA-2 ANOVA results showed CS (mean change=0.010 logCSWeber) and VA (mean change, -0.016 logMAR) between pre-test and 11-day test phases; however, neither VA nor CS changes were statistically significant. There were non-statistically significant declines in VA across test phases. DISCUSSION: Overall, visual field performance did not exhibit clinically significant changes (3 lines or greater change in LogMar chart) during exposure to the mild hypoxic exploration atmosphere environment compared to pre-test baseline. The consistency and stability of VA data during EA-2 suggests that the mild hypobaric hypoxic environment did not cause clinically significant negative impacts to participants’ visual field performance.

Visual Acuity↗

DOE OSTI2022

Real-Time Distribution System State Estimation with Asynchronous Measurements

We report state estimation is a fundamental task in power systems. Although distribution systems are increasingly equipped with sensing devices and smart meters, measurements are typically reported at different rates and asynchronously; these aspects pose severe strains on workhorse state estimation algorithms, which are designed to process batches of data collected in a synchronous manner from all the measurement units. In this paper, we develop a novel state estimation algorithm to continuously update the estimate of the state based on measurements received in an asynchronous manner from measurement units. The synthesis of the algorithm hinges on a proximal-point type method, implemented in an online fashion, and capable of processing measurements received sequentially from sensors. A performance analysis is presented by providing bounds on the estimation error in terms of the mean and variance that hold at each iteration and asymptotically. The scheme is also compared with a more traditional Weighted Least Squares estimator that compensates for the lack of measurement data by using, as pseudo measurements, the measurement retrieved during a certain time window. Numerical simulations on the IEEE 37-bus feeder corroborate the analytical findings.

24 POWER TRANSMISSION AND DISTRIBUTION↗

NASA NTRS1996

Expendable Launch Vehicle Studies

Analytical support studies of expendable launch vehicles concentrates on the stability of the dynamics during launch especially during or near the region of maximum dynamic pressure. The in-plane linearized dynamic equations of a generic launch vehicle with multiple flexible bending and fuel sloshing modes are developed. The design of a robust LQR controller based on the reduced order system is accomplished using the parameter perturbation technique. The ELV modeling and analysis team has spent the past three years working on the theoretical development and application of sensitivity analysis to solve eigenvalue problems associated with structural dynamics. Specific application areas include stochastic vibrations, viscously damped vibrations and nonlinear dynamics. In stochastic linear vibrations, sensitivity analysis methods have been developed which determined the expectation and variance of the response for given probability density functions of various stochastic parameters. Examples worked include beam vibrations with up to six stochastic parameters; and the eigensensitivity results differ little from those obtained, with much effect, from Monte Carlo techniques. For viscously damped vibrations, eigensensitivity analysis has given excellent results for homogeneous beams, modeled by either quadratic or quartic eigenvalue equations. Specific applications include Kevin and Maxwell-type viscoelastic beams.

Bainum, P. M.↗