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All sourcesNASA NTRSDOE OSTI
NASA NTRS1996

[?]Nonlinear Issues in the Aerothermochemistry of Gases and Materials and the Associated Physics and Dynamics of Interfaces

Our research and technology are focused on nonlinear issues in the aerothermochemistry of gases and materials and the associated physics and dynamics of interfaces. Our program is now organized to aggressively support the NASA Aeronautics Enterprise so as to: (a) develop a new generation of environmentally compatible, economic subsonic aircraft; (b) develop the technology base for an economically viable and environmentally compatible high-speed civil transport; (c) develop the technology options for new capabilities in high-performance aircraft; (d) develop hypersonic technologies for air-breathing flight; and (e) develop advanced concepts, understanding of physical phenomena, and theoretical, experimental, and computational tools for advanced aerospace systems. The implications from our research for aeronautical and aerospace technology have been both broad and deep. For example, using advanced computational techniques, we have determined exact solutions for the Schrodinger equation in electron-molecule scattering allowing us to evaluate atmospheric models important to reentry physics. We have also found a new class of exact solutions for the Navier Stokes equations. In experimental fluid dynamics, we have found explicit evidence of turbulence modification of droplet sizes in shock tube flow with condensation. We have developed a new diagnostic tool for the direct estimation of flow velocities at MHz sampling rates in quasi-one dimensional turbulent flow. This procedure suggests an unexpected confirmation of the possibility of 'natural' closure in Reynolds stresses with deep implications for the development of turbulent models. A transient increase is observed in both the spectral energy decay rate and the degree of chaotic complexity at the interface of a shock wave and a turbulent ionized gas. Even though the gas is apparently brought to rest by the shock wave, no evidence is found either of the expected relaminarization. A unique diamond-shaped nozzle has been designed for a detailed investigation of the effect of significant streamwise vorticity on the acoustic and IR characteristics of supersonic jets. Our results provide convincing evidence of the significant effect of vorticity on the far-field noise for the diamond jet as compared to the conventional round jets. We have found that the countercurrent shear layer mixes much more efficiently than conventional coflowing shear layers. We also developed the fluid thrust vectoring procedures which use counter flow to vector a jet. Our materials research has shown that the steep stress gradients at the fiber-matrix interface could be the primary cause of interface cracks after the processing of metallic and intermetallic matrix composites. New techniques have been evolved for: the microcharacterization of materials including microplastic strain and, point by point, the misorientation and plasticity for matrix composites; thermally induced stress measurements and load relaxation; the growth and characterization of metallic matrix composite interfaces; and for the growth of ferrite materials by pulsed laser deposition. The FAMU commitment to the HBCU Research Center also continues to be broad and deep.

Johnson, Joseph A., III↗
NASA NTRS1994

Infinite possibilities: Computational structures technology

Computational Fluid Dynamics (or CFD) methods are very familiar to the research community. Even the general public has had some exposure to CFD images, primarily through the news media. However, very little attention has been paid to CST--Computational Structures Technology. Yet, no important design can be completed without it. During the first half of this century, researchers only dreamed of designing and building structures on a computer. Today their dreams have become practical realities as computational methods are used in all phases of design, fabrication and testing of engineering systems. Increasingly complex structures can now be built in even shorter periods of time. Over the past four decades, computer technology has been developing, and early finite element methods have grown from small in-house programs to numerous commercial software programs. When coupled with advanced computing systems, they help engineers make dramatic leaps in designing and testing concepts. The goals of CST include: predicting how a structure will behave under actual operating conditions; designing and complementing other experiments conducted on a structure; investigating microstructural damage or chaotic, unpredictable behavior; helping material developers in improving material systems; and being a useful tool in design systems optimization and sensitivity techniques. Applying CST to a structure problem requires five steps: (1) observe the specific problem; (2) develop a computational model for numerical simulation; (3) develop and assemble software and hardware for running the codes; (4) post-process and interpret the results; and (5) use the model to analyze and design the actual structure. Researchers in both industry and academia continue to make significant contributions to advance this technology with improvements in software, collaborative computing environments and supercomputing systems. As these environments and systems evolve, computational structures technology will evolve. By using CST in the design and operation of future structures systems, engineers will have a better understanding of how a system responds and lasts, more cost-effective methods of designing and testing models, and improved productivity. For informational and educational purposes, a videotape is being produced using both static and dynamic images from research institutions, software and hardware companies, private individuals, and historical photographs and drawings. The extensive number of CST resources indicates its widespread use. Applications run the gamut from simpler university-simulated problems to those requiring solutions on supercomputers. In some cases, an image or an animation will be mapped onto the actual structure to show the relevance of the computer model to the structure. Transferring the digital files to videotape presents a number of problems related to maintaining the quality of the original image, while still producing a broadcast quality videotape. Since researchers normally do not create a computer image using traditional composition theories or video production requirements, often the image loses some of its original digital quality and impact when transferred to videotape. Although many CST images are currently available, those that are edited into the final project must meet two important criteria: they must complement the narration, and they must be broadcast quality when recorded on videotape.

Beam, Sherilee F.↗
NASA NTRS1995

Steady-state dynamic behavior of an auxiliary bearing supported rotor system

This paper investigates the steady-state responses of a rotor system supported by auxiliary bearings in which there is a clearance between the rotor and the inner race of the bearing. A simulation model based upon the rotor of a production jet engine is developed and its steady-state behavior is explored over a wide range of operating conditions for various parametric configurations. Specifically, the influence of rotor imbalance, support stiffness, and damping is studied. It is found that imbalance may change the rotor responses dramatically in terms of frequency contents at certain operating speeds. Subharmonic responses of 2nd order through 10th order are all observed except the 9th order. Chaotic phenomenon is also observed. Jump phenomena (or double-valued responses) of both hard-spring type and soft-spring type are shown to occur at low operating speeds for systems with low auxiliary bearing damping or large clearance even with relatively small imbalance. The effect of friction between the shaft and the inner race of the bearing is also discussed.

Xie, Huajun↗
NASA NTRS1988

The dynamics of coorbital satellite systems

The dynamical behavior of N coorbital satellites moving with the same average mean motion around a primary has been studied both analytically and by numerical integrations for N from 2 to 9 satellites. Simplified dynamical equations are used to determine the different stationary configurations and their local stability against infinitesimal perturbations. The ring of equally spaced identical satellites is found to be locally unstable for N of six or less, while for N from two to eight there exists another stable compact stationary configuration with separations of 60 deg or less between adjacent satellites. For N of seven or more the equally spaced configuration becomes locally stable, and for N of nine or more it is the only stationary configuration. The motion becomes chaotic for large-amplitude perturbations. The chaotic motion fills a restricted region of phase space whose outer boundary is determined by the maximum velocity curve.

Salo, H.↗
NASA NTRS2006

Weather from the Stratosphere?

Is the stratosphere, the atmospheric layer between about 10 and 50 km, important for predicting changes in weather and climate? The traditional view is that the stratosphere is a passive recipient of energy and waves from weather systems in the underlying troposphere, but recent evidence suggests otherwise. At a workshop in Whistler, British Columbia (1), scientists met to discuss how the stratosphere responds to forcing from below, initiating feedback processes that in turn alter weather patterns in the troposphere. The lowest layer of the atmosphere, the troposphere, is highly dynamic and rich in water vapor, clouds, and weather. The stratosphere above it is less dense and less turbulent (see the figure). Variability in the stratosphere is dominated by hemispheric-scale changes in airflow on time scales of a week to several months. Occasionally, however, stratospheric air flow changes dramatically within just a day or two, with large-scale jumps in temperature of 20 K or more. The troposphere influences the stratosphere mainly through atmospheric waves that propagate upward. Recent evidence shows that the stratosphere organizes this chaotic wave forcing from below to create long-lived changes in the stratospheric circulation. These stratospheric changes can feed back to affect weather and climate in the troposphere.

Baldwin, Mark P.↗
NASA NTRS1993

The role of the margins in ice stream dynamics

At first glance, it would appear that the bed of the active ice stream plays a much more important role in the overall force balance than do the margins, especially because the ratio of the half-width to depth for a typical ice stream is large (15:1 to 50:1). On the other hand, recent observations indicate that at least part of the ice stream is underlain by a layer of very weak till (shear strength about 2 kPa), and this weak basal layer would then imply that some or all of the resistive drag is transferred to the margins. In order to address this question, a detailed velocity profile near Upstream B Camp, which extends from the center of the ice stream, across the chaotic shear margin, and onto the Unicorn, which is part of the slow-moving ice sheet was measured. Comparison of this observed velocity profile with finite-element models of flow shows several interesting features. First, the shear stress at the margin is on the order of 130 kPa, while the mean value along the bed is about 15 kPa. Integration of these stresses along the boundaries indicates that the margins provide 40 to 50 percent, and the bed, 60 to 40 percent of the total resistive drag needed to balance the gravitational driving stress in this region. (The range of values represents calculations for different values of surface slope.) Second, the mean basal stress predicted by the models shows that the entire bed cannot be blanketed by the weak till observed beneath upstream B - instead there must be a distribution of weak till and 'sticky spots' (e.g., 85 percent till and 15 percent sticky spots of resistive stress equal to 100 kPa). If more of the bed were composed of weak till, then the modeled velocity would not match that observed. Third, the ice must exhibit an increasing enhancement factor as the margins are approached (E equals 10 in the chaotic zone), in keeping with laboratory measurements on ice under prolonged shear strain. Also, there is either a narrow zone of somewhat stiffer ice (E equals 5) outward of the shear margin, or the bed is frozen there. And last, the high shear stress and strain rate found at the margin are likely to cause significant viscous heating (q) in the marginal ice. The increase in temperature is proportional to qX/u, where X is the width of the shear zone and u is the transverse velocity component bringing cold ice in from the ice sheet outside the shear zone. Near upstream B, this heating is likely to cause an increase in temperature of 4 to 10 K. Plans are to measure this temperature increase in a series of bore holes near the margin during the 1992-93 field season, as well as to provide a more detailed description of the velocity field there.

Echelmeyer, Keith↗
NASA NTRS2006

Dynamic Theory of Relativistic Electrons Stochastic Heating by Whistler Mode Waves with Application to the Earth Magnetosphere

In the Hamiltonian approach an electron motion in a coherent packet of the whistler mode waves propagating along the direction of an ambient magnetic field is studied. The physical processes by which these particles are accelerated to high energy are established. Equations governing a particle motion by group symmetries of the problem were transformed in to a closed pair of nonlinear difference equations. The solutions of these equations have shown there exists the energetic threshold below that the electron motion is regular, and when the initial energy is above the threshold an electron moves stochastically. It is proved that the upper boundary of particle stochastic heating is conditioned by intrinsic property of the particle chaotic motion. Particle energy spectra and pitch angle electron scattering are described by the Fokker-Planck-Kolmogorov equations. It is shown that significant pitch angle diffusion occurs for the Earth radiation belt electrons with energies from a few keV up to a few MeV.

Khazanov, G. V.↗
NASA NTRS1990

Interfacial dynamics of two liquids under an oscillating gravitational field

The evolution of two miscible liquids meeting at an initially sharp interface inside a cavity under microgravity g-jitter conditions is studied numerically. The response of the interface and kinematics of the flowfield to various g-jitter accelerations and aspect ratio variations is shown. The interface region acts like a vortex source sheet, and it can be unstable to Kelvin-Helmholtz and Rayleigh-Taylor instabilities. The vortices produced along the interface can serve as a stirring mechanism to promote local mixing. Below the critical Stokes-Reynolds number, the destabilization of the interface results in deformation into wavy structures. In some parameter regions, these structures oscillate in time; in others, they are quasi-steady. Above the critical Stokes-Reynolds number, 'chaotic' instability results, and the interface breaks into concentration pockets. The morphology of the initial breakup is similar to observed Rayleigh-Taylor instability. Subsequent mixing of the two fluids after the breakup of the interface is then very rapid.

Duval, Walter M. B.↗
NASA NTRS1992

Long-term evolution of 1991 DA: A dynamically evolved extinct Halley-type comet

The long-term dynamical evolution of 21 variational orbits for the intermediate-period asteroid 1991 DA was followed for up to +/-10(exp 5) years from the present. 1991 DA is close to the 2:7 resonance with Jupiter; it has avoided close encounters, within 1 AU, with this planet for at least the past 30,000 years, even at the node crossing. The future evolution typically shows no close encounters with Jupiter within at least 50,000 years. This corresponds to the mean time between node crossings with either Jupiter or Saturn. Close encounters with Saturn and Jupiter lead to a chaotic evolution for the whole ensemble, while secular perturbations cause large-amplitude swings in eccentricity and inclination (the latter covering the range 15 deg approximately less than i approximately less than 85 deg) which correlate with deep excursions of the perihelion distance to values much less than 1 AU. These variations are similar to those found in P/Machholz and a variety of other high-inclination orbits, e.g., P/Hartley-IRAS. We emphasize the connection between the orbital evolution of 1991 DA and that of Halley-type comets. If 1991 DA was once a comet, it is not surprising that it is now extinct.

Hahn, Gerhard↗
NASA NTRS1995

Wavenumber selection and hysteresis in nonlinear baroclinic flow

Wavenumber transition and hysteresis in a highly unstable baroclinic flow are investigated using a high-resolution spectral numerical model. As the flow becomes more supercritical, the dominant wave gradually shifts from the most unstable wave predicted by the linear theory to a longer wave with a larger time-averaged amplitude, while the rectified mean flow attains a stronger shear at the center of the channel. The numerical results display a complex hysteresis behavior, which occurs not only between the states of different dominant wavenumbers, but also between the states of identical dominant wavenumber but of different dynamic characteristics. In a certain parameter range three stable states, each with different dominant wavenumber, are possible, and in another parameter range four stable states are possible, among them three stable states with an identical dominant wave. The numerical results suggest that a multiple weather regime exists even without external forcing in which the flow aperiodically varies between two distinct behaviors. The effects of stable higher harmonics are assessed and it is found that their presence contributes not only to the better approximation of the model solutions but also to the selection of the final equilibrium state, due to the chaotic nature of the initial transient period.

Chou, Shih-Hung↗
NASA NTRS1992

Symplectic maps for the n-body problem - Stability analysis

The stability of new symplectic n-body maps is examined from the point of view of nonlinear dynamics. The resonances responsible for the principal artifacts are identified. These are resonances between the stepsize and the difference of mean motions between pairs of planets. For larger stepsizes resonant perturbations are evident in the variation of the energy of the system corresponding to these stepsize resonances. It is shown that the principal instability of the method can be predicted and corresponds to the overlap of the stepsize resonances. It is noted that the analysis suggests that other artifacts will occur. For example, the overlap of a stepsize resonance with a resonance of the actual system may also give a region of chaotic behavior that is an artifact. It is pointed out that the fact that the principal artifacts corresponds to a particular set of stepsize resonances suggests that it may be possible to perturbatively remove the effect when the stepsize resonances are nonoverlapping.

Wisdom, Jack↗
NASA NTRS2006

A Unique Perspective from Space on our Planet: Science, Technologies and Applications

The study of Planet earth is a very complex problem. It has many non-linear and chaotic systems operating in parallel and have interdependencies. In reality, these systems/phenomena s are not well understood or mathematically modeled because of our lack of knowledge of such intricate processes. However, in order to further the subject of Earth as an integrated system, space provides excellent vantage points to look at these processes in multidimensional framework. For example, we can make strives to understand the global water cycle, carbon cycle, atmospheric chemistry, biomass changes, oceans and solid Earth variations by making multitude of global measurements such as soil moisture, precipitation, tropospheric and stratospheric gases, aerosols, tropospheric winds, ocean salinity, ocean color, vegetation cover, crustal dynamics and many more. Such suites of measurements derive the coupled models so we may predict the changes due to natural and anthropogenic forcing. NASA along with other international space agencies have made tremendous investments in recent years in developing and flying remote sensing space borne sensors to enable these measurements. These science measurements and products are further used to address pressing issues such as coastal zone erosion, air quality, severe weather, water availability and quality, public health, fires, earthquakes, land slides and others for societal benefits. This presentation provides a comprehensive overview of NASA s science investigations, related technologies and satellites/sensors and applications.

Habib, Shaid↗
NASA NTRS1988

The influence of and the identification of nonlinearity in flexible structures

Several models were built at NASA Langley and used to demonstrate the following nonlinear behavior: internal resonance in a free response, principal parametric resonance and subcritical instability in a cantilever beam-lumped mass structure, combination resonance in a parametrically excited flexible beam, autoparametric interaction in a two-degree-of-freedom system, instability of the linear solution, saturation of the excited mode, subharmonic bifurcation, and chaotic responses. A video tape documenting these phenomena was made. An attempt to identify a simple structure consisting of two light-weight beams and two lumped masses using the Eigensystem Realization Algorithm showed the inherent difficulty of using a linear based theory to identify a particular nonlinearity. Preliminary results show the technique requires novel interpretation, and hence may not be useful for structural modes that are coupled by a guadratic nonlinearity. A literature survey was also completed on recent work in parametrically excited nonlinear system. In summary, nonlinear systems may possess unique behaviors that require nonlinear identification techniques based on an understanding of how nonlinearity affects the dynamic response of structures. In this was, the unique behaviors of nonlinear systems may be properly identified. Moreover, more accutate quantifiable estimates can be made once the qualitative model has been determined.

Zavodney, Lawrence D.↗
NASA NTRS1988

Dynamics of two fluids under periodic acceleration

The evolution of the interface between two fluids confined in a rectangular cavity is investigated numerically to predict its transient behavior. These computations address mixing characteristics of fluids under microgravity conditions, particularly g-jitter conditions resulting from aircraft vibration or crew motion, and have applications in solution crystal growth. The two-dimensional formulation employs the Boussinesq approximation and treats the mixing of two fluids as an initial value problem with a prescribed concentration field. For fluid mixing of practical applications inside a cavity, it is shown that nonlinear convective transport can dominate over both viscous and molecular diffusion. However, viscous diffusion can become important for low Reynolds number or certain cavity sizes. A stacking phenomenon is shown to occur for aspect ratios (Ar) approximately between 5 to 10. For aspect ratios in the neighborhood of 0.1 to 0.2, inner cells evolve at the interface with increasing time. For a square cavity of Ar = 1, chaotic mixing of the fluid appears to occur for a Stokes-Reynolds number greater than about 5.

Duval, Walter M. B.↗
NASA NTRS2000

Section on Observed Impacts on El Nino

Agricultural applications of El Nino forecasts are already underway in some countries and need to be evaluated or re-evaluated. For example, in Peru, El Nino forecasts have been incorporated into national planning for the agricultural sector, and areas planted with rice and cotton (cotton being the more drought-tolerant crop) are adjusted accordingly. How well are this and other such programs working? Such evaluations will contribute to the governmental and intergovernmental institutions, including the Inter-American Institute for Global Change Research and the US National Ocean and Atmospheric Agency that are fostering programs to aid the effective use of forecasts. As El Nino climate forecasting grows out of the research mode into operational mode, the research focus shifts to include the design of appropriate modes of utilization. Awareness of and sensitivity to the costs of prediction errors also grow. For example, one major forecasting model failed to predict the very large El Nino event of 1997, when Pacific sea-surface temperatures were the highest on record. Although simple correlations between El Nino events and crop yields may be suggestive, more sophisticated work is needed to understand the subtleties of the interplay among the global climate system, regional climate patterns, and local agricultural systems. Honesty about the limitations of an forecast is essential, especially when human livelihoods are at stake. An end-to-end analysis links tools and expertise from the full sequence of ENSO cause-and-effect processes. Representatives from many disciplines are needed to achieve insights, e.g, oceanographers and atmospheric scientists who predict El Nino events, climatologists who drive global climate models with sea-surface temperature predictions, agronomists who translate regional climate connections in to crop yield forecasts, and economists who analyze market adjustments to the vagaries of climate and determine the value of climate forecasts. Methods include historical studies to understand past patterns and to test hindcasts of the prediction tools, crop modeling, spatial analysis and remote sensing. This research involves expanding, deepening, and applying the understanding of physical climate to the fields of agronomy and social science; and the reciprocal understanding of crop growth and farm economics to climatology. Delivery of a regional climate forecast with no information about how the climate forecast was derived limits its effectiveness. Explanation of a region's major climate driving forces helps to place a seasonal forecast in context. Then, a useful approach is to show historical responses to previous El Nino events, and projections, with uncertainty intervals, of crop response from dynamic process crop growth models. Regional ID forecasts should be updated with real-time weather conditions. Since every El Nino event is different, it is important to track, report and advise on each new event as it unfolds. The stability of human enterprises depends on understanding both the potentialities and the limits of predictability. Farmers rely on past experience to anticipate and respond to fluctuations in the biophysical systems on which their livelihoods depend. Now scientists are improving their ability to predict some major elements of climate variability. The improvements in the reliability of El Nino forecasts are encouraging, but seasonal forecasts for agriculture are not, and will probably never be completely infallible, due to the chaotic nature of the climate system. Uncertainties proliferate as we extend beyond Pacific sea-surface temperatures to climate teleconnections and agricultural outcomes. The goal of this research is to shed as a clear light as possible on these inherent uncertainties and thus to contribute to the development of appropriate responses to El Nino and other seasonal forecasts for a range of stakeholders, which, ultimately, includes food consumers everywhere.

Rosenzweig, Cynthia↗
NASA NTRS2015

Effective Power-Law Dependence of Lyapunov Exponents on the Central Mass in Galaxies

Using both numerical and analytical approaches, we demonstrate the existence of an effective power-law relation L alpha m(sup p) between themean Lyapunov exponent L of stellar orbits chaotically scattered by a supermassive black hole (BH) in the centre of a galaxy and the mass parameter m, i.e. ratio of the mass of the BH over the mass of the galaxy. The exponent p is found numerically to obtain values in the range p approximately equals 0.3-0.5. We propose a theoretical interpretation of these exponents, based on estimates of local 'stretching numbers', i.e. local Lyapunov exponents at successive transits of the orbits through the BH's sphere of influence. We thus predict p = 2/3 − q with q approximately equaling 0.1-0.2. Our basic model refers to elliptical galaxy models with a central core. However, we find numerically that an effective power-law scaling of L with m holds also in models with central cusp, beyond a mass scale up to which chaos is dominated by the influence of the cusp itself. We finally show numerically that an analogous law exists also in disc galaxies with rotating bars. In the latter case, chaotic scattering by the BH affects mainly populations of thick tube-like orbits surrounding some low-order branches of the x(sub 1) family of periodic orbits, as well as its bifurcations at low-order resonances, mainly the inner Lindblad resonance and the 4/1 resonance. Implications of the correlations between L and m to determining the rate of secular evolution of galaxies are discussed.

chaos aEuro" galaxies:↗
NASA NTRS1997

The Influence of Planetary Mass on the Dynamical Lifetime of Planetary Systems

Recent numerical and analytic studies of planetary orbits have demonstrated the importance of resonances and chaos in destabilizing planetary systems. Newton's "clockwork" description of regular, predictable planetary orbits has been replaced by a view in which many systems can have long but finite lifetimes. This new knowledge has altered our perceptions of the later stages of planetary growth and of the stability of planetary systems. Stability criteria are inexact and time dependent. Most previous studies have focused on the effects in initial planetary orbits on the stability of the system. We are conducting an investigation which focuses on the dependence of stability criteria on planetary mass. Synthetic systems are created by increasing the masses of the planets in our Solar System or of the moons of a particular planet; these systems are then integrated until orbit crossing occurs. We have found that over some ranges, the time until orbit crossing varies to a good approximation as a power clothe factor by which the masses of the secondaries arc increased; some scatter occurs as a consequence of vie chaotic nature of orbital evolution. The slope of this power law varies substantially from system to system, and for moons it is mildly dependent on the inclusion of the planet's quadrupole moment in the gravitational potential.

Lissauer, J. J.↗
NASA NTRS1986

Multiple grid problems on concurrent-processing computers

Three computer codes were studied which make use of concurrent processing computer architectures in computational fluid dynamics (CFD). The three parallel codes were tested on a two processor multiple-instruction/multiple-data (MIMD) facility at NASA Ames Research Center, and are suggested for efficient parallel computations. The first code is a well-known program which makes use of the Beam and Warming, implicit, approximate factored algorithm. This study demonstrates the parallelism found in a well-known scheme and it achieved speedups exceeding 1.9 on the two processor MIMD test facility. The second code studied made use of an embedded grid scheme which is used to solve problems having complex geometries. The particular application for this study considered an airfoil/flap geometry in an incompressible flow. The scheme eliminates some of the inherent difficulties found in adapting approximate factorization techniques onto MIMD machines and allows the use of chaotic relaxation and asynchronous iteration techniques. The third code studied is an application of overset grids to a supersonic blunt body problem. The code addresses the difficulties encountered when using embedded grids on a compressible, and therefore nonlinear, problem. The complex numerical boundary system associated with overset grids is discussed and several boundary schemes are suggested. A boundary scheme based on the method of characteristics achieved the best results.

Eberhardt, D. S.↗