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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.

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All sourcesNASA NTRSDOE OSTI
NASA NTRS1998

Structure and Dynamics of Coronal Plasma

Brief summaries of the four published papers produced within the present performance period of NASA Grant NAGW-4081 are presented. The full text of the papers are appended to the report. The first paper titled "Coronal Structures Observed in X-rays and H-alpa Structures" was published in the Kofu Symposium proceedings. The study analyzes cool and hot behavior of two x-ray events, a small flare and a surge. It was found that a large H-alpha surge appears in x-rays as a very weak event, while a weak H-alpha feature corresponds to the brightest x-ray emission on the disk at the time of the observation. Calculations of the heating necessary to produce these signatures, and implications for the driving and heating mechanisms of flares vs. surges are presented. The second paper "Differential Magnetic Field Shear in an Active Region" has been published in The Astrophysical Journal. The study compared the three dimensional extrapolation of magnetic fields with the observed coronal structure in an active region. Based on the fit between observed coronal structure throughout the volume of the region and the calculated magnetic field configurations, the authors propose a differential magnetic field shear model for this active region. The decreasing field shear in the outer portions of the AR may indicate a continual relaxation of the magnetic field with time, corresponding to a net transport of helicity outward. The third paper "Difficulties in Observing Coronal Structure" has been published in the journal Solar Physics. This paper discusses the evidence that the temperature and density structure of the corona are far more complicated than had previously been thought. The discussion is based on five studies carried out by the group on coronal plasma properties, showing that any one x-ray instrument does see all of the plasma present in the corona, that hot and cool material may appear to be co-spatial at a given location in the corona, and that simple magnetic field extrapolations provide only a poor fit to the observed structure. The fourth paper "Analysis and Comparison of Loop Structures Imaged with NIXT and Yohkoh/SXT" has been published in Astronomy and Astrophysics. This paper analyzes and compares a variety of coronal loops, deriving loop pressure and emission measure from loop models. They are able to determine the volume filling factor in the corona, which is found to be in the range 0.001 to 0.01 for compact loops, and of order 1 for large structures. The small values suggest highly filamented structures, especially at lower temperatures.

Golub, Leon↗
DOE OSTI2025

Resolving electrochemically triggered topological defect dynamics and structural degradation in layered oxides

Understanding topological defects-controlled structural degradation of layered oxides—a key cathode material for high-performance lithium-ion batteries—plays a critical role in developing next-generation cathode materials. Here, by constructing a nanobattery in an electron microscope enabling atomic-scale monitoring of electrochemcial reactions, we captured the electrochemically driven atomistic dynamics and evolution of dislocations—a most important topological defect in material. We deciphered how dislocations nucleate, move, and annihilate within layered cathodes at the atomic scale. Specifically, we found two types of dislocation configurations, i.e., single dislocations and dislocation dipoles. Both pure dislocation glide/climb and mixed motions were captured, and the dislocation glide and climb velocities were first experimentally measured. Moreover, dislocation activity-mediated structural degradation such as crack nucleation, phase transformation, and lattice reorientation was unraveled. Our work provides deep insights into the atomistic dynamics of electrochemically driven dislocation activities in layered oxides.

Wang, Chunyang (ORCID:0000000184613952)↗
NASA NTRS2002

Engine System Loads Analysis Compared to Hot-Fire Data

Early implementation of structural dynamics finite element analyses for calculation of design loads is considered common design practice for high volume manufacturing industries such as automotive and aeronautical industries. However with the rarity of rocket engine development programs starts, these tools are relatively new to the design of rocket engines. In the NASA MC-1 engine program, the focus was to reduce the cost-to-weight ratio. The techniques for structural dynamics analysis practices, were tailored in this program to meet both production and structural design goals. Perturbation of rocket engine design parameters resulted in a number of MC-1 load cycles necessary to characterize the impact due to mass and stiffness changes. Evolution of loads and load extraction methodologies, parametric considerations and a discussion of load path sensitivities are important during the design and integration of a new engine system. During the final stages of development, it is important to verify the results of an engine system model to determine the validity of the results. During the final stages of the MC-1 program, hot-fire test results were obtained and compared to the structural design loads calculated by the engine system model. These comparisons are presented in this paper.

Frady, Gregory P.↗
DOE OSTI2023

Elucidating Structural Transition Dynamics in the Magnesium Cathode MgCr 2 O 4

Multivalent batteries, e.g., those based on magnesium (Mg), are promising candidates for next-generation energy storage due to their high volumetric energy densities and low cost. However, the corresponding ion migration and structural transition mechanisms are often linked and difficult to observe directly. Here, in this paper, we report the direct investigation of atomic transport pathways of cations in spinel magnesium chromate (MgCr 2 O 4 ) by using aberration-corrected scanning transmission electron microscopy (STEM). Cr atoms are directly observed to reversibly occupy the otherwise vacant octahedrally coordinated interstitial sites, passing through tetrahedral sites normally occupied by Mg. Furthermore, imaging and electron energy loss spectroscopy show that electron irradiation induces the formation of Mg and O vacancies, facilitating the migration of Cr and leading to an irreversible phase transition. These results demonstrate the ability of STEM to capture the pathway of deleterious point defects that can result in undesirable phase transitions.

25 ENERGY STORAGE↗
NASA NTRS2004

Testing of Composite Fan Vanes With Erosion-Resistant Coating Accelerated

The high-cycle fatigue of composite stator vanes provided an accelerated life-state prior to insertion in a test stand engine. The accelerated testing was performed in the Structural Dynamics Laboratory at the NASA Glenn Research Center under the guidance of Structural Mechanics and Dynamics Branch personnel. Previous research on fixturing and test procedures developed at Glenn determined that engine vibratory conditions could be simulated for polymer matrix composite vanes by using the excitation of a combined slip table and electrodynamic shaker in Glenn's Structural Dynamics Laboratory. Bench-top testing gave researchers the confidence to test the coated vanes in a full-scale engine test.

Bowman, Cheryl L.↗
DOE OSTI2021

Nutrient availability and senescence spatially structure the dynamics of a foundation species

Significance Landscapes are spatially heterogeneous, and the measurement of spatial pattern is dependent upon observation scale. Understanding plant populations requires assessing their extrinsic interactions with the environment as well as intrinsic biological processes and has been difficult because of the inability to track both plant abundance and health on appropriate scales. We introduce remote sensing observations that assess the abundance and health of giant kelp, an important ecosystem-structuring species, over regional and local scales. We find that both extrinsic nutrient availability and intrinsic senescence processes regulate population dynamics but on regional and local scales, respectively. This suggests that future satellite missions will be able to assess plant abundance and health and their interactions with the environment on local to global scales.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2022

Structural and dynamic mechanisms of GABA A receptor modulators with opposing activities

γ-Aminobutyric acid type A (GABA A ) receptors are pentameric ligand-gated ion channels abundant in the central nervous system and are prolific drug targets for treating anxiety, sleep disorders and epilepsy. Diverse small molecules exert a spectrum of effects on γ-aminobutyric acid type A (GABA A ) receptors by acting at the classical benzodiazepine site. They can potentiate the response to GABA, attenuate channel activity, or counteract modulation by other ligands. Structural mechanisms underlying the actions of these drugs are not fully understood. Here we present two high-resolution structures of GABA A receptors in complex with zolpidem, a positive allosteric modulator and heavily prescribed hypnotic, and DMCM, a negative allosteric modulator with convulsant and anxiogenic properties. These two drugs share the extracellular benzodiazepine site at the α/γ subunit interface and two transmembrane sites at β/α interfaces. Structural analyses reveal a basis for the subtype selectivity of zolpidem that underlies its clinical success. Molecular dynamics simulations provide insight into how DMCM switches from a negative to a positive modulator as a function of binding site occupancy. Together, these findings expand our understanding of how GABA A receptor allosteric modulators acting through a common site can have diverging activities.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2020

Real-time Atomic-scale Visualization of Reversible Copper Surface Activation during the CO Oxidation Reaction

Dynamic structural and chemical information of catalyst surface during a catalytic reaction is critical for deciphering the mechanisms governing the properties of catalysts. Transition-metal based catalysts have emerged as an important alternative to noble metal catalysts, but their catalytic mechanism remains largely elusive, complicated by dynamic structure and valence evolution. Herein, by using in situ aberration-corrected environmental transmission electron microscopy, we capture, for the first time at atomic level, the dynamic evolution of Cu surface during CO oxidation reaction. We discovered that, under reaction condition, the Cu surface is activated, typically involving 2-3 atomic layers with the formation of a reversible meta-stable phase that only exists during catalytic reactions. Further, we reveal the distinctive role of CO and O2 in the surface activation, featuring CO exposure to lead to surface roughening and consequently formation of low-coordinated Cu atoms, while O2 exposure to induce quasi-crystalline CuOx phase. Corrugated by DFT calculations, we rationalize crystalline CuOx reversibly transforms into the amorphous phase, acting as an active species to facilitate the interaction of gas reactants and catalyzing CO oxidation.

Luo, Langli↗
DOE OSTI2018

Direct Visualization of Li Dendrite Effect on LiCoO 2 Cathode by In Situ TEM

Li dendrites are known to cause deterious, and in many cases catastrophic, effect on the performance of Li rechargeable batteries. However, the mechanistic failure mechanism of cathodes upon in contact with Li metal is far from clear. In this study, using in situ transmission electron microscopy, we directly visualize at an atomic scale the interaction of Li dendrites with well-defined, epitaxial thin films of LiCoO2, the most widely used cathode material. We show that a spontaneous and prompt chemical reaction is trigged once the Li contact is made, leading to expansion and amorphization of the LiCoO2 structure, with Li2O and Co metal being the final reaction products. A topotactic phase transition is identified close to the reaction front, resulting in the formation of CoO as a metastable intermediate. Dynamic structural and chemical imaging in combination with ab inito simulations reveal that a high density of antiphase grain boundaries are formed at the reaction front, which are critical for enabling the short-range topotactic reactions and long-range Li propagation. The fundamental insights are of general importance in mitigating Li dendrites related issues and guiding the design principle for more robust energy materials.

In situ transmission electron microscopy (TEM), Li↗
NASA NTRS1993

Extending substructure based iterative solvers to multiple load and repeated analyses

Direct solvers currently dominate commercial finite element structural software, but do not scale well in the fine granularity regime targeted by emerging parallel processors. Substructure based iterative solvers--often called also domain decomposition algorithms--lend themselves better to parallel processing, but must overcome several obstacles before earning their place in general purpose structural analysis programs. One such obstacle is the solution of systems with many or repeated right hand sides. Such systems arise, for example, in multiple load static analyses and in implicit linear dynamics computations. Direct solvers are well-suited for these problems because after the system matrix has been factored, the multiple or repeated solutions can be obtained through relatively inexpensive forward and backward substitutions. On the other hand, iterative solvers in general are ill-suited for these problems because they often must restart from scratch for every different right hand side. In this paper, we present a methodology for extending the range of applications of domain decomposition methods to problems with multiple or repeated right hand sides. Basically, we formulate the overall problem as a series of minimization problems over K-orthogonal and supplementary subspaces, and tailor the preconditioned conjugate gradient algorithm to solve them efficiently. The resulting solution method is scalable, whereas direct factorization schemes and forward and backward substitution algorithms are not. We illustrate the proposed methodology with the solution of static and dynamic structural problems, and highlight its potential to outperform forward and backward substitutions on parallel computers. As an example, we show that for a linear structural dynamics problem with 11640 degrees of freedom, every time-step beyond time-step 15 is solved in a single iteration and consumes 1.0 second on a 32 processor iPSC-860 system; for the same problem and the same parallel processor, a pair of forward/backward substitutions at each step consumes 15.0 seconds.

Farhat, Charbel↗
NASA NTRS2019

A Summary of NASA Rotary Wing Research: Circa 2008–2018

The general public may not know that the first “A” in NASA stands for Aeronautics. If they do know, they will very likely be surprised that in addition to airplanes, the “A” includes research in helicopters, tiltrotors, and other vehicles adorned with rotors. There is, arguably, no subsonic air vehicle more difficult to accurately analyze than a vehicle with lift-producing rotors. No wonder that NASA has conducted rotary wing research since the days of the NACA and has partnered, since 1965, with the U.S. Army in order to overcome some of the most challenging obstacles to understanding the behavior of these vehicles. Since 2006, NASA rotary wing research has been performed under several different project names [Gorton et al., 2015]: Subsonic Rotary Wing (SRW) (2006–2012), Rotary Wing (RW) (2012–2014), and Revolutionary Vertical Lift Technology (RVLT) (2014–present). In 2009, the SRW Project published a report that assessed the status of NASA rotorcraft research; in particular, the predictive capability of NASA rotorcraft tools was addressed for a number of technical disciplines. A brief history of NASA rotorcraft research through 2009 was also provided [Yamauchi and Young, 2009]. Gorton et al. [2015] describes the system studies during 2009–2011 that informed the SRW/RW/RVLT project investment prioritization and organization. The authors also provided the status of research in the RW Project in engines, drive systems, aeromechanics, and impact dynamics as related to structural dynamics of vertical lift vehicles. Since 2009, the focus of research has shifted from large civil VTOL transports, to environmentally clean aircraft, to electrified VTOL aircraft for the urban air mobility (UAM) market. The changing focus of rotorcraft research has been a reflection of the evolving strategic direction of the NASA Aeronautics Research Mission Directorate (ARMD). By 2014, the project had been renamed the Revolutionary Vertical Lift Technology Project. In response to the 2014 NASA Strategic Plan, ARMD developed six Strategic Thrusts. Strategic Thrust 3B was defined as the “Ultra-Efficient Commercial Vehicles—Vertical Lift Aircraft.” Hochstetler et al. [2017] uses Thrust 3B as an example for developing metrics usable by ARMD to measure the effectiveness of each of the Strategic Thrusts. The authors provide near-, mid-, and long-term outcomes for Thrust 3B with corresponding benefits and capabilities. The importance of VTOL research, especially with the rapidly expanding UAM market, eventually resulted in a new Strategic Thrust (to begin in 2020): Thrust 4—Safe, Quiet, and Affordable Vertical Lift Air Vehicles. The underlying rotary wing analysis tools used by NASA are still applicable to traditional rotorcraft and have been expanded in capability to accommodate the growing number of VTOL configurations designed for UAM. The top-level goal of the RVLT Project remains unchanged since 2006: Develop and validate tools, technologies and concepts to overcome key barriers for vertical lift vehicles. In 2019, NASA rotary wing/VTOL research has never been more important for supporting new aircraft and advancements in technology. 2 A decade is a reasonable interval to pause and take stock of progress and accomplishments. In 10 years, digital technology has propelled progress in computational efficiency by orders of magnitude and expanded capabilities in measurement techniques. The purpose of this report is to provide a compilation of the NASA rotary wing research from ~2008 to ~2018. Brief summaries of publications from NASA, NASA-funded, and NASA-supported research are provided in 12 chapters: Acoustics, Aeromechanics, Computational Fluid Dynamics (External Flow), Experimental Methods, Flight Dynamics and Control, Drive Systems, Engines, Crashworthiness, Icing, Structures and Materials, Conceptual Design and System Analysis, and Mars Helicopter. We hope this report serves as a useful reference for future NASA vertical lift researchers.

Yamauchi, Gloria K.↗
DOE OSTI2019

Learning-based Computational Study of the Thermodynamic, Structural, and Dynamic Properties of Molten Salt at the Atomic and Electronic Scale and Experimental Validations (Final Report for Project 19-16909)

The project completed several milestones related to the work scope to determine thermophysical properties of the molten salt FLiNaK. A combined approach to integrate advanced computations with neutron scattering experimental interrogation was used to quantify the structure and mass transport of pure FLiNaK and FLiNaK with additives. The additives investigated included Ce, Cs, Zr, and Cr. The primary outcome of the computational work was the development of a neutral network force field interatomic potential (referred to as the NNFF model in this document). The computational work also involved ab initio calculations to determine the ground truth and molecular dynamics simulations to provide computational analogs to the experimental campaign. The experimental work used enriched F 7 LiNaK prepared by Argonne National Laboratory. The use of enriched F 7 LiNaK reduced parasitic neutron absorption and significantly improved data collection and data quality. Several graduate students were supported during the course of the project, which included a one year no-cost extension. These students were trained in advanced computational methods, advanced neutron scattering methods, neutron scattering data analysis and interpretation, and molten salt sample preparation.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
NASA NTRS1991

PDEMOD: Software for control/structures optimization

Because of the possibility of adverse interaction between the control system and the structural dynamics of large, flexible spacecraft, great care must be taken to ensure stability and system performance. Because of the high cost of insertion of mass into low earth orbit, it is prudent to optimize the roles of structure and control systems simultaneously. Because of the difficulty and the computational burden in modeling and analyzing the control structure system dynamics, the total problem is often split and treated iteratively. It would aid design if the control structure system dynamics could be represented in a single system of equations. With the use of the software PDEMOD (Partial Differential Equation Model), it is now possible to optimize structure and control systems simultaneously. The distributed parameter modeling approach enables embedding the control system dynamics into the same equations for the structural dynamics model. By doing this, the current difficulties involved in model order reduction are avoided. The NASA Mini-MAST truss is used an an example for studying integrated control structure design.

Taylor, Lawrence W., Jr.↗
NASA NTRS2002

Aeroelastic Calculations of Quiet High- Speed Fan Performed

An advanced high-speed fan was recently designed under a cooperative effort between the NASA Glenn Research Center and Honeywell Engines & Systems. The principal design goals were to improve performance and to reduce fan noise at takeoff. Scale models of the Quiet High-Speed Fan were tested for operability, performance, and acoustics. During testing, the fan showed significantly improved noise characteristics, but a self-excited aeroelastic vibration known as flutter was encountered in the operating range. Flutter calculations were carried out for the Quiet High-Speed Fan using a three-dimensional, unsteady aerodynamic, Reynolds-averaged Navier-Stokes turbomachinery code named "TURBO." The TURBO code can accurately model the viscous flow effects that can play an important role in various aeroelastic problems such as flutter with flow separation, flutter at high loading conditions near the stall line (stall flutter), and flutter in the presence of shock and boundary-layer interaction. Initially, calculations were performed with no blade vibrations. These calculations were at a constant rotational speed and a varying mass flow rate. The mass flow rate was varied by changing the backpressure at the exit boundary of the computational domain. These initial steady calculations were followed by aeroelastic calculations in which the blades were prescribed to vibrate harmonically in a natural mode, at a natural frequency, and with a fixed interblade phase angle between adjacent blades. The AE-prep preprocessor was used to interpolate the in-vacuum mode shapes from the structural dynamics mesh onto the computational fluid dynamics mesh and to smoothly propagate the grid deformations from the blade surface to the interior points of the grid. The aeroelastic calculations provided the unsteady aerodynamic forces on the blade surface due to blade vibrations. These forces were vector multiplied with the structural dynamic mode shape to calculate the work done on the blade during one vibration period, then this result was converted to an aerodynamic damping. Flutter occurs when the aerodynamic damping becomes negative, if structural damping is ignored. The results of these aeroelastic calculations are summarized in this plot of aerodynamic damping versus mass flow rate at a constant rotational speed. As the backpressure is increased, the mass flow rate through the fan decreases and the fan operating point moves towards the stall line. The aeroelastic calculations showed that the aerodynamic damping decreases as the stall line is approached, as observed during testing. In addition, the aeroelastic calculations with the TURBO code correctly predicted the aeroelastic parameters: the most unstable vibration mode and interblade phase angle, as observed during testing. The Quiet High-Speed Fan demonstrated significant noise reductions during testing, but flutter imposed limits on its operating range. The accurate calculation of the aeroelastic characteristics using the TURBO code is a significant step toward eliminating flutter from the operating range and toward realizing the benefits of reduced fan noise. The aeroelastic calculations described here were performed under a grant by University of Toledo researchers in collaboration with Glenn's researchers.

Bakhle, Milind A.↗
DOE OSTI2018

DNA-binding induced conformational change of c-Myb R2R3 analyzed using diffracted X-ray tracking

Highlights: • DNA-binding protein, c-Myb R2R3, largely fluctuates in solution. • We evaluated structural dynamics of R2R3 using diffracted X-ray tracking (DXT). • The flexibility of R2R3 is decreased upon the DNA binding. • We could determine the DNA-binding energies of R2R3 using DXT. Previous structural analyses have shown that R2R3, the minimum unit of the DNA-binding domain of the transcriptional factor c-Myb, is largely flexible in solution, and changes to a more rigid structure upon DNA binding. In this study, we evaluated the structural dynamics using the diffracted X-ray tracking method, in correlation with DNA-binding abilities under different salt conditions, and compared them with the previous results. The resultant curve of the mean square angular displacements (MSD) clearly showed that the flexibility of R2R3 was decreased upon DNA binding, and the DNA-binding energies determined using the angular diffusion coefficients were in good agreement with those determined using isothermal titration calorimetry. The results of the MSD curves also indicate that the translational length reduces by approximately half upon DNA binding.

60 APPLIED LIFE SCIENCES↗
NASA NTRS1997

Structure and Dynamics of Coronal Plasmas

During the past year this grant has funded research in the interaction between magnetic fields and the hot plasma in the solar outer atmosphere. The following is a brief summary of the published papers, abstracts and talks which have been supported. The paper 'Coronal Structures Observed in X-rays and H-alpha Structures' was published in the Kofu Symposium proceedings. The study analyzes cool and hot behavior of two x-ray events, a small flare and a surge. We find that a large H-alpha surge appears in x-rays as a very weak event, while a weak H-alpha feature corresponds to the brightest x-ray emission on the disk at the time of the observation. Calculations of the heating necessary to produce these signatures, and implications for the driving and heating mechanisms of flares vs. surges are presented. A copy of the paper is appended to this report. The paper 'Differential Magnetic Field Shear in an Active Region' has been published in The Astrophysical Journal. We have compared the 3D extrapolation of magnetic fields with the observed coronal structure in an active region. Based on the fit between observed coronal structure throughout the volume of the region and the calculated magnetic field configurations, we propose a differential magnetic field shear model for this active region. The decreasing field shear in the outer portions of the AR may indicate a continual relaxation of the magnetic field with time, corresponding to a net transport of helicity outward. The paper 'Difficulties in Observing Coronal Structure' has been accepted for publication in the journal Solar Physics. In this paper we discuss the evidence that the temperature and density structure of the corona are far more complicated than had previously been thought. The discussion is based on five studies carried out by our group on coronal plasma properties, showing that any one x-ray instrument does see all of the plasma present in the corona, that hot and cool material may appear to be co-spatial at a given location in the corona, and that simple magnetic field extrapolations provide only a poor fit to the observed structure. A copy of the paper is appended to this report.

Golub, Leon↗
DOE OSTI2025

Imaging the photochemistry of cyclobutanone using ultrafast electron diffraction: Experimental results

We investigated the ultrafast structural dynamics of cyclobutanone following photoexcitation at λ = 200 nm using gas-phase megaelectronvolt ultrafast electron diffraction. Our investigation complements the simulation studies of the same process within this special issue. It provides information about both electronic state population and structural dynamics through well-separable inelastic and elastic electron scattering signatures. We observe the depopulation of the photoexcited S 2 state of cyclobutanone with n3s Rydberg character through its inelastic electron scattering signature with a time constant of (0.29 ± 0.2) ps toward the S 1 state. The S 1 state population undergoes ring-opening via a Norrish Type-I reaction, likely while passing through a conical intersection with S 0 . The corresponding structural changes can be tracked by elastic electron scattering signatures. These changes appear with a delay of (0.14 ± 0.05) ps with respect to the initial photoexcitation, which is less than the S 2 depopulation time constant. This behavior provides evidence for the ballistic nature of the ring-opening once the S 1 state is reached. The resulting biradical species react further within (1.2 ± 0.2) ps via two rival fragmentation channels yielding ketene and ethylene, or propene and carbon monoxide. Furthermore, our study showcases the value of both gas-phase ultrafast diffraction studies as an experimental benchmark for nonadiabatic dynamics simulation methods and the limits in the interpretation of such experimental data without comparison with such simulations.

Carbon monoxide↗
NASA NTRS1972

Automated design optimization of supersonic airplane wing structures under dynamic constraints

The problems of the preliminary and first level detail design of supersonic aircraft wings are stated as mathematical programs and solved using automated optimum design techniques. The problem is approached in two phases: the first is a simplified equivalent plate model in which the envelope, planform and structural parameters are varied to produce a design, the second is a finite element model with fixed configuration in which the material distribution is varied. Constraints include flutter, aeroelastically computed stresses and deflections, natural frequency and a variety of geometric limitations.

Fox, R. L.↗