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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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At least 343 records · Page 19
All sourcesNASA NTRSDOE OSTI
NASA NTRS1992

Recent results on Mars and Venus; Proceedings of Symposium 3 and the Topical Meetings of the Interdisciplinary Scientific Commission C (Meeting C1) and D (Meeting D1) of the COSPAR 28th Plenary Meeting, The Hague, Netherlands, June 25-July 6, 1990

In this volume, latest results on Mars and Phobos studies are discussed along with recent results on Venus and on Martian plasma environment. Papers are presented on the characteristics of aerosol phenomena in Martian atmosphere from KRFM-experiment data, Phobos map and Phobos globe, submicron-sized dust grains in the Martian environment, remote sensing of Venus atmospheric dynamics, the latitude structure of upper clouds of Venus, and Venus thermospheric response to short-term solar variations. Attention is also given to a comparison between the bow shocks and magnetotails of Venus and Mars, comparative studies of the solar wind interaction with weakly magnetized planets, the magnetic field environment of Mars as studied by Phobos-2, waves and cold plasmas near Mars, and multiple-ion effects at Martian plasma boundaries.

Shorthill, R. W.↗
NASA NTRS2022

Stormtime Ring Current Heating of the Ionosphere and Plasmasphere

The energy deposition from ring current ions into the high density “cold” plasma of the ionosphere and plasmasphere is analyzed, based on a Comprehensive Inner Magnetosphere-Ionosphere simulation of the 2015 October 7 storm. In addition, the Naval Research Laboratory Sami3 is Also a Model of the Ionosphere ionosphere/plasmasphere code is used to simulate the effect of Coulomb-collision heating of plasmasphere and ionosphere electrons by ring current ions. We find that, during stormtime peaks in the Dst index, energy is deposited at altitudes as low as 100 km. Heating along the entirety of any given field line, both in the ionosphere and plasmasphere, contributes to increased temperatures in the ionosphere F layer and inner magnetosphere and to subsequent cold O + outflows. However, relative to the heating of the plasmasphere, the direct heating of the ionosphere by ring current ions produces only small effects. Qualitative model-data agreement on the N + /O + density ratio is consistent with the hypothesis that these outflows are driven by thermal forcing.

J. Krall↗
NASA NTRS2017

Is Diffuse Aurora Driven from Above or Below?

Abstract In the diffuse aurora, magnetospheric electrons, initially precipitated from the inner plasma sheet via wave-particle interaction processes, degrade in the atmosphere toward lower energies, and produce secondary electrons via impact ionization of the neutral atmosphere. These initially precipitating electrons of magnetospheric origin can also be additionally reflected back into the magnetosphere, leading to a series of multiple reactions by the two magnetically conjugate atmospheres that can greatly impact the initially precipitating flux at the upper ionospheric boundary (700-800 km). The resultant population of secondary and primary electrons cascades toward lower energies and escape back to the magnetosphere. Escaping upward electrons traveling from the ionosphere can be trapped in the magnetosphere, as they travel inside the loss cone, via Coulomb collisions with the cold plasma, or by interactions with various plasma waves. Even though this scenario is intuitively transparent, this magnetosphere-ionosphere coupling element is not considered in any of the existing diffuse aurora research. Nevertheless, as we demonstrate in this letter, this process has the potential to dramatically affect the formation of electron precipitated fluxes in the regions of diffuse auroras.

aurora↗
DOE OSTI2018

Hybrid iterative approach for simulation of radio-frequency fields in plasma

In this work, a novel iterative approach for solving discretized linear wave equations in a frequency domain, which combines time evolution with iterative relaxation schemes, is presented. In this hybrid approach, each iteration cycle consists of evolution of electromagnetic (EM) fields in time over a specified number of field periods followed by several iterative relaxations. Provided that there is sufficient dissipation, both the time evolution and the iterative relaxations contribute to the convergence of the EM fields to the solution of the formulated full wave boundary value problem. Time evolution rapidly distributes EM fields, propagating with group velocity, over the simulation domain, while the iterative relaxations smooth the fields, reducing the numerical errors such that iteration cycles converge to a steady state solution, approximating the solution of the formulated problem. This approach is intended for large scale simulations which are beyond the capabilities of direct solvers presently used for solving wave equations in the frequency domain. Lastly, the technique is demonstrated for solving wave equations on a regular grid using a cold plasma dielectric model with collisions for 2D modeling of EM fields in tokamak in an electron cyclotron frequency range.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2021

Simulating the effects of warm O + ions on the growth of electromagnetic ion cyclotron (EMIC) waves

Electromagnetic ion cyclotron (EMIC) waves are believed to play a crucial role in the dynamics of the Earth’s magnetosphere. It has been widely accepted that plasma compositions can influence the growth rate of EMIC waves in the inner magnetosphere, but how warm O + ions change the wave growth rate is not well known. In this study, we investigate the impact of ring current O + concentration on the EMIC growth rate during a specific storm when the O + ion flux significantly increases. We calculate the growth rate of EMIC waves by using physics-based ion distributions output from the Ring current–Atmosphere interactions Model with Self-Consistent magnetic (B) and Electric (E) fields (RAM-SCBE) model. The percentage of warm O + is parametrically varied to examine how the maximum EMIC growth rate changes over time and how the global distribution of the maximum EMIC growth rate is affected. We found that the maximum growth rate of H-band appears in the dusk-to-midnight sector near the plasmapause, while O-band is excited at a slightly outer region. The maximum growth rate of He-band is closely related to the cold plasma density. With the increase of warm O + density, the maximum growth rate of H-band and He-band EMIC wave is reduced, while that of O-band EMIC wave is increased, and the region with this wave excitation is widened. Furthermore, such variation in the maximum EMIC growth rate implies a potential impact on the associated wave–particle interactions and change of the decay rate in the ring current.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

Design workflow of a symmetric traveling wave antenna for fast ion production on DD tokamaks

Initial computational plasma physics scoping and a finite element method antenna modeling design workflow for a symmetric center-fed high-field side high harmonic fast wave traveling wave array (TWA) antenna are reported here. The TWA is designed to generate a test population of fast deuterium ions in an existing D–D tokamak by heating neutral beam deuterium ions, accelerating them from 80 keV to several hundred keV. The resulting fast particles are tailored to mimic key reactor energetic particle parameters with regards to exciting Alfven eigenmode instabilities, allowing for a D–D tokamak like DIII-D or ASDEX-U to replicate reactor-relevant conditions experimentally. Initial scenario scoping for high single-pass absorption as well as good preferential fast ion damping relative to electron damping was completed using the ray-tracing/Fokker–Planck codes GENRAY and CQL3D. Python RF network analysis packages were used to create a custom TWA optimization tool to inform a COMSOL flat antenna design, and Petra-M was used to study cold plasma effects. The TWA produced by this workflow has several novel features when compared to previous TWA studies, including symmetric center feeding, and passive end straps for image current cancellation for reduced impurity production. We show here that the antenna design workflow can readily produce TWA antennas optimized for reflection coefficient, image current cancellation, and launched power spectrum shape; and that a population of fast ions can be generated in the correct region of parameter space, warranting future more detailed studies.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2023

Disruption Mitigation Solutions for Long Pulse Tokamaks (Final Technical Report)

The project "Disruption Mitigation Solutions for Long Pulse Tokamaks" was a collaborative effort between ORNL, GA, and UCSD. The main goal of the project was to study the injection of shattered cryogenic pellets (SPI) as a method of reducing wall damage from global tokamak plasma instabilities called disruptions. This work was primarily carried out in the JET tokamak in the UK. The team already had extensive experience studying PSI in the medium sized DIII-D tokamak in the US, but adding data from the larger JET tokamak enabled more confident extrapolation to the even larger planned ITER tokamak in France. Each of the collaborating institutions focused on different aspects of SPI disruption mitigation. The UCSD collaboration focused on the volume recombination of post-disruption runaway electron (RE) beams by massive hydrogen injection. Disruptions can sometimes result in the formation of toroidal (ring shaped) beams of highly energetic (relativistic) electron beams, which can cause serious local wall damage, even in present-day tokamaks. A promising method to reduce wall damage from the RE beams is to inject massive amounts of H 2 or D 2 into the beams using SPI. although the precise method by which the injected H 2 reducses RE wall damage is still being investigated, it seems clear that a necessary and required step for the reduction to occur is for the H 2 injection to cause volume recombination (free electron density reduction) of the background cold plasma surrounding the fast electron beam. The UCSD collaboration focused on answering the question of if this recombination could be achievable in the future reactor-size tokamak ITER using realistic levels of H 2 injection. A model capturing the most essential physics was developed and was constrained using both DIII-D and JET data. This model was then used to extrapolate to ITER. It was predicted that RE beam recombination should be achievable in ITER, although in the worst-case scenari (a 10 MA Re beam in ITER), recombination is barely achieved within the maximum allowed levels of H 2 injection.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2023

Plasma production and ion heating systems for the Material Plasma Exposure eXperiment

Plasma production and ion heating in the Material Plasma Exposure eXperiment (MPEX), whose design is nearing completion, is accomplished using continuous wave RF power with average power density up to 1.6 MW/m2 at the plasma interface. Plasma is produced using helicon waves coupled through a single helical antenna at 13.56 MHz with power supplied by three 100 kW fixed-frequency RF generators feeding a power combiner network followed by the matching network and launcher. The helicon source can utilize various gasses including hydrogen, deuterium, and helium, with a magnetic field strength in the source region up to 0.2 T, and maximum device |B| of 2.5 T. Power is coupled to ions via ion cyclotron heating at the fundamental resonance using a pair of phased helical antennas operating in the frequency range 4-9 MHz, that launch waves towards the resonance from the antenna region where ω > ωci. ICH power is supplied by a single 500 kW tunable RF transmitter through a 90° power splitter and matching/decoupling network. In the case of both the helicon and ICH systems the antennas are located external to the vacuum, with power transferred through novel water-cooled coaxial vacuum windows consisting of fused quartz outer cylinders and silicon nitride inner cylinders with forced convection water cooling between them. The antenna enclosures are pressurized with dry air to 3 bar absolute for voltage standoff.Several 3-D COMSOL models have been created to simulate the two systems. A model of the helicon region utilizing a cold plasma dielectric tensor with accurate magnetic field and realistic plasma density profiles has been used to calculate the plasma loading/complex antenna input impedance at the launcher feed for various ne and |B| values, for the purpose of estimating power handling. It also incorporates the geometry of all launcher structures relevant to this determination. A still more detailed model of the launcher together with a lossy dielectric plasma surrogate has been used to determine RF electric field values and power losses in the device components.Similar models have been produced to predict the performance and power handling of the ICH launcher. For this device the impedance matrix of the two-element antenna array is calculated using a warm plasma model, necessary to properly determine the wave propagation and absorption.

Goulding, Richard↗
NASA NTRS1996

Modeling of Jovian Hectometric Radiation Source Locations: Ulysses Observations

The Unified Radio and Plasma Wave (URAP) experiment on Ulysses has provided unique high latitude measurements of Jovian hectometric radiation (HOM) during its encounter with Jupiter in February 1992. URAP was the first radio instrument in the Jovian environment with radio direction-finding capability, which was previously used to determine the HOM source locations in the Jovian magnetosphere. These initial source location determinations were based on several assumptions, including the neglect of refractive effects, which may be tested. We have, for the first time, combined the measured incident ray-direction at the spacecraft with a model magnetosphere to directly trace the rays back to the HOM source. We concentrate on the observations of HOM from high northern latitudes when Ulysses was at distances less than 15 R(sub j). The three- dimensional ray-tracing calculations presented here indicate that the HOM sources probably lie on L shells in the range 3 less than or approximately equal to L less than 7 (tilted dipole magnetic field model) consistent with previous determinations that ignored the effects of refraction. The ray-tracing results, however, indicate that wave refraction due to the Io torus and the magnetic field can significantly influence the precise source location. We show that constraints on the locations imposed by the gyroemission mechanism suggest that the lo torus density may have experienced temporal and/or spatial fluctuations during the Ulysses observations of HOM. Finally, in the cold plasma approximation we demonstrate that even if the emission were nearly linearly polarized near the source region, almost circular polarization will be observed at Ulysses, in agreement with observations.

Menietti, J. D.↗
DOE OSTI2014

Effect of spatial density variation and O+ concentration on the growth and evolution of electromagnetic ion cyclotron waves

We simulate electromagnetic ion cyclotron (EMIC) wave growth and evolution within three regions, the plasmasphere (or plasmaspheric plume), the plasmapause, and the low-density plasmatrough outside the plasmapause. First, we use a ring current simulation with a plasmasphere model to model the particle populations that give rise to the instability for conditions observed on 9 June 2001. Then, using two different models for the cold ion composition, we do a full scale hybrid code simulation in dipole coordinates of the EMIC waves on a meridional plane at MLT = 18 and at 1900 UT within a range of L shell from L = 4.9 to 6.7. EMIC waves were observed during June 9, 2001 by Geostationary Operational Environmental Satellite (GOES) spacecraft. While an exact comparison between observed and simulated spectra is not possible here, we do find significant similarities between the two, at least at one location within the region of largest wave growth. We find that the plasmapause is not a preferred region for EMIC wave growth, though waves can grow in that region. The density gradient within the plasmapause does, however, affect the orientation of wave fronts and wave vector both within the plasmapause and in adjacent regions. There is a preference for EMIC waves to be driven in the He+ band (frequencies between the O+ and He+ gyrofrequencies) within the plasmasphere, although they can also grow in the plasmatrough. If present, H+ band waves are more likely to grow in the plasmatrough. This fact, plus L dependence of the frequency and possible time evolution toward lower frequency waves, can be explained by a simple model. Large O+ concentration limits the frequency range of or even totally quenches EMIC waves. This is more likely to occur in the plasmatrough at solar maximum. Such large O+ concentration significantly affects the H+ cutoff frequency and hence the width in frequency of the stop band above the He+ gyrofrequency. EMIC wave surfaces predicted by cold plasma theory are altered by the finite temperature of the ring current H+.

58 GEOSCIENCES↗
DOE OSTI2016

Measurement and modeling of the refilling plasmasphere during 2001

The Naval Research Laboratory SAMI3 (Sami3 is Also a Model of the Ionosphere) and the RAM-CPL (Ring current Atmosphere interaction Model-Cold PLasma) codes are used to model observed plasmasphere dynamics during 25 November 2001 to 1 December 2001 and 1–5 February 2001. Model results compare well to plasmasphere observations of electron and mass densities. Comparison of model results to refilling data and to each other shows good agreement, generally within a factor of 2. We find that SAMI3 plasmaspheric refilling rates and ion densities are sensitive to the composition and temperature of the thermosphere and exosphere, and to photoelectron heating. Furthermore, results also support our previous finding that the wind-driven dynamo significantly impacts both refilling rates and plasmasphere dynamics during quiet periods.

58 GEOSCIENCES↗
DOE OSTI2016

In situ evidence of the modification of the parallel propagation of EMIC waves by heated He + ions

With observations of the Van Allen Probe B, we report in situ evidence of the modification of the parallel propagating electromagnetic ion cyclotron (EMIC) waves by heated He + ions. In the outer boundary of the plasmasphere, accompanied with the He + ion heating, the frequency bands of H + and He + for EMIC waves merged into each other, leading to the disappearance of a usual stop band between the gyrofrequency of He + ions (Ω He+ ) and the H + cutoff frequency (ω H+co ) in the cold plasma. Moreover, the dispersion relation for EMIC waves theoretically calculated with the observed plasma parameters also demonstrates that EMIC waves can indeed parallel propagate across Ω He+ . Therefore, this work provides an in situ evidence of the modification of the parallel propagation of EMIC waves by heated He + ions.

58 GEOSCIENCES↗
DOE OSTI2016

Cross-scale observations of the 2015 St. Patrick's day storm: THEMIS, Van Allen Probes, and TWINS

In this paper, we present cross-scale magnetospheric observations of the 17 March 2015 (St. Patrick's Day) storm, by Time History of Events and Macroscale Interactions during Substorms (THEMIS), Van Allen Probes (Radiation Belt Storm Probes), and Two Wide-angle Imaging Neutral-atom Spectrometers (TWINS), plus upstream ACE/Wind solar wind data. THEMIS crossed the bow shock or magnetopause 22 times and observed the magnetospheric compression that initiated the storm. Empirical models reproduce these boundary locations within 0.7 R E . Van Allen Probes crossed the plasmapause 13 times; test particle simulations reproduce these encounters within 0.5 R E . Before the storm, Van Allen Probes measured quiet double-nose proton spectra in the region of corotating cold plasma. About 15 min after a 0605 UT dayside southward turning, Van Allen Probes captured the onset of inner magnetospheric convection, as a density decrease at the moving corotation-convection boundary (CCB) and a steep increase in ring current (RC) proton flux. During the first several hours of the storm, Van Allen Probes measured highly dynamic ion signatures (numerous injections and multiple spectral peaks). Sustained convection after ~1200 UT initiated a major buildup of the midnight-sector ring current (measured by RBSP A), with much weaker duskside fluxes (measured by RBSP B, THEMIS a and THEMIS d). A close conjunction of THEMIS d, RBSP A, and TWINS 1 at 1631 UT shows good three-way agreement in the shapes of two-peak spectra from the center of the partial RC. A midstorm injection, observed by Van Allen Probes and TWINS at 1740 UT, brought in fresh ions with lower average energies (leading to globally less energetic spectra in precipitating ions) but increased the total pressure. Finally, the cross-scale measurements of 17 March 2015 contain significant spatial, spectral, and temporal structure.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2018

Nonlinear Electrostatic Steepening of Whistler Waves: The Guiding Factors and Dynamics in Inhomogeneous Systems

Whistler mode chorus waves are particularly important in outer radiation belt dynamics due to their key role in controlling the acceleration and scattering of electrons over a very wide energy range. The efficiency of wave-particle resonant interactions is defined by whistler wave properties which have been described by the approximation of plane linear waves propagating through the cold plasma of the inner magnetosphere. However, recent observations of extremely high-amplitude whistlers suggest the importance of nonlinear wave-particle interactions for the dynamics of the outer radiation belt. Oblique chorus waves observed in the inner magnetosphere often exhibit drastically nonsinusoidal (with significant power in the higher harmonics) waveforms of the parallel electric field, presumably due to the feedback from hot resonant electrons. We have considered the nature and properties of such nonlinear whistler waves observed by the Van Allen Probes and Time History of Events and Macroscale Interactions define during Substorms in the inner magnetosphere, and we show that the significant enhancement of the wave electrostatic component can result from whistler wave coupling with the beam-driven electrostatic mode through the resonant interaction with hot electron beams. Being modulated by a whistler wave, the electron beam generates a driven electrostatic mode significantly enhancing the parallel electric field of the initial whistler wave. Finally, we confirm this mechanism using a self-consistent particle-in-cell simulation. The nonlinear electrostatic component manifests properties of the beam-driven electron acoustic mode and can be responsible for effective electron acceleration in the inhomogeneous magnetic field.

58 GEOSCIENCES↗
DOE OSTI2023

Energetic Electron Precipitation Driven by Electromagnetic Ion Cyclotron Waves from ELFIN’s Low Altitude Perspective

We review comprehensive observations of electromagnetic ion cyclotron (EMIC) wave-driven energetic electron precipitation using data collected by the energetic electron detector on the Electron Losses and Fields InvestigatioN (ELFIN) mission, two polar-orbiting low-altitude spinning CubeSats, measuring 50-5000 keV electrons with good pitch-angle and energy resolution. EMIC wave-driven precipitation exhibits a distinct signature in energy-spectrograms of the precipitating-to-trapped flux ratio: peaks at >0.5 MeV which are abrupt (bursty) (lasting ~17 s, or ΔL ~ 0.56) with significant substructure (occasionally down to sub-second timescale). We attribute the bursty nature of the precipitation to the spatial extent and structuredness of the wave field at the equator. Multiple ELFIN passes over the same MLT sector allow us to study the spatial and temporal evolution of the EMIC wave - electron interaction region. Case studies employing conjugate ground-based or equatorial observations of the EMIC waves reveal that the energy of moderate and strong precipitation at ELFIN approximately agrees with theoretical expectations for cyclotron resonant interactions in a cold plasma. Using multiple years of ELFIN data uniformly distributed in local time, we assemble a statistical database of ~50 events of strong EMIC wave-driven precipitation. Most reside at L ~ 5 – 7 at dusk, while a smaller subset exists at L ~ 8 – 12 at post-midnight. The energies of the peak-precipitation ratio and of the half-peak precipitation ratio (our proxy for the minimum resonance energy) exhibit an L-shell dependence in good agreement with theoretical estimates based on prior statistical observations of EMIC wave power spectra. The precipitation ratio’s spectral shape for the most intense events has an exponential falloff away from the peak (i.e., on either side of ~ 1.45 MeV). It too agrees well with quasi-linear diffusion theory based on prior statistics of wave spectra. It should be noted though that this diffusive treatment likely includes effects from nonlinear resonant interactions (especially at high energies) and nonresonant effects from sharp wave packet edges (at low energies). Sub-MeV electron precipitation observed concurrently with strong EMIC wave-driven >1 MeV precipitation has a spectral shape that is consistent with efficient pitch-angle scattering down to ~ 200-300 keV by much less intense higher frequency EMIC waves at dusk (where such waves are most frequent). At ~100 keV, whistler-mode chorus may be implicated in concurrent precipitation. These results confirm the critical role of EMIC waves in driving relativistic electron losses. Nonlinear effects may abound and require further investigation.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

Leveraging artificial intelligence and advanced food processing techniques for enhanced food safety, quality, and security: a comprehensive review

Artificial intelligence is emerging as a transformative force in addressing the multifaceted challenges of food safety, food quality, and food security. This review synthesizes advancements in AI-driven technologies, such as machine learning, deep learning, natural language processing, and computer vision, and their applications across the food supply chain, based on a comprehensive analysis of literature published from 1990 to 2024. AI enhances food safety through real-time contamination detection, predictive risk modeling, and compliance monitoring, reducing public health risks. It improves food quality by automating defect detection, optimizing shelf-life predictions, and ensuring consistency in taste, texture, and appearance. Furthermore, AI addresses food security by enabling resource-efficient agriculture, yield forecasting, and supply chain optimization to ensure the availability and accessibility of nutritious food resources. This review also highlights the integration of AI with advanced food processing techniques such as high-pressure processing, ultraviolet treatment, pulsed electric fields, cold plasma, and irradiation, which ensure microbial safety, extend shelf life, and enhance product quality. Additionally, the integration of AI with emerging technologies such as the Internet of Things, blockchain, and AI-powered sensors enables proactive risk management, predictive analytics, and automated quality control. By examining these innovations' potential to enhance transparency, efficiency, and decision-making within food systems, this review identifies current research gaps and proposes strategies to address barriers such as data limitations, model generalizability, and ethical concerns. These insights underscore the critical role of AI in advancing safer, higher-quality, and more secure food systems, guiding future research and fostering sustainable food systems that benefit public health and consumer trust.

AI↗
DOE OSTI2017

Surface tension models for a multi-material ALE code with AMR

A number of surface tension models have been implemented in a 3D multi-physics multi-material code, ALE–AMR, which combines Arbitrary Lagrangian Eulerian (ALE) hydrodynamics with Adaptive Mesh Refinement (AMR). ALE–AMR is unique in its ability to model hot radiating plasmas, cold fragmenting solids, and most recently, the deformation of molten material. The surface tension models implemented include a diffuse interface approach with special numerical techniques to remove parasitic flow and a height function approach in conjunction with a volume-fraction interface reconstruction package. These surface tension models are benchmarked with a variety of test problems. In conclusion, based on the results, the height function approach using volume fractions was chosen to simulate droplet dynamics associated with extreme ultraviolet (EUV) lithography.

42 ENGINEERING↗
DOE OSTI2018

Continuously differentiable PIC shape functions for triangular meshes

In this study, a new class of continuously-differentiable shape functions is developed and applied to two-dimensional electrostatic PIC simulation on an unstructured simplex (triangle) mesh. It is shown that troublesome aliasing instabilities are avoided for cold plasma simulation in which the Debye length is as small as 0.01 cell sizes. These new shape functions satisfy all requirements for PIC particle shape. They are non-negative, have compact support, and partition unity. They are given explicitly by cubic expressions in the usual triangle logical (areal) coordinates. The shape functions are not finite elements because their structure depends on the topology of the mesh, in particular, the number of triangles neighboring each mesh vertex. Nevertheless, they may be useful as approximations to solution of other problems in which continuity of derivatives is required or desired.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗