Search NASA⌕ Search

SEARCH · Search NASA

Results for “Corner Unit”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 127 records · Page 7
All sourcesNASA NTRSDOE OSTI
NASA NTRS2023

Sewage Treatment Plant #1 Area, SWMU 117 Per- and Polyfluoroalkyl Substances (PFAS) Site Assessment Progress Report

This Per- and Polyfluoroalkyl Substances (PFAS) Site Assessment (SA) Progress Report (SAPR) discusses the investigation activities and findings for the Sewage Treatment Plant #1 (STP1) Area located at Kennedy Space Center (KSC), Florida (Figure 1-1). This site has been designated Solid Waste Management Unit (SWMU) 117 under KSC’s Resource Conservation and Recovery Act (RCRA) Corrective Action Program, as the sewage treatment plant and associated areas were identified as a potential source of PFAS to the environment. This PFAS SAPR was prepared by Tetra Tech, Inc., for the National Aeronautics and Space Administration (NASA) under Indefinite Delivery Indefinite Quantity Contract 80KSC019D0011-80KSC019F0070. This is the first progress report to document on-going SA activities; supplemental progress reports will be provided as additional data is collected. PFAS SA activities were conducted between April 2020 and March 2022 to collect additional data to supplement the existing datasets to better understand the extent of PFAS impacts to the environment in the STP1 Area. The SA for the STP1 Area covers an approximately 130-acre investigation area with multiple structures and buildings. The focus of the SA is the STP1 Complex and associated structures, including the former Polishing Pond, former Sludge Disposal8 Area, and former Spray Field. The STP1 Complex is located in the KSC Industrial Area, at the southwest corner of 4th Street SE and C Avenue SE. The STP1 Complex is located approximately ¼-mile south and downgradient of the Fire Station #1 site (SWMU 116), which is also currently undergoing a PFAS SA because of potential releases of PFAS-containing aqueous film-forming foam (AFFF). During the SA, a total of seven soil, 131 groundwater direct push technology (DPT), 24 groundwater monitoring well, and 11 surface water samples were collected between April 2020 and March 2022. Monitoring well samples were analyzed for 18 PFAS compounds, with all other samples analyzed for 28 PFAS compounds. The SA sample results were used along with historical results to evaluate the extent of PFAS impacts to the environment in the STP1 Area. Data generated to date and prior results were screened against the United States Environmental Protection Agency (USEPA) May 2022 Tap Water Regional Screening Levels (RSL) for groundwater and residential RSLs for soil (hazard quotient of 0.1). Surface water results were screened against the State of Florida Human Health Surface Water Screening Levels (SWSLs). Results from the SA showed exceedances of the applicable screening criteria for groundwater and surface water. Considering the current and historical dataset, PFOS is the prevalent PFAS compound. Based on these results, additional groundwater DPT and surface water sampling should be considered for PFAS analysis, focused on evaluating surface water bodies in the southeast portion of the Industrial Area, which discharge into the Banana River. Additionally, installation of monitoring wells should be considered to evaluate the interaction between the groundwater and surface water at the site. Collection of additional samples for TOC analysis should also be considered from representative groundwater (saturated soils) and surface water locations to further evaluate potential correlations between PFAS and TOC to provide a more comprehensive dataset to assist in fate and transport analyses.

Sewage Treatment Plant↗
DOE OSTI2023

Tail Dependence as a Measure of Teleconnected Warm and Cold Extremes of North American Wintertime Temperatures

Current models for spatial extremes are concerned with the joint upper (or lower) tail of the distribution at two or more locations. Such models cannot account for teleconnection patterns of 2-m surface air temperature ( T 2m ) in North America, where very low temperatures in the contiguous United States may coincide with very high temperatures in Alaska in the wintertime. This dependence between warm and cold extremes motivates the need for a model with opposite-tail dependence in spatial extremes. This work develops a statistical modeling framework that has flexible behavior in all four pairings of high and low extremes at pairs of locations. In particular, we use a mixture of rotations of common Archimedean copulas to capture various combinations of four-corner tail dependence. We study teleconnected T 2m extremes using ERA5 of daily average 2-m temperature during the boreal winter. Further, the estimated mixture model quantifies the strength of opposite-tail dependence between warm temperatures in Alaska and cold temperatures in the midlatitudes of North America, as well as the reverse pattern. These dependence patterns are shown to correspond to blocked and zonal patterns of midtropospheric flow. This analysis extends the classical notion of correlation-based teleconnections to considering dependence in higher quantiles.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2025

Simulation-driven design optimization of reaction injection molding (RIM) process for polydicyclopentadiene (pDCPD): Minimizing cycle time, defects, and warpage

Replacing metal components in trucks, trailers, and buses with lightweight polymer composites is challenging due to high temperatures and complex manufacturing. The Reaction Injection Molding (RIM) process using Dicyclopentadiene (DCPD) resin offers a solution by producing robust parts with excellent stiffness, impact strength, and resistance properties. Simulations are essential for optimizing this process, predicting defects, and improving quality. However, most commercial software is tailored for thermoplastics, requiring thermoset users to generate their own datasets. In this study, a material data card for DCPD was developed to perform RIM simulations. Design of Experiments (DOE) was used to identify key factors affecting filling, curing, and warpage, aiming to minimize cycle time and defects. The simulations explored varying injection gate parameters (size, location, number) and process conditions (mold/resin temperature, injection/curing pressure). Results showed that gate design significantly impacts filling behavior and defects. A single central gate provided balanced flow with fewer defects, while two corner gates led to more defects. Additionally, lower injection pressure increased filling time, while higher mold temperature accelerated curing but led to more warpage. In conclusion, this optimization framework aims to enhance DCPD part performance and promote sustainable manufacturing by reducing waste and energy consumption.

42 ENGINEERING↗
DOE OSTI2025

Synthesis, Crystal Structure, and Optical Properties of the Barium meta -Pertechnetate Ba[TcO 4 ] 2

High-quality single crystals of the colorless title compound Ba[ 99 TcO 4 ] 2 were prepared from radioactive, isotope-pure 99 TcO 2 by oxidation with H 2 O 2 in an aqueous ammonia solution and subsequent cation metathesis using an ion-exchange column. Single-crystal X-ray diffraction revealed Ba[TcO 4 ] 2 to crystallize isotopically to Ba[MnO 4 ] 2 in the orthorhombic space group Fddd with the lattice parameters a = 747.34(5) pm, b = 1244.08(7) pm, and c = 1511.23(9) pm for Z = 8. Its crystal structure features [BaO 12 ] 22- icosahedra surrounded by ten [TcO 4 ] - tetrahedra connected via two edges and eight corners. The optical properties of Ba[TcO 4 ] 2 investigated by UV-vis spectroscopy show the broad O 2- → Tc 7+ ligand-to-metal charge transfer (2p → 4d) process within these discrete [TcO 4 ]- anions at an absorption maximum of around 350 nm.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

The crystal structure of Fe2S at 90 GPa based on single-crystal X-ray diffraction techniques

The Fe-S system was explored in a laser-heated diamond-anvil cell at 89(2) GPa and 2380(120) K to better understand the phase stability of Fe 2 S. Upon temperature quenching, crystallites of Fe 2 S were identified, and their structure was investigated using single-crystal X-ray diffraction techniques. At these conditions, Fe 2 S adopts the C23 structure (anti-PbCl 2 , Co 2 P) with space group Pnma (Z = 4). This structure consists of columns of corner-sharing, FeS 4 tetrahedra, and columns of edge-sharing FeS 5 square pyramids linked along edges in the b direction. Sulfur is in ninefold coordination with Fe. This work marks the first high-pressure structural solution and refinement of Fe 2 S synthesized in a multigrain Fe+FeS sample at 90 GPa and 2400 K and establishes the stability of C23 Fe 2 S at these conditions. A previous powder diffraction study reports an orthorhombic Fe 2 S phase with a C37, Co 2 Si-like unit cell above 190 GPa. A C23–C37 structural transition is inferred to explain the previously observed unit-cell parameters at higher pressures and temperatures. These results highlight the utility of applying single-crystal X-ray diffraction techniques to high P-T multigrain samples to explore the structural properties of iron-rich phases in Earth and planetary cores.

36 MATERIALS SCIENCE↗
NASA NTRS1986

A study of 35-ghz radar-assisted orbital maneuvering vehicle/space telescope docking

An experiment was conducted to study the effects of measuring range and range rate information from a complex radar target (a one-third scale model of the Edwin P. Hubble Space Telescope). The radar ranging system was a 35-GHz frequency-modulated continuous wave unit developed in the Communication Systems Branch of the Information and Electronic Systems Laboratory at Marshall Space Flight Cneter. Measurements were made over radar-to-target distances of 5 meters to 15 meters to simulate the close distance realized in the final stages of space vehicle docking. The Space Telescope model target was driven by an antenna positioner through a range of azimuth and elevation (pitch) angles to present a variety of visual aspects of the aft end to the radar. Measurements were obtained with and without a cube corner reflector mounted in the center of the aft end of the model. The results indicate that range and range rate measurements are performed significantly more accurately with the cooperative radar reflector affixed. The results further reveal that range rate (velocity) can be measured accurately enough to support the required soft docking with the Space Telescope.

Mcdonald, M. W.↗
NASA NTRS2008

New Techniques in Characterization of Ferroelectric Materials

Two new techniques have been developed to characterize Pb(Mg1/3Nb2/3)O3-PbTiO3 (PMN-PT) based ferroelectric single crystals: (i) electro-thermal imaging, and (ii) single crystal x-ray diffraction in the transmission mode. (i) Electro-thermal imaging is a remote sensing technique that can detect the polarization direction and poling state of a whole crystal slice. This imaging technique utilizes an IR camera to determine the field induced temperature change and does not require any special or destructive sample preparation. In the resulting images it is possible to distinguish regions of 180 deg domains. This powerful technique can be used remotely during poling to determine the poling state of the crystal to avoid over-poling that can result in inferior properties and/or cracking of the crystals. Electro-thermal imaging produced the first direct observations of polarization rotation. Under bipolar field, the domains near the corners were the first to switch direction. As the field increased above the coercive field, domains at the center part of the crystals switched direction. (ii) X-ray diffraction in the transmission mode has long been used in structure determination of organic crystals and proteins; however, it is not used much to characterize inorganic systems. 0.7Pb(Mg1/3Nb2/3)O3-0.3PbTiO3 single crystals were examined by this XRD technique for the first time, and a never-before-seen super-lattice was revealed with a doubling of the unit cell in all three directions, giving a cell volume eight times that of a traditional perovskite unit cell. The significance of the super-lattice peaks increased with poling, indicating a structural contribution to ordering. Lack of such observations by electron diffraction in the transmission electron microscope examinations suggests the presence of a bulk effect.

Sehirlioglu, Alp↗
DOE OSTI2025

Cocrystals combining order and correlated disorder via colloidal crystal engineering with DNA

Colloidal cocrystallization enables the formation of multicomponent materials with unique physicochemical properties, yet the role of nanoparticle (NP) shape and specific ligand interactions to cocrystallize anisotropic and isotropic NPs, with order and correlated disorder, remains underexplored. Here, geometry-inspired strategies along with programmable DNA interactions are combined to achieve structural control of colloidal cocrystal assemblies. Coassembling polyhedral and spherical NPs with complementary DNA yields two classes of cocrystals: one where both components order, and another where polyhedral NPs form a periodic lattice, while spherical NPs remain disordered but spatially correlated with polyhedral edges and corners. The size ratio of the building blocks can be used to control the ordering of spherical NPs—smaller octahedral-to-sphere size ratios favor fully ordered cocrystals. Molecular dynamics simulations further elucidate the role of NP shapes and dimensions in the structural outcome of the cocrystal. This work provides a framework for deliberately targeting and accessing crystals with exotic multicomponent structures.

Li, Yuanwei [Northwestern Univ., Evanston, IL (Uni↗
DOE OSTI2024

Hydrogenation of calcite and change in chemical bonding at high pressure: Diamond formation above 100GPa

Synchrotron X-ray diffraction (XRD) and Raman spectroscopy in laser heated diamond anvil cells and first principles molecular dynamics (FPMD) calculations have been used to investigate the reactivity of calcite and molecular hydrogen (H 2 ) at high pressures up to 120 GPa. We find that hydrogen reacts with calcite starting below 0.5 GPa at room temperature forming chemical bonds with carbon and oxygen. This results in the unit cell volume expansion; the hydrogenation level is much higher for powdered samples. Single-crystal XRD measurements at 8–24 GPa reveal the presence of previously reported III, IIIb, and VI calcite phases; some crystallites show up to 4% expansion, which is consistent with the incorporation of ≤ 1 hydrogen atom per formula unit. At 40–102 GPa XRD patterns of hydrogenated calcite demonstrate broadened features consistent with the calcite VI structure with incorporated hydrogen atoms. Above 80 GPa, the C–O stretching mode of calcite splits suggesting a change in the coordination of C–O bonds. Laser heating at 110 GPa results in the formation of C–C bonds manifested in the crystallization of diamond recorded by in situ XRD at 300 K and 110 GPa and by Raman spectroscopy on recovered samples commenced with C 13 calcite. We explored several theoretical models, which show that incorporation of atomic hydrogen results in local distortions of CO 3 groups, formation of corner-shared C–O polyhedra, and chemical bonding of H to C and O, which leads to the lattice expansion and vibrational features consistent with the experiments. In conclusion, the experimental and theoretical results support recent reports on tetrahedral C coordination in high-pressure carbonate glasses and suggest a possible source of the origin of ultradeep diamonds.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

Enabling Next Generation Reaction Injection Molding (RIM) for Lightweight Structures

Replacing metal components in trucks, trailers, and buses with lightweight polymer composites is challenging due to high temperatures and complex manufacturing. The Reaction Injection Molding (RIM) process using Dicyclopentadiene (DCPD) resin offers a solution by producing robust parts with excellent stiffness, impact strength, and resistance properties. Simulations are essential for optimizing this process, predicting defects, and improving quality. However, most commercial software is tailored for thermoplastics, requiring thermoset users to generate their own datasets. In this project, a material data card for DCPD was developed to perform RIM simulations. Design of Experiments (DOE) was used to identify key factors affecting filling, curing, and warpage, aiming to minimize cycle time and defects. The simulations explored varying injection gate parameters (size, location, number) and process conditions (mold/resin temperature, injection/curing pressure). Results showed that gate design significantly impacts filling behavior and defects. A single central gate provided balanced flow with fewer defects, while two corner gates led to more defects. Additionally, lower injection pressure increased filling time, while higher mold temperature accelerated curing but led to more warpage. This optimization framework aims to enhance DCPD part performance and promote sustainable manufacturing by reducing waste and energy consumption. This research has been performed in collaborations with McClarin Composites. The research outcome has been submitted to the Journal of Manufacturing Processes.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Data for "Land-based Resources for Engineered Carbon Dioxide Removal in the United States Exceed the Expected Needs"

Gigatonne-scale atmospheric carbon dioxide removal (CDR), alongside deep emission cuts, is critical to stabilizing the climate. However, some of the most scalable CDR technologies are also the most land intensive. Here, we examine whether adequate land resources exist in the contiguous United States to meet CDR targets when prioritizing grid emissions reduction, food production, and the protection of sensitive ecosystems. We focus on biomass carbon removal and storage (BiCRS) and direct air capture and storage (DACS) and show that suitable lands exceed the expected needs: 37.6 million hectares of land are available for BiCRS, resulting in 0.26 GtCO2 of CDR/year, and 34 million hectares are suitable for wind- and solar-powered DACS, resulting in 4.8 GtCO2 of CDR/year if facilities are co-located with geologic CO2 storage. We identify biomass and energy supply hotspots to meet CDR targets while ensuring land protection and minimizing land competition.

carbon↗
DOE OSTI2020

Long-range and local crystal structures of the S r 1 - x C a x Ru O 3 perovskites

Here, the crystal structures of the Sr 1–x Ca x RuO 3 (0 ≤ x ≤ 1) perovskites are investigated using both long-range and local structural probes. High-resolution synchrotron powder x-ray-diffraction characterization at ambient temperature shows that the materials have orthorhombic long-range crystal structures to high precision, and we support previous work showing that Ca 2+ substitution for Sr 2+ primarily changes the tilting of rigid corner-sharing RuO 6 octahedra at their shared oxygen vertices. X-ray pair-distribution function analysis for SrRuO 3 , CaRuO 3 , and one intermediate composition show them to display monoclinic symmetry at the local level; when averaged over different domain orientations this yields orthorhombic symmetry in the long range, and no long-range or local phase transitions are observed between 80 and 300 K for materials with intermediate compositions. High-resolution transmission electron microscopy shows that the Sr/Ca distribution is random at the nanoscale. Lastly, we plot magnetic characteristics such as the ferromagnetic T c , Curie-Weiss theta, effective moment, and ambient temperature susceptibility vs the octahedral tilt and unit-cell volume.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
NASA NTRS2012

Debris Impact Detection Instrument for Crewed Modules

When micrometeoroid or debris impacts occur on a space habitat, crew members need to be quickly informed of the likely extent of damage, and be directed to the impact location for possible repairs. This is especially important because the outer walls of pressurized volumes are often not easily accessible, blocked by racks or cabinets. The goal of the Habitat Particle Impact Monitoring System (HIMS) is to develop a fully automated, end-to-end particle impact detection system for crewed space exploration modules. The HIMS uses multiple passive, thin film piezo-polymer vibration sensors to detect impacts on a surface, and computer processing of the acoustical signals to characterize the impacts. Development and demonstration of the HIMS is proceeding in concert with NASA's Habitat Demonstration Unit (HDU) Project. The HDU Project is designed to develop and test various technologies, configurations, and operational concepts for exploration habitats. This paper describes the HIMS development, initial testing, and HDU integration efforts. Initial tests of the system on the HDU were conducted at NASA s 2010 and 2011 Desert Research and Technologies Studies (Desert-RATS or D-RATS). The HDU lab module, as seen from above, has an open circular floorplan divided into eight wedge-shaped Segments. The side wall of the module -- the surface used for this technology demonstration -- is a hard fiberglass composite covered with a layer of sprayed-on foam insulation. Four sensor locations were assigned near the corners of a rectangular pattern on the wall of one segment of the HDU lab module. The flat, self-adhesive sensors were applied to the module during its initial outfitting. To study the influence of the wall s construction (thickness and materials), three sets of four sensors were installed at different layer depths: on the interior of the module s wall, on the exterior of the same wall, and on the exterior of the foam insulation. The signal produced when a vibration passes through a sensor is first sent through a pre-amplifier. The amplified signal then is sent to the data acquisition and data processing systems. The vibration data from the sensors are then processed and reduced to a form suitable for presentation to the crew.

Opiela, J.↗
DOE OSTI2025

Role of Intersections in Fracture Connectivity

Networks of intersecting fractures often provide the flow paths through subsurface reservoirs. Assessing network connectivity is challenging because fracture intersections compose a vanishingly small fraction of the network void volume. In this paper, motivated by 3D X-ray imaging of the simplest element of fracture network, that is, two orthogonal fractures, we perform a percolation and finite-size scaling analysis to study the connectivity provided by fracture intersections. The conditions when an intersection enhances connectivity across a sample depend on spatial correlations in the fracture aperture distributions, on the stress state, and on the direction of flow. Here we consider three flow directions: (a) across intersections, (b) parallel to intersections and (c) around corners. For (a), intersections provide minimal enhancement of connectivity because they contribute little additional void area. For (b), intersections increase the probability of a connected path near threshold by enabling 3D connected pathways that are not possible in parallel fractures. Flow around corners, (c), is fundamentally the result of the intersection connecting two fractures in series and spatial correlations are broken around corners, suppressing the connectivity relative to (a). When the connected fractures are stressed equally, a joint percolation threshold emerges that continues to have scale invariance. However, when the fractures are stressed unequally, the system has mixed percolation without clearly defined percolation thresholds. In all cases, percolation probabilities are found to be scale dependent which has important consequences for the connectivity of larger fracture networks composed of the fundamental element studied here.

02 PETROLEUM↗
DOE OSTI2025

Local structure of zinc–indium–tin oxide films via grazing-incidence x-ray pair-distribution functions and theoretical methods

A detailed experimental and theoretical study on the local (r ≤ 4.5 Å) atomic structure of amorphous and crystalline zinc–indium–tin oxide (ZITO) thin films using grazing-incidence x-ray Pair-Distribution Functions (PDFs), ab initio Molecular Dynamics (MD), and Empirical Potential Structure Refinement (EPSR) Monte Carlo simulations is presented. High-energy synchrotron x rays, a two-dimensional detector, and different incident angles were used to probe the depth uniformity of five (ZnO) 0.15 (In 2 O 3 ) 0.70 (SnO 2 ) 0.15 films that were deposited via pulsed-laser deposition at growth temperatures (T G ) ranging from 25 to 300 °C. Films deposited at T G ≤ 150 °C were amorphous. The partially crystalline (T G = 200 °C) and fully crystalline (T G = 300 °C) films were highly textured. Both crystalline and amorphous structures were investigated using ab initio MD and EPSR Monte Carlo simulations. The density of the amorphous films determined from the experimental data agreed with MD calculations. Coordination numbers, bond lengths, and distortion for metal–oxygen and for both the edge- and corner-shared In–metal shells up to 4.5 Å obtained from PDF analysis closely agreed with MD and EPSR simulations. There is a pronounced decrease in the edge- and corner-shared In–Zn distances arising from the shorter Zn–O bond length, Zn–O tetrahedral coordination, and In–O–Zn angle in amorphous ZITO compared to its crystalline counterpart. A maximum in electrical mobility was observed for the amorphous film just before crystallization occurred. While the peak is broad, consistent with nearly unchanged overall cation–oxygen coordination in the amorphous films, ESPR results indicate that the tetrahedral coordination follows the conductivity trend.

Grazing Incidence X-ray↗
DOE OSTI2024

Electrochemically-Formed Disordered Rock Salt ω-Li x V 9 Mo 6 O 40 as a Fast-Charging Li-Ion Electrode Material

Electrochemically-formed disordered rock salt compounds are an emerging class of Li-ion electrode materials for fast-charging energy storage. However, the specific factors that govern the formation process and the resulting charge storage performance are not well understood. Here, we characterize the transformation mechanism and charge storage properties of an electrochemically-formed disordered rock salt from V 9 Mo 6 O 40 (VMO). The crystal structure of VMO has similar motifs to that of α-V 2 O 5 , a well-studied analogue, but VMO has less mechanical flexibility due to additional corner-sharing octahedra in its structure. As a result, VMO undergoes a single-step transformation pathway, which we characterize through operando X-ray diffraction, and forms an unusual highly distorted lamellar microstructure, as we show with high-resolution transmission electron microscopy. The resulting Li x VMO material shows fast charging and other electrochemical characteristics and performance typical of many nanomaterials, even though the material is composed of relatively large particles.

25 ENERGY STORAGE↗
DOE OSTI2025

Control simulations of many-body quantum systems by a synergism of discrete real-time learning and optimal control theory

We present a self-consistent algorithm for optimal control simulations of many-body quantum systems. The algorithm features a two-step synergism that combines discrete real-time machine learning (DRTL) with Quantum Optimal Control Theory (QOCT) using the time-dependent Schrödinger equation. Specifically, in step (1), DRTL is employed to identify a compact working space (i.e., the important portion of the Hilbert space) for the time evolution of the many-body quantum system in the presence of a control field (i.e., the initial or previously updated field), and in step (2), QOCT utilizes the DRTL-determined working space to find a newly updated control field for a chosen objective. Steps 1 and 2 are iterated until a self-consistent control objective value is reached such that the resulting optimal control field yields the same targeted objective value when the corresponding working space is systematically enlarged. Furthermore, to demonstrate this two-step self-consistent DRTL-QOCT synergistic algorithm, we perform optimal control simulations of strongly interacting 1D as well as 2D Heisenberg spin systems. In both scenarios, only a single spin (at the left end site for 1D and the upper left corner site for 2D) is driven by the time-dependent control fields to create an excitation at the opposite site as the target. It is found that, starting from all spin-down zero excitation states, the synergistic method is able to identify working spaces and convergence of the desired controlled dynamics with just a few iterations of the overall algorithm. In the cases studied, the dimensionality of the working space scales only quasi-linearly with the number of spins.

Artificial neural networks↗
NASA NTRS1977

Numerical determination of the fundamental eigenvalue for the Laplace operator on a spherical domain

Methods for obtaining approximate solutions for the fundamental eigenvalue of the Laplace-Beltrami operator (i.e., the membrane eignevalue problem for the vibration equation) on the unit spherical surface are developed. Two types of spherical surface domains are considered: the interior of a spherical triangle, and the exterior of a great circle arc extending for less than pi radians (a spherical surface with a slit). In both cases, zero boundary conditions are imposed. In order to solve the resulting second-order elliptic partial differential equations in two independent variables, a finite difference approximation is employed. The fundamental eigenvalue is approximated by iteration utilizing the power method and point successive overrelaxation. Some numerical results are given and compared, in certain special cases, with analytical solutions to the eigenvalue problem. The significance of the numerical eigenvalue results is discussed in terms of the singularities in the solution of three-dimensional boundary-value problems near a polyhedral corner of the domain.

Walden, H.↗