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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
DOE OSTI2026

Photoelectrode Durability in Two- versus Three-Electrode Configurations: Understanding the Impact of Circuit Configuration on Water-Splitting Stability

Device durability remains a significant challenge in photoelectrochemical (PEC) water splitting under ambient conditions. Yet, a lack of understanding of the test configuration and applied bias effects continue to hinder progress. In this study, we differentiate two-electrode (2E) and three-electrode (3E) configurations for evaluating PEC material durability, focusing particularly on their impacts on photoabsorber solid-state operating conditions. Our results underscore the fallacy of inferring 2E device stability from durability measurements performed solely in 3E configurations. Unmeasured and often misunderstood total circuit bias in 3E tests moderates material degradation, leading to the overestimation of photoelectrode stability compared to short-circuit operation. We demonstrate how the photoabsorber's operating voltage critically governs charge separation, surface stability, and degradation mechanisms during PEC operation. With these findings, we propose a standardized framework for conducting more reliable 3E durability experiments that simulate unassisted performance to help accelerate the development of robust, stable materials for solar-driven water splitting.

08 HYDROGEN↗
DOE OSTI2025

Snow ALbedo eVOlution (SALVO) Campaign Broadband Albedo from April - June, 2024 in Utqiagivk, AK level a1

A field-portable broadband (285 – 2800 nm) albedometer was used to make spatially distributed albedo measurements on tundra and sea ice surfaces. The albedometer consists of paired upward-looking and downward-looking pyranometers, which were both connected to a data logger. The instrument was mounted approximately 1 m above the surface using a tripod and was placed on a 1.4 m-long boom to minimize the impacts of shading from the operator and to observe surfaces undisturbed by footprints (see Appendix for photos of measurement setup and uncertainty assessment). Albedo measurements were taken parallel to the 200-m albedo lines at 5-m increments (41 measurements) ~1.2 m south of the line. On the operator’s end of the boom, there was a bubble level that was aligned with the bubble level on the upward-looking pyranometer. To take a measurement, the operator first relocated the tripod to the measurement location, then leveled the instrument and held it level for at least twice the pyranometers’ response time (5 or 15 seconds, see below), and finally depressed a trigger on the data logger. The data logger recorded the instantaneous voltage on both pyranometers, the measurement number, and the time. The data logger also converted the voltages to irradiances, and from these computed the ratio (outgoing/incoming) for albedo, which could be checked in the field. The operator recorded in a field notebook the measurement number that corresponded with the locations on the line and any pertinent notes (e.g., invalid measurements). With this setup, a trained operator could measure a 200-m albedo line (41 measurements) in approximately 30 minutes. Measurements were made within 3 hours of solar noon. The data logger had sufficient storage capacity to record all measurements from the campaign, but data were downloaded to a computer after each measurement day.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2026

In situ catalyst activation and regeneration enable energy-efficient high-current CO 2 reduction to ethanol-rich C 2+ mixtures

Electrochemical conversion of dissolved CO 2 in bicarbonate electrolytes, i.e., bicarbonate electrolysis, offers distinct advantages over gas diffusion electrode systems by enabling direct utilization of the CO 2 capture electrolyte while bypassing the energy-intensive CO 2 release step. However, bicarbonate electrolysis faces challenges such as CO 2 mass-transfer limitation, local pH-driven CO 2 depletion, and high cathodic potentials. The higher potential often causes catalyst surface reorganization, leading to a gradual loss of active sites and variations in selectivity during CO 2 reduction. Here, we report a directed, in situ activation and regeneration method that allows precatalysts to equilibrate under dynamic (pulsed) electrolysis conditions. We demonstrate in situ activation of a scalable Cu 2 O/Cu mesh that, under short-width (t = 4 s) pulsed electrolysis, provides stable mixed oxidation states of Cu, favoring the formation of an ethanol-rich crude mixture. The pulsed electrolysis waveform, consisting of six distinct segments, is tuned to form Cu + oxides, which are then reduced to generate local alkaline conditions favoring C–C coupling. This synergistic effect results in FEs of 73% for C2+ products and 39% for ethanol at an applied current density of −150 mA cm −2 and a cathodic potential of −1.45 V (vs. RHE). The overall half-cell energy efficiency is ∼30% for C 2+ products. The in situ Raman experiments confirm the role of pCO 2 R in dynamically regenerating Cu+-containing surface species during pulsed operation, thereby steering selectivity towards C 2+ products. A comprehensive multiscale, multiphysics model is developed to investigate the dynamic behavior of copper surface species (Cu, Cu + , and Cu 2+ ) and local microenvironmental conditions during the pCO 2 R. The results reveal that the coexistence of different copper oxidation states, especially the Cu+ intermediate, is critical in steering selectivity towards multicarbon (C 2+ ) products. The dynamic modulation of surface redox states via tailored pulsing strategies favors C–C coupling pathways by inducing localized alkaline conditions and stabilizing reactive intermediates. This work establishes a predictive modeling platform that links pulse waveform design with mechanistic insights into catalyst state evolution and product selectivity. Overall, this study provides valuable insights into the synergistic effect of in situ activation of pre-catalysts and pulsed electrolysis for higher selectivity towards C 2+ products.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2024

Survey of tungsten gross erosion from main plasma facing components in WEST during a L-mode high fluence campaign

An initial high fluence campaign was performed in WEST, in 2023, on the newly installed actively cooled tungsten divertor composed of ITER-grade monoblocks. The campaign consisted in the repetition of a 60 s long Deuterium L-mode pulse in attached divertor conditions, cumulating over 10000s of plasma exposure. A maximum deuterium fluence of approximately 5⋅1⁢026 m−2 was reached in the outer strike point region, representative of a few high performance ITER pulses. Gross tungsten erosion inferred from visible spectroscopy shows that the most eroded plasma facing component is the inner divertor target with rates ten times larger than on the outer divertor target. The outer midplane tungsten bumpers, located a few centimeters from the plasma, show gross erosion rates two times lower than at the outer divertor. We conclude that the outer midplane bumpers have a negligible contribution to the long range tungsten migration and deposition onto the lower divertor. The cumulated gross erosion rate on the inner divertor translates in an effective gross erosion thickness of about 20μ⁢m, while it is about 2μ⁢m for the outer divertor. Strikingly, these orderings coincide with the thickness of deposits found locally on the divertor: the exposed surfaces of high field side monoblocks are covered with several tens of μ⁢m tungsten deposits, while on the lower field side, few μ⁢m thin tungsten deposits are only found on the magnetically shadowed parts of monoblocks. The strong impact of those deposits on WEST operation, namely perturbation of surface temperature measurement with infra-red thermography, and the emission of flakes causing radiative perturbation of the confined plasma, calls for anticipating similar issues in ITER. In particular, the start of research operation shall consider the definition of a divertor erosion budget in order to anticipate the formation of deleterious deposits.

Fedorczak, N.↗
DOE OSTI2024

Perovskite/Perovskite Tandem Photoelectrodes for Low-Cost Unassisted Photoelectrochemical Water Splitting

In this project, we aim to address the challenges of achieving efficient and cost-effective unassisted photoelectrochemical water splitting using perovskite/perovskite tandem photoelectrodes. We utilized the following unique approaches to advance our innovation. (1) We developed stable and efficient low-E g (1.2 – 1.4 eV) perovskites as the bottom electrodes. We demonstrated approaches to improve the photovoltaic performance and photothermal stability of low-E g Sn-Pb iodide perovskite solar cells. (2) We developed a low-temperature synthesis route to fabricate wide-E g (>1.8 eV) perovskite top electrodes. By combining theoretical and experimental investigation, we found methods to reduce the formation of defects and dislocations in the wide-E g mixed halide perovskites and suppress halide segregation. (3) We developed a robust metal oxide-based interconnecting layer to integrate two perovskite layers into tandem photoelectrodes. Our monolithically integrated tandem devices feature a high V OC of more than 2 V and a high J SC of more than 15 mA/cm 2 . We showed the new design of the tandem photoelectrodes that is critical to the stable operation of tandem photoelectrodes for unassisted PV/PEC water splitting. (4) We demonstrated a water-impermeable barrier of carbon paste/epoxy/metal foil composite, which can prevent photocorrosion and water ingress of perovskite active layers. This surface protection enabled the operation of perovskite photoelectrodes in water and significantly enhanced the long-term stability of our tandem devices. (5) We conducted standardized PEC characterization in collaboration with NREL and reported accurate determination of solar-to-hydrogen conversion efficiencies of perovskite/perovskite tandem photoelectrodes. At the end of the project, we demonstrated perovskite/perovskite tandem photoelectrodes with STH efficiencies of up to 18% and less than 20% efficiency loss after continuous operation for more than 500 hours in water. Our results demonstrate the potential to develop a low-cost, durable, and efficient water-splitting system that meets the DOE 2026 and 2031 cost targets for hydrogen production.

08 HYDROGEN↗
DOE OSTI2025

Enhancing the Survivability of Power Systems With Grid-Edge DERs Against DoS Attacks

Power system survivability, defined as the ability of a system to maintain steady-state functionality under varying operational conditions, reflects its resilience against disturbances. While existing research primarily focuses on physical-layer disturbances, the increasing prevalence of grid-edge DERs, which are primarily used for integrating renewable energy, has significantly expanded the cyber attack surface. As a result, operational disruptions caused by cyber threats are posing significant challenges to system survivability and cannot be overlooked. To fill this gap, we redefine system survivability to incorporate the cyber layer’s status and propose a Distributionally Robust Optimization (DRO) approach to enhance power system survivability against potential cyber-physical threats. In this paper, we first analyze the operational guidelines of systems with a high penetration of DERs under various cyber network conditions and redefine survivability in this context. Next, we focus on the most common cyber threat, Denial-of-Service (DoS) attacks, and develop a corresponding attack model. This model allows for the creation of a kernel-based ambiguity set that captures attack uncertainties using historical data. Finally, we transform the proposed DRO model as a tractable optimization problem, with its solution providing an optimal cyber redundancy plan to enhance system survivability in DoS attack scenarios. Simulation results on the IEEE 13-node and 123-node test feeders demonstrate the effectiveness of our proposed model in improving system survivability. This model can also be expanded to include other types of common attacks and serve as a comprehensive planning tool to improve overall cyber physical survival of the system.

cybersecurity↗
DOE OSTI2026

Unravelling chemical pathways of H 2 on Ga 2 O 3 surfaces with spectro-electrochemistry

This work highlights the capability of coupled spectroscopic and electrochemical techniques to probe dynamic surface processes under realistic operating conditions. By simultaneously employing in situ diffuse reflectance infrared Fourier transform spectroscopy (DRIFTS) and electrochemical impedance spectroscopy (EIS), we elucidate the mechanistic interaction between Ga 2 O 3 and hydrogen under elevated temperatures in a low-oxygen environment. This novel spectro-electrochemical approach allows chemistry to be correlated with the surface charge density of Ga 2 O 3 . Our results reveal a concentration-dependent transition in reaction pathway. At low concentrations, hydrogen reacts with ambient oxygen to form surface hydroxyls. At intermediate concentrations, hydrogen interacts with surface adsorbed oxygen to generate hydroxyl groups along with reducing the surface. Finally, at high H 2 concentrations, hydrogen reduces both hydroxyls and surface oxygen, leading to a highly conductive grain surface. As a result, hydrides form on the reduced Ga 2 O 3 surface. The gained insights are relevant for heterogeneous catalysis and gas sensing.

08 HYDROGEN↗
DOE OSTI2024

WEC-Sim Modeling of Laminar Scientific's Patented Seesaw Wave Energy Converter: Preprint

Laminar Scientific's patented seesaw wave energy converter was modeled in WEC-Sim to predict performance. The device operates by utilizing ocean surface waves to rotate a truss in pitch about a pivot. The pivot is located at the top of two pylons, which are embedded in the seafloor. The seesaw has a float on either end, and the buoyancy forces from each float cause the system to rise or fall with passing waves. Device performance relies upon seesaw length and ocean wavelength creating an antiphase effect. The seesaw truss has an adjustable length intended to achieve this effect. The operation method enforces a narrow band of wavelengths which induce the largest rotational motion from the device. The hydrodynamic analysis of the device was performed using Capytaine, and the results confirmed that the device operates best in a narrow frequency band. Four float-to-float spacing cases and three pylon radii were examined. The hydrodynamic results indicate a match between the model and the physical expectations for the device, and that varying the pylon radii by 0.1-m increments for three instances creates minimal changes in hydrodynamic properties. Power matrices for three float spacing cases of the device were simulated with Joint North Sea Wave Project spectra waves and optimal power take-off damping in WEC-Sim. The maximum average power production for the 15-m spacing case was 14.1 kW with a 5.0-s peak wave period and 4-m significant wave heights. Plots of capture-width ratios indicated that the device performance was linear and confirmed that the device is optimal in a narrow frequency band. The maximum percentage of the available wave power produced by the 15-m device was approximately 16%. Simulations of the device in regular waves were used to produce plots of average power compared to a ratio of float spacing to wavelength. These plots indicate that the power production is maximized at a ratio of 0.5, and further confirm that the device has a narrow frequency response. The device was simulated at an example field location, where the device produced an annual average power rating of 1.6 kW given an average omnidirectional wave climate of 10.3 kW m-1 and an optimal, linearized power take-off model. While the maximum predicted device performance is reliant upon a narrow band of wave frequencies, the conducted analysis provides an opportunity to improve device design prior to prototyping and testing. Modifying the design to respond to a broader frequency range would improve device performance.

numerical modeling↗
DOE OSTI2026

Theoretical background for a fast flow liquid metal divertor experiment

Tokamak Energy has developed a liquid metal experiment featuring a lithium loop that circulates lithium through the HIDRA stellarator at the University of Illinois at Urbana-Champaign. The loop includes a replaceable divertor module, designed to demonstrate fast, steady, open-surface liquid metal flow. The first-generation divertor module was deliberately kept simple—an open-surface chute—to enable modelling validation and establish a benchmark for future design iterations. This paper presents the theoretical foundations of the experimental design. Specifically, we identify module overflooding as the primary challenge and determine the limits of fluid velocity and magnetic field strength necessary to prevent it. The steady-state flow patterns are expected to exhibit relatively slow surface velocities, which do not fully align with the fast-flow concept. Nevertheless, the experiment is designed to achieve controllable, continuous open-surface flow within an operational fusion device, representing a significant step toward the development of liquid metal divertor technology.

Experiment↗
DOE OSTI2025

Particle-based modelling of axisymmetric tandem mirror devices

In this work, we describe the use of a 1D-2V quasi-neutral hybrid electrostatic PIC with Monte-Carlo Coulomb collisions and non-uniform magnetic field to model the parallel transport and confinement in an axisymmetric tandem mirror device. End-plugs, based on simple-mirrors, are positioned at each end of the device and fueled with neutral beams (25 and 100 keV) to produce a sloshing ion population and increase the density of the end-plugs relative to the central cell. Results show the formation of a potential difference barrier between the central cell and the end-plugs. This potential confines a large fraction of the low energy thermal ions in the central cell which would otherwise be lost in a simple mirror, demonstrating the advantage of the beam-driven tandem mirror configuration relative to simple mirrors. In addition, we explore the effect of end-plug electron temperature on the confinement time of the device and compare it with theoretical estimates. Finally, we discuss the limitations of the code in its present form and describe the next logical steps to improve its predictive capability such as a fully nonlinear Fokker–Planck collision operator, multiply nested flux surface solutions and modeling the exhaust region up to the wall.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

Solution-Processed Temperature-Adaptive Radiative Paint as a Thermal Imaging Sensitizer

Thermography detects mid-infrared radiation from surfaces based on the Stefan-Boltzmann law, mapping surface temperatures and potentially revealing subsurface thermal activity. Recent developments offer an alternative strategy to traditional camera-based improvements: a thermal imaging sensitizer (TIS) coating whose emissivity increases sharply with local temperature, amplifying small thermal variations into high-contrast signals. Existing TIS structures are nanofabricated solid membranes and face significant challenges in fabrication complexity. Here, in this study, we present a solution-processed, liquid form of TIS, termed temperature-adaptive radiative paint (TARP), to address these limitations. TARP offers drastically reduced fabrication costs, scalability to large areas, applicability to curved surfaces, and an extended operating temperature range, while maintaining the function of temperature amplification. Application of TARP enhances small temperature contrast by more than 3 times, substantially improving ambient thermography and enabling broader applications such as detection of structural defects and hot spots in electronic components.

paint↗
DOE OSTI2025

Characterization of electron spin polarization from positive electron affinity GaAs photocathodes

Negative Electron Affinity (NEA) GaAs photocathodes are widely used to generate spin-polarized electron beams, typically achieving Electron Spin Polarizations (ESPs) in the range of 35%–40%. However, when operated in a Positive Electron Affinity (PEA) state, where a potential barrier inhibits low-energy electrons from escaping into vacuum, measured ESPs can exceed 50%. This effect can occur naturally during photocathode operation, as the NEA surface activation layers can degrade easily over time, increasing the electron affinity. In this work, we investigate and characterize the behavior of enhanced ESPs under PEA conditions. We present experimental measurements of ESP using a retarding-field Mott polarimeter on GaAs photocathodes with controlled and varying electron affinities. These results are complemented by theoretical explanations considering the material band structure, the light excitation profile, and spin depolarization mechanisms.

Electronic band structure↗
DOE OSTI2025

Broadband optical modulation and control at millikelvin temperatures

A universal experimental challenge in studying radiation effects on cryogenic devices is precisely and accurately characterizing the position-dependent device response near the energy detection threshold. Here, we have developed a compact cryogenic optical beam steering system that can be used to generate O (μs) pulses of small numbers of photons over the energy range of 1.3–3.4 eV at room temperature, and deliver those photons via fiber optic to any specified location on the surface of a detector operating at cryogenic temperatures. This new system will allow for robust calibration of any photon-sensitive detector, including superconducting devices. The system can be used efficiently to explore the physics of target materials, quantify the position sensitivity of different sensor designs, measure phonon transport, and study the effects of quasiparticle poisoning on detector operation. We describe the design of this pulsed calibration method and present first results obtained with a second-generation system operated at room temperature and sub-Kelvin temperatures.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2025

Surface deformation coupled with self-organized pattern on a liquid anode of 1 atm DC glow discharge

Intricate, self-organized plasma structures observed above the surface of a liquid anode of atmospheric DC glow discharge were found to give rise to coherent, organized surface deformation and mechanical wave formation at the plasma–liquid interface. This new phenomenon indicates that the liquid is closely coupled to the plasma by the anode sheath’s electrohydrodynamic (EHD) force. A scientific question then arises: Do surface perturbations, coupled with the nonuniform surface charge distribution, enhance the electric field and induce self-organization? Using the reflective background-oriented schlieren technique, the liquid surface profile under the plasma pattern was measured for the first time. The results show that surface distortions are driven by the repulsive Coulomb force of nonuniform net-negative surface charge acted by the anode sheath field. The impacts of various operating parameters on the patterns and surface waves were examined, revealing the significance of gas heating and liquid charge relaxation time in the pattern formation mechanism. Time-resolved dynamics of a pulsed DC discharge indicated that the surface deformation only appeared after the establishment of plasma patterns. Statistically, the surface wave under the plasma has high wave numbers (8000–16000 m −1 ) and small amplitudes ($<$ 10 µm), generally found in the capillary wave regime. Yet the motion of surface deformations is in tandem with the plasma pattern and exhibits a nondispersive nature of constant phase velocity (0.1–0.4 m s −1 ), suggesting the dominant role of EHD force over the surface tension in the observed surface wave. These results indicate that the deformed liquid surface is driven by the EHD force of the plasma sheath. Although they share a similar geometry, the deformation and wave dynamics of the liquid surface do not stimulate a plasma pattern. Importantly, the complex EHD coupling in the plasma–liquid system raises awareness and new challenges for plasma control engineering, and the quantitative characteristics of the nondispersive surface wave are informative for advancing relevant theory and modeling.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2026

Decoupling plasma, catalyst, and gaseous mechanisms for non-oxidative methane conversion

Direct non-oxidative methane (CH 4 ) conversion to value-added hydrogen (H 2 ) and C 2 products remains hindered by fundamental catalytic scaling constraints and rapid surface deactivation at elevated temperatures. Plasma-enabled catalysis offers a promising route to overcome the thermodynamic and kinetic barriers of direct non-oxidative methane upgrading at mild conditions, yet control over C–C product selectivity and catalyst stability remains elusive. Here, we establish a unified mechanistic framework including Langmuir–Hinshelwood (L–H) and Langmuir–Rideal (L–R) mechanisms that disentangles the roles of plasma excitation (including vibrationally activated species and radicals), surface temperature (T sur ), and catalyst binding energy in steering CH 4 conversion to H 2 and C 2 hydrocarbons. Through a combination of density functional theory (DFT) informed microkinetic modeling, in situ and ex situ surface characterization, and product quantification under dielectric barrier discharge conditions, we show that vibrationally excited CH 4 lowers activation barriers selectively for dissociative chemisorption, enabling surface activation across a wide range of transition metal catalysts at low thermal energy input. We find that once CH 4 is dissociatively chemisorbed, the branching between C 2 H 2 , C 2 H 4 , and C 2 H 6 is governed by surface properties (carbon binding energy, T sur , etc), regardless of plasma excitation. The DFT informed microkinetic model decouples the effects of molecular activation from surface properties and indentifies operating windows that maximize target yields while suppressing carbon accumulation and subsequent catalytic inactivation. Experiments on polycrystalline Cu/Al 2 O 3 , Ni/Al 2 O 3 , and Pt/Al 2 O 3 validate these predictions, revealing catalyst-dependent branching toward ethane or ethylene and distinct deactivation profiles. We unify these trends into a generalized three-dimensional plasma-thermal-catalytic design space, from which reduced descriptors such as T vib /T sur in the limit of vibrationally excited L–H pathways emerge as predictive metrics. These results enable rational tuning of methane conversion pathways and unlock selective C 2 formation using earth-abundant metals under mild plasma conditions.

catalyst inactivation↗
DOE OSTI2024

Performance and Durability of Heavy-Duty Fuel Cell Systems with an Advanced Ordered Intermetallic ORR Alloy Catalyst and Novel Support

Ordered PtCo intermetallic (OIM) catalyst (L1 0 -PtCo/C) is a promising candidate as the oxygen reduction reaction (ORR) catalyst in hybrid fuel cell systems (FCS) for class-8 heavy duty (HD) trucks. Compared to a baseline annealed Pt on high surface area carbon (a-Pt/HSC) catalyst, its mass activity (MA) is 71% higher initially and 144% higher after 90,000 potential cycles in an accelerated stress test (AST). Analysis of the AST data indicates that the ORR kinetic constants do not change with aging and the degradation in the OIM catalyst activity is linearly proportional to the loss in the electrochemically active surface area (ECSA). Several operational strategies are investigated to mitigate catalyst degradation and achieve 25,000-h electrode lifetime and 2.5 kW g −1 Pt utilization on a HD truck duty cycle including load sharing with the hybrid battery, regulating the radiator fan power to maintain the coolant temperature close to 60 °C, clipping the maximum cell voltage below 850 mV, limiting the ECSA loss to 55%, and oversizing the active area of the membrane electrode assemblies by 20%. Drive cycle simulations indicate that the lifetime average voltage degradation rate is about 1.8 μV h −1 and the integrated stack and FCS drive cycle efficiencies decrease by 3.5 to 3.9%.

25 ENERGY STORAGE↗
DOE OSTI2024

The VTK-m User's Guide (V. 2.2)

High-performance computing relies on ever finer threading. Advances in processor technology include ever greater numbers of cores, hyperthreading, accelerators with integrated blocks of cores, and special vectorized instructions, all of which require more software parallelism to achieve peak performance. Traditional visualization solutions cannot support this extreme level of concurrency. Extreme scale systems require a new programming model and a fundamental change in how we design algorithms. To address these issues we created VTK-m: the visualization toolkit for multi-/many-core architectures. VTK-m supports a number of algorithms and the ability to design further algorithms through a top-down design with an emphasis on extreme parallelism. VTK-m also provides support for finding and building links across topologies, making it possible to perform operations that determine manifold surfaces, interpolate generated values, and find adjacencies. Although VTK-m provides a simplified high-level interface for programming, its template-based code removes the overhead of abstraction.

97 MATHEMATICS AND COMPUTING↗