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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 235 records · Page 13
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Considerations for using Privacy Preserving Machine Learning Techniques for Safeguards

In international nuclear safeguards, the International Atomic Energy Agency (IAEA) is tasked with inspecting and verifying nuclear facilities and their activities. Data analytics and machine learning to support inspections require large amounts of data that nuclear facility operators may consider proprietary or sensitive, so the IAEA may not have full access. Allowing computation over private data without compromising its security therefore has value for safeguards inspections and analysis. Privacy-preserving machine learning (PPML) consists of security-focused techniques that allow data analytics and machine learning algorithms to run on sensitive data without revealing it. This includes ideas like homomorphic encryption (HE), secure multiparty computation (SMPC), and secure enclaves. HE allows algorithms and mathematical operations to be conducted directly on the encrypted data instead of first decrypting it. With SMPC, multiple entities collaboratively compute over distributed data such that no party is able to directly view any others’ original data. Secure enclaves allow computation to take place in a separate and heavily blocked-off section of a CPU. Techniques like these allow for several potential use cases in which the security of data is essential. With SMPC, machine learning models can be trained over the input data from multiple entities, resulting in a model that all users can benefit from without leaking the input data from any particular entity. With SMPC or a zero-knowledge proof (ZKP), an algorithm returning some single answer or truth value can be run on someone else’s data without ever needing to see that data, potentially allowing for verification or proof of some underlying question. HE can allow for outsourcing computation on data to a hostile or untrusted environment. Although most of the research in this field resides within the health and financial domains, tools from PPML may have similar applications in nuclear safeguards. Allowing the IAEA to compute over proprietary information, such as process models and raw sensor data using PPML techniques, provides the baseline for running complex analytics without needing direct unencrypted access to the underlying data, maintaining its privacy. Important limitations to consider for these techniques include the efficiency and level of security required. The security of HE and SMPC come at the cost of speed—the significant amount of overhead means that algorithms implemented in these protocols and encryption schemes are slower than when run on plaintext. Additionally, several important parameters determine what techniques or protocols are used based on the security requirements. SMPC protocols may need to be selected for resistance against a party that attempts to deviate from the protocol to distort the result or gain access to additional information, and a protocol secure against these attacks may further increase the overhead of the algorithm.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Protein-activated atomic layer deposition for robust crude-oil-repellent hierarchical nano-armored membranes

Atomic layer deposition (ALD) offers unique capabilities to fabricate atomically engineered porous materials with precise pore tuning and multi-functionalization for diverse applications like advanced membrane separations towards sustainable energy-water systems. However, current ALD technique is inhibited on most non-polar polymeric membranes due to lack of accessible nucleation sites. Here, in this study, we report a facile method to efficiently promote ALD coating on hydrophobic surface of polymeric membranes via novel protein activation/sensitization. As a proof of concept, TiO 2 ALD-coated membranes activated by bovine serum albumin exhibit remarkable superhydrophilicity, ultralow underwater crude oil adhesion, and robust tolerance to rigorous environments including acid, alkali, saline, and ethanol. Most importantly, excellent cyclable crude oil-in-water emulsion separation performance can be achieved. The mechanism for activation/sensitization is rooted in reactivity for a particular set of amino acids. Furthermore, the universality of protein-sensitized ALD is demonstrated using common egg white, promising numerous potential usages in biomedical engineering, environmental remediation, lowcarbon manufacturing, catalysis, and beyond.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Preparation of 241 Am/ 243 Am gravimetric mixtures and development of Am isotopic and assay measurement techniques using thermal ionization mass spectrometry

Three gravimetric mixtures with 241 Am/ 243 Am isotope ratios at nominally 1:1, 20:1, and 200:1 were prepared for calibration of the thermal ionization mass spectrometer instruments used for americium isotopic and assay measurements by isotope dilution mass spectrometry. The total evaporation analytical technique was developed for high-accuracy and precision measurements of Am isotopic ratios. The technique was also applied to Am assay measurements using isotope dilution mass spectrometry. The Am isotope ratio and assay measurement techniques were utilized to characterize batches of americium oxide separated from plutonium materials in storage at Los Alamos National Laboratory in support of commercial use of 241 AmO 2 .

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

Nonlinear mode coupling and energetics of driven magnetized shear-flow turbulence

To comprehensively understand the saturation of two-dimensional (2D) magnetized Kelvin–Helmholtz-instability-driven turbulence, energy transfer analysis is extended from the traditional interaction between scales to include eigenmode interactions, by using the nonlinear couplings of linear eigenmodes of the ideal instability. While both kinetic and magnetic energies cascade to small scales, a significant fraction of turbulent energy deposited by unstable modes in the fluctuation spectrum is shown to be re-routed to the conjugate-stable modes at the instability scale. They remove energy from the forward cascade at its inception. The remaining cascading energy flux is shown to attenuate exponentially at a small scale, dictated by the large-scale stable modes. Guided by a widely used instability-saturation assumption, a general quasi-linear model of instability is tested by retaining all nonlinear interactions except those that couple to the large-scale stable modes. These complex interactions are analytically removed from the magnetohydrodynamic equations using a novel technique. Observations are an explosive large-scale vortex separation instead of the well-known merger of 2D, a dramatic enhancement in turbulence level and spectral energy fluxes, and a reduced small-scale dissipation length scale. These show the critical role of the stable modes in instability saturation. Possible reduced-order turbulence models are proposed for fusion and astrophysical plasmas, based on eigenmode-expanded energy transfer analyses.

Physics↗
DOE OSTI2023

Quantum interferometry and pathway selectivity in the nonlinear response of photosynthetic excitons

We propose a time–frequency resolved spectroscopic technique which employs nonlinear interferometers to study exciton–exciton scattering in molecular aggregates. A higher degree of control over the contributing Liouville pathways is obtained as compared to classical light. We show how the nonlinear response can be isolated from the orders-of-magnitude stronger linear background by either phase matching or polarization filtering. Both arise due to averaging the signal over a large number of noninteracting, randomly oriented molecules. We apply our technique to the Frenkel exciton model which excludes charge separation for the photosystem II reaction center. We show how the sum of the entangled photon frequencies can be used to select two-exciton resonances, while their delay times reveal the single-exciton levels involved in the optical process.

Science & Technology - Other Topics↗
DOE OSTI2020

How accurate is a machine learning-based wind speed extrapolation under a round-robin approach?

As the size of commercial wind turbines keeps increasing, having accurate ways to vertically extrapolate wind speed is essential to obtain a precise characterization of the wind resource for wind energy production. Recently, machine learning has been proposed and applied to extrapolate wind speed to hub heights. However, previous studies trained and tested the machine learning methods at the same site, giving them an unfair advantage over the conventional extrapolation techniques, which are instead more universal. Here, we use data from four sites in Oklahoma to test a round-robin validation approach for machine learning, under which we train a random forest at a site, and test it at a different site, where the model has no prior knowledge of the wind resource. We quantify how the accuracy of this technique varies with distance from the training site, and we find that it outperforms conventional techniques for wind extrapolation at all the considered spatial separations. We then assess how the accuracy of the machine-learning based approach varies when it is used to predict wind speed in a wind farm far wake. Finally, we explore as case study the performance of the random forest in extrapolating winds during a low-level jet event.

17 WIND ENERGY↗
DOE OSTI2022

Atom Interferometry with Floquet Atom Optics

Floquet engineering offers a compelling approach for designing the time evolution of periodically driven systems. We implement a periodic atom-light coupling to realize Floquet atom optics on the strontium 1 S 0 - 3 P 1 transition. These atom optics reach pulse efficiencies above 99.4% over a wide range of frequency offsets between light and atomic resonance, even under strong driving where this detuning is on the order of the Rabi frequency. Moreover, we use Floquet atom optics to compensate for differential Doppler shifts in large momentum transfer atom interferometers and achieve state-of-the-art momentum separation in excess of 400 ℏ⁢k. In conclusion, this technique can be applied to any two-level system at arbitrary coupling strength, with broad application in coherent quantum control.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2024

Enabling Ultralow Volume Analysis with a High-Resolution Ion Mobility Mass Spectrometry Platform

Of all the molecules thought to exist in the universe, it is estimated that researchers only know the chemical structures of 5% of them. Identifying the chemical structures of the remaining 95% has proven extremely challenging because many molecules exhibit low abundance, are contained in small volumes (e.g., <10 nL), do not readily ionize, exhibit similar structures to other molecules, etc. No single analytical technique exists to definitively identify the structure of an unknown molecule, and thus multiple different molecular measurements are typically made (i.e., multi-modal approach). Ion mobility (IMS) and mass spectrometry (MS) are two key tools that researchers use to determine the chemical structures of unknown molecules, and recently high-resolution and ultrahigh resolution IMS-MS instruments have provided greater confidence than ever before. However, HR-IMS-MS instruments typically exhibit low ion utilization efficiency, meaning they require large amounts of sample for an analysis (e.g., >10 µL). Unfortunately, this limitation prohibits the analysis of small volume samples where many unknown molecules exist. Described herein are the efforts made to enable the analysis of ultralow volumes with an HR-IMS-MS platform. A new scanning technique, termed a ‘stuttered traveling wave scan’, was developed as a replacement for the dual-gated scanning technique and works by halting the traveling waves after allowing ions to separate and then repeatedly restarting and stopping the traveling waves to incrementally move ions from the SLIM to the Orbitrap. Ions were stored inside the SLIM while the TWs were stopped, allowing the Orbitrap to perform high-resolution mass analysis. When the Orbitrap was ready, the TWs were restarted for short periods of time (<10 ms) to move ions from the SLIM to the Orbitrap. It was discovered that lower TW amplitudes and speeds than used during IMS separation were required to produce IMS peaks with the highest signal intensities and best resolving powers. The stuttered TW scan was found to produce similar resolutions and signal intensities compared to the dual-gated scanning technique. A new IMS design possessing an intersecting ‘tee’ with a reversible traveling wave was also developed to improve ion utilization efficiency during cyclic operation, which is necessary when only a single IMS spectrum can be acquired, such as when analyzing ultralow volume samples. The new capabilities described in this report lay the groundwork for acquiring HR-IMS-MS spectra of ultralow volume biological samples, such as single cells, where HR-IMS-MS can help elucidate the structures of unknown compounds.

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

Preliminary Assessment of Alternate Chlorination Process

Recent advancements towards low-temperature chlorination with disulfur dichloride and thionyl chloride of Al-clad used nuclear fuel (UNF) have been summarized for work control technical expertise development at the Savannah River National Laboratory (SRNL). This technique aims to provide a more effective method for separating U from cladding or alloying elements. A preliminary assessment determined needs for unit operation developments in regard to reaction kinetics and reagent quality. Furthermore, an analysis of an Uruguay fuel plate (U-Al x fuel meat), with low-enriched uranium (LEU) and located at SRNL, was a potential candidate for future experiments with lightly irradiated (0.08% burnup) fuel to test for chlorination. Future proposed work would involve kinetic studies, reagent quality assessments, and the machining of the fuel plate down to an appropriate bench-scale size and testing the effectiveness of low-temperature chlorination on the fuel plate.

Chlorination↗
DOE OSTI2023

Optimized purification methods for metallic contaminant removal from directly recycled Li-ion battery cathodes

Metallic contaminants pose a significant challenge to the viability of directly recycling Li-ion batteries. To date, few strategies exist to selectively remove metallic impurities from mixtures of shredded end-of-life material (black mass; BM) without concurrently damaging the structure and electrochemical performance of the target active material. We herein present tailored methods to selectively ionize two major contaminants—Al and Cu—while retaining a representative cathode (LiNi 0.33 Mn 0.33 Co 0.33 O 2 ; NMC-111) intact. This BM purification process is conducted at moderate temperatures in a KOH-based solution matrix. We rationally evaluate approaches to increase both the kinetic corrosion rate and the thermodynamic solubility of Al 0 and Cu 0 , and evaluate the impact of these treatment conditions on the structure, chemistry, and electrochemical performance of NMC. Specifically, we explore the impacts of chloride-based salts, a strong chelating agent, elevated temperature, and sonication on the rate and extent of contaminant corrosion, while concurrently evaluating the effects on NMC. The reported BM purification process is then demonstrated on samples of “simulated BM” containing a practically relevant 1 wt% concentration of Al or Cu. Increasing the kinetic energy of the purifying solution matrix through elevated temperature and sonication accelerates the corrosion of metallic Al and Cu, such that ∼100% corrosion of 75 μm Al and Cu particles is achieved within 2.5 hr. Further, we determine that effective mass transport of ionized species critically impacts the efficacy of Cu corrosion, and that saturated Cl – hinders rather than accelerates Cu corrosion by increasing solution viscosity and introducing competitive pathways for Cu surface passivation. The purification conditions do not induce bulk structural damage to NMC, and electrochemical capacity is maintained in half-cell format. Testing in full cells suggests that a limited quantity of residual surface species are present after treatment, which initially disrupt electrochemical behavior at the graphite anode but are subsequently consumed. Process demonstration on simulated BM suggests that contaminated samples—which prior to treatment show catastrophic electrochemical performance—can be recovered to pristine electrochemical capacity. The reported BM purification method offers a compelling and commercially viable solution to address contamination, particularly in the “fine” fraction of BM where contaminant sizes are on the same order of magnitude as NMC and where traditional separation approaches are unfeasible. Thus, this optimized BM purification technique offers a pathway towards viable direct recycling of BM feedstocks that would otherwise be unusable.

25 ENERGY STORAGE↗
DOE OSTI2024

Understanding Europium and Terbium Speciation and Ion Pairing in Carbonate Complexes Using Advanced Spectroscopy Techniques

Lanthanide (Ln) elements are critical materials that are typically extracted/mined together. Their separation by solvent extraction from acidic media is well known; however, there are few studies in basic media with carbonate anions. We investigated the complexation of Eu(III) and Tb(III) carbonates as solids and solutions in alkaline K2CO3, wherein we sought to access a Tb(IV) carbonate complex through ozonolysis. L3-edge XANES of Eu and Tb carbonate solids, colorless solutions, and a red-hued Tb solution (obtained by ozonolysis) all showed Ln(III) cations. The absence of evidence for a Tb(IV) complex was confirmed through XAS and EPR analyses, despite the solution exhibiting a deep red color. For solids and solutions, EXAFS results indicate molecular Ln(III)-carbonato anions. In terms of the Eu(III) carbonate coordination number, the coordination does not change upon dissolution of the solid sample. Furthermore, EXAFS for the solutions revealed evidence for the association of potassium cations with the Ln(III)-carbonato anions. Furthermore, this direct observation of contact ion pairing by EXAFS at room temperature is rare. The insights into Ln(III) carbonate complexation and solution speciation afforded by XANES-EXAFS, FT-IR, and EPR provides perspectives that serve as benchmarks for future computational and experimental efforts focused on caustic-side solvent extraction of Ln(III) ions.

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

Correlative SIP-FISH-Raman-SEM-NanoSIMS links identity, morphology, biochemistry, and physiology of environmental microbes

Microscopic and spectroscopic techniques are commonly applied to study microbial cells but are typically used on separate samples, resulting in population-level datasets that are integrated across different cells with little spatial resolution. To address this shortcoming, we developed a workflow that correlates several microscopic and spectroscopic techniques to generate an in depth analysis of individual cells. By combining stable isotope probing (SIP), fluorescence in situ hybridization (FISH), scanning electron microscopy (SEM), confocal Raman microspectroscopy (Raman), and nano-scale secondary ion mass spectrometry (NanoSIMS), we illustrate how individual cells can be thoroughly interrogated to obtain information about their taxonomic identity, structure, physiology, and metabolic activity. Analysis of an artificial community demonstrated that our correlative approach was able to resolve the activity of single cells using heavy water SIP in conjunction with Raman and/or NanoSIMS and establish their taxonomy and morphology using FISH and SEM. We then applied this workflow to a sample of yet uncultured multicellular magnetotactic bacteria. In addition to establishing their identity and activity, backscatter electron microscopy (BSE), NanoSIMS, and energy-dispersive X-ray spectroscopy (EDS) were employed to characterize the magnetosomes within the cells. By integrating these techniques, we demonstrate a cohesive approach to thoroughly study environmental microbes at single cell resolution.

Schaible, George↗
DOE OSTI2023

Influencing Bonding Interactions of the Neptunyl (V, VI) Cations with Electron-donating and -withdrawing Groups

Neptunium makes up the largest percentage of minor actinides found in spent nuclear fuel, yet separations of this element have proven difficult due to its rich redox chemistry. Developing new reprocessing techniques should rely on understanding how to control the Np oxidation state and its interactions with different ligands. Designing new ligands for separations requires understanding how to properly tune a system toward a desired trait through functionalization. Emerging technologies for minor actinide separations focus on ligands containing carboxylate or pyridine functional groups, which are desirable due to their high degree of functionalization. So we use DFT calculations to study the interactions of carboxylate and polypyridine ligands with the neptunyl cation [Np(V/VI)O2] +/2+ . A systematic study is performed by varying the electronic properties of the carboxylate and polypyridine ligands through the inclusion of different electron-withdrawing and electron-donating R groups. We focus on how these groups can affect geometric properties, electronic structure, and bonding characterization as a function of the metal oxidation state and ligand character and discuss how these factors can play a role in neptunium ligand design principles.

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

Developing aqueous solubilizing agents as an alternative to solvent extraction

Here, advancing separations science is important for the entire field of chemistry. One partitioning technique that would benefit from improvement is solvent extraction. Despite its effective and widespread use, the method suffers from some problems: generation of flammable organic waste, lengthy process times, and safety concerns associated with contacting organic solvents with acidic aqueous solutions. We developed an alternative separation method inspired by solvent extraction that side-steps those issues. Toward this end, we identified that the functionality of an extractant (an agent used in solvent extraction to pull analytes from the aqueous phase into the organic phase) would change if it was modified for water solubility. In this alternative scenario, the extractant transforms into an “aqueous solubilizing agent.” We discovered that adding this aqueous solubilizing agent alongside a precipitating agent caused the contaminants to precipitate, but not the analyte. This separation concept was demonstrated within the bounds of one of the most difficult partitioning problems, separating minor actinides (Am 3+ ) from lanthanides (Ln 3+ ). We discovered that the (HSO 3 Ph) 4 BTP (aq) aqueous solubilizing agent prevented Am 3+ (aq) from precipitating with Ln 3+ (aq) when f-element precipitating agents (NaF (aq) or HF (aq) ) were added. This separation boasts impressive Am 3+ (aq) recovery yield (90 ± 2 %), near quantitative Ln 3+ (aq) removal, and high separation factors [>3000, Am 3+ (aq) vs. Nd 3+ (aq) ]. It seems likely – given the large number of candidate extractants that could be modified for aqueous solubility and the numerous combinations of existing solubilizing and precipitating agents – that this alternative approach could be used broadly in place of solvent extraction and solve other important separation problems.

(HSO3Ph)4BTP(aq)↗
DOE OSTI2026

Paramagnetic Properties of Rare Earth Hydroxides, Oxalates, and Dibutyl Phosphates

Magnetic separations present a promising approach for the recovery of rare earth (RE) elements from mixtures, particularly when coupled with precipitation techniques that enhance process efficiency and selectivity. However, the magnetic properties of RE materials typically encountered in the separations industry are not well characterized. Here, we synthesized a series of RE materials, combining four REs – praseodymium (Pr), neodymium (Nd), terbium (Tb), and dysprosium (Dy) – with three precipitating agents – hydroxide, oxalate, and dibutyl phosphate. All materials showed typical paramagnetic behavior consistent with the Curie-Weiss law. At low temperatures, we observed weak antiferromagnetic correlations, which were more pronounced for the lighter RE ions (Pr, Nd). Particle morphology and crystallography analyses showed that the dibutyl phosphates were well-ordered metal-organic frameworks, the oxalates were crystals with mixed hydration states, and the hydroxides were mostly amorphous. Regardless of the degree of crystallinity, the effective magnetic moments aligned well with theoretical estimates based on Hund's rules and 4f electron configurations of the corresponding RE ions. For a given RE ion, the magnetophoretic response was thus determined by the size of the particles rather than specific metal-ligand coupling, with dibutyl phosphate promoting the formation of larger crystals that were the most strongly attracted to magnets.

Wang, Qingpu (ORCID:0000000156041571)↗
DOE OSTI2025

Wafer-Free Crystalline Silicon Solar Cells (CRADA Final Report)

This CRADA project, based on the DOE Solar Energy Technologies Office (SETO) Solar Prize Voucher program, helped Leap Photovoltaics to develop methodologies to immobilize Si particles by permanently attaching them to an Al-coated substrate and thereby forming carrier-selective electrical contacts to the Si particles. The bigger goal was to help Leap Photovoltaics develop these immobilized and contacted particle arrays into relatively efficient, inexpensive, and industrially relevant solar cells. By using Si particles instead of wafers in a solar cell absorber layer, one can avoid costs associated with growing monocrystalline Si ingots, then diamond-sawing them into wafers, then processing wafers into cells – a mainstream practice in today's high-efficiency Si cell and module technology. Monocrystalline or polycrystalline Si particles can be obtained in various ways: for example, Si kerf from wafer sawing is monocrystalline; recycled Si cell wafers can be ball-milled into particles; particles can be grown using various gas-phase techniques (mostly from SiH4). These Si particles can be assembled onto a substrate and serve as an absorber layer for the solar cell, absorbing photons to generate photocarriers. The challenge with this technique is to collect photocarriers from individual Si particles, with separation of photogenerated electrons to the negative cell’s electrode and positive photogenerated holes to the positive electrode. Therefore, each particle must have two isolated, carrier-selective contacts: one for electrons and one for holes. Plus, particles need to be immobilized onto a solid substrate. The goal of this work was focused on the immobilization of Si particles and creating hole-selective contact to them at the same time, using industrially relevant Si photovoltaic (PV) cell technology: screen printing of Al back-surface field electrodes. This is used in the mainstream Propane Education and Research Council (PERC) technology for hole-collecting contacts at the back of the cell. The work performed at NREL consisted of screen printing of Al metal paste on substrates, spreading Si particles onto it, and thermally processing the structures to form hole-collecting contacts. The final structures were investigated by scanning electron microscopy (SEM) after focused ion beam (FIB) cross-sectioning and polishing. The work was done jointly by NREL staff and Leap Photovoltaics (Leap PV) employees stationed at NREL. The samples were then taken to Leap PV for further processing. Training the Leap PV employee on various NREL techniques (laser cutting, screen printing, thermal processing, characterization) was part of the scope.

14 SOLAR ENERGY↗
DOE OSTI2020

On electrolyte wetting through lithium-ion battery separators

The wettability by electrolyte is a critical characteristic of lithium-ion battery separators since electrolyte absorption is essential for ionic transport. Most importantly, fast absorption of electrolyte facilitates the electrolyte filling and wetting processes during the battery assembly. However, theoretical or experimental investigation of separator wettability has rarely been reported, and thus very little quantitative data can be found in the existing literature. Here, innovative experimental techniques and associated analytical models have been developed to fully characterize the electrolyte wetting behavior through battery separators. Both in-plane and through-plane imbibition processes are investigated. The results show that, electrolyte needs significantly longer time to wet separators than it does for commercial electrodes. The wetting rate depends significantly on the pore size, porosity, and micro-structure of the separators. Electrolyte in-plane wetting is only observed in the separators manufactured by the wet process. Increasing separator porosity and pore size is shown to be substantially increasing through-plane wetting rate. In addition, ceramic coating is shown to increase in-plane wetting but retard through-plane wetting.

25 ENERGY STORAGE↗
DOE OSTI2020

Space-energy separated representations for multigroup neutron diffusion using proper generalized decompositions

Proper generalized decomposition (PGD) can be described as a numerical extension of separation of variables and thus be employed as a solution technique for multi-dimensional problems, where dimensions should be understood in the phase-space context of the governing law at hand. This paper presents a PGD approach on efficiently solving problems involving multigroup neutron diffusion. Two PGD approaches are described: space-only and space-energy decompositions. Numerical results include 2-D and 3-D examples with two-, seven-, and 145-group structures. The PGD solutions are compared against traditional multi-dimensional finite element discretization. For few-group problems, both PGD approaches prove effective for mildly heterogeneous geometries, but showed reduced performance with increasing heterogeneity. The space-energy representation was found to be slower than the space-only approach for two-group problems, but proved more effective for seven-group problems. For even larger numbers of groups, the space-energy PGD decomposition was very effective at reducing the computational time.

42 ENGINEERING↗