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At least 415 records · Page 23
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

What Is the Limit of Quantification for the Minor Phase in Time-of-Flight Neutron Diffraction? A Case Study on Fe and Ni Powder Mixtures at VULCAN

A phase present in small quantities within materials may not simply serve as a secondary component; it can play a crucial role in determining the integrity, properties, and performance of the material. These minor but important phases usually draw attention in material design and processing for fundamental understanding as well as material quality control. Accurately quantifying a minor phase amid a majority phase, especially at extremely low fractions, remains a challenging task. Time-of-flight neutron diffraction, coupled with advanced pattern analysis techniques like Rietveld refinement, is a powerful tool for crystal structure identification and phase quantification. The deep penetrating capability of neutrons enables the detection and quantification of trace phases within materials. In this study, the quantification limits of time-of-flight neutron diffraction were explored using the VULCAN diffractometer at the Spallation Neutron Source, using Fe–Ni powder mixtures as a sample system. By comparing the refinement results to the known weighed values, it was determined that the reliable quantification of a minor Ni phase is achievable down to about 0.1 wt% while a Ni fraction as low as 0.02 wt% is difficult to trace. Effective control of the refinement parameters, especially the profile function parameters, are found to significantly influence the convergence of fittings and the accuracy of phase quantification.

Rietveld refinement↗
DOE OSTI2020

The arbitrary‐order virtual element method for linear elastodynamics models: convergence, stability and dispersion‐dissipation analysis

Abstract We design the conforming virtual element method for the numerical approximation of the two‐dimensional elastodynamics problem. We prove stability and convergence of the semidiscrete approximation and derive optimal error estimates under h ‐ and p ‐refinement in both the energy and the L 2 norms. The performance of the proposed virtual element method is assessed on a set of different computational meshes, including nonconvex cells up to order four in the h ‐refinement setting. Exponential convergence is also experimentally observed under p ‐refinement. Finally, we present a dispersion‐dissipation analysis for both the semidiscrete and fully discrete schemes, showing that polygonal meshes behave as classical simplicial/quadrilateral grids in terms of dispersion‐dissipation properties.

Antonietti, Paola F.↗
DOE OSTI2025

Modeling Protein–Protein and Protein–Ligand Interactions by the ClusPro Team in CASP16

ABSTRACT In the CASP16 experiment, our team employed hybrid computational strategies to predict both protein–protein and protein–ligand complex structures. For protein–protein docking, we combined physics‐based sampling—using ClusPro FFT docking and molecular dynamics—with AlphaFold (AF)‐based sampling, followed by AF‐based refinement. Our method produced numerous high‐accuracy complex models, including cases where AF alone failed, underscoring the critical role of physics‐based sampling alongside deep learning‐based refinement. For protein–ligand docking, we integrated the ClusPro LigTBM template‐based approach with a machine learning‐based confidence model for rescoring. The method preserves conserved interaction fragments derived from homologous complexes, followed by local resampling using physics‐based sampling and a diffusion model. Our template‐based strategy achieved a mean lDDT‐PLI of 0.69 across 233 targets, which was highly competitive. These results demonstrate that combining physics‐based modeling with AI‐driven refinement can significantly enhance the accuracy of both protein–protein and protein–ligand structure predictions.

Ashizawa, Ryota [Department of Applied Mathematics↗
DOE OSTI2024

Influence of neutron cross-section resonances on organ/tissue equivalent and effective dose coefficients for the ICRP voxel phantoms

The materials which compose the ICRP Voxel phantoms used in the computation of conversion coefficients involve neutron interaction cross-sections that have resonances at specific energies. Depending on the energy bin structure used in the computations, these cross-section resonances may occur at energies that fall between energies at which dose coefficients are computed, thus their effects may not be completely accounted for in the reported coefficients. Here In this study, a highly refined energy grid that closely follows the resonance structure in the phantom material cross-sections was identified and used to calculate dose coefficients. Both the equivalent organ/tissue doses for male and female voxel phantoms were computed as well as their summation to obtain the effective dose coefficients. The used refined energy grid tracks very closely the cross-sections in the vicinity of the resonances. The resulting refined energy grid coefficients are compared to coefficients for the coarser energy grid used in ICRP Publication 116. Additionally, reference spectra have been folded with both the fine and coarse sets of conversion coefficients. The resulting total effective doses for these reference spectra are used to assess the adequacy of the dose coefficients calculated on the original ICRP 116 energy grid. The dose coefficients were similarly computed for the local skin dose on the trunk of the body using the ICRU Report 95 phantom. The overall impact of the resonances on the organ/tissue equivalent dose, the effective dose, and the local skin dose are presented and discussed. In general, it was found that resonances can impact neutron dose coefficients, but in most cases the wide range of neutron energies encountered minimized this effect. The impact of resonances was further limited when computing effective dose due to organ/tissue summing and sex-averaging. For the neutron fields studied here, the impact was below 5%.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Structural determination of a full-length plant cellulose synthase informed by experimental and in silico methods

Three-dimensional structure determination and prediction of proteins with intrinsically disordered regions, unstructured regions, conformational flexibility, and lacking homologous structures are challenging. We previously predicted and refined an in silico structure of a plant cellulose synthase from cotton (GhCESA1), and more recently, cryo-electron microscopy (cryo-EM) has resolved a majority of the lengths of two CESA structures from poplar (PttCESA8) and cotton (GhCESA7). However, 26–30% of these cryo-EM structures remain unresolved, including the N-terminal domain, half of the class-specific region, the gating loop region, and the C-terminal domain. Here, we describe the generation and evaluation of a full-length hybrid GhCESA1 model based on this cryo-EM PttCESA8 structure, with unresolved regions completed using this in silico refined GhCESA1 model. All-atom molecular dynamics simulations and subsequent energy minimizations were performed for the in silico and hybrid GhCESA1 models in a lipid bilayer-water-ion environment, and structural stability, dynamics, energetics, contacts, and quality were evaluated. The unresolved regions were found to be the most dynamic, in agreement with their poor electron density with cryo-EM. The hybrid model exhibited a higher total secondary structure content, more favorable intra-protein and protein-lipid interaction energies, and improved quality metrics. Moreover, hydrogen bonding was revealed to be a primary mechanism for intra-protein and protein-lipid contacts. These results demonstrate that in silico structure prediction and refinement may be useful to augment experimental structure determination, especially for disordered and unstructured regions. Furthermore, this hybrid model can serve as a steppingstone to derive full-length homology models of other CESAs found in more experimentally tractable organisms.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2023

Impact of artificial topological changes on flow and transport through fractured media due to mesh resolution

Abstract We performed a set of numerical simulations to characterize the interplay of fracture network topology, upscaling, and mesh refinement on flow and transport properties in fractured porous media. We generated a set of generic three-dimensional discrete fracture networks at various densities, where the radii of the fractures were sampled from a truncated power-law distribution, and whose parameters were loosely based on field site characterizations. We also considered five network densities, which were defined using a dimensionless version of density based on percolation theory. Once the networks were generated, we upscaled them into a single continuum model using the upscaled discrete fracture matrix model presented by Sweeney et al. (2019). We considered steady, isothermal pressure-driven flow through each domain and then simulated conservative, decaying, and adsorbing tracers using a pulse injection into the domain. For each simulation, we calculated the effective permeability and solute breakthrough curves as quantities of interest to compare between network realizations. We found that selecting a mesh resolution such that the global topology of the upscaled mesh matches the fracture network is essential. If the upscaled mesh has a connected pathway of fracture (higher permeability) cells but the fracture network does not, then the estimates for effective permeability and solute breakthrough will be incorrect. False connections cannot be eliminated entirely, but they can be managed by choosing appropriate mesh resolution and refinement for a given network. Adopting octree meshing to obtain sufficient levels of refinement leads to fewer computational cells (up to a 90% reduction in overall cell count) when compared to using a uniform resolution grid and can result in a more accurate continuum representation of the true fracture network.

58 GEOSCIENCES↗
DOE OSTI2023

Strontium Effects on the Formation of Iron-Intermetallic Phases in Secondary Al–9Si–0.6Fe Alloys

The presence and morphology of Fe-containing intermetallic phases affect the mechanical properties of aluminum alloys, especially in secondary Al–Si-based cast alloys. Although strontium (Sr) addition of 50 to 500 ppm is known to refine the needle-type eutectic silicon structure, the influence of Sr on the formation of Fe-intermetallic phases remains unclear. The present work investigates the combined additions of Sr and Mn to Al–9Si–0.6Fe–0.35Mg (All compositions are in wt pct except otherwise stated.) alloys on the formation of Fe-intermetallic phases at different solidification rates from ~ 1.5 to ~ 60 °C/s. Long and branched-type AlFeSi phase with size ranging from 50 to 120 µm are more common when solidified at the rate of 1.5 °C/s regardless of Sr and Mn additions. However, at the fast solidification rate of 60 °C/s, a 60 ppm Sr addition significantly reduced the average length of needle-shaped AlFeSi phase to less than 3 to 5 µm. Thermodynamic simulations have been performed using CALculation of PHAse Diagrams (CALPHAD) models to predict the formation of various phases and their possible interactions during solidification. The results indicated that the combination of a high solidification rate and about 60 ppm of Sr is beneficial to refining the δ-Al 3 FeSi 2 phase in Al–Si–Mg alloys containing 0.6 pctFe. As a result, this unexpected finding of Fe-intermetallic refinement by low Sr addition (~60 ppm) provides an important guide in designing secondary alloys for sustainable casting applications.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Martensitic transformation induced strength-ductility synergy in additively manufactured maraging 250 steel by thermal history engineering

Maraging steels are known for their exceptional strength but suffer from limited work hardening and ductility. Here, in this study, we report an intermittent printing strategy to tailor the microstructure and mechanical properties of maraging 250 steel via tuning the thermal history during wire-arc directed energy deposition. By introducing a dwell time between adjacent layers, the maraging 250 steel is cooled below the martensite start temperature, triggering thermally-driven martensitic transformation during the printing process. Thermal cycling during subsequent layer deposition results in the formation of reverted austenite which shows a refined microstructure and induces elemental segregation between martensite and reverted austenite. The Ni enrichment in the austenite promotes stabilization of the reverted austenite upon cooling to room temperature. The reverted austenite is metastable during deformation, leading to strain-induced martensitic transformation under loading. Specifically, a 3 min interlayer dwell time produces a maraging 250 steel with approximately 8% reverted austenite, resulting in improved work hardening via martensitic transformation induced plasticity during deformation. Meanwhile, the higher cooling rate and refined prior austenite grains lead to substantially refined martensitic grains (by approximately fivefold) together with an increased dislocation density. With 3 min interlayer dwell time, the yield strength of the printed maraging 250 steel increases from 836 MPa to 990 MPa, and the uniform elongation is doubled from 3.2% to 6.5%. This intermittent deposition strategy demonstrates the potential to tune the microstructure of maraging steels for achieving strength-ductility synergy by engineering the thermal history during additive manufacturing.

Additive manufacturing↗
DOE OSTI2023

Microstructural evolution in shear-punch tests: A comparative study of pure Cu and Cu-Cr alloy

Understanding the mechanisms behind microstructural evolution during shear deformation has been a long-standing area of interest. However, establishing a connection between microstructure, mechanical properties, and extent of shear deformation is challenging and requires refined experimental approaches. Shear-punch testing (SPT) provides a controlled method to introduce shear into small volumes of material that later can be subjected to detailed microstructural characterization. In this study, we utilize an SPT device to induce shear deformation to pure copper and a binary copper-chromium alloy. Electron backscatter diffraction and transmission electron microscopy were used to study the mechanisms of plastic deformation after SPT. Our results indicate that shear deformation of pure Cu produces a dense network of intercepting microshear bands upon sustained deformation. Twin boundaries undergo degradation into high angle grain boundaries due to simultaneous deviation from the axis-angle pair condition of 60° misorientation on [111] direction. The presence of 50% volume Cr particles in the soft Cu matrix fundamentally altered the shear deformation mechanism. Preferential deformation of the Cu matrix led to accelerated shear-induced formation of low and high angle grain boundaries, and subsequent grain refinement. Comparatively, no grain refinement occurred in pure Cu at a strain ~10 times larger (ϵ = 4.73) than that of the copper-chromium case (ϵ = 0.42). Overall, our study sheds light on the microstructural evolution during shear deformation and highlights the significant influence of particle reinforcements on the shear deformation mechanisms of metals.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Microstructural evolution and phase stability in Nb-containing interstitial Fe-Mn-Co-Cr-C high-entropy alloys: An in-situ synchrotron X-ray diffraction study during laser melting

The influence of Nb on phase stability and microstructural evolution in an interstitial Fe-Mn-Co-Cr-C high-entropy alloy was investigated using in-situ synchrotron X-ray diffraction (SXRD) during laser melting. Scheil-Gulliver simulations predict the formation of σ and γ-f.c.c. phases in all three alloys, along with NbC in Nb-containing compositions. SXRD confirmed the presence of most predicted phases, but the σ phase was absent. Nb promotes crystallite refinement and increases dislocation density, though excessive additions reduce refinement efficiency due to solubility limits and secondary phase formation. Furthermore, Nb addition also enhances ε-h.c.p. phase formation by reducing stacking fault energy through NbC-induced carbon depletion. Analysis of intensity peak evolution reveals that Nb alters preferred grain orientations, reducing {111} γ intensity while enhancing {220} γ , leading to a more isotropic grain distribution. Overall, Nb plays a key role in phase selection, microstructure refinement, and preferred orientation evolution, allowing the tailored microstructure of high-entropy alloys via rapid solidification.

Interstitial high entropy alloys↗
DOE OSTI2023

Annealing influence on the microstructure of irradiated U-Mo monolithic fuel foils

In this study we compared the microstructure evolution of U-Mo fuel foils produced with and without heat treatment at low burn-up via advanced post-irradiation examination. The aim of this study is to observe after irradiation the ways in which the fabrication processes have influenced fuel behavior at early-stage irradiation, as for very low burn up microstructural studies are lacking. In this work it was observed that the larger grain size detected in the heat-treated samples before irradiation led to decreased grain refinement after irradiation. Grain refinement was associated with the presence of small nano-size bubbles and precipitates. This phenomenon is hypothesized to influence early fuel swelling during reactor irradiation. Grain refinement was also observed to increase in regions where γ-U decomposition was present. Thus, it was enhanced in the samples fabricated without heat treatment. The heat treatment also increased the thickness of the U-Mo/Zr interface, namely of the UZr 2 layer. However, the influence of this layer on fuel performance needs further investigation. On one side, it may contribute to better mechanical bonding; on the other, it may influence swelling and blistering in the interaction layer as porosity increases when this layer is increased. This was observed especially in the presence of increased area containing low Mo concentration, and thus containing a higher fraction of the α-U phase, which is highly susceptible to irradiation induced swelling. Strong evidence of reverse transformation under irradiation (α-U + γ'-U2Mo → bcc γ-U) was observed in these samples. While the precipitates (carbides and oxide) seem to be unaffected by the irradiation at these low burnups. However, further analyses are necessary at higher burn-up to assess the exact impact different heat treatments have on fuel performance.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2021

Water heating energy use reductions from EPA WaterSense lavatory plumbing fittings

Hot water savings from water-efficient lavatory fittings lead to reductions in water heating energy consumption, and ultimately to decreases in carbon emissions. This paper characterizes existing and proposed approaches used to estimate hot water savings and carbon emissions reductions stemming from the U.S. Environmental Protection Agency's WaterSense program. Also described are refinements that improve the accuracy of residential hot water use percentage estimates of lavatory fittings. The authors conclude that (1) hot water percentages for showers and faucets calculated using up-to-date, publicly available national data are consistent with those found by regional studies and household-level models of water use; (2) the accuracy of heating energy savings estimates attributable to WaterSense-labeled lavatory products, as well as associated emissions reductions, can be refined by modifying the existing energy factor/uniform energy factor (EF/UEF)-based estimation approach with available data. The refined approach accounts for more nuanced conditions than the EF/UED-based approach but depends on data not always available; and (3) the approaches described and intermediate outputs can be generalized for other water conservation programs or estimating purposes.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2020

Powder X-ray diffraction of fluorometholone, C 22 H 29 FO 4

Commercial fluorometholone, CAS #426-13-1, crystallizes in the monoclinic space group P 2 1 (#4) with a = 6.40648(2), b = 13.43260(5), c = 11.00060(8) Å, β = 92.8203(5)°, V = 945.517(5) Å 3 , and Z = 2. A reduced cell search in the Cambridge Structural Database yielded one previous structure determination, using single-crystal data at 292 K. In this work, the sample was ordered from the United States Pharmacopeial Convention (Lot # R032K0) and analyzed as-received. The room temperature (295 K) crystal structure was refined using synchrotron ( λ = 0.412826 Å) powder diffraction data and optimized using density functional theory (DFT) techniques. Hydrogen positions were included as a part of the structure and were re-calculated during the refinement. The diffraction data were collected on beamline 11-BM at the Advanced Photon Source, Argonne National Laboratory, and the powder X-ray diffraction pattern of the compound has been submitted to ICDD® for inclusion in the Powder Diffraction File™. The agreement of the Rietveld-refined and DFT-optimized structures is excellent; the root-mean-square Cartesian displacement is 0.060 Å. In addition to the O–H⋯O hydrogen bonds observed by Park et al. (Park, Y. J., Lee, M. Y., and Cho, S. I. ( 1992 ). “Fluorometholone,” J. Korean Chem. Soc. 36 , 812–817), C–H⋯O hydrogen bonds contribute to the crystal energy.

Gonzalez, Diana↗
DOE OSTI2022

Multifunctional Cu 2 TSiS 4 (T = Mn and Fe): Polar Semiconducting Antiferromagnets with Nonlinear Optical Properties

In this work, Cu 2 TSiS 4 (T = Mn and Fe) polycrystalline and single-crystal materials were prepared with high-temperature solid-state and chemical vapor transport methods, respectively. The polar crystal structure (space group Pmn2 1 ) consists of chains of corner-sharing and distorted CuS 4 , Mn/FeS 4 , and SiS 4 tetrahedra, which is confirmed by Rietveld refinement using neutron powder diffraction data, X-ray single-crystal refinement, electron diffraction, energy-dispersive X-ray spectroscopy, and second harmonic generation (SHG) techniques. Magnetic measurements indicate that both compounds order antiferromagnetically at 8 and 14 K, respectively, which is supported by the temperature-dependent (100–2 K) neutron powder diffraction data. Additional magnetic reflections observed at 2 K can be modeled by magnetic propagation vectors k = (1/2,0,1/2) and k = (1/2,1/2,1/2) for Cu 2 MnSiS 4 and Cu 2 FeSiS 4 , respectively. The refined antiferromagnetic structure reveals that the Mn/Fe spins are canted away from the ac plane by about 14°, with the total magnetic moments of Mn and Fe being 4.1(1) and 2.9(1) μ B , respectively. Both compounds exhibit an SHG response with relatively modest second-order nonlinear susceptibilities. Density functional theory calculations are used to describe the electronic band structures.

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

Experimental X-ray Charge-Density Studies–A Suitable Probe for Superconductivity? A Case Study on MgB 2

Case studies of 1T-TiSe 2 and YBa 2 Cu 3 O 7-δ have demonstrated that X-ray diffraction (XRD) studies can be used to trace even subtle structural phase transitions which are inherently connected with the onset of superconductivity in these benchmark systems. However, the utility of XRD in the investigation of superconductors like MgB 2 lacking an additional symmetry-breaking structural phase transition is not immediately evident. Nevertheless, high-resolution powder XRD experiments on MgB 2 in combination with maximum entropy method analyses hinted at differences between the electron density distributions at room temperature and 15 K, that is, below the T c of approx. 39 K. The high-resolution single-crystal XRD experiments in combination with multipolar refinements presented here can reproduce these results but show that the observed temperature-dependent density changes are almost entirely due to a decrease of atomic displacement parameters as a natural consequence of a reduced thermal vibration amplitude with decreasing temperature. Our investigations also shed new light on the presence or absence of magnesium vacancies in MgB 2 samples–a defect type claimed to control the superconducting properties of the compound. Here, we propose that previous reports on the tendency of MgB 2 to form non-stoichiometric Mg 1–x B 2 phases (1 – x ~ 0.95) during high-temperature (HT) synthesis might result from the interpretation of XRD data of insufficient resolution and/or usage of inflexible refinement models. Indeed, advanced refinements based on an Extended Hansen–Coppens multipolar model and high-resolution X-ray data, which consider explicitly the contraction of core and valence shells of the magnesium cations, do not provide any significant evidence for the formation of non-stoichiometric Mg 1–x B 2 phases during HT synthesis.

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

A Molecular View of Methane Activation on Ni(111) through Enhanced Sampling and Machine Learning

A combination of machine learned interatomic potentials (MLIPs) and enhanced sampling simulations is used to investigate the activation of methane on a Ni(111) surface. The work entails the development and iterative refinement of MLIPs, initially trained on a dataset constructed via ab initio molecular dynamics (AIMD) simulations, supplemented by adaptive biasing forces, to enrich the sampling of catalytically relevant configurations. Our results reveal that by incorporating collective variables that capture the behavior of the reactant molecule, as well as additional frames that describe the dynamic response of the catalytic surface, it is possible to enhance considerably the accuracy of predicted energies and forces. By employing enhanced sampling schemes in the refinement of the MLIP, we systematically explore the potential energy surface, leading to a refined MLIP capable of predicting DFT-level energies and forces and replicating key geometric characteristics of the catalytic system. The resulting free energy landscapes at several temperatures provide a detailed view of the thermodynamics and dynamics of methane activation. Specifically, as methane approaches and dissociates on the catalytic surface, the process involves the dynamic interplay of CH 4 and the Ni catalyst that includes both enthalpic and entropic contributions. The progression towards the transition state involves an CH 4 moiety that is increasingly restrained in its ability to rotate or translate, while the stage following the transition state is characterized by a notable rise of the Ni atom that interacts with the cleaved C–H bond. Furthermore, this leads to an increase in the mobility of the adsorbed species, a feature that becomes more pronounced at higher temperatures.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2026

Ultrafast population and structural dynamics of a Ni-bipyridine photoredox catalyst reveal a significant deactivation pathway

The ultrafast excited state pathways and dynamics of NiII-bipyridine complexes influence the yield of photochemical processes involved in their catalytic cross-coupling reactions. Here we present ultrafast Ni K x-ray emission spectroscopy (XES) and x-ray solution scattering (XSS) of a NiII-bipyridine aryl halide complex, [Ni(t-Bubpy)(o-tol)Br], to quantify the excited state population dynamics and structural changes of the pre-catalyst. Due to the local spin-sensitivity of XES, the population dynamics of metal-to-ligand charge transfer (MLCT) and metal-centered (MC) excited states is established. A rapid ground state recovery pathway is newly identified, representing a significant deactivation pathway during photocatalysis. Furthermore, the pseudotetrahedral structure of the long-lived MC excited state is unambiguously identified and refined by XSS. The results advance our understanding of the ultrafast relaxation mechanisms that impact the photocatalytic mechanism and yield for NiII-bipyridine aryl halide cross-coupling catalysts.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Separating Physics and Dynamics Grids for Improved Computational Efficiency in Spectral Element Earth System Models

Previous studies have shown that atmospheric models with a spectral element grid can benefit from putting physics calculations on a relatively coarse finite volume grid. Here we demonstrate an alternative high-order, element-based mapping approach used to implement a quasi-equal-area, finite volume physics grid in E3SM. Unlike similar methods, the new method in E3SM requires topology data purely local to each spectral element, which trivially allows for regional mesh refinement. Simulations with physics grids defined by 2 × 2, 3 × 3, and 4 × 4 divisions of each element are shown to verify that the alternative physics grid does not qualitatively alter the model solution. The model performance is substantially affected by the reduction of physics columns when using the 2 × 2 grid, which can increase the throughput of physics calculations by roughly 60%–120% depending on whether the computational resources are configured to maximize throughput or efficiency. A pair of regionally refined cases are also shown to highlight the refinement capability.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗