SEARCH · Search NASA
Results for “Partition of unity”
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.
PhILMs: Collaboratory on Mathematics and Physics-Informed Learning Machines for Multiscale and Multiphysics Problems
The landscape of computational science and engineering is continually evolving, with the challenge of high-dimensional regression problems standing as a significant hurdle in numerous scientific endeavors. Addressing this challenge, our research, funded by this award, has led to the development of an innovative computational framework known as Probabilistic Partition of Unity Networks (PPOU-Nets). This initiative represents a collaborative effort to harness the potential of mathematics and physics-informed machine learning in tackling multiscale and multiphysics problems prevalent in high-dimensional spaces. Through this work, we have proposed a novel methodology that seamlessly integrates adaptive dimensionality reduction and a mixture of experts model, thereby facilitating a more efficient and accurate approximation of complex functions. This research effort has not only advanced the state of computational science but also opened new avenues for exploration in quantum computing and beyond. This report outlines the motivation, methodology, key findings, and implications of our work, underscoring our contributions to the broader scientific community and the potential pathways for future research.
Data-driven Whitney forms for structure-preserving control volume analysis
Control volume analysis models physics via the exchange of generalized fluxes between subdomains. Here, we introduce a scientific machine learning framework adopting a partition of unity architecture to identify physically-relevant control volumes, with generalized fluxes between subdomains encoded via Whitney forms. The approach provides a differentiable parameterization of geometry which may be trained in an end-to-end fashion to extract reduced models from full field data while exactly preserving physics. The architecture admits a data-driven finite element exterior calculus allowing discovery of mixed finite element spaces with closed form quadrature rules. An equivalence between Whitney forms and graph networks reveals that the geometric problem of control volume learning is equivalent to an unsupervised graph discovery problem. The framework is developed for manifolds in arbitrary dimension, with examples provided for H(div) problems in $\mathbb{R}$ establishing convergence and structure preservation properties. Finally, we consider a lithium-ion battery problem where we discover a reduced finite element space encoding transport pathways from high-fidelity microstructure resolved simulations. The approach reduces the 5.89M finite element simulation to 136 elements while reproducing pressure to under 0.1% error and preserving conservation.
Partitioning of Nb, Mo, Ba, Ce, Pb, Th and U between immiscible carbonate and silicate liquids: Evaluating the effects of P2O5,F, and carbonate composition
Previously we have reported carbonate liq./silicate liq. partition coefficients (D) for a standard suite of trace elements (Nb, Mo, Ba, Ce, Pb, Th, and U) and Ra and Pa as well. In brief, we have found that immiscible liquid partitioning is a strong function of temperature. As the critical temperature of the carbonate-silicate solvus is approached, all partition coefficients approach unity. Additionally, for the overwhelming majority of the partitioning elements, InD is a linear function of 'ionic field strength,' z/r, where z is the charge of the partitioned cation and r is its ionic radius.
Experimental Partitioning of Cr(3+) and Sc(3+) into Olivine: Mechanisms and Implications
Olivine (Mg, Fe)Si2O4 does not, by stoichiometry, accept cations such as Sc(3+) or Cr(3+). However, the partition coefficients of Sc and Cr between olivine and liquid are significant 0.2-1.0. We have measured Cr(3+) partition coefficients of near unity and have grown olivines with nearly 3 wt.% Sc2O3. Therefore, there must be a simple means of charge balancing 3+ ions in a crystal structure that was obviously not designed to receive other than 2+ ions on the olivine M sites. The simplest explanation is that two 3+ ions enter the olivine structure by displacing three 2+ ions and creating an M site vacancy. Even this explanation has difficulties. For minor elements in our experiments (~1 wt.%) the odds of a minor element 3+ ion finding a second 3+ for charge balance are of the order of 100:1 against. Because of the reducing conditions of our experiments, Fe(3+) will not suffice; and Al(3+) is not in sufficient quantity in olivine for charge balance. Therefore, Cr or Sc must, in effect, charge balance itself. For true trace elements, the problem is compounded many times. For an ion at the 10 ppm level the chances of finding a second (for example) Sc ion is approx.10(exp 5):1 against. Of course, any other 3+ ion would suffice but comparisons between percent level doping experiments and trace level partitioning indicate that Henry s law is obeyed. This implies that the same substitution mechanism occurs at both the percent and tens of ppm levels. There are two simple solutions to this problem: (i) The electrical conductivity of olivine is such that charge balance need not be local. This requires substantial domains within the olivine crystal in electrical contact by migration of vacancies or electronic defects. (ii) The 3+ cation brings along its own charge-balancing ion because it existed as a dimer in the silicate liquid. Olivine is not a true insulator but is actually a p-type semiconductor. Even so, electrical communication by this means is unlikely over the tens or hundreds of unit cells that would be required for charge balance to be local. Therefore, we cautiously favor the idea that melt speciation is the means by which 3+ ions enter the olivine structure. Possibly this model might be tested by in situ XAFS measurements or by molecular dynamical calculations.
Trace element partitioning between taenite and kamacite - Relationship to the cooling rates of iron meteorites
Instrumental neutron activation analysis (INAA) was used to determine Ni, Co, Cu, Ga, As, Au, W, Re and Ir in taenite lamellae isolated by acid dissolution from eight iron meteorites from groups IA, IIIAB and IVA. Taenite is enriched in Ni, Cu, Ga, As, Au, W, Re and Ir relative to kamacite, whereas taenite is depleted in Co. Taenite/kamacite partition ratios in slowly cooled IAB meteorites are farther from unity than those in rapidly cooled IVA meteorites. Taenite/kamacite partition ratios for Cu, Ir, Au and Co may be sensitive cooling rate indicators.
Compositional trends and cooling rates of group IVB iron meteorites
It is found, on the basis of new neutron activation data for group IVB, that log-element - log-Ni trends are best understood in terms of core formation and fractional crytstallizaion. The limited compositional range found in gorup IVB is thought to reflect the fact that, owing to the low concentrations of S, P, and C and the high concentration of Ni, the k sub x (the solid/liquid partition ratio) values are closer to unity than are those in other magmatic groups. Mean volatile abundances in group IVB are significantly lower than those found in any group of chondritic meteorites, indicating that these low abundances are not entirely the result of nebular processes and that planetary outgassing was also involved.
Global Simulations of Phase State and Equilibration Time Scales of Secondary Organic Aerosols with GEOS-Chem
The phase state of secondary organic aerosols (SOA) can range from liquid through amorphous semisolid to glassy solid, which is important to consider as it influences various multiphase processes including SOA formation and partitioning, multiphase chemistry, and cloud activation. In this study, we simulate the glass transition temperature and viscosity of SOA over the globe using the global chemical transport model, GEOS-Chem. The simulated spatial distributions show that SOA at the surface exist as liquid over equatorial regions and oceans, semisolid in the midlatitude continental regions, and glassy solid over lands with low relative humidity. The predicted SOA viscosities are mostly consistent with the available measurements. In the free troposphere, SOA particles are mostly predicted to be semisolid at 850 hPa and glassy solid at 500 hPa, except over tropical regions including Amazonia, where SOA are predicted to be low viscous. Phase state also exhibits seasonal variation with a higher frequency of semisolid and solid particles in winter compared to warmer seasons. We calculate equilibration time scales of SOA partitioning (τ eq ) and effective mass accommodation coefficient (α eff ), indicating that τ eq is shorter than the chemical time step of GEOS-Chem of 20 min and αeff is close to unity for most locations at the surface level, supporting the application of equilibrium SOA partitioning. However, τ eq is prolonged and α eff is lowered over drylands and most regions in the upper troposphere, suggesting that kinetically limited growth would need to be considered for these regions in future large-scale model studies.
Trapped Melt in IIIAB Irons: Solid/Liquid Elemental Partitioning During the Fractionation of the IIIAB Magma
Group IIIAB, the largest iron-meteorite group, shows compositional trends (including a three-order-of-magnitude It concentration range) indicating that it formed by fractional crystallization of a metallic magma. Because about 200 irons are available, and all degrees of crystallization are well represented, IIIAB offers an excellent set of samples for the study of crystallization at all depths of the asteroidal core. On log-log Ir-Au, and Ir-As diagrams IIIAB forms a broad band; the breadth represents real meteorite-to-meteorite variations, far outside experimental or sampling uncertainties. A successful model must explain the width of this band; I suggest that it mainly resulted from the trapping of parental magma within the crystallizing solid. Because S is essentially insoluble in metal, the abundance of FeS is a measure of the fraction of trapped liquid. The trapped-melt model is supported by the observation that irons having higher S contents plot closer to the inferred composition of the magmatic parental liquid. The lowest S values are found in the irons occupying the left envelope of the IIIAB Ir-Au or Ir-As compositional fields, thus it is this set of irons that should be interpreted as the solid products of a fractionating magma. This simplifies the modeling of the crystallization process and allows inferences regarding the distribution ratios for other elements in the evolved IIIAB system. The large (multiton) Cape York irons show wide variations in their trapped-melt fractions; their compositions seem best understood in terms of a low initial S content of the IIIAB magma, about 20 mg/g. The inferred initial IIIAB distribution coefficient for Ir, 4.6, is much higher than published values based on laboratory studies of low-S systems; I suggest that low-S (and low-P) partition-ratio measurements tend to err in the direction of unity. In IIIAB distribution coefficients for Au, As, and Ni were still < 1 when the most evolved IIIAB irons formed, another indication of a low initial S content.
Dentritic morphology and microsegregation in directionally solidified superalloy, PWA-1480, single crystal: Effect of gravity; center director's discretionary fund report
Primary dendrite spacings, secondary dendrite spacings, and microsegregation have been examined in PWA-1480 single crystal specimens which were directionally solidified during parabolic maneuvers on the KC-135 aircraft. Experimentally observed growth rate and thermal gradient dependence of primary dendrite spacings are in good agreement with predictions from dendrite growth models for binary alloys. Secondary dendrite coarsening kinetics show a reasonable fit with the predictions from an analytical model proposed by Kirkwood for a binary alloy. The partition coefficients of tantalum, titanium, and aluminum are observed to be less than unity, while that for tungsten and cobalt are greater than unity. This is qualitatively similar to the nickel base binaries. Microsegregation profiles experimentally observed for PWA-1480 superalloy show a good fit with Bower, Brody, and Flemings model developed for binary alloys. Transitions in gravity levels do not appear to affect primary dendrite spacings. A trend of decreased secondary arm spacings with transition from high gravity to the low gravity period was observed at a growth speed of 0.023 cm s(exp -1). However, definite conclusions can only be drawn by experiments at lower growth speeds which make it possible to examine the side-branch coarsening kinetics over a longer duration. Such experiments, not possible due to the insufficient low-gravity time of the KC-135, may be carried out in the low-gravity environment of space.
Crystal Field Effects and Siderophile Element Partitioning: Implications for Mars HSE Geochemistry
Analyses of martian (SNC) meteorites indicate that Pt abundances do not vary much compared to other highly siderophile elements (HSE). Therefore, Jones et al. [1] inferred that D(Pt) during basalt petrogenesis was of order unity. This inference was at odds with previously published experiments that gave a D(sub ol/liq) for Pt of approx. 0.01 [2]. Because olivine is likely to be an important constituent of any reasonable martian mantle, the implication of these findings is that minor minerals must have D(Pt) much greater than 1, which seemed improbable. However, not only did the SNC evidence point to a D(sub ol/liq) approx. equal to 1, but so did plots of D(sub ol/liq) vs. ionic radius (Onuma diagram). The ionic radius of Pt(2+) suggested that D(sub ol/liq) for Pt was of order unity, in agreement with the inferences from SNC meteorites. New experiments have failed to detect measurable Pt in olivine, even at high oxygen fugacities [3]. Therefore, some other parameter, other than ionic charge and radius, must hold sway during olivine liquid partitioning of Pt.
Effect of reactive intermediates on the propagation of turbulent premixed flames
The most simple version of the Bray and Libby model of turbulent flame propagation has been extended to include a two-step reaction mechanism in which reactants decompose into reactive intermediates which then form products. A transport equation for the joint pdf of intermediate and product concentrations was derived, the pdf parameterized, and transport equations for the parameters obtained. The one-dimensional concentration equations were then solved for the parameters of the pdf and the turbulent kinetic energy. Flame speed and angle are recovered as eigenvalues of the problem. The results show the essential features observed in experiments. The product pdf is composed of peaks at zero and unity product concentration, with a continuous structure representing partially reacted gas. The intermediates pdf has a peak at zero concentration and a broad bell-shaped distribution at nonzero concentration, representing partially reacted gas. The results also show the effect of partitioning of energy release on the flame thickness and speed.
Primary arm spacing in directionally solidified Pb-10 wt percent Sn alloys
The dependence of primary arm spacings on growth speed was investigated for cellular and dendritic arrays in Pb-10 wt percent Sn samples directionally solidified under a constant positive thermal gradient in the melt. The gradient of constitutional supercooling was varied from almost zero (near the break-down of the planar liquid-solid interface at small growth speeds, cellular morphology) to near unity (large growth speeds, dendritic morphology). The spatial arrangements of cells and dendrites, as given by their coordination number, are not very different from each other. It appears that primary arm spacing maxima and the cell to dendrite transition are strongly influenced by the magnitude of the solute partition coefficient. The planar to cellular bifurcation is supercritical in Pb-Sn which has a high partition coefficient, as compared to the subcritical behavior reported in Al-Cu and succinonitrile-acetone, both of which have low partition coefficients. The primary arm spacing model due to Hunt agrees with the experimentally observed trend for the whole growth regime. There is a good quantitative agreement at higher gradients of supercooling. However, the model overpredicts the primary arm spacings at low gradients of constitutional supercooling.
Primary arm spacing in directionally solidified Pb-10 wt pct Sn alloys
The dependence of primary arm spacings on growth speed was investigated for cellular and dendritic arrays in Pb-10 wt percent Sn samples directionally solidified under a constant positive thermal gradient in the melt. The gradient of constitutional supercooling was varied from almost zero (near the break-down of the planar liquid-solid interface at small growth speeds, cellular morphology) to near unity (large growth speeds, dendritic morphology). The spatial arrangements of cells and dendrites, as given by their coordination number, are not very different from each other. It appears that primary arm spacing maxima and the cell to dendrite transition are strongly influenced by the magnitude of the solute partition coefficient. The planar to cellular bifurction is supercritical in Pb-Sn which has a high partition coefficient, as compared to the subcritical behavior reported in Al-Cu and succinonitrile-acetone, both of which have low partition coefficients. The primary arm spacing model due to Hunt agrees with the experimentally observed trend for the whole growth regime. There is a good quantitative agreement at higher grdients of supercooling. However, the model overpredicts the primary arm spacings at low gradients of constitutional supercooling.
Southern hemisphere ground based measurements of Carbonyl Fluoride (COF2) and Hydrogen Fluoride (HF): Partitioning between Fluoride reservoir species
We report infrared ground based total column measurements of the stratospheric fluorine reservoir gases COF2 and HF above Lauder, New Zealand (45 deg S, 167.8 deg E) obtained between April 1993 and January 1994. The average retrieved COF2 and HF total columns are 2.81(+/- 0.56) x 10(exp 14) and 9.91(+/- 1.09) x 10(exp 14) molecules/sq cm respectively. The daily average COF2 and HF columns are correlated; this correlation is likely the result of dynamics. The average HF/COF2 column ratio on days with measurements of both HF and COF2 is 3.63 (+/- 0.55). Comparison of this ratio with model calculations implies that the quantum yield for COF2 photolysis is near unity. Our measured COF2 columns are higher than all previously reported values, but inconsistencies among the earlier measurements and uncertainty in the latitudinal gradient of the COF2 column preclude an accurate determination of the long-term COF2 trend.
Approximations for the Thermodynamic and Transport Properties of High-Temperature Air
The thermodynamic and transport properties of high-temperature air are found in closed form starting from approximate partition functions for the major components in air and neglecting all minor components. The compressibility, enthalpy, entropy, the specific heats, the speed of sound, the coefficients of viscosity and of thermal conductivity, and the Prandtl numbers for air are tabulated from 500 degrees to 15,000 degrees Kelvin over a range of pressure from 0.0001 to 100 atmospheres. The energy of air and the mol. fractions of the major components of air can be found from the tabulated values for compressibility and enthalpy. It is predicted that the Prandtl number for fully ionized air, which is in complete equilibrium, will become small compared to unity, the order of 0.01, and this implies that boundary layers in such flow will be very transparent to heat flux.