Atomic oscillations in the statistical approximation.
Atomic dipolar spectral distribution function in statistical approximation derived from dynamic equations, describing atom density fluctuations induced by external field
INTEGRAL EQUATION↗
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Results for “Function approximation”
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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Fast approximation to supershell partition functions: Explicit forms of the coefficients
In a previous work, we derived a formula for supershell partition functions, which are the cornerstone of the Super-Transition-Array approach to radiative-opacity calculations. The new expression takes the form of a functional of the distribution of energies within the supershell and allows for fast and accurate computations, truncating the number of terms in the expansion. The latter involves coefficients (denoted Γ $k$ ) for which we obtained a recursion relation. In the present short paper, as a complement of the previous one, we give an explicit formula for the coefficients Γ $k$ . Another recursion relation is also provided, as well as an alternative expression involving Bell polynomials. In conclusion, the connections with cycle indexes of permutation groups and with elementary symmetric polynomials are outlined.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Applicability of the thin-film approximation in terahertz photoconductivity measurements
Thin mesoporous photoconductive layers are critically important for efficient water-spitting solar cells. A detailed understanding of photoconductivity in these materials can be achieved via terahertz transient absorption measurements. Such measurements are commonly interpreted using the thin-film approximation. We compare this approximation with a numerical solution of the transfer function without approximations using experimental results for thin-film mesoporous tin oxide (SnO 2 ) samples which range in thickness from 3.3 to 12.6 µm. These samples were sensitized with either a ruthenium polypyridyl complex or a porphyrin dye. The two sensitizers have markedly different absorption coefficients, resulting in penetration depths of 15 µm and 1 µm, respectively. The thin-film approximation results are in good agreement with the numerical work-up for the short penetration length dye. For the longer penetration length samples, the thin-film formula fails even for thicknesses of only 3 µm ≈ λ/100. The imaginary part of the conductivity calculated using the thin-film formula was significantly larger in magnitude than the value without approximations. This discrepancy between the commonly used thin-film approximation and the numerical solution demonstrates the need for a careful analysis of the thin-film formula.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Assessment of exchange-correlation functionals on oxygen vacancy formation energies of metal oxides
Oxygen vacancies are a common defect in oxides and play vital roles in many technological applications such as oxygen separation, catalytic reactors, solid oxide fuel cells, and solar thermochemical water splitting. The oxygen vacancy formation energy is directly related to the oxide reduction enthalpy and the vacancy concentration and, therefore, is a key quantity underlying these applications. The complexity of measuring oxygen vacancy formation energies experimentally suggests the utility of calculations based on density functional theory (DFT). However, the calculated results are strongly dependent on the exchange-correlation functionals and other parameters such as the Hubbard U . In this work, we compare the performance of the meta-GGA strongly constrained and appropriately normed (SCAN) functional and the commonly used semilocal generalized gradient approximation (GGA) functionals with experimental values of structural parameters, band gaps, magnetic structures, and oxygen vacancy formation energies for six representative oxides, i.e., CaMnO 3 , SrMnO 3 , LaMnO 3 , YMnO 3 , LaFeO 3 , and CeO 2 . Further, our results show that SCAN usually has better agreement with the experimental lattice constants and band gaps and larger magnetic moments than the commonly used GGA functionals. Although SCAN overestimates the oxygen vacancy formation energies of transition metal oxides and therefore requires unusually large Hubbard U values to reproduce the experimental reduction enthalpies, it does predict the correct oxygen vacancy formation energy of CeO 2 , which challenges the commonly used GGA and hybrid functionals. Our results underscore the challenges that exist in describing these complex oxides by DFT and may shed light on developing more accurate exchange-correlation functionals for oxygen vacancy formation energy calculations.
36 MATERIALS SCIENCE↗
Adaptive Radial Basis Function Methods for Data Approximation
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Torchinsky, Jason Louis [Sandia National Laborator↗
Spectral methods for the Navier-Stokes equations with one infinite and two periodic directions
The time-dependent, 3D incompressible Navier-Stokes equations in (1) boundary layers, the semiinfinite domain, and (2) mixing layers or wakes, the fully infinite domain, are respectively solved by two numerical methods which employ rapidly decaying spectral basis functions to approximate the vertical dependence of the solutions. These functions are then combined with one, for method (1), and two, for method (2), slowly decaying 'extra functions' for each wave vector. Each extra function can exactly represent the solution's irrotational component at large distances. The two methods have been applied to extensive direct-numerical simulation of transition and turbulence.
Spalart, Philippe R.↗
A six degree of freedom, plume-fuel optimal trajectory planner for spacecraft proximity operations using an A* node search
Spacecraft proximity operations are complicated by the fact that exhaust plume impingement from the reaction control jets of space vehicles can cause structural damage, contamination of sensitive arrays and instruments, or attitude misalignment during docking. The occurrence and effect of jet plume impingement can be reduced by planning approach trajectories with plume effects considered. An A* node search is used to find plume-fuel optimal trajectories through a discretized six dimensional attitude-translation space. A plume cost function which approximates jet plume isopressure envelopes is presented. The function is then applied to find relative costs for predictable 'trajectory altering' firings and unpredictable 'deadbanding' firings. Trajectory altering firings are calculated by running the spacecraft jet selection algorithm and summing the cost contribution from each jet fired. A 'deadbanding effects' function is defined and integrated to determine the potential for deadbanding impingement along candidate trajectories. Plume costs are weighed against fuel costs in finding the optimal solution. A* convergence speed is improved by solving approach trajectory problems in reverse time. Results are obtained on a high fidelity space shuttle/space station simulation. Trajectory following is accomplished by a six degree of freedom autopilot. Trajectories planned with, and without, plume costs are compared in terms of force applied to the target structure.
Jackson, Mark Charles↗
Challenges of Sustaining the International Space Station Through 2020 and Beyond: Reassessing Confidence Targets for System Availability
The International Space Station (ISS) was originally designed to operate until 2015 with a plan for deorbiting the ISS in 2016. Currently, the international partnership has agreed to extend the operations until 2020 and discussions are underway to extend the life even further to 2028. Each partner is responsible for the sustaining engineering, sparing, and maintenance of their own segments. National Aeronautics and Space Administration's (NASA's) challenge is to purchase the needed number of spares to maintain the functional availability of the ISS systems necessary for the United States On-Orbit Segment s contribution. This presentation introduces an analytical approach to assessing uncertainty in ISS hardware necessary to extend the life of the vehicle. Some key areas for consideration are: establishing what confidence targets are required to ensure science can be continuously carried out on the ISS, defining what confidence targets are reasonable to ensure vehicle survivability, considering what is required to determine if the confidence targets are too high, and whether sufficient number of spares are purchased. The results of the analysis will provide a methodological basis for reassessing vehicle subsystem confidence targets. This analysis compares the probability of existing spares exceeding the total expected unit demand of the Orbital Replacement Unit (ORU) in functional hierarchies approximating the vehicle subsystems. In cases where the functional hierarchies' availability does not meet subsystem confidence targets, the analysis will further identify which ORUs may require additional spares to extend the life of the ISS. The resulting probability is dependent upon hardware reliability estimates. However, the ISS hardware fleet carries considerable epistemic uncertainty which must be factored into the development and execution of sparing risk postures. In addition, it is also recognized that uncertainty in the assessment is due to disconnects between modeled functions and actual subsystem operations. Perhaps most importantly, it is acknowledged that conservative confidence targets per subsystem are currently accepted. This presentation will also discuss how subsystem confidence targets may be relaxed based on calculating the level of uncertainty for each corresponding ORU-function. The presentation will conclude with the various strengths and limitations for implementing the analytical approach in sustaining the ISS through end of life; 2020 and beyond.
Lutomski, Michael G.↗
Challenges of Sustaining the International Space Station through 2020 and Beyond: Including Epistemic Uncertainty in Reassessing Confidence Targets
This paper introduces an analytical approach, Probability and Confidence Trade-space (PACT), which can be used to assess uncertainty in International Space Station (ISS) hardware sparing necessary to extend the life of the vehicle. There are several key areas under consideration in this research. We investigate what sparing confidence targets may be reasonable to ensure vehicle survivability and for completion of science on the ISS. The results of the analysis will provide a methodological basis for reassessing vehicle subsystem confidence targets. An ongoing annual analysis currently compares the probability of existing spares exceeding the total expected unit demand of the Orbital Replacement Unit (ORU) in functional hierarchies approximating the vehicle subsystems. In cases where the functional hierarchies availability does not meet subsystem confidence targets, the current sparing analysis further identifies which ORUs may require additional spares to extend the life of the ISS. The resulting probability is dependent upon hardware reliability estimates. However, the ISS hardware fleet carries considerable epistemic uncertainty (uncertainty in the knowledge of the true hardware failure rate), which does not currently factor into the annual sparing analysis. The existing confidence targets may be conservative. This paper will also discuss how confidence targets may be relaxed based on the inclusion of epistemic uncertainty for each ORU. The paper will conclude with strengths and limitations for implementing the analytical approach in sustaining the ISS through end of life, 2020 and beyond.
Anderson, Leif↗
A42F-03: Small-Scale Variability in Tropical Tropopause Layer Humidity
Recent advances in statistical parameterizations of cirrus cloud processes for use in global models are highlighting the need for information about small-scale fluctuations in upper tropospheric humidity and the physical processes that control the humidity variability. To address these issues, we have analyzed high-resolution airborne water vapor measurements obtained in the Airborne Tropical TRopopause EXperiment (ATTREX) over the tropical Pacific between 14 and 20 km. Using accurate and precise 1-Hz water vapor measurements along approximately-level aircraft flight legs, we calculate structure functions spanning horizontal scales ranging from about 0.2 to 50 km, and we compare the water vapor variability in the lower (about 14 km) and upper (16-19 km) Tropical Tropopause Layer (TTL). We also compare the magnitudes and scales of variability inside TTL cirrus versus in clear-sky regions. The measurements show that in the upper TTL, water vapor concentration variance is stronger inside cirrus than in clear-sky regions. Using simulations of TTL cirrus formation, we show that small variability in clear-sky humidity is amplified by the strong sensitivity of ice nucleation rate to supersaturation, which results in highly-structured clouds that subsequently drive variability in the water vapor field. In the lower TTL, humidity variability is correlated with recent detrainment from deep convection. The structure functions indicate approximately power-law scaling with spectral slopes ranging from about minus 5 divided by 3, to minus 2.
water vapor↗
Approximating Density Probability Distribution Functions Across Cosmologies
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Huanqing Chen↗
Natively Periodic Fast Multipole Method: Approximating the Optimal Green’s Function
The fast multipole method (FMM) obeys periodic boundary conditions "natively" if it uses a periodic Green's function for computing the multipole expansion in the interaction zone of each FMM oct-tree node. One can define the "optimal" Green's function for such a method that results in the numerical solution that converges to the equivalent particle-mesh (PM) solution in the limit of sufficiently high order of multipoles. A discrete functional equation for the optimal Green's function can be derived, but is not practically useful as methods for its solution are not known. Instead, this paper presents an approximation for the optimal Green's function that is accurate to better than 10 -3 in ${L}_{\mathrm{MAX}}$ norm and 10 -4 in L 2 norm for practically useful multipole counts. Such an approximately optimal Green's function offers a practical way for implementing the FMM with periodic boundary conditions natively, without the need to compute lattice sums or to rely on hybrid FMM-PM approaches.
46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
On Formulations of Discontinuous Galerkin and Related Methods for Conservation Laws
A formulation for the discontinuous Galerkin (DG) method that leads to solutions using the differential form of the equation (as opposed to the standard integral form) is presented. The formulation includes (a) a derivative calculation that involves only data within each cell with no data interaction among cells, and (b) for each cell, corrections to this derivative that deal with the jumps in fluxes at the cell boundaries and allow data across cells to interact. The derivative with no interaction is obtained by a projection, but for nodal-type methods, evaluating this derivative by interpolation at the nodal points is more economical. The corrections are derived using the approximate (Dirac) delta functions. The formulation results in a family of schemes: different approximate delta functions give rise to different methods. It is shown that the current formulation is essentially equivalent to the flux reconstruction (FR) formulation. Due to the use of approximate delta functions, an energy stability proof simpler than that of Vincent, Castonguay, and Jameson (2011) for a family of schemes is derived. Accuracy and stability of resulting schemes are discussed via Fourier analyses. Similar to FR, the current formulation provides a unifying framework for high-order methods by recovering the DG, spectral difference (SD), and spectral volume (SV) schemes. It also yields stable, accurate, and economical methods.
Huynh, H. T.↗
Meshfree Methods
Meshfree methods have undergone substantial development and have received much attention in the last two decades. This new family of numerical methods is designed to inherit the main advantages of the finite element method such as compact supports of shape functions and good approximation properties while, at the same time, overcome the main disadvantages of the finite element method caused by the mesh dependence. The meshfree methods share a common feature that no mesh is needed and shape functions are constructed from sets of points, thus eliminating the need for time consuming mesh generation. The most significant advantage of meshfree methods is the flexibility in customizing approximation functions for desired regularity and for capturing essential physics and features of the particular problems of interest. Adaptivity formulation and multiple-scale solution strategies also can be implemented with relative ease. It has become clear that the meshfree methods provide considerable advantages over the conventional finite element methods in solving problems involving moving discontinuities, evolving material interfaces, multiple-scale phenomena, large material distortion and structural deformation, and fracture and damage processes. This Chapter gives an overview of many classes of meshfree methods, with more detailed discussions on Smoothed Particle Hydrodynamics (SPH), the Reproducing Kernel Particle Method (RKPM), Peridynamics (PD), the Material Point Method (MPM), as well as their applications in various challenging engineering problems.2
Chen, Jiun-Shyan↗
Approximation of the dielectric properties of Maxwellian plasmas - Dispersion functions and physical constraints
The dielectric properties of Maxwellian plasmas are approximated using high Pade approximants to the dispersion function and direct approximation of the distribution. Physical constraints on permissible approximations are discussed, and it is found that some previously published results can lead to predictions of qualitatively incorrect wave properties, including unphysical negative damping. Approximate dispersion functions for Maxwellian distributions are given explicitly, and some of the effects of these approximations on the resulting dispersion are discussed. The approximations discussed here are of use both in analytical work and in accelerating large-scale numerical computations.
Robinson, P. A.↗
Benchmark Data Set of Crystalline Organic Semiconductors
This work reports a Benchmark Data set of Crystalline Organic Semiconductors to test calculations of the structural and electronic properties of these materials in the solid state. The data set contains 67 crystals consisting of mostly rigid molecules with a single dominant conformer, covering the majority of known structural types. The experimental crystal structure is available for the entire data set, whereas zero-temperature unit cell volume can be reliably estimated for a subset of 28 crystals. Using this subset, we benchmark r 2 SCAN-D3 and PBE-D3 density functionals. Then, for the entire data set, we benchmark approximate density functional theory (DFT) methods, including GFN1-xTB and DFTB3(3ob-3-1), with various dispersion corrections against r 2 SCAN-D3. Our results show that r 2 SCAN-D3 geometries are accurate within a few percent, which is comparable to the statistical uncertainty of experimental data at a fixed temperature, but the unit cell volume is systematically underestimated by 2% on average. The several times faster PBE-D3 provides an unbiased estimate of the volume for all systems except for molecules with highly polar bonds, for which the volume is substantially overestimated in correlation with the underestimation of atomic charges. Considered approximate DFT methods are orders of magnitude faster and provide qualitatively correct but overcompressed crystal structures unless the dispersion corrections are fitted by unit cell volume.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Why is MP2-Water "Cooler" and "Denser" than DFT-Water?
To maintain water in the liquid phase at the correct (1 g/cm3) density during first-principles simulations, density-functional theory (DFT) with a dispersionless generalized-gradient-approximation (GGA) functional requires a much higher temperature and pressure than the ambient conditions. Conversely, ab initio second-order many-body perturbation (MP2) calculations of liquid water performed by Del Ben et al. [J. Chem. Phys. Lett. 4, 3753 (2013); J. Chem. Phys. 143, 054506 (2015)] and by us [Willow et al., Sci. Rep. 5, 14358 (2015)] required a lower temperature and a negative pressure than DFT to keep water liquid. Here, we present a unifying explanation of these trends derived from classical water simulations using a polarizable force field with different sets of parameters. We show that the calculated temperature and pressure of the liquid phase are strongly correlated with the polarizability of water and the dispersion interaction, respectively. In DFT/GGA, the polarizability and thus the induced dipole moments and the hydrogen-bond strength are all overestimated. This hinders the rotational motion of molecules and requires a higher temperature for water to be liquid. In MP2 and DFT/GGA, the dispersion interaction is stronger and weaker (or lacking), respectively. This explains why liquid water contracts uniformly and becomes too dense in MP2, whereas the opposite is the case for dispersionless DFT/GGA.
Willow, Soohaeng Y.↗
The role of decomposition reactions in assessing first-principles predictions of solid stability
Abstract The performance of density functional theory approximations for predicting materials thermodynamics is typically assessed by comparing calculated and experimentally determined enthalpies of formation from elemental phases, Δ H f . However, a compound competes thermodynamically with both other compounds and their constituent elemental forms, and thus, the enthalpies of the decomposition reactions to these competing phases, Δ H d , determine thermodynamic stability. We evaluated the phase diagrams for 56,791 compounds to classify decomposition reactions into three types: 1. those that produce elemental phases, 2. those that produce compounds, and 3. those that produce both. This analysis shows that the decomposition into elemental forms is rarely the competing reaction that determines compound stability and that approximately two-thirds of decomposition reactions involve no elemental phases. Using experimentally reported formation enthalpies for 1012 solid compounds, we assess the accuracy of the generalized gradient approximation (GGA) (PBE) and meta-GGA (SCAN) density functionals for predicting compound stability. For 646 decomposition reactions that are not trivially the formation reaction, PBE (mean absolute difference between theory and experiment (MAD) = 70 meV/atom) and SCAN (MAD = 59 meV/atom) perform similarly, and commonly employed correction schemes using fitted elemental reference energies make only a negligible improvement (~2 meV/atom). Furthermore, for 231 reactions involving only compounds (Type 2), the agreement between SCAN, PBE, and experiment is within ~35 meV/atom and is thus comparable to the magnitude of experimental uncertainty.
36 MATERIALS SCIENCE↗