Search NASA⌕ Search

SEARCH · Search NASA

Results for “Functional equations”

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.

At least 217 records · Page 12

HfTi_BCC_SolidSolution_128atoms_VASP6

We performed density functional theory (DFT) calculations for body-centered-cubic (BCC) structures with 128 lattices sites of solid solution binary alloys hafnium-titanium (Hf-Ti). The electronic structures of alloys have been calculated using Vienna Ab initio Simulation Package (VASP). Within this package the DFT approach is used to reduce many-body Schrodinger equation to set of single particle Kohn-Sham (KS) equations. The generalized electronic exchange-correlation functional is described by generalized gradient approximation with the Perdew-Burke-Ernzerhof parametrization. The electron-ion interactions is described by pseudopotentials developed within the plane-wave basis projector augmented-wave (PAW) approach. These pseudopotentials are available at the VASP portal (http://cms.mpi.univie.ac.at/vasp/). Our calculations have been run with the pseudopotentials treating s and p semi-core states as valence in case for the elements Hf and Ti. The electronic densities and potentials are expanded over plane-waves with energy cutoff of 350 eV. 2x2x2 k-mesh and normal precision were used. The alloys were modeled by supercell containing 128 randomly distributed atoms. At initial step the atoms occupy perfect bcc lattice cites. This initial structure was optimized until energy changes less than 1e-6 eV, while forces acting on atoms don't exceed 1e-2 eV/angstrom. The electron-ion interaction is described by PAW pseudopotentials. The calculations have been collected by sampling chemical compositions across the entire compositional range. The chemical compositions have been sampled by progressively changing the number of atoms per constituent by 4. For each chemical composition of binaries and ternaries, the first-principle calculations have been run for 100 randomized arrangements of the constituents on the BCC lattice sites. We collected data for a total of 3,100 randomized atomic structures over 31 chemical compositions. The calculations have been collected on Air Force HPC11 cluster using the VASP 6.5.1. Additional methodology and file structure information is available in the dataset README.txt file.

36 MATERIALS SCIENCE↗

TiV_BCC_SolidSolution_128atoms_VASP6

We performed density functional theory (DFT) calculations for body-centered-cubic (BCC) structures with 128 lattices sites of solid solution binary alloys titanium-vanadium (Ti-V). The electronic structures of alloys have been calculated using Vienna Ab initio Simulation Package (VASP). Within this package the DFT approach is used to reduce many-body Schrodinger equation to set of single particle Kohn-Sham (KS) equations. The generalized electronic exchange-correlation functional is described by generalized gradient approximation with the Perdew-Burke-Ernzerhof parametrization. The electron-ion interactions is described by pseudopotentials developed within the plane-wave basis projector augmented-wave (PAW) approach. These pseudopotentials are available at the VASP portal (http://cms.mpi.univie.ac.at/vasp/). Our calculations have been run with the pseudopotentials treating s and p semi-core states as valence in case for the elements Ti and V. The electronic densities and potentials are expanded over plane-waves with energy cutoff of 350 eV. 2x2x2 k-mesh and normal precision were used. The alloys were modeled by supercell containing 128 randomly distributed atoms. At initial step the atoms occupy perfect bcc lattice cites. This initial structure was optimized until energy changes less than 1e-6 eV, while forces acting on atoms don't exceed 1e-2 eV/angstrom. The electron-ion interaction is described by PAW pseudopotentials. The calculations have been collected by sampling chemical compositions across the entire compositional range. The chemical compositions have been sampled by progressively changing the number of atoms per constituent by 4. For each chemical composition of binaries and ternaries, the first-principle calculations have been run for 100 randomized arrangements of the constituents on the BCC lattice sites. We collected data for a total of 3,100 randomized atomic structures over 31 chemical compositions. The calculations have been collected on Air Force HPC11 cluster using the VASP 6.5.1. Additional methodology and file structure information is available in the dataset README.txt file.

36 MATERIALS SCIENCE↗

HfTa_BCC_SolidSolution_128atoms_VASP6

We performed density functional theory (DFT) calculations for body-centered-cubic (BCC) structures with 128 lattices sites of solid solution binary alloys hafnium-tantalum (Hf-Ta). The electronic structures of alloys have been calculated using Vienna Ab initio Simulation Package (VASP). Within this package the DFT approach is used to reduce many-body Schrodinger equation to set of single particle Kohn-Sham (KS) equations. The generalized electronic exchange-correlation functional is described by generalized gradient approximation with the Perdew-Burke-Ernzerhof parametrization. The electron-ion interactions is described by pseudopotentials developed within the plane-wave basis projector augmented-wave (PAW) approach. These pseudopotentials are available at the VASP portal (http://cms.mpi.univie.ac.at/vasp/). Our calculations have been run with the pseudopotentials treating s and p semi-core states as valence in case for the elements Hf and Ta. The electronic densities and potentials are expanded over plane-waves with energy cutoff of 350 eV. 2x2x2 k-mesh and normal precision were used. The alloys were modeled by supercell containing 128 randomly distributed atoms. At initial step the atoms occupy perfect BCC lattice sites. This initial structure was optimized until energy changes less than 1e-6 eV, while forces acting on atoms don't exceed 1e-2 eV/angstrom. The electron-ion interaction is described by PAW pseudopotentials. The calculations have been collected by sampling chemical compositions across the entire compositional range. The chemical compositions have been sampled by progressively changing the number of atoms per constituent by 4. For each chemical composition of binaries and ternaries, the first-principle calculations have been run for 100 randomized arrangements of the constituents on the BCC lattice sites. We collected data for a total of 3,040 randomized atomic structures over 31 chemical compositions. Further methodological and structural information is contained in the dataset README.txt file.

36 MATERIALS SCIENCE↗

Source function from two-particle correlation function through entropy-regularized Richardson-Lucy deblurring

Source functions are obtained from p – p and d – α correlation functions by applying the Richardson-Lucy (RL) deblurring to the Koonin-Pratt (KP) equation. To prevent fitting of noise in the correlation function, total-variation (TV) regularization is employed that has been effective in ordinary image restoration. TV alone cannot ensure normalization of the source functions. To ensure the latter, we propose a maximum-entropy regularized RL algorithm (MEM-RL). We outline the MEM-RL formalism and optimization strategy for the KP equation, demonstrating its effectiveness on both simulated and experimental data, including the p – p and d – α correlation functions.

62 RADIOLOGY AND NUCLEAR MEDICINE↗

NbTi_BCC_SolidSolution_128atoms_VASP6

We performed density functional theory (DFT) calculations for body-centered-cubic (BCC) structures with 128 lattices sites of solid solution binary alloys niobium-titanium (Nb-Ti). The electronic structures of alloys have been calculated using Vienna Ab initio Simulation Package (VASP), https://github.com/ORNL/ScalableWorkflow_VASP_Calculations. Within this package the DFT approach is used to reduce many-body Schrodinger equation to set of single particle Kohn-Sham (KS) equations. The generalized electronic exchange-correlation functional is described by generalized gradient approximation with the Perdew-Burke-Ernzerhof parametrization. The electron-ion interactions is described by pseudopotentials developed within the plane-wave basis projector augmented-wave (PAW) approach. These pseudopotentials are available at the VASP portal (http://cms.mpi.univie.ac.at/vasp/). Our calculations have been run with the pseudopotentials treating s and p semi-core states as valence in case for the elements Nb and Ti. The electronic densities and potentials are expanded over plane-waves with energy cutoff of 350 eV. 2x2x2 k-mesh and normal precision were used. The alloys were modeled by supercell containing 128 randomly distributed atoms. At initial step the atoms occupy perfect bcc lattice cites. This initial structure was optimized until energy changes less than 1e-6 eV, while forces acting on atoms don't exceed 1e-2 eV/angstrom. The electron-ion interaction is described by PAW pseudopotentials. The calculations have been collected by sampling chemical compositions across the entire compositional range. The chemical compositions have been sampled by progressively changing the number of atoms per constituent by 4. For each chemical composition of binaries and ternaries, the first-principle calculations have been run for 100 randomized arrangements of the constituents on the BCC lattice sites. We collected data for a total of 3,100 randomized atomic structures over 31 chemical compositions. The calculations have been collected on Air Force HPC11 cluster using the VASP 6.5.1. Additional methodology and file structure information is available in the dataset README.txt file.

36 MATERIALS SCIENCE↗

A physical basis for cosmological correlators from cuts

Significant progress has been made in our understanding of the analytic structure of FRW wavefunction coefficients, facilitated by the development of efficient algorithms to derive the differential equations they satisfy. Moreover, recent findings indicate that the twisted cohomology of the associated hyperplane arrangement defining FRW integrals overestimates the number of integrals required to define differential equations for the wave-function coefficient. We demonstrate that the associated dual cohomology is automatically organized in a way that is ideal for understanding and exploiting the cut/residue structure of FRW integrals. Utilizing this understanding, we develop a systematic approach to organize compatible sequential residues, which dictates the physical subspace of FRW integrals for any n -site, ℓ-loop graph. In particular, the physical subspace of tree-level FRW wavefunction coefficients is populated by differential forms associated to cuts/residues that factorize the integrand of the wavefunction coefficient into only flat space amplitudes. After demonstrating the validity of our construction using intersection theory, we develop simple graphical rules for cut tubings that enumerate the space of physical cuts and, consequently, differential forms without any calculation.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

An effective field theory approach to quarkonium at small transverse momentum

In this work we apply effective field theory (EFT) to observables in quarkonium production and decay that are sensitive to soft gluon radiation, in particular measurements that are sensitive to small transverse momentum. Within the EFT framework we study χQ decay to light quarks followed by the fragmentation of those quarks to light hadrons. We derive a factorization theorem that involves transverse momentum distribution (TMD) fragmentation functions and new quarkonium TMD shape functions. We derive renormalization group equations, both in rapidity and virtuality, which are used to evolve the different terms in the factorization theorem to resum large logarithms. This theoretical framework will provide a systematic treatment of quarkonium production and decay processes in TMD sensitive measurements.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Kinetics-Informed Neural Networks

Chemical kinetics and reaction engineering consists of the phenomenological framework for the disentanglement of reaction mechanisms, optimization of reaction performance and the rational design of chemical processes. Here, we utilize feed-forward artificial neural networks as basis functions to solve ordinary differential equations (ODEs) constrained by differential algebraic equations (DAEs) that describe microkinetic models (MKMs). We present an algebraic framework for the mathematical description and classification of reaction networks, types of elementary reaction, and chemical species. Under this framework, we demonstrate that the simultaneous training of neural nets and kinetic model parameters in a regularized multi-objective optimization setting leads to the solution of the inverse problem through the estimation of kinetic parameters from synthetic experimental data. We analyze a set of scenarios to establish the extent to which kinetic parameters can be retrieved from transient kinetic data, and assess the robustness of the methodology with respect to statistical noise. Furthermore, this approach to inverse kinetic ODEs can assist in the elucidation of reaction mechanisms based on transient data.

36 MATERIALS SCIENCE↗

Co-training of multiple neural networks for simultaneous optimization and training of physics-informed neural networks for composite curing

This paper introduces a Physics-Informed Neural Network (PINN) technique that co-trains neural networks (NNs) that represent each function in a system of equations to simultaneously solve equations representing an out-of-autoclave (OOA) cure process while conducting optimization in adherence to process requirements. Specifically, this co-training approach benefits from using NNs to represent OOA inputs (air temperature profile) and outputs (part and tool temperature profiles and degree of cure). Production requirements can then be levied on the inputs, such as maximum air temperature and minimum cure cycle, and simultaneously on the outputs, such as degree of cure, maximum part temperature, and part temperature rate limits. The technique is validated with finite element (FE) simulations and physical experiments for curing a Toray T830H-6 K/3900-2D composite panel. Furthermore, this novel approach efficiently models and optimizes the OOA cure process.

Composite curing↗

Time-discretization of a plasma-neutral MHD model with a semi-implicit leapfrog algorithm

The semi-implicit leapfrog time-discretization is a workhorse algorithm for initial-value MHD codes to bridge between vastly separated time scales. Inclusion of atomic interactions with neutrals breaks the functional structure of the MHD equations that exploited by the leapfrog. In this work, we address how to best integrate atomic physics into the semi-implicit leapfrog. Following the Crank-Nicolson method, one approach is to time-center the atomic interactions in the linear solver and use a Newton method to include the nonlinear contributions. Alternatively, another family of methods are based on operator-splitting the terms associated with the atomic interactions using a Strang-splitting technique. These methods naturally break equations into constituent ODE and PDE parts and preserve the structure exploited by the semi-implicit leapfrog. We study the accuracy and efficiency of these methods through a battery of 0D and 1D cases and show that a second-order-in-time Douglas-Rachford inspired coupling between the ODE and PDE advances is effective in reducing the time-discretization error to be comparable to that of Crank-Nicolson with Newton iteration of the nonlinear terms. Splitting ODE and PDE parts results in independent matrix solves for each field which reduces the computational cost considerably and provides parallelization over species relative to Crank-Nicolson.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

A Semi-Empirical Density Law for Ternary, Homogeneous PuCl 3 /HCl/H 2 O Solutions

Nuclear material operations pose unique hazards that are not encountered in other chemical, energy, or manufacturing industries. One of these hazards is the potential for a nuclear criticality accident when handling fissile isotopes such as 235 U and 239 Pu. These hazards are particularly high when fissile material is dissolved in solution as the neutron behaviors of the system can change rapidly with the physical and chemical changes accessible in solution. Current estimates of solution density used for criticality safety are outdated and hinder fissile material handling. Developing new estimates for these safety calculations requires experimental characterization and the derivation of empirical density models. We have derived a density law describing PuCl 3 /HCl/H 2 O solutions from experimental data characterizing solution density. Density data was treated using a Pitzer-derived eight-parameter equation, defining density as a function of analyte concentrations, temperature, and interactions between these variables. The model is predictive across the concentration and temperature ranges from which it was derived. The potential effects of varying oxidation states of plutonium, which are easily accessible in aqueous media, on the bulk solution density of the ternary system were also investigated. The resulting Pitzer-derived density law was applied to a nuclear criticality safety model, and the impact of the experimental characterization of solution density relative to previous estimates was demonstrated to be significant and suggest that the current approach to estimating density in nuclear criticality safety calculations may lead to overly conservative controls.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Thermal-fluctuation effects on small-scale statistics in turbulent gas flow

Kolmogorov's theory of turbulence assumes that the small-scale turbulent structures in the energy cascade are universal and are determined by the energy dissipation rate and the kinematic viscosity alone. However, thermal fluctuations, absent from the continuum description, terminate the energy cascade near the Kolmogorov length scale. Here, we propose a simple superposition model to account for the effects of thermal fluctuations on small-scale turbulence statistics. Furthermore, for compressible Taylor–Green vortex flow, we demonstrate that the superposition model in conjunction with data from direct numerical simulation of the Navier–Stokes equations yields spectra and structure functions that agree with the corresponding quantities computed from the direct simulation Monte Carlo method of molecular gas dynamics, verifying the importance of thermal fluctuations in the dissipation range.

42 ENGINEERING↗

Designing a quantum-accurate machine-learning potential to enable large-scale simulations of deuterium under shock

Large-scale molecular dynamics of deuterium under shock can elucidate kinetic processes vital to the target design in inertial confinement fusion and high-energy-density experiments. However, modeling the complex evolution of this material from an insulating molecular state at ambient pressure to an ionized, atomic fluid under strong shock is beyond the capability of simple pair and even bond order potentials. We thus train a quantum-accurate and broadly transferable machine-learning interatomic potential for deuterium using the Chebyshev Interaction Model for Efficient Simulations framework. We show that due to an improved description of the molecular-to-atomic transition, our model is able to better reproduce the ab initio equation of state, radial distribution functions, and principal Hugoniot than bond order potentials. This represents an important step toward large-scale quantum-accurate and nonequilibrium simulations of complicated systems under dynamic changes including phase transitions.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Phonon Olympics: Phonon property and lattice thermal conductivity benchmarking from open-source packages

Three widely used open-source packages for determining phonon properties and lattice thermal conductivities (ALAMODE, phono3py, and ShengBTE) are benchmarked by teams of expert users and the package developers. The phonons for Ge, RbBr, monolayer MoSe 2 , and AlN are modeled at zero temperature, and they scatter through three-phonon and phonon-isotope processes, with thermal conductivities obtained from the linearized Peierls–Boltzmann transport equation with input from density functional theory calculations. Over a wide range of temperatures, the thermal conductivities calculated by the teams fall within at most ±15% of their mean values for each of the four materials. The phonon frequencies, obtained from the harmonic force constants, do not show large differences between the calculations, indicating that the modal heat capacities and group velocities are not responsible for the thermal conductivity variations. It is the lifetimes associated with three-phonon scattering, obtained from the cubic force constants, that drive the variations. The many decisions required to calculate the cubic force constants (e.g., supercell size, atomic displacement, neighbor cutoff, and application of symmetries) make identification of the precise origin of the thermal conductivity variations challenging. The calculated thermal conductivities do not generally show agreement with experimental measurements, which is attributed to the limitations of the density functional theory calculations. Guidance for the development of best practices is provided, which will help to standardize protocols needed for building thermal conductivity databases. The results provide a baseline for future benchmarking of other packages and more advanced calculations.

McGaughey, Alan J. H. [Carnegie Mellon Univ., Pitt↗

Atomic forces from Dirac-Kohn-Sham equations: Implementation in flexible (APW+lo/LAPW)+LO basis set

The atomic forces formulation based on the Dirac-Kohn-Sham equation and the flexible (APW+lo/LAPW)+LO basis set is presented here. The formulation was implemented in the code FlapwMBPT and allows a user to easily switch between different basis functions of the augmentation type (APW or LAPW) and between different kind of local orbitals. Similar to the work (Phys.Rev.B 91 (2015) 035105), the implementation takes into account small discontinuities of the wave functions, density, and potential at the muffin-tin sphere boundaries. Applications to the materials with strong relativistic effects, such as $\alpha$-Uranium, PuCoGa$_{5}$, and FePt, demonstrate robustness of the method. The comparison of the calculated forces with the ones obtained by numerical differentiation of free energy shows close agreement with deviations about 0.1% or less.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Phase-space sensitivity (weight functions) of 3 MeV proton diagnostics

Abstract The weight function W ( X ) is a diagnostic sensitivity to phase-space variables X that relates the measured signal C to the distribution function F ( X ) through the equation C = ∫ W ( X ) F ( X ) d X . In the present work, an algorithm to calculate W for a diagnostic that measures 3 MeV protons produced in d(d,p)t fusion reactions between a fast ion and a thermal deuteron is developed. The emitted protons escape the tokamak on curved orbits and are detected. These curved orbits constitute effective diagnostic ‘sightlines’. The presented algorithm accounts for the complications associated with these curved sightlines. An initial calculation of time-reversed proton orbits computes effective solid angles and sightlines for the relevant range of incident proton velocity vectors. These precomputed orbits are then used within the framework of FIDASIM (2020 Plasma Phys. Cont. Fusion 62 105008) to calculate the reactivity averaged over the thermal distribution of the ‘target’ deuterons and the probability that a fast ion of specified energy and pitch has a gyroangle that is consistent with the kinematic equations along each of the sightlines. Comparisons with analytic formulas and with independent calculations for the Mega Amp Spherical Tokamak 3 MeV proton diagnostic verify the algorithm.

Physics↗

Muon-induced fission as a probe of the underlying dynamics in nuclear fission

Muon-induced fission could be utilized as a probe to study the underlying dynamics of nuclear fission. Here, the probability of muon attachment to the light asymmetric fission fragment is sensitive to fission dynamics, such as the timescale and friction of the fission event, charge asymmetry, and possibly the shape of the fission fragments. We focus on muonic atoms that are formed with actinide nuclei. A relativistic approach is employed, solving the Dirac equation for the muonic wave function in the presence of a time-dependent electromagnetic field generated by the fissioning nucleus. Computations are carried out on a three-dimensional Cartesian lattice with no symmetry assumptions. The results show a strong dependence of the attachment probability on the fission charge asymmetry and a more modest dependence on friction.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

Elimination of the linearization error in APW/LAPW basis set: Dirac-Kohn-Sham equations

Here, a detailed account of the implementation of equations of the relativistic density functional theory (RDFT) using basis sets of APW/LAPW type with flexible extensions provided by local orbitals is given. Earlier discoveries of the importance of the high derivative local orbital (HDLO) extension of the APW/LAPW basis set for enhancing the accuracy of DFT calculations are confirmed using a fully relativistic approach and α – U as an example. High energy local orbitals (HELO's), however indispensable for GW calculations, are considerably less efficient in enhancing the accuracy of DFT applications. It is shown that a simplified approach to the relativistic effects, namely considering them only inside the muffin-tin (MT) spheres, produces basically identical results (as compared to fully relativistic approach) for the electronic free energy of the five materials considered in this work. By comparing the effect of the simplified approach on the electronic free energy with its effect on the electronic kinetic energy we conclude that the insensitivity of the free energy to the way we describe the relativistic effects in the interstitial region is related to the variational property of this quantity.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗