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NASA NTRS2006

Semi-Analytic Reconstruction of Flux in Finite Volume Formulations

Semi-analytic reconstruction uses the analytic solution to a second-order, steady, ordinary differential equation (ODE) to simultaneously evaluate the convective and diffusive flux at all interfaces of a finite volume formulation. The second-order ODE is itself a linearized approximation to the governing first- and second- order partial differential equation conservation laws. Thus, semi-analytic reconstruction defines a family of formulations for finite volume interface fluxes using analytic solutions to approximating equations. Limiters are not applied in a conventional sense; rather, diffusivity is adjusted in the vicinity of changes in sign of eigenvalues in order to achieve a sufficiently small cell Reynolds number in the analytic formulation across critical points. Several approaches for application of semi-analytic reconstruction for the solution of one-dimensional scalar equations are introduced. Results are compared with exact analytic solutions to Burger s Equation as well as a conventional, upwind discretization using Roe s method. One approach, the end-point wave speed (EPWS) approximation, is further developed for more complex applications. One-dimensional vector equations are tested on a quasi one-dimensional nozzle application. The EPWS algorithm has a more compact difference stencil than Roe s algorithm but reconstruction time is approximately a factor of four larger than for Roe. Though both are second-order accurate schemes, Roe s method approaches a grid converged solution with fewer grid points. Reconstruction of flux in the context of multi-dimensional, vector conservation laws including effects of thermochemical nonequilibrium in the Navier-Stokes equations is developed.

Gnoffo, Peter A.↗

DOE OSTI2016

Molecular dynamics simulations of hydrogen diffusion in aluminum

In this study, hydrogen diffusion impacts the performance of solid-state hydrogen storage materials and contributes to the embrittlement of structural materials under hydrogen-containing environments. In atomistic simulations, the diffusion energy barriers are usually calculated using molecular statics simulations where a nudged elastic band method is used to constrain a path connecting the two end points of an atomic jump. This approach requires prior knowledge of the “end points”. For alloy and defective systems, the number of possible atomic jumps with respect to local atomic configurations is tremendous. Even when these jumps can be exhaustively studied, it is still unclear how they can be combined to give an overall diffusion behavior seen in experiments. Here we describe the use of molecular dynamics simulations to determine the overall diffusion energy barrier from the Arrhenius equation. This method does not require information about atomic jumps, and it has additional advantages, such as the ability to incorporate finite temperature effects and to determine the pre-exponential factor. As a test case for a generic method, we focus on hydrogen diffusion in bulk aluminum. We find that the challenge of this method is the statistical variation of the results. However, highly converged energy barriers can be achieved by an appropriate set of temperatures, output time intervals (for tracking hydrogen positions), and a long total simulation time. Our results help elucidate the inconsistencies of the experimental diffusion data published in the literature. The robust approach developed here may also open up future molecular dynamics simulations to rapidly study diffusion properties of complex material systems in multidimensional spaces involving composition and defects.

36 MATERIALS SCIENCE↗

DOE OSTI2017

Two-dimensional Radiative Magnetohydrodynamic Simulations of Partial Ionization in the Chromosphere. II. Dynamics and Energetics of the Low Solar Atmosphere

We investigate the effects of interactions between ions and neutrals on the chromosphere and overlying corona using 2.5D radiative MHD simulations with the Bifrost code. We have extended the code capabilities implementing ion–neutral interaction effects using the generalized Ohm’s law, i.e., we include the Hall term and the ambipolar diffusion (Pedersen dissipation) in the induction equation. Our models span from the upper convection zone to the corona, with the photosphere, chromosphere, and transition region partially ionized. Our simulations reveal that the interactions between ionized particles and neutral particles have important consequences for the magnetothermodynamics of these modeled layers: (1) ambipolar diffusion increases the temperature in the chromosphere; (2) sporadically the horizontal magnetic field in the photosphere is diffused into the chromosphere, due to the large ambipolar diffusion; (3) ambipolar diffusion concentrates electrical currents, leading to more violent jets and reconnection processes, resulting in (3a) the formation of longer and faster spicules, (3b) heating of plasma during the spicule evolution, and (3c) decoupling of the plasma and magnetic field in spicules. Our results indicate that ambipolar diffusion is a critical ingredient for understanding the magnetothermodynamic properties in the chromosphere and transition region. The numerical simulations have been made publicly available, similar to previous Bifrost simulations. This will allow the community to study realistic numerical simulations with a wider range of magnetic field configurations and physics modules than previously possible.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2023

Automated Symbolic Upscaling: 1. Model Generation for Extended Applicability Regimes

Abstract In porous media theory, upscaling techniques are fundamental to deriving rigorous Darcy‐scale models for flow and reactive transport in subsurface systems. Due to limitations in classical techniques, a number of ad hoc approaches have been proposed to address physical regimes in which reactive time scales are similar to, or faster than, diffusive time scales. In Part 1 of this two part series, we present a strategy for expanding the applicability of classical homogenization theory by generalizing the assumed closure form. We detail the implementation of this strategy on two reactive mass transport problems with moderately reactive physics. The strategy produces nontrivial homogenized models with emergent terms and effective parameters that couple reactive, diffusive, and advective transport. The differences in equation forms between the macroscopic and pore‐scale descriptions advise caution to further studies where the forms of macroscopic equations are assumed, as opposed to rigorously derived. Numerical validation is provided for each problem to show that the error estimates of homogenization theory are satisfied, and to justify the implemented strategy. In Part 2, the presented strategy is automated using symbolic computing to expedite its implementation.

Pietrzyk, Kyle↗

DOE OSTI2014

Simulations of singlet exciton diffusion in organic semiconductors: a review

Our review describes the various aspects of simulation strategies for exciton diffusion in condensed phase thin films of organic semiconductors. Several methods for calculating energy transfer rate constants are discussed along with procedures for how to account for energetic disorder. Exciton diffusion can be modelled by using kinetic Monte-Carlo methods or master equations. Recent literature on simulation efforts for estimating exciton diffusion lengths of various conjugated polymers and small molecules are introduced. Moreover, these studies are discussed in the context of the effects of morphology on exciton diffusion and the necessity of accurate treatment of disorder for comparison of simulation results with those of experiment.

14 SOLAR ENERGY↗

DOE OSTI2020

Theoretical formulation of phase space microbunching instability in the presence of intrabeam scattering for single-pass or recirculation accelerators

Microbunching instability (MBI) has been one of the most challenging issues in designs of high-brightness beam transport lines for single-pass or recirculating accelerators. Although the intrabeam scattering (IBS) has long been studied in lepton or hadron storage rings as a slow diffusion process or in high-intensity proton linear accelerators as one mechanism for the beam halo, the effects of IBS on single-pass or recirculating electron accelerators have drawn attention only in the recent two decades due to emergence of linac-based or energy-recovery-linac-based fourth-generation light sources, which require high-quality electron beams during the beam transport. In this paper we develop a theoretical formulation of microbunching instability in the presence of IBS for single-pass or recirculation accelerators. To quantify MBI with inclusion of IBS, we start from the Vlasov-Fokker-Planck (VFP) equation, combining both collective interactions and incoherent IBS effects. The linearized VFP equation and the corresponding friction and diffusion coefficients are derived. The evolutions of the resultant density and energy modulations are formulated as a set of coupled integral equations. The theoretical formulation is then applied to a recirculating beamline design. The results from the semianalytical calculation are compared and show good agreement with massive particle tracking simulations.

43 PARTICLE ACCELERATORS↗

DOE OSTI2015

Theory of activated penetrant diffusion in viscous fluids and colloidal suspensions

We heuristically formulate a microscopic, force level, self-consistent nonlinear Langevin equation theory for activated barrier hopping and non-hydrodynamic diffusion of a hard sphere penetrant in very dense hard sphere fluid matrices. Penetrant dynamics is controlled by a rich competition between force relaxation due to penetrant self-motion and collective matrix structural (alpha) relaxation. In the absence of penetrant-matrix attraction, three activated dynamical regimes are predicted as a function of penetrant-matrix size ratio which are physically distinguished by penetrant jump distance and the nature of matrix motion required to facilitate its hopping. The penetrant diffusion constant decreases the fastest with size ratio for relatively small penetrants where the matrix effectively acts as a vibrating amorphous solid. Increasing penetrant-matrix attraction strength reduces penetrant diffusivity due to physical bonding. For size ratios approaching unity, a distinct dynamical regime emerges associated with strong slaving of penetrant hopping to matrix structural relaxation. A crossover regime at intermediate penetrant-matrix size ratio connects the two limiting behaviors for hard penetrants, but essentially disappears if there are strong attractions with the matrix. Activated penetrant diffusivity decreases strongly with matrix volume fraction in a manner that intensifies as the size ratio increases. We propose and implement a quasi-universal approach for activated diffusion of a rigid atomic/molecular penetrant in a supercooled liquid based on a mapping between the hard sphere system and thermal liquids. Calculations for specific systems agree reasonably well with experiments over a wide range of temperature, covering more than 10 orders of magnitude of variation of the penetrant diffusion constant.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2016

Exact results on diffusion in a piecewise linear potential with a time-dependent sink

The Smoluchowski equation with a time-dependent sink term is solved exactly. In this method, knowing the probability distribution P(0, s) at the origin, allows deriving the probability distribution P(x, s) at all positions. Exact solutions of the Smoluchowski equation are also provided in different cases where the sink term has linear, constant, inverse, and exponential variation in time.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2014

Diffusion Processes Satisfying a Conservation Law Constraint

We investigate coupled stochastic differential equations governing N non-negative continuous random variables that satisfy a conservation principle. In various fields a conservation law requires that a set of fluctuating variables be non-negative and (if appropriately normalized) sum to one. As a result, any stochastic differential equation model to be realizable must not produce events outside of the allowed sample space. We develop a set of constraints on the drift and diffusion terms of such stochastic models to ensure that both the non-negativity and the unit-sum conservation law constraint are satisfied as the variables evolve in time. We investigate the consequences of the developed constraints on the Fokker-Planck equation, the associated system of stochastic differential equations, and the evolution equations of the first four moments of the probability density function. We show that random variables, satisfying a conservation law constraint, represented by stochastic diffusion processes, must have diffusion terms that are coupled and nonlinear. The set of constraints developed enables the development of statistical representations of fluctuating variables satisfying a conservation law. We exemplify the results with the bivariate beta process and the multivariate Wright-Fisher, Dirichlet, and Lochner’s generalized Dirichlet processes.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2017

Numerical method for angle-of-incidence correction factors for diffuse radiation incident photovoltaic modules

Here, a numerical method is provided for solving the integral equation for the angle-of-incidence (AOI) correction factor for diffuse radiation incident photovoltaic (PV) modules. The types of diffuse radiation considered include sky, circumsolar, horizon, and ground-reflected. The method permits PV module AOI characteristics to be addressed when calculating AOI losses associated with diffuse radiation. Pseudo code is provided to aid users in the implementation, and results are shown for PV modules with tilt angles from 0° to 90°. Diffuse AOI losses are greatest for small PV module tilt angles. Including AOI losses associated with the diffuse irradiance will improve predictions of PV system performance.

14 SOLAR ENERGY↗

DOE OSTI2022

Assessment of diffuse-interface methods for compressible multiphase fluid flows and elastic-plastic deformation in solids

This work describes three diffuse-interface methods for the simulation of immiscible, compressible multiphase fluid flows and elastic-plastic deformation in solids. The first method is the localized-artificial-diffusivity approach of Cook, Subramaniam et al., and Adler and Lele, in which artificial diffusion terms are added to the individual phase mass fraction transport equations and are coupled with the other conservation equations. The second method is the gradient-form approach that is based on the quasi-conservative method of Shukla et al., in which the diffusion and sharpening terms (together called regularization terms) are added to the individual phase volume fraction transport equations and are coupled with the other conservation equations. The third approach is the divergence-form approach that is based on the fully conservative method of Jain et al., in which the regularization terms are added to the individual phase volume fraction transport equations and are coupled with the other conservation equations. In the present study, all three diffuse-interface methods are used in conjunction with a four-equation, multicomponent mixture model, in which pressure and temperature equilibria are assumed among the various phases. The primary objective of this work is to compare these three methods in terms of their ability to: maintain constant interface thickness throughout the simulation; conserve mass, momentum, and energy; and maintain accurate interface shape for long-time integration. The second objective of this work is to consistently extend these methods to model interfaces between solid materials with strength. To assess and compare the methods, they are used to simulate a wide variety of problems, including (1) advection of an air bubble in water, (2) shock interaction with a helium bubble in air, (3) shock interaction and the collapse of an air bubble in water, and (4) Richtmyer–Meshkov instability of a copper–aluminum interface. The current work focuses on comparing these methods in the limit of relatively coarse grid resolution, which illustrates the true performance of these methods. In conclusion, this is because it is rarely practical to use hundreds of grid points to resolve a single bubble or drop in large-scale simulations of engineering interest.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2016

Characteristics of a DC discharge with a water cathode in argon

The characteristics of a dc discharge excited between a metal anode and a water cathode in argon were studied experimentally. The dimensions of the positive column and the electric field in it were measured, and the vibrational temperature in the positive column was determined from the N{sub 2}C{sup 3}Π{sub u} → B{sup 3}Π{sub g} (0–2) emission band. It is shown that the power deposited in the positive column is almost entirely spent on gas heating. The obtained dependence of the reduced electric field on the gas pressure and the ionization frequencies calculated by solving the Boltzmann equation indicate that electrons are lost diffusively, whereas ionization proceeds in a stepwise manner via the lower metastable states of argon atoms.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2014

Electrical transport properties and interaction of heteropolyacid in N,N-dimethylformamide by dielectric spectroscopy

We present a dielectric spectroscopy study on binary electrolyte solutions composed of N,N-Dimethylformamide (DMF) and heteropolyacid (HPA). The HPA is composed of H{sup +} and Keggin polyoxometalate anion, [XW{sub 12}O{sub 40}]{sup n−}(X = P, Si, with n = 3, 4). Conductivity was used to calculate the transport parameters such as diffusion coefficient and hydrodynamic radius of polyoxometalate (POM) anions. The hydrodynamic radius of P{sub 12}W{sub 40}{sup 3−} and Si{sub 12}W{sub 40}{sup 4−} are 5.552 Å and 5.526 Å, respectively, indicating that these POM anions remain unsolvated in DMF. The temperature dependences of conductivity obeyed the Arrhenius equation, suggesting that the activation energy of proton conduction process is independent with temperature. The calculated activation energy shows that the conduction process follows Grotthuss mechanism, i.e., proton is transported by hydrogen bond between DMF molecules. One dielectric relaxation observed around 10 MHz is considered to be associated with the POM-DMFH{sup +} ion-pairs formed by POM and protonated DMF. The interaction between anion and cation in POM-DMFH{sup +} ion-pairs are explained theoretically based on the Bruggeman's mixture theories and dipole rotation model. From this, the distance between POM and DMFH{sup +} are estimated as about 1.1 nm.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2012

A GENERALIZED DIFFUSION TENSOR FOR FULLY ANISOTROPIC DIFFUSION OF ENERGETIC PARTICLES IN THE HELIOSPHERIC MAGNETIC FIELD

The spatial diffusion of cosmic rays in turbulent magnetic fields can, in the most general case, be fully anisotropic, i.e., one has to distinguish three diffusion axes in a local, field-aligned frame. We reexamine the transformation for the diffusion tensor from this local to a global frame, in which the Parker transport equation for energetic particles is usually formulated and solved. Particularly, we generalize the transformation formulae to allow for an explicit choice of two principal local perpendicular diffusion axes. This generalization includes the 'traditional' diffusion tensor in the special case of isotropic perpendicular diffusion. For the local frame, we describe the motivation for the choice of the Frenet-Serret trihedron, which is related to the intrinsic magnetic field geometry. We directly compare the old and the new tensor elements for two heliospheric magnetic field configurations, namely the hybrid Fisk and Parker fields. Subsequently, we examine the significance of the different formulations for the diffusion tensor in a standard three-dimensional model for the modulation of galactic protons. For this, we utilize a numerical code to evaluate a system of stochastic differential equations equivalent to the Parker transport equation and present the resulting modulated spectra. The computed differential fluxes based on the new tensor formulation deviate from those obtained with the 'traditional' one (only valid for isotropic perpendicular diffusion) by up to 60% for energies below a few hundred MeV depending on heliocentric distance.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2026

Regularization by denoising diffusion models for solving inverse PDE problems with application to full waveform inversion

Partial differential equation (PDE)-governed inverse problems are fundamental across various scientific and engineering applications; yet they face significant challenges due to nonlinearity, ill-posedness, and sensitivity to noise. Here, we introduce a computational framework, regularization by denoising using diffusion models for partial differential equations (RED-DiffEq), by integrating physics-driven inversion and data-driven learning. RED-DiffEq leverages pretrained diffusion models as a regularization mechanism for PDE-governed inverse problems. We apply RED-DiffEq to solve the full waveform inversion problem in geophysics, a challenging seismic imaging technique that seeks to reconstruct high-resolution subsurface velocity models from seismic measurement data. Our method shows enhanced accuracy and robustness compared to benchmark methods. Additionally, it exhibits strong generalization and domain decomposition capacity, enabling the inversion of more complex velocity models with larger domains than those used in training the diffusion model. Our framework can also be directly applied to diverse PDE-governed inverse problems.

Shan, Siming [Yale University, New Haven, CT (Unit↗

DOE OSTI2019

A Fokker–Planck approach to a moment closure for mixing in variable-density turbulence

We develop a theory for the cascade mixing terms in a moment closure approach to binary active scalar mixing in variable-density turbulence. To address the variable-density complications we apply, as a principle and constraint, the conservation of the probability density function (PDF) through a Fokker–Planck equation with bounded sample space whose attractor is the beta PDF with skewness. Mixing is related to a single-point PDF as a realisability principle to provide mathematically rigorous expressions for the small scale statistics in terms of largescale moments. The problem of the unknown small-scale mixing is replaced with the determination of the drift and diffusion terms of a Fokker–Planck equation in a beta-PDF-convergent stochastic process. We find that realisability of a beta-convergent process requires the mixing time-scale ratio, taken as a constant in passive scalar mixing, to be a function of the mean mass fraction, mean fluid density, the Atwood number, the density-volume correlation and moments of the density field. We develop and compare the new model with direct numerical simulations data of non-stationary homogeneous variable-density turbulence.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2011

Grating formation by a high power radio wave in near-equator ionosphere

The formation of a volume grating in the near-equator regions of ionosphere due to a high power radio wave is investigated. The radio wave, launched from a ground based transmitter, forms a standing wave pattern below the critical layer, heating the electrons in a space periodic manner. The thermal conduction along the magnetic lines of force inhibits the rise in electron temperature, limiting the efficacy of heating to within a latitude of few degrees around the equator. The space periodic electron partial pressure leads to ambipolar diffusion creating a space periodic density ripple with wave vector along the vertical. Such a volume grating is effective to cause strong reflection of radio waves at a frequency one order of magnitude higher than the maximum plasma frequency in the ionosphere. Linearly mode converted plasma wave could scatter even higher frequency radio waves.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2021

Challenging Practices of Algebraic Battery Life Models through Statistical Validation and Model Identification via Machine-Learning

Various modeling techniques are used to predict the capacity fade of Li-ion batteries. Algebraic reduced-order models, which are inherently interpretable and computationally fast, are ideal for use in battery controllers, technoeconomic models, and multi-objective optimizations. For Li-ion batteries with graphite anodes, solid-electrolyte-interphase (SEI) growth on the graphite surface dominates fade. This fade is often modeled using physically informed equations, such as square-root of time for predicting solvent-diffusion limited SEI growth, and Arrhenius and Tafel-like equations predicting the temperature and state-of-charge rate dependencies. In some cases, completely empirical relationships are proposed. However, statistical validation is rarely conducted to evaluate model optimality, and only a handful of possible models are usually investigated. This article demonstrates a novel procedure for automatically identifying reduced-order degradation models from millions of algorithmically generated equations via bi-level optimization and symbolic regression. Identified models are statistically validated using cross-validation, sensitivity analysis, and uncertainty quantification via bootstrapping. On a LiFePO 4 /Graphite cell calendar aging data set, automatically identified models utilizing square-root, power law, stretched exponential, and sigmoidal functions result in greater accuracy and lower uncertainty than models identified by human experts, and demonstrate that previously known physical relationships can be empirically "rediscovered" using machine learning.

25 ENERGY STORAGE↗