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All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Robust Implicit Adaptive Low Rank Time-Stepping Methods for Matrix Differential Equations

In this work, we develop implicit rank-adaptive schemes for time-dependent matrix differential equations. The dynamic low rank approximation (DLRA) is a well-known technique to capture the dynamic low rank structure based on Dirac–Frenkel time-dependent variational principle. In recent years, it has attracted a lot of attention due to its wide applicability. Our schemes are inspired by the three-step procedure used in the rank adaptive version of the unconventional robust integrator (the so called BUG integrator) (Ceruti et al. in BIT Numer Math 62(4):1149–1174, 2022) for DLRA. First, a prediction (basis update) step is made computing the approximate column and row spaces at the next time level. Second, a Galerkin evolution step is invoked using an implicit solves for the small core matrix. Finally, a truncation is made according to a prescribed error threshold. Since the DLRA is evolving the differential equation projected on to the tangent space of the low rank manifold, the error estimate of the BUG integrator contains the tangent projection (modeling) error which cannot be easily controlled by mesh refinement. This can cause convergence issue for equations with cross terms. To address this issue, we propose a simple modification, consisting of merging the row and column spaces from the explicit step truncation method together with the BUG spaces in the prediction step. In addition, we propose an adaptive strategy where the BUG spaces are only computed if the residual for the solution obtained from the prediction space by explicit step truncation method, is too large. Here, we prove stability and estimate the local truncation error of the schemes under assumptions. We benchmark the schemes in several tests, such as anisotropic diffusion, solid body rotation and the combination of the two, to show robust convergence properties.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

A fully implicit, asymptotic-preserving, semi-Lagrangian algorithm for the time dependent anisotropic heat transport equation

In this paper, we extend the operator-split asymptotic-preserving, semi-Lagrangian algorithm for time dependent anisotropic heat transport equation proposed in Chacón et al. (2014) [18] to use a fully implicit time integration with backward differentiation formulas. The proposed implicit method can deal with arbitrary heat-transport anisotropy ratios $\mathcal{X}$∥ /$ \mathcal{X}$⟂ $\ggg$ 1 (with $\mathcal{X}$∥, $ \mathcal{X}$⟂ the parallel and perpendicular heat diffusivities, respectively) in complicated magnetic field topologies in an accurate and efficient manner. Further, the implicit algorithm is second-order accurate temporally and demonstrates an accurate treatment at boundary layers (e.g., island separatrices), which was not ensured by the operator-split implementation. The condition number of the resulting algebraic system is independent of the anisotropy ratio, and is inverted with preconditioned GMRES. We propose a simple preconditioner that renders the finite-dimensional linear operator compact, resulting in mesh-independent convergence rates for topologically simple magnetic fields, and convergence rates scaling as ~ (NΔt) 1/4 (with N the total mesh size and Δt the timestep) in topologically complex magnetic-field configurations. We demonstrate the accuracy and performance of the approach with test problems of varying complexity, including an analytically tractable boundary-layer problem in a straight magnetic field, and a topologically complex magnetic field featuring magnetic islands with extreme anisotropy ratios $\mathcal{X}$∥ /$ \mathcal{X}$⟂ = 10 10 ) .

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Inertial and anisotropic pressure effects on cross-field electron transport in low-temperature magnetized plasmas

In this paper, a one-dimensional (1D) particle-in-cell Monte Carlo collision (PIC-MCC) model is developed to investigate the effects of anisotropic pressure and inertial terms due to non-Maxwellian velocity distribution functions on cross-field electron transport. The conservation of momentum is evaluated by taking the moments of the first-principles gas-kinetic equation. A steady-state discharge is obtained without any low-frequency ionization oscillations by considering an anomalous electron scattering profile. Further, the results obtained from the 1D PIC-MCC model are compared with fluid models, including the quasi-neutral drift-diffusion (DD), non-neutral DD, and full fluid moment models. The discharge current obtained from the PIC-MCC model is in good agreement with the fluid models. The cross-field electron transport due to the inertial terms, i.e. the gradient of axial and azimuthal drift, is evaluated. Moreover, PIC-MCC simulation results show non-zero, anisotropic, off-diagonal pressure tensor terms due to asymmetric non-Maxwellian electron velocity distribution function, potentially contributing to cross-field electron transport.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2022

Mesoscopic finite-size effects of unconventional electron transport in PdCoO 2

A wide range of unconventional transport phenomena has recently been observed in single-crystal delafossite metals. Here, we present a theoretical framework to elucidate electron transport using a combination of first-principles calculations and numerical modeling of the anisotropic Boltzmann transport equation. Using PdCoO 2 as a model system, we study different microscopic electron and phonon scattering mechanisms and establish the mean free path hierarchy of quasiparticles at different temperatures. We treat the anisotropic Fermi surface explicitly to numerically obtain experimentally-accessible transport observables, which bridge between the “diffusive,” “ballistic,” and “hydrodynamic” transport regime limits. Further, we illustrate that the distinction between the “quasiballistic” and “quasihydrodynamic” regimes is challenging and often needs to be quantitative in nature. From first-principles calculations, we populate the resulting transport regime plots and demonstrate how the Fermi surface orientation adds complexity to the observed transport signatures in micrometer-scale devices. Our work provides key insights into microscopic interaction mechanisms on open hexagonal Fermi surfaces and establishes their connection to the macroscopic electron transport in finite-size channels.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Aggregation of solutes in bosonic versus fermionic quantum fluids

Quantum fluid droplets made of helium-3 ( 3 He) or helium-4 ( 4 He) isotopes have long been considered as ideal cryogenic nanolabs, enabling unique ultracold chemistry and spectroscopy applications. The droplets were believed to provide a homogeneous environment in which dopant atoms and molecules could move and react almost as in free space but at temperatures close to absolute zero. Here, we report ultrafast x-ray diffraction experiments on xenon-doped 3 He and 4 He nanodroplets, demonstrating that the unavoidable rotational excitation of isolated droplets leads to highly anisotropic and inhomogeneous interactions between the host matrix and enclosed dopants. Superfluid 4 He droplets are laced with quantum vortices that trap the embedded particles, leading to the formation of filament-shaped clusters. In comparison, dopants in 3 He droplets gather in diffuse, ring-shaped structures along the equator. The shapes of droplets carrying filaments or rings are direct evidence that rotational excitation is the root cause for the inhomogeneous dopant distributions.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

Superior discretizations and AMG solvers for extremely anisotropic diffusion via hyperbolic operators [Slides]

Diffusion in magnetic confinement fusion is extremely anisotropic in the direction of field lines. Rewrote diffusion system based on directional gradients, apply discretization and solver techniques developed for advection. Orders of magnitude decrease in error and solve wallclock time vs. traditional methods. The next steps include: (1) incorporate into larger MHD simulations, (2) better solvers for closed field lines or mixed regimes, and (3) possibly other extremely anisotropic equations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Mechanochemical topological defects in an active nematic

We propose a reaction-diffusion system that converts topological information of an active nematic into chemical signals. We show that a curvature-activated reaction dipole is sufficient for creating a system that dynamically senses topology by producing a concentration field possessing local extrema coinciding with ±$\frac{1}{2}$ defects. The enabling term is analogous to polarization charge density seen in dielectric materials. We demonstrate the ability of this system to identify defects in both passive and active nematics. Our results illustrate that a relatively simple feedback scheme, expressed as a system of partial differential equations, is capable of producing chemical signals in response to inherently nonlocal structures in anisotropic media. Here, we posit that such coarse-grained systems can help generate testable hypotheses for regulated processes in biological systems, such as morphogenesis, and motivate the creation of bio-inspired materials that utilize dynamic coupling between nematic structure and biochemistry.

42 ENGINEERING↗
DOE OSTI2022

A model of thermal creep and annealing in finite domains based on coupled dislocation climb and vacancy diffusion

Here, we develop a framework to investigate thermal creep and annealing in finite domains, where the climb motion of discrete dislocations is coupled to the diffusion of a continuum vacancy field. The model is first formulated in a continuum finite-deformation setting. All governing equations and boundary conditions are obtained from a unified irreversible thermodynamics principle. The resulting model couples a mechanical boundary value problem (BVP), a vacancy diffusion BVP, and the climb and glide motion of the discrete dislocation network within the crystal. The framework is then linearized for implementation in three-dimensional (3D) discrete dislocation dynamics (DDD) simulations for arbitrary anisotropic crystals. A solution scheme is developed based on the superposition principle, which is imposed weakly on the dislocation network to obtain a Galerkin solution for the nodal climb velocities. The framework includes diffusional (Nabarro–Herring) creep deformation as well as dislocation creep by climb-assisted-glide. The method is applied to simulate the annealing of vacancy loops in Al, with good agreement to experimental measurements by Silcox and Hirsch. We further consider the effects of annealing under stress, and of the proximity of the vacancy loops to loaded and free boundaries Simulations in polycrystalline materials are carried out to highlight the effects of the grain size on dislocation climb and vacancy loop annealing. The method is also applied to estimate the creep rate due to climb-assisted glide of jogged-screw dislocations in γ-TiAl, and results are compared to experiments by Viswanathan et al. Finally, we discuss the effects of uniaxial and hydrostatic stresses on the two diffusive deformation pathways of the material, namely Nabarro–Herring creep and dislocation climb.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Monte-Carlo modeling of phonon thermal transport using DFT-based anisotropic dispersion relations over the full Brillouin zone

In this work, we present a Monte Carlo (MC) approach to solve the phonon Boltzmann transport equation (BTE) in which the anisotropic phonon dispersion relations over the full Brillouin zone (BZ) are used. In this approach, the discretization of the BZ used to compute the phonon relaxation time places constraints on the direction of scattered phonons in the real-space simulation domain. The phonon dispersion and phonon relaxation times are calculated using the density functional theory (DFT) approach. The modified MC approach is validated by a close examination of its ability to simulate phonon transport in both the ballistic and diffusive regimes for multiple materials including GaAs, InAs, ThO 2 , and α-U. In doing so, the phonon thermal conductivities from 100 K to 1000 K are calculated and compared with traditional non-transport solution of the phonon BTE. It is found that the phonon thermal conductivities of α-U and ThO 2 obtained from MC simulations using isotropic dispersion are larger than the values obtained using anisotropic phonon dispersion relations over the full BZ. The effect of phonon-defect scattering on the thermal conductivity of ThO 2 is also studied as an application of the current MC approach and found to agree with previously computed values in the literature. The MC solver developed here has been parallelized as a step to demonstrate its potential to solving computationally intensive phonon thermal transport problems at the mesoscale.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Fourier-like Thermal Relaxation of Nanoscale Explosive Hot Spots

Hot spots are local regions of high temperature that are widely considered to govern explosive initiation. Hot spot dynamics rests on a delicate balance between heat generation due to chemical reactions and heat loss through thermal conduction, making accurate determinations of the conductivity under extreme conditions a key component of predictive explosive models. We develop here an approach to directly determine the thermal transport properties of explosive hot spots with realistic initial structures through a combination of molecular dynamics (MD) and diffusive heat equation (HEq) modeling. Effective thermal conductivity values are determined by fitting HEq models to MD predictions of long timescale hot spot relaxation. The approach is applied to model hot spots in the molecular crystalline explosive 1,3,5-triamino-2,4,6-trinitrobenzene (TATB) for a range of shock strengths and two limiting cases for impact orientation. Isotropic and anisotropic HEq models yield similar results, despite TATB exhibiting some of the largest and most anisotropic thermal conductivity values for explosive near normal conditions. The conductivity is found to be a strong function of density, which parametrically captures dependence on temperature, pressure, and material state. The associated root-mean-square errors of the fitted HEq models are approximately 5% of MD predicted final equilibrium temperatures. Here, the conductivity values determined here for TATB hot spots are considerably larger than those used in a prior hot spot criticality study, which may significantly impact predictions for critical hot spot sizes. The approach provides a convenient foundation for determining the effective thermal conductivity for hot spot problems in other explosives and directly yields information on reasonable approximations that might be taken in higher-level models for those materials.

36 MATERIALS SCIENCE↗
DOE OSTI2022

On heat conduction in an irregular magnetic field. Part 1

Anisotropic heat conduction in a plasma embedded in a magnetic field with irregular, possibly chaotic, field lines is discussed. If the collisional mean free path exceeds the electron gyroradius, the heat conductivity is much larger along the field lines than across them, and this enhances the transport across a domain where good flux surfaces do not exist. Recognising that anisotropic heat conduction may be cast in a variational form, and by constructing increasingly sophisticated trial functions that are based on invariant and almost-invariant structures under the magnetic field-line flow, bounds are derived on this enhancement and on the temperature variation along the magnetic field. In this way, remarkably accurate approximations for the temperature can be rapidly constructed without solving the diffusion equation, even in the small perpendicular-diffusion limit when the solution for the temperature is dominated by the fractal structure the magnetic field lines.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

Error analysis of numerical methods for thick diffusive neutron transport problems on Shishkin mesh

A thin layer will develop at the boundary if the incoming angular flux is anisotropic in thick diffusive neutron transport problems. Solving such singularly perturbed problems, which have non-smooth solutions with singularity near the boundary, is computationally challenging. Standard finite difference schemes on a uniform mesh cannot yield ε-uniform convergence, where ε is a small parameter, while it can be achieved on a suitable piecewise-uniform Shishkin mesh. We present a formal error analysis of the diamond difference (DD) method and step difference (SD) method for solving the S{sub N} neutron transport equation. The analysis can be extended to other finite difference methods. Numerical results are presented to confirm the error estimates and the advantages of the Shishkin mesh. (author)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2020

Wide bandwidth frequency-domain thermoreflectance: Volumetric heat capacity, anisotropic thermal conductivity, and thickness measurements

A frequency-domain thermoreflectance (FDTR) system with a frequency range of 1 Hz to 75 MHz is presented. The wide bandwidth of pump modulation frequency enables enhanced simultaneous thermophysical property characterization of bulk and thin-film materials. The wide bandwidth FDTR system is demonstrated with simultaneous measurements of in-plane and cross-plane thermal conductivities of sapphire and muscovite mica, thickness and thermal conductivity of gold/titanium thin films, and isotropic thermal conductivity and volumetric heat capacity of lithium niobate and silicon. Thermophysical properties measured with FDTR are compared to literature values or independent measurements for verification. Finally, at low frequencies, a negligible error was achieved in the numerical integration of the heat diffusion equation with a Gauss–Legendre quadrature method solved with 400 integration points and an upper integration limit of 16weff, where weff is the effective pump–probe 1/e2 radius.

Ziade, Elbara (ORCID:0000000249020497)↗
DOE OSTI2021

Generation of mean flows in rotating anisotropic turbulence: The case of solar near-surface shear layer

Results from helioseismology indicate that the radial gradient of the rotation rate in the near-surface shear layer (NSSL) of the Sun is independent of latitude and radius. Theoretical models using the mean-field approach have been successful in explaining this property of the NSSL, while global direct or large-eddy magnetoconvection models have so far been unable to reproduce this. We investigate the reason for this discrepancy by measuring the mean flows, Reynolds stress, and turbulent transport coefficients under conditions mimicking those in the solar NSSL. Simulations with as few ingredients as possible to generate mean flows were studied. These ingredients are inhomogeneity due to boundaries, anisotropic turbulence, and rotation. The parameters of the simulations were chosen such that they matched the weakly rotationally constrained NSSL. The simulations probe locally Cartesian patches of the star at a given depth and latitude. The depth of the patch was varied by changing the rotation rate such that the resulting Coriolis numbers covered the same range as in the NSSL. We measured the turbulent transport coefficient relevant for the nondiffusive (Λ-effect) and diffusive (turbulent viscosity) parts of the Reynolds stress and compared them with predictions of current mean-field theories. A negative radial gradient of the mean flow is generated only at the equator where meridional flows are absent. At other latitudes, the meridional flow is comparable to the mean flow corresponding to differential rotation. We also find that the meridional components of the Reynolds stress cannot be ignored. Additionally, we find that the turbulent viscosity is quenched by rotation by about 50% from the surface to the bottom of the NSSL. Our local simulations do not validate the explanation for the generation of the NSSL from mean-field theory where meridional flows and stresses are neglected. However, the rotational dependence of the turbulent viscosity in our simulations agrees well with theoretical predictions. Moreover, our results agree qualitatively with global convection simulations in that an NSSL can only be obtained near the equator.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

Learning Optimal Multigrid Smoothers via Neural Networks

Multigrid methods are one of the most efficient techniques for solving large sparse linear systems arising from partial differential equations (PDEs) and graph Laplacians from machine learning applications. One of the key components of multigrid is smoothing, which aims at reducing high-frequency errors on each grid level. However, finding optimal smoothing algorithms is problem-dependent and can impose challenges for many problems. In this paper, we propose an efficient adaptive framework for learning optimized smoothers from operator stencils in the form of convolutional neural networks (CNNs). Here, the CNNs are trained on small-scale problems from a given type of PDEs based on a supervised loss function derived from multigrid convergence theories and can be applied to large-scale problems of the same class of PDEs. Numerical results on anisotropic rotated Laplacian problems and variable coefficient diffusion problems demonstrate improved convergence rates and solution time compared with classical hand-crafted relaxation methods.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Thermodynamically consistent Cahn–Hilliard–Navier–Stokes equations using the metriplectic dynamics formalism

Cahn–Hilliard–Navier–Stokes (CHNS) systems describe flows with two-phases, e.g., a liquid with bubbles. Obtaining constitutive relations for general dissipative processes for such systems, which are thermodynamically consistent, can be a challenge. We show how the metriplectic 4-bracket formalism (Morrison and Updike, 2024) achieves this in a straightforward, in fact algorithmic, manner. First, from the noncanonical Hamiltonian formulation for the ideal part of a CHNS system we obtain an appropriate Casimir to serve as the entropy in the metriplectic formalism that describes the dissipation (e.g. viscosity, heat conductivity and diffusion effects). General thermodynamics with the concentration variable and its thermodynamics conjugate, the chemical potential, are included. Having expressions for the Hamiltonian (energy), entropy, and Poisson bracket, we describe a procedure for obtaining a metriplectic 4-bracket that describes thermodynamically consistent dissipative effects. The 4-bracket formalism leads naturally to a general CHNS system that allows for anisotropic surface energy effects. Furthermore, this general CHNS system reduces to cases in the literature, to which we can compare.

Cahn–Hilliard↗
DOE OSTI2023

Neural Network Enhanced RKPM for Electrochemical-Mechanical Coupled Damage Modeling of Energy Storage Materials

Energy storage materials undergo significant charge cycling, which makes understanding their reliability and durability fundamental in predicting performance and service life. Strong electrochemical-mechanical coupling and highly anisotropic material properties contribute to the formation and propagation of micro-cracking, largely along material interfaces and grain boundaries. For Li-ion batteries, for example, lithium moving between electrodes during charging and discharging process causes expansion and contraction of grains, and the strongly anisotropic and nonlinearly [Li]-dependent grain material properties can cause grains to expand into and contract away from each other, leading to chemo-mechanical cracking. In the first part of this work, a RKPM based computational framework for solving the coupled solid-phase lithium conservation with Fickian diffusion and the lithium concentration dependent anisotropic mechanical problem subjected to a highly nonlinear Butler-Volmer boundary condition is introduced. The choice of RKPM completeness conditions for lithium concentration and mechanical deformation fields, and the variational consistency condition for the domain integration of the coupled problem is first determined. In the second part of this work, a neural network-enhanced reproducing kernel particle method (NN-RKPM) [1] is leveraged to accurately capture damage and crack propagation throughout the material, by learning the location, orientation, and sharpness of discontinuity while allowing for a coarser nodal distribution than that is necessary for capturing sharp solution transitions using traditional mesh-based methods. NN-RKPM is used to inform how crack opening and closure in turn affect the coupled chemical equations and material microstructure.

damage modeling↗
DOE OSTI2020

Effects of dissipative baryon current in heavy-ion collisions at RHIC-BES energies

The CLVisc (3+1)D viscous hydrodynamic model is extended to include the equation of net baryon conservation and the Israel-Stewart-like equations for dissipative baryon current. Using the NEOSB equation of state, we simulate the dynamical evolution and collectivity of the quark-gluon plasma with finite chemical potential, assuming smooth energy density and net baryon density distributions at the initial proper time. Numerical results are shown for the impact of net-baryon dissipation on particle yields and p T spectra in heavy-ion collisions at beam energy scan energies.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗