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Results for “Anisotropic diffusion equation”

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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At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

A provably stable numerical method for the anisotropic diffusion equation in confined magnetic fields

We present a novel numerical method for solving the anisotropic diffusion equation in magnetic fields confined to a periodic box which is accurate and provably stable. We derive energy estimates of the solution of the continuous initial boundary value problem. A discrete formulation is presented using operator splitting in time with the summation by parts finite difference approximation of spatial derivatives for the perpendicular diffusion operator. Weak penalty procedures are derived for implementing both boundary conditions and parallel diffusion operator obtained by field line tracing. We prove that the fully-discrete approximation is unconditionally stable. Discrete energy estimates are shown to match the continuous energy estimate given the correct choice of penalty parameters. A nonlinear penalty parameter is shown to provide an effective method for tuning the parallel diffusion penalty and significantly minimises rounding errors. Several numerical experiments, using manufactured solutions, the “NIMROD benchmark” problem and a single island problem, are presented to verify numerical accuracy, convergence, and asymptotic preserving properties of the method. Finally, we present a magnetic field with chaotic regions and islands and show the contours of the anisotropic diffusion equation reproduce key features in the field.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
NASA NTRS1985

The expected cosmic ray density and stream distributions at the heliolatitudinal asymmetry of solar wind

The results of the spatial distribution of cosmic ray density, gradients, and anisotropy obtained on the basis of the numerical solution of the anisotropic diffusion equation with an account of solar wind velocity change depending on the latitudinal angle theta of the form U=u sub oe sup alpha theta and the diffusion coefficient depending on the spatial coordinates and the particle rigidity are presented. It is shown that the increase of the solar wind velocity and the diffusion coefficient with heliolatitude leads to gradient distributions that are in accord with experimental data observed in space. The results of the energetic spectrum of 11 and 22-year cosmic ray variations obtained with an account of direction of the general magnetic field of the Sun are presented are given.

Alania, M. V.↗
NASA NTRS1985

Spatial distribution of cosmic rays in magnetic cycles of the Sun

The space-time distribution of cosmic ray (CR) density is determined from the solution of the anisotropic diffusion equation. The boundary problem is solved in a quasistationary axisymmetric case, taking into account the delay of cosmic rays relative to the cyclic variations of solar activity. The specificty of the modulation model is the use of the empirical dependence of the free path of particles for scattering on the observed latitudinal distribution of solar activity. Another specific feature of the model is an account of an additional effect due to the drift of charged particles towards the helio-equator plane or backward, depending on the direction of the total magnetic field of the Sun.

Alania, M. V.↗
NASA NTRS1985

The modulation features of the long-period cosmic ray variations in connection with the sign change of the general magnetic field of the Sun

On the basis of the model and experimental investigations, the spatial distribution of cosmic ray anisotropy for the different epochs of solar activity is studied. A solution is offered to the anisotropic diffusion equation with regard to the electromagnetic conditions for the periods of minimum and maximum solar activity and cosmic ray particle drift in a regular interplanetary magnetic field. It is shown that the long period changes amplitude and phase of the diurnal variations of cosmic rays is limited not only by convection and diffusion of particles but also by the drift effect before and after the sing change of the general magnetic field of the Sun. The calculated model is compared with the results obtained on the basis of an analysis of the experimental data from the neutron super monitor station, and it is shown that for the periods when the lines of magnetic force of the Sun come from the Northern Hemisphere the phase of the first harmonic diurnal variation is shifted forwards to an earlier time.

Iskra, K.↗
NASA NTRS1985

The theoretical and experimental investigation of cosmic ray Forbush-effects

The theoretical results of analysis of the expected spatial distributions of density, gradients and anisotropy of cosmic rays, obtained on the basis of the numerical solution of anisotropic diffusion equation in the presence of the disturbances of shock wave type in the interplanetary space are presented. The theoretical calculations on the definition of the energy spectrum and anisotropy of galactic cosmic rays during Forbush effect are compared with the experimental data.

Alania, M. V.↗
DOE OSTI2023

An Efficient High-Order Solver for Diffusion Equations with Strong Anisotropy on Non-Anisotropy-Aligned Meshes

This paper concerns numerical solution of the diffusion equation with strong anisotropy on meshes not aligned with the anisotropic vector field. In order to resolve the numerical pollution for simulations on a non-anisotropy-aligned mesh and reduce the associated high computational cost we propose an effective preconditioner, extending our previous work. Similar to the anisotropy-aligned mesh case, we apply the auxiliary space preconditioning framework to design a preconditioner where a continuous finite element space is used as the auxiliary space for the discontinuous finite element space. The key component is an effective line smoother that can mitigate the high-frequency errors perpendicular to the magnetic field. We design a graph-based approach to find such a line smoother that is approximately perpendicular to the vector fields when the mesh does not align with the anisotropy. Finally, numerical experiments for several benchmark problems are presented, demonstrating the effectiveness and robustness of the proposed preconditioner when applied to Krylov iterative methods.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Electron kinetics in a positive column of AC discharges in a dynamic regime

Abstract We have performed hybrid kinetic-fluid simulations of a positive column in alternating current (AC) argon discharges over a range of driving frequencies f and gas pressure p for the conditions when the spatial nonlocality of the electron energy distribution function (EEDF) is substantial. Our simulations confirmed that the most efficient conditions of plasma maintenance are observed in the dynamic regime when time modulations of mean electron energy (temperature) are substantial. The minimal values of the root mean square electric field and the electron temperature have been observed at f/p values of about 3 kHz Torr −1 in a tube of radius R = 1 cm. The ionization rate and plasma density reached maximal values under these conditions. The numerical solution of a kinetic equation allowed accounting for the kinetic effects associated with spatial and temporal nonlocality of the EEDF. Using the kinetic energy of electrons as an independent variable, we solved an anisotropic tensor diffusion equation in phase space. We clarified the role of different flux components during electron diffusion in phase space over surfaces of constant total energy. We have shown that the kinetic theory uncovers a more exciting and rich physics than the classical ambipolar diffusion (Schottky) model. Non-monotonic radial distributions of excitation rates, metastable densities, and plasma density have been observed in our simulations at pR > 6 Torr cm. The predicted off-axis plasma density peak in the dynamic regime has never been observed in experiments so far. We hope our results stimulate further experimental studies of the AC positive column. The kinetic analysis could help uncover new physics even for such a well-known plasma object as a positive column in noble gases.

Physics↗
DOE OSTI2022

An efficient high-order numerical solver for diffusion equations with strong anisotropy

In this paper, we present an interior penalty discontinuous Galerkin finite element scheme for solving diffusion problems with strong anisotropy arising in magnetized plasmas for fusion applications. Additionally, we demonstrate the accuracy produced by the high-order scheme and develop an efficient preconditioning technique to solve the corresponding linear system, which is robust to the mesh size and anisotropy of the problem. Several numerical tests are provided to validate the accuracy and efficiency of the proposed algorithm.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Heat conduction in an irregular magnetic field. Part 2. Heat transport as a measure of the effective non-integrable volume

Given the large anisotropy of transport processes in magnetized plasmas, the magnetic field structure can strongly impact heat diffusion: magnetic surfaces and cantori form barriers to transport while chaotic layers and island structures can degrade confinement. When a small but non-zero amount of perpendicular diffusion is included, the structure of the magnetic field becomes less important, allowing pressure gradients to be supported across chaotic regions and island chains. We introduce a metric for the effective volume over which the local parallel diffusion dominates based on the solution to the anisotropic heat diffusion equation. To validate this metric, we consider model fields with a single island chain and a strongly chaotic layer for which analytic predictions of the relative parallel and perpendicular transport can be made. We also analyse critically chaotic fields produced from different sets of perturbations, highlighting the impact of the mode number spectrum on the heat transport. Our results indicate that this metric coincides with the effective volume of non-integrability in the limit $\kappa _{\perp } \rightarrow 0$ , where $\kappa_{\perp}$ is the perpendicular diffusion coefficient. We propose that this metric be used to assess the impact of non-integrability on the heat transport in stellarator equilibria.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2026

Energy-conserving contact dynamics of nonspherical rigid-body particles

Understanding the contact dynamics of nonspherical particles is crucial for accurately modeling colloidal and granular systems where shape anisotropy dictates structural organization and transport properties. We here introduce an energy-conserving contact dynamics framework for arbitrary convex rigid-body particles by implementing vertex–boundary interactions in 2D and vertex–surface and edge–edge detection in 3D. The established formulation enables continuous force evaluation and prevents particle overlap while conserving total energy during translational and rotational motion. We demonstrate the framework’s stability and its utility to capture packing behavior, anisotropic diffusion, and equations of state of polygonal and polyhedral particles as examples. The framework establishes a robust and extensible foundation for investigating nonequilibrium dynamics of complex nonspherical particulate systems, enabling enhanced understanding of phenomena across spatiotemporal scales in self- and directed-assembly, granular flows, and hydrodynamics, potentially coupled with interactions that represent underlying physical mechanisms.

Discrete element method↗
DOE OSTI2026

Overlapping Schwarz Methods Are Not Anisotropy‐Robust Multigrid Smoothers

We analyze overlapping multiplicative Schwarz methods as smoothers in the geometric multigrid solution of two-dimensional anisotropic diffusion problems. For diffusion equations, it is well known that the smoothing properties of point-wise smoothers, such as Gauss Seidel, rapidly deteriorate as the strength of anisotropy increases. On the other hand, global smoothers based on line smoothing are known to generally provide good smoothing for diffusion problems, independent of the anisotropy strength. Here, a natural question is whether global methods are really necessary to achieve good smoothing in such problems, or whether it can be obtained with locally overlapping block smoothers using sufficiently large blocks and overlap. Through local Fourier analysis and careful numerical experimentation, we show that global methods are indeed necessary to achieve anisotropy-robust smoothing. Specifically, for any fixed block size bounded sufficiently far away from the global domain size, we find that the smoothing properties of overlapping multiplicative Schwarz rapidly deteriorate with increasing anisotropy, irrespective of the amount of overlap between blocks. Moreover, our results indicate that anisotropy-robust smoothing requires blocks of diameter 𝒪⁡(𝜖 −1/2 ) for anisotropy ratio 𝜖 ∈(0,1] .

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1989

Interpretation and modeling of the averaged equations for a fluid-solid flow

A self-consistent derivation of the conservation laws is given for flows of a fluid-solid mixture. A unified analytical framework for obtaining constitutive relations is provided. This analysis uses a control volume/control surface approach that is widely used in fluid mechanics. All terms in the governing equations and the constitutive relations are written in terms of the mass-weighted averages except solid concentration. It is believed that the mass-weighted average is the natural bridge between micromechanics and constitutive relations. The derived momentum equations contain terms that differ from all existing models except that of Prosperetti and Jones (1984). However, their assumptions are not needed here. Special attention is given to the solid phase pressure. The physical basis of the previously assumed form for this pressure (Givler 1987) becomes clear. A number of related phenomena are also discussed. These include the anti-diffusion and anisotropic normal stresses. The energy equations are also different from existing models.

Shen, Hayley H.↗
NASA NTRS1987

Modeling scalar flux and the energy and dissipation equations

Closure models derived from the Two-Scale Direct-Interaction Approximation were compared with data from direct simulations of turbulence. Attention was restricted to anisotropic scalar diffusion models, models for the energy dissipation equation, and models for energy diffusion.

Yoshizawa, A.↗
NASA NTRS1985

Pitch Angle Distributions of Solar Energetic Particles and the Local Scattering Properties of the Interplanetary Magnetic Field

An approximate solution of the Fokker-Planck equation containing pitch angle scattering and adiabatic focusing is discussed. For modest focusing effects the omnidirectional density obeys an ordinary diffusion equation with a modified diffusion coefficient. The anisotropic part of the distribution function is properly normalized and split into an even and an odd part. The even part is determined by the ratio between the scattering mean free path and the focusing length and by the degree of polarization of the magnetic field fluctuations. The odd part is determined by the deviation of the pitch angle scattering from isotropic scattering. The method supplies a powerful tool to obtain the local characteristics of pitch angle scattering. It is insensitive to long lasting solar injections and to moderate radial variations of the mean free path. The method is applied to solar particle events observed on Helio-1 and -2.

Beeck, J.↗
NASA NTRS1991

Convective instability in superposed fluid and anisotropic porous layers

The onset of thermal convection due to heating from below in a system consisting of a fluid layer overlying a porous layer with anisotropic permeability and thermal diffusivity is studied. The linear perturbation equations are solved numerically. It is found that the effects of anisotropy on the onset of thermal convection are most profound for small values of the depth ratio xi (ratio of fluid layer thickness to porous layer thickness), since in that case, the onset of convection corresponds to significant motion in both layers. For fixed vlues of the vertical permeability in the porous medium, decreasing the value of zeta (ratio of horizontal to vertical permeability) leads to stabilization of the superposed layer configuration because of increased resistance to motion in the porous medium. For larger values of xi, the onset of motion is increasingly confined to the fluid layer, with the transport of heat through the porous layer occurring primarily by conduction. Accordingly, the influence of zeta on the stability characteristics for larger xi is less significant than the effects of an anisotropic thermal conductivity.

Chen, Falin↗
NASA NTRS2012

Modifications Of Discrete Ordinate Method For Computations With High Scattering Anisotropy: Comparative Analysis

A numerical accuracy analysis of the radiative transfer equation (RTE) solution based on separation of the diffuse light field into anisotropic and smooth parts is presented. The analysis uses three different algorithms based on the discrete ordinate method (DOM). Two methods, DOMAS and DOM2+, that do not use the truncation of the phase function, are compared against the TMS-method. DOMAS and DOM2+ use the Small-Angle Modification of RTE and the single scattering term, respectively, as an anisotropic part. The TMS method uses Delta-M method for truncation of the phase function along with the single scattering correction. For reference, a standard discrete ordinate method, DOM, is also included in analysis. The obtained results for cases with high scattering anisotropy show that at low number of streams (16, 32) only DOMAS provides an accurate solution in the aureole area. Outside of the aureole, the convergence and accuracy of DOMAS, and TMS is found to be approximately similar: DOMAS was found more accurate in cases with coarse aerosol and liquid water cloud models, except low optical depth, while the TMS showed better results in case of ice cloud.

Korkin, Sergey V.↗
NASA NTRS1971

Transport of solar flare protons: Comparison of a new analytic model with spacecraft measurements

An analytic solution was obtained to the complete Fokker-Planck equation including the effects of convection, interplanetary deceleration and acceleration, corotation, and anisotropic diffusion. The solution yields a time to maximum for the particle flux of about 10 h and an exponential decay time of about 5 h. Several solar flare particle events were observed with the solar and galactic cosmic ray experiment on OGO-6. Comparisons of the calculated time dependence of the fluxes with these observations of 1 to 70 MeV protons show that the model adequately describes both the rise and decay times.

Lupton, J. E.↗
NASA NTRS1993

Mass loading and velocity diffusion models for heavy pickup ions at comet Grigg-Skjellerup

We compare model predictions of cometary water group ion densities and the solar wind slow down with measurements made by the Giotto Johnstone plasma analyzer implanted ion sensor at the encounter with comet Grigg-Skjellerup (G-S) on July 10, 1992. The observed slope of the ion density profile on approach to the comet is unexpectedly steep. Possible explanations for this are discussed. We present also a preliminary investigation of the quasilinear velocity-space diffusion of the implanted heavy ion population at G-S using a transport equation including souce, convection, adiabatic compression, and velocity diffusion terms. Resulting distributions are anisotropic, in agreement with observations. We consider theoretically the waves that may be generated by the diffusion process for the observed solar wind conditions. At initial ion injections, waves are generated at omega approximately Omega(sub i) the ion gyrofrequency, and lower frequencies are predicted for diffusion toward a bispherical shell.

Huddleston, D. E.↗