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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↗
DOE OSTI2014

Application of Perona Malik anisotropic diffusion on digital radiographic image

Perona Malik Anisotropic Diffusion (PMAD) is a very useful and efficient denoising technique if the parameters are properly selected. Overestimating the parameters may cause oversmoothed and underestimating it may leave unfiltered noise. This makes the selection of parameters a crucial process. In this paper the PMAD model is solved using a finite difference scheme The discretized model is evaluated using different diffusion coefficient of exponential and quadratic on defective radiographic images in terms of quality and efficiency. In the application of the PMAD model on image data, a set of defective radiographic images of welding is used as input data. Peak Signal to Noise Ratio (PSNR), Structural Similarity Measure (SSIM) and temporal time are used to evaluate the performance of the model. The implementation of the experiment has been carried out using MATLAB R2009a. In terms of quality, results show that the Quadratic Diffusion Coefficient Function (QDCF) provides better results compared with the Exponential Diffusion Coefficient Function (EDCF). In conclusion, the denoising effect using PMAD model based on finite difference scheme shows able to improve image quality by removing noise in the defective radiographic image.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2016

Finite-volume scheme for anisotropic diffusion

In this paper, we apply a special finite-volume scheme, limited to smooth temperature distributions and Cartesian grids, to test the importance of connectivity of the finite volumes. The area of application is nuclear fusion plasma with field line aligned temperature gradients and extreme anisotropy. We apply the scheme to the anisotropic heat-conduction equation, and compare its results with those of existing finite-volume schemes for anisotropic diffusion. Also, we introduce a general model adaptation of the steady diffusion equation for extremely anisotropic diffusion problems with closed field lines.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Effect of charge and anisotropic diffusivity on ion exchange kinetics in nuclear waste form materials

Motifs of radionuclide absorption particles (absorbers or fillers) for the nuclear waste treatment usually have crystal structures with nano-sized tunnels for fast ion diffusion. On the other hand, the accumulation of ions generates an electric field which affects the ion diffusion. Therefore, the strong anisotropic kinetic properties and the electric field affect ion exchange kinetics, especially in an assembly of these particles which may block the fast diffusion tunnels for each other. In this work, we developed a mesoscale model to describe the effect of anisotropic kinetic properties and electric field on ion exchange kinetics. With the model we simulated the effect of anisotropic diffusivity, electrochemical potential, particle morphology, size and aggregation on ion exchange kinetics during batch experiments. Here, tt is found that 1) ion exchange results in charge accumulation inside the particle because different ions have different diffusivities and different diffusion driving forces. The electric neutrality is not preserved; 2) the electric field speeds up the slower diffusion ions (either uptake or release) and slows down the faster diffusion ions to reduce charge accumulation and maintains charge neutrality; 3) the ion exchange becomes very slow at the later stages because a) diffusion driving force (electrochemical gradient, Cs + concentration in the solution and Na + inside the particle) continuously decreases, as observed in batch experiments; and b) the diffusivity decreases due to the Cs + concentration dependence of ion diffusivity. This can explain the observed experimental phenomena that the radionuclide capacity usually cannot be reached; 4) for needle-like particle and their clusters, both surface area and diffusion distance along the direction with the fastest diffusivities have significant impact on the ion exchange kinetics.

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗
DOE OSTI2017

Anisotropic diffusion in media with absorbing traps (application to the Keller–Dykhne theorem)

Anisotropic diffusion in multidimensional media with absorbing traps is investigated. It is shown that the diffusion asymptotic forms of the survival probability over long time intervals are determined by a new effective diffusion coefficient. Exact expressions derived for the effective diffusion coefficient generalize the well-known Dykhne–Keller 2D results to the 3D case. In other words, the dynamic diffusion approach is used for obtaining a generalization of the Dykhne–Keller theorem to the 3D case.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2014

Finite-difference schemes for anisotropic diffusion

In fusion plasmas diffusion tensors are extremely anisotropic due to the high temperature and large magnetic field strength. This causes diffusion, heat conduction, and viscous momentum loss, to effectively be aligned with the magnetic field lines. This alignment leads to different values for the respective diffusive coefficients in the magnetic field direction and in the perpendicular direction, to the extent that heat diffusion coefficients can be up to 10{sup 12} times larger in the parallel direction than in the perpendicular direction. This anisotropy puts stringent requirements on the numerical methods used to approximate the MHD-equations since any misalignment of the grid may cause the perpendicular diffusion to be polluted by the numerical error in approximating the parallel diffusion. Currently the common approach is to apply magnetic field-aligned coordinates, an approach that automatically takes care of the directionality of the diffusive coefficients. This approach runs into problems at x-points and at points where there is magnetic re-connection, since this causes local non-alignment. It is therefore useful to consider numerical schemes that are tolerant to the misalignment of the grid with the magnetic field lines, both to improve existing methods and to help open the possibility of applying regular non-aligned grids. To investigate this, in this paper several discretization schemes are developed and applied to the anisotropic heat diffusion equation on a non-aligned grid.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2018

Ensemble Grouping Strategies for Embedded Stochastic Collocation Methods Applied to Anisotropic Diffusion Problems

Previous work has demonstrated that propagating groups of samples, called ensembles, together through forward simulations can dramatically reduce the aggregate cost of sampling-based uncertainty propagation methods [E. Phipps, M. D'Elia, H. C. Edwards, M. Hoemmen, J. Hu, and S. Rajamanickam, SIAM J. Sci. Comput., 39 (2017), pp. C162--C193]. However, critical to the success of this approach when applied to challenging problems of scientific interest is the grouping of samples into ensembles to minimize the total computational work. For example, the total number of linear solver iterations for ensemble systems may be strongly influenced by which samples form the ensemble when applying iterative linear solvers to parameterized and stochastic linear systems. In this paper we explore sample grouping strategies for local adaptive stochastic collocation methods applied to PDEs with uncertain input data, in particular canonical anisotropic diffusion problems where the diffusion coefficient is modeled by truncated Karhunen--Loève expansions. Finally, we demonstrate that a measure of the total anisotropy of the diffusion coefficient is a good surrogate for the number of linear solver iterations for each sample and therefore provides a simple and effective metric for grouping samples.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2014

Perona Malik anisotropic diffusion model using Peaceman Rachford scheme on digital radiographic image

In image processing, it is important to remove noise without affecting the image structure as well as preserving all the edges. Perona Malik Anisotropic Diffusion (PMAD) is a PDE-based model which is suitable for image denoising and edge detection problems. In this paper, the Peaceman Rachford scheme is applied on PMAD to remove unwanted noise as the scheme is efficient and unconditionally stable. The capability of the scheme to remove noise is evaluated on several digital radiography weld defect images computed using MATLAB R2009a. Experimental results obtained show that the Peaceman Rachford scheme improves the image quality substantially well based on the Peak Signal to Noise Ratio (PSNR). The Peaceman Rachford scheme used in solving the PMAD model successfully removes unwanted noise in digital radiographic image.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2018

Anisotropic Diffusion of a Charged Tritium Interstitial in Li 2 Ti O 3 from First-Principles Calculations

Tritium diffusion in the nuclear fusion reactor breeder material Li 2 TiO 3 is a fundamentally important process for the tritium release kinetics. The energy barrier of tritium diffusion in Li 2 TiO 3 was reported with a considerable uncertainty in previous experiments. Here, we perform the systematic density-functional-theory (DFT) studies for the diffusion processes of a positively-charged tritium, which is the most preferable charge state of the tritium interstitial in a single Li 2 TiO 3 crystal. By calculating various local-diffusion minimum-energy paths, we find that the diffusion of the tritium is strongly anisotropic along different crystalline directions. The most favorable diffusion paths appear within a Li 6 atomic single layer of Li 2 TiO 3 and the corresponding DFT diffusion barrier is 0.281 eV, while the diffusion barrier for most favorable diffusion paths crossing a Li 2 Ti 4 atomic layer is 0.868 eV.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
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 OSTI2023

Mesh refinement for anisotropic diffusion in magnetized plasmas

Highly accurate simulation of plasma transport is needed to drive the successful design and operation of magnetically confined fusion reactors. Unfortunately, the extreme anisotropy present in magnetized plasmas results in thin boundary layers that are expensive to resolve. Here, this work investigates how various mesh refinement strategies might reduce that expense to allow for more efficient simulation by comparing standard variable refinement approaches that use a field quantity to an adaptive approach that uses an error estimator. It is first verified that higher order discretization only realizes the proper rate of convergence once the mesh resolves the thin boundary layer, therefore motivating the focusing of refinement on the boundary layer. For three two-dimensional test cases that contain characteristic features of tokamak magnetic fields, an exponential refinement strategy based on the magnetic flux function, which is the standard refinement approach in the field, is compared to an adaptive strategy utilizing the established Zienwiekicz and Zhu error estimator. The adaptive mesh refinement strategy consistently achieves the same accuracy using orders of magnitude less degrees of freedom than either exponential or uniform refinement. This result makes the adaptive refinement strategy more efficient than the exponential refinement strategy while also being more generalizable to problems with complex magnetic geometries. Scaling laws are derived that quantify the improvement in cost of the adaptive refinement strategy over other refinement approaches in 2D and 3D.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Revisiting Anisotropic Diffusion of Carbon Dioxide in the Metal–Organic Framework Zn 2 (dobpdc)

The diffusion of gases confined in nanoporous materials underpins membrane and adsorption-based gas separations, yet relatively few measurements of diffusion coefficients in the promising class of materials, metal–organic frameworks (MOFs), have been reported to date. Recently we reported self-diffusion coefficients for 13 CO 2 in the MOF Zn 2 (dobpdc) (dobpdc 4– = 4,4'-dioxidobiphenyl-3,3'-dicarboxylate) which has one-dimensional channels with a diameter of approximately 2 nm [Forse, A. C.; J. Am. Chem. Soc. 2018, 140, 1663-1673]. By analyzing the evolution of the residual 13 C chemical shift anisotropy line shape at different gradient strengths, we obtained self-diffusion coefficients both along (D ∥ ) and between (D ⊥ ) the one-dimensional MOF channels. The observation of nonzero D ⊥ was unexpected based on the single crystal X-ray diffraction structure and flexible lattice molecular dynamics simulations, and we proposed that structural defects may be responsible for self-diffusion between the MOF channels. In this paper we revisit this analysis and show that homogeneous line broadening must be taken into account to obtain accurate values for D ⊥ . In the presence of homogeneous line broadening, intensity at a particular NMR frequency represents signal from crystals with a range of orientations relative to the applied magnetic field and magnetic gradient field. To quantify these effects, we perform spectral simulations that take into account homogeneous broadening and allow improved D ⊥ values to be obtained. Finally, our new analysis best supports nonzero D ⊥ at all studied dosing pressures and shows that our previous analysis overestimated D ⊥ .

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗