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Results for “nonlocal diffusion 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.

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At least 37 records · Page 2
All sourcesNASA NTRSDOE OSTI
NASA NTRS1989

An evaluation of the role of eddy diffusion in stratospheric interactive two-dimensional models

An interactive two-dimensional model of the stratosphere, consisting of a primitive equation dynamics module, a simplified HO(x) ozone model, and a full radiative transfer scheme, is used to study the effect of eddy diffusion in the model. Consideration is given to the effects of nonlocal forcing from dissipation in the model troposphere and frictional drag at mesospheric levels, mechanical damping in the stratosphere itself, and potential vorticity flux due to large scale waves. It is found that the ozone distributions generated with the model are very sensitive to the choice of values for the friction and the eddy diffusion coefficients. It is shown that reasonable latitudinal gradients of ozone may be obtained by using small values for the mechanical damping for the mid- and high-latitude stratopsphere.

Schneider, Hans R.↗
NASA NTRS1986

Solvability condition for needle crystals at large undercooling in a nonlocal model of solidification

It is explicitly shown that, in a realistic model of diffusion-controlled dendritic solidification, Ivantsov's continuous family of steady-state needle crystals is destroyed by the addition of surface tension. The starting point is in the exact integro-differential equation for the one-sided model, in two dimensions, in a moving frame of reference. In the limit of large undercooling, where the range of the diffusion field is much smaller than the radius of curvature of the tip of the needle, this problem is reduced to a linear, inhomogeneous differential equation of infinite order. A solvability condition for this equation is derived and it is shown that solutions cease to exist for arbitrarily small but finite isotropic surface tension.

Caroli, B.↗
DOE OSTI2020

Analysis of Tempered Fractional Operators

Tempered fractional operators provide an improved predictive capability for modeling anomalous effects that cannot be captured by standard partial differential equations. These effects include subdiffusion and superdiffusion (i.e. the mean square displacement in a diffusion process is proportional to a fractional power of the time), that often occur in, e.g., geoscience and hydrology. We analyze tempered fractional operators within the nonlocal vector calculus framework in order to assimilate them to the rigorous mathematical structure developed for nonlocal models. First, we show they are special instances of generalized nonlocal operators by means of a proper choice of the nonlocal kernel. Then, we present a plan for showing tempered fractional operators are equivalent to truncated fractional operators. These truncated operators are useful because they are less computationally intensive than the tempered operators.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

DiscoFluxM version 1.x

DiscoFluxM is software for modeling the deformation behavior of crystalline materials within the third-party finite-element based software, MOOSE. The software enables full-field crystal plasticity finite element simulations of deformation processes of metals and other crystalline materials. DiscoFluxM is an implementation of the DiscoFlux theoretical and algorithmic framework for coupling single-crystal plasticity with dislocation transport. The continuum mesoscale modeling framework is founded upon a nonlocal dislocation-density based crystal plasticity theory. The nonlocal theory couples continuum dislocation transport with an otherwise spatially-local single-crystal model employing nonlinear thermoelasticity and crystallographic plasticity. Dislocation transport is modeled by enforcing dislocation conservation at a slip-system level through the solution of advection-diffusion equations. The configuration of geometrically necessary dislocation density gives rise to a back-stress that inhibits or accentuates the flow of dislocations.

Luscher, Darby↗
DOE OSTI2021

An optimization-based approach to parameter learning for fractional type nonlocal models

Nonlocal operators of fractional type are a popular modeling choice for applications that do not adhere to classical diffusive behavior; however, one major challenge in nonlocal simulations is the selection of model parameters. In this study we propose an optimization-based approach to parameter identification for fractional models with an optional truncation radius. We formulate the inference problem as an optimal control problem where the objective is to minimize the discrepancy between observed data and an approximate solution of the model, and the control variables are the fractional order and the truncation length. For the numerical solution of the minimization problem we propose a gradient-based approach, where we enhance the numerical performance by an approximation of the bilinear form of the state equation and its derivative with respect to the fractional order. Several numerical tests in one and two dimensions illustrate the theoretical results and show the robustness and applicability of our method.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

An optimization-based approach to parameter learning for fractional type nonlocal models

Nonlocal operators of fractional type are a popular modeling choice for applications that do not adhere to classical diffusive behavior; however, one major challenge in nonlocal simulations is the selection of model parameters. In this work we propose an optimization-based approach to parameter identification for fractional models with an optional truncation radius. We formulate the inference problem as an optimal control problem where the objective is to minimize the discrepancy between observed data and an approximate solution of the model, and the control variables are the fractional order and the truncation length. For the numerical solution of the minimization problem we propose a gradient-based approach, where we enhance the numerical performance by an approximation of the bilinear form of the state equation and its derivative with respect to the fractional order. Several numerical tests in one and two dimensions illustrate the theoretical results and show the robustness and applicability of our method.

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↗
NASA NTRS1983

The nonlocal theory of periodic density drift instabilities

The nonlocal theory of electrostatic density drift instabilities is developed for an arbitrary, periodic variation in plasma density. Linear Vlasov theory is applied to Fourier series representations of the plasma density, and the resulting equations are solved numerically. The growth rates of the universal drift instability are compared by using local and nonlocal theories. It is demonstrated that the local theory is inadequate if the nonlocal eigenfunctions of the electrostatic modes have significant amplitude over spatial regions wider than the regions of large plasma density gradients. Nonlocal effects can enhance wave-particle diffusion of a triangular-shaped variation and reduce wave-particle dissipation of a square-shaped irregularity.

Bernhardt, P. A.↗
DOE OSTI2021

Experimental test of a predicted dynamics–structure–thermodynamics connection in molecularly complex glass-forming liquids

Significance The physical origin of cooperative activated relaxation in glass-forming liquids, which underlies the huge growth of viscosity and suppression of diffusion with cooling, remains elusive. Understanding this problem is of fundamental scientific interest and can assist in the design of new materials. We combine experimental relaxation time and equation-of-state data of molecular liquids to provide strong evidence for the recent theoretical prediction that activated relaxation, though causally driven by dynamically coupled cage-scale hopping and nonlocal collective elasticity quantified by local structural correlations, is well captured based on a thermodynamic description with dimensionless compressibility as the key variable. A mechanistic deduction is the crossover from fragile to strong (e.g., inorganic networks) relaxation behavior arises from the irrelevance of long-range elasticity effects.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Ionization waves in low-current dc discharges in noble gases obtained with a hybrid kinetic-fluid model

A hybrid kinetic-fluid model is used to study ionization waves (striations) in a low-current plasma column of dc discharges in noble gases. Coupled solutions of a kinetic equation for electrons, a drift-diffusion equation for ions, and a Poisson equation for the electric field are obtained to clarify the nature of plasma stratification in the positive column. A simplified two-level excitation-ionization model is used for the conditions when the nonlinear effects due to stepwise ionization, gas heating, and Coulomb interactions among electrons are negligible. It is confirmed that the nonlocal effects are responsible for the formation of moving striations in dc discharges at low plasma densities and low values of pR (the product of gas pressure and tube radius). Here, the calculated properties of self-excited waves of S–, P–, and R types in neon and S type in argon agree with available experimental data. The reason for helium plasma stability to stratification is clarified. It is shown that sustaining stratified plasma is more efficient than striation-free plasma when the ionization rate is a nonlinear function of the electric field. However, the nonlinear dependence of the ionization rate on the electric field is not required for plasma stratification. Striations of S–, P–, and R types in neon exist with minimal or no ionization enhancement. Effects of the column length and plasma density on the wave properties are demonstrated.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
NASA NTRS1992

Shadowing and the role of small diffusivity in the chaotic advection of scalars

Using techniques from shadowing theory, the solution of the scalar advection-diffusion equation is studied. It is shown that, under certain circumstances, the effect of small scalar diffusivity is to smooth the zero-diffusivity solution by averaging local fine-scaled structure against a Gaussian. The method of study depends on shadowing and thus fails for nonuniformly stretching systems, its failure suggesting the ways in which the effects of asymptotically small molecular diffusion can become nonlocal in chaotic fluid flows.

Klapper, I.↗
DOE OSTI2021

Fractional Modeling in Action: A Survey of Nonlocal Models for Subsurface Transport, Turbulent Flows, and Anomalous Materials

Modeling of phenomena such as anomalous transport via fractional-order differential equations has been established as an effective alternative to partial differential equations, due to the inherent ability to describe large-scale behavior with greater efficiency than fully-resolved classical models. In this review article, we first provide a broad overview of fractional-order derivatives with a clear emphasis on the stochastic processes that underlie their use. We then survey three exemplary application areas – subsurface transport, turbulence, and anomalous materials – in which fractional-order differential equations provide accurate and predictive models. For each area, we report on the evidence of anomalous behavior that justifies the use of fractional-order models, and survey both foundational models as well as more expressive state-of-the-art models. We also propose avenues for future research, including more advanced and physically sound models, as well as tools for calibration and discovery of fractional-order models.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Pattern formation in a coupled membrane-bulk reaction-diffusion model for intracellular polarization and oscillations

Reaction-diffusion systems have been widely used to study spatio-temporal phenomena in cell biology, such as cell polarization. Coupled bulk-surface models naturally include compartmentalization of cytosolic and membrane-bound polarity molecules. Here we study the distribution of the polarity protein Cdc42 in a mass-conserved membrane-bulk model, and explore the effects of diffusion and spatial dimensionality on spatio-temporal pattern formation. We first analyze a one-dimensional (1-D) model for Cdc42 oscillations in fission yeast, consisting of two diffusion equations in the bulk domain coupled to nonlinear ODEs for binding kinetics at each end of the cell. In 1-D, our analysis reveals the existence of symmetric and asymmetric steady states, as well as anti-phase relaxation oscillations typical of slow-fast systems. We then extend our analysis to a two-dimensional (2-D) model with circular bulk geometry, for which species can either diffuse inside the cell or become bound to the membrane and undergo a nonlinear reaction-diffusion process. We also consider a nonlocal system of PDEs approximating the dynamics of the 2-D membrane-bulk model in the limit of fast bulk diffusion. In all three model variants we find that mass conservation selects perturbations of spatial modes that simply redistribute mass. In 1-D, only anti-phase oscillations between the two ends of the cell can occur, and in-phase oscillations are excluded. In higher dimensions, no radially symmetric oscillations are observed. Instead, the only instabilities are symmetry-breaking, either corresponding to stationary Turing instabilities, leading to the formation of stationary patterns, or to oscillatory Turing instabilities, leading to traveling and standing waves. Codimension-two Bogdanov—Takens bifurcations occur when the two distinct instabilities coincide, causing traveling waves to slow down and to eventually become stationary patterns. Our work clarifies the effect of geometry and dimensionality on behaviors observed in ma.ss-conserved cell polarity models.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Perturbative treatment of nonlocal chiral interactions in auxiliary-field diffusion Monte Carlo calculations

Nuclear many-body systems, ranging from nuclei to neutron stars, are some of the most interesting physical phenomena in our universe, and quantum Monte Carlo (QMC) approaches are among the most accurate many-body methods currently available to study them. In recent decades, interactions derived from chiral effective field theory (EFT) have been widely adopted in the study of nuclear many-body systems. One drawback of the QMC approach is the requirement that the nuclear interactions need to be local, whereas chiral EFT interactions usually contain nonlocalities. In this work, we leverage the capability of computing second-order perturbative corrections to the ground-state energy in order to develop a self-consistent approach to including nonlocal operators in QMC calculations. In conclusion, we investigate both the deuteron and the neutron-matter equation of state in order to show the robustness of our technique and pave the way for future QMC calculations at higher orders in the EFT, where nonlocal operators cannot be avoided.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
NASA NTRS1986

Magnetized thermal conduction fronts

The evolution of planar thermal conduction fronts in the presence of a dynamically weak, but otherwise self-consistent, magnetic field is considered. The field is assumed to be connected and untangled. In the diffusion limit for the thermal conductivity, these fronts exhibit self-similar behavior, even in the presence of a field. The role of the field is restricted to channeling the heat flux along its lines of force, and it enters into the problem as a dimensionless angle variable. 'Combing' (or opening) of insulating field lines by the evaporative flow is explicitly demonstrated. Unless the field is nearly perpendicular to the front normal in the hot gas, insulating effects are not profound. Self-similarity breaks down if the front becomes saturated, and under certain conditions magnetized saturated conduction fronts cannot propagate: the solution characteristics of the wave equation form caustics. The physical resolution is the advent of two-fluid (nonlocal) heating. Such Coulomb-heated fronts are expected to be relatively rare in typical astrophysical systems. The large-scale effects of a magnetic field on cloud evaporation in the interstellar medium are briefly discussed, and it is suggested that these fields preclude the presence of time-independent evaporative solutions. Thermal interfaces may then continue to evolve until radiative cooling halts their development; large tracts of warm 10,000 K gas may result.

Balbus, S. A.↗
DOE OSTI2024

A splice method for local-to–nonlocal coupling of weak forms

Here, we propose a method to couple local and nonlocal diffusion models. By inheriting desirable properties such as patch tests, asymptotic compatibility and unintrusiveness from related splice and optimization-based coupling schemes, it enables the use of weak (or variational) formulations, is computationally efficient and straightforward to implement. We prove well-posedness of the coupling scheme and demonstrate its properties and effectiveness in a variety of numerical examples.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker--Planck Equation and Physics-Informed Neural Networks

The Fokker--Planck (FP) equation governing the evolution of the probability density function (PDF) is applicable to many disciplines, but it requires specification of the coefficients for each case, which can be functions of space-time and not just constants and hence require the development of a data-driven modeling approach. When the data available is directly on the PDF, there exist methods for inverse problems that can be employed to infer the coefficients and thus determine the FP equation and subsequently obtain its solution. Herein, we address a more realistic scenario, where only sparse data are given on the particles' positions at a few time instants, which are not sufficient to accurately construct directly the PDF even at those times from existing methods, e.g., kernel estimation algorithms. To this end, we develop a general framework based on physics-informed neural networks (PINNs) that introduces a new loss function using the Kullback--Leibler divergence to connect the stochastic samples with the FP equation to simultaneously learn the equation and infer the multidimensional PDF at all times. In particular, we consider two types of inverse problems, type I, where the FP equation is known but the initial PDF is unknown, and type II, in which, in addition to the unknown initial PDF, the drift and diffusion terms are also unknown. In both cases, we investigate problems with either Brownian or Lévy noise or a combination of both. Here, we demonstrate the new PINN framework in detail in the one-dimensional (1D) case, but we also provide results for up to five dimensions demonstrating that we can infer both the FP equation and dynamics simultaneously at all times with high accuracy using only very few discrete observations of the particles.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1981

Magnetic field diffusion and dissipation in reversed-field plasmas

A diffusion equation is derived which describes the evolution of a magnetic field in a plasma of arbitrary beta and resistivity. The equation is valid for a one-dimensional slab geometry, assumes the plasma remains in quasi-equilibrium throughout its evolution and does not include thermal transport. Scaling laws governing the rate of change of the magnetic energy, particle drift energy, and magnetic flux are calculated. It is found that the magnetic free energy can be substantially larger than the particle drift energy and can be an important energy reservoir in driving plasma instabilities (e.g., the lower-hybrid-drift instability). In addition, the effect of a spatially varying resistivity on the evolution of a reversed-field plasma is studied. The resistivity model used is based upon the anomalous transport properties associated with the nonlocal mode structure of the lower-hybrid-drift instability. The relevance of this research to laboratory plasmas (e.g., theta pinches, reversed-field theta pinches) and space plasmas (e.g., the earth's magnetotail) is discussed.

Drake, J. F.↗