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Results for “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 1,117 records · Page 62

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NASA NTRS1976

Climate stability for a Sellers-type model

We study a diffusive energy-balance climate model governed by a nonlinear parabolic partial differential equation. Three positive steady-state solutions of this equation are found; they correspond to three possible climates of our planet: an interglacial (nearly identical to the present climate), a glacial, and a completely ice-covered earth. We consider also models similar to the main one studied, and determine the number of their steady states. All the models have albedo continuously varying with latitude and temperature, and entirely diffusive horizontal heat transfer. The diffusion is taken to be nonlinear as well as linear. We investigate the stability under small perturbations of the main model's climates. A stability criterion is derived, and its application shows that the 'present climate' and the 'deep freeze' are stable, whereas the model's glacial is unstable. A variational principle is introduced to confirm the results of this stability analysis. For a sufficient decrease in solar radiation (about 2%) the glacial and interglacial solutions disappear, leaving the ice-covered earth as the only possible climate.

Ghil, M.↗

DOE OSTI2025

An end-to-end deep learning method for solving nonlocal Allen–Cahn and Cahn–Hilliard phase-field models

Here, we propose an efficient end-to-end deep learning method for solving nonlocal Allen–Cahn (AC) and Cahn–Hilliard (CH) phase-field models. One motivation for this effort emanates from the fact that discretized partial differential equation-based AC or CH phase-field models result in diffuse interfaces between phases, with the only recourse for remediation is to severely refine the spatial grids in the vicinity of the true moving sharp interface whose width is determined by a grid-independent parameter that is substantially larger than the local grid size. In this work, we introduce non-mass conserving nonlocal AC or CH phase-field models with regular, logarithmic, or obstacle double-well potentials. Because of non-locality, some of these models feature totally sharp interfaces separating phases. The discretization of such models can lead to a transition between phases whose width is only a single grid cell wide. Another motivation is to use deep learning approaches to ameliorate the otherwise high cost of solving discretized nonlocal phase-field models. To this end, loss functions of the customized neural networks are defined using the residual of the fully discrete approximations of the AC or CH models, which results from applying a Fourier collocation method and a temporal semi-implicit approximation. To address the long-range interactions in the models, we tailor the architecture of the neural network by incorporating a nonlocal kernel as an input channel to the neural network model. We then provide the results of extensive computational experiments to illustrate the accuracy, predictive capabilities, and cost reductions of the proposed method.

42 ENGINEERING↗

DOE OSTI2021

Time-lapse approach to correct deficiencies of 2D soil zymography

Membrane zymography is commonly used in rhizosphere ecology to map enzyme activities in intact soil samples and plant roots. The method consists of incubating a membrane saturated with an enzyme-specific fluorogenic substrate on the soil/root surface followed by measurements of fluorescence intensity of the product in the membrane. The traditional zymography is based on assumptions that fluorescence on membrane images is linearly increasing with time during zymography and an increase in product content is numerically equal to enzyme activity in soil below the membrane. These assumptions are unlikely to hold in experimental settings. In this work we introduce a new zymography technique, time-lapse zymography (TLZ); the approach that eliminates the need for assumptions of the traditional zymography and provides more realistic estimates of enzymatic activities. We assessed the new technique in a series of laboratory and modeling experiments, including quantification of the fluorescent product diffusion (e.g. MUF: 4-methylumbelliferone) from plant roots, enzyme activity measurements on the roots, and HYDRUS-2D & HP2 software calibration with obtained data. The calibrated model was used to analyze the processes governing spatial and temporal dynamics of MUF contents in the membrane. The results indicated that the enzyme diffusion within the membrane-soil system was negligible, and measured zymograms were adequately reproduced solely by accounting for substrate and product diffusions and for catalytic enzyme reaction described by the Michaelis-Menten equation. TLZ enabled identifying and using linear parts on MUF time series and accounting for MUF losses in each zymogram pixel, considerably improving the accuracy as compared to the traditional method. Results demonstrated that enzymatic activity from only a thin soil layer (~0.2 mm) is adequately represented in zymograms.

54 ENVIRONMENTAL SCIENCES↗

NASA NTRS1992

Computation of relaminarization in scramjet nozzles

The sudden expansion of a turbulent, Mach 1.76 flow through a 12 degree turning angle is computed with two low-Reynolds number, compressible turbulence models - a Reynolds stress model and a two-equation model. The two models give significantly different predictions of the wall shear stress downstream of the expansion. The Reynolds stress model predicts a 35 percent decrease in the wall shear stress whereas the two-equation model predicts only a 13 percent decrease. Furthermore, the two-equation model gives less decrease in the wall shear stress for the 12 degree expansion than for a 6 degree expansion, unlike the Reynolds stress model. Although a detailed explanation of these results is not attempted in this paper, possible causes have to be found in one or more of the many differences between the two models, including the treatment of the production and diffusion terms, and the absence of the redistribution term in the two-equation model.

Kamath, Pradeep S.↗

DOE OSTI2021

Hierarchical Fractional Advection-Dispersion Equation (FADE) to Quantify Anomalous Transport in River Corridor over a Broad Spectrum of Scales: Theory and Applications

Fractional calculus-based differential equations were found by previous studies to be promising tools in simulating local-scale anomalous diffusion for pollutants transport in natural geological media (geomedia), but efficient models are still needed for simulating anomalous transport over a broad spectrum of scales. This study proposed a hierarchical framework of fractional advection-dispersion equations (FADEs) for modeling pollutants moving in the river corridor at a full spectrum of scales. Applications showed that the fixed-index FADE could model bed sediment and manganese transport in streams at the geomorphologic unit scale, whereas the variable-index FADE well fitted bedload snapshots at the reach scale with spatially varying indices. Further analyses revealed that the selection of the FADEs depended on the scale, type of the geomedium (i.e., riverbed, aquifer, or soil), and the type of available observation dataset (i.e., the tracer snapshot or breakthrough curve (BTC)). When the pollutant BTC was used, a single-index FADE with scale-dependent parameters could fit the data by upscaling anomalous transport without mapping the sub-grid, intermediate multi-index anomalous diffusion. Pollutant transport in geomedia, therefore, may exhibit complex anomalous scaling in space (and/or time), and the identification of the FADE’s index for the reach-scale anomalous transport, which links the geomorphologic unit and watershed scales, is the core for reliable applications of fractional calculus in hydrology.

97 MATHEMATICS AND COMPUTING↗

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↗

NASA NTRS1983

Efficient solution of the Euler and Navier-Stokes equations with a vectorized multiple-grid algorithm

A multiple-grid algorithm for use in efficiently obtaining steady solutions to the Euler and Navier-Stokes equations is presented. The convergence of the explicit MacCormack algorithm on a fine grid is accelerated by propagating transients from the domain using a sequence of successively coarser grids. Both the fine and coarse grid schemes are readily vectorizable. The combination of multiple-gridding and vectorization results in substantially reduced computational times for the numerical solution of a wide range of flow problems. Results are presented for subsonic, transonic, and supersonic inviscid flows and for subsonic attached and separated laminar viscous flows. Work reduction factors over a scalar, single-grid algorithm range as high as 76.8.

Navier Stokes simulation↗

DOE OSTI2022

Accelerated Deterministic Phonon Transport With Consistent Material Temperature and Intensities

Abstract We present a method for deterministically solving the frequency and temperature dependent phonon radiative transport (PRT) equation in the single-mode relaxation time (SMRT) approximation in the self-adjoint angular flux (SAAF) form. To handle the nonlinear coupling between the phonon intensities and the material temperature, we apply a linearization approach that is similar to one in thermal radiative transport. This procedure leads to the PRT equation with pseudo-scattering. The method presented includes acceleration of both the inner pseudo-scattering source iterations and outer temperature iteration with a gray diffusion synthetic acceleration (DSA) and Anderson acceleration, respectively. We use the finite-element method to discretize the PRT equation in space and the method of discrete ordinates (SN) for angular discretization. The proposed method is verified by a gray method of manufactured solutions problem and demonstrated on a problem using temperature and direction dependent multigroup data from lithium aluminate (LiAlO2). The iterative performance of the acceleration method in each test is then compared to the unaccelerated method.

Engineering↗

NASA NTRS1990

A numerical simulation of the flow in the diffuser of the NASA Lewis icing research tunnel

The flow in the diffuser section of the Icing Research Tunnel at the NASA Lewis Research Center is numerically investigated. To accomplish this, an existing computer code is utilized. The code, known as PARC3D, is based on the Beam-Warming algorithm applied to the strong conservation law form of the complete Navier-Stokes equations. The first portion of the paper consists of a brief description of the diffuser and its current flow characteristics. A brief discussion of the code work follows. Predicted velocity patterns are then compared with the measured values.

Addy, Harold E., Jr.↗

NASA NTRS1990

A numerical simulation of the flow in the diffuser of the NASA Lewis Icing Research Tunnel

Addy, Harold E., Jr.↗

DOE OSTI2021

Preconditioners for multiphase poromechanics with strong capillarity

This paper aims to enhance the performance of Newton–Krylov solvers for coupled poromechanical problems with two-phase flow. In particular, we investigate the impact of capillary pressure on preconditioning strategies. Capillarity complicates the coupling between the solid deformation and fluid pressure degrees of freedom, as well as increases the nonlinearity of the system. Depending on the capillary pressure relation used in the constitutive formulation, the flow equations may exhibit a spectrum of advection-dominated to diffusion-dominated behavior. We propose preconditioning approaches that account for this behavior and lead to robust numerical performance within a broad range of regimes.

42 ENGINEERING↗

DOE OSTI2022

A multi-resolution approach to hydraulic fracture simulation

Abstract We present a multi-resolution approach for constructing model-based simulations of hydraulic fracturing, wherein flow through porous media is coupled with fluid-driven fracture. The approach consists of a hybrid scheme that couples a discrete crack representation in a global domain to a phase-field representation in a local subdomain near the crack tip. The multi-resolution approach addresses issues such as the computational expense of accurate hydraulic fracture simulations and the difficulties associated with reconstructing crack apertures from diffuse fracture representations. In the global domain, a coupled system of equations for displacements and pressures is considered. The crack geometry is assumed to be fixed and the displacement field is enriched with discontinuous functions. Around the crack tips in the local subdomains, phase-field sub-problems are instantiated on the fly to propagate fractures in arbitrary, mesh independent directions. The governing equations and fields in the global and local domains are approximated using a combination of finite-volume and finite element discretizations. The efficacy of the method is illustrated through various benchmark problems in hydraulic fracturing, as well as a new study of fluid-driven crack growth around a stiff inclusion.

58 GEOSCIENCES↗

DOE OSTI2020

An Efficient Numerical Algorithm for Solving Data Driven Feedback Control Problems

The goal of this paper is to solve a class of stochastic optimal control problems numerically, in which the state process is governed by an Itô type stochastic differential equation with control process entering both in the drift and the diffusion, and is observed partially. The optimal control of feedback form is determined based on the available observational data. In this work, we call this type of control problems the data driven feedback control. The computational framework that we introduce to solve such type of problems aims to find the best estimate for the optimal control as a conditional expectation given the observational information. To make our method feasible in providing timely feedback to the controlled system from data, we develop an efficient stochastic optimization algorithm to implement our computational framework.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Tackling the curse of dimensionality in fractional and tempered fractional PDEs with physics-informed neural networks

Fractional and tempered fractional partial differential equations (PDEs) are effective models of long-range interactions, anomalous diffusion, and non-local effects. Traditional numerical methods for these problems are mesh-based, thus struggling with the curse of dimensionality (CoD). Physics-informed neural networks (PINNs) offer a promising solution due to their universal approximation, generalization ability, and mesh-free training. In principle, Monte Carlo fractional PINN (MC-fPINN) estimates fractional derivatives using Monte Carlo methods and thus could lift CoD. However, this may cause significant variance and errors, hence affecting convergence; in addition, MC-fPINN is sensitive to hyperparameters. In general, numerical methods and specifically PINNs for tempered fractional PDEs are under-developed. Herein, we extend MC-fPINN to tempered fractional PDEs to address these issues, resulting in the Monte Carlo tempered fractional PINN (MC-tfPINN). To reduce possible high variance and errors from Monte Carlo sampling, we replace the one-dimensional (1D) Monte Carlo with 1D Gaussian quadrature, applicable to both MC-fPINN and MC-tfPINN. We validate our methods on various forward and inverse problems of fractional and tempered fractional PDEs, scaling up to 100,000 dimensions. Our improved MC-fPINN/MC-tfPINN using quadrature consistently outperforms the original versions in accuracy and convergence speed in very high dimensions.

42 ENGINEERING↗

DOE OSTI2021

A multi-phase-field model of topological pattern formation during electrochemical dealloying of binary alloys

Porous metal structures produced by electrochemical dealloying have been extensively studied for use in various applications that require high surface areas. A new comprehensive multi-phase-field (MPF) model is proposed to study topological porous patterns formed by spontaneously etching a bulk binary alloy that involves electrochemical reactions, bulk and surface diffusion, ion transport, applied electrode potential, and charge conservation. The governing equations for the alloy-porous cluster-electrolyte system account for a generalized Butler-Volmer electrochemical reaction and are in accordance with the classical nucleation theory. Based on a quantitative examination of the effects of electrode potential and precursor composition, the simulation results reproduce typical phenomena including passive surface dealloying, active porosity evolution, critical potential, and characteristic length scale in two-dimension (2-D) and three-dimension (3-D). To consider more complicated dealloying systems, the evolutions of a bimodal porous metal structure, nanocomposite, and nested porous network with a structured hierarchy are investigated. Finally, the proposed model can be a useful tool for understanding and predicting the morphology evolution of diverse porous structures during electrochemical dealloying.

Hierarchical porous structure↗

DOE OSTI2020

From synchrotrons for XFELs: the soft x-ray near-edge spectrum of the ESCA molecule

A predictive understanding of soft x-ray near-edge absorption spectra of small molecules is an enduring theoretical challenge and of current interest for x-ray probes of molecular dynamics. We report the experimental absorption spectrum for the electron spectroscopy for chemical analysis (ESCA) molecule (ethyl trifluoroacetate) near the carbon 1s absorption edge between 285-300 eV. Furthermore, the ESCA molecule with four chemically distinct carbon sites has previously served as a theoretical benchmark for photoelectron spectra and now for photoabsorption spectra. We report a simple edge-specific approach for systematically expanding standard basis sets to properly describe diffuse Rydberg orbitals and the importance of triple excitations in equation-of-motion coupled-cluster calculations of the energy interval between valence and Rydberg excitations.

74 ATOMIC AND MOLECULAR PHYSICS↗

DOE OSTI2024

Predicting the Slowing of Stellar Differential Rotation by Instability-driven Turbulence

Abstract Differentially rotating stars and planets transport angular momentum (AM) internally due to turbulence at rates that have long been a challenge to predict reliably. We develop a self-consistent saturation theory, using a statistical closure approximation, for hydrodynamic turbulence driven by the axisymmetric Goldreich–Schubert–Fricke instability at the stellar equator with radial differential rotation. This instability arises when fast thermal diffusion eliminates the stabilizing effects of buoyancy forces in a system where a stabilizing entropy gradient dominates over the destabilizing AM gradient. Our turbulence closure invokes a dominant three-wave coupling between pairs of linearly unstable eigenmodes and a near-zero frequency, viscously damped eigenmode that features latitudinal jets. We derive turbulent transport rates of momentum and heat and provide them in analytic forms. Such formulae, free of tunable model parameters, are tested against direct numerical simulations; the comparison shows good agreement. They improve upon prior quasi-linear or “parasitic saturation” models containing a free parameter. Given model correspondences, we also extend this theory to heat and compositional transport for axisymmetric thermohaline-instability-driven turbulence in certain regimes.

Astronomy & Astrophysics↗

NASA NTRS1966

Gunn domain dynamics.

Carrier motion equation in Gunn diode in terms of average drift velocity and diffusion coefficient

DIFFUSION COEFFICIENT↗