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At least 163 records · Page 9
All sourcesNASA NTRSDOE OSTI
DOE OSTI2018

Lumping Techniques for Mixed Finite Element Diffusion Discretizations

We are interested in positivity preserving spatial discretizations applicable to arbitrary order mixed finite element spatial discretizations of the radiative diffusion equations. Though not guaranteed to yield strict positivity, matrix lumping is appealing as a method of increasing solution positivity because unlike ad hoc fixups or strictly non-negative, non-linear solution representations, matrix lumping keeps the mixed finite element diffusion equations as a linear system of equations. Self-lumping schemes have previously been shown to be a generalization of more traditional matrix lumping techniques. Self-lumping schemes restrict integration quadrature points to the interpolation points used to define interpolatory finite element basis functions. As shown previously, the positivity and order of accuracy of self-lumping schemes are directly related to the choice of finite element interpolation point. Here, we develop self-lumping schemes for arbitrary order, mixed finite element spatial discretizations of the diffusion equation in Cartesian geometry. After introducing several possible self-lumping schemes appropriate for mixed finite elements, we test the ability of each scheme to increase solution positivity relative to an unlumped discretization using a simple test problem designed to induce negative solutions, evaluating efficacy as a function of finite element order and finite element interpolation point type. The method of manufactured solutions is then used to evaluate order of convergence for promising self-lumping schemes.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2022

FEDIS (The “Fresnel Equations” for Diffuse radiation on Inclined photovoltaic Surfaces) [SWR-22-64]

The well-known Fresnel equations solve for the reflection and transmission of light for precise incident angles. The transmission of diffuse radiation incident on a planar or domed surface is often needed for real-world applications. Due to the complexity of the Fresnel equations, the analytical solution of the integration has hitherto been unobtainable over the last centuries. Therefore, this problem was numerically solved by integrating the angular transmittances in space often leading to substantial computing burden and bias in the results. To efficiently estimate the solar energy resource for a glass-covered photovoltaic (PV) module, we derive an analytical solution of diffuse transmission based on the rigorous integration of an alternate form of the Fresnel equations. The approach leads to a simple yet accurate relative transmittance model that reconciles the solar energy sensed by pyranometers and PV panels. With limited and clearly stated approximations, the complex mathematical derivation resulted in an elegant solution.

Xie, Yu↗
NASA NTRS1981

Diffusion of cold magnetospheric ions

The paper investigates conditions under which a diffusion equation is a valid description of the transport across magnetic field lines of low energy magnetospheric plasma. The case in which the guiding center drift and its fluctuating component are comparable in size to the particle thermal speed is considered. It is concluded that a two-dimensional diffusion equation is valid in the alpha-beta space of the Euler potentials, provided that the fluctuations have a short enough correlation time so that the guiding center drifts remain negligible, and ions in Jupiter's Io plasma torus can thus be described by a diffusion equation if the correlation time is no more than a few rotation periods of the planet.

Birmingham, T. J.↗
NASA NTRS1993

Multilevel methods for transport equations in diffusive regimes

We consider the numerical solution of the single-group, steady state, isotropic transport equation. An analysis by means of the moment equations shows that a discrete ordinate S(sub N) discretization in direction (angle) with a least squares finite element discretization in space does not behave properly in the diffusion limit. A scaling of the S(sub N) equations is introduced so that the least squares discretization has the correct diffusion limit. For the resulting discrete system a full multigrid algorithm was developed.

Manteuffel, Thomas A.↗
DOE OSTI2016

PolyPole-1: An accurate numerical algorithm for intra-granular fission gas release

This paper describes the development of a new numerical algorithm (called PolyPole-1) to efficiently solve the equation for intra-granular fission gas release in nuclear fuel. The work was carried out in collaboration with Politecnico di Milano and Institute for Transuranium Elements. The PolyPole-1 algorithms is being implemented in INL's fuels code BISON code as part of BISON's fission gas release model. The transport of fission gas from within the fuel grains to the grain boundaries (intra-granular fission gas release) is a fundamental controlling mechanism of fission gas release and gaseous swelling in nuclear fuel. Hence, accurate numerical solution of the corresponding mathematical problem needs to be included in fission gas behaviour models used in fuel performance codes. Under the assumption of equilibrium between trapping and resolution, the process can be described mathematically by a single diffusion equation for the gas atom concentration in a grain. In this work, we propose a new numerical algorithm (PolyPole-1) to efficiently solve the fission gas diffusion equation in time-varying conditions. The PolyPole-1 algorithm is based on the analytic modal solution of the diffusion equation for constant conditions, with the addition of polynomial corrective terms that embody the information on the deviation from constant conditions. The new algorithm is verified by comparing the results to a finite difference solution over a large number of randomly generated operation histories. Furthermore, comparison to state-of-the-art algorithms used in fuel performance codes demonstrates that the accuracy of the PolyPole-1 solution is superior to other algorithms, with similar computational effort. Finally, the concept of PolyPole-1 may be extended to the solution of the general problem of intra-granular fission gas diffusion during non-equilibrium trapping and resolution, which will be the subject of future work.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2025

Space-Time Finite Element Tensor Network Approach for the Time-Dependent Convection–Diffusion–Reaction Equation with Variable Coefficients

In this paper, we present a new space-time Galerkin-like method, where we treat the discretization of spatial and temporal domains simultaneously. This method utilizes a mixed formulation of the tensor-train (TT) and quantized tensor-train (QTT) (please see Section Tensor-Train Decomposition), designed for the finite element discretization (Q1-FEM) of the time-dependent convection–diffusion–reaction (CDR) equation. We reformulate the assembly process of the finite element discretized CDR to enhance its compatibility with tensor operations and introduce a low-rank tensor structure for the finite element operators. Recognizing the banded structure inherent in the finite element framework’s discrete operators, we further exploit the QTT format of the CDR to achieve greater speed and compression. Additionally, we present a comprehensive approach for integrating variable coefficients of CDR into the global discrete operators within the TT/QTT framework. The effectiveness of the proposed method, in terms of memory efficiency and computational complexity, is demonstrated through a series of numerical experiments, including a semi-linear example.

convection–diffusion–reaction equation↗
DOE OSTI2014

Intrinsic delay of permeable base transistor

Permeable base transistors (PBTs) fabricated by vacuum deposition or solution process have the advantages of easy fabrication and low power operation and are a promising device structure for flexible electronics. Intrinsic delay of PBT, which characterizes the speed of the transistor, is investigated by solving the three-dimensional Poisson equation and drift-diffusion equation self-consistently using finite element method. Decreasing the emitter thickness lowers the intrinsic delay by improving on-current, and a thinner base is also preferred for low intrinsic delay because of fewer carriers in the base region at off-state. The intrinsic delay exponentially decreases as the emitter contact Schottky barrier height decreases, and it linearly depends on the carrier mobility. With an optimized emitter contact barrier height and device geometry, a sub-nano-second intrinsic delay can be achieved with a carrier mobility of ∼10 cm{sup 2}/V/s obtainable in solution processed indium gallium zinc oxide, which indicates the potential of solution processed PBTs for GHz operations.

36 MATERIALS SCIENCE↗
DOE OSTI2023

1-D coupled surface flow and transport equations revisited via the physics-informed neural network approach

The de Saint-Venant equation (SVE) and advection–diffusion equation (ADE) are commonly employed to solve solute transport problems in surface water. In this work, we propose a mesh-free method based on the physics-informed neural network (PINN) to solve the one dimensional (1-D) SVE, ADE, and the coupled SVE and ADE (SVE-ADE) under various initial and boundary conditions. The PINN model extends the architecture of deep neural networks (DNNs) with implementation of loss function, which are additionally subject to constraints imposed by the physical laws of SVE and ADE, along with their initial and boundary conditions. In such a manner, PINNs can be quickly steered to the true solution while obeying the physical laws. The results of PINN model are compared with the analytical and/or numerical solutions under various conditions to investigate its accuracy and efficiency in solving the SVE, ADE, and SVE-ADE. Our results indicate PINN can accurately simulate the shock wave morphology and avoid numerical dissipation in unsteady flow condition. The PINN method outweighed traditional numerical methods in several aspects, including its ability to function with small amounts of data, no grid discretization, and random selection of sampling points, etc. Additionally, the PINN method is also suitable for solving inverse problems with sparse and noisy data. With 1% noise and 2000 initial and boundary condition points (N u ), the errors of the estimated flow rate (v) and diffusion coefficient (D) are 0.003% and 0.105%, respectively, which indicate the accuracy and robustness of the proposed method. Finally, our results indicate the capability and robustness of the proposed PINN methodology for solving multi-physics problems, irrespective of the presence of sparse and noisy data in the training dataset.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2018

Optimal design of a model energy conversion device

Fuel cells, batteries, and thermochemical and other energy conversion devices involve the transport of a number of (electro-) chemical species through distinct materials so that they can meet and react at specified multi-material interfaces. Therefore, morphology or arrangement of these different materials can be critical in the performance of an energy conversion device. In this paper, we study a model problem motivated by a solar-driven thermochemical conversion device that splits water into hydrogen and oxygen. We formulate the problem as a system of coupled multi-material reaction-diffusion equations where each species diffuses selectively through a given material and where the reaction occurs at multi-material interfaces. In conclusion, we introduce a phase-field formulation of the optimal design problem and numerically study selected examples.

30 DIRECT ENERGY CONVERSION↗
DOE OSTI2019

High-order gradients with the shifted boundary method: An embedded enriched mixed formulation for elliptic PDEs

Here, we propose an extension of the embedded boundary method known as “shifted boundary method” to elliptic diffusion equations in mixed form (e.g., Darcy flow, heat diffusion problems with rough coefficients, etc.). Our aim is to obtain an improved formulation that, for linear finite elements, is at least second-order accurate for both flux and primary variable, when either Dirichlet or Neumann boundary conditions are applied. Following previous work of Nishikawa and Mazaheri in the context of residual distribution methods, we consider the mixed form of the diffusion equation (i.e., with Darcy-type operators), and introduce an enrichment of the primary variable. This enrichment is obtained exploiting the relation between the primary variable and the flux variable, which is explicitly available at nodes in the mixed formulation. The proposed enrichment mimics a formally quadratic pressure approximation, although only nodal unknowns are stored, similar to a linear finite element approximation. We consider both continuous and discontinuous finite element approximations and present two approaches: a non-symmetric enrichment, which, as in the original references, only improves the consistency of the overall method; and a symmetric enrichment, which enables a full error analysis in the classical finite element context. Combined with the shifted boundary method, these two approaches are extended to high-order embedded computations, and enable the approximation of both primary and flux (gradient) variables with second-order accuracy, independently on the type of boundary conditions applied. We also show that the primary variable is third-order accurate, when pure Dirichlet boundary conditions are embedded.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2015

Quantification of the internal quantum efficiency in GaN via analysis of the heat generated by non-radiative recombination processes

The internal quantum efficiency (IQE) in a GaN epilayer is quantified using transient lens (TL) spectroscopy and numerical simulations. TL spectroscopy can optically detect temperature and carrier changes induced in a photo-pumped GaN layer, and the observed temperature change is closely associated with non-radiative recombination processes that create heat. Then numerically solving diffusion equations, which represent the diffusion processes of the photo-generated heat and carriers, provide the spatiotemporal distributions. These distributions are subsequently converted into the refractive index distributions, which act as transient convex or concave lenses. Finally, ray-tracing simulations predict the TL signals. Comparing the experimentally obtained and simulated TL signals quantifies the generated heat and the IQE without the often-adopted assumption that non-radiative recombination processes are negligible at low temperatures.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2014

Explicit solutions of the radiative transport equation in the P{sub 3} approximation

Purpose: Explicit solutions of the monoenergetic radiative transport equation in the P{sub 3} approximation have been derived which can be evaluated with nearly the same computational effort as needed for solving the standard diffusion equation (DE). In detail, the authors considered the important case of a semi-infinite medium which is illuminated by a collimated beam of light. Methods: A combination of the classic spherical harmonics method and the recently developed method of rotated reference frames is used for solving the P{sub 3} equations in closed form. Results: The derived solutions are illustrated and compared to exact solutions of the radiative transport equation obtained via the Monte Carlo (MC) method as well as with other approximated analytical solutions. It is shown that for the considered cases which are relevant for biomedical optics applications, the P{sub 3} approximation is close to the exact solution of the radiative transport equation. Conclusions: The authors derived exact analytical solutions of the P{sub 3} equations under consideration of boundary conditions for defining a semi-infinite medium. The good agreement to Monte Carlo simulations in the investigated domains, for example, in the steady-state and time domains, as well as the short evaluation time needed suggests that the derived equations can replace the often applied solutions of the diffusion equation for the homogeneous semi-infinite medium.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2014

Transport of an interacting Bose gas in 1D disordered lattices

We use ultracold atoms in a quasiperiodic lattice to study two outstanding problems in the physics of disordered systems: a) the anomalous diffusion of a wavepacket in the presence of disorder, interactions and noise; b) the transport of a disordered superfluid. a) Our results show that the subdiffusion, observed when interaction alone is present, can be modelled with a nonlinear diffusion equation and the peculiar shape of the expanding density profiles can be connected to the microscopic nonlinear diffusion coefficients. Also when noise alone is present we can describe the observed normal diffusion dynamics by existing microscopic models. In the unexplored regime in which noise and interaction are combined, instead, we observe an anomalous diffusion, that we model with a generalized diffusion equation, where noise- and interaction-induced contributions add each other. b) We find that an instability appearing at relatively large momenta can be employed to locate the fluid-insulator crossover driven by disorder. By investigating the momentum-dependent transport, we observe a sharp crossover from a weakly dissipative regime to a strongly unstable one at a disorder-dependent critical momentum. The set of critical disorder and interaction strengths for which such critical momentum vanishes, can be identified with the separation between a fluid regime and an insulating one and can be related to the predicted zero-temperature superfluid-Bose glass transition.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Alluvial fan morphology: A self-similar free boundary problem description

In this work, we examine approximate geometrically self-similar solutions to a parabolic free boundary value problem applied to alluvial fan surface morphology and growth. Alluvial fans are fan- or cone-shaped sedimentary deposits caused by the rapid deposition of sediment from a canyon discharging onto a flatter plain. Longitudinal, topographic profiles of fans can be readily described by a seemingly time independent dimensionless profile (DeChant et al., 1999). However, because an alluvial fan can be expected to grow over time, it is clear that this “steady” profile is certainly time dependent and can be described using a space-time self-similar solution. In an experimental and theory-based study, Guerit et al. (2014) developed a self-similar (or as they describe it a self-affine) linear solution based upon an approximate first order small parameter expansion solution for a 1-d homogeneous nonlinear diffusion equation. Direct substitution of this result into a linear diffusion equation suggests that this first order expression may not fully satisfy the associated governing equation. In contrast, we develop a more complete solution based upon a modeled approximation for the axi-symmetric formulation such that the associated temporal behavior is consistent with a 1/3 time power-law as described by Reitz and Jerolmack (2014). The resulting expression is an exact solution to a linear heat equation. Furthermore, we emphasize that a small parameter is not inherent to the resulting profile result and is not included in our model development. Though developed using rather different approaches, the formal solution developed here is in good agreement with the simple polynomial described by DeChant et al. (1999) suggesting that this self-similar solution is a suitable time dependent representation of alluvial fan longitudinal profile form and improves on earlier work.

58 GEOSCIENCES↗
DOE OSTI2019

PVRD-FASP: A Unified Solver for Modeling Carrier and Defect Transport in Photovoltaic Devices

In our paper, we present a simulator for modeling transport of charge carriers and electrically active defect centers in solar cells by treating them on an equal footing, which allows us to address metastability and reliability issues. The exact nonlinear differential equations set solved by our solver is presented. The formulation of such differential equations, namely the continuity equations, drift-diffusion equation, and Poisson equation, for studying charge and defect transport is explained. The parameters needed for forming the differential equations are taken from first principle calculations. The solver is verified with test cases built on PN heterojunctions, Cu diffusion in single crystal CdTe and comparing Silvaco simulations with our numerical results.

14 SOLAR ENERGY↗
DOE OSTI2018

Initial-Boundary Value Problem for the Nonstationary Radiative Transfer Equation with Diffuse Reflection and Refraction Conditions

We consider the initial-boundary value problem for the nonstationary radiative transfer equation in a system of semitransparent bodies with the conditions of diffuse reflection and refraction of radiation. We establish the unique solvability of the problem with boundary and initial data in the complete scale of Lebesgue spaces. We obtain estimates for solutions.

42 ENGINEERING↗
DOE OSTI2018

An Investigation of Chaotic Diffusion in a Family of Hamiltonian Mappings Whose Angles Diverge in the Limit of Vanishingly Action

The chaotic diffusion for a family of Hamiltonian mappings whose angles diverge in the limit of vanishingly action is investigated by using the solution of the diffusion equation. The system is described by a two-dimensional mapping for the variables action, I, and angle, θ and controlled by two control parameters: (i) ϵ, controlling the nonlinearity of the system, particularly a transition from integrable for ϵ=0 to non-integrable for ϵ≠0 and; (ii) γ denoting the power of the action in the equation defining the angle. For ϵ≠0 the phase space is mixed and chaos is present in the system leading to a finite diffusion in the action characterized by the solution of the diffusion equation. The analytical solution is then compared to the numerical simulations showing a remarkable agreement between the two procedures.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗