Numerical Solution Of Transport Equation Research Articles

Unified Framework for the Finite Element Modeling of Li-Based Batteries with Liquid Electrolyte

Despite the high demand for accurate electrochemical simulations tools that can compute the d.c. (charge and discharge), a.c. (i.e. electrochemical impedance spectroscopy), and transient characteristics of electrochemical devices, there is currently a lack of such simulation environments in the literature. Most often, the different research groups and institutions are writing their own custom simulators that usually work only for specific battery configurations and chemistries. Even more advanced commercial simulators, such as Comsol Multiphysics, have a few modules that can simulate batteries, however, these modules can simulate only specific chemistries such as Li-ion or Li-air batteries and do not have a more general electrochemical device simulator that can simulate any general, user-designed device.In this presentation, we summarize our efforts to build such a tool, starting from the numerical implementation of the transport models to problems associated with the discretization and numerical solution of transport equations. We focus on how to design a unified framework for designing electrolytes, modeling ion transport in the electrolyte and electron transport in the electron conductive materials, and interface phenomena such as double layer effects, SEI layers, and the multitude of possible chemical and electrochemical reactions that can take place once the system is designed. In our tool, the transport in the electrolyte is governed by a modified concentrated solution theory written in the form of a generalized Nernst-Planck model. The a.c. analysis is performed by linearizing the discretized system of differential equations and using the linear perturbations theory on the finite element grid. More details about the formulation of the mathematical model, state variables, electrochemical reactions, microstructure modeling, and numerical implementation will be discussed at the conference. Specific examples for the simulation of Li-ion, Li-air, Li-S and other Li-based systems (symmetric and asymmetric cells) will also be presented.

On the relation between deep level compensation, resistivity and electric field in semi-insulating CdTe:Cl radiation detectors

A compensation model for semi-insulating CdTe:Cl based on a single dominant deep level 0.725 eV above the valence band is proposed. The model is corroborated by experimental evidence: resistivity measurements as a function of temperature on bulk crystals and stationary electric field distributions in Ohmic/Schottky radiation detectors, obtained by the Pockels effect. The latter are in close agreement with the numerical solutions of transport equations when considering the deep centre concentration in the range 2 − 4 × 1012 cm−3, and a compensation ratio R = 2.1, this one being consistent with an original ambipolar analysis of resistivity. More generally, the approach elucidates the role of electrical contacts and deep levels in controlling the electric fields in devices based on compensated materials.

Solving Parker’s transport equation with stochastic differential equations on GPUs

The numerical solution of transport equations for energetic charged particles in space is generally very costly in terms of time. Besides the use of multi-core CPUs and computer clusters in order to decrease the computation times, high performance calculations on graphics processing units (GPUs) have become available during the last years. In this work we introduce and describe a GPU-accelerated implementation of Parker’s equation using Stochastic Differential Equations (SDEs) for the simulation of the transport of energetic charged particles with the CUDA toolkit, which is the focus of this work. We briefly discuss the set of SDEs arising from Parker’s transport equation and their application to boundary value problems such as that of the Jovian magnetosphere. We compare the runtimes of the GPU code with a CPU version of the same algorithm. Compared to the CPU implementation (using OpenMP and eight threads) we find a performance increase of about a factor of 10–60, depending on the assumed set of parameters. Furthermore, we benchmark our simulation using the results of an existing SDE implementation of Parker’s transport equation.

Quasi-three-dimensional modelling of penetration and influence of impurities in plasma

In fusion devices strongly localized intensive sources of impurities may arise unexpectedly from plasma–wall interactions or can be created deliberately through impurity injection. The spreading of impurities from such sources is essentially a three-dimensional and non-stationary phenomenon involving physical processes of extremely different time scales. Numerical modelling of such events is still a very challenging task, even using most modern computers. To diminish the calculation time drastically, a new quasi-three-dimensional description is proposed, combining a ‘shell’ model for the impurity penetration process and a two-zone approximation for the main plasma components. This approach allows us to reduce fluid equations for particle, parallel momentum and energy balances of the main and impurity plasma species to one-dimensional equations describing the time evolution of radial profiles of the most characteristic parameters. The assumptions in the approach are verified by comparing its predictions with direct numerical solutions of transport equations. The results of modelling for the penetration process of argon species into hot H-mode plasma during impurity puffing experiments are presented.

Thermal conductive and radiative properties of solid foams: Traditional and recent advanced modelling approaches

The current paper presents an overview of traditional and recent models for predicting the thermal properties of solid foams with open- and closed-cells. Their effective thermal conductivity has been determined analytically by empirical or thermal-resistance-network-based models. Radiative properties crucial to obtain the radiative conductivity have been determined analytically by models based on the independent scattering theory. Powerful models combine three-dimensional (3D) foam modelling (by X-ray tomography, Voronoi tessellation method, etc.) and numerical solution of transport equations. The finite-element method (FEM) has been used to compute thermal conductivity due to solid network for which the computation cost remains reasonable. The effective conductivity can be determined from FEM results combined with the conductivity due to the fluid, which can be accurately evaluated by a simple formula for air or weakly conducting gas. The finite volume method seems well appropriate for solving the thermal problem in both the solid and fluid phases. The ray-tracing Monte Carlo method constitutes the powerful model for radiative properties. Finally, 3D image analysis of foams is useful to determine topological information needed to feed analytical thermal and radiative properties models. Cet article présente une vue globale des modèles traditionnels et récents de prédiction des propriétés thermiques et radiatives des mousses solides ayant des cellules ouvertes ou fermées. Leur conductivité thermique effective est déterminée par des modèles empiriques ou analytiques basés sur le réseau de résistances. Les propriétés radiatives nécessaires pour remonter à la conductivité radiative sont déterminées par des modèles analytiques basés sur la théorie de diffusion indépendante. Les approches robustes couplent la modélisation tridimensionnelle (3D) de mousses (par exemple, par la tomographie à rayons X, la mosaïque de Voronoï, etc.) et la résolution numérique des équations de transport. La conductivité thermique due à la phase solide est directement calculée par la méthode des éléments finis (EF), avec un coût de calcul raisonnable. La conductivité thermique effective, quant à elle, peut être déterminée à partir des calculs EF combinés avec la conductivité thermique due à la phase fluide. Cette dernière peut être évaluée de façon précise par des formules simples dans le cas de l'air ou d'un gaz faiblement conducteur thermique. Cependant, la méthode des volumes finis apparaît la mieux appropriée pour résoudre le problème thermique, à la fois dans la phase solide et la phase fluide. La méthode de Monte Carlo et de tracé de rayons constitue une approche solide pour calculer les propriétés radiatives. Enfin, la reconstruction d'image 3D des mousses est essentielle pour déterminer les informations topologiques nécessaires pour alimenter les modèles analytiques de conductivité thermique et de propriétés radiatives.

Generation and evaluation of gamma-ray buildup factors prepared as the standard data in the Atomic Energy Society of Japan

c A data set of gamma-ray buildup factors for point isotropic sources has been generated by the method of invariant embedding. Specific features of the data set as compared with the ANSI/ANS-6.4.3 standard published by the American Nuclear Society are 1) the data corresponding to effective dose with AP and ISO irradiation geometries, and ambient dose equivalents are added, 2) the distance from the source is extended up to 100 mean free paths (mfp) from 40 mfp in the ANS data, 3) the method of invariant embedding and the photon cross section PHOTX are used consistently for all materials, and 4) treatment of bremsstrahlung is improved by using the bremsstrahlung production data calculated by the EGS4 code. The accuracy of the transport calculations for exposure buildup factors by the IE method was evaluated quantitatively by works including 1) comparison of buildup factors with those obtained by the moments method in the ANS data, 2) comparison of the buildup factors with those obtained by the EGS4 code, and 3) self evaluation of influence of mesh width in space, energy and angle in the numerical solution of transport equation by the IE method. It is concluded that the magnitude of error in the present data is estimated about 10% or less up to 40 mfp.

Numerical solution of the neutron transport equation using cellular neural networks

Various methods have been used for solving the neutron transport equation in the past, and a number of computer codes have been developed based on these solution methods. This paper describes a novel method for the solution of the steady-state and time-dependent neutron transport equation using the duality between neutronic parameters in the method of characteristic (MOC) and the electrical parameters in the cellular neural networks (CNN). The relevant electrical circuit can be simulated by professional electrical circuit simulator software, HSPICE. This software is used for numerical solution of the transport equation only by preparation of appropriate inputs. This method does not need inner and outer iterations, which is a necessary step in the other deterministic methods. One of the main applications of the proposed method may be the development of a new hardware by VLSI technology for online spatio-temporal calculations of the transport equation for nuclear reactor core. The accuracy and capability of this method are examined in a 2D steady-state problem for a BWR fuel assembly, and a 2D time-dependent TWIGL seed/blanket problem.

Geometric observers for dynamically evolving curves.

This paper proposes a deterministic observer framework for visual tracking based on non-parametric implicit (level-set) curve descriptions. The observer is continuous-discrete, with continuous-time system dynamics and discrete-time measurements. Its state-space consists of an estimated curve position augmented by additional states (e.g., velocities) associated with every point on the estimated curve. Multiple simulation models are proposed for state prediction. Measurements are performed through standard static segmentation algorithms and optical-flow computations. Special emphasis is given to the geometric formulation of the overall dynamical system. The discrete-time measurements lead to the problem of geometric curve interpolation and the discrete-time filtering of quantities propagated along with the estimated curve. Interpolation and filtering are intimately linked to the correspondence problem between curves. Correspondences are established by a Laplace-equation approach. The proposed scheme is implemented completely implicitly (by Eulerian numerical solutions of transport equations) and thus naturally allows for topological changes and subpixel accuracy on the computational grid.

Numerical solution of transport equations for plasmas with transport barriers

An approach to solve numerically transport equations for plasmas with spontaneously arising and arbitrarily located transport barriers, regions with a strongly reduced transfer of energy, is proposed. The transport equations are written in a form conserving heat flux and solved numerically by using piecewisely exact analytical solutions of linear differential equations. Compared to standard methods, this approach allows to reduce significantly the number of operations required to obtain a converged solution with a heat conductivity changing abruptly at a critical temperature gradient and to use large time steps in modeling the formation and dynamics of transport barriers. Computations for the tokamak JET are done.

Solutal convection around growing protein crystals and diffusional purification in Space

Convective and diffusional mass transport to an isolated crystal growing from solution, with slow linear interface kinetics, is considered analytically as a generic scaling model. We focus on the interface kinetics which is slow as compared to the diffusion mass transport which is typical of protein crystal growth. Independently, full-scale numerical solutions of transport equations around a cylindrical crystal, at the center of the bottom of a cylindrical cell filled with the solution, are found. The two approaches give results that agree over a wide range of parameters, providing dimensionless relationships that allow predictions of the contribution of convection and diffusion to mass transport. Requirements for microgravity level in Space experiments to achieve diffusional mass transport are estimated on the basis of these relationships. Coefficients of impurity distribution between a growing crystal and its solution, under the influences of convection and diffusion around the crystal, were numerically evaluated as functions of time. The results provide further support for the hypothesis concerning the role of the impurity depletion zone in the purification of a growing crystal. They also reveal that in general, the impurity distribution within the crystal is not homogeneous due to convection. The effects of various factors on growth kinetics and crystal purity are considered.

A priori error estimates for interior penalty versions of the local discontinuous Galerkin method applied to transport equations

AbstractThe local discontinuous Galerkin method has been developed recently by Cockburn and Shu for convection‐dominated convection‐diffusion equations. In this article, we consider versions of this method with interior penalties for the numerical solution of transport equations, and derive a priori error estimates. We consider two interior penalty methods, one that penalizes jumps in the solution across interelement boundaries, and another that also penalizes jumps in the diffusive flux across such boundaries. For the first penalty method, we demonstrate convergence of order k in the L∞(L2) norm when polynomials of minimal degree k are used, and for the second penalty method, we demonstrate convergence of order k+1/2. Through a parabolic lift argument, we show improved convergence of order k+1/2 (k+1) in the L2(L2) norm for the first penalty method with a penalty parameter of order one (h−1). © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 545–564, 2001

Groundwater pollution by organic compounds: a two-dimensional analysis of contaminant transport in stratified porous media with multiple sources of non-equilibrium partitioning

Transport of pollutants in soil and groundwater often occurs in stratified media under non-equilibrium conditions. Confined aquifers are usually bounded by low-permeability layers of soil which have been shown to exert a significant influence on the fate of contaminants in groundwater. Numerical solutions of transport equations have usually been restricted to single layers and have included single sources of non-equilibrium processes or none at all. The effect of soil stratification itself has sometimes been reduced to a transport-based non-equilibrium process. A boundary element solution of the transport equations in the Laplace domain is extended to include multiple sources of non-equilibrium processes in saturated media under the assumption of rate-limited mass transfer. Green functions accurately model infinite and semi-infinite domains such as soils and Laplace transforms remove the need for time-stepping and the associated numerical complexity. The proposed numerical technique is validated by comparing its results to analytical solutions. Its scope is illustrated through a case study of a sand aquifer bounded by less permeable layers of silt, and infiltrated by pollutants from a neighbouring lake. Copyright © 1999 John Wiley & Sons, Ltd.

The analytic approach in the modelling of one-dimensional electron concentration distribution in some two-valley semiconductor electron devices

The conventional approach to the modelling of semiconductor devices operation is based on a numerical solution of transport equations. This paper exposes a complete analytical treatment of transport equations for a two-valley semiconductor. We consider the case when the geometry of the analysed structure permits us to assume that the transport in one of the dimensions of the structure is dominant, and when the electric field is homogeneous and stationary. The obtained solution for the case of generally set initial conditions is reduced to the quadratures. We used the example of a p-i-n diode to demonstrate the superiority of the analytical treatment as compared with the numerical one. The proposed approach offers the possibility to determine exactly and quickly the spatial-temporal distribution of the electron concentration within the structure. Considering the accuracy of the analytical procedure, compared with that of the numerical one, the obtained solution could also be used as a kind of standard for the analysis of particular approximate solutions of the considered type of partial differential equation systems.

Numerical solution of transport equation for applications in environmental hydraulics and hydrology

The advective term in the one-dimensional transport equation, when numerically discretized, produces artificial diffusion. To minimize such artificial diffusion, which vanishes only for Courant number equal to unity, transport owing to advection has been modeled separately. The numerical solution of the advection equation for a Gaussian initial distribution is well established; however, large oscillations are observed when applied to an initial distribution with sleep gradients, such as trapezoidal distribution of a constituent or propagation of mass from a continuous input. In this study, the application of seven finite-difference schemes and one polynomial interpolation scheme is investigated to solve the transport equation for both Gaussian and non-Gaussian (trapezoidal) initial distributions. The results obtained from the numerical schemes are compared with the exact solutions. A constant advective velocity is assumed throughout the transport process. For a Gaussian distribution initial condition, all eight schemes give excellent results, except the Lax scheme which is diffusive. In application to the trapezoidal initial distribution, explicit finite-difference schemes prove to be superior to implicit finite-difference schemes because the latter produce large numerical oscillations near the steep gradients. The Warming-Kutler-Lomax (WKL) explicit scheme is found to be better among this group. The Hermite polynomial interpolation scheme yields the best result for a trapezoidal distribution among all eight schemes investigated. The second-order accurate schemes are sufficiently accurate for most practical problems, but the solution of unusual problems (concentration with steep gradient) requires the application of higher-order (e.g. third- and fourth-order) accurate schemes.

Numerical Solution of Transport Equations for Bacterial Chemotaxis: Effect of Discretization of Directional Motion

A mathematical model proposed by Alt [J. Math. Biol., 9 (1980), pp. 147–177] to describe chemotactic bacterial migration is studied, and solutions to this model are compared to solutions of a simpler model proposed by Rivero et al. [Chemical Engineering Science, 44 (1989), pp. 2881–2897]. It is found that a discretized version of Alt’s model produces solutions similar to the continuous model, even at very coarse discretization. The relationship between the discretized Alt model and the model of Rivero et al. is elucidated, and it is found that a slightly modified version of the Rivero et al. model produces solutions similar to those of the Alt model for systems with one- and two-dimensional attractant gradients, suggesting that the simple and easy-to-use Rivero et al. model (after slight modification) is adequate for modeling bacterial behavior within the parameter ranges investigated. A preliminary investigation of the use of the lattice Boltzmann method for the study of bacterial migration is reported.

Development of Method Based on Perturbation Theory for Estimating and Correcting Errors in Numerical Solution of Transport Equation

Development of Method Based on Perturbation Theory for Estimating and Correcting Errors in Numerical Solution of Transport Equation

Development of method based on perturbation theory for estimating and correcting errors in numerical solution of transport equation.

The numerical solution of the transport equation has the errors caused by the approximations used in the computational method. In the past estimations of these errors have been performed experimentally. In the present study, formulas to estimate the errors have been derived on the basis of the perturbation theory. This method enables us to deterministically estimate the numerical errors due to the iteration, spatial discretization and Legendre polynomial expansion of scattering transfer cross sections. Using the error estimation method developed in the present study, two examples of error analyses were carried out to confirm its validity and applicability to error estimation for a practical purpose. The errors of the calculated tritium breeding ratio for 7Li in a infinite slab geometry were estimated, and they agreed well with the values predicted from direct calculation. As the second example, error analysis was carried out for one-dimensional nuclear calculations on two types of commercial fusion reacto...

Effects of diffusion on the electrophoretic behavior of associating systems: The Gilbert-Jenkins theory revisited

The Gilbert-Jenkins theory predicts the asymptotic shape of moving-boundary sedimentation and electrophoretic patterns and broad zone molecular sieve chromatographic elution profiles for the class of interacting systems, A + B ⇄ C, in which two dissimilar macromolecules react reversibly to form a complex. A particularly provocative case is the one in which the complex has a greater migration velocity than that of either reactant, each of which has a different velocity. Depending upon conditions, this case predicts, for example, that in the asymptotic limit an ascending electrophoretic pattern or a frontal gel chromatographic elution profile can show two hypersharp reaction boundaries separated by a plateau. This prediction is now confirmed by numerical solution of transport equations which retain the second-order diffusional term and extrapolation of the computed patterns to zero diffusion coefficient. For finite diffusion coefficient, however, the two hypersharp reaction boundaries are separated by a weak negative gradient. These calculations are extended to an examination of the transitions between the three types of patterns admitted by the case under consideration in order to gain physical understanding and to define criteria for recognizing the transitions. Studies of this kind not only establish confidence in the Gilbert-Jenkins theory, but, in addition, they provide new insights which make for more effective application of the theory to real systems.