Finite-element Flux-corrected Transport Research Articles

Development of a total variation diminishing (TVD) sea ice transport scheme and its application in an ocean (SCHISM v5.11) and sea ice (Icepack v1.3.4) coupled model on unstructured grids

Abstract. As the demand for increased resolution and complexity in unstructured sea ice models is growing, higher demands are also placed on the sea ice transport scheme. In this study, we couple the Semi-implicit Cross-scale Hydro-science Integrated System Model (SCHISM, v5.11) with Icepack (v1.3.4), the column physics package of the Los Alamos sea ice model (CICE); a key step is to implement a total variation diminishing (TVD) transport scheme for the multi-class sea ice module in the coupled model. Compared with the second-order upwind scheme and the finite-element flux-corrected transport (FEM-FCT) scheme, the TVD transport scheme is overall superior when evaluated based on conservation, accuracy, efficiency (even with very high resolution), and strict monotonicity. Although it is slightly weaker than FEM-FCT in terms of accuracy alone, the TVD scheme still outperforms the other two schemes in comprehensive performance. The new coupled model outperforms the existing single-class ice model of SCHISM in the case of Lake Superior. For the Arctic Ocean case, it successfully reproduces the long-term changes in the sea ice extent, sea ice boundary, concentration observations from satellites, and thickness from in situ measurement.

A cross-diffusion system modeling rivaling gangs: Global existence of bounded solutions and FCT stabilization for numerical simulation

In this paper, we study a gang territorial model consisting of two parabolic and two ordinary differential equations, where a taxis-type mechanism models that the two rivaling gangs are repelled by each other’s graffiti. Our main analytical finding shows the existence of global, bounded classical solutions. By making use of quantitative global estimates, we prove that these solutions converge to homogeneous steady states if the initial data are sufficiently small. Moreover, we perform numerical experiments which show that for different choices of parameters, the system may become diffusion- or convection-dominated, where in the former case the solutions converge toward constant steady states while in the latter case nontrivial asymptotic behavior such as segregation is observed. In order to perform these experiments, we apply a nonlinear finite element flux-corrected transport method (FEM-FCT) which is positivity-preserving. Then we treat the nonlinearities in both the system and the proposed nonlinear scheme simultaneously using fixed-point iteration.

Existence of solutions of a finite element flux-corrected-transport scheme

The existence of a solution is proved for a nonlinear finite element flux-corrected-transport (FEM-FCT) scheme with arbitrary time steps for evolutionary convection–diffusion–reaction equations and transport equations.

3-D numerical analysis of EHD turbulent flow and mono-disperse charged particle transport and collection in a wire-plate ESP

The present study attempts to develop a detailed numerical approach and a simulation procedure to predict the motion of gas, ions and particles inside a simple parallel plate channel containing a single corona wire. A hybrid Finite Element (FEM)-Flux Corrected Transport (FCT)-Finite Volume (FVM) method is used: the FEM–FCT numerical algorithm is applied for modeling the steady-state corona discharge, while the turbulent gas flow and the particle motion under electrostatic forces are modeled using the commercial CFD code FLUENT. Calculations for the gas flow are carried out by solving the Reynolds-averaged Navier–Stokes equations and turbulence is modeled using the k– ɛ turbulence model. An additional source term is added to the gas flow equation to include the effect of the electric field, obtained by solving a coupled system of the electric field and charge transport equations using User-Defined Functions (UDFs). The particle phase is simulated based on the Lagrangian approach, where a large number of particles is traced with their motion affected by the gas flow and electrostatic forces using the Discrete Phase Model (DPM) in FLUENT. The developed model is useful to gain insight into the particle collection phenomena that take place inside an ESP.

A Finite Element Flux-Corrected Transport Method for Wave Propagation in Heterogeneous Solids

When moving discontinuities in solids need to be simulated, standard finite element (FE) procedures usually attain low accuracy because of spurious oscillations appearing behind the discontinuity fronts. To assure an accurate tracking of traveling stress waves in heterogeneous media, we propose here a flux-corrected transport (FCT) technique for structured as well as unstructured space discretizations. The FCT technique consists of post-processing the FE velocity field via diffusive/antidiffusive fluxes, which rely upon an algorithmic length-scale parameter. To study the behavior of heterogeneous bodies featuring compliant interphases of any shape, a general scheme for computing diffusive/antidiffusive fluxes close to phase boundaries is proposed too. The performance of the new FE-FCT method is assessed through one-dimensional and two-dimensional simulations of dilatational stress waves propagating along homogeneous and composite rods.

기체 방전의 시뮬레이션을 위한 FE-FCT를 이용한 준 2차원적 수치 모델

기체 방전의 수치적 시뮬레이션을 위하여 FE-FCT(Finite-Element Flux-Corrected Transport) 방법을 이용한 준 2차원 수치적 모델을 제시한다. 이 모델에서는 전자와 이온에 대한 1차원 연속방정식을 풀어 시 공간적으로 변하는 전하 분포를 계산하고, 공간 전하 분포에 의한 전기장의 변화는 2차원적 전하 분포를 고려하는 디스크 방법을 적용하여 푸아송(Poisson)방정식을 풀어 계산한다. 다양한 벤치마크 문제에 대해 계산한 결과는 이 모델의 정확성과 적용성을 잘 보여준다. 또 스트리머 방전에 대해 계산한 결과는 앞선 연구 결과와 잘 일치하는 것을 보여 준다. A quasi two-dimensional model for numerical simulation of gas discharge is presented, based on the finite-element flux-corrected transport method. A one-dimensional continuity convection-diffusion equation coupled Poisson's equation is solved to calculate the charge density variation and the electric field is evaluated by the classical disk method. Results calculated for various benchmark problems verify the accuracy of the proposed model and illustrate its performance. This model has been applied to a streamer simulation, and the results are shown to agree well with previously published results.

Numerical analysis of the stability of the electrohydrodynamic (EHD) electroconvection between two plates

The time evolution of the problem of electrohydrodynamic convection in a liquid between two plates is analysed numerically. The equations are non-dimensionalized using the ion drift velocity and the viscous time scales. Following the non-dimensionalization of the respective model, two different techniques have been used to describe the charge evolution, namely, the finite-element flux-corrected transport method and the particle-in-cell technique. The results obtained with the two schemes, apart from showing good agreement, have revealed the appearance of a two-roll structure not described in previous works. This is investigated in detail for both strong and weak injection.

A non-oscillatory method for spallation studies

This paper introduces a non-oscillatory method, the finite element flux-corrected transport (FE-FCT) method for spallation studies. This method includes the implementation of a one-dimensional FCT algorithm into a total Lagrangian finite element method. Consequently, the FE-FCT method can efficiently eliminate fluctuations behind shock wave fronts without smearing them. In multidimensional simulations, the one-dimensional FCT algorithm is used on each grid line of the structured meshes to correct the corresponding component of nodal velocities separately. The requirement of structured meshes is satisfied by using an implicit function so that arbitrary boundaries of the simulated object can be described. In this paper, the proposed FE-FCT method is applied in spallation studies. One- and two-dimensional examples show this non-oscillatory method could be one of the candidates to accurately predict spallation and the spall thickness. Copyright © 2005 John Wiley & Sons, Ltd.

Streamer branching in transient nonuniform short gap discharges using numerical modeling

Images of streamer propagation and branching are presented in a short-point plane gap in air at atmospheric pressure. The results are obtained by employing a finite-element flux corrected transport (FE-FCT) algorithm in conjunction with a subgridding scheme for the efficient and accurate calculation of secondary effects during streamer development and propagation. The significant reduction in time required for the calculations makes numerical modeling a useful tool for a better understanding of physical processes, such as branching, that are central to the propagation and development of streamers.

Three-dimensional characterization of short-gap streamers in air at atmospheric pressure

Three-dimensional (3-D) images are presented for the propagation of streamers in a short uniform gap in air at atmospheric pressure. Results are obtained through a self-consistent solution of the hydrodynamic diffusion-drift model, which includes Poisson's equation and the continuity equations for the charged particles. A 3-D extension of the finite element-flux corrected transport (FE-FCT) algorithm with the incorporation of a new subgridding scheme for the efficient calculation of secondary effects, such as photoionization, is utilized to obtain these 3-D simulation images.

Numerical simulation of DNA sample preconcentration in microdevice electrophoresis

A numerical model is presented for the accurate and efficient prediction of preconcentration and transport of DNA during sample introduction and injection in microcapillary electrophoresis. The model incorporates conservation laws for the different buffer ions, salt ions, and DNA sample, coupled through a Gaussian electric field to account for the field modifications that cause electromigration. The accuracy and efficiency required to capture the physics associated with such a complex transient problem are realized by the use of the finite element-flux corrected transport (FE-FCT) algorithm in two dimensions. The model has been employed for the prediction of DNA sample preconcentration and transport during electrophoresis in a double-T injector microdevice. To test its validity, the numerical results have been compared with the corresponding experimental data under similar conditions, and excellent agreement has been found. Finally, detailed results from a simulation of DNA sample preconcentration in electrophoretic microdevices are presented using as parameters the electric field strength and the other species concentrations. The effect of the Tris concentration on sample stacking is also investigated. These results demonstrate the great potential offered by the model for future optimization of such microchip devices with respect to significantly enhanced speed and resolution of sample separation.

An FE-FCT method with implicit functions for the study of shock wave propagation in solids

A finite element flux-corrected transport (FE-FCT) method is proposed for the study of shock wave propagation in solids. The FCT algorithm contains two stages: transport and antidiffusion. The total Lagrangian finite element method is used here and the FCT algorithm is only applied to correct the nodal velocities along the lines of the structured mesh. An implicit function is implemented into the finite element method so that the objective with arbitrary boundaries can be modeled with structured mesh. Both one-dimensional and two-dimensional examples show that the FE-FCT method can efficiently eliminate the oscillations behind the shock wave fronts.

Secondary emission effects on streamer branching in transient non-uniform short-gap discharges

A new approach is presented for calculation of photoionization rates, in fully three-dimensional grids, that improves the accuracy of the secondary processes calculation without significantly compromising the efficiency of the numerical algorithm. The method is based on generating a coarser secondary grid and interpolating the photoionization values between the two meshes, in order to overcome the enormous effort required for calculation of photoionization in gas discharge problems.A comprehensive study of the effects of photoionization, photoemission and background ionization in a short point-plane gap in air at atmospheric pressure is then presented, by using the above approach for the secondary processes in two dimensions, in conjunction with the two-dimensional axisymmetric finite-element flux-corrected transport algorithm. The secondary processes are modelled individually within a wide range of parametric values to reflect the uncertainty in the experimental data, and their effect on streamer development and propagation is investigated. The significant reduction in time required for the calculations makes numerical modelling an essential tool for better understanding of the very important yet not well understood physical processes central to the propagation and development of streamers.Finally, numerical branching is observed under certain conditions in the absence of an adequate supply of electrons in high field regions.

A two-dimensional, finite-element, flux-corrected transport algorithmfor the solution of gas discharge problems

An improved finite-element flux-corrected transport (FE-FCT)scheme, which was demonstrated in one dimension by the authors, is nowextended to two dimensions and applied to gas discharge problems. Thelow-order positive ripple-free scheme, required to produce a FCTalgorithm, is obtained by introducing diffusion to the high-orderscheme (two-step Taylor-Galerkin). A self-adjusting variable diffusioncoefficient is introduced, which reduces the high-order scheme to theequivalent of the upwind difference scheme, but without the complexities of anupwind scheme in a finite-element setting. Results are presented which showthat the high-order scheme reduces to the equivalent of upwinding when the newdiffusion coefficient is used. The proposed FCT scheme is shown to givesimilar results in comparison to a finite-difference time-split FCT codedeveloped by Boris and Book. Finally, the new method is applied for the firsttime to a streamer propagation problem in its two-dimensional form.

Simulation of the coronal development in air at radio frequency: the effects of attachment, secondary emission and diffusion

The finite-element flux-corrected transport (FE-FCT) method is used to examine the transient development of corona discharges in a 1 cm hyperboloid (50 /spl mu/m tip radius) point-plane gap in air at a frequency of 40 MHz. The investigation revealed the formation of streamers in the positive part of the voltage cycle during the transient stage. These are suppressed in the steady state due to the formation of negative ions, while a cathode fall region is fully established. The effects of attachment, secondary emission and diffusion are also examined, both in the transient and steady states.

Characterization of point-plane corona in air at radio frequency using a FE-FCT method

The development of corona under RF conditions in point-plane configuration has been investigated in this paper, using the recently developed finite-element flux-corrected transport (FE-FCT) method. The study was made in air for a 1 cm gap with 50 µm tip radius needle at 40 MHz and the effects of different parameters, such as the peak applied voltage (in the range 3-4.5 kV) on the coronal development and particularly on the current and light output, are examined. Polarity and hysteresis effects are observed, as well as a plateau in the current waveform, which are confirmed by previously published experimental results. Finally, a Fourier analysis on the steady-state current waveform is performed.

The theory of short-gap breakdown of needle point-plane gaps in air using finite-difference and finite-element methods

Numerical results are presented for the formation of breakdown streamers which bridge a 1 mm gap between a positive 50 µm radius hyperboloid point and a plane, when a dc voltage is applied. The results show that, for such gaps, no streamers form at voltages lower than or equal to 2.5 kV and that streamers bridge the gap at higher voltages. The streamer speed and radial dimensions of the streamer are found to linearly increase with the applied voltage and this agrees with what is predicted by existing two-dimensional models. The electric field in the streamer channel behind the streamer head is initially found to be much lower than that for longer gaps and becomes comparable only later in the development of the streamer. The results are obtained using a pre-existing finite-difference code and a new finite-element code developed using a new finite-element flux-corrected transport (FE-FCT) method. The finite-element results are shown to be almost identical to the finite-difference results. The finite-element method, however, through the use of unstructured grids, reduces significantly the number of unknowns and makes the modelling of streamers and arbitrarily shaped electrodes in two dimensions a feasible task.

An Improved Finite-Element Flux-Corrected Transport Algorithm

An improved finite-element flux-corrected transport (FE-FCT) method for the numerical solution of hydrodynamic conservation equations is described, based on the method developed by Lohner and his collaborators to solve conservation equations in fluid mechanics, and its application is extended to gas discharge problems. The high- and low-order schemes used are the ones proposed by Lohner who adds diffusion to the high-order scheme by subtracting the lumped-mass matrix from the consistent-mass matrix to give the low-order scheme; the diffusion coefficient is adjusted globally. A variable diffusion coefficient is introduced; it is assumed to be constant in each element and is shown to transform the high-order solution to a scheme equivalent to an upwind scheme which has minimal diffusion but ensures positive results. This avoids the complexity of upwinding in FE, especially in two dimensions. It is also shown that the correct amount of “real” diffusion may be easily added to the algorithm when required, for example, for electrons. Results are presented which show that the high-order scheme reduces to the upwind difference scheme when the new diffusion is used. The proposed FCT scheme is shown to give similar results, in comparison with a fourth-order FD-FCT algorithm. Finally, the new method is applied to a streamer propagation problem in one dimension, and the results obtained are shown to agree well with previously published results.

Turbulent boundary-layer characteristics over a flat-plate/wedge configuration at Mach 6

References ^ack, L. H., and Cuffel, R. F., Detection of Oblique Shocks in a Conical Nozzle with a Circular-Arc Throat, AIAA Journal, Vol. 4, No. 12, 1966, pp. 2219-2221. Thornhill, D., and Ness, R., private communication, Lockheed Missile & Space Co., Palo Alto, CA, 1988-1989. Adamson, T. C., and Nichols, J. A., On the Structure of Jets from Highly Underexpanded Nozzles into Still Air, Journal of Aerospace Sciences, Vol. 26, No. 1, 1959, pp. 16-24. Addy, A. L., Dutton, J. C., and Amatucci, V. A., Nonuniform Nozzle Flow Effects on Base Pressure at Supersonic Flight Speeds, AIAA Journal, Vol. 24, No. 7, 1986, pp. 1209-1212. Serra, J., Determination of Internal Gas Flows by a Transient Numerical Technique, AIAA Journal, Vol. 10, No. 5, 1972, pp. 603-611. L6hner, R., Baum, J. D., Loth, E., and Ramamurti, R., A Element Solver for Axisymmetric Compressible Flows, AIAA Paper 89-1794, Buffalo, NY, June, 1989. L6hner, R., Morgan, K., Peraire, J., and Vahdati, M., Finite Element Flux Corrected Transport (FEM-FCT) for the Euler and Navier-Stokes Equations, International Journal of Numerical Methods in Fluids, Vol. 7, No. 10, 1987, pp. 1093-1109. L6hner, R., Morgan, K., and Zienkiewicz, O. C., An Adaptive Element Procedure for High Speed Flows, Computer Methods in Applied Mechanics and Engineering, Vol. 51, No. 1-3, 1985, pp. 441-465. Roe, P. L., Error Estimates for Cell-Vertex Solvers of the Compressible Euler Equation, Institute for Computer Applications in Science and Engineering, Rept. 87-6, Langley, VA, 1987. Oran, E. S., and Boris, J. P., Numerical Simulation of Reactive Flow, Elsevier, New York, 1987. 11 Loth, E. Baum, J. D., and Lohner, R., Formation of Shocks Within Axisymmetric Flows, AIAA Paper 90-1655, Buffalo, NY, June 1990.