Finite-element Flux-corrected Transport Method Research Articles

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.

A density-dependent FEM-FCT algorithm with application to modeling platelet aggregation.

Upon injury to a blood vessel, flowing platelets will aggregate at the injury site, forming a plug to prevent blood loss. Through a complex system of biochemical reactions, a stabilizing mesh forms around the platelet aggregate forming a blood clot that further seals the injury. Computational models of clot formation have been developed to a study intravascular thrombosis, where a vessel injury does not cause blood leakage outside the blood vessel but blocks blood flow. To model scenarios in which blood leaks from a main vessel out into the extravascular space, new computational tools need to be developed to handle the complex geometries that represent the injury. We have previously modeled intravascular clot formation under flow using a continuum approach wherein the transport of platelet densities into some spatial location is limited by the platelet fraction that already reside in that location, i.e., the densities satisfy a maximum packing constraint through the use of a hindered transport coefficient. To extend this notion to extravascular injury geometries, we have modified a finite element method flux-corrected transport (FEM-FCT) scheme by prelimiting antidiffusive nodal fluxes. We show that our modified scheme, under a variety of test problems, including mesh refinement, structured vs unstructured meshes, and for a range of reaction rates, produces numerical results that satisfy a maximum platelet-density packing constraint.

Numerical stability analysis of two‐dimensional solute transport along a discrete fracture in a porous rock matrix

AbstractThis work reports numerical stability conditions in two‐dimensional solute transport simulations including discrete fractures surrounded by an impermeable rock matrix. We use an advective‐dispersive problem described in Tang et al. (1981) and examine the stability of the Crank‐Nicolson Galerkin finite element method (CN‐GFEM). The stability conditions are analyzed in terms of the spatial discretization length perpendicular to the fracture, the flow velocity, the diffusion coefficient, the matrix porosity, the fracture aperture, and the fracture longitudinal dispersivity. In addition, we verify applicability of the recently developed finite element method‐flux corrected transport (FEM‐FCT) method by Kuzmin () to suppress oscillations in the hybrid system, with a comparison to the commonly utilized Streamline Upwinding/Petrov‐Galerkin (SUPG) method. Major findings of this study are (1) the mesh von Neumann number (Fo) ≥ 0.373 must be satisfied to avoid undershooting in the matrix, (2) in addition to an upper bound, the Courant number also has a lower bound in the fracture in cases of low dispersivity, and (3) the FEM‐FCT method can effectively suppress the oscillations in both the fracture and the matrix. The results imply that, in cases of low dispersivity, prerefinement of a numerical mesh is not sufficient to avoid the instability in the hybrid system if a problem involves evolutionary flow fields and dynamic material parameters. Applying the FEM‐FCT method to such problems is recommended if negative concentrations cannot be tolerated and computing time is not a strong issue.

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.

Finite element methods for time-dependent convection–diffusion–reaction equations with small diffusion

Numerical studies of stabilized finite element methods for solving scalar time-dependent convection–diffusion–reaction equations with small diffusion are presented in this paper. These studies include the streamline-upwind Petrov–Galerkin (SUPG) method with different parameters, various spurious oscillations at layers diminishing (SOLD) methods, a local projection stabilization (LPS) scheme based on enrichment and two finite element method flux corrected transport (FEM-FCT) methods. The focus of the evaluation of the numerical results is on the reduction of spurious oscillations.

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.

구형 집속 빔 핵융합 장치에서 그리드 음극 구조의 최적 설계

Neutron production rate in spherically convergent beam fusion(SCBF) device as a portable neutron source strongly depends on the ion current and the grid cathode structure. In this paper, as the process of design and analysis, Design of Experiment(DOE) based on the results by Finite Element Method-Flux Corrected Transport(FEM-FCT) method is employed to calculate the ion current. This method is very useful to find optimal design conditions in a short time. Number of rings, radius of rings, and distance between the grid cathode and center are selected as control factors. From the results in the optimized model, the higher ion current is calculated and deeper potential well is also observed.

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.

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.

The effect of photoemission on the streamer development andpropagation in short uniform gaps

Results are presented for the time evolution of photoemission ina 0.1 cm parallel-plane gap in atmospheric pressure air when a positive dcvoltage is applied at one of the electrodes. The hydrodynamic set ofequations is solved using the finite-element flux-corrected transportmethod in two dimensions. The time evolution of the electron distributionat the cathode and the variation of the spread of the electrons areexamined during the avalanche, the avalanche-to-streamer transition andstreamer propagation stages. Finally, the effect of the variation of thephotoemission coefficient on the field distribution and the currentwaveform are presented.

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.

Adaptive unstructured finite element method for two-dimensional detonation simulations

A non-equilibrium reacting flow methodology has been added to a conservative, monotonic, compressible flow solver to allow numerical simulations of gas detonations. This flow solver incorporates unstructured dynamically adaptive meshes with the Finite Element Method – Flux Corrected Transport (FEM-FCT) scheme, which has shown excellent predictive capability of various non-reacting compressible flows. A two-step induction parameter model was used to model the combustion of the gas phase coupled with an energy release equation which was simulated with a point implicit finite element scheme. This combustion model was then applied to a two-dimensional detonation test case of a hypothetical fuel:oxygen mixture. The detonation simulation yielded two transverse waves which continued to propagate. This feature and the detonation shock speed mean and fluctuations were found to be grid-independent based on a resolution of about twenty elements within the average induction length. The resolution was efficiently achieved with the unstructured dynamically adaptive finite elements, which were three orders of magnitude less in number then required for uniform discretization.

Unstructured adaptive remeshing finite element method for dusty shock flow

The passage of planar shocks in a dusty gas was investigated to note effects due to particle loading and initial shock Mach number. Two-phase flow equations have been added to a conservative, monotonic flow solver to allow study of compressible particle and droplet flows, which are of importance for shock propagation in two-phase flows and spray propulsion systems. The formulation developed herein employed a conservative Eulerian treatment for the gas and particle phases. The computations were performed using the finite element method-flux corrected transport (FEM-FCT) scheme, which has shown excellent predictive capability of various compressible flows which include both strong and weak shocks. The flux limiting technique was modified to provide monotonic particle velocity fields to increase the scheme's computational stability. Adaptive unstructured methodology based on adapting to high gradients of both the fluid and particle densities was used in conjunction with the conservative shock-capturing scheme to adequately resolve strong flowfield gradients. The shock attenuation of this scheme was then compared with previous experimental and numerical results and was found to yield robust predictions. Various interphase coupling terms were also considered to note their effect on the shock attenuation.

Vorticity produced by shock wave diffraction

Numerical simulations with a monotonicity preserving flow solver have been performed to study shock diffraction phenomena and shock wave generated vorticity. The computations were performed using the conservative Finite Element Method-Flux Corrected Transport (FEM-FCT) scheme, which has been shown to have an excellent predictive capability for various compressible flows with both strong and weak shocks. Adaptive unstructured methodology based on adapting to high density and entropy gradients was used in conjunction with a conservative shock-capturing scheme to adequately resolve storng and weak flowfield gradients. The chief interest was therefore, a topic of vortices (original paper:the formation of vorticity) arising from shock wave propagation over a sharp corner and the high accuracy and resolution of the interacting compressible wave features. Numerical simulations were compared with previous experimental results and exhibited remarkably good agreement in terms of compressible wave propagation, as well as vorticity development and transport. The computations also allowed insight to the fundamental fluid dynamics, spencifically shock diffraction, vortex convection and shock-vortex interactions.

A study of low pressure particle impaction processes

Sonic velocity impactor stages have been investigated by the numerical solution of Euler equations and the equation of motion of the particles using the FEM-FCT (Finite Element Method - Flux Corrected Transport) method. Calculated solutions compare well with published results. The effects of the pressure ratio, χ = P ∞/ P 0 and the jet to plate distance S/ D are examined. It is found that at certain jet to plate distances the impinging jet is unstable and the effects of this instability are examined.