Multi-scale finite element modelling using bubble function method for a convection-diffusion problem

Due to the multi-scale nature of the convection-diffusion equation the standard Galerkin method fails to predict accurate and stable results. A multi-scale finite element method based on the bubble function approach has been developed and applied to this problem. Finite element formulations based on the discontinuous weighting functions including the streamline upwind Petrov-Galerkin (SUPG) method are also evaluated. The results show that the multi-scale method provides stable and accurate results and matches very well the analytical solution.     

Lod generalized trapezoidal formula schemes for parabolic differential equations in two space dimensions

We describe locally one-dimensional (LOD) time integration schemes for parabolic differential equations in two space dimensions, based on the generalized trapezoidal formulas (GTF(α)). We describe the schemes for the diffusion equation with Dirichlet and Neumann boundary conditions, for nonlinear reaction-diffusion equations, and for the convection-diffusion equation in two space dimensions. The obtained schemes are second order in time and unconditionally stable for all α ∈ [0, 1]. Numerical experiments are given to illustrate the obtained schemes and to compare their performance with the better known LOD Crank-Nicolson scheme. While the LOD Crank-Nicolson scheme can give unwanted oscillations in the computed solution, our present LOD-GTF(α) schemes provide both stable and accurate approximations for the true solution.     

Hybrid numerical methods for convection-diffusion problems in arbitrary geometries

The hybrid nodal-integral/finite element method (NI-FEM) and the hybrid nodal-integral/finite analytic method (NI-FAM) are developed to solve the steady-state, two-dimensional convection-diffusion equation (CDE). The hybrid NI-FAM for the steady-state problem is then extended to solve the more general time-dependent, two-dimensional, CDE. These hybrid coarse mesh methods, unlike the conventional nodal-integral approach, are applicable in arbitrary geometries and maintain the high efficiency of the conventional nodal-integral method (NIM). In steady-state problems, the computational domain for both hybrid methods is discretized using rectangular nodes in the interior of the domain and along vertical and horizontal boundaries, while triangular nodes are used along the boundaries that are not parallel to the x or y axes. In time-dependent problems, the rectangular and triangular nodes become space-time parallelepiped and wedge-shaped nodes, respectively. The difference schemes for the variables on the interfaces of adjacent rectangular/parallelepiped nodes are developed using the conventional NIM. For the triangular nodes in the hybrid NI-FEM, a trial function is written in terms of the edge-averaged concentration of the three edges and made to satisfy the CDE in an integral sense. In the hybrid NI-FAM, the concentration over the triangular/wedge-shaped nodes is represented using a finite analytic approximation, which is based on the analytic solution of the one-dimensional CDE. The difference schemes for both hybrid methods are then developed for the interfaces between the rectangular/parallelepiped and triangular/wedge-shaped nodes by imposing continuity of the flux across the interfaces. A formal derivation of these hybrid methods and numerical results for several test problems are presented and discussed.     

An analysis of the convection-diffusion problems using meshless and meshbased methods

The numerical solution of the convection-diffusion equation represents a very important issue in many numerical methods that need some artificial methods to obtain stable and accurate solutions. In this article, a meshless method based on the local Petrov-Galerkin method is applied to solve this equation. The essential boundary condition is enforced by the transformation method, and the MLS method is used for the interpolation schemes. The streamline upwind Petrov-Galerkin (SUPG) scheme is developed to employ on the present meshless method to overcome the influence of false diffusion. In order to validate the stability and accuracy of the present method, the model is used to solve two different cases and the results of the present method are compared with the results of the upwind scheme of the MLPG method and the high order upwind scheme (QUICK) of the finite volume method. The computational results show that fairly accurate solutions can be obtained for high Peclet number and the SUPG scheme can very well eliminate the influence of false diffusion.     

Meshless techniques for convection dominated problems

In this paper, the stability problem in the analysis of the convection dominated problems using meshfree methods is first discussed through an example problem of steady state convection-diffusion. Several techniques are then developed to overcome the instability issues in convection dominated phenomenon simulated using meshfree collocation methods. These techniques include: the enlargement of the local support domain, the upwind support domain, the adaptive upwind support domain, the biased support domain, the nodal refinement, and the adaptive analysis. These techniques are then demonstrated in one- and two-dimensional problems. Numerical results for example problems demonstrate the techniques developed in this paper are effective to solve convection dominated problems, and in these techniques, using adaptive local support domain is the most effective method. Comparing with the conventional finite difference method (FDM) and the finite element method (FEM), the meshfree method has found some attractive advantages in solving the convection dominated problems, because it easily overcomes the instability issues.     

Jacobi galerkin spectral method for cylindrical and spherical geometries

The approximation of the convection-diffusion problem based on the Galerkin method in Cartesian, cylindrical and spherical coordinates is considered with emphasis in the last two cases. In particular, cylindrical and spherical coordinates can lead to a degeneracy in the global system of equations. This difficulty is removed by incorporating the factor r into the weight function which is introduced naturally by using Jacobi polynomials with α=0 and β=1,2. By doing this, an unified framework is obtained for handling the typical geometries required in chemical engineering. Examples are presented based on the Galerkin method for discussing the applicability of this approach.     

Numerical approximation of a flow and transport system in unsaturated-saturated porous media

In this paper we develop an approximation scheme for solving a coupled system of flow and contaminant transport with adsorption in unsaturated-saturated porous media. The flow model is based on nonlinear Richard's equation and is coupled with the transport equation through saturation and Darcy's velocity (discharge) terms. The approximation is based on time stepping, operator splitting and the method of characteristics (see Pironneau (Numer. Math. 38 (1982) 309), Douglas and Russel (SIAM J. Numer. Anal. 19(5) (1982) 871)). The nonlinear terms in Richard's equation are approximated by means of a relaxation scheme (see Jäger and Ka?ur (Math. Modelling Numer. Anal. 29 (1995) 605), Ka?ur (IMA J. Numer. Anal. 19 (1999b) 119), Ka?ur (SIAM J. Numer. Anal. 36(1) (1999a) 290)). The convergence of the approximation scheme is proved. Some numerical experiments in 1D are included.     

高深宽比微细结构电铸时传质过程数值分析

针对高深宽比微细结构电铸时存在严重传质受限的问题,以微深槽特征为分析对象建立液相传质两种数学模型——维扩散模型和二维对流—扩散模型,并分别用Matlab专用工具箱和Fluent 6.2流体仿真软件进行数值求解,依次分析以扩散、强制对流－扩散、复合对流(强制对流和自然对流)－扩散等为主导传质模式作用下微细电铸时,流场和离子浓度场的空间变化规律及其对液相传质效果的影响,并进行试验验证。结果表明：微细结构电铸时,单一扩散作用仅能用于深宽比小于2且电流密度小于2 A/dm2的液相传质场合;槽外强制对流只能对深宽比小于2的微槽内电解液产生一定搅拌作用;强化槽内自然对流作用并与槽外强制对流协同配合时,槽(深宽比为5)内可形成独特的单个或多个占据整个槽空间的涡流循环胞,涡流流速约为强制对流流速的1/20～1/2,明显改善传质效果,试验结果与此相印证。     

The UNIT algorithm for solving one-dimensional convection-diffusion problems via integral transforms

A unified approach for solving convection-diffusion problems using the Generalized Integral Transform Technique (GITT) was advanced and coined as the UNIT (UNified Integral Transforms) algorithm, as implied by the acronym. The unified manner through which problems are tackled in the UNIT framework allows users that are less familiar with the GITT to employ the technique for solving a variety of partial-differential problems. This paper consolidates this approach in solving general transient one-dimensional problems. Different integration alternatives for calculating coefficients arising from integral transformation are discussed. Besides presenting the proposed algorithm, aspects related to computational implementation are also explored. Finally, benchmark results of different types of problems are calculated with a UNIT-based implementation and compared with previously obtained results.     

A comparison of several numerical methods for the solution of the Convection-Diffusion Equation using the method of finite spheres

In this paper we compare several numerical methods for the solution of the convection-diffusion equation using the method of finite spheres; a truly meshfree numerical technique for the solution of boundary value problems. By conducting numerical inf-sup tests on a one-dimensional model problem it is found that a higher order derivative artificial diffusion (Ho DAD) method performs the best among the schemes tested. This method is then applied to the analysis of problems in two-dimensions.