求解非稳态对流占优问题的非标准无网格Galerkin方法  

Nonstandard element free Galerkin method for solving unsteady convection dominated problems

作　　者：欧阳洁[1] 张林[1] 张小华[1] 

机构地区：[1]西北工业大学应用数学系,陕西西安710072

出　　处：《空气动力学学报》2007年第3期287-293,299,共8页Acta Aerodynamica Sinica

基　　金：国家自然基金重大项目(批准号:10590353);陕西省自然基金(批准号:2005A16);2004-2005西北工业大学本科毕业设计重点扶持项目资助

摘　　要：用θ加权法离散时间域,并将四种稳定化方案与无网格Galerkin方法相耦合进行空间域的离散。在无网格Galerkin方法中,采用线性基和具有连续的权函数,基于移动最小二乘法构造了高阶导数连续的形函数,从而避免了有限元方法中采用线性元插值时,因忽略稳定项中二阶导数项而降低计算精度的问题。数值计算表明:本文构造的方法成功地消除了非定常对流扩散方程中对流项占优时的数值伪振荡现象,并具有计算精度高、稳定性好、算法实施简单、前后处理方便的优点。特别是所构造的MFLS方法非常适宜于求解非定常的对流扩散方程。The main advantages of meshless methods compared with traditional mesh-based methods are that they can dispense with the modeling effort dedicated to mesh generation. The element free Galerkin （EFG） method is one of meshless methods. Although it has been widely used for solving several problems of materials mechanics and solid mechanics, it is few in aerodynamics and hydromechanics. It is will known that numerical solutions of conventional methods may be corrupted by non-physical oscillations when the convection action dominates the diffusion action in the transport problems. The similar phenomena will happen if EFG method is directly applied to convection dominated problems. In order to eliminate spurious oscillations, time discretization is here carried by 0 family of methods while spatial discretization is carried by EFG method combined with stabilization schemes such as streamline upwind Petrov-Galerkin method, Galerkin least squares method, sub-grid scale method and least squares method. In above constructed stabilization system based on moving least squares approximation, second-order derivatives of the interpolation are well defined in the whole domain even for linear interpolation. Thus, it avoids neglecting second-order derivatives of shape function needed in the stabilization term. The efficiency of these methods used for unsteady convection dominated problems are observed by several presented numerical examples. It can be seen that these methods have high accuracy and good stabilization since spurious oscillations can be largely restrained. At the same time, the precision of numerical solutions for discontinuity problems can be improved by the supplement of nodes. Specially, least squares method combined with EFG method is the best one among above mentioned method because it can make error and spurious oscillation least for solving linear or nonlinear unsteady convection dominated problems.

关 键 词：无网格方法 对流占优 稳定化 

分 类 号：O35[理学—流体力学]