引入补偿刚度的流体力学标准伽辽金有限元研究

Standard Galerkin Finite Element Method Considering Compensation Stiffness in Fluid Mechanics

摘要:

对一维定常对流扩散方程有限元解的波动问题进行了研究，指出提高单元间形函数一阶导数的连续性是改善有限元解数值波动的有效方法.在标准伽辽金有限元基础上引入考虑补偿刚度的补偿项，即在单元与单元之间的节点上增加一个“不平衡力”，形成补偿标准伽辽金有限元格式.针对线性拉格朗日插值和指数型插值分别探讨补偿项中补偿刚度表达式，并对比了补偿有限元解与解析解结果.研究表明，引入补偿项后能提高单元间形函数的连续性，从而明显改善一维对流扩散方程的数值波动.与标准线性拉格朗日插值相比，补偿指数型插值不仅在单元内可精确给出变量的分布，而且单元之间的连续性更好，因而能更好地控制数值波动现象，取得问题较好的数值解.

关键词:流体力学; 对流扩散方程; 数值波动; 补偿刚度; 补偿标准伽辽金

Abstract:

The wave problems of the finite element solution of one dimensional steady state convection diffusion equation was analyzed. It is indicated that the continuity improvement of the shape function between elements is an effective method to decrease the amplitude of wave. Based on the standard Galerkin (SG) finite element method, the compensation term considering the elastic stiffness was introduced by adding an “unbalanced force” to the nodes between elements, to form a complemental standard Galerkin (CSG) finite element equation. The expressions of compensation stiffness of linear Lagrange based interpolation (LLBI) function and exponential function based interpolation (EFBI) function were discussed. The results obtained by CSG and SG were compared with those of the analytical solution. It shows that the continuity of the shape function is improved and the amplitude of wave is decreased when the compensation term is introduced into the SG finite element. Compared with the SG finite element using LLBI, the CSG finite element using EFBI is more effective to control the wave phenomenon since it has a better continuity both inside the element and between two elements.

Key words:fluid mechanics; convection diffusion equation; numerical wave; compensation stiffness; complemental standard Galerkin (CSG)