矩形空腔内Stokes流的状态空间有限元法

基于Hellinger-Reissner二类变分原理,从平面Stokes流问题的平衡方程、连续性要求和边界条件出发,得到相应的Hamilton函数,建立Hamilton正则方程后,采用分离变量法对场变量进行离散求解:在x方向采用有限元插值,在y方向采用状态空间法给出控制坐标方向的解析解。计算过程中的指数矩阵均采用精细积分法求解,使得本文算法具有高效率、高精度、对步长不敏感的优点。通过对侧边自由液面边界条件的单板驱动矩形空腔Stokes流问题的求解,得到与文献相同的结果,从而验证了本文方法的有效性。本文旨在将弹性力学状态空间有限元法的思想引入到低雷诺数流体力学中,为Hamilton体系下研究复杂边界Stokes流问题提供新的途径。

The state space finite element method for Stokes flow in rectangular cavity

Based on the Hellinger-Reissner variational principle,the Hamilton canonical equation of the plane incompressible Stokes flow was derived from the equilibrium equations,continuity conditions and the force boundary conditions.By the separation of variables,the general finite element method was employed in x direction and was derived by the state space control method.In addition,the precise integration method for the exponential matrix was employed in the calculation.The effectiveness of the state space finite element method is demonstrated by analyzing and comparing the simulation example in case of a single-lid driven cavity with free surface side walls.The study of this paper is to introduce the idea of the semi-analytical method into the low Reynolds number flow problems,and lay a foundation of further study on the Stokes flow with complex boundary in the Hamiltonian system.