基于控制体的分步有限元算法及其在非正交网格上的应用

A two-step control volume based finite element scheme and its application on non-orthogonal grids

利用Newton-Raphson算法的局部收敛特性,建立了一种基于分步思想的有限元算法。该算法首先利用基于流动条件构造差值函数的迎风有限元方法获得流动问题的近似收敛解;然后,在近似解的基础上,引入混合插值格式,进一步计算降低误差。混合插值函数的形式由数值实验获得,数值结果与理论分析相一致。对方腔顶盖驱动流和倾斜空腔驱动流进行了数值模拟,数值解与基准解吻合很好。与传统的基于流动条件构造差值函数的迎风有限元方法相比较,该文算法能在较稀疏的网格条件下获得比较精确的计算结果,尤其针对存在漩涡的流动问题,能大大提高数值解的精度。

The local convergence feature of the Newton-Raphson method was utilized to establish a two-step finite element scheme in which the approximate solution of a flow problem was obtained using flow-condition-based interpolation finite element approach with the mixed interpolation function then introduced to reduce the error.The mixed interpolation was determined by numerical experiments.The results agreed well with theoretical analyses.The lid-driven flows in a square cavity and an inclined cavity were calculated to show that the numerical results agreed well with benchmark solutions. Compared with traditional flow-condition-based interpolation finite element methods,the proposed scheme achieves quite precise results on the coarse mesh,while greatly improving the accuracy of numerical solutions,especially for flows with vortexes.