应用最小二乘有限元法求解应力方程

Solving Stress Formulation of Equations with Least Squares Finite Element Method

为了有效地应用最小二乘有限元法求解层流和湍流问题,获得高精度的数值结果,传统的速度-涡量-压力方法精度低,因此提出速度-应力-压力的新方法。通过采用牛顿线性化方法和预处理共轭梯度法,最终实现了应力形式NavierStokes方程的求解。后台阶流和圆柱绕流的计算结果与实验结果比较表明,与有限体积法相比,最小二乘有限元法的计算结果与实验结果更加接近;与此同时,速度-应力-压力形式的计算结果比传统速度-涡量-压力形式的计算结果更加接近实验结果。尽管传统的速度-涡量-压力形式能够进行层流计算,但无法处理大涡模拟的亚格子应力项,而速度-应力-压力形式能够很好地解决亚格子应力项的问题,为大涡模拟湍流计算打下了基础。

In order to solve laminar and turbulent flow problems efficiently with least squares finite element meth- od and get more accurate results, this paper used a new formulation of velocity-stress-pressure instead of traditional formulation of velocity-vorticity-pressure. With the Newton＇ s linearized method and preconditioned conjugate gradi- ent method, the stress formulation of Navier-Stokes equations was solved. The correspondence between the numerical results and experimental results of backward facing step flow and circular cylinder flow shows that, the least squares finite element method can get more accurate results than finite volume method. Also, the results of stress formulation fit better to the experiment results than those of vorticity formulation. The solution for stress formulation can cope with the subgrid-scale model well, which is very difficult for vorticity formulation. This lays a solid foundation for large eddy simulation to solve turbulent problems.