粘弹流动的间断有限元模拟

Numerical Simulation of Viscoelastic Flows Using Discontinuous Galerkin Finite Element Method

针对传统有限元法求解Oldroyd-B本构方程时需加入稳定化方案的缺点,本文基于非结构网格给出了统一间断有限元求解框架.该框架包含采用IPDG(interior penalty discontinuous Galerkin)求解质量方程和动量方程,与采用RKDG(RungeKutta discontinuous Galerkin)求解本构方程这两个核心.数值结果表明:该方法在求解Oldroyd-B本构方程时无需加入稳定化方案,实施比有限元法简便,且具有较高的计算精度,可有效地模拟包含应力奇异点的复杂粘弹流动问题,进而揭示非牛顿粘弹流动的基本特征.

The traditional finite element method needs to supplement a stabilization scheme to simulate Oldroyd-B viscoelastic flows. To alleviate this issue, a unified discontinuous Galerkin finite element framework based on unstructured grids is proposed in this paper. The system contains two key points： one is using the IPDG （interior penalty discontinuous Galerkin） method to discretize mass and momen-tum equations, and the other is employing the RKDG （Runge- Kutta DG） method to solve the Oldroyd-B constitutive equation. Simulation results reveal the intrinsic characteristics of non-Newtonian viscoelastic fluids and indicate that the approach can effectively overcome the drawback of the traditional finite element method, which redundantly introduces stabilization process in the method. Moreover, these results substantiate that the proposed method is simple to implement, has high accuracy and can be used to simulate complex viscoelastic flows with stress singularity.