非定常Navier-Stokes方程的并行两水平稳定有限元算法

doi:10.19596/j.cnki.1001-246x.8546

摘要/Abstract

摘要：

使用标准的有限元方法求解非定常Navier-Stokes方程所得速度误差常受压力误差影响，且误差随粘性系数的减少而增大。为了增强压力的鲁棒性，本文引入grad-div稳定项，以提高近似解的精度，提出数值求解非定常Navier-Stokes方程的并行两水平grad-div稳定有限元算法，其时间和空间离散分别采用隐式Euler格式和Galerkin有限元方法。首先在全局粗网格上求解非线性grad-div稳定问题，然后在相互重叠的细网格子区域上并行求解grad-div稳定问题，以校正粗网格解。最后给出数值实验验证理论分析的正确性和算法的有效性。

关键词: 非定常Navier-Stokes方程, grad-div稳定项, 两水平方法, 并行算法, 有限元方法

Abstract:

In numerical solution of unsteady Navier-Stokes equations with standard finite element method, errors of computed velocity are usually affected by pressure errors, where smaller viscosity coefficients lead to greater velocity errors. To improve pressurerobustness, in this paper, we introduce a grad-div stabilization term to improve accuracy of approximate solutions. We present parallel two-level grad-div stabilized finite element algorithms for unsteady Navier-Stokes equations, where implicit Euler scheme and Galerkin finite element methods are used for temporal and spatial discretizations, respectively. At each time step, firstly we solve a nonlinear Navier-Stokes problem with grad-div stabilization on a coarse grid, and then linearized and grad-div stabilized problems are solved with Stokes, Oseen and Newton iterations on overlapping fine grid subdomains in a parallel manner to correct the coarse grid solution. Finally, numerical experiments are given to verify correctness of theoretical predictions and demonstrate effectiveness of the algorithms.

Key words: unsteady Navier-Stokes equations, grad-div stabilization, two-level method, parallel algorithm, finite element method

王湛煌, 郑波, 尚月强. 非定常Navier-Stokes方程的并行两水平稳定有限元算法[J]. 计算物理, 2023, 40(1): 14-28.

Zhanhuang WANG, Bo ZHENG, Yueqiang SHANG. Parallel Two-level Stabilized Finite Element Algorithms for Unsteady Navier-Stokes Equations[J]. Chinese Journal of Computational Physics, 2023, 40(1): 14-28.

图/表 11

表1 算法3.1~3.3的近似解误差

Table 1 Errors of approximate solutions by Algorithms 3.1-3.3

算法	h	H	CPU/s	E_u	E_p	收敛阶
算法3.1	1/27	1/18	7.782	2.283 × 10^-3	2.947 × 10^-3
	1/64	1/32	23.472	4.923 × 10^-4	4.444 × 10^-4	2.044
	1/125	1/50	62.289	1.186 × 10^-4	1.595 × 10^-4	2.060
算法3.2	1/27	1/18	7.553	2.832 × 10^-3	2.943 × 10^-3
	1/64	1/32	23.519	4.923 × 10^-4	4.444 × 10^-4	2.043
	1/125	1/50	63.909	1.186 × 10^-4	1.595 × 10^-4	2.060
算法3.3	1/27	1/18	6.86	2.833 × 10^-3	2.931 × 10^-3
	1/64	1/32	23.222	4.934 × 10^-4	4.456 × 10^-4	2.041
	1/125	1/50	63.645	1.191 × 10^-4	1.586 × 10^-4	2.059

表2 不同ν值下两种算法的误差

Table 2 Errors of two algorithms with different ν

ν	算法3.1		无稳定项的并行算法
ν	E_u	E_p	E_u	E_p
1	1.126 3 × 10^-4	6.216 51 × 10^-4	1.119 99 × 10^-4	5.582 96 × 10^-4
10^-1	1.175 85 × 10^-4	2.083 03 × 10^-4	1.126 32 × 10^-4	2.036 05 × 10^-4
10^-2	1.311 98 × 10^-4	1.933 03 × 10^-4	1.420 51 × 10^-4	2.002 67 × 10^-4
10^-3	3.479 11 × 10^-4	1.946 46 × 10^-4	6.041 92 × 10^-4	1.984 52 × 10^-4
10^-4	5.473 91 × 10^-4	1.966 75 × 10^-4	1.536 08 × 10^-3	1.994 52 × 10^-4
10^-5	5.915 42 × 10^-4	1.970 91 × 10^-4	1.937 1 × 10^-3	1.999 92 × 10^-4
10^-6	5.968 42 × 10^-4	1.971 32 × 10^-4	1.989 61 × 10^-3	2.000 63 × 10^-4

表3 不同子区域下算法3.1的误差

Table 3 Errors of Algorithm 3.1 with different subdomains

子区域数	E_u	E_p
2 × 2	2.283 21 × 10^-3	2.947 28 × 10^-3
2 × 4	2.554 87 × 10^-3	2.092 27 × 10^-3
4 × 4	2.566 64 × 10^-3	2.133 74 × 10^-3

图1 圆柱周围的三角形网格(a) 粗网格; (b)细网格

Fig.1 Triangle partition around a cylinder (a) coarse grid; (b) fine grid

图2 圆柱绕流的速度流线u1 (a) T = 2; (b) T = 4; (c) T = 6; (d) T = 8; (e) T = 10

Fig.2 u1－ velocity of a flow around a circular cylinder (a) T = 2; (b) T = 4; (c) T = 6; (d) T = 8; (e) T = 10

图3 圆柱绕流的速度流线u2 (a) T = 2; (b) T = 4; (c) T = 6; (d) T = 8; (e) T = 10

Fig.3 u2－ velocity of a flow around a circular cylinder (a) T = 2; (b) T = 4; (c) T = 6; (d) T = 8; (e) T = 10

图4 圆柱绕流的等压线(a) T = 2; (b) T = 4; (c) T = 6; (d) T = 8; (e) T = 10

Fig.4 Isobars of a flow around a circular cylinder (a) T = 2; (b) T = 4; (c) T = 6; (d) T = 8; (e) T = 10

图5 方腔驱动流

Fig.5 The lid-driven cavity flow

图6 算法3.1计算Re=100时沿垂直中心线的u1速度分布

Fig.6 u1-velocity profile along the vertical centerline at Re = 100 with Algorithm 3.1

图7 算法3.1计算Re=100时沿水平中心线的u2速度分布

Fig.7 u2-velocity profiles along the horizontal centerline at Re = 100 with Algorithm 3.1

图8 算法3.1计算Re=100时方腔驱动流的流线

Fig.8 Streamlines for the lid-driven cavity flow at Re=100 with Algorithm 3.1

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