带阻尼项定常Navier-Stokes方程的并行两水平有限元算法

doi:10.19596/j.cnki.1001-246x.8646

摘要/Abstract

摘要：

基于两重网格离散和区域分解技术, 提出数值求解带阻尼项定常Navier-Stokes方程的三种并行两水平有限元算法。其基本思想是首先在粗网格上求解完全的非线性问题, 以获得粗网格解, 然后在重叠的局部细网格子区域上并行求解Stokes、Oseen和Newton线性化的残差问题, 最后在非重叠的局部细网格子区域上校正近似解。数值算例验证了算法的有效性。

关键词: 阻尼项, Navier-Stokes方程, 两水平方法, 有限元方法, 并行算法

Abstract:

Based on two-grid discretizations and domain decomposition techniques, this paper presents three parallel finite element algorithms for numerically solving the steady Navier-Stokes equations with damping term. The basic idea of the present algorithms is to first solve a fully nonlinear problem on a coarse grid to get a coarse grid solution, then solve Stokes, Oseen, and Newton linearized residual problems in parallel in overlapping local fine grid subdomains, and finally update the approximate solution in non-overlapping fine grid subdomains. The effectiveness of the proposed algorithms is demonstrated by some numerical examples.

Key words: damping term, Navier-Stokes equations, two-grid discretization, finite element method, parallel algorithm

王国梁, 郑波, 尚月强. 带阻尼项定常Navier-Stokes方程的并行两水平有限元算法[J]. 计算物理, 2023, 40(5): 535-547.

Guoliang WANG, Bo ZHENG, Yueqiang SHANG. Parallel Finite Element Algorithms Based on Two-grid Discretizations for the Steady Navier-Stokes Equations with Damping Term[J]. Chinese Journal of Computational Physics, 2023, 40(5): 535-547.

图/表 12

图1 (a) 全局粗网格TH(Ω); (b)局部细网格Th(Ωj); (c)局部细网格Th(Dj)

Fig.1 (a) A global coarse mesh of TH(Ω); (b) a local fine mesh of Th(Ωj); (c) a local fine mesh of Th(Dj)

表1 二维情形下标准两重网格有限元算法与并行有限元算法误差比较：ν=0.1, α=1, r=3

Table 1 Comparison of the errors of the approximate solutions by standard two-grid finite element algorithm and parallel finite element algorithms in 2D: ν=0.1, α=1, r=3

算法	h	H	CPU时间/s	迭代次数	$\|\|\| \nabla\left(\boldsymbol{u}-\boldsymbol{u}^h\right) \|\|\|_{0, \varOmega} $	$ \|\|\| p-p^h \|\|\|_{0, \varOmega}$	收敛阶
标准算法	1/27	1/18	0.958 92	7	0.038 048 80	0.009 489 85	-
	1/64	1/32	3.101 29	7	0.004 081 53	0.001 621 26	2.457 09
	1/125	1/50	10.494 00	7	0.000 904 789	0.000 422 136	2.178 11
	1/216	1/72	31.317 20	7	0.000 288 131	0.000 141 144	2.063 25
算法1	1/27	1/18	0.592 65	7	0.045 058 60	0.010 069 90	-
	1/64	1/32	2.068 82	7	0.005 789 67	0.001 759 92	2.303 67
	1/125	1/50	5.952 02	7	0.001 395 74	0.000 484 417	2.076 6
	1/216	1/72	14.557 60	7	0.000 403 89	0.000 147 814	2.241 64
算法2	1/27	1/18	0.635 46	7	0.044 699 70	0.010 083 90	-
	1/64	1/32	2.118 71	7	0.005 769 56	0.001 762 59	2.299 08
	1/125	1/50	6.034 74	7	0.001 392 48	0.000 485 928	2.074 53
	1/216	1/72	15.046 00	7	0.000 403 328	0.000 148 118	2.240 8
算法3	1/27	1/18	0.801 42	7	0.044 541 70	0.010 090 90	-
	1/64	1/32	2.382 69	7	0.005 759 67	0.001 762 35	2.297 44
	1/125	1/50	6.268 96	7	0.001 391 31	0.000 485 551	2.073 75
	1/216	1/72	15.260 20	7	0.000 403 184	0.000 148 063	2.239 95

表2 二维情形下不同粘性系数下并行有限元算法的数值结果：h=1/216, H=1/72, α=0.1, r=3

Table 2 Numerical results of parallel finite element algorithms with different values of viscosities in 2D: h=1/216, H=1/72, α=0.1, r=3

算法	ν	CPU时间/s	迭代次数	$\|\|\| \nabla\left(\boldsymbol{u}-\boldsymbol{u}^h\right) \|\|\|_{0, \varOmega} $	\|\|\|p-p^h\|\|\|_{0, Ω}
算法1	1	10.203 9	3	0.000 287 601	0.000 160 076
	0.1	12.975 1	5	0.000 403 884	0.000 147 888
	0.01	18.830 3	9	0.002 949 800	0.000 149 033
	0.001	31.991 8	18	0.075 603 900	0.000 243 342
算法2	1	10.519 0	3	0.000 287 593	0.000 160 282
	0.1	13.173 1	5	0.000 403 442	0.000 148 188
	0.01	19.131 7	9	0.002 985 070	0.000 153 978
	0.001	34.256 5	18	0.063 123 700	0.000 247 973
算法3	1	11.052 5	3	0.000 287 593	0.000 160 251
	0.1	13.414 5	5	0.000 403 367	0.000 148 185
	0.01	19.478 3	9	0.002 896 300	0.000 156 669
	0.001	32.931 2	18	0.024 774 600	0.000 262 535

表3 二维情形下不同阻尼系数r时并行有限元算法的数值结果：h=1/125, H=1/50, α=1, ν=0.1

Table 3 Numerical results of parallel finite element algorithms with different values of damping parameter r in 2D: h=1/125, H=1/50, α=1, ν=0.1

算法	r	CPU时间/s	迭代次数	$\|\|\| \nabla\left(\boldsymbol{u}-\boldsymbol{u}^h\right) \|\|\|_{0, \varOmega} $	\|\|\|p-p^h\|\|\|_{0, Ω}
算法1	3	6.159 79	7	0.001 395 74	0.000 484 417
	6	10.308 4	14	0.001 395 72	0.000 484 620
	10	14.147 6	20	0.001 395 72	0.000 484 707
	20	22.345 3	33	0.001 395 71	0.000 484 384
	50	31.687 1	49	0.001 395 70	0.000 484 380
算法2	3	6.279 78	7	0.001 392 48	0.000 485 928
	6	10.539 9	14	0.001 393 04	0.000 485 697
	10	14.293 9	20	0.001 393 26	0.000 485 769
	20	22.024 2	33	0.001 393 43	0.000 485 567
	50	32.155 2	49	0.001 393 58	0.000 485 739
算法3	3	6.456 41	7	0.001 391 31	0.000 485 551
	6	10.631 2	14	0.001 390 47	0.000 483 011
	10	14.335 3	20	0.001 389 31	0.000 481 080
	20	22.170 0	33	0.001 387 42	0.000 478 291
	50	32.220 8	49	0.001 385 24	0.000 475 619

表4 三维情形下标准两重网格有限元算法与并行有限元算法误差比较：ν=1, α=1, r=3

Table 4 Errors of approximate solutions by standard two-grid and parallel finite element algorithms in 3D: ν=1, α=1, r=3

算法	h	H	CPU时间/s	迭代次数	$ \|\|\|\nabla\left(\boldsymbol{u}-\boldsymbol{u}^h\right) \|\|\|_{0, \varOmega} $	\|\|\|p-p^h\|\|\|_{0, Ω}	收敛阶
标准算法	1/8	1/4	4.448 66	2	0.001 550 550	0.004 051 25	-
	1/12	1/6	22.151 90	2	0.000 682 911	0.001 796 71	2.009 99
	1/16	1/8	105.382 00	2	0.000 383 025	0.001 009 80	2.004 90
	1/20	1/10	408.273 00	2	0.000 244 837	0.000 646 003	2.002 85
算法1	1/8	1/4	1.271 54	2	0.002 763 490	0.010 183 20	-
	1/12	1/6	4.485 49	2	0.001 090 180	0.004 401 69	2.115 03
	1/16	1/8	12.701 90	2	0.000 570 030	0.002 510 84	2.009 36
	1/20	1/10	33.003 90	2	0.000 346 007	0.001 609 92	2.036 12
算法2	1/8	1/4	1.397 73	2	0.002 763 490	0.010 183 30	-
	1/12	1/6	4.785 84	2	0.001 090 190	0.004 401 81	2.114 99
	1/16	1/8	13.531 60	2	0.000 570 040	0.002 510 97	2.009 28
	1/20	1/10	34.505 40	2	0.000 346 018	0.001 610 05	2.036 00
算法3	1/8	1/4	1.563 61	2	0.002 763 480	0.010 183 20	-
	1/12	1/6	5.375 81	2	0.001 090 180	0.004 401 69	2.115 03
	1/16	1/8	14.856 20	2	0.000 570 028	0.002 510 85	2.009 36
	1/20	1/10	36.657 40	2	0.000 346 006	0.001 609 93	2.036 12

图2 二维情形下不同子区域时并行有限元算法所得速度与压力解的收敛阶(a)速度梯度的L2误差收敛阶；(b)压力L2误差收敛阶

Fig.2 Rates of convergence of the solutions by parallel finite element algorithms with different numbers of subdomains in 2D (a) order of convergence of L2-error for the gradient of velocity; (b) order of convergence of L2-error for the pressure

图3 在不同的阻尼参数α下圆柱绕流问题的并行有限元算法(左)与标准两重网格有限元算法(右)的速度分量u1(a)α=0；(b)α=0.1；(c)α=1；(d)α=10

Fig.3 Computed u1-velocity for the flow past a circular cylinder by parallel finite element algorithm (left) and standard two-grid finite element algorithm (right) with different α (a) α=0; (b) α=0.1; (c) α=1; (d) α=10

图4 在不同参数α下圆柱绕流问题的并行有限元算法(左)与标准两重网格有限元算法(右)的速度分量u2(a)α=0；(b)α=0.1；(c)α=1；(d)α=10

Fig.4 Computed u2-velocity for the flow past a circular cylinder by parallel finite element algorithm (left) and standard two-grid finite element algorithm (right) with different α (a) α=0; (b) α=0.1; (c) α=1; (d) α=10

图5 在不同参数α下圆柱绕流问题的并行有限元算法(左)与标准两重网格有限元算法(右)的压力p (a)α=0；(b)α=0.1；(c)α=1；(d)α=10

Fig.5 Computed p for the flow past a circular cylinder by parallel finite element algorithm (left) and standard two-grid finite element algorithm (right) with different α (a) α=0; (b) α=0.1; (c) α=1; (d) α=10

图6 在不同参数α下后台阶流问题的并行有限元算法(左)与标准两重网格有限元算法(右)的速度分量u1(a)α=0；(b)α=0.1；(c)α=1；(d)α=10

Fig.6 Computed u1-velocity for the backward-facing step flow by parallel finite element algorithm (left) and standard two-grid finite element algorithm (right) with different α (a) α=0; (b) α=0.1; (c) α=1; (d) α=10

图7 在不同参数α下后台阶流问题的并行有限元算法(左)与标准两重网格有限元算法(右)的速度分量u2 (a)α=0；(b)α=0.1；(c)α=1；(d)α=10

Fig.7 Computed u2-velocity for the backward-facing step flow by parallel finite element algorithm (left) and standard two-grid finite element algorithm (right) with different α (a) α=0; (b) α=0.1; (c) α=1; (d) α=10

图8 在不同参数α下后台阶流问题的并行有限元算法(左)与标准两重网格有限元算法(右)的压力p(a)α=0；(b)α=0.1；(c)α=1；(d)α=10

Fig.8 Computed pressure p for the backward-facing step flow by parallel finite element algorithm (left) and standard two-grid finite element algorithm (right) with different α (a) α=0; (b) α=0.1; (c) α=1; (d) α=10

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