求解压力Poisson方程的混合粗化代数多重网格算法

doi:10.19596/j.cnki.1001-246x.8662

摘要/Abstract

摘要：

针对复杂流动中Navier-Stokes(N-S)方程SIMPLE算法导出的压力Poisson(泊松)离散线性系统, 提出一类基于混合粗化的代数多重网格(AMG)算法。该算法采用一类非光滑聚类粗化和经典C/F粗化结合的方式构造网格层次结构, 希望在不影响收敛性的情况下, 减少AMG算法的启动开销。通过航空发动机燃烧室复杂流动数值模拟应用验证了该算法的有效性。结果表明: 对于典型算例, 相对于经典AMG算法, 该算法可以获得78%的加速。

关键词: 不可压N-S方程, Poisson方程, 线性解法器, 预条件迭代, 代数多重网格

Abstract:

An algebraic multigrid (AMG) algorithm based on hybrid coarsening is proposed for the linear systems of the discrete pressure Poisson which is derived from the SIMPLE algorithm for the Navier-Stokes equations in complex flows. This algorithm combines a class of non-smoothed aggregation coarsening with classical C/F coarsening to construct grid hierarchy, hoping to reduce the cost in the setup phase of the AMG algorithm without affecting convergence. The high performance of the proposed algorithm is verified by numerical simulation of complex flow in the combustion chamber of aero-engine. The results show that the proposed algorithm can achieve 78% acceleration compared with the classical AMG algorithm.

Key words: incompressible Navier-Stokes equations, Poisson equation, linear solver, preconditioned iteration, algebraic multigrid

胡少亮, 许开龙, 徐然, 刘再刚, 徐小文, 安恒斌, 范荣红, 汪振宇, 王伟. 求解压力Poisson方程的混合粗化代数多重网格算法[J]. 计算物理, 2023, 40(5): 527-534.

Shaoliang HU, Kailong XU, Ran XU, Zaigang LIU, Xiaowen XU, Hengbin AN, Ronghong FAN, Zhenyu WANG, Wei WANG. A Algebraic Multigrid Algorithm Based on Hybrid Coarsening for Pressure Poisson Equation[J]. Chinese Journal of Computational Physics, 2023, 40(5): 527-534.

图/表 7

图1 台阶算例的模型结构与网格

Fig.1 The structure of step model and the mesh for numerical simulation

图2 燃烧室的模型结构与网格

Fig.2 The structure of combustion chamber model and the mesh for numerical simulation

表1 台阶流算例，27.9万网格，1核，模拟10个非线性步，Poisson线性系统求解信息

Table 1 Step model case, 0.279 million cells of mesh, one core, 10 nonlinear steps of simulation, results of linear solver for Poisson system

时间/s	C-AMG	SA-AMG	UA-AMG	FV-AMG	HC-AMG
启动时间	5.37	9.67	2.73	1.439	1.443
求解时间	20.59	34.91	26.26	15.92	13.59
总时间	25.96	44.58	28.99	17.36	15.03
平均迭代次数	20.20	24.10	20.60	22.90	19.35

表2 燃烧室算例，2千万网格，100核，模拟10个非线性步，Poisson线性系统求解信息

Table 2 Combustion chamber model case, 20 million cells of mesh, 100 cores, 10 nonlinear steps of simulation, results of linear solver for Poisson system

时间/s	C-AMG	SA-AMG	UA-AMG	FV-AMG	HC-AMG
启动时间	42.27	29.83	8.60	5.12	5.98
求解时间	116.62	151.89	1 373.08	121.11	80.96
总时间	158.89	181.72	1 381.68	126.23	86.85
平均迭代次数	24.35	29.40	32.30	36.10	30.30

表3 燃烧室算例，2千万网格，100核，模拟500个非线性步，Poisson线性系统求解信息

Table 3 Combustion chamber model case, 20 million cells of mesh, 100 cores, 500 nonlinear steps of simulation, results of linear solver for Poisson system

时间/s	C-AMG	HC-AMG	时间减少比重/%
启动时间	2 112.33	300.31	85.80
求解时间	5 330.29	3 876.69	27.27
总时间	7 442.62	4 176.90	43.88

表4 燃烧室算例，2千万网格，模拟500个非线性步，强可扩展测试结果

Table 4 Combustion chamber model case, 20 million cells of mesh, 500 nonlinear steps of simulation, strong scalability results of linear solver

核数	线性解法器时间/s	加速比	并行效率/%
100	3 209.56	1	100
200	1 507.66	2.13	106.50
400	732.92	4.38	109.50
800	384.63	8.34	104.25
1 000	335.91	9.55	95.50
1 600	239.50	13.40	83.75
3 200	202.56	15.85	49.53

表5 燃烧室算例，1.6亿网格，模拟50个非线性步，强可扩展测试结果

Table 5 Combustion chamber model case, 0.16 billion cells of mesh, 50 nonlinear steps of simulation, strong scalability results of linear solver

核数	线性解法器时间/s	加速比	并行效率/%
400	1 033.82	1	100
800	511.14	2.02	101.00
1 600	258.42	4.00	100.00
3 200	137.42	7.52	94.00
6 400	76.76	13.47	84.19
10 000	64.71	16.98	67.92

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