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1. 北京航空航天大学航空科学与工程学院, 北京 100083;
2. 中国科学院计算机网络信息中心超级计算中心, 北京 100190

Improved Jacobi iterative method for hybrid grid and its application
HUANG Yu1, YAN Chao1 , YUAN Wu2
1. School of Aeronautic Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100083, China;
2. Super Computing Center, Computer Network Information Center, Chinese Academy of Sciences, Beijing 100190, China
Received: 2015-04-07; Accepted: 2015-07-10; Published online: 2015-10-14 15:09
Corresponding author. Tel.: 010-82317019 E-mail: yanchao@buaa.edu.cn
Abstract: LU-SGS scheme is widely used today because of its robustness and cheap memory cost. However, the original LU-SGS shows less competitive convergence rate; in order to apply paralleled codes on hybrid unstructured grid, the grid reordering and regrouping procedure must be carried out beforehand. In this paper, an improved implicit method suitable for complex hybrid gird is developed to achieve fast convergence rate and to parallelize the algorithm without grid reordering and regrouping procedure. This method is simple for coding and easy to use OpenMP for code parallelization. The numerical results of Euler and viscous flows show that the method has a reliable performance, and it is able to achieve a significant efficiency improvement over implicit counterparts such as LU-SGS scheme with less requirement of extra memory, and parallel computation produce exactly the same result as serial case.
Key words: hybrid grid     implicit method     parallel computation     OpenMP     Jacobi iteration     grid reorder

1 格心有限体积法

2 改进的时间方法 2.1 原始LU-SGS格式

2.2 改进的雅可比迭代方法

2.2.1 改进的算法流程

2.2.2 通量雅可比矩阵计算

3 算例及讨论 3.1 NACA0012翼型跨声速无黏绕流

 图 1 NACA0012翼型计算网格 Fig. 1 Computational grid of NACA0012 airfoil

 图 2 流场马赫数云图 Fig. 2 Mach contour of flow filed
 图 3 NACA0012翼型壁面压力系数Cp分布 Fig. 3 Pressure coefficient Cp distribution on wall of NACA0012 airfoil

 图 4 不同CFL数下LU-SGS格式残差收敛曲线 Fig. 4 Residual convergence curves of LU-SGS scheme for different CFL numbers
 图 5 不同CFL数下本文方法残差收敛曲线 Fig. 5 Residual convergence curves of proposed method for different CFL numbers
 图 6 LU-SGS格式和本文方法残差随计算时间变化对比 Fig. 6 Comparison of residual convergence histories versus CPU time between LU-SGS scheme and proposed method

 图 7 采用不同内迭代步数时残差随时间步和 计算时间收敛曲线 Fig. 7 Residual convergence curves versus time steps and CPU time for different inner iteration numbers

 图 8 不同通量函数构造雅可比矩阵对残差收敛影响 Fig. 8 Influence of Jacobi matrix constructed by different flux functions on residual convergence
3.2 RAE2822翼型黏性绕流

 图 9 RAE2822翼型计算网格 Fig. 9 Computational grid of RAE2822 airfoil
 图 10 RAE2822翼型壁面压力系数Cp分布 Fig. 10 Pressure coefficient Cp distribution on wall of RAE2822 airfoil

 图 11 网格排序前后系数矩阵非零元素分布 Fig. 11 Coefficient matrix non-zero elements distribution on non-reordered and reordered grid

 图 12 LU-SGS格式在未排序网格上计算残差收敛情况 Fig. 12 Residual convergence histories by LU-SGS scheme computation on non-reordered grid

 图 13 本文方法在排序网格和未排序网格上 计算残差收敛情况 Fig. 13 Residual convergence histories by proposed method computation on non-reordered and reordered grid

 图 14 本文方法与LU-SGS格式残差随时间步和 计算时间的收敛情况对比 Fig. 14 Comparison of residual convergence histories versus time steps and CPU time between proposed method and LU-SGS scheme
3.3 ONERA-M6机翼绕流模拟

 图 15 ONERA-M6机翼对称面及壁面网格 Fig. 15 Grid of ONERA-M6 wing on symmetry and wall surface

 图 16 对称面和壁面上等密度线分布及不同站位处壁面 压力系数Cp与实验值对比 Fig. 16 Density contour on symmetry and wall surface and comparison of pressure coefficient Cp on wall with experimental data on different sections

 图 17 本文方法与LU-SGS格式残差收敛对比 Fig. 17 Comparison of residual convergence histories between proposed method and LU-SGS scheme

 图 18 ONERA-M6机翼在未分组网格上串/并行 计算残差收敛过程 Fig. 18 Residual convergence histories for serial/parallel computation of ONERA-M6 wing on non-reordered grid

 计算方法 LU-SGS串行 本文方法串行 本文方法并行 总内存/GB 2.71 2.98 2.98
3.4 DLR-F6翼身组合体绕流模拟

 图 19 DLR-F6翼身组合体对称面及壁面网格 Fig. 19 Grid of DLR-F6 wing body on symmetry and wall surface

 图 20 对称面和壁面压力系数Cp分布及不同 站位处壁面压力系数Cp与实验值对比 Fig. 20 Distribution of pressure coefficient Cp on symmetry and wall surface and comparison of Cp on wall with experimental data on different sections

 图 21 本文方法与LU-SGS格式残差收敛情况对比 Fig. 21 Comparison of residual convergence histories between proposed method and LU-SGS scheme

 图 22 升力系数CL收敛情况对比 Fig. 22 Comparison of lift coefficient CL convergence histories
3.5 超声速轴对称压缩拐角模拟

 图 23 超声速轴对称压缩拐角计算网格 Fig. 23 Computational grid of supersonic axial symmetry compress hollow corner

 图 24 密度纹影图 Fig. 24 Schlieren of density
 图 25 壁面无量纲压力计算结果与实验结果对比 Fig. 25 Comparison of dimensionless pressure on wall between computaional and experimental results

 图 26 不同CFL数下本文方法与LU-SGS格式 残差收敛情况对比 Fig. 26 Comparison of residual convergence histories between proposed method and LU-SGS scheme for different CFL numbers

 图 27 不同CFL数下本文方法及LU-SGS格式 残差随计算时间对比 Fig. 27 Comparison of residual convergence histories versus CPU time between proposed method and LU-SGS scheme for different CFL numbers
4 结 论

1) 通过改进方法流程,本文方法无需预先进行网格排序即能达到较好的收敛性能;也无需进行网格分组即可实现算法并行化。

2) 采用基于共享内存的OpenMP方法实现并行计算,简单方便,并行结果与串行结果一致。

3) 相对于LU-SGS格式,本文方法内存需求增加不大。采用本文方法计算ONERA-M6机翼复杂外形,共计220万混合网格,内存需求比LU-SGS 格式仅增加了9.9%。

4) 采用基于重构变量的近似通量函数构造通量雅可比矩阵,有利于满足矩阵对角占优,且不会影响方法的收敛性能。算例表明,在各来流条件下,相比于LU-SGS格式,本文方法稳定性及鲁棒性较好,收敛速度更快。

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#### 文章信息

HUANG Yu, YAN Chao, YUAN Wu

Improved Jacobi iterative method for hybrid grid and its application

Journal of Beijing University of Aeronautics and Astronsutics, 2016, 42(3): 551-561.
http://dx.doi.org/10.13700/j.bh.1001-5965.2015.0197