Research on CMFD Preconditioner for Two-dimensional MOC Krylov Subspace Iteration
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摘要: 为了提高二维特征线（MOC）Krylov子空间迭代的效率，提出了基于粗网有限差分（CMFD）矩阵的预条件子。研究首先将CMFD加速方法进行线性化，推导出线性CMFD预条件子；其次将线性CMFD预处理Krylov子空间方法用于求解二维MOC方程；最后利用IAEA LWR和2D C5G7 基准题对线性CMFD预条件子的加速性能进行了测试。结果表明：在应用CMFD预条件子后，IAEA LWR基准题的迭代次数减少了52.7%，计算时间减少了41.8%；2-D C5G7基准题的迭代次数减少了20.3％，计算时间减少了13.2%；研究还发现CMFD预条件子对于局部非均匀性不强的问题效果很好，对于局部非均匀性较强的问题性能下降。
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关键词:
- 特征线方法(MOC) /
- Krylov子空间迭代 /
- 粗网有限差分(CMFD) /
- 预条件子
Abstract: To improve the efficiency of the Krylov subspace iteration for two-dimensional method of characteristics (MOC), a preconditioner based on the coarse-mesh finite difference (CMFD) matrix is proposed. Firstly, the CMFD acceleration method is linearized and the linear CMFD preconditioner is derived. Secondly, the linear CMFD preconditioner is applied to the Krylov subspace method to solve the two-dimensional MOC equation. Finally, the acceleration performance of the linear CMFD preconditioner is tested using the IAEA LWR and 2-D C5G7 benchmarks. The results show that, after applying the CMFD preconditioner, the iteration count for the IAEA LWR benchmark is reduced by 52.7%, and the computational time is decreased by 41.8%. For the 2-D C5G7 benchmark, the iteration count is reduced by 20.3% and the computational time is reduced by 13.2%. The study also finds that the CMFD preconditioner works well for problems with weak local heterogeneities, but its performance decreases for problems with strong local heterogeneities.-
Key words:
- MOC /
- Krylov subspace iteration /
- CMFD /
- Preconditioner
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图 2 IAEA 轻水池式反应堆基准题示意图
Figure 2. Schematic Diagram of IAEA Light Water Pool-Type Reactor Benchmark Problem
图 3 每次外迭代中GMRES子空间迭代减少值
Figure 3. Reduction Value of GMRES Subspace Iterations in Each Outer Iteration
图 5 二维C5G7基准题每次外迭代的Krylov子空间迭代次数
Figure 5. Number of Krylov Subspace Iterations in Each Outer Iteration for 2-D C5G7 Benchmark Problem
图 6 二维C5G7基准题第一次外迭代中每群的内迭代次数
Figure 6. Number of Krylov Subspace Iterations in First Outer Iteration for Each Energy Group
表 1 IAEA 轻水池式反应堆基准题单群截面 cm−1
Table 1. One-Group Cross-Sections for IAEA Light Water Pool-Type Reactor Benchmark Problem cm−1
材料区 总截面 吸收截面 中子产生截面 散射截面 散射比 材料区1 0.60 0.07 0.079 0.53 0.883 材料区2 0.48 0.28 0.0 0.20 0.417 材料区3 0.70 0.04 0.043 0.66 0.943 材料区4 0.65 0.15 0.0 0.50 0.769 材料区5 0.90 0.01 0.0 0.89 0.989 
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表 2 IAEA轻水池式反应堆基准题计算结果
Table 2. Calculation Results for IAEA Light Water Pool-Type Reactor Benchmark Problem
求解器类型 特征值 外迭代
次数GMRES
迭代次数GMRES
迭代时间/s预处理
时间/s总计算
时间/s有预条件子 1.00643 5 74 153.2 0.0 162.6 无预条件子 1.00643 5 35 79.6 5.8 94.6
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表 3 二维C5G7基准题的计算结果
Table 3. Calculation Results for 2D C5G7 Benchmark Problem
迭代类型 特征值 外迭代次数 GMRES迭代次数 GMRES 计算时间/s 预处理时间 /s 总时间/s 含预处理 1.18602 4 202 757.8 0.0 836.8 不含预处理 1.18602 4 161 647.8 23.8 726.4
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