Study on Diffusion Source Cascade Variance Reduction Method for Monte Carlo Deep-penetration Shielding Calculation
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摘要: 提出了一种用于蒙特卡罗屏蔽深穿透计算的弥散源级联降方差方法。该方法的核心思想是通过多层局部相空间的响应关系级联获取全局的降方差参数分布。该方法首先通过弥散源计算相邻相空间之间的与空间和能量相关的通量响应因子，随后从源空间向外逐层正向级联获得全局预估通量分布，再从计数空间反向级联获取重要性分布，最终生成一致性的源偏倚参数和权窗参数。该方法通过局部蒙特卡罗前置计算获取降方差参数，因此无需迭代，可有效降低迭代时间成本，提高计算效率。将该方法应用于单探测器问题和多探测器问题，与蒙特卡罗直接计算的计算值符合良好，品质因子相比直接计算提升了约2~4个数量级。同时，将结果与典型降方差方法（MAGIC方法和CADIS方法）的计算效果进行对比，数值结果表明，弥散源级联方法的整体降方差效果更优，可满足屏蔽深穿透问题的需求。Abstract: This paper proposed the diffusion source cascade variance reduction method for Monte Carlo deep-penetration shielding calculation. The core idea of this method is to obtain the global variance-reducing parameter distribution through the cascade of response relations in multi-layer local phase space. The method first calculates the flux response factors related to space and energy between adjacent phase spaces by the dispersion source. Secondly, it cascades outward from the source space to obtain the global estimated flux distribution. Thirdly, it cascades backward from the count space to obtain the importance distribution. Finally it generates the consistent source bias parameters and weight window parameters. This method obtains the variance-reducing parameters through local Monte Carlo pre-calculation, so there is no need for iteration, which can effectively reduce the iteration time cost and improve the calculation efficiency. This method is applied to the single detector problem and the multi-detector problem. The calculated values are in good compliance with the Monte Carlo direct calculation, and the quality factor is improved by about 2~4 orders of magnitude. At the same time, the results are compared with those of the typical variance-reducing methods MAGIC and CADIS. The numerical results show that the overall variance-reducing effect of the diffusion source cascade method is better, and it can meet the requirements of the deep-penetration shielding calculation.
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图 1 弥散源级联方法原理示意图
$\varPhi $—实际通量；$ g $—通量响应因子。
Figure 1. Schematic Diagram of Diffusion Source Cascade Method
表 1 弥散源级联降方差方法各步骤计算时间
Table 1. CalculationTime for Each Step of Diffusion Source Cascade Method
前置弥散
源计算时间/min级联计算
过程时间/min最终正式
计算时间/min总时间/min 3.86×10−1 16.20 2.03×102 2.20×102 
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表 2 不同计算方法得到FOM对比
Table 2. Comparison of FOM Obtained by Different Calculation Method
计算方法 直接计算 Booth方法 弥散源级联方法 FOM 1.93×10−1 1.18 20.70
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表 3 不同方法计算HBR-2基准题的计算时间
Table 3. Time of Different Methods in Calculating HBR-2 Benchmarking Problem
计算方法 权窗获取
计算时间/min最终正式
计算时间/min总时间/min 直接计算 0 3.31×105 3.31×105 MAGIC方法 1.12×103 4.03×104 4.14×104 弥散源级联降方差方法 6.11×102 1.05×103 1.66×103
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表 4 HBR-2屏蔽问题计算中辐照监督管的归一化剂量（rem）和相对标准偏差
Table 4. Normalized Dose (rem) and Relative Standard Deviation of Surveillance Capsule in HBR-2 Shielding Problem
计算方法 参数 237Np(n,f)137Cs 238U(n,f)137Cs 58Ni(n,p)58Co 54Fe(n,p)54Mn 46Ti(n,p)46Sc 63Cu(n,α)60Co 直接计算 归一化剂量 2.73×10−33 3.61×10−34 1.06×10−34 7.80×10−35 1.03×10−35 5.13×10−37 相对标准偏差 5.80×10−3 8.90×10−3 1.22×10−2 1.32×10−2 2.18×10−2 3.13×10−2 MAGIC方法 归一化剂量 2.72×10−3 3.55×10−34 1.03×10−34 7.52×10−35 9.89×10−36 5.66×10−37 相对标准偏差 1.08×10−2 1.28×10−2 1.60×10−2 1.72×10−2 2.36×10−2 3.10×10−2 弥散源级联降方差方法 归一化剂量 2.76×10−33 3.64×10−34 1.06×10−34 7.75×10−35 1.03×10−35 5.97×10−37 相对标准偏差 9.00×10−4 1.00×10−3 1.20×10−3 1.30×10−3 2.10×10−3 3.00×10−3
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表 5 HBR-2屏蔽问题计算中堆外活化探测片的归一化剂量（rem）和相对标准偏差
Table 5. Normalized Dose (rem) and Relative Standard Deviation of Ex-core Detector in HBR-2 Shielding Problem
计算方法 参数 237Np(n,f)137Cs 238U(n,f)137Cs 58Ni(n,p)58Co 54Fe(n,p)54Mn 46Ti(n,p)46Sc 63Cu(n,α)60Co 直接计算 归一化剂量 1.08×10−34 5.75×10−36 1.10×10−36 7.16×10−37 4.37×10−38 2.21×10−39 相对标准偏差 4.69×10−2 1.31×10−1 1.89×10−1 2.24×10−1 4.39×10−1 7.95×10−1 MAGIC方法 归一化剂量 9.87×10−35 4.22×10−36 8.98×10−37 5.91×10−37 7.52×10−38 4.45×10−39 相对标准偏差 2.47×10−2 2.79×10−2 2.78×10−2 3.21×10−2 5.01×10−2 7.28×10−2 弥散源级联降方差方法 归一化剂量 9.50×10−35 3.98×10−36 8.55×10−37 5.65×10−37 7.54×10−38 4.86×10−39 相对标准偏差 2.46×10−2 1.88×10−2 1.32×10−2 1.37×10−2 1.53×10−2 1.78×10−2
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表 6 HBR-2屏蔽问题计算中辐照监督管的FOM对比
Table 6. FOM Comparasion of Surveillance Capsule in HBR-2 Shielding Problem
计算方法 237Np(n,f)137Cs 238U(n,f)137Cs 58Ni(n,p)58Co 54Fe(n,p)54Mn 46Ti(n,p)46Sc 63Cu(n,α)60Co 直接计算 9.00×10−2 3.80×10−2 2.00×10−2 1.70×10−2 6.40×10−3 3.10×10−3 MAGIC方法 2.07×10−1 1.47×10−1 9.43×10−2 8.16×10−2 4.33×10−2 2.51×10−2 CADIS方法 188.00 60.00 37.00 32.80 10.60 4.75 弥散源级联降方差方法 7.43×102 6.02×102 4.18×102 3.56×102 1.36×102 66.9
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表 7 HBR-2屏蔽问题计算中堆外活化探测片的FOM对比
Table 7. FOM Comparasion of Ex-core Detector in HBR-2 Shielding Problem
计算方法 237Np(n,f)137Cs 238U(n,f)137Cs 58Ni(n,p)58Co 54Fe(n,p)54Mn 46Ti(n,p)46Sc 63Cu(n,α)60Co 直接计算 1.40×10−3 1.70×10−4 8.50×10−5 6.00×10−5 1.60×10−5 4.80×10−6 MAGIC方法 3.96×10−2 3.10×10−2 3.12×10−2 2.34×10−2 9.62×10−3 4.56×10−3 CADIS方法 19.10 1.17 6.14×10−1 4.89×10−1 4.47×10−2 4.34×10−3 弥散源级联降方差方法 9.94×10−1 1.70 3.45 3.21 2.57 1.90
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