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## Global variance reduction method for Monte Carlo simulation of thermal radiation transport

Li Shu, Wang Yang, Ji Zhi-Cheng, Lan Ke
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• #### 摘要

针对隐式蒙特卡罗方法模拟高温热辐射输运问题时存在的“辐射强度计算误差时间空间分布严重不均匀”现象, 通过理论分析和数值模拟手段, 找到决定误差大小的主要因素为“网格内的辐射径迹长度记录数”. 据此提出“隐式蒙特卡罗全局降方差方法”并推导相应的计算公式. 该方法主要包含如下3个关键技术: 1)针对辐射输运蒙特卡罗模拟的自适应动态分配源粒子方法; 2)与自适应动态分配源粒子相匹配的动态权窗设计技术及粒子权无偏估计算法; 3)辐射强度的解析估计降方差方法. 针对这3个关键技术, 设计蒙特卡罗数值模拟方案, 编写相应的数值模拟程序模块. 典型辐射输运问题模拟结果显示: 隐式蒙特卡罗全局降方差方法能够使网格辐射强度计算误差在整个时空范围内分布相对比较均匀, 最大误差可控, 计算效率提升10倍左右. 新方法在激光惯性约束聚变的黑腔辐射输运模拟应用中取得了显著效果.

#### Abstract

The implicit Monte Carlo (IMC) method is an important numerical approximation method of simulating the thermal radiative transfer problems under high temperature condition. However, one problem plaguing the IMC method is that the calculation error distributions of the radiation specific intensities are highly asymmetric in space and time. By theoretical analysis and numerical simulations, we find that the error is affected by the records of track in the tallying mesh. Accordingly, a global variance reduction method for implicit Monte Carlo simulation is developed and the corresponding formulas are derived. This method includes three key techniques: 1) the automated dynamic distribution method for the Monte Carlo simulation source particles; 2) the dynamic weight-window technique and the none-bias weight revise algorithm that is suited to the particle distribution method; 3) the analytical estimation variance reduction method of the radiation specific intensity. In view of the above, a three-dimensional simulation code, named IMC3D, is developed to simulate the thermal radiative transfer phenomena. The typical thermal radiative transport problem, known as Marshak wave, is simulated. The simulation results indicate that the global variance reduction method for implicit Monte Carlo makes the statistical errors much more symmetric in space and time and the maximum of error is controllable, thereby increasing the calculation speed approximately 10 times. The new IMC method and code are used for simulating the radiative transportation in hohlraum of ICF successfully.

#### 作者及机构信息

###### 通信作者: 李树, li_shu@iapcm.ac.cn
• 基金项目: 国家自然科学基金(批准号: 12035002)和中国工程物理研究院于敏基金(批准号: FZ025)资助的课题.

#### Authors and contacts

###### Corresponding author: Li Shu, li_shu@iapcm.ac.cn
• Funds: Project supported by the National Natural Science Foundation of China (Grant No. 12035002) and the Yu Min Foundation of China Academy of Engineering Physics, China (Grant No. FZ025).

#### 施引文献

• 图 1  Marshak波辐射温度分布(104粒子/步)

Fig. 1.  Radiation temperature for the Marshak wave (104 particles/step).

图 2  Marshak波辐射温度误差分布(105粒子/步)

Fig. 2.  Statistical error of radiation temperature for the Marshak wave (105 particles/step).

图 3  Marshak波辐射温度分布(105粒子/步)

Fig. 3.  Radiation temperature for the Marshak wave (105 particles/step).

图 4  Marshak波辐射温度误差分布(105粒子/步)

Fig. 4.  Statistical error of radiation temperature for the Marshak wave (105 particles/step).

图 5  3个时刻的MC模拟源粒子分布

Fig. 5.  MC simulated source particles for three time step.

图 6  3个时刻的辐射强度误差分布

Fig. 6.  Statistical error of radiation intensity for different time step.

图 7  基于权窗的轮盘赌和分裂技术示意图

Fig. 7.  Weight window with splitting and Russian Roulette.

图 8  解析估计示意图

Fig. 8.  Analytical estimation for specific intensity tally.

图 9  源粒子分配数比较(old vs. new 1)

Fig. 9.  Comparison of the source particles (old vs. new 1).

图 10  误差分布比较(old vs. new 1)

Fig. 10.  Comparison of statistical error (old vs. new 1).

图 11  源粒子分配数比较(old vs. new 2)

Fig. 11.  Comparison of the source particles (old vs. new 2).

图 12  误差分布比较(old vs. new 2)

Fig. 12.  Comparison of statistical error (old vs. new 2).

图 13  误差分布比较(new 2 vs. new 3)

Fig. 13.  Comparison of statistical error (new 2 vs. new 3)

图 14  误差分布比较(old vs. new 3)

Fig. 14.  Comparison of statistical error (old vs. new 3).

图 15  六孔球腔网格剖分图

Fig. 15.  Profile of the octahedral spherical hohlraum.

图 16  黑腔壁光斑分布

Fig. 16.  Spots on the wall of the hohlraum.

图 17  激光能源注入强度曲线

Fig. 17.  Curve of laser power inject to the intensity.

图 18  黑腔辐射温度曲线比较(IMC3D vs. Haan[27])

Fig. 18.  Comparison of the radiation temperature (IMC3D vs. Haan[27]).