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Sampling of radiation source particles is important for obtaining a correct result in the thermal radiative transfer simulation with implicit Monte Carlo. When conducting the implicit Monte Carlo simulation of spherical geometry, temperature in a cell (a spherical shell) is generally treated as a spatially independent value. That means that the particles of radiative source are uniformly distributed in a spherical shell. In some cases where the gradient of temperature inside a cell is relatively small, the treatment does not cause too many errors. However, when the opacity of material becomes large enough or the spherical shell becomes thick enough, the temperature of thermal wave head will change sharply and there will be a great temperature gradient even in a single spherical shell. The treatment will make the thermal radiation propagate much faster than the practical one, which is unacceptable in physics. We investigate the physical and numerical reasons for this violation, finding that the simulation results strongly rely on the separation of cell and that the thermal wave propagates faster with the cell number decreasing. In order to yield an accurate result, the cell number has to increase up to a large enough value. Unfortunately, more cells need more particles to reduce the numerical variance, and more particles will cost more computation time and thus causing the simulation efficiency to lower. In our work, temperature is not treated as a constant in space any more. Instead, it is treated as a linear function in a cell. Based on a new temperature function and radiative energy density distribution, a probability density distribution function of emitting position of radiation source particle in spherical geometry is obtained. Then two new spatial sampling methods are proposed and the sampling procedures of radiation source particle are designed. To verify our new sampling methods, we test several typical thermal radiative problems and compare the result with a reference solution. Numerical experiments show that both two new sampling methods can correct the errors of thermal radiative propagation speed and overcome the difficulty that simulation result is strongly dependent on cell number. In addition, both new sampling methods can obtain an accurate result even with less cells and less particles, which can saves plenty of computation time and improves the simulation efficiency.
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Keywords:
- implicit Monte Carlo /
- thermal radiation transport /
- spherical geometry /
- spatial sampling
[1] 彭惠民 2008 等离子体中辐射输运和辐射流体力学 (北京: 国防工业出版社) 第38页
Peng H M 2008 Radiation Transport and Radiation Hydrodynamics in Plasmas (Beijing: National Defense Industry Press) p38 (in Chinese)
[2] Hammersly J M, Handscomb D C 1964 Monte Carlo Method (New York: John Wiley & Sons Press) p76
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Pei L C, Zhang X Z 1980 Monte Carlo Method and Application in Particle Transportation (Beijing: Science Press) p18 (in Chinese)
[4] Fleck J A 1963 Computational Method in the Physical Sciences (Vol. 1) (New York: McGraw-Hill) p43
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[16] 李树, 李刚, 田东风, 邓力 2013 62 249501
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Li S, Li G, Tian D F, Deng L 2013 Acta Phys. Sin. 62 249501
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[17] Huo W Y, Li Z C, Yang D, Lan K, Liu J, Ren G L, Li S W, Yang Z W, Guo L, Hou L F, Xie X F, Li Y K, Deng K L, Yuan Z, Zhan X Y, Yuan G H, Zhang H J, Jiang B B, Huang L Z, Du K, Zhao R C, Li P, Wang W, Su J Q, Ding Y K, He X T, Zhang W Y 2016 Matter Radiat. Extremes 1 2
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Li S, Chen Y H, Ji Z C, Zhang M Y, Ren G L, Huo W Y, Yan W H, Han X Y, Li Z C, Liu J, Lan K 2018 Acta Phys. Sin. 67 025202
Google Scholar
[19] Feng T G, Lai D X 1996 Sci. China: E 39 461
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Li S, Deng L, Tian D F, Li G 2014 Acta Phys. Sin. 63 239501
Google Scholar
[21] Wollaber A B 2008 Ph.D. Dissertation (Ann Arbor: University of Michigan)
[22] 许淑艳 2006 蒙特卡罗方法在实验核物理中的应用 (北京: 原子能出版社) 第46页
Xu S Y 2006 Applications of Monte Carlo Method in Experimental Nuclear Physics (Beijing: Atomic Energy Press) p46 (in Chinese)
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图 1 不同网格数的物质温度空间分布(t = 10 ns)
Figure 1. Material temperature with different cell numbers (t = 10 ns).
图 2 物质温度的收敛情况 (a)不同网格数情况下r = 0.05 cm处的物质温度随时间的变化; (b) r = 0.05 cm处物质温度随网格数的变化(t = 5 ns)
Figure 2. The convergence of material temperature: (a) Material temperature change with time in r = 0.05 cm; (b) material temperature change with cell number in r = 0.05 cm (t = 5 ns).
图 3 网格内温度与空间的关系可近似为线性关系
Figure 3. The dependence of temperature on space is approxi-mately linear.
图 4 辐射波不同位置的辐射源粒子空间分布概率密度 (a)网格9, 波后处; (b)网格18, 波后处; (c)网格22, 波头处; (d)网格26, 波头处
Figure 4. Spatial probability density distribution of radiation source particle in different positions of radiation wave: (a) Cell 9, in the behind of wave; (b) cell 18, in the behind of wave; (c) cell 22, in the head of wave; (d) cell 26, in the head of wave.
图 5 不同网格数时的物质温度空间分布 (t = 10 ns) (a)乘抽样法; (b)阶梯近似抽样法
Figure 5. Material temperature with different cell numbers (t = 10 ns): (a) Multiplying sampling method; (b) stepped approximation sampling method.
图 6 网格数为40时两种新抽样法计算得到的物质温度空间分布(t = 10 ns)
Figure 6. Results of two new sampling methods with 40 cells (t = 10 ns).
表 1 不同网格数时的温度曲线相对参考解的标准偏差和最大误差
Table 1. Relative to the reference solution, the standard deviation and the maximum error of temperature curves with different cell numbers.
Cell number Standard deviation/eV Maximum error/eV 20 80.8 273 40 48.7 254 100 28.9 245 200 13.0 125 
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表 2 不同网格数的计算时间
Table 2. Computation time with different cell numbers.
Cell number Particle number Computation time/s 20 1 × 104 1.80 × 103 40 2 × 104 2.72 × 103 100 5 × 104 5.74 × 103 200 2 × 105 2.28 × 104 400 4 × 105 5.06 × 104
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表 3 不同网格数时的温度曲线相对基准解的标准偏差和最大误差
Table 3. Relative to the reference solution, the standard deviation, and the maximum error of temperature curves with difference cell numbers.
Cell number 乘抽样法 阶梯近似抽样法 Standard deviation/eV Maximum error/eV Standard deviation/eV Maximum error/eV 20 21.60 91.6 18.80 77.8 40 13.60 75.4 12.40 68.7 100 10.40 77.9 10.40 87.6 200 5.64 63.3 5.27 59.5
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表 4 计算问题所花时间
Table 4. Computation time of the problem.
Sampling method Cell number Particle number/104 Computation time/103 s 等温法抽样 20 1 1.80 40 2 2.72 100 5 5.74 200 20 22.80 400 40 50.60 乘抽样法 20 1 1.65 40 2 2.56 100 5 5.64 200 20 22.40 阶梯近似抽样法 20 1 1.61 40 2 2.58 100 5 5.75 200 20 22.40
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[1] 彭惠民 2008 等离子体中辐射输运和辐射流体力学 (北京: 国防工业出版社) 第38页
Peng H M 2008 Radiation Transport and Radiation Hydrodynamics in Plasmas (Beijing: National Defense Industry Press) p38 (in Chinese)
[2] Hammersly J M, Handscomb D C 1964 Monte Carlo Method (New York: John Wiley & Sons Press) p76
[3] 裴鹿成, 张孝泽 1980 蒙特卡罗方法及其在粒子输运问题中的应用 (北京: 科学出版社) 第18页
Pei L C, Zhang X Z 1980 Monte Carlo Method and Application in Particle Transportation (Beijing: Science Press) p18 (in Chinese)
[4] Fleck J A 1963 Computational Method in the Physical Sciences (Vol. 1) (New York: McGraw-Hill) p43
[5] Campbell P M, Nelson R G 1964 Livermore, Calif : Lawrence Radiation Laboratory Report UCRL-7838
[6] Fleck J A, Cummings J D 1971 J. Comput. Phys. 8 313
Google Scholar
[7] Evans T M, Urbatsch T J 2002 MILAGRO : A Parallel Implicit Monte Carlo Code for 3-D Radiative Transfer (Los Alamos: Los Alamos National Laboratory) pNM 87545
[8] Kasen D, Thomas R C, Nugent P 2006 Astrophys. J. 651 366
Google Scholar
[9] Kromer M, Sim S A 2009 Mon. Not. R. Astron. Soc. 398 1809
Google Scholar
[10] Wollaeger R T, van Rossum D R, Graziani C, Couch S M, Jordan G C, Lamb D Q, Moses G A 2013 Astrophys. J. Suppl. Ser. 209 20
Google Scholar
[11] Fleck J A, J R, Canfield E H 1984 J. Comput. Phys. 54 508
Google Scholar
[12] Giorla J, Sentis R 1987 J. Comput. Phys. 70 145
Google Scholar
[13] Gentile N A 2001 J. Comput. Phys. 172 543
Google Scholar
[14] Densmore J D, Urbatsch T J, Evans T M, Buksas M W 2007 J. Comput. Phys. 222 485
Google Scholar
[15] Densmore J D, Thompson K G, Urbatsch T J 2012 J. Comput. Phys. 231 6924
Google Scholar
[16] 李树, 李刚, 田东风, 邓力 2013 62 249501
Google Scholar
Li S, Li G, Tian D F, Deng L 2013 Acta Phys. Sin. 62 249501
Google Scholar
[17] Huo W Y, Li Z C, Yang D, Lan K, Liu J, Ren G L, Li S W, Yang Z W, Guo L, Hou L F, Xie X F, Li Y K, Deng K L, Yuan Z, Zhan X Y, Yuan G H, Zhang H J, Jiang B B, Huang L Z, Du K, Zhao R C, Li P, Wang W, Su J Q, Ding Y K, He X T, Zhang W Y 2016 Matter Radiat. Extremes 1 2
Google Scholar
[18] 李树, 陈耀华, 姬志成, 章明宇, 任国利, 霍文义, 闫威华, 韩小英, 李志超, 刘杰, 蓝可 2018 67 025202
Google Scholar
Li S, Chen Y H, Ji Z C, Zhang M Y, Ren G L, Huo W Y, Yan W H, Han X Y, Li Z C, Liu J, Lan K 2018 Acta Phys. Sin. 67 025202
Google Scholar
[19] Feng T G, Lai D X 1996 Sci. China: E 39 461
Google Scholar
[20] 李树, 邓力, 田东风, 李刚 2014 63 239501
Google Scholar
Li S, Deng L, Tian D F, Li G 2014 Acta Phys. Sin. 63 239501
Google Scholar
[21] Wollaber A B 2008 Ph.D. Dissertation (Ann Arbor: University of Michigan)
[22] 许淑艳 2006 蒙特卡罗方法在实验核物理中的应用 (北京: 原子能出版社) 第46页
Xu S Y 2006 Applications of Monte Carlo Method in Experimental Nuclear Physics (Beijing: Atomic Energy Press) p46 (in Chinese)
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