1.Institute of Applied Physics and Computational Mathematics, Beijing 100094, China 2.Graduate School of China Academy of Engineering Physics, Beijing 100088, China
Abstract: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. Keywords:implicit Monte Carlo/ thermal radiation transport/ spherical geometry/ spatial sampling
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) Figure2. 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).
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 Table1.Relative to the reference solution, the standard deviation and the maximum error of temperature curves with different cell numbers.
3.辐射源粒子抽样的两种新方法假设同一网格内温度与半径r的关系是线性的(以下称为“线性假设”), 这种假设在绝大多数情况下比“等温假设”的近似效果更好. 如图3所示, 某网格的内外边界半径分别为r1, r2, 内外边界物质温度分别为T1, T2, 则温度与r的关系可表示为 图 3 网格内温度与空间的关系可近似为线性关系 Figure3. The dependence of temperature on space is approxi-mately linear.
为分析方程(11)与等温假设下推导出的(8)式在描述辐射源粒子空间分布时的差异, 下面以第2节热辐射输运问题中网格数为40的计算结果为例, 选取了第9, 18, 22, 26个网格(球的中心网格为第1个网格)作为辐射波的代表点. 图4为相应网格的辐射源粒子空间概率密度分布. 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 Figure4. 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.
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 Table3.Relative to the reference solution, the standard deviation, and the maximum error of temperature curves with difference cell numbers.