一种求解三维热辐射输运方程的整体预处理迭代方法及并行计算

doi:10.19596/j.cnki.1001-246x.8238

摘要/Abstract

摘要：

为三维灰体热辐射输运方程的隐式离散纵标方法发展一个整体预处理迭代方法并研制并行程序。该方法采用组装线性代数方程组策略，同时求出所有离散方向上的辐射强度。借助预处理的Krylov子空间迭代法，避免复杂网格上扫描算法可能遇到的死锁问题，能够提高健壮性和计算效率。空间离散上采用一阶迎风有限体积格式。数值实验测试变形六面体网格上的收敛率、评估预处理迭代方法的性能并计算辐射和物质的耦合问题，给出三维弯管和黑腔问题的模拟结果，验证程序的正确性和方法的适应性。

关键词: 三维辐射输运方程, 预处理Krylov子空间方法, 离散纵标方法, 并行计算

Abstract:

We present a monolithic preconditioned iterative solver for implicit discrete ordinate equations of three-dimensional grey thermal radiation transport and parallel codes are developed. A strategy of assembling linear algebraic systems is used to obtain radiation intensity in all discrete directions simultaneously. With preconditioned Krylov subspace iterative methods, the solver avoids possible mesh cycles in complex grids associated with sweep algorithms, which improves robustness and computational efficiency. First order upwind finite volume scheme is used for space discretization. Numerical experiments verify convergence rate on distorted hexahedral grids and assess performance of preconditioned iterative methods. Problems with coupled radiation and matter are simulated. Simulation results of three-dimensional crooked pipe and hohlraum problems are shown. It shows validity of the codes and flexibility of the method.

Key words: three-dimensional radiation transport equations, preconditioned Krylov subspace methods, discrete ordinate method, parallel computing

中图分类号:

O242.1

李凌霄, 翟传磊, 谢辉, 施意. 一种求解三维热辐射输运方程的整体预处理迭代方法及并行计算[J]. 计算物理, 2021, 38(3): 269-279.

Lingxiao LI, Chuanlei ZHAI, Hui XIE, Yi SHI. A Monolithic Preconditioned Iterative Solver and Parallel Computing for Three-dimensional Thermal Radiation Transport Equation[J]. Chinese Journal of Computational Physics, 2021, 38(3): 269-279.

图/表 17

图1 变形六面体网格图示

Fig.1 Illustration of distorted hexahedral mesh

表1 空间收敛率

Table 1 Convergence rate of space discretization

网格	六面体个数	‖ϕ-ϕ_h‖_L²	收敛率	‖T-T_h‖_L²	收敛率
G1	512	4.48 × 10^-2		3.27 × 10^-2
G2	4 096	2.25 × 10^-2	0.994	1.64 × 10^-2	0.996
G3	32 768	1.13 × 10^-2	0.994	8.19 × 10^-3	1.002
G4	262 144	5.68 × 10^-3	0.992	4.10 × 10^-3	0.998

图2 拉伸网格图示

Fig.2 Illustration of stretched mesh

图3 终止时刻物质温度在z= 0.5截面上的分布

Fig.3 Distribution of material temperature on z = 0.5 at end time

图4 一维Marshak波测试：物质温度比较(a) S4, (b) S8

Fig.4 1D Marshak wave test: Comparison of material temperature (a) S4, (b) S8

表2 解法器并行可扩展性测试

Table 2 Parallel scalability of the solver

网格/进程数	S2 build	S2 solve	S4 build	S4 solve
G1/1	9.33 s	1.85 s	59.18 s	8.80 s
G2/8	10.87 s	3.77 s	73.90 s	22.73 s
G3/64	12.12 s	6.63 s	85.37 s	46.06 s
G4/512	14.65 s	11.33 s	103.27 s	70.18 s

图5 三维弯管问题所使用网格

Fig.5 The mesh used in 3D crooked pipe problem

表3 不同预处理Krylov迭代方法的迭代步数对比

Table 3 Comparison of iteration number for different preconditioned Krylov methods

S8角度离散			GMRES		BiCGSTAB
时间步	τ/ns	NonIter	Jacobi	ASM	Jacobi	ASM
88	0.2	3	123	25	70	14
			117	24	62	13
			128	23	70	14
89	0.085 6	3	109	23	61	13
			104	22	55	12
			109	20	55	10

表4 32进程时不同预处理Krylov迭代方法的求解时间对比

Table 4 Comparison of solving time for different preconditioned Krylov methods with 32 MPI ranks

	GMRES+Jacobi	BiCGSTAB+Jacobi	GMRES+ASM	BiCGSTAB+ASM
S4	236.4 s	197.1 s	158.4 s	126.1 s
S8	2 458.2 s	2 285.4 s	2 118.1 s	1 864.7 s

表5 64进程时不同Krylov迭代方法的求解时间对比

Table 5 Comparison of solving time for different Krylov methods with 64 MPI ranks

	GMRES+Jacobi	BiCGSTAB+Jacobi
S4	126.1 s	103.6 s
S8	1 221.9 s	1 029.4 s

图6 三维弯管问题x-z截面上的物质温度分布

Fig.6 Distribution of material temperature on x-z cross section

图7 三维弯管问题x-z截面上的辐射温度分布

Fig.7 Distribution of radiation temperature on x-z cross section

图8 三维黑腔问题计算区域

Fig.8 Computational domain for 3D holaraum problem

图9 三维黑腔问题t= 0.461 138 ns时物质温度

Fig.9 Material temperature at t= 0.461 138 ns

图10 三维黑腔问题t= 1 ns时物质温度

Fig.10 Material temperature at t= 1 ns

图11 三维黑腔问题t= 0.006 105 1 ns时的辐射温度

Fig.11 Radiation temperature at t= 0.006 105 1 ns

图12 三维黑腔问题t= 0.057 275 ns时的辐射温度

Fig.12 Radiation temperature at t= 0.057 275 ns

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