基于一阶中子输运方程的杂交间断有限元方法研究

孙启政; 刘晓晶; 张滕飞

doi:10.13832/j.jnpe.2024.06.0248

基于一阶中子输运方程的杂交间断有限元方法研究

doi: 10.13832/j.jnpe.2024.06.0248

上海交通大学核科学与工程学院，上海，200240

基金项目: 国家自然科学基金（12175138、U20B2011）

详细信息

作者简介:
孙启政（1997—），男，博士研究生，现主要从事中子输运算法相关研究，E-mail: qizhengsun@sjtu.edu.cn

通讯作者:
张滕飞，E-mail: zhangtengfei@sjtu.edu.cn

中图分类号: TL329
计量
- 文章访问数: 19
- HTML全文浏览量: 6
- PDF下载量: 2
- 被引次数: 0
出版历程
- 收稿日期: 2024-07-07
- 修回日期: 2024-08-21
- 刊出日期: 2024-12-17

Investigation on Hybrid Discontinuous Galerkin Method Based on First-Order Neutron Transport Equation

School of Nuclear Science and Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China

摘要

摘要: 复杂构型先进反应堆的不断提出对中子学数值方法提出了更高的要求，为实现对复杂问题的准确、高效模拟，本文提出一种基于一阶双曲型中子输运方程（NTE）的杂交间断有限元（HDG）方法。该方法在角度上采用了离散纵标（S_N）的格式将原始的方程解耦为各个角度方向的相互独立的方程；在空间上采用迎风格式对方程进行离散，整个问题的全局矩阵耦合系统呈现分块下三角的形式，更加适合网格数目较多的复杂几何非均匀中子输运问题场景。选取了TAKEDA1基准题与非均匀组件问题作为分析对象，对所提出的HDG方法的计算性能进行了分析。数值结果表明，HDG在上述算例中均实现稳定收敛，有效增殖系数k_eff与参考解的最大误差为108pcm（1pcm=10⁻⁵）。此外，与传统二阶偶对称形式方法相比，一阶HDG方法空间扫描更为高效，在上述算例中实现了约2倍的加速比。因此，本文研究的HDG方法能够为复杂构型反应堆问题提供可选的解决方案。
- 杂交间断有限元（HDG） /
- 中子输运方程（NTE） /
- 迎风格式 /
- 离散纵标（S_N）
Abstract: The development of advanced reactor designs imposes higher demands on neutronic numerical methods. To achieve accurate and efficient simulation of complex problems, this paper introduces a hybrid discontinuous Galerkin (HDG) method based on the first-order hyperbolic neutron transport equation (NTE). The method decouples the original equation into independent equations for each angular direction using the discrete-ordinates (S_N) method in angular space. In spatial discretization, this paper employs an upwind scheme that results in a blocked-lower-triangular global matrix coupling system, making it well-suited for complex, geometrically heterogeneous neutron transport scenarios with a large number of meshes. The study evaluates the performance of the proposed HDG method using the TAKEDA1 benchmark and a heterogeneous assembly problem. The results demonstrate that the HDG method achieves stable convergence for the aforementioned problems, with a maximum error between the effective multiplication coefficient k_eff and the reference solution of 108 pcm (1pcm = 10⁻⁵). In addition, compared with the traditional second-order even-parity method, the first-order HDG method is more efficient in spatial scanning, and the acceleration ratio is about 2 times in the above examples. Therefore, the proposed HDG method can provide an alternative solution for complex reactor problems.
- Hybrid discontinuous Galerkin (HDG) /
- Neutron transport equation (NTE) /
- Upwind scheme /
- Discrete-ordinates (S_N)

HTML全文

图 1 一阶HDG方法在TAKEDA1问题中的注量率分布

Figure 1. Flux Distribution of First-order HDG Method for the TAKEDA1 Problem

下载: 全尺寸图片幻灯片

图 2 TAKEDA1基准题中一阶与二阶方法计算时间对比

Figure 2. Comparison of Execution Time between First-order and Second-Order Methods for the TAKEDA1 Benchmark

下载: 全尺寸图片幻灯片

图 3 HDG方法在PWR组件问题中的功率与误差分布

Figure 3. Power Distribution and Relative Error Distribution of HDG Method for PWR-Assembly Problem

下载: 全尺寸图片幻灯片

表 1 不同工况下的k_eff误差及计算时间对比

Table 1. Comparison of k_eff Error and Calculation Time under Different Test Cases

方法	空间展开阶数	误差/pcm	总时间/s
二阶	1^st/0^th	1491	83.5
	2^nd/1^st	33	191.0
	4^th/2^nd	−10	649.9
	6^th/4^nd	−13	4099.1
一阶	1^st/0^th	−2429	47.4
	2^nd/1^st	14	100.1
	4^th/2^nd	−18	279.1
	6^th/4^nd	−18	1046.6

下载: 导出CSV

参考文献(26)

[1]	XIAO W, LI X Y, LI P J, et al. High-fidelity multi-physics coupling study on advanced heat pipe reactor[J]. Computer Physics Communications, 2022, 270: 108152. doi: 10.1016/j.cpc.2021.108152
[2]	STERBENTZ J W, WERNER J E, MCKELLAR M G, et al. Special purpose nuclear reactor (5 MW) for reliable power at remote sites assessment report: INL/EXT-16-40741[R]. Idaho Falls: Idaho National Laboratory, 2017.
[3]	JARETEG K, VINAI P, DEMAZIÈRE C. Fine-mesh deterministic modeling of PWR fuel assemblies: Proof-of-principle of coupled neutronic/thermal-hydraulic calculations[J]. Annals of Nuclear Energy, 2014, 68: 247-256. doi: 10.1016/j.anucene.2013.12.019
[4]	FIORINA C, HURSIN M, PAUTZ A. Extension of the GeN-Foam neutronic solver to SP₃ analysis and application to the CROCUS experimental reactor[J]. Annals of Nuclear Energy, 2017, 101: 419-428. doi: 10.1016/j.anucene.2016.11.042
[5]	CHEN J, LIU Z Y, ZHAO C, et al. A new high-fidelity neutronics code NECP-X[J]. Annals of Nuclear Energy, 2018, 116: 417-428. doi: 10.1016/j.anucene.2018.02.049
[6]	ZHAO C, PENG X J, ZHANG H B, et al. A new numerical nuclear reactor neutronics code SHARK[J]. Frontiers in Energy Research, 2021, 9: 784247. doi: 10.3389/fenrg.2021.784247
[7]	PEIRÓ J, SHERWIN S. Finite difference, finite element and finite volume methods for partial differential equations[M]//YIP S. Handbook of Materials Modeling. Dordrecht: Springer, 2005: 2415-2446.
[8]	EKLUND M D, DUPONT M, CARACAPPA P F, et al. Neutronics simulations of the RPI walthousen reactor critical facility (RCF) using proteus-SN[J]. Transactions of the American Nuclear Society, 2017, 117: 114-115.
[9]	WANG Y Q, SCHUNERT S, ORTENSI J, et al. Rattlesnake: a MOOSE-based multiphysics multischeme radiation transport application[J]. Nuclear Technology, 2021, 207(7): 1047-1072. doi: 10.1080/00295450.2020.1843348
[10]	PRINCE Z M, HANOPHY J T, LABOURÉ V M, et al. Neutron transport methods for multiphysics heterogeneous reactor core simulation in Griffin[J]. Annals of Nuclear Energy, 2024, 200: 110365. doi: 10.1016/j.anucene.2024.110365
[11]	GIACOMINI M, SEVILLA R, HUERTA A. HDGlab: an open-source implementation of the hybridisable discontinuous Galerkin method in MATLAB[J]. Archives of Computational Methods in Engineering, 2021, 28(3): 1941-1986. doi: 10.1007/s11831-020-09502-5
[12]	COCKBURN B, QIU W F, SHI K. Conditions for superconvergence of HDG methods for second-order elliptic problems[J]. Mathematics of Computation, 2012, 81(279): 1327-1353. doi: 10.1090/S0025-5718-2011-02550-0
[13]	BUI-THANH T. From Godunov to a unified hybridized discontinuous Galerkin framework for partial differential equations[J]. Journal of Computational Physics, 2015, 295: 114-146. doi: 10.1016/j.jcp.2015.04.009
[14]	ZHANG T F, LI Z P. Variational nodal methods for neutron transport: 40 years in review[J]. Nuclear Engineering and Technology, 2022, 54(9): 3181-3204. doi: 10.1016/j.net.2022.04.012
[15]	SUN Q Z, XIAO W, LI X Y, et al. A variational nodal formulation for multi-dimensional unstructured neutron diffusion problems[J]. Nuclear Engineering and Technology, 2023, 55(6): 2172-2194. doi: 10.1016/j.net.2023.02.021
[16]	DIJOUX L, FONTAINE V, MARA T A. A projective hybridizable discontinuous Galerkin mixed method for second-order diffusion problems[J]. Applied Mathematical Modelling, 2019, 75: 663-677. doi: 10.1016/j.apm.2019.05.054
[17]	CARTIER J, PEYBERNES M. Mixed variational formulation and mixed-hybrid discretization of the transport equation[J]. Transport Theory and Statistical Physics, 2010, 39(1): 1-46. doi: 10.1080/00411450.2010.529630
[18]	AZMY Y, SARTORI E. Nuclear computational science: a century in review[M]. Dordrecht: Springer, 2010, 106-112.
[19]	LEWIS E E, CARRICO C B, PALMIOTTI G. Variational nodal formulation for the spherical harmonics equations[J]. Nuclear Science and Engineering, 1996, 122(2): 194-203. doi: 10.13182/NSE96-1
[20]	SMITH M A, LEWIS E E, PALMIOTTI G, et al. A first-order spherical harmonics formulation compatible with the variational nodal method[C]//Proceedings of the Physics of Fuel Cycles and Advanced Nuclear Systems: Global Developments. Argonne: Argonne National Lab. , 2004: 25-29.
[21]	Smith M A, Lewis E E, Palmiotti G, et al. A first-order integral method developed for the VARIANT code[C]. PHYSOR-2006-American Nuclear Society's Topical Meeting on Reactor Physics. 2006.
[22]	ZHANG T F, XIAO W, YIN H, et al. VITAS: a multi-purpose simulation code for the solution of neutron transport problems based on variational nodal methods[J]. Annals of Nuclear Energy, 2022, 178: 109335. doi: 10.1016/j.anucene.2022.109335
[23]	张滕飞,殷晗,孙启政,等. 通用型中子输运程序VITAS应用研究[J]. 核动力工程,2023, 44(2): 15-23.
[24]	TAKEDA T, IKEDA H. 3-D neutron transport benchmarks[J]. Journal of Nuclear Science and Technology, 1991, 28(7): 656-669. doi: 10.1080/18811248.1991.9731408
[25]	SUN Q Z, LIU X J, CHAI X, et al. A discrete-ordinates variational nodal method for heterogeneous neutron Boltzmann transport problems[J]. Computers & Mathematics with Applications, 2024, 170: 142-160.
[26]	SMITH M A, LEWIS E E, NA B C. Benchmark on DETERMINISTIC 3-D MOX fuel assembly transport calculations without spatial homogenization[J]. Progress in Nuclear Energy, 2006, 48(5): 383-393. doi: 10.1016/j.pnucene.2006.01.002