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

Sun Qizheng; Liu Xiaojing; Zhang Tengfei

doi:10.13832/j.jnpe.2024.06.0248

Volume 45 Issue 6

Dec. 2024

Turn off MathJax

Article Contents

Article Navigation > Nuclear Power Engineering > 2024 > 45(6): 248-253

Sun Qizheng, Liu Xiaojing, Zhang Tengfei. Investigation on Hybrid Discontinuous Galerkin Method Based on First-Order Neutron Transport Equation[J]. Nuclear Power Engineering, 2024, 45(6): 248-253. doi: 10.13832/j.jnpe.2024.06.0248

Citation:

Sun Qizheng, Liu Xiaojing, Zhang Tengfei. Investigation on Hybrid Discontinuous Galerkin Method Based on First-Order Neutron Transport Equation[J]. Nuclear Power Engineering, 2024, 45(6): 248-253. doi: 10.13832/j.jnpe.2024.06.0248

Citation:

PDF( 9425 KB)

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

doi: 10.13832/j.jnpe.2024.06.0248

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

Received Date: 2024-07-07
Rev Recd Date: 2024-08-21
Publish Date: 2024-12-17

Abstract

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)

FullText(HTML)

References(26)

References

[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