Izvestiya vuzov. Yadernaya Energetika

The peer-reviewed scientific and technology journal. ISSN: 0204-3327

Influence of the Spatial Grid Type on the Result of Calculating the Neutron Fields in the Nuclear Power Plant Shielding

9/23/2022 2022 - #03 Modelling processes at nuclear facilities

Nikolaeva O.V. Gaifulin S.A. Bass L.P. Dmitriev D.V. Nikolaev A.A.

DOI: https://doi.org/10.26583/npe.2022.3.13

UDC: 519.6

Purpose. The paper considers the influence of the spatial grid type on the result of solving the equation of neutron transport in the nuclear power plant (NPP) shielding.

Method. Neutron fields have been calculated in a realistic model of a liquid metal cooled fast neutron tank reactor with an integral equipment layout. Structured cubic and unstructured hexahedral grids (PMSNSYS and FRIGATE codes) and unstructured tetrahedral and prismatic grids (RADUGA T code) are used. Limiting values of the group fluxes averaged over the material zones for refined grids have been obtained.

Results. It has been shown that the calculation results depend on the type of approximation of the curvilinear inner boundaries between the material zones rather than on the grid cell type (cube, hexahedron, tetrahedron, prism). Using «toothed» approximations for curvilinear boundaries leads to an increase in the area of the boundaries, as well as to the neutron flux refraction condition arising on them. These effects lead to an overestimation of the transport equation solution, and also in all energy groups.

Conclusion. When solving an equation of neutron transport in the NPP shielding by a grid method, it is necessary to use grids other than leading to «toothed» approximations of the inner boundaries. Tetrahedral or prismatic grids, or grids of arbitrary hexahedrons can be recommended, as well as composite grids in which cubic cells are located inside the material zone, and hexahedron cells are located near the zone boundary.


  1. Alcouffe R.E. Three Dimensional Transport Benchmark Exercise Using THREEDANT. Proc. of Meeting on Three+Dimensional Neutron Transport Benchmarks. Los Alamos, New Mexico. 1990. Available at: https://inis.iaea.org/collection/NCLCollectionStore/_Public/22/031 /22031958.pdf?r=1 (accessed Jun. 10, 2022).
  2. Azmy Y.Y. The Three-Dimensional, Discrete Ordinates Neutral Particle Transport Code TORT: an Overview. Proc. of OECD/NEA Meeting on 3D Deterministic Radiation Transport Computer Programs, Feature, Applications and Perspectives. Paris, France. 1996. Available at: https://www.researchgate.net/publication/255255411 (accessed Jun. 10, 2022).
  3. Dahl J.A. PARTISN Results for the OECD/NEA 3-D Extension C5G7MOX Benchmark. Progress in Nuclear Energy. 2006, v. 48, pp. 401-409; DOI: https://doi.org/10.1016/j.pnucene.2006.01.003 .
  4. Humbert P. Results for the C5G7 3-D Extension Benchmark Using the discrete Ordinated Code PANDA. Progress in Nuclear Energy. 2006, v. 48, pp. 394-400; DOI: https://doi.org/10.1016/j.pnucene.2006.01.009 .
  5. Corau T., Sjoden G. 3D Neutron Transport and HPC: A PWR Full Core Calculation Using PENTRAN SN Code and IBM BLUEGENE/P Computers. Progress in Nuclear Science and Technology. 2011, v. 2, pp. 628-633; DOI: https://doi.org/10.15669/pnst.2.628 .
  6. CNCSN. One, Two- and Three-Dimensional Coupled Neutral and Charged Particle Discrete Ordinates Parallel Multi-Threaded Code System. RSICC code package CCC+726. 2009.
  7. Royston K.E., Johnson S.R., Evans T.M., Mosher S.W., Naish J., Kosand B. JET Contributors. Application of the Denovo Discrete Ordinates Radiation Transport Code to Large-Scale Fusion Neutronics. Fusion Science and Technology. 2018, v. 74 (4), pp. 303- 314; DOI: https://doi.org/10.1080/15361055.2018.1504508 .
  8. Moryakov A.V. LUCKY_TD Code for Solving the Time-Dependent Transport Equation with the Use of Parallel Computations. Physics of Atomic Nuclei. 2017, v. 79 (8), pp. 1242-1245; DOI: https://doi.org/10.1134/S1063778816080159 .
  9. Kovalishin A.A., Moryakov A.V, Raskach K.F. Neutronics Calculation of Fast Reactor Using Modern Computing Systems. Atomic Energy. 2018, v. 124 (2), pp. 75-81; DOI: https://doi.org/10.1007/s10512-018-0378-5 .
  10. PC Software. Calculation of Neutronic Characteristics of Nuclear Reactors. «PMSNSYS». 8624607.00622. OKB «GIDROPRESS», 2011 (in Russian).
  11. Nikolaev A.A. Generalization of Two-Dimensional DDL Schemes of the GQ Method for Three-Dimensional Arbitrary Hexahedral Spatial Mesh. Physics of Atomic Nuclei. 2016, v. 79 (8), pp. 1261–1266; DOI: https://doi.org/10.1134/S1063778816080160 .
  12. Hoagland D.S., Yessayan R.A., Azmy Y.Y. Solution of the Neutron Transport Equation on Unstructured Grids Using the Parallel Block Jacobi-Integral Transport Matrix Method via the Novel Green’s Function ITMM Construction Algorithm on Massively Parallel Computer Systems. Nuclear Science and Engineering. 2021, v. 195, pp. 1036-1064; DOI: https://doi.org/10.1080/00295639.2021.1898309 .
  13. Vassiliev O.N., Wareing T.A., Davis I.M., McGhee J., Barnett D., Horton J.L., Gifford K., Failla G., Titt U., Mourtada F. Feasibility of a multigroup deterministic solution method for 3D radiotherapy dose calculations. International Journal of Radiation Oncology Biology Physics. 2008, v. 72 (1), pp. 220-227; DOI: https://doi.org/10.1016/j.ijrobp.2008.04.057 .
  14. Aristova E.N., Astafurov G.O. The Second Order Short Characteristics Method for the Solution of the Transport Equation on Tetrahedral Grid. Mathematical Models and Computer Simulation. 2017, v. 9, pp. 40-47; DOI: https://doi.org/10.1134/S2070048217010045 .
  15. Chen Y., Zhang B., Zhang L., Zheng J., Zheng Y., Liu C. ARES: A Parallel Discrete Ordinates Transport Code for Radiation Shielding Applications and Reactor Physics Analysis. Hindawi Science and Technology of Nuclear Installations. 2017. Article ID 2596727; DOI: https://doi.org/10.1155/2017/2596727 .
  16. Kim J.W., Lee Y.O. A Deep Penetration Problem Calculation Using AETIUS: an Easy Modeling Discrete Ordinates Transport Code Using Unstructured Tetrahedral Mesh, Shared Memory Parallel. EPJ Web of Conferences. 2017, v. 153, pp. 06025; DOI: https://doi.org/10.1051/epjconf/201715306025 .
  17. Belousov V.I., Grushin N.A., Sychugova E.P., Seleznev E.F. Some Results of Verification of Code ODETTA for Shielding Calculations. VANT. Ser. Fizika Yadernykh Reactorov. 2018, iss. 3, pp. 46-53 (in Russian).
  18. Nikolaeva O.V., Gaifulin S.A, Bass L.P. Code RADUGA T for Simulating Neutrons Fluxes in Nuclear Power Stations. Vestnik YuUrGU. Ser. Vychislitel’naya Matematika i Informatika. 2021. v. 10 (1), pp. 75-89 (in Russian); DOI: https://doi.org/10.14529/cmse210106 .
  19. Orsi R. A General Method of Conserving Mass in Complex Geometry Simulations on Mesh Grids and its Implementation in BOT3P5.0. Nuclear Science and Engineering. 2006, v. 154 (2), pp. 247-259; DOI: https://doi.org/10.13182/NSE06-A2631 .
  20. Gurevich M.I., Russkov A.A., Voloschenko A.M. ConDat 1.0 – Code for Converting by the Tracing Algorithm the Combinatorial Geometry Presentation to the Bit+Mapped One. Users Guide. Preprint No 12. Keldysh Institute of Applied Mathematics, Russian Academy of Sciences Publ., 2007. Available at: https://keldysh.ru/papers/2007/prep12/prep2007_12.html (accessed Jun. 10, 2022) (in Russian).
  21. Dedul A.V., Nikolaev A.A. REBEL – Software of Pre and Post Processing Calculations of the Neutronic Performances of Reactor on High-Velocity Neutrons with Lead Bismuthic Heat Carrier. Tyazhyoloe Mashinostroenie. 2014, v. 8, pp. 39-45 (in Russian).
  22. Carlson B.G. A Method of Characteristics and Other Improvements in Solutions Methods for the Transport Equations. Nuclear Science and Engineering. 1976, v. 61, pp. 408-425; DOI: https://doi.org/10.13182/NSE76-A26927 .
  23. Voronkov A.V., Sinitsa V.V, Dedul A.V., Kalchenko V.V. Libraries of Multigroup Constants for Code REACTOR-GP. VANT. Ser. Yadernye Konstanty. 2009, iss. 24, pp. 100-110 (in Russian).

transport equation grids curvilinear boundaries

Link for citing the article: Nikolaeva O.V., Gaifulin S.A., Bass L.P., Dmitriev D.V., Nikolaev A.A. Influence of the Spatial Grid Type on the Result of Calculating the Neutron Fields in the Nuclear Power Plant Shielding. Izvestiya vuzov. Yadernaya Energetika. 2022, no. 3, pp. 146-157; DOI: https://doi.org/10.26583/npe.2022.3.13 (in Russian).