Three-dimensional variational nodal method parallelization for pin resolved neutron transport calculations

Yongping Wang*, Tengfei Zhang, E. E. Lewis, W. S. Yang, M. A. Smith, Hongchun Wu

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


Hybrid parallelization of the three-dimensional variational nodal method (VNM) for pin-resolved neutron transport calculations is presented. Sets of node-energy response and associated matrices are assigned evenly to the available set of MPI processors and OpenMP is further employed to parallelize integrations over angle in the response matrix formation. Parallelization of the solution algorithm is accomplished using non-overlapping domain decomposition with one MPI processor assigned to each subdomain. Additionally, the red and black subdomain iterations of the within-group equations are parallelized by OpenMP. Only the pre-calculated matrix sets needed in a subdomain are stored in the corresponding processor. This parallel solution strategy preserves the block Gauss-Seidel red-black iterations employed in serial calculation, with data transfer between processors only after each red and black iteration. Partitioned matrix acceleration is implemented in a similar manner. These parallel algorithms are implemented in the code PANX allowing both higher-order space-angle approximations to be employed and larger spatial domains to be treated than was the case with the serial form of the code. The KAIST problem is used to both evaluate the parallel performance and demonstrate the increased accuracy of the higher-order angular approximations. With 27 MPI processors and 4 OpenMP threads, speedups of nearly 39 are obtained. To demonstrate the applicability of parallel algorithms to larger spatial domains, a pin-resolved, three-dimensional seven-group model of the NuScale modular reactor core with sufficiently high order space-angle discretization is employed using 49 MPI processors and 4 OpenMP threads, and completed in 4 h with P3 approximation.

Original languageEnglish (US)
Article number102991
JournalProgress in Nuclear Energy
StatePublished - Nov 2019


  • Finite element method
  • Hybrid parallelization
  • Pin-resolved
  • Spherical harmonics method
  • Variational nodal method

ASJC Scopus subject areas

  • Nuclear Energy and Engineering
  • Safety, Risk, Reliability and Quality
  • Energy Engineering and Power Technology
  • Waste Management and Disposal


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