Nuclear Engineering and Technology

Korean Nuclear Society (한국원자력학회)
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Diffusion synthetic acceleration with the fine mesh rebalance of the subcell balance method with tetrahedral meshes for SN transport calculations

  • Muhammad, Habib (Department of Nuclear Engineering, Kyung Hee University) ;
  • Hong, Ser Gi (Department of Nuclear Engineering, Kyung Hee University)
  • Received : 2019.07.16
  • Accepted : 2019.08.27
  • Published : 2020.03.25
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Abstract

A diffusion synthetic acceleration (DSA) technique for the SN transport equation discretized with the linear discontinuous expansion method with subcell balance (LDEM-SCB) on unstructured tetrahedral meshes is presented. The LDEM-SCB scheme solves the transport equation with the discrete ordinates method by using the subcell balances and linear discontinuous expansion of the flux. Discretized DSA equations are derived by consistently discretizing the continuous diffusion equation with the LDEM-SCB method, however, the discretized diffusion equations are not fully consistent with the discretized transport equations. In addition, a fine mesh rebalance (FMR) method is devised to accelerate the discretized diffusion equation coupled with the preconditioned conjugate gradient (CG) method. The DSA method is applied to various test problems to show its effectiveness in speeding up the iterative convergence of the transport equation. The results show that the DSA method gives small spectral radii for the tetrahedral meshes having various minimum aspect ratios even in highly scattering dominant mediums for the homogeneous test problems. The numerical tests for the homogeneous and heterogeneous problems show that DSA with FMR (with preconditioned CG) gives significantly higher speedups and robustness than the one with the Gauss-Seidel-like iteration.

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References

  1. T.A. Wareing, J.M. McGhee, J.E. Morel, S.D. Pautz, Discontinuous finite element $S_N$ methods on three-dimensional unstructured grids, Nucl. Sci. Eng. 138 (2001) 256-268. https://doi.org/10.13182/NSE138-256
  2. J.E. Morel, J.S. Warsa, An $S_N$ spatial discretization scheme for tetrahedral meshes, Nucl. Sci. Eng. 151 (2005) 157-166. https://doi.org/10.13182/NSE05-A2537
  3. J.S. Warsa, A continuous finite element-based discontinuous finite element method for $S_N$ transport, Nucl. Sci. Eng. 160 (2008) 385-400. https://doi.org/10.13182/NSE160-385TN
  4. S.G. Hong, Two subcell balance methods for solving the multigroup discrete ordinates transport equation with tetrahedral meshes, Nucl. Sci. Eng. 173 (2013) 101-117. https://doi.org/10.13182/NSE11-38
  5. S.G. Hong, Y.O. Lee, Subcell balance methods with linear discontinuous expansion for $S_N$ transport calculation on tetrahedral meshes, Trans. Am. Nucl. Soc. 103 (2010) 354.
  6. S.G. Hong, J.W. Kim, Y.O. Lee, Development of MUST (Multi-Group Unstructured Geometry SN Transport) Code, in: Transaction of Korean Nuclear Society Autumn Meeting, Gyeong-ju, Korea, October 29-30, 2009.
  7. M.L. Adams, E.W. Larsen, Fast iterative methods for discrete ordinates particle transport calculations, Prog. Nucl. Energy 40 (2002) 3-159. https://doi.org/10.1016/S0149-1970(01)00023-3
  8. R.E. Alcouffe, Diffusion synthetic acceleration methods for diamond difference discrete ordinates equations, Nucl. Sci. Eng. 64 (1977) 344-355. https://doi.org/10.13182/NSE77-1
  9. M.L. Adams, W.R. Martin, Diffusion synthetic acceleration of discontinuous finite element transport iterations, Nucl. Sci. Eng. 111 (1992) 145-167. https://doi.org/10.13182/NSE92-A23930
  10. K. Hussein, Effectiveness of a consistently formulated diffusion synthetic acceleration approach, Nucl. Sci. Eng. 98 (1988) 226-243. https://doi.org/10.13182/NSE88-A22324
  11. J.S. Warsa, T.A. Wareing, J.E. Morel, Fully consistent diffusion synthetic acceleration of linear discontinuous $S_N$ transport discretizations on unstructured tetrahedral meshes, Nucl. Sci. Eng. 141 (2002) 236-251. https://doi.org/10.13182/NSE141-236
  12. H. Muhammad, S.G. Hong, Diffusion Synthetic Acceleration (DSA) for the Linear Discontinuous Expansion Method with Subcell Balance (LDEM-SCB), Transaction of Korean Nuclear Society, Jeju, Korea, May 16-18, 2017.
  13. T.A. Wareing, E.W. Larsen, M.L. Adams, Diffusion accelerated discontinuous finite element schemes for the $S_N$ equations in slab and X-Y geometries, in: Proc. International Topical Meetings on Advances in Mathematics, Computations, Reactor Physics, Pittsburgh, Pennsylvania, vol. 3, 28 April - 2 May 1991, 1-1.
  14. Y. Wang, J.C. Ragusa, Diffusion synthetic acceleration for high order discontinuous finite element $S_N$ transport schemes and application to locally refined unstructured meshes, Nucl. Sci. Eng. 166 (2010) 145-166. https://doi.org/10.13182/NSE09-46
  15. A.R. Owens, J. Kophazi, M.D. Eaton, Optimal trace inequality constants for interior penalty discontinuous Galerkin discretisations of elliptic operators using arbitrary elements with non-constant Jacobians, J. Comput. Phys. 350 (2017) 847-870. https://doi.org/10.1016/j.jcp.2017.09.020
  16. J.S. Warsa, T.A. Wareing, J.E. Morel, Krylov iterative methods and the degraded effectiveness of diffusion synthetic acceleration for multidimensional $S_N$ calculations in problems with material discontinuities, Nucl. Sci. Eng. 147 (2004) 218-248. https://doi.org/10.13182/NSE02-14
  17. Y. Saad, Iterative Methods for Sparse Linear Systems, second ed., Society for industrial and applied mathematics, Philadelphia, Pennsylvania, 2003.
  18. H. Muhammad, S.G. Hong, Effective Use of the Linear Fine Mesh Rebalance for the DSA of $S_N$ Transport Equation with Tetrahedral Meshes, vol. 118, Transaction of American Nuclear Society, Philadelphia, Pennsylvania, 2018.
  19. A. Yamamoto, Generalized coarse mesh rebalance method for acceleration of neutron transport calculations, Nucl. Sci. Eng. 151 (2005) 274-282. https://doi.org/10.13182/NSE151-274
  20. A. Yamamoto, Y. Kitamura, T. Ushio, N. Sugimura, Convergence improvement of coarse mesh rebalance method for neutron transport calculations, J. Nucl. Sci. Technol. 41 (2004) 781-789. https://doi.org/10.1080/18811248.2004.9715547
  21. S.G. Hong, K.S. Kim, J.S. Song, Fourier convergence analysis of the rebalance methods for discrete ordinates transport equations in eigenvalue problems, Nucl. Sci. Eng. 164 (2010) 33-52. https://doi.org/10.13182/NSE09-18
  22. S.G. Hong, N.Z. Cho, A rebalance approach to nonlinear iteration for solving the neutron transport equations, Ann. Nucl. Energy 24 (1997) 144-160.
  23. H. Muhammad, S.G. Hong, A three-dimensional fourier analysis of fine mesh rebalance acceleration of linear discontinuous sub-cell balance method for diffusion equation on tetrahedral meshes, Ann. Nucl. Energy 133 (2019) 145-153. https://doi.org/10.1016/j.anucene.2019.05.020