On the Consistent Spatial Homogenization Method in Neutron Transport Theory

Abstract

The recently developed consistent spatial homogenization method in neutron transport theory is based on basis functions in spatial and angular domains and spatial rehomogenization to correct for the core environment effects while reproducing the heterogeneous solution in full phase space resolution and accuracy. In this article, an efficient version of the method is developed that also simplifies implementation and memory requirements for data handling in multidimensional geometries. This is done by (1) replacing the Fourier series with B-spline basis functions for the spatial domain; (2) truncating the angular expansion to first order, and (3) developing a source iteration method to replace the original local fixed-source transport method for on-the-fly rehomogenization. The new method is tested in a 1D benchmark problem characteristic of pressurized water reactors with mixed oxide fuel. The method is found to be very accurate and more efficient than the original consistent spatial homogenization method.

Keywords:

• consistent spatial homogenization method
• transport theory
• rehomogenization
• basis functions

APPENDIX

The implementation of the CSH method, in particular the auxiliary cross section, for a 1D slab geometry using diamond differencing scheme and multi-group approximation is described in this appendix. Furthermore, similar to the problems presented in this paper and in order to simplify the derivations, the scattering kernel is treated as isotropic together with transport-correcting the total cross sections to account for linearly anisotropy in scattering. Assume the mesh boundaries are given as where the half integer subscripts refer to mesh edges and are the mesh midpoints.

In 1D slab geometry, the Legendre polynomials are substituted for spherical harmonics. The auxiliary cross-section defined in Equation (2) over a homogenized region x ∈ [0, a] is simplified to: (1A)

As explained in Section 3.1, a B-spline basis function is used to expand the auxiliary cross section in the spatial domain. Diamond differencing scheme is based on linear spatial dependence of the flux within each mesh. Hence, it would be a natural choice to pick a B-spline basis function that possesses the same spatial dependence within a mesh. The second-order B-spline, defined in Equation (2A), is a piecewise linear function consistent with diamond differencing scheme. (2A)

Using Equations (1A) and (2A), the auxiliary cross section is shown in Equation (3A). (3A) where (4A) (5A) (6A) (7A)

Given the above definition for the auxiliary cross section term, the homogenized transport equation over the jth homogeneous spatial mesh is defined as: (8A)

Additional information

Funding

This work was supported by NEUP Award Number DE-AC07-O5ID14517. Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the Department of Energy Office of Nuclear Energy.