Variational derivation of modal-nodal finite difference equations in spatial reactor physics

Abstract

A class of consistent coarse mesh modal-nodal approximation methods is presented for the solution of the spatial neutron flux in multigroup diffusion theory. The methods are consistent in that they are systematically derived as an extension of the finite element method by utilizing general modal-nodal variational techniques. Detailed subassembly solutions, found by imposing zero current boundary conditions over the surface of each subassembly, are modified by piecewise continuous Hermite polynomials of the finite element method and used directly in trial function forms. Methods using both linear and cubic Hermite basis functions are presented and discussed. The proposed methods differ substantially from the finite element methods in which homogeneous nuclear constants, homogenized by flux weighting with detailed subassembly solutions, are used. However, both schemes become equivalent when the subassemblies themselves are homogeneous. One-dimensional, two-group numerical calculations using representative PWR nuclear material constants and 18-cm subassemblies were performed using entire subassemblies as coarse mesh regions. The results indicate that the proposed methods can yield comparable if not superior criticality measurements, comparable regional power levels, and extremely accurate subassembly fine flux structure with little increase of computational effort in comparison with existing coarse mesh methods.