Calculation of Higher-Order Fluxes in Symmetric Cores—I: Theory

Abstract

In neutron chain systems with material symmetries, various k-eigenvalues of the neutron balance equation beyond the dominant one may be degenerate. Eigenfunctions can be partitioned into several classes according to their invariance properties with respect to the symmetry operations (mirror symmetries and rotations) keeping the material distribution in the system unchanged. Their calculation can be limited to a fraction of the system (sector) provided that innovative boundary conditions matching the symmetry classes are used, and whole-system eigenfunctions can then be unfolded from the solutions obtained over the sector. With power iteration as the method for searching k-eigenvalues, this use of the material symmetries to split the global problem into a variety of smaller-sized problems has several computational advantages: lower computation times and memory requirements, increased dominance ratios, lowered possible degeneracies in each subproblem, and possible parallel (separated) treatment of the subproblems. The implementation is discussed in a companion paper using diffusion and transport theories.

Modal methods for the neutron diffusion equation using different spatial modes

“…, to obtain a biorthonormal basis. For reactors with radial symmetry, it can be proved (see (Tommasi et al, 2016)) that degenerated eigenvalues (i.e. eigenvalues with multiplicity greater than 1) can appear and consequently the adjoint modes computed are not directly biorthogonal.…”

Block hybrid multilevel method to compute the dominant λ-modes of the neutron diffusion equation

“…It means that the spectrum of these modes is very clustered. The third and fourth eigenvalue are equal since they are degenerate due to the reactor radial symmetry (for more details, see (Tommasi et al, 2016) 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 First, we study the performance of the block inverse-free preconditioner Arnoldi method (BIFPAM). Table 2, shows a comparison of the number of iterations (Iterations) and CPU time needed to solve the problem with the BIFPAM using different dimensions, m, of the Krylov subspace without preconditioner, with the ILU preconditioner and with the geometric multigrid preconditioner (GMG).…”

Calculation of Higher-Order Fluxes in Symmetric Cores—II: Implementation

“…The innovative boundary conditions described in the companion paper 10 were implemented in an Sn solver and 2-D Cartesian diffusion solver. Advantage was taken from the currently implemented boundary conditions (reflective, periodic) so that only a few changes were needed (except for the coupled problem).…”