Reduced-order modeling of neutron transport eigenvalue problems separated in energy by Proper Generalized Decomposition

Abstract

In this article, we develop and validate an a priori Reduced-Order Model (ROM) of neutron transport separated in energy by Proper Generalized Decomposition (PGD) as applied to the k-eigenvalue problem. To do so, we devise a novel PGD algorithm for eigenvalue problems, in which the update step is solved as an eigenproblem. Further, we demonstrate this problem can be efficiently solved by a recursive LU factorization which exploits the matrix structure imposed by Progressive PGD. Numerical experiments validate Galerkin and Petrov-Galerkin (Minimax) ROMs, as compared to the full-order model, in approximating the fine- and coarse-group neutron flux, the fission source, and multiplication factor (k-eigenvalue). These benchmarks consider representative light water reactor pins of UO2 or Mixed Oxide fuel with CASMO-70, XMAS-172, and SHEM-361 energy meshes. In all cases, the ROM achieves an L2 error of the angular flux less than 0.1% given fifty modes, or enrichment iterations. At the same number of modes, the eigenvalue error is found to be less than 2×10−4 and 2×10−5 for the Galerkin and Minimax ROMs respectively. Meanwhile, the fine-group ROM surpasses the accuracy of the coarse-group full-order model—comparing both to the fine-group full-order model—in estimating the angular and scalar coarse-group fluxes and k-eigenvalue between roughly ten and twenty modes. The computational cost of the PGD ROM is comparable to that of the otherwise-identical fixed-source ROM presented in previous work. Altogether, we expect this PGD ROM may achieve considerable computational savings in modeling fine-group reactor physics. Moreover, it offers an alternative means of approximation to cross section condensation, preferable in that it requires no reference solution or loss of resolution, yet may achieve superior precision for a comparable computational effort.