Improvements to the discrete generalized multigroup method based on proper orthogonal decomposition and SPH factors

Abstract

The deterministic transport methods that drive much of nuclear reactor design are invariably based on the multigroup method, in which fluxes and cross sections are treated as constant within small intervals of the energy domain called groups. The discrete generalized multigroup (DGM) method provides an alternative way to represent and solve the multigroup neutron transport equation by dividing the group structure into a smaller number of coarse groups and expanding the energy variable within each coarse group in an orthogonal basis. However, the original form of the DGM equations leads to higher memory costs and a larger computational burden than the equivalent multigroup formulation of a given problem. This work presented herein aimed to improve the efficiency of the DGM method while preserving its accuracy (1) by incorporating a basis from proper orthogonal decomposition (POD) that yields highly-accurate, low-order representations of fine-group fluxes and (2) by introducing specialized superhomogénéisation (SPH) factors to mitigate errors related to averaging (i.e., "homogenizing") cross sections over space and energy. By truncating the flux-moment expansions, computational savings are gained, but accuracy is somewhat reduced. POD-driven bases were generated using spectral information extracted from small, representative models, and, therefore, perform well under truncation, i.e., the leading terms capture the majority of the variation in energy. This is in stark contrast to the discrete Legendre polynomials, which incorporate no physics and, therefore, require complete expansions in the basis to preserve the underlying, multigroup physics. A key observation made is that the number of degrees of freedom per coarse group required to obtain a desired accuracy is nearly independent of the total number of energy groups. For example, a POD basis truncated to three degrees of freedom per coarse group resulted in approximately 0.1% error in the fission density for all 1-D problems and group structures tested. The second set of improvements explored the use of spatial homogenization and angular approximation to reduce the memory requirements of DGM. These improvements are again at the cost of some accuracy, but the impact is on the same order of magnitude as that of the truncated basis. Approximating the angular dependence of the total cross section using a linear Legendre expansion introduces approximately 0.5% error into the solution, and homogenizing the cross section moments over coarse-mesh regions increases the error by approximately 2%. These two approximations used in conjunction with a POD basis truncated to three degrees of freedom per coarse group results in a total error of around 2% in the fission densities. The final improvement is the use of superhomogénéisation or SPH factors, which are used to correct homogenized cross sections for use in larger, downstream problems. SPH factors were used to correct spatial homogenization of the cross section moments to preserve the reaction rates. Although traditional SPH factors performed better than the corrected moments for specific problems, the DGM method with SPH factors produced cross section moments with a smaller error for general problems. In other words, correcting the DGM moments provided a way to create cross section moments that were more problem agnostic as compared to the traditional method. In particular, the DGM-SPH cross section moments achieved an error of around 1% in the pincell fission densities using just three degrees of freedom per coarse group which was consistent over several different problems. This can be compared to traditional SPH correction of spatial homogenization that resulted in as large as 10% error for a comparable number of energy degrees of freedom.