Globally Conservative, Hybrid Self-Adjoint Angular Flux and Least-Squares Method Compatible with Voids

Abstract

In this paper, we derive a method for the second-order form of the transport equation that is both globally conservative and compatible with voids, using Continuous Finite Element Methods (CFEM). The main idea is to use the Least-Squares (LS) form of the transport equation in the void regions and the Self-Adjoint Angular Flux (SAAF) form elsewhere. While the SAAF formulation is globally conservative, the LS formulation need a correction in void. The price to pay for this fix is the loss of symmetry of the bilinear form. We first derive this Conservative LS (CLS) formulation in void. Second we combine the SAAF and CLS forms and end up with an hybrid SAAF-CLS method, having the desired properties. We show that extending the theory to near-void regions is a minor complication and can be done without affecting the global conservation of the scheme. Being angular discretization agnostic, this method can be applied to both discrete ordinates (S N ) and spherical harmonics (P N ) methods. However, since a globally conservative and void compatible second-order form already exists for S N (Wang et al. 2014), but is believed to be new for P N , we focus most of our attention on that latter angular discretization. We implement and test our method in Rattlesnake within the Multiphysics Object Oriented Simulation Environment (MOOSE) framework. Results comparing it to other methods are presented.

An Accurate Globally Conservative Subdomain Discontinuous Least-Squares Scheme for Solving Neutron Transport Problems

“…However, the method generally has low accuracy in problems with interfaces between optically thin and thick materials without local refinement near the interface [14,16,15,17]. Additionally, the standard least-squares method does not have particle conservation (except in the limit as the number of zones goes to infinity), which would induce undesirably large k-eigenvalue errors in neutronics simulations [18], unless conservative acceleration schemes, such as nonlinear diffusion acceleration described in [19,20,21], are applied to regain the conservation.…”

“…CFEM-SAAF is globally conservative, yet, not compatible with void and potentially ill-conditioned in near-void situations. Therefore, efforts have been put in alleviating CFEM-SAAF in void [17,18,21,24]. We will specially give a brief review of the treatment developed in [18,24].…”

Application of Mathematical Statistics Analysis Algorithm for Chemical Data

“…The LINEST function [8] computes the statistical value of a line by using the least square method to calculate the line with the best fit of the existing data, then returns an array that describes the line.…”

A geometry conforming isogeometric method for the self-adjoint angular flux (SAAF) form of the neutron transport equation with a discrete ordinate (SN) angular discretisation

“…al. [13] maintains conservation of neutrons and can model near-void regions, but is not symmetric. Void treatments for the SAAF form of the neutron transport equation have similar issues [11,14].…”