Uncollided flux techniques for arbitrary finite element meshes

The uncollided angular flux can be difficult to compute accurately in discrete-ordinate radiation transport codes, especially in weakly-scattering configurations with localized sources. It has long been recognized that an analytical or semi-analytical treatment of the uncollided flux, coupled with a discrete-ordinate solution for the collided flux, can yield dramatic improvements in solution accuracy and computational efficiency. In this paper, we present such an algorithm for the semi-analytical calculation of the uncollided flux. This algorithm is unique in several aspects: (1) it applies to arbitrary polyhedral cells (and can be thus coupled with collided flux solvers that support arbitrary polyhedral meshes without the need for explicit tetrahedral re-meshing), (2) it provides accurate uncollided solutions near sources, (3) it is devised with parallel implementation in mind, and (4) it minimizes the total number of traced rays and maintains a reasonable ray density on each local subdomain. This paper provides a complete derivation of the algorithm and demonstrates its important features on a set of simple examples and a standard transport benchmark. Assessment of its parallel performance will be the subject of a subsequent paper.