Much ado about nothing: response matrices for void regions

Weak form Galerkin approaches are examined for obtaining response matrices for void regions – that is regions where nothing is present. The need arises in spherical harmonics methods based on second-order forms of the neutron transport equation; the methods fail in voids because the cross-section appearing in the equation’s denominator then vanishes. The diffusion approximation, being the lowest-order spherical harmonic method, is first employed as a vehicle for examining response matrices derived from both primal and dual weak forms of the mixed-first-order and second-order transport equations. Those for which discretization results in singular matrix equations as the cross-section goes to zero are rejected. First-order- mixed formulations with modified natural boundary conditions are shown to lead to nonsingular response matrices for voids. The primal method is chosen as the better of the two candidates for generalization from the diffusion to the transport equations, and the transport formulation is presented.