2-DIMENSIONAL GEOMETRICAL CORNER SINGULARITIES IN NEUTRON DIFFUSION -PART-I - ANALYSIS

A novel analysis of the neutron multigroup diffusion equation is prese nted for two-dimensional piecewise homogeneous domains with interior c orners that arise at the intersections between regions with distinct m aterial properties. Using polar coordinates centered at a typical inte rior comet; the solution of the multigroup flux is obtained as an infi nite series of products of pairs of functions such that for every pail ; one of the functions depends solely on the angular variable and a si ngle energy group while the other function depends on the radial varia ble and on all energy groups. The angular functions are shown to be th e eigenfunctions of a Sturm-Liouville system that admits an infinite s et of discrete and, in general, noninteger eigenvalues. On the other h and, the radial functions are the solutions of an infinite system of s econd-order ordinary linear differential equations. Exact explicit sol utions for the multigroup diffusion equation (MODE) for two-dimensiona l disk-like homogeneous domains are also derived and shown to yield an alytic expressions for the group fluxes. This analyticity is shown to stem from the fact that the relevant eigenvalues are positive integers , independent of material properties and/or group structure. The exact expressions for the angular eigenvalues and corresponding eigenfuncti ons for two-region domains are then derived and shown to depend crucia lly on the specific angle between the two regions. This fact is unders cored by deriving the exact expressions for the complete sets of eigen values and eigenfunctions for two geometries of particular importance to nuclear reactors, namely the hexagonal and rectangular geometries, respectively, and by showing that they are fundamentally distinct from one another Of course, these expressions reduce to one and the same f orm for both geometries when the respective two-region domains are red uced to a single-region domain. Finally, the multigroup fluxes are sho wn to be bounded but nonanalytic at the respective interior corners; t he reason underlying this behavior is traced back to the noninteger ch aracter of the relevant eigenvalues. This nonanalyticity is shown to b e the fundamental reason for the failure of conventional (e.g., finite difference, finite element) numerical methods for solving the MODE at and around such corners.