Development of a new two-dimensional cartesian geometry nodal multigroup discrete ordinates method

The purpose of this work is the development and testing of a new family of methods for calculating the spatial dependence of the neutron density in nuclear systems described in two-dimensional Cartesian geometry. The energy and angular dependence of the neutron density is approximated using the multigroup and discrete ordinates techniques respectively. The resulting FORTRAN computer code is designed to handle an arbitrary number of spatial, energy, and angle subdivisions. Any degree of scattering anisotropy can be handled by the code for either external source or fission systems.

The basic approach is to (1) approximate the spatial variation of the neutron source across each spatial subdivision as an expansion in terms of a user-supplied set of exponential basis functions; (2) solve analytically for the resulting neutron density inside each region; and (3) approximate this density in the basis function space in order to calculate the next iteration flux-dependent source terms. In the general case the calculation is iterative due to neutron sources which depend on the neutron density itself, such as scattering interactions.

The three methods which were developed differ in the detail of the spatial description:

1. The first method expands the two-dimensional intranode neutron flux as two separable one-dimensional expansions in the x- and y-dimensions and represents the edge fluxes as constant;

2. The second method is the same as the first in the interior of each node, but represents the edge fluxes as one-dimensional expansions in the basis function set; and

3. The third method is the same as the second on the edges, but represents the interior flux shape in a full two-dimensional expansion in the x- and y-dependent basis functions.

In order to test the accuracy versus computer time of the three methods, five sample problems were run and the results compared with those of the finite-difference code DOT4.2. Three shielding problems were run: a simple "benchmark" calculation, a bore-hole geometry volumetric source problem solved with two and five energy groups, and a boundary source large void shielding problem. In addition, two eigenvalue problems, a simple benchmark reactor calculation and a boiling water reactor core lattice case, were run and again compared with DOT4.2. In all of these cases, the first method showed similar accuracy/cost characteristics as DOT4.2, with the second and third methods performing significantly better than the finite difference code.

The major conclusion of the study is that the new exponential expansion methods show promise of reducing the cost of accurately calculating the neutron density inside nuclear systems. Future research work suggested by the present study include:

1. Extension to other geometric systems, such as two-dimensional curvilinear and three-dimensional Cartesian geometries,

2. Extension to handle time dependent problems,

3. Optimization studies on the choice of basis functions, and

4. Development of acceleration schemes tailored to nodal methods.