A Second Order Linear Discontinuous Cut-Cell Discretization for the SN Equations in RZ Geometry

Abstract

In this dissertation we detail the development, implementation, and testing of a new cut-cell discretization for the discrete ordinates form of the neutron transport equation. This method provides an alternative to homogenization for problems containing material interfaces that do not coincide with mesh boundaries. A line is used to represent the boundary between the two materials in a mixed-cell converting a rectangular mixed-cell into two non-orthogonal, homogeneous cut-cells. The linear-discontinuous Galerkin finite element method (LDGFEM) spatial discretization is used on all of the rectangular cells as well as the non-orthogonal sub-cells. We have implemented our new cut-cell method in a test code which has been used to evaluate its performance relative to homogenization.

We begin by developing the equations and methods associated with the LDGFEM discretization of the transport equation in RZ geometry for a homogenous orthogonal mesh. Next we introduce cut-cell meshes and develop a modification to the LDGFEM equations to account for material interfaces. We also develop methods to account for meshing errors encountered when representing curved material interfaces with linear cell faces.

Finally we present test problems including manufactured solutions, fixed source, and eigenvalue problems for geometries with curvilinear material interfaces. The results of these test problems show the new cut-cell discretization to be second-order convergent for the scalar flux removed from singularities in the solution, as well as significantly more computationally efficient than homogenization.