NURBS-enhanced finite element spatial discretisation methods for the steady-state multigroup neutron diffusion equation

Abstract

The NURBS-enhanced finite element method (NEFEM) is a recent innovation in spatial discretisation methods. The NEFEM combines the conventional FEM with computer-aided geometric design (CAGD) boundary representation (B-rep) approaches based upon Non-Uniform Rational B-spline (NURBS) geometrical representations of the computational domain. The aim of the NEFEM is to streamline the CAGD to computer-aided engineering (CAE) modelling and simulation (M&S) pipeline and provide improved geometrical representations of the underlying curvilinear geometry in nuclear reactor physics and reactor shielding simulations. This eliminates the requirement for local modifications to the underlying computational mesh to preserve the surface areas and volumes of curvilinear geometrical features within the computational domain. Such local mesh modifications are required, within conventional isoparametric Lagrangian FEM approaches, to preserve fissile mass and neutron leakage within curvilinear geometrical and computational domains.

This paper presents the application of the NEFEM to the multigroup neutron diffusion equation (NDE) for three nuclear reactor physics benchmark verification test cases. A further method of manufactured solution (MMS) benchmark verification test case is used to establish the order of convergence of the NEFEM compared to the FEM for both linear and quadratic elements. In addition, an analytical Wigner–Seitz pincell problem is used to further investigate the accuracy of the NEFEM. The results from these benchmark verification test cases demonstrate that the NEFEM yields improved numerical accuracy compared to the conventional FEM. This improved numerical accuracy is primarily achieved through the improved geometrical representation of curvilinear geometries. While the NURBS-enhancement of elements necessitates a small increase to the pre-processing time associated with the method, the increased accuracy of the NEFEM allows it to achieve competitive computational solution times compared to the standard Lagrangian FEM.