Computer implementation of singular function enrichment of finite element methods for multigroup neutron diffusion

Abstract

Angular singularities for the multigroup neutron diffusion equation are examined. Since the solution to this equation as well as the solution to other engineering equations of interest can exhibit unbounded derivatives at angular singularities, the accuracy and convergence rate of the finite element method can be significantly reduced. Accordingly, this study presents computer implementation techniques for singular function enrichment in order to mitigate the effects of angular singularities. Further, in order to assess the efficacy of these techniques, the results of a series of numerical experiments with and without singular function enrichment are reported. The computer implementation techniques presented are sufficient to handle many angular singularities occurring in nuclear reactor diffusion calculations as well as those common to many other engineering problems. The tasks necessary to identify and locate angular singularities in a rectangular configuration are outlined. The form of the compact support singular functions necessary to mitigate the angular singularity effects for boundary corners, the right angle intersection of two material interfaces, and the intersection of a material interface with the boundary are derived. A criterion that specifies the minimum number of singular functions necessary to restore the optimum convergence rate for any order finite element method is obtained. Further, some techniques are developed to handle the case of more than one angular singularity in the domain and the problem of overlapping singular function support. The numerical experiments demonstrate that the techniques developed here can significantly improve the accuracy and convergence rate of the finite element solution in the presence of severe angular singularities. With mild singularities and the one group neutron diffusion equation, singular function enrichment neither significantly improves nor impairs the finite element solution.