Pyramid finite elements for discontinuous and continuous discretizations of the neutron diffusion equation with applications to reactor physics

When using unstructured mesh finite element methods for neutron diffusion problems, hexahedral elements are in most cases the most computationally efficient and accurate for a prescribed number of degrees of freedom. However, it is not always practical to create a finite element mesh consisting entirely of hexahedral elements, particularly when modelling complex geometries, making it necessary to use tetrahedral elements to mesh more geometrically complex regions. In order to avoid hanging nodes, wedge or pyramid elements can be used in order to connect hexahedral and tetrahedral elements, but it was not until 2010 that a study by Bergot established a method of developing correct higher order basis functions for pyramid elements. This paper analyses the performance of first and second-order pyramid elements created using the Bergot method within continuous and discontinuous finite element discretisations of the neutron diffusion equation. These elements are then analysed for their performance using a number of reactor physics benchmarks. The accuracy of solutions using pyramid elements both alone and in a mixed element mesh is shown to be similar to that of meshes using the more standard element types. In addition, convergence rate analysis shows that, while problems discretized with pyramids do not converge as well as those with hexahedra, the pyramids display better convergence properties than tetrahedra.