Abstract
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.
Highlights
For 3D finite element problems the most commonly used element types are tetrahedra and hexahedra (Bathe, 1996; Dhatt et al, 2012)
Results are taken for the diffusion equation solved with both a continuous finite element (FEM) formulation and for discontinuous DG-FEM with an modified interior penalty (MIP) penalty scheme (Wang and Ragusa, 2010)
This paper used an established method for forming the basis functions of pyramid elements, developed by Bergot, with the aim of demonstrating their effectiveness in the solution of neutron diffusion problems in reactor physics
Summary
For 3D finite element problems the most commonly used element types are tetrahedra and hexahedra (Bathe, 1996; Dhatt et al, 2012). An example of the difference between the two may be understood by comparing tri-linear hexahedral elements to linear tetrahedral elements. The tri-linear hexahedral elements have coupling between the different parametric co-ordinates whereas the linear tetrahedral elements do not. This difference means that the tetrahadral elements are less accurate overall than hexahedral elements. While various robust mesh generation techniques, such as advancing front and Delaunay, exist to mesh complex geometrical domains with tetrahedral elements, no general technique exists for hexahedral elements (Frey and George, 2008), due to the geometrically stiff structure of hexahedra (Schneiders, 2000; Puso and Solberg, 2006). Often the only reliable and robust way of systematically generating a fully hexahedral mesh for a complex geometrical domain is to generate a tetrahedral mesh and split each tetrahedron into four hexahedra (García, 2002), a process which substantially increases the cost of generating the mesh
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