Tech Report CS-00-03
Using Pyramids in Mixed Meshes - Point Placement and Basis Functions
Vasiliki Chatzi and Franco P. Preparata
For mixed 3D finite element meshes (meshes that contain cubes and tetrahedra) pyramids are very convenient because they have both square and triangular faces and can therefore be used in implementing conformal transitions between regions that contain cubes and regions that contain tetrahedra. However, pyramids are rarely used, because the specification of basis functions for them is known to be problematic. Of specific importance it the development of Lagrangian basis functions for pyramids, since Lagrangian standard elements are the most commonly used type of elements today.
In this paper, we present basis functions for pyramidal finite elements that can be used in Lagrangian meshes. The functions are ratios of polynomials, but reduce to polynomials on the boundaries of the shape. We also prove that polynomial basis functions for Lagrangian pyramids do not exist, and therefore the basis functions shown here are the simplest ones possible. We describe in detail a convenient and elegant method to obtain the basis functions for the order-k pyramid (for any k>0), which uses the functions for the order-(k-1) pyramid. A valid set of basis functions for the order-1 pyramid (which can be used as the basis of the construction) are also presented.