The Prism  The Elements  The Tetrahedron

The Pyramid 

The reference pyramid is described as the set of points: $$Py_{ref} = \{(r,s,t) \vert -1 \leq r,s,t \leq 1; \hspace{8pt} r+t \leq 0; \hspace{8pt} s+t \leq 0 \}$$ The pyramid will be useful for making bridges between tetrahedra and hexahedra or between prisms and hexahedra. The reference pyramid is mapped to a straight-sided physical pyramid by:

The tensor element is mapped to the reference pyramid by:

Clearly, this mapping has a first order singularity in the r,t and s,t coordinates making it have a second-order singularity at r=s=-1,t=1. However, this is only a singularity due to the choice of coordinates and this singularity will be removed by the volume Jacobian in a volume integral. Also, we will use quadrature for the pyramid that does not include the top vertex, hence avoiding the calculation of derivatives at this point.
 
 

  

Figure 2.4: The tensor coordinates of the pyramid.

  

10/24/1998