The Pyramid  The Elements  The Quadrilateral

The Tetrahedron 

The natural coordinate systems will be given in terms of the ordered triplet ${\bf x} = (x,y,z)$. The coordinate system of the reference elements will be given in terms of (r,s,t). Finally the local tensor space for each element will be given in terms of (a,b,c) coordinates.

The reference tetrahedron is described as the set of points:

$$Te_{ref} = \{(r,s,t) \vert -1 \leq r,s,t \leq 1; \hspace{8pt} r+s+t \leq 1\}$$

The reference tetrahedron is mapped to a straight-sided physical tetrahedron with the following mapping:

$${\bf x} = -\frac{(1+r+s+t)}{2} {\bf v^1} + \frac{(1+r)}{2} ... ... v^2} + \frac{(1+s)}{2} {\bf v^3} + \frac{(1+t)}{2} {\bf v^4}$$

where the ${\bf v^1}$,${\bf v^2}$,${\bf v^3}$ and ${\bf v^4}$ are the physical coordinates of the vertices of the tetahedron.

The tensor element is the set of points:

$$Q_{tensor}^3 = \{(a,b,c) \vert -1 \leq a,b,c \leq 1\}$$

The tensor element is mapped to the reference tetrahedron by:

We notice that this mapping is singular at b=1 and c=1.
 
 

 

Figure 2.3: The tensor coordinates of the tetrahedron.

 

10/24/1998