The Tetrahedron  The Elements  The Triangle

The Quadrilateral 

   

Figure 2.2: The tensor coordinates of the quadrilateral.

 

Next we describe the quadrilateral. The reference quadrilateral, shown in figure 2.2, is mapped to a straight-sided quadrilateral by the following:

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

Again ${\bf v^1}$ is an arbitrary vertex of the quadrilateral and ${\bf v^2}$,${\bf v^3}$,${\bf v^4}$ are the vertices labeled counter-clockwise from this vertex. The Jacobian for this mapping is:

$$\frac{\partial(x,y)}{\partial(r,s)}= \frac{1}{2}\vert \frac... ...f v^4}-{\bf v^1})}{2}\times\frac{({\bf v^2}-{\bf v^3})}{2}\vert$$

This Jacobian includes three terms. If the quadrilateral is rectangular then the last two terms are zero and the first term is just the ratio of the area of the quadrilateral and the reference quadrilateral. If the quadrilateral is not rectangular then the last two terms are proportional to the sine of the angles between the two sets of edges: r=+1,-1 and s=+1,-1. The reference quadrilateral and the tensor element coincide so the mapping between (a,b) and (r,s) is the identity mapping. Thus, $\frac{\partial(r,s)}{\partial(a,b)}$ is identically one. When we compare the overall Jacobian of the mapping from physical element to tensor element we see that the triangle Jacobian is dependent on b whereas the quadrilateral Jacobian is both a function of a and b. The reference quadrilateral is the set of points $Qu_{ref} = \{(r,s) \vert-1 \leq r,s \leq 1\}$ like the reference element for the triangle.

We next consider a set of elements commonly used to discretize a three-dimensional volume. There are now four elements: the tetrahedron, the pyramid, the prism and the hexahedron. Again we will consider the physical, reference and tensor representations for each element which were proposed in [27].

10/24/1998