Algorithm 1

Given a conforming discretization we can generate the local orientation of the tetrahedra using the following algorithm. We assume that we have both a list of vertices and a list of elements which touch each vertex. This list of elements will be called a vertex group and all elements are assumed to have a tag of zero.

For every vertex in the list:

• Orient all elements with a tag of one in this vertex group so that their local base vertex points at this vertex. Then set their tags to two.
• Orient all elements with a tag of zero in this vertex group so that their local top vertex points at this vertex. Then set their tags to one.

This algorithm visits all vertices in the mesh and if this is the first time the elements in the vertex group have been visited the local top vertex is orientated to this vertex. If this is the second time the elements in the vertex group have been visited then set the local base vertex to this vertex. To see how this works we can consider the example shown in figure 2.8.

  

Figure 2.8: Setting up the required connectivity for the discretization of a box as shown in a). Vertex A is given as the first local top vertex as shown in b). In c) vertex B is then given as the local base vertex and the local base vertices from group one are aligned to satisfy connectivity and the final element orientation is shown.

Here we assume that we are given a discretization of a box using six tetrahedra as shown in figure 2.8a. Starting our algorithm we begin with vertex A. Since all elements have a tag of zero at this point we go straight to the second part of the algorithm and orientate all elements that touch this vertex so that their local top vertices point to A. Therefore tetrahedra HBDA and BHEA are orientated as shown in figure 2.8b and now have a tag set to one. Continuing to the next vertex B we see that all elements belong to this vertex group. The first part of the algorithm is to orientate the elements with a tag of one to have their local base vertex pointing at B. So the tetrahedra HBDA and BHEA are rotated as shown in figure 2.8c and their tags are set to two. The second part of the algorithm then orientates all the other tetrahedra to have their local top vertex pointing at B. The connectivity is actually satisfied at this point since the orientation the faces have on the boundaries is irrelevant. However, if we continue the algorithm looping through the vertices consecutively we end up with the tetrahedra orientated as shown in figure 2.8c.

Clearly, the connectivity is not unique since any elements that have their local top vertex pointing at E can be rotated about E. We have demonstrated however that it is possible to satisfy the connectivity requirements imposed by the co-ordinate system and thereby imply that the requirement is non-restrictive.