Page 228 - Basic Structured Grid Generation

Unstructured grid generation 217

A B A B
e 1
e 2
C C
D D

Fig. 8.41 Diagonal swapping.

A A

C
C B B
D D

Fig. 8.42 No diagonal swapping in this case.

Diagonal swapping
This procedure, already referred to in Section 8.2.1, does not change the position of
the nodes, but may change their connectivities. A loop (i.e. a DO loop) over all the
triangle sides, excluding those elements on the boundary, is set up, in which for a
typical side such as AC in Fig. 8.41, which is common to the two triangles ABC,
ACD, the possibility of replacing AC by BD is considered. This diagonal swapping
is performed if the resulting configuration, consisting of the two triangles ABD and
BCD, is preferable according to some specified criterion. One such criterion would be
that the minimum angle in the two triangles in the changed configuration is larger than
the one in the original one.
No diagonal swapping is carried out if the original triangles compose a quadrilateral
which is not convex (Fig. 8.42).

Grid smoothing
In this procedure the positions of interior nodes are altered without changing any con-
nectivities. The idea is to regard the triangle sides as linear springs with identical
stiffnesses and tensions proportional to the lengths of the springs. The overall equilib-
rium position of the nodes is then sought by iteration. In each iteration a loop over
the interior nodes is carried out in which each node is moved to the centroid of the
three nodes to which it is connected. Between three and five iterations are generally
required to arrive at a satisfactory smoothed grid.

8.4 Solving hosted equations using finite elements

Here we present a very brief introduction to the solution of field equations (the hosted
equations) in a domain which has been triangulated, using linear triangular elements.