Unformatted text preview: ond the original mesh is not allowed. The process
is complex. It must be done without introducing irregular nodes. Suppose, for example,
that the quartet of small elements (shown with dashed lines) in the center of the mesh of
Figure 8.2.8 were scheduled for removal. Their direct removal would create three irregular
nodes on the edges of the parent triangle. Thus, we would have to determine if removal
of the elements containing these irregular nodes is justi ed based on error-indication
information. If so, the mesh would be coarsened to the one shown in Figure 8.2.11.
Notice that the coarsened mesh of Figure 8.2.11 di ers from mesh of Figure 8.2.7 that
was re ned to create the mesh of Figure 8.2.8. Hence, re nement and coarsening may
not be reversible operations because of their independent treatment of irregular nodes.
Coarsening may be done without a tree structure. Shephard et al. 38] describe an
\edge collapsing" procedure where the vertex at one end of an element edge is \collapsed"
onto the one at the other end. Ai a 2] describes a two-dimensional variant of this
procedure which we reproduce here. Let P be the polygonal region composed of the union
of elements sharing Vertex V0 (Figure 8.2.12). Let V1 V2 : : : Vk denote the vertices on the
k triangles containing V0 and suppose that error indicators reveal that these elements may 10 Adaptive Finite Element Techniques Figure 8.2.11: Coarsening of a quartet of elements shown with dashed lines in Figure
8.2.8 and the removal of surrounding elements to avoid irregular nodes.
V4 V4 V3 V3 V0
V5 V1 V1 Figure 8.2.12: Coarsening of a polygonal region (left) by collapsing Vertex V0 onto V1
be coarsened. The strategy of collapsing V0 onto one of the vertices Vj , j = 1 2 : : : k, is
done by deleting all edges connected to V0 and then re-triangulating P by connecting Vj
to the other vertices of P (cf. the right of Figure 8.2.12). Vertex V0 is called the collapsed
vertex and Vj is called the target vertex.
Collapsing has to be evaluated for topological compatibility and geometric validity
before it is performed. Checking for geometric validity prevents situations like the one
shown in Figure 8.2.13 from happening. A collapse is topologically incompatible when,
e.g., V0 is on @ and the target vertex Vj is within . Assuming that V0 can be collapsed,
the target vertex is chosen to be the one that maximizes the minimum angle of the
resulting re-triangulation of P . Ai a 2] does no collapsing when the smallest angle that
would be produced by collapsing is smaller than a prescribed minimum angle. This might
result in a mesh that is ner than needed for the speci ed accuracy. In this case, the
minimum angle restriction could be waived when V0 has been scheduled for coarsening
more than a prescribed number of times. Suppose that the edges h1e h2e h3e of an 8.2. h-Re nement 11 element e are indexed such that h1e
e may be calculated as h2e h3e, then the smallest angle
sin 1e where Ae is the area of Element e. of Element = h2Ae
h 2e 3e V4 V4 V5 1e V5
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