Civil Engineering Reference
In-Depth Information
the node elimination process, we have to consider the shape quality of the elements and the
length conformity of all the edges before and after the transformation. Edges shorter than
a certain threshold are identified. For each short edge, the surrounding polyhedron, which
consists of all the tetrahedra connected to it, is determined. The short edge can be collapsed
if there is a point on the edge that is visible to all the faces of the surrounding polyhedron, as
shown in Figure 6.48. A small element can also be eliminated if there is a point within the
element that is visible to all the faces of the bounding polyhedron, as shown in Figure 6.49.
There are many possibilities of face/edge swap or local remeshing (Misztal et al. 2009) for
the optimisation of tetrahedral meshes. The most well-known operations are the transfor-
mations T
23
and T
32
, which swap two tetrahedra for three tetrahedra and vice versa within
the same space. Let P
1
P
2
P
3
J and P
3
P
2
P
1
I be two tetrahedra sharing common face P
1
P
2
P
3
with
nodes I and J on opposite sides of face P
1
P
2
P
3
. If segment IJ intersects the interior of tri-
angle P
1
P
2
P
3
, the convex hull of the two tetrahedra can be transformed by T
23
into another
configuration consisting of three tetrahedra IJP
1
P
2
, IJP
2
P
3
and IJP
3
P
1
, as shown in Figure
6.50. T
32
is the inverse transformation of T
23
. When a line segment IJ in a mesh is shared by
exactly three tetrahedra IJP
1
P
2
, IJP
2
P
3
and IJP
3
P
1
, the ring of these three tetrahedra can be
transformed into a second configuration consisting of two tetrahedra P
1
P
2
P
3
J and P
3
P
2
P
1
I.
Consider a polyhedron formed by putting two pyramids back to back to each other, as
shown in Figure 6.51a. Owing to the symmetry in topology along the three axes, there are
three ways in dividing the octahedron into four tetrahedra by introducing a diagonal joining
any pair of opposite nodes, namely, P
1
P
2
, Q
1
Q
2
or R
1
R
2
, as shown in Figure 6.51b-d. The
transformation T
44
converts these configurations from one to another. Transformation T
56
Figure 6.48
A short edge shrinks to a point.
Figure 6.49
A small element shrinks to a point.
J
J
T
23
P
3
P
3
P
2
P
2
T
32
P
1
P
1
I
I
Figure 6.50
Transformation of tetrahedral elements T
23
and T
32
.
