Civil Engineering Reference
In-Depth Information
(a)
P
1
(b)
P
1
(c)
P
1
(d)
P
1
R
2
R
2
R
2
R
2
Q
2
Q
2
Q
2
Q
2
Q
1
Q
1
Q
1
Q
1
T
44
T
44
R
1
P
2
R
1
P
2
R
1
P
2
R
1
P
2
F ig ur e 6 . 51
Three configurations in transformations T
44
: (a) octahedron; (b) joining P
1
P
2
; (c) joining Q
1
Q
2
;
(d) joining R
1
R
2
.
involves the replacement of a ring of five tetrahedra around line segment IJ by six tetrahedra,
three on each side of the polygon P
1
P
2
P
3
P
4
P
5
made, respectively, with nodes I and J, as shown
in Figure 6.52a. There are five ways in dividing the polygon P
1
P
2
P
3
P
4
P
5
into three triangles
by selecting each vertex and its two opposite vertices in turn, as shown in Figure 6.52b. The
desired configuration is the one in which the sum of the diagonals is minimised.
There are other transformations of higher order involving more tetrahedral elements,
but only transformations T
23
, T
32
, T
44
and T
56
will be considered in the mesh optimisation,
which are relatively simple and effective in removing poorly shaped elements and rapidly
improve the quality of the mesh. The triangular faces in the tetrahedral mesh are examined
one by one; T
23
transformation is performed if the mean shape quality of the two original
elements is less than that of the three new elements. As for the other transformations T
32
, T
44
and T
56
, they are carried out on a line-by-line basis. The edges (line segments) of the mesh
are examined in turn, and the ring of tetrahedral elements around the edge is identified.
The edge will be skipped if there are more than five tetrahedra attached to it; otherwise, the
quality of the elements is evaluated, and appropriate transformations T
32
, T
44
or T
56
will be
performed according to the number of tetrahedra around the edge.
The preferred configurations for local transformations are those whose mean geometric
shape qualities are maximised. Although the resulting mesh would certainly be different,
the optimisation by local transformations can be done with respect to any valid shape mea-
sure μ (μ = γ, ρ or θ). A minimum shape quality, μ
min
, can also be imposed such that no ele-
ment whose μ value is less than μ
min
would be generated by local transformations to assure
the minimum quality of the elements. A single pass consists of scanning through all the
edges in the mesh, and the edges affected by a local transformation will be kept in a separate
list, which has to be updated throughout the transformation process. As the shape quality
of a mesh cannot be improved indefinitely by shifting from one configuration to another, at
a certain stage, no transformation could be done to further improve the quality of the mesh.
(a)
(b)
J
P
4
P
3
P
2
P
5
P
1
I
Figure 6.52
Transformations T
56
and five ways of forming six tetrahedra: (a) ring of five tetrahedra; (b) five
ways of dividing polygon P
1
P
2
P
3
P
4
P
5
.
