Civil Engineering Reference
In-Depth Information
5.3.3.2.7 Step 7: Removing Steiner points on boundary faces
Similar to the removal of Steiner points from a boundary edge, the elimination of Steiner
points from a boundary face is achieved by relocating all the Steiner points towards the inte-
rior of the object. Again, we only have to focus on the tetrahedral elements on one side of the
boundary face. Let's consider face ABC whose geometry has been recovered by the introduc-
tion of four Steiner points, as shown in Figure 5.28a. Following the natural sequence of the
Steiner points, S
1
is raised by an amount δ normal to the face to create tetrahedron ABCS
1
, as
shown in Figure 5.28b. By lifting S
1
, S
2
and S
4
are also raised on the triangular face BCS
1
,
and S
3
is raised on face CAS
1
. When S
2
is lifted, tetrahedra BCS
1
S
2
and CS
3
S
1
S
2
are created,
and S
4
is further raised on the face CS
3
S
2
, as shown in Figure 5.28c. Finally, S
3
and S
4
are
lifted at the same time to create tetrahedra CAS
1
S
3
and CS
3
S
2
S
4
, as shown in Figure 5.28d.
There is a possibility to improve the quality of the tetrahedral elements (Liu et al. 2014).
The Steiner points are to be lifted following the order based on a neighbouring coefficient,
which is equal to the number of adjacent Steiner points. Take the above face recovery with
four Steiner points as an example; the neighbouring coefficients of the Steiner points are,
respectively, N(S
1
) = 2, N(S
2
) = 3, N(S
3
) = 3 and N(S
4
) = 2. Accordingly, Steiner points S
1
and
S
4
are lifted up first to create tetrahedra AS
2
S
3
S
1
, ABS
2
S
1
and CS
3
S
2
S
4
, as shown in
Figure
5.29a. The Steiner points S
3
and S
2
remaining on the face ABC can now be lifted to produce
tetrahedra CAS
2
S
3
and CABS
2
in turn, as shown in Figure 5.29b. Similar to the edge recov-
ery, the lifting of Steiner points off a face can be converted into a mesh optimisation problem
C
C
S
4
B
S
2
B
S
4
S
2
S
3
S
3
S
1
S
1
(a)
(b)
A
A
S
4
C
C
S
2
S
2
S
3
S
4
B
B
S
3
S
1
S
1
(d)
(c)
A
A
Figure 5.28
Lifting Steiner points off a boundary face: (a) Steiner points on triangle ABC; (b) point S
1
lifted;
(c) point S
2
lifted; (d) points S
3
and S
4
are lifted.
C
C
S
4
B
B
S
2
S
3
S
3
S
2
S
1
(b)
(a)
A
A
Figure 5.29
Steiner points prioritised with neighbouring coefficient: (a) points S
1
and S
4
are lifted;
(b) points S
2
and S
3
are lifted.
