Civil Engineering Reference
In-Depth Information
5.3.3.2.8 Step 8: The final mesh within the boundary surface
The final mesh of the object consists of those tetrahedral elements within the boundary sur-
face. Indeed, this should be a trivial process, as a usual practice in MG by DT or Delaunay-
ADF approach is that elements inside and outside the domain or elements belonging and not
belonging to the object are always distinguished by giving different zonal labels. In case label-
ling has not been done, elements within the boundary surface can be conveniently determined
by the adjacency relationship, which is available in any DT. Just start with any seed element,
which will grow in size by attaching adjacent tetrahedral elements on all the non-boundary
faces (see
Algorithm RBR3D
in Section 2.5.11). This procedure will partition all the tetrahe-
dral elements in the mesh into two groups, one inside and the other outside the object sepa-
rated exactly by the domain boundary. If we have started with an element inside the object, we
will automatically get the FE mesh of the object; on the other hand, if we have chosen an ele-
ment outside the boundary, then we shall get all the elements outside the object by connecting
neighbouring elements. Always use topological criteria but not geometrical criteria in separat-
ing elements to avoid numerical error and enhance the overall robustness of the MG scheme.
5.3.4 Worked examples and industrial applications
A number of worked examples and industrial applications are presented in this section to
show the technical details in recovering boundary edges and faces. The results in attaching
priority to Steiner points according to the number of neighbouring Steiner points are studied
with an aeroplane bounded by 141,470 poorly shaped triangles with a lot of sharp angles.
Complicated industrial objects and biological models are also included to show the charac-
teristics and the statistics of the boundary recovery procedure in realistic applications.
5.3.4.1 Worked examples
Four numerical examples are presented to illustrate some key features of the boundary recov-
ery procedure. The first example is a mechanical support model shown in Figure 5.34a. One
hundred fifty-two boundary facets were missing in the initial DT. To recover these missing
boundary facets, 50 Steiner points were inserted including 43 points on the boundary edges.
The largest number of Steiner points inserted on a single edge was 5, as shown in Figure
5.34b and c. The missing edge is drawn in blue colour, and the five Steiner points are marked
with different colours. To remove these Steiner points, only two iterations were required
by the method of dividing the Steiner points into groups. As shown in Figure 5.34d, in the
first iteration, three
unlocked
points are first relocated; then the other two
locked
points are
repositioned in the second iteration. On the other hand, a random procedure would need
five iterations to remove all these five Steiner points as four out of five Steiner points were
locked
. As shown in Figure 5.34e, only one Steiner point could be relocated at a time.
The second example is a screw model shown in Figure 5.35a. In this model, 433 Steiner
points were introduced including 40 points on the boundary faces in which the largest num-
ber of Steiner points inserted on a single face was four. One face marked in blue with four
Steiner points is shown in Figure 5.35b and c, where Figure 5.35c is the mesh of Figure 5.35b
in a different view. To remove these Steiner points, Figure 5.35d depicts the process follow-
ing the suppression order optimisation (i.e. higher priority to points with a small number
of Steiner point neighbours). As expected, the random order required more iterations to
suppress Steiner points on the missing boundary face. As shown in Figure 5.35e, although
it is not the worst case (four iterations), it still needed one more iteration compared to the
prioritised sequence.
