Civil Engineering Reference
In-Depth Information
5.5.2.4 No tetrahedron found on triangle J
1
J
2
J
3
As there is no guarantee that tetrahedral meshes exist for any volume bounded by discre-
tised surface with full conformity in geometry and topology for all the boundary triangles,
ADF meshing over 3D volumes may encounter, from time to time, difficulty in convergence.
A typical scenario is that no valid tetrahedral element can be formed with a frontal node or
an interior node on the selected triangular facet. In this case, a simple remedy is to gener-
ate additional interior nodes normal to the triangular facet. Let
n
be the unit normal at the
centroid M of triangular facet J
1
J
2
J
3
and h be the height of the ideal tetrahedron that could
be formed with the base triangle, as shown in Figure 5.67. The initial position of interior
node I is located at
I = M + h
n
The following are the steps to judge whether node I can be accepted:
1. Verify if tetrahedron J
1
J
2
J
3
I penetrates into the generation front Γ; if yes, lower the
height h by 10% and go back to step 1 to check again; else go to the next step.
2. Compute the λ value of tetrahedron J
1
J
2
J
3
I; if λ
I
= γ
I
ξ
I
η
I
ζ
I
is greater than some given
threshold hold value, point I will be accepted; else further lower the height h to see if
the situation improves.
In case no such interior node I exists, the last resort is to remove some tetrahedral ele-
ments in the vicinity of the troubled facet and reconstruct the local site with additional
interior nodes generated normal to the frontal facets. While there is no guarantee for success
in every case, fairly complicated practical 3D domains can be meshed by this strategy. The
way in placing interior nodes with respect to a base triangle is crucial in producing high-
quality tetrahedral meshes in the ADF scheme. Many strategies have been proposed in the
positioning of interior nodes to enhance stability and efficiency for ADF meshing (Lohner
and Parikh 1988; George and Seveno 1994; Borouchaki et al. 1996; Lohner 1997; Lohner
and Onate 1998; Frey et al. 1998; Zuo et al. 2005).
5.5.2.5 Check for intersections
In the 3D ADF meshing, one important step is to ensure that the tetrahedral element formed
with the base triangle J
1
J
2
J
3
is valid without penetrating into the generation front. To this
end, the intersection between tetrahedron J
1
J
2
J
3
P, P ∈ Σ ∪ Λ, and the generation front has
to be verified. The intersection check between a tetrahedron and the generation front can
be reduced to a series of intersection checks between a line segment and a triangular facet,
I
h
n
J
3
J
1
M
J
2
Figure 5.67
Forming a tetrahedron with an interior node I.
