Civil Engineering Reference
In-Depth Information
Intersections between two meshed objects or their boundary surfaces are intrinsic to their
geometry and topology, even though their determination is based entirely on geometrical
computations between line segments and facets. The topological features of intersection thus
would not be affected by the method of its determination, or by a change of co-ordinates,
or by any continuous mapping such as bending, twisting and scaling. As a result, in the
determination of intersection between two objects, it may not be a good idea to take inter-
section as discrete information of intersection points and line segments, as finite precision
arithmetic will often produce inconsistent results. Instead, we have to focus on the pattern
and form of intersection as a whole with reference to the geometrical characteristics for their
determination. Hence, for the intersection between discretised curved surfaces, we aim at
getting surface patches bounded by loops of intersection segments, and for the intersection
between two solid objects, we would like to see volumes or regions of intersections bounded
by surface patches derived from the boundary of the two objects under consideration. In
summary, the boundary of the regions of intersection is composed of surface patches of
intersection, whose boundary, in turn, is given by the intersection loops, and that's why
intersection loops are fundamental to the mesh-merging process.
The following are the steps of the mesh-merging algorithm:
1. Determine all the intersection loops between the boundary surfaces of the given
meshed solid objects by means of the neighbour-tracing method introduced in Section
4.5.4.
2. Perform 2D triangulation on all the intersected triangular facets on the boundary so
that intersection segments are incorporated on the boundary surfaces of the given
objects.
3. Each tetrahedral element with a triangulated boundary facet is divided into as many
tetrahedra as the number of triangles on the triangulated face. As all the sub-triangles
on the triangulated face are visible to the opposite node, this subdivision exists and is
simple to carry out.
4. As each intersection segment is present as an edge of a tetrahedron on the boundary
surfaces of the given objects, the intersection loops will partition the boundary surface
of the objects into a number of zones.
5. Regions or volumes of intersection could be identified and defined by collecting
bounding surfaces from intersection loops and surface patches of the boundary sur-
face partition.
6. Incompatible tetrahedra intersected by the bounding surfaces of all the regions of
intersection are removed, and mesh compatibility is restored by filling up the void with
new tetrahedra between the bounding surfaces and the cavities so created.
7. All the regions of intersection common to both objects can now be detached freely
from the solid meshes, and the two solid meshes are now ready to be merged with full
compatibility at all intersecting surfaces.
8.6.2.1 Intersection of boundary surfaces
Let Ω and be the given 3D solid objects discretised into tetrahedral meshes, simply or
multi-connected with or without
in
ternal openings. In a broader sense and within the scope
of the meshing algorithm, Ω and could be extended to include a collection of solid objects,
even th
ou
gh it is general enough to c
on
sider the special case of a single object in the two sets
Ω and
. The boundaries of Ω and
are closed surfaces denoted, respectively, by ∂Ω and
