Civil Engineering Reference
In-Depth Information
involved for the identification and the definition of the regions of the intersection are purely
topological, which could be crucial to enhancing the stability of the entire merging process.
A generic algorithm will be presented in Section 8.6.2 to merge arbitrary solid tetrahedral
meshes automatically into one single valid finite element mesh (Lo 2013c). The intersec-
tion segments in the form of distinct non-overlapping loops between the boundary surfaces
of the given solid objects are determined by the neighbour-tracing technique discussed in
Section 4.5.4. Each intersected triangle on the boundary surface will be triangulated to
incorporate the intersection segments onto the boundary surface of the objects. Tetrahedra
on the boundary surface with intersected triangular facets as one of the faces are each
divided into as many tetrahedra as the number of sub-triangles on the triangulated face.
There is a natural partition of the boundary surfaces of the solid objects by the intersection
loops into a number of zones. Volumes of intersection can now be identified by the collected
bounding surfaces from the patches of the surface partition. While mesh compatibility has
already been established on the boundary of the solid objects, mesh compatibility has yet
to be restored on the bounding surfaces of the regions of intersection. Tetrahedral elements
intersected by the cut surfaces are removed, and new tetrahedra can be generated to fill the
resulting cavity to restore mesh compatibility at the cut surfaces. Upon restoring mesh com-
patibility over volumes of intersection, the objects are ready to be combined as all regions of
intersection can now be detached freely from the objects. All operations, besides the deter-
mination of intersection loops, are entirely topological, and no parameter and tolerance is
needed in the entire merging process. The examples of various characteristics are presented
to show the steps and the performance of the mesh-merging procedure.
8.6.2 Algorithm: Merging tetrahedral mesh
Problem statement:
Given two 3D objects Ω and , each of which is a valid finite element
mesh of tetrahedral elements such that Ω = {T
i
, i = 1, NT}
T
} and
=
i
1
where
N
T
and
N
T
are, respectively, the number of tetrahedral elements in Ω and
.
The merging
problem is to find the union of the two given meshes in
to
a single valid finite element mesh
ˆ
consisting of
f
only tetrahedral elements, i.e.
ˆ
{,
Ti
=
,
N
},
{
ˆ
=∪=
Ti
,
=
1
,
N
}
, such that points
ˆ
i
T
.
are found in
ˆ
as well.
found in Ω or
ˆ
∀∈
i
{, ,
12
N
}
andj
∀∈
{, ,
12
NxTorx
},
∈
∈
Tx
∈
TTk
∈{, ,
12
N
}
ˆ
T
T
i
j
k
T
The objects to be dealt with are very general as long as they are valid finite element meshes,
and there is no restriction on the size, shape and the topology of the objects, which could be
convex, concave, simply or multi-connected, etc. The algorithm proposed can even handle
objects that are disjoint; however, without loss of generality, we simply assume that objects
are connected pieces, since Boolean operations can be applied sequentially between different
intersecting objects to form ever more complex objects. To merge two objects together, it is
necessary to consider their intersections, which by definition are the parts common to both
objects. Hence, the union of the two objects/meshes is given by the objects putting together
subtracting the parts in common from either one of the objects. For a robust treatment of the
merging of solid objects, the nature and the fundamentals of intersection have to be care-
fully reviewed. The boundary of a meshed 3D object is a closed discretised surface of trian-
gular facets. The intersections between two objects are regions common to the two objects
bounded by boundary surface parts from the two objects. In other words, the intersection
between two solid objects can be fully determined by considering solely the boundary sur-
faces of the two objects. This idea is crucial to the mesh-merging algorithm to be presented.
