Civil Engineering Reference
In-Depth Information
boundary is increasingly important (Joe 1992; Wright and Jack 1994; George et al. 2003;
Du and Wang 2004; Si and Gaertner 2010). However, while it is rapid and robust, DT of
spatial points, in general, will only define a convex hull of the given points. Hence, for
irregular non-convex objects or objects with interior boundaries, the DT has to be properly
modified so as to recover those missing boundary edges or faces. As the missing edges and
faces are not present in the original DT, they are not Delaunay and they violate the empty
sphere criterion with respect to the system of spatial points. As a result, strictly speaking, the
modified triangulation is no longer a DT, and this problem is also known as constrained DT.
In 2D, boundary edge recovery is quite easy as an edge joining any two points in the mesh
can always be recovered simply by the swap of diagonals, as elucidated in Section 3.5.7.
In 3D, the situation is quite different. First, not all the triangulations of the point set can
be reproduced by a series of valid element transformations of edge/face swaps (Joe 1993,
1995b), and second, there exists a twisted pentahedron for which a conforming triangula-
tion of the boundary points does not exist without introducing additional points, the so-
called Steiner points; in other words, a twisted pentahedron cannot be triangulated without
introducing an interior point. However, there is no formal rigorous proof that a fully con-
formable FE mesh exists for any general boundary surface, even though Steiner points are
allowed at the interior of the domain. Arguably, if tetrahedra of zero volumes (
sliver
) are
permitted, there is always a solution; indeed,
slivers
exist anyhow in a valid DT. In practice,
from the research over the last 20 years or so, we are now able to produce a topologically
and geometrically valid conforming FE mesh from the DT for boundary surfaces of rather
complex industrial objects or biological models with some flat elements of a very small vol-
ume at some critical sites. These flat elements can possibly be opened up to attain a finite
volume provided that relatively poorly shaped tetrahedral elements are still acceptable for
difficult boundary conditions.
In general, the boundary recovery approaches developed so far can be grouped into two
categories: (1) local mesh reconnection and (2) introduction of Steiner points. In the first
approach, no Steiner point will exist in the final tetrahedron mesh. However, there is no
guarantee for the success of such methods due to the existence of a twisted pentahedron in
Schönhardt configuration. In the second approach, Steiner points are introduced to assist
boundary recovery. Although they could recover the geometry and the topology of the miss-
ing quantities, how to open up the flat tetrahedral elements so created systematically to
ensure topological integrity is still an open issue. Nevertheless, effective heuristic schemes
were developed for practical difficult 3D domains, as elaborated in Section 5.3.3.
5.3.2 Boundary recovery by local mesh reconnection
As no Steiner points are inserted in the final mesh, approaches by local mesh reconnection
attract the interest of many researchers. Borouchaki et al.
(2000b) split the constrained
boundary by inserting Steiner points on edges and facets, and then suppressed the inserted
points by locally remeshing the tetrahedral elements linked to them. Nevertheless, local
remeshing could not guarantee a valid boundary-recovered topological structure even
though it exists, and very often, Steiner points remain on some of the boundary faces. Liu et
al. (2007) employed an exhaustive method named
small polyhedron reconnection
to achieve
the boundary recovery. For a
small
polyhedron with no more than 20 triangular facets, all
possible topological structures are evaluated to detect if the missing quantities could be
recovered in one of the configurations. Ghadyani et al. (2010) carved out a hole in the vicin-
ity of missing boundary facets, and the hole along with the missing boundary facets formed
a polyhedron, which is to be meshed by a method known as the LAST RESORT.
