Civil Engineering Reference
In-Depth Information
an indirect approach based on a triangulated surface (Merhof et al. 2007). Discrete fracture
modelling by unstructured triangular mesh was generated by the Delaunay-AFT with data
points being stored in a uniform grid (Sahimi et al. 2010). An Octree partition for parallel
meshing rather than to facilitate local searching has also been proposed (Lohner 2001). The
Octree subdivision has to be updated as the generation front progresses, and the sizes of the
boxes and the smallest element have to be carefully controlled to ensure that the elements
do not protrude into other zones. Octree as a background grid can also be used to discern
boundaries of different material types, and tetrahedral elements are generated at the bound-
ary between materials (Zhang et al. 2010). From this brief survey of recent publications on
the use of background grids, it is clear that most of the applications are made to provide
support for some geometrical issues, and very often the grid will turn into a mesh or part
of the final mesh after some local modifications. The Delaunay-AFT has been developed to
generate high-quality tetrahedral meshes (Frey et al. 1998) and anisotropic meshes speci-
fied by a general metric for computational fluid dynamics (Borouchaki et al. 1997b). In the
Delaunay-ADF meshing, the DT, which has been generated earlier, in fact, serves as an
unstructured background grid (control space) to generate interior points by the AFT. A grid
is often employed to reduce the CPU time for searching and matching of geometrical quanti-
ties; however, it is usually used in a static manner under the condition that the position of
every item is known
a priori
.
After a brief introduction of the basic idea of AFT, the procedure of ADF meshing will be
presented in 10 steps. In Section 3.6.2, the generation of adaptive mesh governed by a speci-
fied nodal spacing by the AFT will be discussed. The implementation details for each step of
the AFT will also be given so as to explain when and why a search on the generation front
would be needed. In order to speed up the check for intersections between the proposed ele-
ment and the generation front, which is perhaps the weakest part in the ADF approach, the
idea of using a dynamic grid is presented in Section 3.6.3. The setting of the background
grid is to facilitate a search on the generation front for possible intersections in which details
will be given on how the size of an individual cell in the partition can be determined, and
formulas and procedure as to how segments are stored and deleted from intersecting cells
will also be given. The dynamic grid has been designed to deal with an unknown number of
quantities assigned to a cell, and those items could be added or deleted from time to time in
a continuous manner throughout the MG process.
Test examples are given in Section 3.6.4 in which two series of meshes are generated with
full statistics on their characteristics depicted in Tables 3.1 and 3.2 to show the characteris-
tics of meshes generated by the AFT. Two examples on adaptive refinement meshing are also
provided to assess the performance of the dynamic grid in practical applications. Finally,
closing remarks and discussions are presented in which a brief note on how a straightfor-
ward extension of the idea to three dimensions is also included.
3.6.2 Adaptive meshing by the AFT
The MG problem is completely defined by the boundary segments
Γ
, which is derived from
the discretised boundary
B
of domain
Ω
, along with the node spacing function ρ, which
governs the size of the elements to be generated. By virtue of the counter-clockwise order
assigned to the nodes on the exterior boundary and clockwise order to the nodes on the
internal boundaries, the domain to be meshed always situates to the left of a boundary
segment. Following this convention, the list of segment in
Γ
need not follow any sequential
or particular order as long as the segments are oriented correctly as exterior and interior
boundary segments. This flexibility allows boundary segments to be entered and prepared
independently in the collection of domain boundary edges for MG.
