Civil Engineering Reference
In-Depth Information
Figure 8.77
Boundary surface refined along three orthogonal planes.
8.4.3 3D refinement in compliance with a
specified node-spacing function
Adaptive unstructured refinement is basically applied only to simplicial meshes, though
hexahedral meshes can also be refined using transition elements, as discussed in Section
5.8.15. As the 8T subdivision is less flexible, the quality of the resulting mesh is very often
inferior to that of the initial mesh. A simple and efficient refinement procedure for tetrahe-
dral meshes based on successive bisection of edges is presented in this section (Laug and
Borouchaki 2003a). The quality of the elements produced can be optimised if the subdivi-
sion is performed in the sequence according to the length of the edges to be divided. Such
an order of priority can be established by a simple sorting process on all the edges for which
refinement are needed. This list of ordered segments has to be updated from time to time to
take into account the new edges created during the subdivision process. As the ring of tetra-
hedra around the LE are all refined at the same time, mesh conformity is always maintained
at each refinement step, which, in general, is the most crucial part for a refinement algo-
rithm. The shape quality of the mesh can be further improved by the standard optimisation
procedures discussed in Chapter 6 during and at the end of mesh refinement.
8.4.3.1 The algorithm
Mesh refinement can be achieved by introducing a new node in a tetrahedron's interior, face
or edge to divide it, respectively, into four, three and two tetrahedra. Insertion of a new node
on an edge is a better choice because the new tetrahedra produced are of better quality, and
the nodes can be placed everywhere in the mesh, both in the interior and on the external
faces. In this section, an algorithm is presented in which edges of the mesh and the associ-
ated tetrahedral elements are bisected until the specified refinement is reached.
The input for the procedure is a coarse mesh
M
of tetrahedral elements, which are defined
by the element vertices and the nodal co-ordinates.
Tetrahedral mesh
M
= {T
i
, i = 1,NT},
T
}, T
i
= {P
ij
, j = 1, 4},
{(x
i
, y
i
, z
i
), i = 1, 1,NT},
P
}
where N
P
and N
T
are, respectively, the number of points and of tetrahedral elements in
mesh
M
. The algorithm will refine the initial mesh
M
according to the given node spacing
