Civil Engineering Reference
In-Depth Information
(a)
(b)
Figure 3.72
Comparing (a) Q-Morph and (b) Lee and Lo's algorithm.
By the indirect method, quadrilateral meshes are generated from triangular meshes, and
thus, it can be very effective for irregular or multi-connect domains. With the advancement
in the techniques for element shape optimisation, fairly good quality quadrilateral elements
can be produced even for a rapid change of element size. Hence, apart from regular domains
where mapping methods can be applied to generate high-quality quadrilateral elements,
for irregular domain and adaptive FE analysis, indirect methods are commonly adopted to
generate the required quadrilateral meshes. In Sections 3.93 to 3.96, general considerations
to produce quadrilaterals from triangular meshes will be discussed, and the detailed proce-
dures to convert a triangular mesh into an all-quad mesh will also be presented.
3.9.3 Quadrilateral-dominated mesh
An algorithm is presented in this section to generate quad elements from a triangular mesh
by selectively removing the diagonals between adjacent triangles. The shape quality of a tri-
angular element is measured by the α-quality coefficient. As for quad elements, a distortion
coefficient β is introduced, with which the quality of the quad can be compared to those of
the two triangles arising from a cut along either diagonal. A parameter γ can be specified by
the user to give higher preference either to triangles or to quads. As a result, a careful selec-
tion of γ would lead to an optimised mixed mesh consisting of both triangular and quad
elements.
The generation of quadrilateral elements is based on the simple fact that a quadrilateral
will be produced whenever a diagonal between two triangles is removed. The quality of the
quadrilateral and hence the quality of the resulting FE mesh will depend highly upon the
way in which the diagonals are removed. In order to facilitate the mesh updating work for
each diagonal removal, prior to the removal of diagonals, some computations on the topol-
ogy of a triangular element mesh have to be done. These include (1) determination of edges
in the triangulation (Section 2.5.4), (2) determination of the neighbouring triangles (Section
2.5.1) and (3) for each triangle, determination of the three edges associated with it.
In order to determine the exact sequence in which diagonals are to be removed, one
should have some global measure of each edge of the mesh on the quality of the quadri-
lateral that it will generate and how the neighbouring edges are affected upon its removal.
A β value will be attached to each edge, which determines the quality of the quadrilateral
that will be generated as a result of its removal. A new β* value for each edge can then be
