Civil Engineering Reference
In-Depth Information
Mesh generation on planar domain
A plane is flat, Euclidean and everywhere visible, so it ought not to be very hard.
3.1 INTRODUCTION
Finite element (FE) mesh generation (MG) probably started its journey from the discretisa-
tion of a bounded region on a planar domain. Many MG algorithms, including the mapping
method, Delaunay triangulation (DT), Quadtree decomposition, advancing-front approach,
etc., had their ideas and development found first on two dimensions. This is a long chapter
as all the basic concepts and ideas in FE MG will be fully explained and illustrated with
examples. Following the classification in Section 2.3.5, the MG schemes developed on 2D
domains can be classified into structured and unstructured meshing methods.
Transformation by FE interpolation and transfinite mapping, drag and sweeping meth-
ods are the common techniques employed at the very early stage in producing structured
FE meshes, which are still useful nowadays to generate regular meshes. These methods can
also be generalised quite naturally into higher dimensions following a similar procedure,
and a short description of these methods can be found in Section 3.2. Section 3.3 presents
unstructured meshing methods developed since the early 1970s; many MG schemes have
been proposed among which the DT, AFT and MG using contour, coring method, Quadtree
and mesh refinement have all been proved to be effective schemes for MG on 2D.
Quadtree decomposition as a spatial partition scheme takes up a unique place in the
generation of quadrilateral elements in compliance with a node spacing function. In fact,
the Quadtree method to be presented in Section 3.4 is apt in generating one-level refine-
ment meshes for which efficient transition quadrilateral elements can be applied for adaptive
refinement analysis. As the boundary treatment is the major weakness of Quadtree meshing,
an enhanced version, in which triangular elements are generated by AFT in filling up the
gap between the Quadtree mesh and the domain boundary, will be presented in Section 3.8.
The DT and the AFT are perhaps the most popular for the generation of unstructured
meshes on planar domains, and a comprehensive account of these two methods will be given,
respectively, in Sections 3.5 and 3.6. The Delaunay-AFT, which is the resulting scheme by
combining these two popular methods, will also be presented in Section 3.7. Triangular and
quadrilateral elements are the 2D FEs employed in FE MG. While it is straightforward to
generate regular quadrilateral meshes by structured meshing, unstructured adaptive quadri-
lateral meshes can be generated through an indirect process by converting triangular meshes
generated by the DT or AFT. Such an indirect approach for the generation of unstructured
quadrilateral meshes is presented in Section 3.9.
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