Civil Engineering Reference
In-Depth Information
Mesh generation in three dimensions
The main challenge of FE mesh generation in 3D comes from the boundary constraints.
5.1 INTRODUCTION
From Chapters 3 and 4, it appears that automatic mesh generation (MG) has reached such
a mature stage that efficient algorithms are available to generate high-quality meshes in
compliance with the metric requirement on arbitrary 2D domains and over general curved
surfaces in a robust manner. However, when we look at the MG for 3D solid objects, we
immediately notice that the problem becomes much more complex, and many skills that
work pretty well in 2D simply cannot be extended to a higher dimension. In 2D, the boundary-
constrained MG is a deterministic process so that solutions are always guaranteed. On the
other hand, in 3D, constrained MG algorithms are more iterative in nature. The fundamen-
tal difference between meshing a 2D domain and a 3D domain is that a 2D boundary can
always be meshed without the need for additional nodes. However, there are geometries
in 3D that cannot be discretised without introducing interior points, and a twisted penta-
hedron is a well-known example. Since there is no systematic way to decide where points
should be introduced, analytical solutions are not available, leading to the development of
iterative algorithms of heuristic nature for specific applications.
The problem becomes even more complicated when meshes of variable element size are
required for isotropic and anisotropic metric specifications. Nevertheless, after years of
dedicated research, many practical algorithms for meshing 3D solid objects are quite reli-
able such that the integrity of the domain boundary can be preserved, unless extremely
poor boundary conditions are encountered, characterised by the presence of many elongated
facets with a large aspect ratio and sharp dihedral angles between adjacent faces. The two
popular techniques, namely, the Delaunay triangulation (DT) (George 1997; Johnson and
Tezduyar 1997) and the AFT (Lo 1991b; Lohner 1996c; Rassineux 1997), again play an
important role in 3D FE MG and will be discussed in detail in this chapter. Other tech-
niques such as the Octree method, the adaptive refinement, the medial surface method, the
plastering method, the whisker weaving method and the H-morph algorithm for the genera-
tion of hex meshes will be briefly described at the end of this chapter.
Over a 3D domain, it is necessary to employ different strategies for the MG of solid objects
and for the fluid mechanics problems (Baker 1997; Mavriplis 1997; Morgan and Peraire 1998;
Johnson and Tezduyar 1999). As stress and strain are likely to concentrate on the solid boundar-
ies and solid objects are often meshed by components, boundary surface conformity in terms of
both geometry and topology is strictly required leading to a (fully) constrained boundary MG
problem. On the other hand, for fluid mechanics problems, emphasis is more on the speed of
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