Introduction - Finite Element Mesh Generation - page 8

(recovery) will also be interesting research topics as the scale and complexity of practical
scientific computations and engineering problems are ever increasing.
1.9 TOPICS DISCUSSED IN THE CHAPTERS
Following this chapter, Chapter 2 presents the fundamentals in finite element mesh genera-
tion. Notations, symbols and abbreviations used in this text will first be listed out in Section
2.2, and terminologies and data structures pertinent to the finite element mesh generation
are elaborated in Section 2.3. Geometrical operations and formulas are given in Section 2.4,
whereas various topological operations and algorithms are provided in Section 2.5. Popular
data-sorting methods such as bubble sort , insertion sort , quick sort and bin sort are all
described and compared in Section 2.6. Background grids, namely, regular/irregular grids,
Quadtree/Octree grids and kd-tree partitions as effective means to speed up searching and
matching of various geometrical quantities are discussed with examples in Section 2.7.
Methods for finite element mesh generation are formally introduced in Chapter 3 in which
various 2D mesh generation algorithms are presented. Following an introduction in Section
3.1, structured and unstructured meshes on planar domain are described, respectively, in
Sections 3.2 and 3.3. Meshing by Quadtree decomposition is discussed in Section 3.4, and
Delaunay triangulation over 2D domain is presented in Section 3.5. The ADF approach
and its extension to combine with Delaunay triangulation will be explored, respectively,
in Sections 3.6 and 3.7. As the Quadtree method may not be very effective in handling
irregular boundaries, an enhanced scheme coupled with advancing-front technique (AFT) is
proposed in Section 3.8, and finally, generation of quadrilateral meshes on a planar domain
is presented with details and examples in Section 3.9.
Mesh generation on curved surfaces is discussed in Chapter 4. The parametric mapping
method, surface curvatures and metric tensor specifications and mesh generation by the
Delaunay-ADF scheme are described in Section 4.2. Mesh generation by packing of ellipse
following an anisotropic curved surface metric and direct mesh generation on analytical
curved surfaces are presented, respectively, in Sections 4.3 and 4.4. Mesh generation by
means of a mesh-merging process through surface intersections is introduced in Section
4.5, and a brief account on the generation of quadrilateral meshes by schematic merging of
triangles is given in Section 4.6.
Finite element mesh generation over three dimensions will be explored in Chapter 5. A
detailed algorithm of Delaunay triangulation by a point inserted in 3D is described in Section
5.2. Boundary recovery procedures to achieve fully constrained Delaunay triangulations are
discussed in Section 5.3, whereas boundary-protection techniques for geometry-conforming
meshes are presented in Section 5.4. Classical ADF approach along with programming
details are given in Section 5.5, and its extension to Delaunay-ADF meshing in 3D can
be found in Section 5.6. Similar to ellipse packing in 2D, sphere packing in 3D as a means
of generating tetrahedral meshes of variable element sizes is discussed in Section 5.7. The
chapter ends with the introduction of various methods in Section 5.8 for the generation of
structured and unstructured hexahedral meshes.
Chapter 6 is about mesh optimisation in which geometrical and topological operations
for the enhancement of 2D and 3D finite element meshes are presented. In order to have an
objective view apart from aesthetic judgements, various shape measures for simplices are
discussed in Section 6.2. Mesh optimisation by means of shifting of nodes and topological
operations such as face/edge swaps are described, respectively, in Sections 6.3 and 6.4.
Mesh generation by means of concurrent parallel processing is explored in Chapter 7.
Before the development of any parallel meshing algorithms, the fundamentals and strategies