Civil Engineering Reference
In-Depth Information
Quadtree subdivision can be used to produce triangles with better shape quality without caus-
ing numerical difficulties in the Delaunay insertion, as shown in Figure 3.36b.
3.5.6.5.2.3 RANDOM POINT GENERATION
Elements of uniform size will be generated using grid points of a regular grid, and elements of
variable size can be created by a recursive subdivision of a rectangular or a triangular grid, as
discussed in Section 3.5.6.5.2.2. MG by means of a random point was proposed many years
ago by Fukuda and Suhara (1972) and Cavendish (1974) to generate triangles of roughly equal
size but of rather arbitrary shapes. However, to generate elements of variable size using a regu-
lar grid without recursive spatial partition, the technique of random point generation can also
be employed. Points are generated by a random process within a cell, which will be rejected if
they are too close to the existing points within the cell or with respect to points of the adjacent
cells. Each cell of the grid will be considered in turn, and the point insertion process will pass
on to the next cell when an expected number of points have been generated. When all the cells
of the grid have been considered, the domain will be filled up with randomly generated points
in compliance with the element size requirement in a statistical sense. A grid with recursive
spatial partition can also be used in conjunction with the random generation process to pro-
duce elements with size in better compliance with the specified node spacing.
3.5.6.5.2.4 'VARIOGRAM' - DISTANCE FROM BOUNDARY
For a planar domain bounded by line segments, points are inserted one by one within the
given domain. An inserted point has to be located at a position that is farthest away from
all the existing points, including the boundary points and the previously inserted points
(Tacher and Parriaux 1996). In the actual implementation, a background grid with grid-
point spacing compatible with the smallest element is prepared. The distance of each grid
point from the boundary is computed by means of an induction formula based on the pre-
vious grid point. A
variogram
is formed by listing the distance of all the grid points in an
ascending order of magnitude. Points are accepted as long as the distance to the boundary
is greater than some prescribed threshold value.
3.5.6.5.3 Other techniques
Apart from the refinement methods and those based on a background grid, many other tech-
niques have been proposed to generate interior points within a given domain. An exhaustive
account on all these methods is quite impossible and impractical; apart from the packing of
circles, ellipses and spheres (Lo and Wang 2005b,c,d), two more methods that make use of
entirely different concepts in point creation are included for a detailed discussion in Sections
3.5.6.5.3.1 and 3.5.6.5.3.2.
3.5.6.5.3.1 CONTOUR LINE
Right from the beginning in the development of FE MG, two major trends were the struc-
tured and unstructured meshes. While various mapping methods were available for the
generation of structured mesh, the generation of unstructured mesh was still at the infant
stage. Apart from the mesh subdivision techniques (Thacker 1980), an early attempt for
the automatic generation of unstructured mesh is by means of random point generation
(Fukuda and Suhara 1972; Cavendish 1974), which was later modified by Shaw and Pitchen
(1978) to generate nodes in a more systematic manner over rectangular cells. When the AFT
