Civil Engineering Reference
In-Depth Information
5.8.8 Grid-based or voxel-based method
The grid-based or voxel-based approach presented by Schneiders (1997), Lee and Yang (2000),
Kim and Swan (2003a,b), Kaminsky et al. (2005), Teran et al. (2005) and Owen and Shepherd
(2009) involves the fitting of a 3D grid of hex elements into the interior of a volume, as shown
in Figure 5.100. More hex or tetrahedral elements have to be added to fill the gaps between the
regular grid and the boundary surface of the solid. Very much limited by the geometry of the
solid, poor elements of irregular shapes are almost inevitable in the boundary fitting process.
As shown in Figure 5.100a and b, the hex elements retained within the same ellipsoid are quite
different if the major principal axis of the ellipsoid is pointing at a different direction. Hex
elements are, in general, not in good alignment with the domain boundary, and the resulting
mesh is rather sensitive to the orientation and the positioning of the interior grid, which can,
however, be improved by the
marching cube
and other smoothing techniques (Samani et al.
2001; Boyd and Muller 2006; Labelle and Shewchuk 2006; Zhang et al. 2006, 2007; Young
et al. 2008). Owing to the use of a regular grid, the element size at the interior of the volume
will be approximately the same. Frey et al. (1994), Weiler and Schneiders (1996), Greaves and
Borthwick (1999), Schneiders (1997) and Zhang et al. (2005) have made modifications that
allow for significant change in the element size based on an Octree decomposition, which can
also be combined with DT (Schroeder and Shephard 1990; Jung and Lee 1993) or with AFT
to mesh regions with cracks (Neto et al. 2001). While the grid-based or voxel-based method is
less convenient for domains with prescribed discretised boundary, it is pretty effective to mesh
biomedical volumes bounded by smooth surfaces or scattered data points from MRI images.
5.8.9 Medial surface method
The
medial axis
of an object is the set of all points having more than one closest point
on the object's boundary (Gursoy and Patrikalakis 1992; Sherbrooke et al. 1996a,b). In 2D,
the medial axis of a planar object is given by the locus of the largest circle rolling along the
boundary of the object, as shown in Figure 5.101. As a direct extension of the medial axis
method for quad meshing, the domain is subdivided by a set of medial surfaces, which can
be thought of as surfaces generated from the mid-point of a maximal sphere allowed to roll
through the volume. The medial surface can also be constructed with the aid of a DT of the
boundary points of the solid object (Sheehy et al. 1996; Turkiyyah et al. 1997). The medial
Void to be filled by hexahedra or tetrahedra
Ellipsoid
(a)
(b)
Figure 5.100
Fitting hexahedral element within a bounding surface: (a) major principal axis normal to a face;
(b) major principal axis along a diagonal.
