Civil Engineering Reference
In-Depth Information
or hex elements. Along with the FE technology, there are significant advances in the
generation of tetrahedral and hex meshes (Cook 1974; Benzley et al. 1995; Schneiders
and Bunten 1995; Melander et al. 1997; Pavlakos et al. 1997; Tautges et al. 1997). Hex
elements are superior to tetrahedral elements in the following aspects: (i) high-quality
structured regular hex elements can be generated rapidly by means of mapping or trans-
formation; (ii) hex elements could be generated following the domain boundary in layers
and in alignment with important geometrical features for easy visualisation and inspec-
tion, whereas the validity of tetrahedral meshes can only be verified by a computer; and
(iii) low-order high-performance hybrid-stress hex elements are available (Sze 1992; Sze
and Ghali 1993a,b; Lee and Lo 1997a,b; Sze and Lo 1999; Sze et al. 2002; Ramos and
Simoes 2006), which are considered to be more computationally efficient than the tetra-
hedral element counterparts.
Hex meshes of various characteristics can be generated by different approaches as pro-
posed by researchers working in diverse scientific and engineering fields, and a compre-
hensive account of the current status and difficulties of hex meshing can be found in the
works of Shepherd and Johnson (2008) and Staten et al. (2010a,b). Tautges (2001b), on
the other hand, not only reviewed the hex meshing strategies but also laid down some
evaluation criteria for hex meshing. Automatic unstructured MG algorithms usually refer
to the generation of tetrahedral meshes, as mapping techniques based on regular grid
will, in general, give rise to structured meshes. While most of the literature and software
on unstructured meshes are about triangulation methods, there have been continuous
research efforts on structured and unstructured hex MG (Li and Cheng 1998; Mitchell
1998; Muller-Hannemann 1998; Kraft 1999; Shepherd et al. 1999; Staten et al. 1999;
Tautges 1999; Trease and Barrett 1999; Wada et al. 1999; White and Tautges 1999;
Dhondt 2001; Lu et al. 2001; Baker 2005; Zhao et al. 2008; Ruiz-Girones et al. 2009,
2012; Ran et al. 2012). Advanced meshing algorithms are available for the generation of
isotropic and anisotropic tetrahedral meshes over complex 3D industrial and biomechani-
cal objects; a versatile mesh generator capable of producing high-quality unstructured
hex elements is yet to be developed. Compared to tetrahedral meshing, these meshing
techniques are still at the development stage; further improvement and refinement are
expected, and verifications are required before any conclusion can be drawn on the merits
and drawbacks of the various schemes put forward.
The situation is compounded by the fact that answers to many fundamental questions are
still outstanding. For instance, what are the generally acceptable boundary conditions for
a hex meshing problem? Given a closed surface meshed in quads, what can you say about
the possibility of having an all-hex mesh? What would hex meshes of variable element sizes
as specified by a node spacing function look like? Anyway, hex meshing will remain as one
of the most interesting topics in the world of MG, and answers to some of these questions
along with better meshing algorithms will emerge as time goes by. Unlike triangular meshes
and tetrahedral meshes, extension from a 2D quad mesh to a 3D hex mesh is by no means
straightforward, and very often, a completely novel approach has to be adopted. Owing to
the methods employed and since the boundary condition requirements are quite different, it
is necessary to categorise hex meshing into two broad types: (I) conforming mesh bounded
by smooth surfaces and (II) constrained mesh bounded by surfaces discretised into quads.
Most hex meshing problems and the methods developed today only address problem type
I. Similar to the quad MG, there are both direct and indirect methods for the construction
of structured and unstructured hex meshes. Very often, several methods can be applied in a
combined mode or one after the other to generate hex meshes of various characteristics for
objects of difficult geometry and topology. In summary, hex meshing is still a challenging
task, and no single method appears to dominate over all situations.
