Civil Engineering Reference
In-Depth Information
In view of the diverse possibilities in mesh generation, mesh generation using simplices
and/or non-simplices on planar domain, on curved surfaces and over volumes bounded or
unbounded will all be investigated and discussed in this text. According to Lohner (1997),
there are only two basic ways to fill up a general bounded domain with elements: (i) filling
the
empty
, i.e. an as-yet-unmeshed region, with elements, and (ii) modifying an existing
mesh that is already covered with elements. However, there is perhaps a third way (iii) in
which the mesh is refined, modified and stretched while its boundary is snapped onto the
boundary of the object. A typical method for the first technique is the advancing-front
approach (ADF); that for the second technique is the Delaunay triangulation; and exam-
ples for the third are the Meccano and grid/voxel methods. Finite element meshes can be
broadly divided into two main types, namely, the structured mesh and the unstructured
mesh. Structured meshes can be generated over smooth regular domains based on some
deterministic procedures, whereas unstructured meshes are for complex irregular domains
possibly with additional requirements such as element size variation and mesh directional
properties, which in general can only be generated by means of some heuristic approaches.
1.5 GENERAL STRATEGIES, ROBUSTNESS,
DIFFICULTIES AND METHODOLOGIES
As far as the existence of a solution is concerned, the most difficult problem or perhaps the
only difficulty in mesh generation is the construction of a fully constrained finite element
mesh for an arbitrary three-dimensional domain with irregular geometry and complicated
boundary constraints. The difficulty is due to the fact that there exist polyhedra that can
only be meshed with the introduction of interior points, the so-called Steiner points. As there
is no systematic way to determine the number of Steiner points needed and their locations,
we have to resort to heuristic means in an attempt to obtain a solution without degenerate
elements. Since the first valid finite element mesh has special significance in indicating that
the given domain is
meshable
, robustness is therefore of primary concern to a mesh genera-
tion algorithm, followed by mesh quality and speed of mesh generation. Moreover, based
on the first finite element mesh, mesh quality can be further improved, and adaptive meshes
with gradation in element size and directional characteristics can all be created by means
of refinement and various mesh optimisation techniques. In general, simplicial meshes can
have better adaptation to the more difficult boundary conditions and allow a progressive
change in element size within the mesh, whereas quadrilateral and hexahedral meshes can
be generated rapidly using mapping techniques over regular domains with simple bound-
aries. Popular mesh generation methods so far developed include Delaunay triangulation,
ADF, Quadtree/Octree decomposition, Meccano transformation, refinement and coarsen-
ing, mapping and modification, optimisation by iterations, intersection and merging based
on Boolean operations, etc.
1. 6 M ATH E M ATIC S
No doubt, in mesh generation, mathematics plays a vital role in providing values to various
geometric quantities such as distance, angle, volume, mappings, shape measures and metric
tensors in quantifying element shape and size, etc. However, mesh generation is more con-
cerned with the number of nodes, where to place them and how they should be connected
to form elements - topological operations in terms of nodal combinations for which there
is no direct relationship with geometrical computations, though some estimations can be
