Civil Engineering Reference
In-Depth Information
loading conditions and physics - static or dynamic, mechanical, thermal or coupled multi-
field problems, etc. To meet all these requirements, we have to devise algorithms to generate
rapidly finite element meshes of various characteristics on a planar domain, on curved sur-
faces and over three-dimensional volumes in a robust manner. Hybrid and mixed meshes,
which are meshes consisting of various types of elements in different dimensions, may some-
times be required for certain problem types. Many mesh generation techniques, for instance,
the Delaunay triangulation and the advancing-front technique (AFT), can also have appli-
cations in many other fields including data visualisation, terrain modelling, surface recon-
struction, structural networking for arbitrary point sets, etc.
1.4 PROBLEM DEFINITION, SCOPE AND PHILOSOPHY:
SCIENCE OR ART?
Owing to diverse applications for various disciplines under different situations, there are no
formal universal rules as to how finite element mesh generation problems should be defined.
However, domains represented by boundary specification are quite a common practice for
meshing engineering objects, in which a planar domain is well defined by a series of bound-
ary line segments, and three-dimensional volumes are bounded by triangular and/or quad-
rilateral facets without ambiguity. Other possibilities include volumes defined implicitly by
a system of spatial points for which the boundary of the object can only be detected by
means of some
in-or-out
inquiry mechanisms and meshing of a computational domain,
which is large enough to contain the physical object or event under consideration. In sum-
mary, broadly speaking, there are three types of boundary settings for finite element mesh
generation:
1. No boundary is defined, and just a large interior part extensive enough to cover the
object or the event under consideration needs to be meshed, e.g. a background grid or
the convex hull of a Delaunay triangulation, etc.
2. Geometrically conforming meshes, the boundary nodes of the mesh have to be on the
boundary surface of the object.
3. Fully constrained meshes: apart from points, the boundary edges and faces of the mesh
should all have a perfect match with those specified on the boundary surface of the
object. As mesh generation is very sensitive to boundary requirements, even for the
same physical domain, the mesh generation problem could be quite different subject
to various boundary constraints, and very often, different mesh generation strategies
have to be employed accordingly.
'Mesh generation: Art or science?' is a review paper written by Timothy J. Baker in 2005 in
which no definite conclusion on whether the subject belongs to art or science has been given,
except the comment 'Some of the advances were based on a sound theoretical understanding;
many others were heuristic in nature, guided by an intuitive feel for what seemed like the
right approach'. Mesh generation is a science in the sense that there are deterministic ways in
producing certain mesh types, and there are systematic optimisation procedures in improving
the quality of a finite element mesh; however, it is also an art in the sense that a solution may
not exist, and there is freedom in choosing the element types, in using a different number and
size of elements and in placing nodes at various positions to arrive at a solution. Boundary
and internal constraints and optimal mesh quality further impose additional difficulties in
the theoretical approach to the mesh generation problem, and what is the expected quality
of the mesh satisfying all the boundary constraints being most likely an open question.
