Civil Engineering Reference
In-Depth Information
Fundamentals
The fundamentals allow you to go through the other chapters with clarity and comfort.
2.1 INTRODUCTION
The basic concepts, notations, terminologies, geometrical and topological operations, sorting
methods and background grids for mesh generation (MG) will all be presented in this chapter
to pave the way for a formal discussion of finite element (FE) MG in the following chapters.
The notations, symbols and abbreviations commonly employed in MG are given in Section
2.2. Although the symbols and abbreviations used are more or less those usually adopted in
the MG community, however, the notations are quite unique in a way that the counter and
index are employed to specify systematically a particular node, neighbour, edge or face of an
FE. Terminologies related to MG and data structures for an FE mesh are presented in Section
2.3. For terminologies, instead of a formal mathematical definition, their characteristics are
highlighted; in particular, those features intimately related to MG will be discussed in detail.
Again, data structures are not those generally encountered in computer science but rather the
data format and arrangement to specify an MG problem, i.e. how nodes of common FEs in 2D
and 3D are labelled, how to store and address an FE mesh in a computer, etc.
Geometrical operations, namely, those for computing the distance between two simplices
of the same or different dimensions, i.e. point-to-line segment and line segment to triangular
facet, etc., frequently required in MG along with other useful formulas (for instance, nor-
mal at a point, solid angles, determination of intersection points, etc.), are given in Section
2.4. In MG, topological operations are equally if not more important than geometrical
operations, as enquiries such as how many elements are connected to a particular node and
how to list all the nodes or segments on the boundary of a mesh are always faced. As the
number of nodes and the number of elements in a mesh can be very large, efficient topol-
ogy computation algorithms are crucial to a robust MG scheme. Accordingly, algorithms
for the common topological operations including elements connected to a node, adjacency
relationship of a mesh, etc., in the form of detailed pseudo-code readily translated into C++
or FORTRAN programs are given in Section 2.5.
The ability to sort a large amount of data in an efficient and reliable manner is always a
great asset in numerical computations. In mesh refinement based on the bisection of the lon-
gest edge, the edges in a mesh have to be sorted repeatedly such that the longest edge is always
bisected in the refinement process, and very often, an MG process has to be carried out follow-
ing a sequence according to the size of the elements, etc. Common sorting methods are intro-
duced in Section 2.6; their performance on large data sets is compared, and the pseudo-codes
of the sorting algorithms are also given. Perhaps the background grid is the most important
11
