Civil Engineering Reference
In-Depth Information
Mesh generation over curved surfaces
A surface is curved and non-Euclidean, and not every point is visible, but all these are
negated by the fact that locally it has a topological structure that resembles that of a plane.
4.1 INTRODUCTION
There is a growing demand for robust and efficient discretisation algorithms for general
curved surfaces into finite elements of variable sizes and shapes. The boundary of a physical
object is generally made up of patches of curved surfaces, which can be typically represented
by NURBS surfaces generated in a commercial CAD environment. The surface discretisa-
tion itself is a surface model of the object for rapid visualisation, and it can also be used
for the FE analysis, or it serves as an input for FE mesh generation within a volume. While
many techniques developed in planar mesh generation are still applicable to surface mesh-
ing, there are difficulties and different geometric characteristics over curved surfaces that
require modification, extension and special considerations.
There are, in general, two ways to represent a surface for the purpose of visualisation
and computation. One way is to use analytical surface patches via functions such as Coon's
patches, B-splines or NURBS surfaces (Kobbelt et al. 1997; Frey and George 2000). Another
way is to use discrete data structure such as quadrilateral facets and triangular facets
(Lohner 1996a). For engineering applications, surfaces defined by discrete data are widely
adopted, for instance, in the presentation of complex molecule (Akkiraju and Edelsbrunner
1996; Laug and Borouchaki 2000, 2001), engineering design (Shostko et al. 1999), visuali-
sation (Bruyns and Senger 2001; Dillard et al. 2007) and computational models for analysis
by the FE method (Lo 1988c, 1995). According to the type of surface representation, many
surface mesh generation schemes are available, and they can be broadly classified into either
parametric mapping approaches or direct surface mesh generation in 3D space.
4.1.1
Parametric meshing for curved surfaces
By this approach, the surface to be discretised is represented by a bivariate analytical func-
tion such that any point on the 3D surface is mapped to a 2D parametric space (Canann et al.
1997; Cuillière 1998; Lee and Hobbs 1998; Shimada and Gossard 1998; Borouchaki et al.
2000a; Sherwin and Peiro 2002; Lee 2003a,b; Cherouat et al. 2010). The mesh generation
process is carried out entirely on the parametric space by a 2D general-purpose mesh genera-
tor. The final surface mesh is obtained by proper transformation of the mesh generated on
the parametric space back to the 3D space, as shown in Figure 4.1. This method gives rea-
sonably good meshes for simple surfaces that are sufficiently smooth with respect to a planar
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