Civil Engineering Reference
In-Depth Information
and the circum-radius. Without storing the circumcentre and the circum-radius, 50 million
3D points can be inserted quite comfortably by a parallel insertion based on 16 MB on a PC.
3.5.6 Generation of interior points
The point insertion algorithm and the insertion kernel provide a powerful tool for the con-
struction of the convex hull of DT of a given point set. However, for MG over 2D domains
with a well-defined boundary, two more issues are needed to be addressed: (i) the generation
of interior points in compliance with the required element-size specification; and (ii) the
boundary edge recovery so that all the boundary edges of the given object are present in the
triangulation. These two important aspects are to be discussed in Sections 3.5.6.1-3.5.6.5.
Before describing the various methods for the creation of interior points, the specific
requirements in the generation of internal points have to be made clear. The geometrical
requirement is to create points inside the given domain to produce triangles as equilateral as
possible. However, there are other physical requirements and constraints requiring points to
be distributed according to a specified nodal spacing, which can be isotropic or anisotropic.
3.5.6.1 Specification of nodal spacing
Nodal spacing is also known as point or element size distribution, which specifies the distance
between two points or the size of the resulting FEs anywhere within the given domain to be
meshed. The simplest mesh required is the boundary mesh in which an FE mesh is produced
based solely on the boundary points, and this is the coarsest (minimum) mesh that can be gen-
erated with respect to the given boundary of the domain. In case more triangular elements are
needed over empty parts within the domain, a mesh of uniform element size can be generated
with the aid of a background grid. For more complicated node distributions with an isotropic
or anisotropic variation of element size over the domain arising from, for instance, adaptive
refinement analysis, a more general way to specify the element size distribution is required.
3.5.6.2 Control space
The idea of
control space
introduced in Section 2.3.8 is to facilitate MG and the creation of
internal points within the problem domain under consideration. The control space provides
information whether a given point is inside or outside the problem domain and about the
nodal spacing requirement at that point. For this purpose, the control space is a cover of
the problem domain so that any point of the problem domain can be found also in the con-
trol space. It is natural to use a previous mesh or the minimum mesh as the control space;
however, a background grid large enough to cover the problem domain can be used as the
control space (Lo and Lee 1994). Based on the work of George and Borouchaki (1998), a
control space is defined as follows.
Definition
(
, ρ) is a control space of a given domain Ω if
i. Ω is covered by
, i.e.
.
ii. A function ρ to specify the nodal spacing over the control space
∀∈
x
CCR
,()
ρ
:
→
