Civil Engineering Reference
In-Depth Information
2D planar domain Ω can be readily meshed by the methods, as discussed in Chapter 3, and
when points on Ω are mapped by ϕ onto S in 3D space, an FE mesh is created on the surface S.
However, in general ϕ is not isometric (length-preserving), and the edges and hence the shape
and the area of the triangular and quadrilateral elements would change by the mapping ϕ.
As a result, the quality of well-shaped triangles and quadrilaterals in Ω may deteriorate upon
the transformation from Ω to S. The geometrical approximation may not be adequate either,
as straight line segments are used in places of large curvatures on the surface.
4.2.1.2 Gap between a triangular facet and the curved surface
Let S be the curved surface of parameterisation ϕ(u,v) with parametric variables (u,v) on
planar domain Ω, and T be a triangle in Ω with vertices mapped on the surface S. The gap
between the triangle and the surface S, denoted by δ, is given by Filip et al. (1986)
δ
=
SupuvTuv
uv T
φ
(,)
−
( ,)
(,)
∈
where T(u,v) is a point on triangle T, and ϕ(u,v) is the corresponding point on surface S.
2
9
2
δ≤ ++
hI
(
2
I
I
)
1
2
3
in which h is the longest edge of triangle T, and I
1
, I
2
and I
3
are given by
I
=
up
φ
(,),
uv
I
=
up
φ
(,)
uv
,
I
3
=
up
φ
( ,)
uv
1
uu
2
uv
vv
(,)
uv T
∈
( ,)
uv T
∈
(,)
uv T
∈
Hence, the gap between the surface and a triangular facet is governed by the longest edge
of the triangle and the second derivatives of the surface.
4.2.1.3 Metric for curved surface geometry
The idea to be explored in detail is the design of a mesh generated on the parametric planar
domain such that the resulting mesh on surface S upon the transformation by ϕ is a close
geometrical approximation to the surface. This characteristic can be achieved by controlling
the mesh construction on the parametric space ensuring that the gap between an edge and
the surface is within any specified tolerance.
4.2.2 Fundamental forms and the related metric
ϕ is a mapping from Ω to S and is of class C
2
. For curved surfaces with ridges, the C
2
require-
ment can be relaxed for a finite number of points, where the surface characteristics can be
estimated or interpolated from the neighbouring points (Lee and Lee 2003; Clemencon et
al. 2006). Two fundamental forms are defined at every point
p
∈ S allowing the length of a
curve on S and the curvature along the curve to be determined.
4.2.2.1 Tangent and normal vectors
p
= ϕ(u,v). The tangent plane T
p
at
p
is spanned by the basis (co-ordinate) vectors.
