Mesh generation over curved surfaces - Finite Element Mesh Generation - page 217

of triangle ABC I is inferior to the best triangle ABC Γ that could be formed with a node C Γ
on the generation front.

4.4.2.3 Space-to-surface projection
The purpose of space-to-surface projection is to map a point C I in the 3D space onto the
surface such that the size and the shape of triangle ABC I are preserved after projection. Two
algorithms, one using the closest point projection for simple surfaces and the other using a
more general scheme based on surface derivatives, will be described.

4.4.2.3.1 Closest-point projection
For simple surfaces such as cylinder and sphere whose closest-point projection is available
in explicit form or surfaces whose closest-point projection can be computed readily, this
mapping is a convenient choice for the space-to-surface projection. Although the size and
the shape of the resulting triangle need not be the same as those of the original one, the
closest-point projection does give a reasonably close approximation of the required triangle.
Depending on how far the spatial point C I is from the surface, the quality of the projected
triangle may differ from the required one by an unacceptable amount if relatively large ele-
ments are generated at places of small radius of curvature. A remedy for this is to construct a
better spatial point C I * closer to the surface. This can be achieved by replacing vector e 2 with
a vector e * , which better approximates the direction of the new node relative to the surface.
Let C I be the projection of C I ; the updated direction vector e * is given by

ee
ee

+
+

v
v

( e 11

* =

2

e where unitvector e = withMCM

I

v = − C I ⋅

2

2

As shown in Figure 4.61, a spatial point closer to the surface is given by

* *

CMh
I

=+ e

2

v is the projection of vector MC I onto the plane spanned by vectors n and e 2 normal to e 1 ,
and the updated direction e * is the average of the original direction e 2 and the projected
vector v . Since the new direction is on the plane of vectors n and e 2 , the size and the shape
of triangle ABC I are preserved. Geometrically, it is equivalent to rotating the direction of
the vector towards the surface. The process can be repeated in an iterative manner until the
projected point on the surface is of a distance close to the specified element size.

4.4.2.3.2 Projection for general surface
Let p = p (u,v) be a parametric representation of curved surface S, where p is the position
vector of the points on S. Assuming that S is regular, i.e. for all the points on S,

∂
∂

pp uv0
uv

× ∂

∂ =×≠

where u and v are, respectively, basis vectors associated with co-ordinates u and v.