Civil Engineering Reference
In-Depth Information
Even though the two peaks are not identical, it can be seen that the unit metric plot is
more or less symmetric about these two peaks, as shown in Figure 4.24. Owing to the large
curvature at the peaks and the trough, ellipses of very small size are found at these positions;
that is to say that triangles of relatively small size have to be used at these points in order
to have a close approximation of the surface geometry. The third example is a wavy surface
with a sharp peak at the middle, whose parametric equations are given by
sin(
3
u
)sin()
;
3
v
xu yv z
=
;
=
;
=
uv
,
∈ −
[,]
11
uv
There are sharp changes of curvature from the peak towards the rest of the domain. As
shown in Figure 4.25, a small ellipse of unit metric is found at the peak, and elongated
ellipses are found on the four sides of the peak showing a large difference in the principal
curvatures along the side of the peak.
The fourth example is a wavy surface characterised with a series of peaks of more or less
equal height, and the surface parameterisation is given by
x = u
3
+ 10u;
y = v
3
+ 10v;
z = sin(2u)cos(2v);
u,v ∈ [−5,5]
Owing to the wavy nature of the surface, the surface curvatures vary substantially from
peak to trough and across ridges, as shown in Figure 4.26. Small ellipses are found at peak
and trough positions, and elongated ellipses are found along ridges, as shown in Figure 4.27.
As this wavy surface will be taken as an example for anisotropic mesh generation of para-
metric surfaces in Section 4.2.12, contour lines around points are shown in Figure 4.28,
Figure 4.24
Curved surface of double peak and its unit metric plot.
Figure 4.25
Ripple surface of single peak and its unit metric field.
