Civil Engineering Reference
In-Depth Information
(e)
(f)
(g)
(h)
Figure 8.145
(Continued) Generation of p2 mesh by refinement and optimisation of p1 meshes: (e) p2 ele-
ments subdivided into p1 triangles; (f) optimised p1 mesh; (g) further refined and optimised p1
mesh; (h) completed p2 mesh of 148 elements.
points {P
1
, P
2
, . . . , P
n
}, there are n curve segments S
k
= S
k
(P
k
, P
k+1
, P
k+2
, P
k+3
), k = 1, n, in
which if k + j > n, k + j ↦ k + j − n. A point P(t) on curve segment S
k
is given by
P
P
P
P
−
13 31
3630
30 30
1410
−
k
1
6
−
k
+
1
32
S
=
S
withS Pt
:()[
=
t
t
t
1
]
k
k
−
k
+
2
kn
=
1
,
k
+
3
Fifty-nine and 31 control points are put in to define, respectively, the exterior and the inte-
rior boundary loops, on which 59 and 31 p2 (parabolic) curved boundary segments are gen-
erated, as shown in Figure 8.145a. The control points are placed in such a way that curved
segments of unequal lengths and large curvatures are created. At this moment onwards, the
MG problem is fully defined by a domain boundary of 90 parabolic curved segments, and
the control points and the B3 spline will not be referred to anymore. Based on the ending
points of the boundary segments, a p1 triangular mesh of 148 elements is generated, as
shown in Figure 8.145b, in which there are potential invalid elements where the boundary
segments cut into the p1 mesh. As shown in Figure 8.145c, the p1 mesh is converted to a p2
mesh by adding 264 mid-side nodes to the edges, out of which 90 mid-side nodes have to be
snapped onto the boundary to create curved p2 triangles. The first p2 mesh of 148 curved
T6 triangles is thus generated, as shown in Figure 8.145d; the MG can be stopped at this
