Civil Engineering Reference
In-Depth Information
1
3
(
We have
OOOO
A
=
+
+
)
.
B
C
Alternatively, the centre of the circumdisk can also be determined by the Newton-
Raphson iteration (Appendix A.6). Define F = d
A
- d
B
and G = d
A
- d
C
. Then compute
numerically the derivatives of F and G with respect to x and y.
=
∂
∂
F
x
Fx hFx
h
(
+−
)
()
,
=
∂
∂
F
y
Fy hFy
h
(
+−
)
()
,
F
≈
F
≈
x
y
=
∂
∂
G
x
Gx hGx
h
(
+−
)
()
,
=
∂
∂
G
y
Gy hGy
h
(
+−
)
()
,
G
≈
G
≈
x
y
where (x,y) are the co-ordinates of O, and h can be taken as 0.0001 relative to the size
of the triangle. The new position of centre O is given by
−1
FF
GG
x
y
Δ
Δ
x
y
x
y
F
G
xy
O
=+=−
xy
Iteration cycles are repeated as necessary until F ≈ 0 and G ≈ 0. Starting from the
base triangle, which contains the inserted point P, triangles are deleted following the
adjacency relationship. Let O be the centre of the circumdisk of triangle P
1
P
2
P
3
; tri-
angle P
1
P
2
P
3
will be deleted if ‖OP‖
M
≤
max
i
‖OP
i
‖
M
i = 1, 2, 3.
6.
Updating the generation front.
The generation front can be updated in exactly the
same manner as the classical ADF approach before the next point is introduced; refer
to Section 3.6.2 for details. The mesh generation is completed when there are no more
segments on the generation front; otherwise, go to step 4.
4.2.12.2 The completed mesh
Figure 4.37 shows the construction of the first 1000 triangular elements by the Delaunay-
ADF method for the anisotropic meshing according to the wavy surface metric. The pattern
and sequence of element formation are similar to those of the ADF meshing of isotropic
meshes except that elongated triangles are generated over some parts of the domain as
required. Figure 4.38 shows another intermediate stage of anisotropic mesh near comple-
tion in which 5000 triangles have been generated. Smaller and elongated elements are cre-
ated at the centre of the domain where there are many peaks clustered closely together.
The α-quality of triangles on parametric curved surfaces has been discussed and defined in
Section 4.2.10, whereas the conformity coefficient of a unit metric mesh is given by
1
δ=
min
AB
,
,
length of edge AB
,
M
AB
M
1
∫
0
T
AB
=
AB M
(
A AB AB dt
+
)
M
