Civil Engineering Reference
In-Depth Information
Polygon
P
E
k
= T
x
E
k
= Q
Figure 6.18
Surrounding polygon for a mixed mesh of quadrilaterals and triangles.
where
α
η
if E satriangle
if E saquadrilateral
k
k
=
k
k
k
6.3.2.2 GETMe transformation for quadrilaterals
Let {X
1
, X
2
, X
3
, X
4
} be the vertices of quadrilateral Q. The transformation of quadrilateral
Q will be done in two steps. In the first step, quadrilateral Q will be changed to quadrilat-
eral Q′ = {Y
1
, Y
2
, Y
3
, Y
4
} by the following vertex transformation, as shown in Figure 6.19a:
YX
L
k
)
(
u
+
tan
θ
)
kk
(
+
1
k
k
=+
(6.5)
+
1
k
2
where
u
k
is a unit vector along X
k
X
k+1
,
v
k
is a unit vector normal to X
k
X
k+1
,
L
k(k+1)
is the
length between vertices X
k
and X
k+1
and k + 1 follows the modulus arithmetic such that
k + 1 ≡ mod(k,4) + 1. Comparing Q′ with Q, we can see that the shape of the quadrilateral
has been much improved by the transformation. Vartziotis and Wipper (2009) showed that
if we keep applying transformation 6.5 to quadrilateral Q, it will converge to a square, i.e.
Q ↦ Q
n
≈ square for large n by transformation 6.5
(a)
(b)
Quadrilateral Q˝
v
3
v
2
Quadrilateral Q
Z
3
Y
3
Y
3
X
3
θ
θ
u
3
Z
2
θ
u
2
X
2
θ
Y
4
Y
4
Y
2
Y
2
X
1
Quadrilateral Q´
X
4
Y
1
Z
4
Z
1
Y
1
Figure 6.19
Transforming quadrilateral Q to Q
ʹʹ
in two steps: (a) step 1; (b) step 2.
