Civil Engineering Reference
In-Depth Information
6.3.2 Optimisation of quadrilateral and mixed meshes
The three optimisation schemes for triangular meshes by node shifting, namely, QL smooth-
ing, local quality optimisation and GETMeT3, are generic and can be applied to quadri-
lateral or mixed meshes of triangles and quadrilaterals. Nothing needs to be changed for
the QL smoothing and the LO schemes except a proper shape measure for quadrilateral
elements. As for the GETMe, only a slight modification is needed for the transformation of
quadrilaterals.
6.3.2.1 Shape quality of a mixed mesh of triangles and quadrilaterals
The shape measure η for a quadrilateral element Q can be defined as follows.
14
/
2det(F )
∏
14
k
η
()
Q
=
η
with
η
=
k
k
2
F
k
=
,
k
where
F
k
is the transformation matrix at the k
th
vertex. Let
a
k
be the vector along edge
X
k
X
k+1
and
b
k
be the vector along edge X
k
X
k-1
, as shown in Figure 6.17; then
F
k
is given by
F
k
= [
a
k
b
k
]
det(
F
k
) = area of the parallelogram spanned by vectors
a
k
and
b
k
, and ‖
F
k
‖
2
= ‖
a
k
‖
2
+ ‖
b
k
‖
2
.
Without changing the magnitude of vectors
a
k
and
b
k
, det(
F
k
) attains the largest value when
a
k
and
b
k
are perpendicular to each other forming a rectangle. Moreover, η
k
will have the
largest value of 1 when vectors
a
k
and
b
k
are of the same magnitude forming the sides of
a square. Hence, quadrilateral Q will be regular (a square) if η
k
at the four vertices are all
equal to 1.
Given a mixed mesh of N triangular and quadrilateral elements,
M
= {E
i
, i = 1,N}, for a
given node
x
, the patch of elements surrounding node
x
, as shown in Figure 6.18, is given by
P
(
x
) = {E
k
∈
M
;
x
∈ E
k
}
x
∈ E
k
means that
x
is a vertex of element E
k
To measure the quality of the elements in
P
, the geometric mean μ-quality and the mini-
mum μ-value of the elements can be used:
1/
n
∏
k
E
=
;
=
min
min
k
E
∈
P
k
∈
P
k
X
k-1
b
k
X
k+1
X
k
a
k
Figure 6.17
Transformation matrix at vertex K.
