Biomedical Engineering Reference
In-Depth Information
1
0.8
0.6
N
1
1
0.4
0.2
0
η
0
-1
-0.5
0
-1
0.5
1
ξ
Figure 16.5
Shape function associated with node 1.
∂
dx
=
∂
N
i
N
i
∂ξ
∂ξ
∂
x
+
∂
N
i
∂η
∂η
∂
x
∂
N
i
dy
=
∂
N
i
∂ξ
∂
y
+
∂
N
i
∂η
∂
y
.
(16.45)
∂ξ
∂η
These relations can be rewritten in the following matrix form:
⎡
⎤
⎡
⎤
⎡
⎤
∂
N
i
∂
x
∂
∂ξ
∂
x
∂η
∂
x
∂
N
i
∂ξ
∂
⎣
⎦
=
⎣
⎦
⎣
⎦
.
(16.46)
N
i
∂
∂ξ
∂
∂η
∂
N
i
∂η
y
y
y
The derivatives
are readily available, but the terms in the
matrix cannot be directly computed since the explicit expressions
∂
N
i
/∂ξ
and
∂
N
i
/∂η
ξ
=
ξ
(
x
,
y
) and
η
=
η
(
x
,
y
) are not known. However, due to the isoparametric formulation the
inverse relations are known, so the following matrix can be calculated easily:
⎡
⎣
⎤
⎦
∂
x
∂ξ
y
∂ξ
∂
x
,
ξ
=
,
(16.47)
∂
x
∂η
∂
y
∂η
