Biomedical Engineering Reference
In-Depth Information
(a) 3-node triangular element, linear interpolation:
N
1
=
λ
1
N
2
=
λ
2
N
3
=
λ
(17.21)
3
(b) 6-node triangular element, quadratic interpolation:
N
1
=
λ
1
(2
1)
N
2
=
λ
2
(2
λ
2
−
1)
N
3
=
λ
3
(2
λ
1
−
λ
3
−
1)
N
4
=
4
λ
1
λ
2
N
5
=
4
λ
λ
2
3
N
6
=
4
λ
1
λ
3
(17.22)
(c) 7-node triangular element, bi-quadratic interpolation:
N
1
=
λ
1
(2
λ
1
−
1)
+
3
λ
1
λ
2
λ
3
N
2
=
λ
2
(2
λ
2
−
1)
+
3
λ
1
λ
2
λ
3
N
3
=
λ
3
(2
λ
3
−
1)
+
3
λ
1
λ
2
λ
3
N
4
=
4
λ
1
λ
2
−
12
λ
1
λ
2
λ
3
N
5
=
λ
2
λ
3
−
λ
1
λ
2
λ
3
4
12
N
6
=
4
λ
λ
−
12
λ
λ
λ
1
3
1
2
3
N
7
=
27
λ
1
λ
2
λ
3
.
(17.23)
λ
1
λ
2
λ
3
is called a 'bubble' function giving zero contributions along the
boundaries of the element.
There are two major differences between the method used in Section
17.2
and
the method with triangle coordinates:
•
Determining the derivatives of these shape functions with respect to the global coor-
dinates is not trivial. Consider the derivative of a shape function
N
i
with respect to
x
.
Applying the chain rule for differentiation would lead to
The factor
∂
N
i
∂
x
=
∂
∂λ
1
∂λ
1
N
i
∂
x
+
∂
∂λ
2
∂λ
2
N
i
∂
x
+
∂
∂λ
3
∂λ
3
N
i
∂
x
.
(17.24)
But, by definition a partial derivative as to one variable implies that the other variables
have to be considered as constant. In this case a partial derivative with respect to
λ
1
means that, when this derivative is determined,
λ
2
and
λ
3
have to be considered constant.
However, the
λ
i
's are related by Eq. (
17.17
). Only two variables can be considered as
