Biomedical Engineering Reference
In-Depth Information
The shape functions are simple polynomial expressions in terms of the coor-
dinate
x
. For instance for a linear interpolation the shape functions are linear
polynomials, according to
x
−
x
1
x
−
x
1
N
1
=
−
N
2
=
1
x
1
,
x
1
,
(14.66)
x
2
−
x
2
−
where
x
1
and
x
2
denote the position of the nodes of the element. In this case the
shape functions are linear functions of the global coordinate
x
. It is appropriate
in the context of a generalization to more-dimensional problems to introduce a
local coordinate
ξ
=
1 correspond to the edges of the element. With respect to this local coordinate
system, the shape functions may be written as
−
1
≤
ξ
≤
1 within each element such that
ξ
=−
1 and
1
2
(
1
2
(
N
1
=−
ξ
−
1) ,
N
2
=
ξ
+
1) .
(14.67)
This is visualized in Fig.
14.5
.
Computation of components of the element coefficient matrix and the element
load array requires the evaluation of integrals of the form
f
(
x
)
dx
.
(14.68)
e
1
N
1
N
2
x
x
1
x
2
Figure 14.5
Shape functions with respect to the global
x
-coordinate.
N
1
N
2
1
ξ
1
-1
Figure 14.6
Shape functions with respect to the local
ξ
-coordinate.
