Biomedical Engineering Reference
In-Depth Information
N
N
element 1
element 2
element 1
element 2
x
x
dN
dx
dN
dx
element 1
element 2
element 1
element 2
x
x
−∞
(a) Admissible shape functions
(b) Non admissible shape functions
Figure 17.1
Continuity of shape functions.
an adequate approximation can be achieved, the shape functions have to contain
all constant and linear functions. Assuming that the exact solution is described by
an arbitrary linear polynomial, the element interpolation has to be able to exactly
describe this field. In mathematical terms this means the following. Assume
u
to
be approximated by
n
u
h
(
x
)
=
N
i
(
x
)
u
i
,
(17.1)
i
=
1
with
N
i
(
x
) the interpolation functions and
u
i
the nodal values of
u
h
(
x
). Con-
sider the case, where the nodal values
u
i
are selected to be related to the nodal
coordinates
x
i
,
y
i
,
z
i
by
u
i
=
c
0
+
c
1
x
i
+
c
2
y
i
+
c
3
z
i
,
(17.2)
according to a linear field. Substitution of Eq. (
17.2
) into (
17.1
) reveals:
n
n
n
n
u
h
=
c
0
N
i
(
x
)
+
c
1
N
i
(
x
)
x
i
+
c
2
N
i
(
x
)
y
i
+
c
3
N
i
(
x
)
z
i
.
(17.3)
i
=
1
i
=
1
i
=
1
i
=
1
