Biomedical Engineering Reference
In-Depth Information
u
x
x
i
-1
x
i
x
i
+1
x
i
+2
Figure 14.2
Solid line:
u
(
x
), dashed line: polynomial approximation of
u
(
x
).
n
u
h
(
x
)
=
N
i
(
x
)
u
i
,
(14.17)
i
=
1
where the functions
N
i
(
x
) are polynomial expressions of order
n
−
1 in terms of
the coordinate
x
. These functions
N
i
(
x
) are called
shape functions
because they
define the shape of the interpolation of
u
h
, for instance linear, quadratic etc. To
illustrate this, consider a first-order polynomial on the domain [
x
1
,
x
2
], with
u
(
x
)
known at
x
1
and
x
2
. In that case
1
u
1
+
x
1
x
2
−
x
1
x
−
x
1
x
2
−
x
1
u
2
,
x
−
u
h
(
x
)
=
−
(14.18)
implying that
1
,
x
−
x
1
x
2
−
x
1
x
−
x
1
x
2
−
x
1
.
N
1
=
−
N
2
=
(14.19)
Rather than approximating
u
(
x
) with a single polynomial of a certain degree over
the entire domain
, the domain
may also be divided into a number of non-
overlapping subdomains, say
e
a
local
polynomial
approximation of
u
may be constructed. A typical example is a piecewise lin-
ear approximation within each subdomain
e
, see Fig.
14.3
. Consider one of the
subdomains
e
=
[
x
i
,
x
i
+
1
]. Then within
e
the function
u
(
x
) is approximated by
e
. Within each subdomain
u
h
(
x
)
|
e
=
N
1
(
x
)
u
i
+
N
2
(
x
)
u
i
+
1
,
(14.20)
with, in conformity with Eq. (
14.19
):
1
x
i
x
i
+
1
−
x
i
x
−
x
i
x
i
+
1
−
x
i
.
x
−
N
1
=
−
N
2
=
(14.21)
More generally, if within a subdomain
e
an
n
-th order polynomial approximation
of
u
is applied, the subdomain should have
n
+
1 points at which the function
u
is
