Biomedical Engineering Reference
In-Depth Information
u
u
i
u
i
+1
x
x
i
-1
x
i
x
i
+1
x
i
+2
Ω
e
Figure 14.3
Solid line:
u
(
x
), dashed line: piecewise linear approximation of
u
(
x
).
known. For instance, to use a second-order (quadratic) polynomial, the subdomain
should cover at least three consecutive points, for example
e
=
[
x
i
,
x
i
+
2
], such
that
u
h
(
x
)
|
e
=
N
1
(
x
)
u
i
+
N
2
(
x
)
u
i
+
1
+
N
3
(
x
)
u
i
+
2
.
(14.22)
The subdomain
e
within which a certain polynomial approximation is used is
referred to as an
element
. The points at which the values of
u
are defined are
called
nodes
.
The shape functions
N
i
may not be chosen arbitrarily. The most stringent
requirement is that
u
h
must be interpolated continuously over the total domain
(the first-order derivative of
u
h
(
x
) should exist). Suppose that the nodes
x
j
within
an element are numbered
j
=
1,
...
,
n
and that the associated shape functions
N
i
are numbered
i
=
1,
...
,
n
. In that case, for consistency, the shape functions must
be chosen such that
N
i
(
x
j
)
=
δ
ij
,
(14.23)
with
δ
ij
=
0if
i
=
j
=
1if
i
=
j
.
(14.24)
14.5
Galerkin approximation
To transform the weak form into a linear set of equations in order to derive an
approximate solution the following steps are taken.
Step 1. Element division
As shown in the previous section the domain
may
be split into a number of subdomains
e
, elements, and within each element a
