Biomedical Engineering Reference
In-Depth Information
polynomial interpolation can be made of the function
u
. An example of such an
element division is given in Fig.
14.4
. This distribution of elements is called a
mesh
. Then, the integration over the domain
can be performed by summing up
the integrals over each element. Consequently, Eq. (
14.14
) yields:
N
el
N
el
dw
dx
c
du
dx
dx
=
wf dx
+
B
,
(14.25)
e
e
e
=
1
e
=
1
where
N
el
denotes the number of elements.
Step 2. Interpolation
Suppose that the domain
has been divided into three
linear elements, as depicted in Fig.
14.4
. Then the nodal values
u
i
may be collected
in an array
∼
:
⎡
⎣
⎤
⎦
u
1
u
2
u
3
u
4
∼
=
.
(14.26)
The unknowns associated with each of the elements
e
are collected in the arrays
∼
e
, such that
u
1
u
2
,
u
2
u
3
,
u
3
u
4
.
∼
1
=
∼
2
=
∼
3
=
(14.27)
So, it is important to realize that each particular element array
∼
e
contains a subset
of the total, or global, array
∼
. Within each element array
∼
e
a local numbering
may be used, such that for this particular example with linear elements:
u
1
u
1
u
2
u
2
u
3
u
3
u
1
u
4
Ω
1
u
2
Ω
2
u
3
Ω
3
x
1
x
2
x
3
x
4
Figure 14.4
Element distribution and unknowns at local and global levels.
