Biomedical Engineering Reference
In-Depth Information
Figure 16.3
Example of a two-dimensional finite element mesh using triangular elements. One element has
been highlighted.
The integration over the domain
can be performed by a summation of the
integrals over each element:
.
N
el
N
el
e
∇
(
c
∇
c
∇
w
·
u
)
d
=
wf d
+
w
n
·
ud
(16.15)
e
e
e
=
e
=
1
1
The boundary part
e
denotes the intersection of element
e
with the boundary
, hence
e
=
e
∩
. Clearly, not each element will have an intersection with
.
Step 2
If a Cartesian coordinate system is used for two-dimensional problems
(extension to the three-dimensional case is straightforward), the inner product
∇
w
·
(
c
∇
u
) yields
∂
w
∂
x
e
y
c
∂
u
e
y
e
x
+
∂
w
e
x
+
∂
u
∇
(
c
∇
w
·
u
)
=
∂
y
·
∂
x
∂
y
c
∂
w
.
∂
x
∂
u
∂
x
+
∂
w
∂
y
∂
u
=
(16.16)
∂
y
Step 3
Introduce a discretization for both the weighting function
w
and the
unknown
u
, so within each element
n
T
(
x
,
y
)
∼
e
w
h
|
e
=
N
i
(
x
,
y
)
w
e
,
i
=
∼
(16.17)
i
=
1
n
T
(
x
,
y
)
∼
e
.
u
h
|
e
=
N
i
(
x
,
y
)
u
e
,
i
=
∼
(16.18)
i
=
1
