Biomedical Engineering Reference
In-Depth Information
Subsequently, use the divergence theorem to convert the left-hand side into the
boundary integral:
∇
(
φψ
)
d
=
n
φψ
d
.
(16.10)
This yields the desired result.
16.4
Weak form
Following the same steps as in Chapter
14
, the differential equation Eq. (
16.1
)is
multiplied with a weighting function
w
and integrated over the domain
:
w
∇·
(
c
∇
u
)
+
f
d
=
0,
for all
w
.
(16.11)
Next the integration by parts rule according to Eq. (
16.7
) is used:
w n
·
c
∇
ud
−
∇
w
·
(
c
∇
u
)
d
+
wf d
=
0.
(16.12)
The boundary integral can be split into two parts, depending on the essential and
natural boundary conditions:
w n
·
c
∇
ud
=
w n
·
c
∇
ud
+
wP d
,
(16.13)
u
p
where
c
∇
u
·
n
=
P
at
p
is used. It will be clear that, similar to the derivations
in Chapter
14
, the first integral on the right-hand side of Eq. (
16.13
) is unknown,
while the second integral offers the possibility to incorporate the natural bound-
ary conditions. For the time being we keep both integrals together (to limit the
complexity of the equations and rewrite Eq. (
16.12
) according to:
∇
w
·
(
c
∇
u
)
d
=
w n
·
c
∇
ud
wf d
+
.
(16.14)
16.5
Galerkin discretization
Step 1
Introduce a mesh by splitting the domain
into a number of non-
e
. In a two-dimensional configuration the elements typ-
ically have either a triangular (in this case the mesh is sometimes referred to as a
triangulation) or a quadrilateral shape. A typical example of triangulation is given
in Fig.
16.3
. Each triangle corresponds to an element.
overlapping elements
