Biomedical Engineering Reference
In-Depth Information
Step 6
Writing the force per unit volume vector
f
as
f
=
f
x
e
x
+
f
y
e
y
,
(18.54)
and the weighting function
w
within element
e
as
T
∼
xe
e
x
+
∼
T
∼
ye
e
y
,
w
|
e
=
∼
(18.55)
the second integral on the right-hand side of Eq. (
18.24
) can be written as
e
·
fd
T
∼
xe
f
x
+
T
∼
ye
f
y
)
d
w
=
(
∼
∼
e
e
f
∼
v
=
∼
e
,
(18.56)
where
⎡
⎣
⎤
⎦
N
1
f
ex
N
1
f
ey
N
2
f
ex
N
2
f
ey
.
N
n
f
ex
N
n
f
ey
v
f
∼
e
=
d
.
(18.57)
e
The first integral on the right-hand side of Eq. (
18.24
) may, if applicable, be
elaborated in exactly the same manner. This results in
e
f
∼
p
e
w
·
(
σ
·
n
)
d
=
∼
e
,
(18.58)
p
e
v
where
f
∼
is structured analogously to
f
∼
e
. The contribution of the right-hand side
triggers the abbreviation:
v
p
f
∼
e
=
f
∼
e
+
f
∼
e
,
(18.59)
often referred to as the element load contribution.
Step 7
Putting all
the pieces together,
the discrete weak formulation of
Eq. (
18.24
) is written as:
N
el
N
el
e
K
e
∼
e
=
e
f
∼
∼
∼
e
.
(18.60)
e
=
1
e
=
1
Following an equivalent assembling procedure as outlined in Chapter
14
the
following result may be obtained:
T
K
∼
=
T
f
∼
∼
∼
,
(18.61)
