Biomedical Engineering Reference
In-Depth Information
15.4
Spatial discretization
Following a similar derivation as in the previous chapter, the weak form is
obtained by multiplication of Eq. (
15.1
) with a suitable weighting function
w
,
performing an integration over
=
[
a
,
b
], followed by an integration by parts:
w
∂
u
∂
wv
∂
u
dw
dx
c
∂
u
dx
+
x
dx
+
x
dx
=
B
,
(15.19)
t
∂
∂
where the right-hand side term
B
results from the integration by parts:
x
=
b
−
x
=
a
w
(
b
)
c
∂
u
∂
x
w
(
a
)
c
∂
u
∂
x
B
=
.
(15.20)
Notice that no partial integration of the convective term has been performed. The
discrete set of equations, according to Eq. (
15.19
), is derived by subdivision of the
domain in elements and by discretization at element level according to
T
(
x
)
∼
e
(
t
),
T
(
x
)
∼
e
(
t
) .
u
h
(
x
,
t
)
|
e
=
∼
w
h
(
x
)
|
e
=
∼
(15.21)
Note that the shape functions
∼
are a function of the spatial coordinate
x
only and
not of the time
t
. The nodal values of
u
h
, at element level stored in the column
∼
e
, however, do depend on the time
t
. Substitution of Eq. (
15.21
) into Eq. (
15.19
)
yields
∼
e
e
e
N
el
T
T
dx
d
∼
e
∼
a
d
∼
T
T
dt
+
∼
∼ ∼
dx
∼
e
dx
e
e
=
1
dx
∼
e
e
e
T
d
∼
dx
c
d
∼
+
∼
=
B
.
(15.22)
dx
With
T
dx
,
M
e
=
∼ ∼
(15.23)
e
and
T
T
∼
a
d
∼
d
∼
dx
c
d
∼
K
e
=
dx
+
dx
,
(15.24)
dx
dx
e
e
Eq. (
15.22
) can be written as
e
M
e
d
∼
e
K
e
∼
e
N
el
∼
dt
+
=
B
.
(15.25)
e
=
1
After the usual assembly process this is written in global quantities:
T
M
d
dt
+
K
∼
T
f
∼
∼
=
∼
,
(15.26)
