Biomedical Engineering Reference
In-Depth Information
FIGURE 3.5
A control volume for scalar variables.
(Eqn
3.37
) and its redistancing (Eqn 3.38) are spatially discretized with WENO5
[20]
and advected
using TVD-RK2
[21]
. These schemes are computationally intensive. To reduce the computational
effort, the level-set method is implemented in a narrow-band procedure
[9]
where the level-set
function is solved only within a band of certain thickness from the interface. This reduces one
order of computational effort.
3.4.3
Solution algorithm
The overall solution procedure is as follows: Given known
c
n
,
u
n
,
p
n
,
T
n
,
4
e
,
4
m
, and
f
n
at time
t
,
evolve the solution to
c
nþ
1
,
u
nþ
1
,
p
nþ
1
,
T
nþ
1
,
4
nþ
1
e
,
4
nþ
1
m
and
f
nþ
1
at time
t
þ D
t
.
u
n
.
2.
Solve for the new interface
f
nþ
1
from Eqn
(3.37)
and perform redistancing via Eqns (3.38).
3.
Calculate the properties of the fluid using Eqn (3.34).
4.
Solve for concentration field
c
nþ
1
from Eqn
(3.1)
.
5.
Solve for
u
nþ
1
and
p
nþ
1
from Eqns
(3.3)
and
(3.4)
using the SIMPLER algorithm
6.
Solve for the temperature field
T
nþ
1
from Eqn
(3.6)
.
7.
Solve for the electric field
4
nþ
e
from Eqn
(3.23)
.
8.
Solve for the magnetic field
4
nþ
m
from Eqn
(3.30)
.
9.
Repeat steps (2) and (8) until the solution converges.
10.
Repeat steps (1) and (8) for all time steps.
1.
Set
u
nþ
1
¼
Some of the above steps can be skipped if the relevant fields are not involved in the problem of interest.
For example, if an electric field is not considered, step (7) can be skipped.
3.5
VERIFICATIONS AND VALIDATIONS
The presented solution procedure has to be validated before it can be used. The first verification is
made for the solution of the general transient convection-diffusion equation (Eqn
3.42
). The
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