Biomedical Engineering Reference
In-Depth Information
holds. As boundary conditions, at
x
=
0,
u
=
0 and at
x
=
1,
u
=
1are
specified.
(a)
Prove that the exact solution is given by
1
e
c
x
).
u
=
e
c
(1
−
−
1
(b) Verify this by means of the script
demo_fem1dcd
, to solve the one-
dimensional convection-diffusion problem, which can be found in the
directory
oned
of the programme library
mlfem_nac
. Use five ele-
ments and select
c
=
1, while
v
is varied. Choose
v
=
0,
v
=
1,
v
=
10 and
v
=
20. Explain the results.
(c) According to Section
15.4
, the solution is expected to be oscillation
free if the element Peclet number is smaller than 1:
ah
2
c
<
1.
Pe
h
=
Verify that this is indeed the case.
15.2
Investigate the unsteady convection-diffusion problem:
∂
c
∂
,
u
dt
+
v
∂
u
∂
x
=
∂
∂
x
u
∂
x
on the domain
=
[ 0 1] subject to the initial condition:
u
ini
(
x
,
t
=
0)
=
0,
inside the domain
and the boundary conditions:
u
=
0at
x
=
0,
u
=
1at
x
=
1.
The
-scheme for time integration is applied. Modify the
m
-file
demo_fem1dcd
accordingly. Use ten linear elements.
(a) Choose
v
θ
=
c
=
1 and solve the problem with different values of
0.25. For each problem start
with a time step of 0.01 and increase the time step with 0.01 until a
maximum of 0.05. Describe what happens with the solution.
(b) In the steady state case, what is the maximum value of the convective
velocity
v
such that the solution is oscillation free for
c
=
1?
(c) Does the numerical solution remain oscillation free in the unsteady
case for
θ
=
0.5 and
t
=
0.001, 0.01, 0.1? What happens?
(d) What happens at
t
=
0.001, if the convective velocity is reduced?
θ
.Use
θ
=
0.5,
θ
=
0.4 and
θ
=
15.3
Investigate the unsteady convection problem:
∂
c
∂
∂
t
+
v
∂
u
u
∂
x
=
∂
∂
x
u
∂
x
