Biomedical Engineering Reference
In-Depth Information
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
x
Figure 15.4
Solution of the steady convection-diffusion problem using 20 elements at
v
= 25.
The set of equations that results after discretization is, as usual:
K
∼
=
f
∼
.
If only two linear elements of equal length
h
are used the coefficient matrix
K
may
be written as
⎡
⎤
⎡
⎤
−
110
−
101
0
1
−
10
−
12
−
1
0
v
2
c
1
h
⎣
⎦
+
⎣
⎦
K
=
,
−
11
−
11
where the first, asymmetric, part corresponds to the convective term and the sec-
ond, symmetric, part to the diffusion term. In the absence of a source term the
second component of
f
∼
is zero. Let
u
1
and
u
3
be located at the ends of the domain
such that
u
1
=
0 and
u
3
=
1, then
u
2
is obtained from
vh
2
c
.
2
u
2
=
1
−
An oscillation becomes manifest if
u
2
<
0. To avoid this, the
element Peclet
number
should be smaller than one:
vh
2
c
<
Pe
h
=
1.
Consequently, at a given convective velocity
v
and diffusion constant
c
, the mesh
size
h
can be chosen such that an oscillation-free solution results. In particular
for large values of
v
/
c
this may result in very fine meshes. To avoid the use of
