Biomedical Engineering Reference
In-Depth Information
the current choice of parameters, the nodal solutions are exact. The right part of
the figure shows the computed flux, say flux
p
dx
. Again, the solid line
denotes the computed flux
p
, which is clearly discontinuous from one element
to the next, and the dashed line denotes the exact solution. The discontinuity of
the computed flux field is obvious: the field
u
is piecewise linear, therefore the
derivative
du
=
cdu
/
dx
is piecewise constant. The flux
p
does not necessarily have to be
piecewise constant: if the parameter
c
is a function of
x
, the flux
p
will be varying
within an element.
Mesh refinement leads to an improved approximate solution. For instance,
using ten rather than five elements, yields the results depicted in Fig.
14.10
. The
impact of a varying
c
, say
c
=
1
+
x
is depicted in Fig.
14.11
.
Changing the interpolation order of the shape functions
N
i
from linear to
quadratic, also has a significant impact on the results, in particular on the qual-
ity of the flux prediction. For the constant
c
case, the solution becomes exact.
Also for
c
/
=
1
+
x
a significant improvement can be observed, as depicted in
Fig.
14.13
.
0.14
0.5
0.4
0.12
0.3
0.1
0.2
0.1
0.08
u
p
0
0.06
-0.1
-0.2
0.04
-0.3
0.02
-0.4
0
-0.5
0
0
0.2
0.4
0.6
0.8
1
0.2
0.4
0.6
0.8
1
x
x
Figure 14.10
Ten element solution. Left: (solid line) approximate solution
u
h
(
x
), (dashed line) exact solution
u
(
x
).
Right: (solid line) approximate flux
p
h
(
x
), (dashed line) exact flux
p
(
x
).
