Biomedical Engineering Reference
In-Depth Information
cases 10 contour lines are shown ranging from 0.1 to 1 with increments of 0.1.
The effect of an increasing velocity is clearly demonstrated.
Exercises
16.1 The weak form of the two-dimensional convection-diffusion equation is
given by
w
∂
u
u
)
d
∇
+ ∇
(
c
∇
∂
t
+
w
v
·
u
w
·
c
∇
=
wf d
+
w
n
·
u
=
d
.
After discretization, the element inertia matrix is defined as
T
d
,
M
e
=
∼ ∼
e
while the element matrix related to the convective part is given by
v
x
d
∼
d
.
T
T
dx
+
v
y
d
∼
C
e
=
∼
dy
e
Consider the element as depicted in the figure below.
y
4
3
x
1
2
This is a bi-linear element. The element spans the spatial domain
−
2
≤
x
≤
2 and 2
≤
y
≤
2.
(a) Compute the element inertia matrix
M
e
using a 2
×
2 Gauss integration
rule. It is recommended to use MATLAB for this computation.
(b) Suppose that the location of the integration points coincides with the
nodes of the element. What is the element inertia matrix
M
e
in this
case?
(c) Compute the matrix
C
e
=
=
×
if
v
x
1 and
v
y
0, using a 2
2 Gauss
integration rule.
(d)
Suppose that along the edge of the element located at
x
=
2 a constant
c
∇
flux
q
=
n
·
u
is prescribed. Then compute the column
