Biomedical Engineering Reference
In-Depth Information
A
0
A
4
is the speed
of the wave
p
ropagation along the vessel. Condition (
14
) corresponds to a CFL
number of
√
3
.
The final finite element solution for the area and flow rate is obtained by finding,
at each time step
t
n
+
1
,
U
n
+
1
h
β
where
h
is the size of the spatial mesh and
c
=
c
(
A
;
A
0
;
β
)=
2
ρ
N
+
0
U
n
+
1
1
is the
space of
P
1 finite elements in 1D for the uniform mesh associated to
h
spacing, and
{
ψ
i
}
(
z
)=
∑
ψ
i
(
z
)
∈
V
h
(
a
,
b
)
,where
V
h
(
a
,
b
)
i
=
i
N
i
1
its basis, satisfying the following expression for the interior nodes:
=
t
F
n
t
B
n
,
ψ
j
,
∂ψ
j
∂
−
Δ
t
2
−
Δ
t
2
U
n
+
1
h
U
h
,
ψ
j
)+
Δ
H
n
B
n
B
U
B
n
(
,
ψ
j
)=(
−
Δ
z
H
n
∂
B
U
∂
z
,
ψ
j
t
2
2
F
n
∂
t
2
2
F
n
∂
−
Δ
z
,
∂ψ
j
+
Δ
,
j
=
1
,···,
N
,
n
=
0
,···,
M
−
1
.
∂
z
(15)
=
a
Here
U
h
is a suitable approximation of the initial data,
(
u
,
v
)
:
u
·
v
d
z
represents
,
ψ
j
N
j
the inner product in
V
h
(
a
,
b
)
are the basis functions of
V
h
(
a
,
b
)
,
Δ
t
=
=
1
t
n
+
1
t
n
,and
−
0
1
0
0
H
=
,
B
U
=
,
Q
2
A
2
ρ
A
0
A
2
2
β
Q
A
2
Q
A
1
A
K
r
−
K
r
−
α
+
α
2
Q
0
=
,
=
.
F
B
Q
2
A
β
A
0
A
2
K
r
A
+
−
α
3
ρ
System (
15
) must be supplemented with proper initial,
U
h
, and boundary
conditions for the solution
U
n
+
1
h
, at the left and right boundary points,
z
=
a
and
z
=
b
, respectively. In the present work, the initial conditions were taken to be
A
0
A
0
and
Q
0
(
z
)=
(
z
)=
0.
Compatibility Conditions for the 1D Model
By choosing relation (
12
), the pressure may be eliminated from the momentum
equation, and system (
11
) becomes hyperbolic, with two distinct eigenvalues (see
[
6
,
17
] for the characteristic analysis of system (
11
))
=
±
,
λ
u
c
(16)
1
,
2
where
c
is the speed of the propagation of waves along the artery, defined above. The
eigenfunctions, or characteristic variables, corresponding to the eigenvalues
λ
2
,are
1
,
defined by
Search WWH ::
Custom Search