Biomedical Engineering Reference
In-Depth Information
−
p
1
(
r
,
z
|
∂Σ
a
)
n
+
2
με
r
,
z
(
u
1
(
r
,
z
|
∂Σ
a
))
n
=
2π
−
n
d
1
2
p
2
(
r
,
φ
,
z
|
Γ
a
)
n
+
2
με
(
u
2
(
r
,
φ
,
z
|
Γ
a
))
φ
for
r
∈
∂Σ
a
.
(9.47b)
π
0
In this case the opposite situation occurs, as expressed by Eqs. (9.37) and (9.39),
respectively.
9.3.2 Coupling 3D and 1D models
In this case let us assume that over
Ω
1
a 1D model is set up, so we have that
Ω
1
≡
I
(
I
an interval), and we keep a full 3D model in
Ω
2
(see Fig. 9.3). Specifically, for all
t
we consider the following kinematics
u
1
→
u
1
,
z
(
z
)
∈Q
1
u
2
→
u
2
(
x
)
∈Q
2
,
(9.48)
where
u
1
,
z
is the component of
u
1
in the axial direction
e
z
, so it would be equivalent
to define
u
1
as
u
1
(
z
)=
u
1
,
z
(
z
)
e
z
. In this case, the spaces
Q
i
and
T
Γ
a
(
Q
i
)
,
i
=
1
,
2
are
H
1
Q
=
(
)
T
Γ
a
(
Q
)=R
,
I
(9.49)
1
1
H
1
H
1
/
2
Q
2
=
(
Ω
2
)
T
Γ
a
(
Q
2
)=
(
Γ
a
)
.
(9.50)
Here the coupling interface corresponds to
Γ
a
when viewed from
Ω
2
, and to the
point
ζ
a
when viewed from
I
, as seen in Fig. 9.3. As in the previous case, we write
x
|
Γ
a
→
(
,
,
z
|
Γ
a
)
γ
=
x
y
. For this case, noting that
n
coincides with
e
z
over
Γ
a
,
1 yields
1
|
Γ
a
|
u
1
,
z
(
z
|
ζ
a
)=
u
2
(
x
,
y
,
z
|
Γ
a
)
·
n
d
Γ
a
at
ζ
a
,
(9.51a)
Γ
a
μ
∂
u
1
,
z
∂
−
p
1
(
z
|
ζ
a
)+
2
z
(
z
|
ζ
a
)=
−
p
2
(
x
,
y
,
z
|
Γ
a
)+
2
με
(
u
2
(
x
,
y
,
z
|
Γ
a
))
n
·
n
for
(
x
,
y
)
∈
Γ
a
,
(9.51b)
0
=
2
με
(
u
2
(
x
,
y
,
z
|
Γ
a
))
n
·
t
for
(
x
,
y
)
∈
Γ
a
,
(9.51c)
where
t
is any vector tangent to the coupling interface
Γ
a
. In Eq. (9.51a) the de-
pendence on
is eliminated, establishing the continuity in a mean sense from
the point of view of the entire area
(
x
,
y
)
Γ
a
, whereas in Eq. (9.51b) an extension is made
such that the value of the quantity from the 1D model, a real value, is mapped to all
the points
(
x
,
y
)
over
Γ
a
from the 3D model. Finally, Eq. (9.51c) establishes that the
traction over
Γ
a
, when viewed from the 3D model, has no component tangent to the
coupling interface. Here the continuity of the velocity field is given in the sense of
R
(mean value) while the dual variable is continuous in a strong sense, that is the
sense of
H
−
1
/
2
.
In the same manner, for
(
Γ
)
a
γ
=
0 we obtain the following coupling equations
u
1
(
z
|
ζ
a
)
n
=
u
2
(
x
,
y
,
z
|
Γ
a
)
for
(
x
,
y
)
∈
Γ
a
,
(9.52a)
