Biomedical Engineering Reference
In-Depth Information
Fig. 9.3.
One-dimensional kinematics on Ω
1
, yielding 3D-1D models
Since
Q
1
⊂Q
|
Ω
1
then
T
Γ
a
(
Q
1
)
⊂T
Γ
a
(
Q
|
Ω
1
)=
T
Γ
a
(
Q
|
Ω
2
)=
T
Γ
a
(
Q
2
)
,
and it
follows that
T
Γ
a
(
Q
2
)
can be described by the following direct decomposition
T
Γ
a
(
Q
2
)=
T
Γ
a
(
Q
1
)
⊕
(
T
Γ
a
(
Q
1
)
)
⊥
,
(9.14)
(
T
Γ
a
(
Q
1
)
)
⊥
are orthogonal, in the sense of the duality prod-
uct defined by Eq. (9.12), to any element belonging to
where the elements of
T
Γ
a
(
Q
1
)
. In other words, for
all fields
u
2
∈Q
2
the trace over
Γ
a
,
u
2
|
Γ
a
∈T
Γ
a
(
Q
2
)
, can be uniquely decomposed
as
u
2
|
Γ
a
∈T
Γ
a
(
Q
2
)
⇔
u
2
|
Γ
a
=
u
21
+
u
2
f
,
(9.15)
where
u
21
∈T
Γ
a
(
Q
1
)
and
)
.
s
1
,
u
2
f
T
Γ
a
(
Q
1
)
×T
Γ
a
(
Q
1
)
=
0
∀
s
1
∈T
Γ
a
(
Q
(9.16)
1
It is worth noting that by using the above decomposition the following expression is
well defined for
u
2
|
Γ
a
∈T
Γ
a
(
Q
)
2
)
.
,
T
Γ
a
(
Q
1
)
×T
Γ
a
(
Q
1
)
=
,
T
Γ
a
(
Q
1
)
×T
Γ
a
(
Q
1
)
∀
∈T
Γ
a
(
Q
s
1
u
2
s
1
u
21
s
1
(9.17)
1
Analogously to (9.14), we introduce the following decomposition
T
Γ
a
(
Q
2
)
=
T
Γ
a
(
Q
1
)
⊕
(
T
Γ
a
(
Q
1
))
⊥
,
(9.18)
