Biomedical Engineering Reference
In-Depth Information
(
T
Γ
a
(
Q
1
))
⊥
is the orthogonal space to
where
in the sense of the duality
product defined by (9.13). That is, for all
s
2
∈T
Γ
a
(
Q
2
)
we can write
T
Γ
a
(
Q
1
)
s
2
∈T
Γ
a
(
Q
2
)
⇔
s
2
=
s
21
+
s
2
f
,
(9.19)
)
and
where
s
21
∈T
Γ
a
(
Q
1
s
2
f
,
T
Γ
a
(
Q
2
)
×T
Γ
a
(
Q
2
)
=
∀
∈T
Γ
a
(
Q
)
.
u
1
0
u
1
(9.20)
1
This yields
s
2
,
u
1
T
Γ
a
(
Q
2
)
×T
Γ
a
(
Q
2
)
=
s
21
,
u
1
T
Γ
a
(
Q
1
)
×T
Γ
a
(
Q
1
)
∀
u
1
∈T
Γ
a
(
Q
)
.
(9.21)
1
)
, decompositions (9.14)
∈T
Γ
a
(
Q
)
∈T
Γ
a
(
Q
For two arbitrary elements
u
2
and
s
2
2
2
and (9.18) lead to
s
2
,
u
2
T
Γ
a
(
Q
2
)
×T
Γ
a
(
Q
2
)
=
s
21
,
u
21
T
Γ
a
(
Q
1
)
×T
Γ
a
(
Q
1
)
+
s
2
f
,
u
2
f
T
Γ
a
(
Q
2
)
×T
Γ
a
(
Q
2
)
.
(9.22)
On the other hand, since now the kinematics at both sides of
Γ
a
are different, it is
necessary to redefine the sets
U
i
of kinematically admissible fields
u
i
,
i
=
1
,
2
U
i
=
{
u
i
∈Q
i
;
u
i
|
Γ
D
i
=
u
i
}.
(9.23)
From the above considerations, we can formulate a
new
extended varia-
tional statement which we named
Non-Classical Extended Variational Formu-
lation
.
Problem 3.
Non-Classical
Extended Variational Formulation. For
γ
∈
[
0
,
1
]
find
t
2
)
∈U
1
×U
2
×T
Γ
a
(
Q
1
)
×T
Γ
a
(
Q
2
)
such that
(
u
1
,
u
2
,
t
1
,
R
1
(
u
1
)
,
v
1
Q
1
+
R
2
(
u
2
)
,
v
2
Q
2
+
γ
t
1
,
(
v
1
−
v
2
)
T
Γ
a
(
Q
1
)
×T
Γ
a
(
Q
1
)
+(
1
−
γ
)
t
2
,
(
v
1
−
v
2
)
T
Γ
a
(
Q
2
)
×T
Γ
a
(
Q
2
)
+
γ
s
1
,
(
u
1
−
u
2
)
T
Γ
a
(
Q
1
)
×T
Γ
a
(
Q
1
)
+(
1
−
γ
)
s
2
,
(
u
1
−
u
2
)
T
Γ
a
(
Q
2
)
×T
Γ
a
(
Q
2
)
s
2
)
∈V
1
×V
2
×T
Γ
a
(
Q
1
)
×T
Γ
a
(
Q
2
)
.
∀
(
v
1
,
v
2
,
s
1
,
(9.24)
In the above formulation,
R
i
is the
equilibrium
operator associated to the kinematics
characterized by
Q
i
,
i
=
1
,
2 (recalling that
Q
=
Q
|
Ω
2
,so
R
=
R
).
2
2
now playing a fundamental
role from the mechanical point of view. In fact, this parameter governs how the con-
tinuity over
As we will show, Problem 3 is
γ
-dependent with
γ
a
of the primal variables (
u
1
and
u
2
) and of the dual ones (
t
1
and
t
2
)
is established. Here it is important to notice the difference between Problem 2 and
Problem 3 with respect to
Γ
defines the kinematics
that will prevail on the coupling interface. This is not the case for Problem 2 since
the kinematics are the same on both sub-domains.
γ
. In Problem 3 the parameter
γ
