Biomedical Engineering Reference
In-Depth Information
[11, 58]) and, basically, consists in adding to the original VPP, given by Eq. (9.1),
the virtual powers (duality products), over the coupling interface
Γ
a
, between the dis-
continuity and admissible variations of the traction, and between the traction and ad-
missible variations of the discontinuity (see Problem 2 below). Following the above
considerations, the new extended variational formulation takes the following form.
Problem 2.
Classical
Extended Variational Formulation. For
γ
∈
[
0
,
1
]
find
(
u
1
,
u
2
,
×T
Γ
a
(
Q
)
×T
Γ
a
(
Q
)
such that
t
1
,
t
2
)
∈U
×U
1
2
R
(
)
,
Q
|
Ω
1
+
R
(
)
,
Q
|
Ω
2
u
1
v
1
u
2
v
2
+
γ
t
1
,
(
v
1
−
v
2
)
T
Γ
a
(
Q
)
×T
Γ
a
(
Q
)
+(
1
−
γ
)
t
2
,
(
v
1
−
v
2
)
T
Γ
a
(
Q
)
×T
Γ
a
(
Q
)
+
γ
s
1
,
(
u
1
−
u
2
)
T
Γ
a
(
Q
)
×T
Γ
a
(
Q
)
+(
1
−
γ
)
s
2
,
(
u
1
−
u
2
)
T
Γ
a
(
Q
)
×T
Γ
a
(
Q
)
s
2
)
∈V
1
×V
2
×T
Γ
a
(
Q
)
×T
Γ
a
(
Q
)
,
∀
(
v
1
,
v
2
,
s
1
,
(9.8)
where
2.
It is worth noting that the above problem is consistent with Problem 1 in the sense
that for any
V
i
is the kinematical virtual variation space associated to
U
i
,
i
=
1
,
the solutions of both problems are exactly the same provided
the data are regular enough. In fact, for any
γ
∈
[
0
,
1
]
γ
∈
[
0
,
1
]
and from Eq. (9.8) we obtain
that
s
2
∈T
Γ
a
(
Q
)
.
s
γ
,
(
u
1
−
u
2
)
T
Γ
a
(
Q
)
×T
Γ
a
(
Q
)
=
0
∀
s
γ
=
γ
s
1
+(
1
−
γ
)
(9.9)
In other words, the above equation guarantees the continuity of the solution as re-
quired by Problem 1 (see Eq. (9.3)). However, in this classical extended variational
formulation the continuity is
not imposed
but is
automatically
satisfied by the so-
lution
of Problem 2.
On the other hand, denoting
t
γ
=
γ
(
u
1
,
u
2
)
t
1
+(
1
−
γ
)
t
2
,
γ
∈
[
0
,
1
]
, and from Eq. (9.8) we
also obtain
t
γ
+
R
(
)
|
Γ
a
,
T
Γ
a
(
Q
)
×T
Γ
a
(
Q
)
=
∀
∈V
,
u
1
v
1
0
v
1
(9.10)
1
t
γ
−R
(
u
2
)
|
Γ
a
,
v
2
T
Γ
a
(
Q
)
×T
Γ
a
(
Q
)
=
0
∀
v
2
∈V
2
.
(9.11)
The above equations together with (9.9) ensure that the jump equation required by
Problem 1 (see Eq. (9.4)) is satisfied, and works as natural boundary condition. Fur-
thermore, the above expression gives the mechanical meaning of the dual variable
t
γ
(a generalized force over
Γ
a
).
Here we want to highlight that:
since the kinematics adopted on both parts of the
domain are the same, the solution of
Problem 2 (i)
does not depend on the value
of the parameter
and
(ii)
is exactly the same as the solution of
Problem 1
pro-
vided the data are regular enough
. Despite the fact that this is well known in the
literature, it is important to note that Problem 2 gives us a new insight for the case
when
different kinematics
are taken on each partition of the domain, as is the case
when coupling dimensionally-heterogeneous models. In what follows we are going
to develop this approach, which, to the best of our knowledge, is new.
γ