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
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