Biomedical Engineering Reference
In-Depth Information
From Eq. (9.33) we have
u
1
=
u
21
in the sense of
T
Γ
a
(
Q
)
,
1
(9.36)
(
T
Γ
a
(
Q
1
)
)
⊥
,
u
2
f
=
0
in the sense of
that is
u
1
=
u
2
in the sense of
T
Γ
a
(
Q
2
)
.
(9.37)
In this case the continuity condition of the primal variables is satisfied in exactly the
same sense as in the original problem, noting that now
u
2
f
is identically null over
Γ
a
.
From the mechanical point of view this is a stronger restriction over the behaviour of
u
2
on
Γ
a
that certainly impacts on the behaviour of the solution in
Ω
2
. On the other
hand, from (9.34)-(9.35) we obtain
t
21
=
−R
1
(
T
Γ
a
(
Q
1
)
,
u
1
)
|
Γ
a
=
R
21
(
u
2
)
|
Γ
a
in the sense of
(9.38)
t
2
f
=
R
2
f
(
u
2
)
|
Γ
a
=
0
obtained after solving (9.24)
,
implying that
T
Γ
a
(
Q
1
)
.
−R
1
(
u
1
)
|
Γ
a
=
R
2
(
u
2
)
|
Γ
a
in the sense of
(9.39)
Again, due to the difference in the governing kinematics, the continuity of the dual
variables is not satisfied as in the original Problem 9.1 (see Eq. (9.4)). In fact there is
a fluctuation of the traction coming from
Ω
2
, and defined by the variational equation
(9.24), which is not identically null as expressed by (9.38)
2
.
Before closing this section it is worth summarizing the results already obtained
with the non-classical extended variational formulation given by Problem 3.
1. The non-classical extended variational formulation given by Problem 3 gives an
appropriate framework to deal with the coupling of dimensionally-heterogeneous
models through the concept of kinematically incompatible models.
2. With this variational formulation the coupling conditions to be prescribed on the
coupling interface are already accounted for, and are satisfied by the solution as
natural boundary conditions.
3. The selection criterion for choosing the value of
is dictated by purely mechani-
cal considerations, since it defines the sense in which the continuity of the primal
and dual variables is addressed by the formulation at the continuous level.
4. The continuity equations for
γ
γ
=
1 are (see (9.30) and (9.32))
u
1
=
u
2
in the sense of
T
Γ
a
(
Q
1
)
,
(9.40)
T
Γ
a
(
Q
2
)
,
−R
1
(
u
1
)
|
Γ
a
=
R
2
(
u
2
)
|
Γ
a
in the sense of
and for
γ
=
0 are (see (9.37) and (9.39))
u
1
=
u
2
in the sense of
T
Γ
a
(
Q
2
)
,
(9.41)
T
Γ
a
(
Q
1
)
.
−R
1
(
u
1
)
|
Γ
a
=
R
2
(
u
2
)
|
Γ
a
in the sense of
