Biomedical Engineering Reference
In-Depth Information
9.2.2 A partition with different kinematics
Now let us turn our attention to the case in which a different kinematics is adopted
in a part of the domain, for example, in
Ω
1
. This kinematics is characterized by the
space
Q
1
. Since this kinematics is obtained by introducing known restrictions on the
behaviour and/or by increasing the regularity of the original kinematics (3D model)
characterized by the space
Q
, it follows that
Q
⊂Q
|
Ω
1
and therefore
T
Γ
a
(
Q
)
⊂
1
1
T
Γ
a
(
Q
|
Ω
1
)
1
has a symmetry of revolution and the
boundary conditions guarantee that the fields have the same property, then we can
adopt a kinematics with symmetry of revolution, and the associated model will be
defined in a 2-dimensional domain (see Fig. 9.2). On the other hand, if
. For example, if the domain
Ω
1
has a
characteristic length
L
that in one dimension is much greater than the length
l
orth
in
any orthogonal (transversal) direction (
L
Ω
l
orth
) and the boundary conditions al-
low us to consider a known behaviour over the transversal direction, then we can
adopt a kinematics with this property, and the associated model will be defined in a
1-dimensional domain (see Fig. 9.3).
Given
T
Γ
a
(
Q
1
)
is
automatically defined as well, as is the duality pairing between elements of this two
spaces, which is denoted by
Q
1
we have that
T
Γ
a
(
Q
1
)
is defined, and then its dual space
s
1
,
v
1
T
Γ
a
(
Q
1
)
×T
Γ
a
(
Q
1
)
.
(9.12)
Moreover, and for the sake of simplicity, let us introduce similar notations for the
partition
Ω
2
where the original kinematics has been maintained: (i)
Q
2
(
Q
2
=
Q
|
Ω
2
);
T
Γ
a
(
Q
2
)
and the corresponding duality pairing is
(ii)
T
Γ
a
(
Q
2
)=
T
Γ
a
(
Q
|
Ω
2
)
; (iii)
denoted by
s
2
,
v
2
T
Γ
a
(
Q
2
)
×T
Γ
a
(
Q
2
)
.
(9.13)
Fig. 9.2.
Two-dimensional kinematics on
Ω
1
(symmetry of revolution), yielding 3D-2D models
