Biomedical Engineering Reference
In-Depth Information
these WECCs are
natural boundary conditions
that are satisfied automatically by
the solution of the extended variational formulation, the desired coupling conditions
are derived in a consistent way with respect to the kinematical restrictions adopted
for each dimensionally-reduced model.
Therefore our aim in this section is to develop this extended variational formula-
tion. At first this will be done in a compact (abstract) form. In doing this we believe
that the basic ideas and mechanical concepts behind this new variational principle
will emerge more clearly without being obscured by excessive mathematical details.
The application of such ideas to the specific problem of our concern is presented in
the forthcoming section.
Let
3
. Then, the compact form of a Clas-
sical Primal (kinematical) Variational Formulation of a generalized problem defined
in
Ω
be an open and bounded domain in
R
Ω
governed by a kinematics whose regularity is characterized by the space
Q
can
be cast as follows.
Problem 1.
Virtual Powers Principle. Find
u
∈U
such that
R
(
u
)
,
v
Q
=
0
∀
v
∈V .
(9.1)
The above expression corresponds to the well-known Virtual Power Principle
(VPP) in which
represents the set of kinematically admissible fields
u
.In
general, this set is a linear manifold given by a translation in
u
o
(an arbitrary element
of
U ⊂Q
U
V
Q
defining the kinematically admissible virtual variation
fields
v
. Taking into account essential boundary conditions prescribed on
) of the subspace
of
⊂
∂Ω
Γ
,
D
this means that
U
=
{
u
∈Q
;
u
|
Γ
D
=
u
},
(9.2)
V
=
{
v
∈Q
;
v
|
Γ
D
=
0
}.
Q
, generally non-linear
In Eq. (9.1),
R
(
·
)
is the
equilibrium
operator from
Q
into
and time-dependent, and
·,·
Q
, which is linear with respect to
v
, stands for the vir-
tual power between the kinematical virtual variation field
v
and elements belonging
to
Q
, the corresponding dual space of
.
Problem (9.1) is well-posed provided that (see [44])
Q
Q
is a separable, reflexive
Banach space and
R
satisfies the following properties: (i)
R
is weakly sequentially
continuous, (ii) the restriction of
R
to finite-dimensional supspaces of
Q
is contin-
uous, (iii)
is bounded.
For the problems we are aiming at in this work it is important to remark that we
choose the space
R
is coercive and (iv)
R
to guarantee the continuity of the fields
u
through any (suffi-
ciently smooth) internal interface
Q
Γ
a
in the sense defined by the space of traces of
Q
T
Γ
a
(
Q
)
elements in
over that boundary,
. We will denote this continuity property
by
u
|
Γ
a
=
0
in the sense given by
T
Γ
a
(
Q
)
,∀
Γ
.
(9.3)
a
Furthermore, in the sense of the duality product associated to
, the varia-
tional Problem 1 gives, as a natural boundary condition (known in the literature as
jump equation
or WECC), the continuity of the traction on any sufficiently smooth
T
Γ
a
(
Q
)
