Civil Engineering Reference
In-Depth Information
of freedom (dof) and keeping only the dof attached to the master structure and the acous-
tic cavity. In addition, to allow for incompatible meshes, Equation (13.53) is augmented
using Lagrange multipliers to enforce the continuity of the pressure and the displacement
at the interface
a
and
e
, respectively. The resulting condensed impedance matrix is
added to enrich and couple the modal impedance matrices of the master structure and
the cavity. This formulation has the advantage of considerably simplifying the modelling
effort by: (i) not increasing the size of the global master structure-cavity model and (ii)
allowing for the direct and independent modelling of the porous component, the mas-
ter structure and the acoustic cavity. Using this methodology, the accurate solution of
a fully trimmed vehicle is possible (Anciant
et al
. 2006). An example is discussed in
Section 13.9.7.
13.7
Dissipated power within a porous medium
The classical vibroacoustic indicators (kinetic energy, quadratic velocity, quadratic pres-
sure, absorption coefficient, transmission loss, etc.) can be easily calculated from the
solution of the formulations presented. One important feature of the FE implementation
is the ability to break down the dissipated power within a poroelastic material in terms of
the relative contributions of the viscous, thermal and structural effects. The expressions of
these contributions are well known for the
(
u
s
,
u
f
)
formulations (Dauchez
et al
., 2002).
For the
(
u
s
,p)
formulations, an initial derivation of the expression of these powers was
given by Sgard
et al
. (2000). Recently, a formal derivation was given by Dazel
et al
.
(2008). Dazel started from the
(
u
s
,
u
f
)
-based expressions to derive both the dissipated
powers and stored energy expressions for the
(
u
s
,p)
formalism. He highlighted in par-
ticular the proper interpretation of Sgard
et al
. (2000) initial expressions. In this section,
and for the sake of conciseness, the simple derivation of Sgard
et al
. (2000) is recalled.
With the following particular choice for the admissible functions,
δ
u
s
jω
u
s
∗
for
the solid-phase displacement vector and
δp
=−
jωp
∗
for the fluid-phase interstitial pres-
sure, where
f
∗
denotes the complex conjugate of
f
, the weak integral form is rewritten
=−
jω
jω
3
σ
ij
(
u
s
)
:
s
(
u
s
∗
)
d
u
s
i
u
s
i
d
−
+
ρ
˜
s
elas
s
iner
jω
jω
φ
2
R
pp
∗
d
φ
2
αρ
0
ω
2
∂p
∗
∂x
i
d
∂p
∂x
i
+
−
f
elas
f
iner
∂p
∂x
i
u
s
i
d
φ
1
+
(pu
s
∗
jω
jω
Q
R
∂p
∗
φ
α
∂x
i
u
s
∗
p
∗
u
s
i,i
)
d
+
+
+
+
i
i,i
s
coup
+
jω
σ
ij
n
j
u
s
i
d
S
d
S
+
jω
φ(u
n
−
u
s
n
)p
∗
d
S
=
0
(13.54)
s
exc
