Civil Engineering Reference
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et al
. 2002; Dazel 2005), plane wave decomposition (Sgard 2002), adaptive meshing
(Castel 2005), axisymmetric implementations (Kang
et al
. 1999, Pilon 2003) and finally
use of hierarchical elements (Horlin
et al
. 2001, Horlin 2004, Rigobert 2001, Rigobert
et al
. 2003, Langlois 2003). The latter authors show that
p
-implementations allow for
quick convergence and accurate results. Still, the classical linear and quadratic elements
are the widely used due to the simplicity of their implementations and their generality (for
instance handling complex geometries with discontinuities; examples include foams with
embedded masses or cavities and domains made up of a patchwork of several materials;
see Sgard
et al
. 2007).
An interesting implementation based on the use of the mixed formulation is given
by Hamdi
et al
. (2000). Consider the practical configuration wherein a typical porous
component is attached to a master structure along a part
e
of its boundary and is
coupled on the remaining part
a
to an acoustic cavity, as shown in Figure 13.3. Using
the notations of Section 13.4, the weak integral form of the porous component accounting
for the coupling with the master structure and the cavity is given by
φ
ω
2
αρ
0
R
pδp
d
φ
2
∂p
∂x
i
∂(δp)
∂x
i
p
[
σ
ij
δe
s
ij
−
ω
2
˜
u
s
i
δu
s
i
]d
ρ
+
−
p
α
δ
∂p
∂x
i
u
s
i
d
−
p
φ
1
+
δ(pu
s
i,i
)
d
Q
R
φ
−
(13.53)
p
e
T
i
δu
s
i
d
S
(δu
s
i
,δp)
+
a
δ(u
n
p(
x
))
d
S
x
−
a
W
n
δp
d
S
−
=
∀
0
σ
ij
n
j
are the components of the total stress vector at the
structure - porous interface
e
and
W
n
is the normal acoustic displacement at the inter-
face
a
with the cavity. Note that the last two terms of Equation (13.53) represent the
energy exchanged between the porous component and its surrounding environment. The
first term represents the acoustic energy absorbed by the porous component and the sec-
ond term the mechanical energy absorbed by the porous component from the master
structure. The FEM discretization of Equation (13.53) allows for the computation of the
mixed impedance matrix of the porous component by eliminating all its internal degrees
In this equation,
T
i
=
Master structure
e
Γ
e
Γ
Acoustic Cavity
Ω
n
Porous component
Figure 13.3
Porous component attached to a master structure - cavity system.
