Civil Engineering Reference
In-Depth Information
patches of solid and porous elements,
...)
, the coupling boundary term of the elastic
medium, Equation (13.10), reads
p
a
δu
n
d
S
=
jω
I
i
=
u
n
(
y
)Z(
x
,
y
)δu
n
(
x
)
d
S
y
d
S
x
i
i
i
+
ε
(13.42)
p
p
b
(
y
)δu
n
(
x
)
d
S
y
d
S
x
=
0
p
Z(
x
,
y
)
is an impedance operator given by
ρ
0
ω
k
mn
N
mn
ϕ
mn
(x
1
,x
2
)ϕ
mn
(y
1
,y
2
)
Z(
x
,
y
)
=
(13.43)
mn
Equation (13.38) depicts the coupling with the waveguide in terms of radiation
impedance in the emitter and receiver media, together with a blocked-pressure loading.
Note that this equation holds if the elastic structure is replaced by a septum.
13.5
Other formulations in terms of mixed variables
To eliminate the drawbacks of the
(
u
s
,
u
f
)
formulation, G oransson (1998) presented a
symmetric
(
u
s
,p,ϕ)
formulation wherein the fluid is described by both the fluid pressure
and the fluid displacement scalar potential. In particular, this formulation assumes the fluid
displacement to be rotational free which is not true due to the strong inertial and viscous
coupling between the two phases. Using a simple plate - foam - plate example, Horlin
(2004) showed, as a consequence of this rotational free assumption, that this formulation
overestimates the dissipation associated with the relative motion between the fluid and
the frame and in consequence the viscous damping. To eliminate this limitation, Horlin
(2004) proposed an extension based on four variables: frame displacement, acoustic pore
pressure, scalar potential and the vector potential of the fluid displacement. The associated
coupling conditions and convergence studies in the context of both
h
and
p
elements
implementation are also given. Numerical examples show that this formulation leads to
the same solutions as the previously described
(
u
s
,
u
f
)
and
(
u
s
,p)
formulations. In
consequence, the latter is preferred for its computational efficiency. Note that variants on
the
(
u
s
,p)
formulation can be found in Hamdi
et al
. (2000) and Dazel (2005).
13.6
Numerical implementation
The numerical implementation of the presented formulations presented, using the finite
element method, is classical and is discussed in several references (e.g. Atalla
et al
.
1998). For example, using classical finite element notations, the discretized form of
Equation (13.7) leads to the following linear system
u
s
p
F
s
F
f
−
[
C
]
[
Z
]
=
(13.44)
−
[
C
]
T
[
A
]
where
{
u
}
and
{
p
}
represent the solid phase and the fluid phase global nodal variables,
ω
2
[
M
]
[
K
] is the mechanical impedance matrix of the skeleton
respectively. [
Z
]
=−
+
