Civil Engineering Reference
In-Depth Information
with [
M
]and[
K
] equivalent mass and stiffness matrices:
u
s
i
δu
s
i
d
δu
s
[
M
]
u
s
ρ
˜
=
{
}
(13.45)
[
K
]
σ
ij
δe
s
ij
d
δu
s
u
s
=
{
}
(13.46)
[
Q
] is the acoustic admittance matrix of the interstitial fluid with [
H
]
and [
Q
] equivalent kinetic and compression energy matrices for the fluid phase
[
H
]
/ω
2
[
A
]
=
−
φ
2
ρ
22
∂p
∂x
i
∂(δp)
∂x
i
[
H
]
d
=
δp
{
p
}
(13.47)
d
φ
2
R
pδp
d
[
Q
]
=
δp
{
p
}
(13.48)
[
C
]
[
C
1
]
[
C
2
] is a volume coupling matrix between the skeleton and the interstitial
=
+
fluid variables
α
δ
∂p
∂x
i
u
s
i
d
φ
[
C
1
]
[
C
1
]
T
δu
s
u
s
=
{
p
}+
δp
{
}
(13.49)
φ
1
+
δ(pu
s
i,i
)
d
S
=
δu
s
Q
R
[
C
2
]
{
p
}+
δp
[
C
2
]
T
{
u
s
}
(13.50)
F
s
{
}
is the surface loading vector for the skeleton
T
i
δu
s
i
d
S
=
δu
s
{
F
s
}
(13.51)
σ
ij
n
j
represent the components on the coordinate axis
x
i
of the total stress
vector T. Finally
where
T
i
=
F
f
{
}
is the surface kinematic coupling vector for the interstitial fluid
φ(u
n
−
u
s
n
)δp
d
S
F
f
=
δp
{
}
(13.52)
In the above equations,
{}
denotes a vector and
its transpose.
The system of equations (13.44) is first solved in terms of the porous solid phase
nodal displacements and interstitial nodal pressures. Next, the vibroacoustic indicators of
interest are calculated.
The main limitation of the numerical implementations of these formulations resides in
their computational cost. Typical implementations use linear and quadratic elements. Due
to the highly dissipative nature of the domains and the existence of different wavelength
scales, classical mesh criteria used for elastic domains and modal techniques are not
strictly applicable. A discussion of meshing criteria and convergence can be found in
(Panneton 1996, Dauchez
et al
. 2001, Rigobert
et al
. 2003, Horlin, 2004). As an example,
it is found that, for linear elements, as many as 12 elements for the smallest 'Biot'
wavelength may be required, in some applications, for convergence! Several authors
studied alternatives to alleviate the computational cost. Examples of techniques include
the use of selective modal analysis (Sgard
et al
. 1997), use of complex modes (Dazel
