Civil Engineering Reference
In-Depth Information
structures, this formulation has the disadvantage of requiring cumbersome calculations
for large finite element models and spectral analyses. To alleviate these difficulties, a
mixed displacement - pressure formulation has been developed (Appendix 6.A). Note in
passing that similar variants of Equation (13.1), using the solid phase displacement
u
s
and the displacement flux
w
=
−
u
s
)
, have been developed and implemented in
the finite element context (Coyette and Pelerin 1994, Coyette and Wynendaele 1995,
Johansen
et al
. 1995, Dazel 2005). The use of the displacement flux simplifies certain
coupling conditions compared with the classical
(
u
s
,
u
f
)
formulation, but their numerical
performances are similar.
φ(
u
f
13.3
The mixed displacement - pressure formulation
Recall Equation (6.A.22):
#
$
div σ
s
(
u
s
)
+
ω
2
ρ
u
s
+
γ
grad
p
=
0
ω
2
ρ
22
ω
2
ρ
22
φ
2
(13.3)
˜
γdiv
u
s
%
p
+
R
p
−
=
0
The weak integral equations associated with Equation (13.3) are given by Atalla
et al
.
(1998)
ω
2
∂p
∂x
i
δu
s
i
d
σ
ij
(
u
s
)e
s
ij
(δ
u
s
)
d
u
s
i
δu
s
i
d
σ
ij
(
u
s
)
n
j
δu
s
i
d
S
−
ρ
−
γ
−
=
0
φ
2
ω
R
pδp
d
φ
2
∂p
∂x
i
∂(δp)
∂x
i
−
∂(δp)
∂x
i
u
s
i
d
−
γ
2
ρ
22
γ
u
n
−
δp
d
S
=
0
φ
2
ρ
22
ω
2
∂p
∂n
+
(δ
u
s
,δp)
∀
(13.4)
Here,
and
refer to the poroelastic domain and its bounding surface (Figure 13.1).
δ
u
s
and
δp
are admissible variations of the solid phase displacement vector and the
interstitial fluid pressure of the poroelastic medium, respectively.
n
is the unit outward
normal vector around the bounding surface
, and subscript
n
denotes the normal com-
ponent of a vector. To simplify the formulation and its coupling with elastic, porous and
acoustic media, Atalla
et al
. (1998) used the fact that, for the majority of porous materials
used in acoustics, the bulk modulus of the porous material is negligible compared with
the bulk modulus of the material from which the skeleton is made:
K
b
/K
S
1, thus:
Q/ R)
K
b
/K
s
=
1. Using this assumption, the coupling of the (
u
s
,p
)for-
mulation with elastic, acoustic and porous elastic media becomes simple. A detailed
derivation and discussion of these coupling conditions are given by Debergue
et al
.
(1999). It is shown that in this formulation, a poroelastic medium couples naturally
with acoustic and poroelastic media, and couples through a classical fluid - structure cou-
pling matrix with elastic media (solids, septum, etc.). A variant of this formulation that
eliminates recourse to the approximation
K
b
/K
S
φ(
1
+
=
−
1
1 and couples naturally with elastic
