UNIVERSITE DU MAINE
Le Mans - France
HABILITATION A DIRIGER DES RECHERCHES
Spécialité: Acoustique
Numerical methods
for the Biot theory in acoustics
par
Olivier DAZEL
soutenue devant le jury composé de
Jean-François Allard (Pr. em.)
LAUM (Le Mans)
Président
Noureddine Atalla (Pr)
GAUS (Canada)
Rapporteur
Jean-François Deü (Pr)
LMSSC-CNAM (Paris)
Rapporteur
Peter Göransson (Pr)
MWL (Suède)
Rapporteur
Jean-Pierre Coyette (Pr)
UCLouvain (Belgique)
Claude Depollier (Pr)
LAUM (Le Mans)
Marie-Annick Galland (Pr)
LMFA-ECL (Lyon)
Claude-Henri Lamarque (Dr-HDR)
ENTPE-DGCB (Lyon)
Franck Sgard (Dr-HDR)
IRSST (Canada)
Laboratoire d’Acoustique de l’Université du Maine - UMR CNRS 6613
2
To the memory of Walter Lauriks,
In all chaos, there is a cosmos
In all disorder, a secret order.
Carl Gustav Jung
4
Contents
1 Biot model for sound propagation in poroelastic materials
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Homogenized fields . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Displacement fields . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Continuity relations for displacements . . . . . . . . . . . . .
1.2.3 Stress tensors . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.4 Continuity relations of stress tensors at interfaces . . . . . . .
1.3 Lagrangian approach . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Former Biot models . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.1 Original formulation . . . . . . . . . . . . . . . . . . . . . . .
1.4.2 Modified Biot Formulation . . . . . . . . . . . . . . . . . . . .
1.5 Strain decoupled formulation [A3] . . . . . . . . . . . . . . . . . . . .
1.5.1 Fields of interest . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.2 Gedanken experiments . . . . . . . . . . . . . . . . . . . . . .
1.5.3 Introduction of dissipation . . . . . . . . . . . . . . . . . . . .
1.6 {us , p} mixed formulations . . . . . . . . . . . . . . . . . . . . . . . .
1.7 Biot’s waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8 Equivalent fluid and limp models . . . . . . . . . . . . . . . . . . . .
1.9 From {us , uW } to {us , ut } formulation for sound absorbing materials
1.10 Normal incidence surface impedance . . . . . . . . . . . . . . . . . .
1.11 Transversely isotropic porous material [A8] . . . . . . . . . . . . . .
1.11.1 Wave equations . . . . . . . . . . . . . . . . . . . . . . . . . .
1.11.2 Plane waves propagating in TIPM . . . . . . . . . . . . . . .
1.11.3 Theoretical and experimental results . . . . . . . . . . . . . .
1.12 Nonlinear Biot waves in non consolidated granular media [A12] . . .
1.13 Biot model for double porosity media [A19] . . . . . . . . . . . . . .
1.A Gedanken experiments for {us , uf } formulation . . . . . . . . . . . .
1.B Equivalent fluid models . . . . . . . . . . . . . . . . . . . . . . . . .
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17
17
18
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19
21
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22
23
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25
26
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28
29
30
32
33
34
34
35
35
37
39
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44
49
50
2 Numerical methods for porous materials
51
2.1 Plane waves methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.1.2 Principle of transfer matrices . . . . . . . . . . . . . . . . . . . . . . 52
2.1.3 Expression of reflexion and transmission coefficients . . . . . . . . . 53
2.1.4 Expression of the Transfer Matrix of a transversely isotropic poroelastic material [A8] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.1.5 Stroh formalism for materials with gradient of properties [A16] . . . 57
2.1.6 Divergence of TMM and alternative method [C16] . . . . . . . . . . 59
2.1.7 Coupled Mode Model (CMM) . . . . . . . . . . . . . . . . . . . . . . 62
5
2.2
Finite-element models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.2.1 Outline of the finite-element method . . . . . . . . . . . . . . . . . . 67
2.2.2 Implementation of the FEM for poroealstic materials: example of
the {us , ut } formulation [A10] . . . . . . . . . . . . . . . . . . . . . . 69
2.2.3 Expression of energies and dissipated powers [A7] . . . . . . . . . . . 71
2.2.4 Periodic finite-elements . . . . . . . . . . . . . . . . . . . . . . . . . 76
2.2.5 Inversion method for recovering EF parameters . . . . . . . . . . . . 81
2.3 Resolution methods for PE FEM . . . . . . . . . . . . . . . . . . . . . . . . 83
2.3.1 Principle of modal decomposition techniques [A13] . . . . . . . . . . 84
2.3.2 Automatic selection of the modes [A13] . . . . . . . . . . . . . . . . 87
2.3.3 Semi-analytical method for compression problems [A10] . . . . . . . 88
2.3.4 Direct and iterative resolution methods [A10] . . . . . . . . . . . . . 91
2.4 Component Mode Synthesis with normal modes . . . . . . . . . . . . . . . . 97
2.5 CMS with normal modes for the {us , p} formulation . . . . . . . . . . . . . . 101
2.A Variational forms for poroelastic problems . . . . . . . . . . . . . . . . . . . 105
6
List of Figures
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11
1.12
1.13
Decomposition of the Representative Elementary Volume . . . .
Continuity relations on displacements . . . . . . . . . . . . . . .
Continuity relations on stresses . . . . . . . . . . . . . . . . . .
Porous layer bonded on a rigid impervious wall . . . . . . . . .
Slowness curves for a mineral wool . . . . . . . . . . . . . . . .
Measurement of phase velocities of a mineral wool . . . . . . .
Transmission of a glass bead layer . . . . . . . . . . . . . . . . .
Transmission of a glass bead layer, 2nd harmonic generation . .
Double porosity material (Courtesy of MatelysAcV) . . . . . .
Normal incidence sound absorption coefficient for material W1 .
Detail of Fig. 1.10 around the solid frame resonance . . . . . .
Absorption coefficient for material W2 . . . . . . . . . . . . . .
Absorption coefficient for material F . . . . . . . . . . . . . . .
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2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
2.15
2.16
2.17
2.18
2.19
2.20
2.21
2.22
2.23
2.24
2.25
2.26
Example of a multilayered structure . . . . . . . . . . . . . . . . . . . .
Surface impedance of a transverse isotropic mineral wool . . . . . . . . .
Reflection coefficient of a bilayered structure for two incident angle . . .
Modulus of the reflection coefficient of a bilayered structure . . . . . . .
Reflection coefficient for a highly absorbing material . . . . . . . . . . .
Porous layer with corrugations . . . . . . . . . . . . . . . . . . . . . . .
Reflexion coefficient for a corrugated semi-infinite porous layer . . . . . .
Reflexion coefficient for a corrugated porous layer on a rigid-backing . .
A two layer structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Convergence of the Mean Total stored energy . . . . . . . . . . . . . . .
Convergence of the Mean Total dissipated power . . . . . . . . . . . . .
Evolution of the fitted coefficients . . . . . . . . . . . . . . . . . . . . . .
Porous corrugated layer: FE mesh . . . . . . . . . . . . . . . . . . . . .
Specular reflection coefficient for different meshes . . . . . . . . . . . . .
Evolution of error between CMM and FEM . . . . . . . . . . . . . . . .
Configurations considered in the inversion procedure of fluid parameters
Sound absorption coefficient for material W1 (Eq. Fluid model) . . . . .
Sound absorption coefficient for material W2 (Eq. Fluid model) . . . . .
Validation of the {us , ut } formulation . . . . . . . . . . . . . . . . . . . .
Amplitude of polar functions . . . . . . . . . . . . . . . . . . . . . . . .
Phase of polar functions . . . . . . . . . . . . . . . . . . . . . . . . . . .
Amplitude of modal contributions . . . . . . . . . . . . . . . . . . . . . .
Phase of modal contributions . . . . . . . . . . . . . . . . . . . . . . . .
Transfer function at the node of excitation . . . . . . . . . . . . . . . . .
Ratio of computational times . . . . . . . . . . . . . . . . . . . . . . . .
Total displacement of two porous structures . . . . . . . . . . . . . . . .
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. 52
. 58
. 59
. 60
. 60
. 62
. 66
. 67
. 74
. 75
. 76
. 77
. 78
. 80
. 81
. 81
. 82
. 83
. 90
. 90
. 91
. 92
. 92
. 96
. 96
. 100
7
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19
20
21
34
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40
43
44
45
47
48
48
49
2.27
2.28
2.29
2.30
2.31
Absorption for 1 mode in each porous substructure
Error vs truncation criterion . . . . . . . . . . . . .
Number of selected modes vs frequency . . . . . . .
Total displacement at 2300 Hz . . . . . . . . . . . .
Error on the absorption coefficient vs frequency . .
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101
101
102
102
103
List of Tables
1
2
3
4
5
Saturating fluid parameters . . . . . . . . . . .
Acoustical Parameters of porous materials . . .
Mechanical Parameters of poroelastic materials
Frequency dependent coefficients . . . . . . . .
Displacement and stress fields . . . . . . . . . .
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13
13
14
14
15
1.1
Parameters for the porous medium . . . . . . . . . . . . . . . . . . . . . . .
46
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
Number of waves in each medium . . . . . . . . . . . . . . . . . . . . . .
State vectors for each medium . . . . . . . . . . . . . . . . . . . . . . . .
Relations between state vectors at interfaces . . . . . . . . . . . . . . . .
Definition of the stored energies and dissipated powers . . . . . . . . . . .
Energy expressions in the case of {us , uf } formulation . . . . . . . . . . . .
Energy expressions in the case of modified {us , p} formulation . . . . . . .
Material 1 and 2 properties . . . . . . . . . . . . . . . . . . . . . . . . . .
Parameters of porous material A and B . . . . . . . . . . . . . . . . . . .
Computational time for the example of Fig. 2.24 for the different methods
52
53
54
71
72
73
75
89
95
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10
Preface
This document presents an overview of my work since my PhD. I was recruited six months
after my doctoral defense in Laboratoire d’Acoustique de l’Université du Maine (LAUM)
as assistant professor. This position was somewhat particular as there was only recommendation for teaching and courses: Numerical methods in physics. Concerning research, the
only comment was: should integrate one of the research group of LAUM. I have been the
last assistant professor recruited with this profile as the council of the laboratory realized
that such profiles prevent it from developing a long-term research policy. I was then very
lucky and it was a chance for me: for a young acoustician, there is no real difficulty to find
a subject of interest in the large spectrum of research topics of LAUM.
The subject of my PhD was to develop resolution techniques for vibroacoustic problems
involving porous materials. I was not very inclined to pursue in this direction for three
main reasons: I spent three years on this subject; it was not really in the priorities of LAUM
and the context and the research thematics had changed in ENTPE where I prepared my
PhD. Moreover, I was not really interested by continuing to study porous media but I
have had the opportunity to interact with several colleagues (among them Denis Lafarge,
Claude Depollier Bruno Brouard and Jean-François Allard...), specialists in this field, and
this collaboration was a pleasure and of great interest for myself. I really discovered the
physics of porous media with them and I want to sincerely thank them. In september 2010
Jean-Philippe Groby joined the laboratory with its experience on sonic cristals and it was
an implicit restart [?]. I have had also the great chance and honor to work with Pr Walter
Lauriks from the Katholieke Universiteit Leuven.
In the ocean of research on sound absorbing materials, I mainly work on the Biot theory
which models the wave propagation in deformable frame porous media. It is an interesting
subject as it is at the intersection of several worlds. It is both mechanics (solid phase)
and acoustics (saturating fluid). It mixes primal and dual variables (displacement and
pressure) dissipation and dispersion. I spent several years to tame Biot’s wild equations
but now I have to confess that the model has tamed me[?].
Researches on porous materials have strongly changed in this last decade. Anisotropy,
inhomogeneities and inclusions have succeeded to isotropic poroelastic solids and new scientific research topics have emerged. On the other hand, it is always a challenge to measure
mechanical parameters of poroelastic materials. I am still convinced that there is still
work to do and nice results to obtain on sound absorbing materials but it is necessary to
imagine new concepts to improve their performance and methods to model them accurately.
When I decided to present my work and to write this habilitation thesis, I asked to my
university administration and my colleagues what I was supposed to write and to explain
11
to the committee. I encountered various answers from which the synthesis was an impossible goal to reach. What really impressed me was the number of colleagues, who recently
passed this diploma, who said to me that I will waste my time reading their dissertation.
Habilitation is not a second PhD, and even if it is an interesting and necessary work for a
researcher to make a synthesis of his own work, it is more a personal work than something
which is supposed to be diffused to the community. On the other hand, teaching is supposed to be half of my work. I am coordinator, for the four last years, of a master course
on sound absorbing materials. I have then imagined that an original way to present my
results could be to explain my scientific results as lecture notes for master and/or doctoral
students. Of course, as it is a personal work submitted to a committee, most of the results
presented in this manuscript are original ones based on my personal bibliography1 .
This document is divided in four chapters. The first one corresponds to a synthesis
of my scientific results and my activities as assistant professor in Le Mans. Chapter 2 is
an extended CV. Chapter 3 is a presentation of the Biot theory based on my works and
publications. Chapter 4 finally presents numerical methods for poroelastic materials. A
separate report corresponding to a selection of scientific papers is added to this manuscript.
Olivier DAZEL
Le Mans, July 1st 2011
You are reading the web document. Chapter 1 and 2 have been skipped. If
you want the full version, send me an email
1
Personal references will be noted with a letter and a number; the letter is associated to the type
of publication and the references correspond to those of the extended CV of chapter 2. Non personal
references will be just denoted by a number associated to the bibliography at the end of the document.
12
Tables of Notations
Name
Symbol
Unity
Fluid density
ρ0
kg.m−3
Fluid viscosity
η0
P a.s
Ambiant pressure
P0
Pa
Celerity
c0
ms−1
Ratio of specific heats
γ0
1
Table 1: Saturating fluid parameters
Porosity
φ
1
Flow resistivity
σ
N.s.m−4
Low frequency tortuosity
α0
1
Geometric tortuosity
α∞
1
Viscous characteristic length
Λ
m
Thermal characteristic length
Λ0
m
Thermal permeability
k00
m
Thermal tortuosity
α00
1
Table 2: Acoustical Parameters of porous materials
13
Name
Symbol
Unity
Solid density
ρ1
kg.m−3
Young Modulus
E
Pa
Shear Modulus
N
Pa
In-vacuo Lamé first coefficient
A
Pa
In-vacuo solid elastic coefficient
P̂
Pa
Poisson ratio
ν
1
Loss factor
ηs
1
Table 3: Mechanical Parameters of poroelastic materials
Name
Symbol
Unity
Equivalent fluid compressibility
e eq
K
Pa
Equivalent fluid density
ρeeq
kg.m−3
Solid phase apparent density
ρe11
kg.m−3
Fluid phase apparent density
ρe22
kg.m−3
Coupling apparent density
ρe12
kg.m−3
Solid phase elastic coefficient
Pe
Pa
Fluid phase elastic coefficient
e
R
Pa
Coupling elastic coefficient
e
Q
Pa
Solid-Pressure coupling coefficient
γ
e
1
Table 4: Frequency dependent coefficients
14
Name
Symbol
Unity
Solid phase displacement
us
m
Fluid phase displacement
uf
m
Total displacement
ut = (1 − φ)us + φuf
m
Relative flow
w = φ(uf − us )
m
Intersticial pressure
p
Pa
Solid partial stress tensor
s
σij
Pa
In-vacuo solid stress tensor
s
σ̂ij
Pa
Fluid partial stress tensor
f
σij
Pa
Total stress tensor
t
σij
Pa
Table 5: Displacement and stress fields
15
16
Chapter 1
Biot model for sound propagation in
poroelastic materials
1.1
Introduction
The theory of deformation of a porous elastic solid saturated by a compressible fluid has
been established by Biot [?] in 1956. These materials can be separated in two categories:
geomaterials and acoustical (or sound absorbing) materials. Geomaterials are a subject
of interest for scientists and lots of works have been undertaken in the particular scope
of oil prospection. Sound absorbing materials are attractive because of their properties of
sound and mechanical energy dissipation; they are key points for the acoustical research
community in order to find answers to the social demand of minimizing noise annoyances.
In order to model the mechanical behavior of such materials, different approaches can
be undertaken. If numerical aspects are disregarded, the microstructural approach seems
interesting [?, ?]. The principle of this method is that each media of the aggregate is
considered as a continuum. Hence, each part is governed by partial differential equations
(PDE) which are motion equations and constitutive relations. Continuity relations between the solid and fluid continuum can also be written. This approach is mathematically
and physically very graceful, nevertheless it is impossible to solve this type of equations
without a prohibitive numerical cost out of some academical examples.
A second approach, considered here, is to use continuum mechanics at a mesoscopic
scale, i.e. on homogenized fields of the porous materials. This approach is only possible if
there exists a Representative Elementary Volume (REV) (denoted by Ω in this document)
whose size is sufficiently small regarding to the wavelength of the phenomena but also sufficiently high compared to the characteristic size of the heterogeneities to be representative.
The motion of the solid (resp. fluid) part is averaged on the REV and the homogenized
medium is called the solid (resp. fluid) phase.
In this scope, three classes of model were developed. The first set of them is referred to
as equivalent fluid models. This set brings together all models in which the solid phase is
assumed to be motionless. Under this assumption, only one compressional wave can propagate. The main difference between propagation in a free fluid and in a fluid saturating an
immobile porous solid is the relative importance of viscous and thermal effects in the second
case[?, ?, ?, ?, ?, ?, ?]. The second set of models is called limp models which consider that
the solid phase is moving but elastic effects of the solid phase can be neglected regarding
17
to the other mechanisms. The propagation is still governed by the fluid phase and only
one compression wave is exhibited in this model. The last set is called poroelastic models
and exhibits several propagation waves both in solid and fluid phase. Among poroelastic
models, a central place is given to Biot theory [?, ?] whose main aspects are here remained.
In a paper published in 1956[?], Biot represented the homogenized medium with 6
fields which are the three displacements of each homogenized phase (solid and fluid). This
paper and formulation are called original in the following. In a second time and in a paper
published in 1962[?], the theory was reformulated in order to model inhomogeneous media
(i.e. with non constant macroscopic properties). The fields of interest are the solid displacements and the flow of the fluid relative to the solid measured in terms of volume per
unit area of the bulk medium. This second formulation is refereed to as modified formulation. In a different scope and more recently, Atalla et al. proposed a mixed formulation of
Biot’s equations[?] whose fields are the solid displacement and the interstitial pressure of
the fluid. This mixed formulation is obtained from Biot’s displacement equations through
linear combinations and differentiations. It exhibits 4 fields instead of 6 for the displacement formulations. An other interest of this formulation is the introduction of an in-vacuo
stress tensor of the solid phase which exhibits some advantages compared to the partial
stress tensor of the solid phase used in the original formulation. More recently Dazel et
al.[?] proposed a strain decoupled formulation which is equivalent to 1962 representation.
This formulation simplifies the Biot formalism thereby leading to shorter analytical expressions without making physical assumptions. In this document, the presentation of the
Biot theory is mainly done through this representation.
1.2
1.2.1
Homogenized fields
Displacement fields
Ω can be decomposed in two volumes, one associated to the skeleton Ωs and another one
associated to the fluid part Ωf :
Ω = Ωs ∪ Ωf .
(1.1)
The porosity is defined as the ratio of the fluid volume over the total one:
φ=
V (Ωf )
.
V (Ω)
(1.2)
Let usµ be the microscopical displacement of the skeleton (µ denotes microscopic fields).
It is only defined in Ωs and one first approximation of the Biot theory is that it is assumed
to be constant on this volume. It is then straightforward to define the solid macroscopical
displacement us :
us =< usµ >Ωs .
(1.3)
Similarly, let ufµ (defined in Ωf ) be the microscopical displacement of the fluid. It is
not supposed to be uniform on Ωf which does not prevent us from defining the fluid
macroscopical displacement uf :
uf =< ufµ >Ωf .
(1.4)
The total displacement ut is defined as the average of the displacement fields over the
whole volume Ω
V (Ωs ) s V (Ωf ) f
u +
u = (1 − φ)us + φ uf .
(1.5)
ut =
V (Ω)
V (Ω)
18
Figure 1.1: Decomposition of the Representative Elementary Volume
In several conditions, the solid can be assumed to be motionless (i.e us = 0). The
porous material is then considered as an equivalent fluid. The porosity should then be
taken into account and the equivalent displacement ueq can be defined as1 :
ueq =< ufµ >Ω = φuf .
(1.6)
In case of a solid motion and as usµ is uniform on Ω it is interesting to consider the motion
of the fluid relative to the solid:
w =< ufµ − us >Ω = φ(uf − us ).
(1.7)
Note that this last expression seems to mix microscopic and macroscopic fields which could
seem abusive. In fact, this is not due to the uniformity of the solid displacement on the
elementary volume Ω.
For all displacement fields, the derivative with respect to space are expressed with the
∂ui
1
generic notation ui,j =
. The deformation is εij = (ui,j + uj,i ). The deformation
∂xj
2
tensor is also denoted ε in tensor form.
1.2.2
Continuity relations for displacements
In case of an interface between two porous materials or in the case of inhomogeneity of a
porous sample, it is necessary to express continuity relations between the displacements.
1
This expression is equivalent to the total displacement (1.5) when us = 0
19
n
Medium 1
(1 − φ1 )
(1 − φ2 )
φ1
φ2
Medium 2
n
us1 = us2
ut1 .n = ut2 .n
uf1 .n �= uf2 .n
Figure 1.2: Continuity relations on displacements
Let consider two porous medium (or an inhomogeneous porous medium with non uniform
porosity) denoted by indices 1 and 2. Deresiewicz and Skalak[?] expressed these relations.
Concerning the solid part; the two parts of the skeleton phase are supposed to remain in
contact at the interface so that the solid displacements are continuous in every direction
us (1) = us (2).
(1.8)
Concerning the fluid displacement, it should first be mentioned that continuity relations
can only be expressed in terms of normal component. In addition, there is no continuity of
the normal component of the fluid displacement at the interface. The conservation at the
interface correspond to the fluid mass. As it is supposed incompressible at the microscopic
scale, this relation can be expressed in terms of volume and finally through the normal
total displacements:
ut (1).n = ut (2).n.
(1.9)
n denotes the normal vector to the interface. Relations (1.8) and (1.10) while combined
also induce the continuity of the normal component of w;
w(1).n = w(2).n.
(1.10)
It is the straightforward that the relation between the normal fluid displacements reads:
uf (1).n = (1 −
φ2 s
φ2
) u (2) .n + uf (2).n.
φ1 | {z }
φ1
=us (1)
20
(1.11)
1.2.3
Stress tensors
Let now define the partial stress tensor of the solid phase σ s defined as the stress in the
skeleton measured by unit area of bulk:
σ s =< σ sµ >Ω .
(1.12)
The partial stress tensor of the fluid phase σ f can be analogously as the stress in the fluid
measured by unit area of bulk. This stress is associated to the microscopic pressure which
is assumed to be constant in Ωf . σ f is then a diagonal tensor defined by:
σ f = − < pµ >Ω δ = −
V (Ωf )
< pµ >Ωf δ = −φpδ.
V (Ω)
(1.13)
p is the macroscopic interstitial pressure defined as the average pressure in the fluid volume.
δ is the identity tensor.
1.2.4
Continuity relations of stress tensors at interfaces
n
Medium 1
Medium 2
σ t1 .n = σ t2 .n
n
p 1 = p2
Figure 1.3: Continuity relations on stresses
Continuity relations for stress tensors are presented in Fig. 1.3. First relation is the
continuity of the homogenized fluid pressure. This is due to the fact that the pores are
opened and completely connected:
p(1) = p(2).
(1.14)
At each point of the interface, the total stress continuity is required for any displacements.
Hence, one has
f
f
s
s
σij
(1)usi (1) + σij
(1)ufi (1) = σij
(2)usi (2) + σij
(2)ufi (2).
21
(1.15)
This form is not very convenient due to the non-conservation of the fluid displacement. It
is then necessary to express this relation with pressure:
t
t
σij
(1)usi (1) − p(1)wi (1) = σij
(2)usi (2) − p(2)wi (2).
(1.16)
One then has the continuity of the total stress tensor at the interface.
t
t
∀ i, j ∈ {x, y, z} , σij
(1).i = σij
(2).i.
1.3
(1.17)
Lagrangian approach
Biot proposes to obtain the dynamical equations by the way of lagrangian formalism.
Hence, the methodology is to express the lagrangian density L and then derives the motion
equations by the way of Euler-Lagrange’s relations:
3
−
d ∂L X d
−
dt ∂ η̇
dxk
k=1
∂L
∂L
+
= 0.
∂η
∂η
∂
∂xk
(1.18)
In this last equation η denotes each continuous field used for the description of the mechanical behavior of the system. The difference between the different representations of
the Biot theory lies in the choice of these fields. General comments can first be made for
displacement formulations:
• The problem correspond to a wave-like motion, then the lagrangian density contains
terms up to the first order in terms of time and spatial derivatives.
• Homogeneous media are considered so that there is no dependance of the lagrangian
density on position.
• The material is assumed to be isotropic; the lagrangian is then invariant by every
rotation.
• Small deformations are considered leading to linear equations of motion. The lagrangian density should be quadratic in the field variables.
The general form of the lagrangian density is first investigated. We are looking for
the description of wave propagation in a medium represented by two displacement fields
(denoted by u1 and u2 for the moment so as to keep generality).
As a mechanical system is involved, this density is the difference of kinetic and potential
densities. The first one is associated to time derivative terms and the second one to spatial
derivatives.
L = T − V,
2
(1.19)
2
T = αu˙1 + β u˙1 u˙2 + γ u˙2 ,
(1.20)
W = δ( ∇.u1 )2 + ε ∇.u1 . ∇.u2 + γ( ∇.u2 )2
+ ι | ∇ ∧ u1 |2 + κ ∇ ∧ u1 . ∇ ∧ u2 + µ | ∇ ∧ u2 |2 .
(1.21)
In fact, it can be shown that additional terms can appear in expressions of densities
T and W but they can be cancelled as they do not contribute to dynamical equations
22
while derived from Euler-Lagranges’s equations (1.18). Two additional remarks should be
made. First, all the formulations consider the solid phase displacement us as first field2 .
Second, the hypothesis that the fluid does not restore shear forces automatically implies
that κ = µ = 0.
1.4
1.4.1
Former Biot models
Original formulation
In the original paper [?], Biot proposed to use us and uf as set of description fields. Expression of the strain energy of the porous medium (denoted W56 ) in {us , uf } formulation
derives from (1.21) and reads:
W56 = A
( ∇.us )2
( ∇.uf )2
+R
+ 2N εsij εsi,j + Q ∇.us ∇.uf .
2
2
(1.22)
A, R, N and Q are the constitutive coefficients of the homogenized porous medium[?].
N is the shear modulus of the skeleton. Q is a coupling coefficient between the dilatation
and stress of the two phases, R may be interpreted as the bulk modulus of the air occupying a fraction φ of a unit volume of aggregate. The elastic coefficients A, Q and R can
be obtained by the Biot and Willis gedanken experiments[?] from Kb , the bulk modulus
of the skeleton in vacuo, from Ks , the bulk modulus of the elastic solid from which the
skeleton is made and from Kf , the bulk modulus of the fluid in the pores. A and N
are the Lamé coefficient of the solid partial stress tensor. The expression[?] of A shows a
dependance on Kf . Hence, this apparent solid parameter depends on the intersticial fluid
property. The complete expressions of these quantities through the gedanken experiments
are given Annex 1.A.
The stress-strain relations of the porous media are obtained from Helmholtz relations.
For the original formulation, the two partial stress tensor are[?]:
s
σij
=
f
σij
=
∂W56
= A ∇.us δij + 2N εsij + Q ∇.uf δij
∂εsij
(1.23a)
∂W56
= R ∇.uf δij + Q ∇.us δij .
∂ ∇.uf
(1.23b)
Hence, the two partial stress tensors depend on both solid and fluid displacements.
In order to obtain the equations of motion, it is necessary to also express kinetic energies:
T56 =
ρ11 s 2 ρ22 f 2
u̇ +
u̇ + ρ12 u̇s u̇f .
2
2
(1.24)
Apparent densities ρ11 , ρ12 and ρ22 can be deduced from densities of the frame ρs and
the saturating fluid ρ0 , porosity φ and the geometric tortuosity α∞ through expressions
(1.25) and (1.26). Expression of the total density ρt and the density of the equivalent fluid
medium ρeq are also given.
Densities associated to mass
ρ1 = (1 − φ)ρs , ρ2 = φρ0 , ρt = ρ1 + ρ2 .
2
It is demonstrated section 1.5
23
(1.25)
Densities including geometric tortuosity
ρ12 = −φρf (α∞ − 1), ρ11 = ρ1 − ρ12 , ρ22 = ρ2 − ρ12 , ρeq =
ρ22
.
φ2
(1.26)
From (1.18), (1.22) and (1.24), it is possible to express in the time domain, the motion
equation of a poroelastic medium in the {us , uf } formulation:
Conservative porous medium Original formulation[?]: {us , uf }
∇.σ s (us , uf ) = ρ11 üs + ρ12 üf ,
f
s
f
s
s
f
(1.27a)
f
s
∇.σ (u , u ) = ρ
h 12 ü + ρ22 ü ,
s
(1.27b)
f
i
s
s
σ (u , u ) = A ∇.u + Q ∇.u δ + 2N ε (u ),
h
i
σ f (us , uf ) = Q ∇.us + R ∇.uf δ.
(1.27c)
(1.27d)
Dissipative effects should now be introduced. It is reminded that three mechanisms of
dissipation occurs in the porous medium; the first one is due to viscous dissipation; the
second and third one are respectively associated with structural and thermal effects.
Viscous dissipation was introduced by Biot in 1956[?], with the assumption that the
flow of the fluid relative to the solid through the pores is of Poiseuille type. These effects
were introduced through a dissipation function which is an homogeneous quadratic form
of the six velocities. Because of isotropy, orthogonal directions are uncoupled. As viscous
effects are considered they should vanish when there is no relative motion. This dissipation
function then reads:
2
σφ2 G f
(1.28)
D56 =
u̇ − u̇s ,
2
where σ is the flow resistivity of the porous sample and G is a non-dimensional correction
function. This function is useful to represent the variation of apparent viscosity versus
frequency. This function is first assumed to be a constant.
It is then necessary to modify Euler-Lagrange’s equations (1.18) and to add an addi∂D
tional term
to take this term into account. Equations (1.27) should then be modified
∂ η̇
and reads:
∇.σ s (us , uf ) = ρ11 üs + ρ12 üf + σφ2 G u̇s − u̇f ,
(1.29a)
∇.σ f (us , uf ) = ρ12 üs + ρ22 üf − σφ2 G u̇s − u̇f
(1.29b)
In fact, it is not correct to assume a constant correction term G as it depends on the
frequency. In addition it can be shown that it is a complex and frequency dependent term
whose expression can be deduced from the equivalent fluid density ρeeq . Various model of
viscosity of air saturating an immobile porous solid (i.e. of ρeeq or G(ω)) have been proposed
in the past[?, ?]. Expressions of the equivalent fluid densities are summarized in Annex
1.B and inertial coefficients of Biot’s equations are given by:
24
Equivalent densities including visco-inertial effects
ρe22 = φ2 ρeeq , ρe12 = ρ2 − ρe22 , ρe11 = ρ1 − ρe12 .
(1.30)
The structural dissipation in the skeleton is taken into account by modifying the elastic
coefficients of the jacketed stress-tensor. As a temporal dependance is assumed, complex
es, K
e b and N
e can be used in the gedanken experiments
frequency dependent extensions K
instead of constant and real ones.
The thermal effects are taken in account by modifying Kf which is now:
e f = K0 ,
K
β(ω)
(1.31)
with K0 the adiabatic compressibility coefficient of air and β(ω) the thermal dynamic
susceptibility. Like the viscous function G, various model have been proposed in order to
explicit this function and the reader can refer to these models[?, ?].
Finally, one has:
Original formulation[?]: {us , uf }
∇.σ s (us , uf ) = −ω 2 ρe11 us − ω 2 ρe12 uf ,
2
s
2
f
∇.σ f (us , uf ) = −ω
h ρe12 u − ω ρe22iu ,
e ∇.us + Q
e ∇.uf δ + 2N εs (us ),
σ s (us , uf ) = A
i
h
e ∇.us + R
e ∇.uf δ.
σ f (us , uf ) = −φpδ = Q
1.4.2
(1.32a)
(1.32b)
(1.32c)
(1.32d)
Modified Biot Formulation
In the modified formulation[?], Biot proposed to use us and w as fields of description for
the porous material. The strain energy is then written as:
( ∇.us )2
Q+R
( ∇.w)2
+ 2N εsij εsi,j +
∇.us ∇.w + Keq
,
2
φ
2
ρt s 2 ρeq 2 ρ2 s
=
u̇ +
ẇ + u̇ ẇ.
2
2
φ
W62 = A0
T62
with
(1.33)
(1.34)
R
.
(1.35)
φ2
and W62 ) are equivalent for homo-
A0 = A + 2Q + R , Keq =
It can easily be checked that W56 and W62 (resp. T56
geneous materials.
The stress associated to this formulation are the total stress tensor σ t and the pressure.
They are defined by:
∂W62
Q+R
t
0
s
σij =
= A e + 2N ε −
ε δij ,
(1.36)
∂εsij
φ
∂W62
Q+R
p=
= −
e − Keq ∇.w δij .
(1.37)
∂ ∇.w
φ
Equations of the modified Biot formulation reads:
25
Modified formulation[?]: {us , w}
∇.σ̂ t (us , w) = −ω 2 ρt us − ω 2 ρ0 w,
2
s
2
− ∇p = −ω
h ρ0 u − ω ρeeq w,
i
s
e eq ) ∇.us + γ 0 K
e eq ∇.w δ + 2N εs (us )
σ (u , w) = (Â + γ 02 K
e eq ∇.w + γ 0 ∇.us
p(us , w) = −K
t
(1.38a)
(1.38b)
(1.38c)
(1.38d)
with
Kb
Q2
γ =1−
, Â = A −
.
(1.39)
Ks
R
Note that the notations used in this document ensures the coherence of all the document.
The correspondences with Ref. [?] are:
0
e eq , α = γ 0 .
M =K
1.5
1.5.1
(1.40)
Strain decoupled formulation [A3]
Fields of interest
This section presents the strain decoupled formulation. A preliminary remark must first be
made: the solid and fluid phase are coupled and no mathematical trick can let forget this
physical consideration. Nevertheless, as a n degrees of freedom mechanical system can be
represented by n decoupled oscillators based on its vibrating modes, a diagonalization of
the expression of the strain energy can be proposed without making any additional physical
assumption.
The objective of the method is not to postulate at the beginning the adapted choice of
representation fields and then to replace us and uf by any free linear combination. The
objective of this new choice is to avoid coupled terms in the expression of the strain energy.
The conservative porous medium is first considered.
Let u1 and u2 be an adapted set of generalized coordinates. Without loss of generality
the following linear relations can be written:
us = au1 + bu2 , uf = cu1 + bu2 . , ad − bc 6= 0.
(1.41)
The strain energy can then be written as:
( ∇.u1 )2
Aa2 + Rc2 + 2Qac
2
( ∇.u2 )2
+
Ab2 + Rd2 + 2Qbd
2
+ ∇.u1 ∇.u2 (Aab + Rcd + Q(ad + bc))
+ 2N a2 ε1ij ε1ij + b2 ε2ij ε2ij + 2abε1ij ε2ij .
W07 =
For the sake of simplicity it seems natural to avoid ε1ij ε2ij and ε2ij ε2ij terms. This implies
Q
that b = 0 is an appropriate choice. Hence, e1 e2 term is avoided if c = − a and:
R
2
2
( ∇.u1 )
( ∇.u2 )
W07 = a2 Â
+ d2 R
+ 2N a2 ε1ij ε1ij ,
(1.42)
2
2
26
All choices of a and d are mathematically equivalent. The adequate choice is a = 1 so
that u1 = us and d = φ−1 in order to limit the influence of porosity on the model in the
following. The new fields are now totally determined and the strain decoupled formulation
is called {us , uW } with
Q s
W
f
(1.43)
u =φ u + u ,
R
The strain energy density reads:
W07 = Â
( ∇.us )2
( ∇.uW )2
+ Keq
+ 2N εsij εsij .
2
2
(1.44)
Constitutive laws for {us , uW } formulation are:
s
σ̂ij
= 2N εsij + Â ∇.us δij , pf = −Keq ∇.uW .
(1.45)
s corresponds to the in-vacuo stress tensor of the solid phase. Unlike the solid partial
σ̂ij
s which is a function of both solid and fluid phase displacements, σ̂ s only
stress tensor σij
ij
depends on the motion of the solid phase.
The term in-vacuo can be understood through the link between the σ t and σ̂ s :
t
σij
= σ̂ij − γ 0 p.
(1.46)
Hence, these two tensor are equal, only if there is no pressure (i.e. under vacuum condition)
Hence, the strain energy is the sum of two terms (and not three like for W56 and W62 ).
Relations (1.45) can be compared to (1.23) and (1.36). A first remark is that each stress
is associated to its corresponding displacement then avoiding coupling terms. A second
remark is that the pressure p can be expressed as a divergence in (1.45). It gives an interpretation for uW which is the apparent displacement for the pressure of the fluid phase
taking into account the motion of the solid phase. In the case of a motionless solid, one
has uW = φuf . This is the equivalent fluid displacement introduced (1.6).
The kinetic energy density is obtained through a change of variables:
T07 =
ρ3 s 2 ρeq W 2
u̇ +
u̇
+ ρeq γ u̇s u̇W ,
2
2
with
ρ3 = ρ1 + ρ2
Q
R
2
− ρ12
γ02
.
φ2
(1.47)
(1.48)
For a conservative poroelastic medium, equations of motion in {us , uW } formulation
read:
∇.σ̂(us ) = ρs üs + ρeq γ üW ,
s
W
Keq ∇ζ = ρeq γ ü + ρeq ü .
(1.49a)
(1.49b)
These equations are equivalent to those proposed by Biot as no additional assumptions
have been made. Unlike the original ones, there is no stress coupling terms in them and
each stress tensor is a function of only the corresponding displacement. The symmetry is
also preserved for inertial terms.
27
1.5.2
Gedanken experiments
Biot and Willis[?] presented three gedanken experiments which provide expressions for the
elastic coefficient appearing in Biot’s original model: A, N Q and R. In the case of the
strain decoupled formulation it is now shown that the gedanken experiments provide the
three elastic coefficients of the model Â, N and Keq and an expression of γ 0 . Biot and Willis
gedanken experiments assume quasistatic deformation. In a recent contribution Lafarge
[?] shows that this assumption is not restrictive and that these gedanken experiments can
be extended to harmonic excitations.
The first gedanken experiment is a measure of the shear modulus N of the material
and consequently the shear modulus of the frame since the fluid does not contribute to the
shear force.
In the second gedanken experiment (called jacketed experiment), the material is surrounded by a flexible jacket that is subjected to a pressure pjac . The fluid inside the jacket
remains at the ambient pressure. It follows that p = 0 and σ̂ij = −pjac . The deformation
of the solid phase is denoted by ej . The stress-strain relation (1.45) implies that
2N
−pjac = Â +
ejac .
(1.50)
3
This last relation must be linked to the definition of the bulk modulus Kb of the frame at
pjac
and one obtains:
constant pressure in the air Kb = −
ejac
 = Kb −
2N
.
3
(1.51)
The last gedanken experiment is called unjacketed experiment and provides two additional equations. The material is subjected to an increase of pressure pu in the fluid
t = −p δ The divergence of the us (resp. uf and uW ) is
inducing a total stress equal σij
u ij
called eu (resp εu and ζu ). Concerning this experiment, Biot introduced two coefficients:
Kf = −
pu
pu
, Ks = − .
εu
eu
(1.52)
The stress-strain relations (1.46) and (1.45) are now expressed:
−pu = Kb eu − γ 0 pu , Keq = −
pu
.
ζu
(1.53)
The first equation of (1.53) enables to find an expression of γ 0 as a function of Ks and Kb :
γ0 = 1 −
Kb
.
Ks
(1.54)
This last result is linked to the second equation of (1.53) and the expression of Keq is
provided:
Keq =
Kf
.
Kb Kf
Kf
−
φ + (1 − φ)
Ks
Ks2
(1.55)
Biot and Willis results for A, N Q and R can rigorously be obtained from the expressions of Â, N γ 0 and Keq . Nevertheless the expressions of the latter are simpler. In
28
particular, it is interesting to notice that  does not depend on Kf unlike the constitutive
coefficient A of Biot’s original formulation.
A second and fundamental remark is that γ 0 is independent on the compressibility of
the fluid through (1.54) and it is possible to express uW as:
Kb
uW = φuf + (1 − φ)us − us .
|
{z
} Ks
(1.56)
ut
ut is the called total displacement of the porous material. Hence uW is the total displacement from which is subtracted a term corresponding to the motion of the solid matter
constituting the solid. In particular, it is here shown that uW is in fact independent on
porosity. It is the a-posteriori justification of the particular choice d = φ−1 considered in
the preceding section.
1.5.3
Introduction of dissipation
This section deals with the introduction of dissipative effects in the formulation. It is shown
that the symmetry of equations (1.49) is preserved even if dissipation is considered. The
dissipation is taken into account for harmonic excitation by modifying the constitutive and
inertial coefficients of the model.
In order to integrate this dissipation in the lagrangian formulation, a dissipation function D56 was defined as a homogeneous quadratic form with the six generalized velocities
in (1.28). This function is first rewritten through us and uW :
2 σG σφ2 G f
u̇W − γ 0 u̇s 2 ,
u̇ − u̇s =
2
2
Modified Euler-Lagrange’s equations then reads:
2
∇.σ̂(us ) = ρs üs + ρeq γ üW + σG γ 0 u̇s − γ 0 u̇W ,
Keq ∇ζ = ρeq γ üs + ρeq üW + σG u̇W − γ 0 u̇s .
D07 =
The right-hand sides of these two equations are rewritten:
"
2 #
Q
ρ2 (φ − γ 0 ) W γ 0 2
γ0
s
s
∇.σ̂(u ) = ρ1 + ρ2
üs +
ü
−
V
(u
)
+
V (uW ),
R
φ2
φ2
φ2
Keq ∇ζ =
s ρ2 W
ρ2 γ0
1
0
φ
−
γ
ü
+
ü
+
V (us ) − 2 V (uW ),
φ2
φ2
φ2
φ
(1.57)
(1.58a)
(1.58b)
(1.59a)
(1.59b)
with the time differential operator V defined by the functional relation:
V (u) = ρ12 ü − φ2 σ G u̇.
(1.60)
In the case of harmonic excitation, one obtains:
V (u) = −ω 2 ρe12 u , ρe12 = ρ12 −
φ2 σ G(ω)
.
jω
ρe12 is the apparent coupling density and has been introduced in (1.30).
Finally , the frequency equations including dissipation are:
29
(1.61)
Strain decoupled formulation [A3]: {us , ut }
∇.σ̂(us ) = −ω 2 ρes us − ω 2 ρeeq γ
e uW ,
2
s
2
(1.62a)
W
− ∇p = −ω ρeeq γ
e u − ω ρeeq u ,
σ̂(us ) = Â ∇.us + 2N εs (us ),
e eq ∇.uW .
p(uW ) = −K
with
φ
ρe22
,γ
e = − γ0 = φ
α
e=
ρ2
α
e
e
ρe12 Q
−
e
ρe22 R
(1.62b)
(1.62c)
(1.62d)
!
.
(1.63)
The main advantage of the proposed formulation is that the three dissipative mechanisms are well separated: the viscous effects are taken into account by modifying the
densities; the structural effects by modifying the jacketed stress tensors and the thermal
effects by modifying Keq .
1.6
{us , p} mixed formulations
Mixed formulations were introduced by Atalla et al. [?] through a variable change in the
equations of motion in the frequency domain. It is shown in this section that they can be
obtained in the time domain for a conservative porous material. Dissipative mechanisms
are then introduced in the frequency domain which shows an other method to obtain Atalla
et al. equations [?].
Each displacement (or each combination of displacement) can be expressed as the sum
of a gradient and a rotational. Let introduce φ and ψ so that:
γus + uW = ∇φ + ∇ ∧ ψ , ∇.ψ = 0.
(1.64)
uW can then be replaced in (1.47) and one has:
2 2
s2
2T98 = ρu̇ + ρeq
∇ φ̇ + ∇ ∧ ψ̇ + 2 ∇ φ̇. ∇ ∧ ψ̇
Concerning strain energy, it is expressed by the way of the pressure p which is defined
by:
p = −Keq ∇.uW ⇐⇒ ∇.uW = −
1
p,
Keq
(1.65)
consequently:
2W98 = σ̂ s (us ) : ε̂s (us ) +
p2
.
Keq
From T98 and W98 it is possible to express the lagrangian density from us , φ, ψ and p but
all these fields are dependent. One then has the holonomic relations:
p = −Keq ∇. ( ∇φ + ∇ ∧ ψ − γus ) , ∇.ψ = 0.
(1.66)
Hence Lagrange multipliers should be considered to modify the Lagrangian density:
L098 = T98 − W98 − λ1 [p + Keq ∇. ( ∇φ + ∇ ∧ ψ − γus )] − λ2 ∇.ψ,
30
(1.67)
On has the following differentiation relations:
2
δ ∇ φ̇ = ∇2 φ̈δφ,
2
δ ∇ ∧ ψ̇ = δψ. ∇ ∧ ∇ ∧ ψ̈ ,
(1.68)
δ(λ1 ∇2 φ) = ∇2 φδλ1 + ∇2 λ1 δφ,
(1.70)
δ(λ2 ∇.ψ) = ∇.ψδλ2 − ∇λ 2 .δψ.
(1.69)
(1.71)
6 equations can then be derived. The first one is associated to the variation of the
displacements and reads:
∇.σ̂ s (us ) = ρüs − γKeq ∇λ 1 .
(1.72a)
Equation associated to the pressure provides a relation between λ1 and p:
p
= λ1 .
Keq
(1.72b)
Equation relative to φ is
ρeq ∇2 φ̈ = ∇2 λ1
(1.72c)
ρeq ∇ ∧ ∇ ∧ ψ̈ = ∇λ 2 .
(1.72d)
and the one associated to ψ reads:
Finally equations relative to Lagrange multipliers are:
p + Keq ∇2 φ − ∇.γus = 0,
∇.ψ = 0.
(1.72e)
(1.72f)
λ1 can be eliminated by mixing (1.72a) and (1.72b):
∇.σ̂ s (us ) = ρüs + γ ∇p .
(1.73)
Equation (1.72d) shows that ψ and λ2 do not have an influence on the dynamical response
of the system. Finally, the three equations (1.72c), (1.72e) and (1.73) describe the propagation of poroelastic waves in the time domain for a conservative medium.
In the case of dissipative medium, real coefficients can be replaced by complex and frequency dependent ones, similarly to the substitutions done for displacement formulations.
In addition and in the frequency domain, there is a simple link between φ and p which
allows to have 4 equations and four fields of representations.
Mixed formulation[?]: {us , p}
∇.σ̂(us ) = −ω 2 ρe us − γ
e ∇p ,
−
∇2 p
p
= −e
γ ∇.us +
.
e
ρeeq ω 2
Keq
(1.74a)
(1.74b)
with
ρe = ρes − γ
e2 ρeeq .
(1.75)
In order to illustrate links between {us , uW } and {us , p} formulation, it can be seen that
can be obtained by replacing uW in (1.62a) by its expression in (1.62b) and that (1.74b) is
the divergence of (1.62b) where the divergence of uW has been canceled through (1.62d).
31
1.7
Biot’s waves
This section deals with the expression of the wave numbers of the three Biot’s waves. The
methodology is the same that the one proposed in reference [?]. The two compressional
waves are first studied. Two scalar potentials ϕs and ϕW are defined for the compressional
waves. Hence, equations of motion of the strain decoupled formulation (1.49) are written
as:
 












s
s
 ϕ 

ρes ρeeq γ
e   ϕ   P̂
0 



 ∇2
−ω 2 
=
,
(1.76)











W
W




e
ρeeq γ
e ρeeq
ϕ
0 Keq
ϕ
with P̂ = Â+2N . Let δ12 and δ22 be the eigenvalues of the eigen problem. An elementary
algebraic calculation gives that:
q 2
2 + δ2 ±
2 2 − 4δ 2 δ 2
δs2 + δeq
δs2
eq s1
eq
,
(1.77)
δi2 =
2
with
s
δeq = ω
ρeeq
,δ =ω
e eq s1
K
s
ρe
P̂
s
, δs2 = ω
ρes
P̂
.
(1.78)
These expressions are equivalent to the classical expressions of these two wave numbers
which can be found in reference [?]. It is quite evident that the proposed expressions are
more condensed than the classical ones.
The following symmetric relations exist between the wave numbers:
2 2
2
2
δ12 δ22 = δs1
δeq , δ12 + δ22 = δs2
+ δeq
.
(1.79)
δeq is the wave number of the equivalent fluid model i.e. when the solid phase is immobile and more details will be found in section 1.8. Dually to the equivalent fluid model
which assumes that us = 0, an equivalent solid model can be considered which postulate
that uW = 0. In this model only one compression wave propagates whose wave number is
δs2 . Even if there is a perfect mathematical symmetry between these two cases, the first
one is physically realistic (and has been often used in the past) while the second is not.
δs1 is the wave number of the wave propagating in the solid if the porous medium is in
vacuum (and not saturated by air).
s
One obtains two possible and equivalent expressions for µ0i = ϕW
i /ϕi :
µ0i = γ
e
2
2 )
δeq
(δi2 − δs2
=
γ
e
.
2 − δ2
2
δs2
δi2 − δeq
s1
(1.80)
As symmetric relations (1.79) were obtained for the wave numbers, orthogonality relations can be obtained on µ0i :
e eq µ01 µ02 = 0,
P̂ + K
(1.81a)
ρs + ρeq γ(µ01 + µ02 ) + ρeq µ01 µ02 = 0.
(1.81b)
It is also interesting to introduce the following ratios
νi =
2
δj2 − δeq
µ0i
=
µ0i − µ0j
δj2 − δi2
with
32
(i, j) ∈ {1, 2}.
(1.82)
The shear wave is now considered by using a vector potential:
us = ∇ ∧ Ψ
so
uW = µ03 ∇ ∧ Ψ.
Substituting these expressions in the motion equations (1.49), one obtains:
r
ρe
and
µ03 = −e
γ.
δ3 = ω
N
1.8
(1.83)
(1.84)
Equivalent fluid and limp models
The equivalent fluid model corresponds to a motionless solid phase (us = 0 ). (1.49b) is
then assimilated as fluid propagation equation which reads:
e eq ∇2 uW = ρeeq üW .
K
(1.85)
Is is straightforward to find that the wave number of the equivalent fluid model is
δeq . This remark validates the definition of this quantity proposed in section 1.7. The
characteristic impedance of the equivalent fluid in the strain decoupled formulation is
defined as:
q
p
e eq .
Zeq = W = ρeeq K
(1.86)
kv k
Let now consider the limp model which also exhibits one compressional wave. Unlike
equivalent fluid model, the fluid is not motionless and this model takes into account the
inertia of the solid phase. It is associated with materials whose rigidity is negligible (light
mineral wools, cotton ..). The jacketed strain energy of the solid phase is negligible compared to those of the other mechanisms of the propagation so that  ' 0 ' N . Hence,
the compressional term ∇.σ̂ can be neglected in (1.49a) which gives a relation between us
and uW :
ρes us = −e
ρeq γ
euW .
(1.87)
This relation is now inserted in (1.49b) and a propagation equation on uW is obtained as:
2
ρ
e
γ
e
eq
2
e eq ∇ζ = −ω ρeeq 1 −
K
uW
(1.88)
ρs
The limp model is a one compression wave model whose difference with the equivalent fluid
is the definition of the density:
2
δs1
ρlimp = ρeeq
.
(1.89)
δs2
The wave number of the limp model can now be expressed as a function of the three
intrinsic wave numbers (1.78) of the porous medium:
r
ρlimp
δs1
δlimp = ω
= δeq
.
(1.90)
Keq
δs2
The characteristic impedance of the limp model is :
Zlimp =
p
pf
= ρlimp Keq .
W
ku k
33
(1.91)
1.9
From {us , uW } to {us , ut } formulation for sound absorbing
materials
This section is devoted to the study of a special category of porous materials. these are
materials with a very stiff skeleton. All the sound absorbing materials are in this
category. This assumption induces additional simplifications which are now detailed.
The high stiffness of the solid matter induces that:
K
K
eb ef 1, 1.
es es K
K
(1.92)
This assumption implies simplifications in both expressions of γ 0 (1.54) and uW (1.56):
γ 0 ' 1 , uW ' ut .
(1.93)
Hence uW corresponds to the total displacement. This is an interesting result: firstly it
gives a direct physical interpretation of uW and secondly, it greatly simplifies the continuity relations.
e and K
e eq in (1.55):
It also possible to simplify R
e = φK
ef
R
1.10
,
e
e eq = Kf .
K
φ
(1.94)
Normal incidence surface impedance
This section is devoted to the obtention of the surface impedance of a porous layer bonded
on to a rigid impervious wall under normal incidence (depicted Fig. 1.4). The normal
Figure 1.4: Porous layer bonded on a rigid impervious wall
incidence surface impedance Z is defined by:
Z=
pa (x = −l)
,
v a (x = −l)
34
(1.95)
where pa (resp. v a ) is the pressure (resp. normal velocity) at the interface (x = −l)
between the air and the porous material. Two incident and two reflected compressional
waves propagate in the direction parallel to the x-axis. By taking into account that the
porous material is bonded onto a rigid wall, the longitudinal displacements us and ut are
written in the following form:
us (x) = A sin(δ1 x) + B sin(δ2 x),
ut (x) = µ01 A sin(δ1 x) + µ02 B sin(δ2 x).
Concerning stresses, one has:
s
σ̂xx
(x) = P̂ [δ1 A cos(δ1 x) + δ2 B cos(δ2 x)] ,
e eq µ01 δ1 A cos(δ1 x) + δ2 µ02 B cos(δ2 x) .
p(x) = −K
At the surface of the porous material (x = −l), 3 continuity conditions need to be
written. The first (resp. second) one is the continuity of the pressure (resp. total displacement). The last one is the nullity of the in-vacuo normal stress. These relations
respectively read:
s
σ̂xx
(x) = 0 , pa (−l) = p(−l) , v a (−l) = v t (−l).
(1.96)
The first one provides a relation between A and B:
A=−
δ2 cos(δ2 l)
B.
δ1 cos(δ1 l)
(1.97)
With this relation, it is easy to express pressure and velocity in x = −l which are both
proportional to A. Their ration (surface impedance) is independant of A and one has:
Z=
e eq
K
δ1 δ2 (µ02 − µ01 )
,
×
jωφ δ1 µ02 tan(δ2 l) − δ2 µ01 tan(δ1 l)
(1.98)
Z=
e eq
K
δ1 δ2
×
,
0
jω
δ1 ν2 tan(δ2 l) + δ2 ν10 tan(δ1 l)
(1.99)
or equivalently:
with νi0 defined in equation (1.82). This expression is simpler and equivalent to the classical
one provided in [?].
1.11
1.11.1
Transversely isotropic porous material [A8]
Wave equations
We are interested in this section by a transverse isotropic porous material (i.e. anisotropic
material for which there exists an isotropy plane). Equations presented in the previous
section should then be modified. Anisotropy consists in a modification of constitutive laws
(rigidities of the frame) as well as the one of acoustic parameters (flow resistivity, tortuosity and viscous characteristic length). Biot extended the isotropic theory for poroelastic
materials to the anisotropic case. In this work, he only took into account the anisotropy
in mechanical parameters (rigidities). It has been shown in a recent contribution [A8] that
a refinement of this work is possible so as to consider the anisotropy acoustic parameters.
35
Under harmonic excitation, the Biot motion equations for a transverse isotropic material (with {x, y} as isotropy plane) in {us , uf } formulation become:
∇.σ s (us , uf ) = −ω 2 [ρ̃11 ]us − ω 2 [ρ̃12 ]uf ,
∇.σ f (us , uf ) = −ω 2 [ρ̃12 ]us − ω 2 [ρ̃22 ]uf .
(1.100a)
(1.100b)
[ρ̃ij ] {i, j}∈ {1, 2} are diagonal matrices defined by
[ρ̃ij ] = diag(ρ̃xij , ρ̃xij , ρ̃zij ).
(1.101)
In these expressions, the Biot densities ρ̃iij , with i replaced by x (for x and y direction)
or z, are given by:
ρei22 = φ2 ρeieq , ρei12 = φρ0 − ρei22 , ρ̃i11 = (1 − φ)ρs − ρ̃i12 ,
(1.102)
where ρeieq is the equivalent fluid density in the direction x or z. Its expression is obtained
through equivalent fluid models or Appendix 1.B by replacing the isotropic properties by
the one of the considered direction.
Algebraic manipulations can be do so as to express transverse isotropic Biot’s equations
(1.100) in {us , ut } formulation.
(1.100a) −→ (1.100a) +
(1.100b) −→
e
Q
(1.100b)
e
R
(1.100b)
.
φ
The equations of motion for the {us , ut } formulation are
∇.σ̂ s = −ω 2 [ρ̃s ]us − ω 2 [γ̃][ρ̃eq ]ut ,
K̃eq ∇.(∇.ut [I]) = −ω 2 [γ̃][ρ̃eq ]us − ω 2 [ρ̃eq ]ut ,
(1.104a)
where [γ̃], [ρ̃eq ],and [ρ̃s ] are diagonal matrices given by:
[γ̃] = φ([ρ̃22 ]−1 [ρ̃12 ] −
1−φ
[I]),
φ
[ρ̃eq ] = [ρ̃22 ]/φ2 ,
[ρ̃s ] = [ρ̃] + [γ̃]2 [ρ̃eq ].
In these Eqs. the matrix [I] is the identity matrix of size 3 and the matrix [ρ̃] = [ρ̃11 ] −
[ρ̃12 ]2 [ρ̃22 ]−1 . The equations of motion obtained with the new {us , ut } formulation [A3] are
simpler than the ones in the previous representations.
On the other hand, the in-vacuo stress-strain relations for a transverse isotropic porous
material are similar to those of a solid and can be written:
σ̂xx = (2N + Â)εxx + Âεyy + F̂ εzz ,
σ̂yy = Âεxx + (2N + Â)εyy + F̂ εzz ,
σ̂zz = F̂ εxx + F̂ εyy + Ĉεzz ,
σ̂yz = 2Lεyz , σ̂xz = 2Lεxz , σ̂xy = 2N εxy .
36
(1.105)
1.11.2
Plane waves propagating in TIPM
Quasi-plane waves propagating in a transverse isotropic porous material are now investigated. The acoustic field is created in a transversely isotropic medium by an incident
air wave. Without loss of generality (choice of the y axis so that it does not contain the
incident field), the incidence plane is the xz plane. Let θ be the angle of incidence. The
space dependence can then be written:
us = a exp(iqωx), ut = b exp(iqωx),
(1.106)
where a = {ax , ay , az } and b = {bx , by , bz } are the polarization vectors for the solid and
total displacements. q ={qx = sin θ/c0 , qy = 0, qz } is the slowness vector. qx is imposed
by the incident field.
Substituting the expressions for displacement given by Eq. (1.106) in Eq. (1.104)
provide an homogeneous linear system of 6 equations which can be split in 2 sets. One
set corresponds to the 2 y−direction equations for the solid displacement and the total
displacement and concerns the qSH waves. The following relations are obtained for these
waves
1
by = −γ̃y ay , qz2 = (ρ̃x − N qx2 ).
(1.107)
L
Hence 2 qSH waves are obtained by taking the square root of qz2 , a down-going (Re(qz ) > 0)
wave and an upgoing (Re(qz ) < 0) wave. These 2 waves are not investigated as they are
not excited by the incident field. The four remaining equations from (1.104) relate the
polarizations in the x and the z directions and can be written in the following form
T
[A]
ax
az
bx
= {0},
bz
(1.108)
with


 Lqx2 − ρ̃xs + qx2 P̂



 (L + F̂ )qx qz

[A] = 


−γx ρ̃xeq



0
(F̂ + L)qx qz
−γx ρ̃xeq
0
(Ĉqz2 + Lqx2 ) − ρ̃zs
0
−γz ρ̃zeq
0
−ρ̃xeq + K̃eq qx2
K̃eq qx qz
−γz ρzeq
K̃eq qx
K̃eq qz2 − ρ̃zeq






 . (1.109)





Values of qz associated to quasi plane waves correspond to |A| = 0. It can then be shown
that they are the roots of a square polynom:
T3 qz6 + T2 qz4 + T1 qz2 + T0 = 0,
(1.110)
T3 = −LĈ K̃eq ρ̃xeq ,
(1.111)
T2 = T2,2 qx2 + T2,0 ,
(1.112)
T2,2 = −K̃eq [LĈ ρ̃zeq + ρ̃xeq (P̂ Ĉ + L2 − (F̂ + L)2 )],
(1.113)
T2,0 = ρ̃xeq [ρ̃x Ĉ K̃eq + L(Ĉ ρ̃zeq + ρ̃s,z K̃eq )],
(1.114)
with
37
T1 = T1,4 qx4 + T1,2 qx2 + T1,0 ,
(1.115)
T1,4 = −K̃eq [LP̂ ρ̃xeq + ρ̃zeq (P̂ Ĉ + L2 − (F̂ + L)2 ],
(1.116)
T1,2 = K̃eq [L(ρ̃zeq ρ̃z + ρ̃xeq ρ̃x ) + P̂ ρ̃xeq ρ̃zs + Ĉ ρ̃zeq ρ̃xs ]
(1.117)
+ρ̃zeq ρ̃xeq [L2 + P̂ Ĉ − (F + L)2 − 2(F + L)K̃eq γ̃x γ̃z ,
T1,0 = −ρ̃xeq [Lρ̃xeq ρ̃z + Ĉ ρ̃zeq ρ̃x + K̃eq ρ̃x ρ̃zs ],
(1.118)
T0 = T0,6 qx6 + T0,4 qx4 + T0,2 qx2 + T0,0 ,
(1.119)
T0,6 = −ρ̃zeq LP̂ K̃eq ,
(1.120)
T0,4 = ρ̃zeq [L(K̃eq ρ̃xs + P̂ ρ̃xeq ) + ρ̃x P̂ K̃eq ],
(1.121)
T0,2 = −ρ̃zeq [Lρ̃x ρ̃xeq + ρ̃z (K̃eq ρ̃xs + P̂ ρ̃xeq )],
(1.122)
T0,0 = ρ̃zeq ρ̃z ρ̃x ρ̃xeq .
(1.123)
There are no odd terms in this cubic polynomial in qz2 . Each root provides 2 square roots.
They can be numbered with the index k = 1, 2, 3 for the downgoing waves and k + 3 for
the upgoing waves. For each qz , the polarization of the wave can be normalized so that
bz = 1
{ax , az , bx , bz } ∝ {µx,s , µz,s , µx,t , 1}.
(1.124)
This normalization does not allow a null z total displacement component, but there is no
restriction to use it in our context. The coefficients µ can be written:
µx,t =
2 q q )[(P q 2 ) + L q 2 − q 2 ] − (γ q 2 q q )(q 2 − q 2 )
−(γz qc,z
x c,x x z
x z
0 x
0 z
s,x
z
eq,z
,
2
2
2
2
2
2
2
2
2
γz qc,z [(P0 qx + L0 qz − qs,x )(qx − qeq,x ) − γx qc,x qe,x ] + γz qc,z qz2 qx2
(1.125)
where P0 = P̂ /(L + F̂ ) and L0 = L/(L + F̂ ),
µz,s =
2 )+q q µ ]
K̃eq [(qz2 − qeq,z
z x x,t
µx,s =
2 )µ )
K̃eq (qz qx + (qx2 − qeq,x
x,t
2
(L + F̂ )γ̃z qc,z
2
(L + F̂ )γ̃z qc,z
,
,
with the slownesses of the material
s
s
s
ρ̃ieq
ρ̃ieq
ρ̃is
qeq,i =
, qc,i =
, qs,i =
.
K̃eq
L + F̂
L + F̂
(1.126)
(1.127)
(1.128)
The following parity/imparity relations are used for the upgoing waves
µx,t (k + 3) = −µx,t (k), µz,s (k + 3) = µz,s (k), µx,s (k + 3) = −µx,s (k).
38
(1.129)
Figure 1.5: Slowness curves for a mineral wool
1.11.3
Theoretical and experimental results
Fig. 1.5 presents the slowness curves for a mineral foam and Fig. 1.6 represents the
experiments. Measurements have been performed on a layer of thickness l = 2cm of mineral
foam in the plane and transverse direction. To obtain a uniform boundary condition doublefaced tape was used to glue the porous layer onto a rigid impervious backing. Rayleigh
waves were excited by means of a magnetic transducer putting in motion a circular plate of
diameter 1 cm bonded on the layer. The magnetic transducer was consecutively fed with
a sine burst signal of 2, 3, and 3.5 kHz, respectively. The normal velocity of the frame is
measured with a laser Doppler vibrometer. Measurement points are located on a radius
through the excitation point. The radius was scanned up to 2.5 cm and with a typical step
size of 5 mm. The damping and the time of flight between two locations are evaluated
from the intercorrelations. For this mineral wool a strong anisotropy is observed which is
accurately predicted by the theory.
1.12
Nonlinear Biot waves in non consolidated granular media [A12]
This section is dedicated to the application of the Biot theory to unconsolidated granular media. The acoustics of these materials has been widely investigated among different
fields such as geophysics, underwater or airborne acoustics (sound proofing, shock wave
absorption ...). Recently in physics, granular materials have attracted a strong interest
because they exhibit unusual behaviors as gas, liquids or solids[?, ?]. These behaviors are
sometimes comparable to glassy media, from the point of view of the ageing process for
instance [?], or regarding the prediction of the soft modes and their relation to the boson
peak [?, ?]. The transition from one state to another, and in particular the unjamming
39
Figure 1.6: Measurement of phase velocities of a mineral wool
transition from a solid-like to a fluid-like phase, is intensively studied [?].
Most of the studies on acoustic wave propagation through granular media either consider the propagation through the solid network, neglecting the saturating fluid, or consider
the propagation through the fluid phase saturating the rigid bead packing, which is assumed
to be motionless. These assumptions have been shown to be invalid by experiments.
By considering two coupled continuous media, Biot theory seemed to be suitable to
model this type of material by including solid and fluid behavior. This means that it is
necessary to extend the classical Biot theory so as to catch non-linear behavior. This nonlinearity is mainly due to the contact between solid beads and it is reasonable to consider
that the saturating fluid can be considered as linear. Moreover, its density and compressibility can be modelled by classical equivalent fluid model as far as the acoustic parameters
are known. Similarly, viscous and inertial effects are not supposed to be influenced by the
properties of the fluid.
It seems then natural to modify only the in-vacuo stress-strain relation of the solid:
s
σij
= 2N εsij + λεδij + ∂ εij H,
(1.130a)
e eq ∇.uW + K
e eq θδij ∂ ε H.
p = −K
ij
(1.130b)
θ is the fluid compliance coefficient and H is a non-linear potential defined by:
!
3
I1
H=D
− I 1 I 2 + I 3 + F (I 1 I 2 − 3I 3 ) + GI 3
3
40
(1.131)
where the I i are the strain invariants associated to the modified strain[?] εij defined by
εij = εij + pθδij .
(1.132)
D, F and G are the coefficients of nonlinearity. Strain invariants are recalled:
I 1 = tr(εij ) , I 2 = ε11 ε22 +ε11 ε33 +ε22 ε33 −ε12 ε21 −ε13 ε31 −ε23 ε32 , I 3 = det(εij ). (1.133)
Nonlinear contributions of H to the in-vacuo stress and pressure can be then expanded
and read:
2
∂ εij H = F I 1 δij + (F − D)(δij I 2 − I 1 εij ) + (G + D − 3F )cof (εij ),
2
2
(1.134a)
2
δij ∂ εij H = 3F I 1 + (F − D)(3I 2 − I 1 ) + (G + D − 3F )I 2 = P I 1 + (G − 2D)I 2 , (1.134b)
with P = D + 2F and cof (εij ) corresponds to the matrix of cofactors.
In the case of a monodimensional problem along x axis, it is possible to simplify the
problem. One has:
I 1 = us,x + 3θp , I 2 = (2us,x + 3θp)θp , I 3 = (us,x + θp)θ2 p2 ,
(1.135)
and the nonlinear contributions to the stress are
∂ εij H = Dus,x 2 + 2P (θpus,x ) + K(θp)2 ,
(1.136a)
δij ∂ εij H = P us,x 2 + 2Kθpus,x + 3K(θp)2 ,
(1.136b)
with K = D + 6F + G. In the following, the method of successive approximation is used
and the forces corresponding to the nonlinear terms are built with the linear solutions
previously computed (and denoted with a l index) usl and uW
l . This induces that the
e eq uW . The forces induced
pressure pl in the nonlinear terms is defined by relation pl = −K
l,x
by the nonlinear potential then read:
e 2 θ2 uW uW .
e eq P θ(uW us + uW us ) + 2K K
∂ εij Hl,x = 2Dusl,x usl,xx − 2K
eq
l,x l,xx
l,x l,xx
l,xx l,x
(1.137a)
e eq Kθ(uW us + uW us ) + 6K K
e 2 θ2 uW uW .
δij ∂ εij Hl,x = 2P usl,x usl,xx − 2K
eq
l,xx l,x
l,x l,xx
l,x l,xx
(1.137b)
Additional simplifications can be made in the case of granular media taken here as an
illustration, and for other porous media supporting Biot waves as long as the following
conditions on compressibility are fulfilled. The bead material is much stiffer than the contacts between beads (and consequently the solid frame)[?]. Approximations of subsection
1.9 can then simplify the expressions. The two main consequences are that uW = ut ,
the only remaining nonlinear term in (1.137a) is the one containing usl,x usl,xx and that the
constitutive law (1.130b) can be approximated by its linear part.
For model granular media, the quadratic elastic nonlinearity is a reasonably good first
approximation[?, ?, ?] as for other porous media, in order to explain the harmonic generation process, at least for some moderate range of excitation amplitude [?, ?]. Consequently,
we keep this type of nonlinearity as the starting point. The value of the parameter of
quadratic nonlinearity has been measured to be several orders of magnitude higher than
in homogeneous fluid or solids [?, ?], but can vary over a wide range depending on the
41
external conditions of static stress or the particular bead arrangement for instance [?]
P̂ us,xx + 2Dus,x us,xx = −ω 2 ρes us − ω 2 γ
eρeeq ut ,
e eq ut = −ω 2 γ
K
eρeeq us − ω 2 ρeeq ut ,
,xx
(1.138a)
(1.138b)
The linear part (D = 0) is first considered on the slab configuration.It consists of an
excitation boundary at x = −d and a rigid boundary at x = 0. The solid and total
displacements are imposed at x = −d and vanish at the rigid boundary x = 0. They can
be written using trigonometric functions as:
us = (A1 sin(δ1 x) + A2 sin(δ2 x) + B1 cos(δ1 x) + B2 cos(δ2 x)) ejωt ,
(1.139)
ut = µ01 A1 sin(δ1 x) + µ02 A2 sin(δ2 x) + µ01 B1 cos(δ1 x) + µ01 B2 cos(δ2 x) ejωt .
(1.140)
As both displacements are zero at x = 0, one has B1 = B2 = 0. At x = −d, one has
u0 = −A1 sin(δ1 d) − A2 sin(δ2 d),
(1.141)
u0 = −µ01 A1 sin(δ1 d) − µ02 A2 sin(δ2 d),
(1.142)
and hence:
A1 =
(µ01
(µ0 − 1)u0
(µ02 − 1)u0
, A2 = 0 1 0
.
0
− µ2 )sin(δ1 d)
(µ2 − µ1 )sin(δ2 d)
(1.143)
The transmission of a layer (thickness=8 cm) of glass beads of 2 mm diameter is
considered. Fig. 1.7 presents the confrontation between the model and the experimental
curves. It can be observed that the model is able to catch the transition between the solid
controlled behavior (low frequency range) and the transmission through the fluid phase
(high frequency range).
Concerning non-linear terms, solid and fluid displacements can be written as the sum
of a general and particular solution. The particular solution is obtained by considering
the non-linear term as a forcing in which the displacement is the solution of the linear
problem. The spatial derivatives of the solid displacement field ul , solution of the problem
when non-linear terms are cancelled are:
∂ul
= −jδ1 u1 e−jδ1 x − jδ2 u2 e−jδ2 x + jδ1 u01 ejδ1 x + jδ2 u02 ejδ2 x ,
∂x
∂ 2 ul
= −δ12 u1 e−jδ1 x − δ22 u2 e−jδ2 x − δ12 u01 ejδ1 x − δ22 u02 ejδ2 x .
∂x2
(1.144)
(1.145)
Hence, with 2δ3 = δ1 + δ2 and 2δ4 = δ1 − δ2 ,
∂ul ∂ 2 ul
2jδ1 x
=jδ13 δ12 e−2jδ1 x − jδ13 u02
1e
∂x ∂x2
2jδ2 x
+ jδ23 u22 e−2jδ2 x − jδ23 u02
2e
+ 2jδ1 δ2 δ3 u1 u2 e−2jδ3 x − 2jδ1 δ2 δ3 u01 u02 e2jδ3 x
− 2jδ1 δ2 δ4 u01 u2 e−2jδ4 x + 2jδ1 δ2 δ4 u02 u1 e2jδ4 x .
42
(1.146)
So
lid
Air
Transmission(dB)
Frequency (Hz)
Figure 1.7: Transmission of a glass bead layer
The particular solution is calculated at frequency 2ω. Tilde coefficients are replaced by
double dot ones. One can then write the particular solution under the following form:
usp
8
X
= α0 +
αi e−2jδi x ,
utp = β0 +
i=1
8
X
βi e−2jδi x .
(1.147)
(1.148)
i=1
For i > 4, δi = −δi−4 , αi = αi0 and βi = βi0 . As the product (1.146) do not involve constant
terms, one has:
ρ̈s α0 + γ
eρ̈eq β0 = 0 , γ̈ ρeeq α0 + ρ̈eq β0 = 0,
(1.149)
and thereby α0 = β0 = 0. The additional amplitudes are the solutions of the following
system:








α























0
2
2
4ω 2 γ̈ ρ̈eq [I8 ]
 4ω ρ̈s [I8 ] − 4P̂ [k]
 α   ν 


=
,
(1.150)









 08 


β 
4ω 2 γ̈ ρ̈eq [I8 ]
4ω 2 ρ̈eq [I8 ] − 4K̈eq [k]2 










 0 

 β 

where [δ] is the diagonal matrix of the δi wave numbers and ν is defined by
t
3 02
0 0
0
ν = j δ13 u21 , δ23 u22 , 2δ1 δ2 δ3 u1 u2 , −2δ1 δ2 δ4 u01 u2 , −δ13 u02
.
1 , −δ2 u2 , −2δ1 δ2 δ3 u1 u2 , 2δ1 δ2 δ4 u2 u1
(1.151)
43
Stress level (dB, norm.)
0
−20
−40
−60
−80
−100
−120
1st harmonic exp.
−140
−160
0
2nd harmonic exp.
0.5
1
1.5
2
2.5
Frequency (Hz)
3
3.5
4
4.5
4
x 10
Figure 1.8: Transmission of a glass bead layer, 2nd harmonic generation
It is possible to find analytical expressions of the solutions (j = 1..8):
νi
αi =
4ω 2 ρ̈eq −
βi = −
4P̂ δi2
4ω 2 γ̈ ρ̈eq
−
4ω 2 ρ̈eq − 4K̈eq δi2
,
4ω 2 γ̈ ρ̈eq
αi .
4ω 2 ρ̈eq − 4K̈eq δi2
(1.152)
(1.153)
The general solution corresponds to a zero nonlinear displacement at x = −d and x = 0
and its amplitudes are solutions of the system:


4

X
 




0




α
+
α
i




i








1
1
1 
γ
 1



1
i=1







 


4


X









0





β
+
β




i
i
  γ2  
 µ̈1

µ̈
µ̈
µ̈
2
1
2


i=1
=
.
(1.154)


4
X








2jki d
0 −2jki d 
0
0
0

 e1





e2
e1
e2  
γ  
αi e
+ αi e



 1 
 











i=1









4




X
0
0
0




µ̈1 e1 µ̈2 e2 µ̈1 e1 µ̈2 e2
γ2


2jk
d
0
−2jk
d
i
i


β
e
+
β
e


i
i


i=1
It is straightforward to find numerical values of the amplitudes γi using Cramer’s method.
1.13
Biot model for double porosity media [A19]
Double porosity have mostly be modeled through equivalent fluid models. Experimental
results have shown that this assumption is not valid as peaks associated to Biot’s effects
have been observed. A theory to extend Biot’s theory to double porosity materials is then
44
Figure 1.9: Double porosity material (Courtesy of MatelysAcV)
proposed.
One of the main assumptions of Biot’s theory is to assume that the possible frame
deformation does not influence the visco-thermal dissipation inside the double porosity
material. That is the reason why, for homogeneous porous material, it is legitimate to use
the equivalent-fluid properties in the Biot theory. Double porosity models have been shown
to be in the family of equivalent fluids[?]. In this framework, the main idea of the model is
to use double porosity densities and compressibilities in the Biot theory. A particular care
should be held with porosity.
The material is supposed to be homogenizable under the same assumptions than the
ones of Olny and Boutin[?]. The REV Ω is then divided in three parts Ωs , Ωf and Ωp
respectively associated to the solid, fluid part of the microporous domain and the mesopore
V (Ωf )
be the porosity of the single porosity material and and
volume. Let φm =
V (Ωs ∪ Ωf )
V (Ωp )
φp =
the porosity of the mesopores. The volume of air fraction (i.e. Ωf ) which is
V (Ω)
also called mesopore, is supposed to be aligned with the x direction.
First, φdp = (1 − φp )φm + φp is the actual porosity of the double porosity material (i.e.
the ratio of the total volume of air to the total material one). The solid displacement field
45
SP model
DP model
φ
φdp = (1 − φp )φm + φp
us
usm
ut
utdp = (1 − φp )utm + φp up
ρ1
ρ1dp = (1 − φp )ρ1
ρ2
ρ2dp = [φp + (1 − φp )φm ]ρ0
ρeeq
dp
ρeeq
= ρedp
ρes
ρ0
γ
e dp =
γ
e
ρesdp
dp
ρ̃eq
Remark
in each direction
− (1 − 2φdp )
= (1 − φdp )ρs − (ρ0 −
dp
φdp ρ̃eq
)
2φdp −
in each direction
ρ0
!
dp
ρ̃eq
in each direction
Table 1.1: Parameters for the porous medium
usdp =< usµ >Ωs is defined as the average microscopical displacement on Ωs (µ index is
associated to microscopical fields). For the fluid displacement uf , one has:
ufdp =< ufµ >Ωp ∪Ωf =
φp
(1 − φp )φm
< ufµ >Ωp +
< ufµ >Ωf .
φdp
φdp
(1.155)
The total displacement is defined as
utdp = (1 − φp )(1 − φm ) usdp + φdp ufdp .
|
{z
}
(1.156)
1−φdp
ρ0 and ρs remain unchanged as they correspond to densities of air and of the frame material.
It is generally common (even if unused in the present representation) to use the solid frame
density ρ1 which correspond to the density of the porous solid and ρ2 equivalent density of
air. Their expressions are given in Table 1.1. Assumption usdp = 0 leads to the equivalent
fluid model and utdp is then the displacement of the homogenized medium. Parameters
e eq of the equivalent fluid should then be respectively replaced by ρedp and K
e dp .
ρeeq and K
dp
Expressions of [e
γ ] and [e
ρs ] are deduced while replacing φ (resp. ρeeq ) by φdp (resp. ρedp ).
We will compare the model with impedance tube measurements which are associated to
a plane wave normal incidence absorption measurements and monodimensional problems.
We are then interested by wave propagation along z dimension and the motion equations
read:
∂ 2 usdp
(Cdp + Fdp )
= −ω 2 ρesdp usdp − ω 2 γ
e dp ρedp utdp ,
(1.157a)
{z
} ∂z 2
|
Pdp
46
1
0.9
0.8
Abs. coefficient
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
200
400
600
800
Frequency [Hz]
1000
1200
Figure 1.10: Normal incidence sound absorption coefficient for material W1; ’Solid’: eqFEM, ’o’: ’Biot-FEM, ’+’: Analytical proposed model, ’Dashed’: Experimental data.
e dp
K
∂ 2 utdp
∂z 2
= −ω 2 γ
e dp ρedp usdp − ω 2 ρedp utdp .
(1.157b)
Cdp and Fdp are the Lamé coefficients in direction z of the sample. Pdp is the Young’s
modulus of the double porosity material. This system is of Biot type and it can be shown
that two compression waves can propagate. Calculation of the absorption coefficient is
then straightforward.
Models accounting for the porous frame deformations are now compared to experiments
for three double porosity materials (F, W1 and W2). Results are presented in Fig. 1.10 to
1.13. In each one of the following graphs, absorption curve are obtained through eq-FEM,
Biot-FEM3 and analytically through expression (1.99) and using the inversion procedure
proposed above (see section 2.2.5) for equivalent fluid parameters.
Fig. 1.10 presents results for material W1. Solid frame resonance is around in the 5001000 Hz frequency range. A good agreement is obtained between the proposed analytical
model (1.99) and experimental results. In particular, the proposed model captures the
curve inflexion visible on the measured data between 600 Hz and 800 Hz. Absorption levels
at other frequencies are unchanged compared to the model of the previous section which
assumed a rigid and motionless frame. For material W2, presented Fig. 1.12, the proposed
model also shows to be robust and compares well with the Biot-FEM model and measured
data. For the foam, material F, results are presented Fig. 1.13. Biot effects are visible on
the entire frequence range tested in this case. Therefore, for this material, the eq-FEM is
not suited to model the absorption correctly. The analytical proposed model enables to
catch the frame resonance and the correspondence between the obtained results and the
measured data is satisfactory. These results compare also well with results obtained with
the Biot-FEM model, though with larger deviations between these two models compared
to the previously tested materials. The reason for this is still unclear and will be subject
of further research.
3
eq-FEM and Biot-FEM are Finite elements model of the problem. For the first one, the porous sample
is modelled through an equivalent fluid model. For the second one it is a Biot model.
47
Abs. coefficient
0.9
0.85
0.8
0.75
0.7
500
600
700
800
Frequency [Hz]
900
1000
1100
Figure 1.11: Absorption coefficient for material W1 -Detail of Fig. 1.10 around the
solid frame resonance-; ’Solid’: eq-FEM, ’o’: ’Biot-FEM, ’+’: Analytical proposed model,
’Dashed’: Experimental data.
1
0.9
0.8
Abs. coefficient
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
200
400
600
800
Frequency [Hz]
1000
1200
Figure 1.12: Absorption coefficient for material W2; ’Solid’: EFEM, ’o’: ’BFEM, ’+’:
Analytical proposed model, ’Dashed’: Experimental data.
48
1
0.9
0.8
Abs. coefficient
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1000
2000
3000
Frequency [Hz]
4000
5000
Figure 1.13: Absorption coefficient for material F; ’Solid’: eq-FEM, ’o’: ’Biot-FEM, ’+’:
Analytical proposed model, ’Dashed’: Experimental data.
1.A
Gedanken experiments for {us , uf } formulation
Kb
Ks Kb
(1 − φ) 1 − φ −
Ks + φ
ef
Ks
K
2N
e=
A
−
,
Kb
Ks
3
1−φ−
+φ
ef
Ks
K
(1.158a)
Kb
1−φ−
φKs
Ks
e
,
Q=
Kb
Ks
1−φ−
+φ
ef
Ks
K
(1.158b)
e=
R
φ2 Ks
.
Kb
Ks
1−φ−
+φ
ef
Ks
K
49
(1.158c)
1.B
Equivalent fluid models
Johnson-Allard Model (5 parameters: φ, σ, α∞ , Λ, Λ0 )
"
#
r
ρ0
ωc
M jω
ρeeq (ω) = α∞ 1 +
1+
φ
jω
2 ωc
ωc =
ν0 φ
8k0 α∞
µ0
,M=
, k0 =
.
2
k0 α∞
φΛ
σ
γ0
e eq (ω) =
K
ω0
γ0 − (γ0 − 1) 1 − j c
ω
ωc0 =
(1.160)
P0
φ
"
s
M 0 jω
1+
2 ωc0
(1.159)
(1.161)
#
8ν0
P r Λ02
(1.162)
Johnson-Lafarge Model (6 parameters: φ, σ, α∞ , Λ, Λ, k00 )
"
#
r
ρ0
ωc
M jω
ρeeq (ω) α∞ 1 − j
1+
(1.163)
φ
ω
2 ωc
ωc =
8k0 α∞
ν0 φ
,M=
k0 α∞
φΛ2
γ0
e eq (ω) =
K
P0
φ
"
ω0
γ0 − (γ0 − 1) 1 − j c
ω
ωc0 =
(1.164)
s
M 0 jω
1+
2 ωc0
(1.165)
#
8k00
ν0 φ
0
,
M
=
k00
φΛ02
(1.166)
Pride-Lafarge Model (8 parameters: φ, σ, α∞ , Λ, Λ, k00 , M, M 0 )
s
"
(
)#
ρ0
ωc
M jω
ρeeq (ω) α∞ 1 − j
1−p+p 1+ 2
(1.167)
φ
ω
2p ωc
ωc =
e eq (ω) =
K
ν0 φ
8k0 α∞
,M=
,p=
k 0 α∞
φΛ2
"
ω0
γ0 − (γ0 − 1) 1 − j c
ω
ωc0 =
M
α0
−1
α∞
4
P0
γ0
φ
(
1 − p0 + p0
s
M 0 jω
1 + 02 0
2p ωc
ν0 φ
8k00
M0
0
0
,
M
=
,
p
=
k00
φΛ02
4(α00 − 1)
50
(1.168)
)#
(1.169)
(1.170)
Chapter 2
Numerical methods for porous
materials
Numerical methods for porous materials can be divided in two categories. The first one
are the semi-analytical and are concerned with simplified problems (generally associated
to plane or quasi plane waves) and the second one are based on discretization techniques
(generally finite-difference of finite-element methods). This chapter is divided in three
parts. The first two ones are associated to these two types of methods and the last one
correspond to modal techniques of resolution of finite-element problems.
2.1
2.1.1
Plane waves methods
Introduction
In this section, we are interested with multilayered panels presented Fig. 2.1. Each layer
is generally composed of an homogeneous (i.e. with constant properties) material. In the
most general case, the different layers can be associated to several types of medium (elastic
solid, fluid, isotropic or transverse isotropic poroelastic material ...). The layers are also
assumed to be infinite in the lateral dimensions; the only geometric information is then the
thickness of the layer.
The whole structure is excited by an airborne plane wave with an angle of incidence
denoted by θ. This implies that the common transversal wave number reads
kx = k0 sin(θ),
(2.1)
where k0 is the wave number in air. Two configurations can be considered at the right-hand
side: the stratified structure can be bonded on a rigid wall or radiating in a semi-infinite
medium.
This type of problem does not correspond to practical application but is very popular
as it is not very numerically consuming and its results are good estimates of the acoustical behavior of the system. In particular, it can be interesting to first customize a sound
package with this technique and then simulate the response of the whole structure in only
a limited number of cases associated to the optimized structures.
Propagation of sound inside a layer of material can be modelized by a square matrix
whose size depends on the material. Two types of matrices can be considered:
51
Figure 2.1: Example of a multilayered structure
medium
n
fluid
1
isotropic solid
2
poroelastic
3
Table 2.1: Number of waves in each medium
• Transfert matrices which give the relation between stresses and velocities at each side
of the layer.
• Impedance matrices which provide the relation between stresses on both side of the
layer to velocities on both side of the layer.
Transfert matrix are preferred for their convenience as they provide a simple way for
modelling acoustical properties of multilayers whereas impedance matrix are very interesting to emphasis reciprocical properties. In this chapter, we focus on the first class.
2.1.2
Principle of transfer matrices
Let now consider a layer of arbitrary type. Let n be the number of waves propagating in
this medium. Table 2.1 presents the number of waves propagating in the media of interest.
It can be easily shown that all the fields (displacements, stresses) can be expressed
through 2 × n parameters. These ones are associated to the amplitudes of the waves of
each type propagating in the increasing or decreasing z. The principle of transfer matrices
is then to express the relation between the state vectors at the two opposite sides of the
layer. It can also be shown that for each type of medium, the state vector is the aggregation
of n velocities and n stresses. Table 2.2 presents the expression of the state vectors W
depending of the actual medium. The state vector depends on the geometrical parameter
52
medium
fluid
State vector



f 

 vz 

medium
State vector
isotropic solid



 pf 





vxe








 vze















e


σxz







e
 σzz












medium
poroelastic
State vector









vzs 














s


v

x 









 t 


 vz 








p 














s


σ̂


xz













s 
 σ̂zz

Table 2.2: State vectors for each medium
z, and it is possible to express it with the amplitudes of the different waves
W(z) = [M(z)]A.
(2.2)
A is the 2n vector of the amplitudes of the waves and [M(z)] is the (2n × 2n) matrix
associated to expressions of velocities and stresses as function of the amplitudes. It can
be shown that these matrix are invertible for every position z. It is then straightforward
to formally express the relation between the state vectors at the two opposite side of the
layer (z = 0 and z = H)
W(H) = [M(H)][M(0)]−1 W(H).
|
{z
}
(2.3)
[T]
[T] is the transfer matrix of the layer. It should be highlighted that the state vectors are
expresses inside the layer (i.e. one has W(0+ ) and W(H − )). A second step consists in
expressing the relations between the state vectors at an interface.
Let now consider an interface between two homogeneous media. Let W− and W+ be
the two state vectors associated to the two materials. These two vectors can be of different
length but the most easy case correspond to layers of same type where it can be shown that
vectors defined in table 2.2 are continuous. In this case W− = W+ and it is straightforward
that the transfer matrix of the multilayered system can be obtained by multiplication of the
transfer matrices of each layer. In the case of different media interface, physical relations
should first be written. One can then write relations between the component of the two
state vectors or exhibits unknowns. It is summarized Table 2.3. In the case of the original
TMM technique these type of interface was accounted for through interface matrices based
on these relations. It was then possible to express the relation of the state vectors at the
two opposite sides of any multilayered structure.
2.1.3
Expression of reflexion and transmission coefficients
This subsection presents how to calculate the response of the package to an incident wave
from the transfer matrix. Only the case of a (single or multilayer) porous structure is
53
Interface
solid-porous
vze
=
vze
Relations
=
vxe
=
e
σzz
porous-fluid
vzt
vze
vzt
e
σzz
vxs
e
σxz
= σ̂zz − p
e
σxz
=
Unknowns
solid-fluid
=
vzf
= −pf
=
vzt
= vzf
p
= pf
σ̂xz =
0
σ̂xz =
0
0
σ̂xz
vxs = ?
vxe = ?
? = p
vzs = ?
Table 2.3: Relations between state vectors at interfaces
considered but the method can be easily adapted for the other types of structures. The
structure is considered to be between z = 0 and z = H and let [T] be the transfer matrix
between the two interfaces. It can be shown that the state vector at the z = 0 interface
can be written in the following form:


 0 0



 0 0



 1 0

W(0) = 


 0 1



 0 0


 jkz
0
ρ0
|
{z
[Ω0 ]






















0 
0




0 
0






R 









0 
1


s
+
  u1 (0)  






0 
0











s
  u2 (0)  





{z
}
|
1 
0




X(0)






jk

 − z
0
ρ
}
{z0
|
S






















(2.4)





















}
usx (0), usz (0) are the unknowns solid phase displacements at z = 0. R is the reflexion
coefficient.
In the case of a transmission problem, the state vector Wt (H) can be expressed through
usz (H) and the transmission coefficient T .
usx (H),
54














































0
 0



 0
0
0



 1
0
T

Wt (H) =
=
 



s (H) 


1
u


 0
x












 


s (H) 


 0
0
u


z





 





  jk2

jk


 − zT 
0
−
ρ0
ρ0
|
{z
0
[ΩH ]
0 




0 




 T












0 

s
  ux (H) 



0 







s
  uz (H) 



{z
}
1 |
X(H)


0
}
(2.5)
One then has
[T][ΩH ]X(H) = [Ω0 ]X(0) + S.
(2.6)
Unknowns X(H) and X(0) are solution of a 6 × 6 linear system:
[[T][ΩH ]| − [Ω0 ]]





 X(H) 

=S
(2.7)



 X(0) 

In the case of a rigid backing, one has:

 



s (H) 





σ̂xx

















s


σ̂zz (H) 



















 p(H)  

Wb (H) =
=











0








 










0





















0

1 0 0 




0 1 0 



s (H) 


 σ̂xx









0 0 1 

s
  σ̂zz (H) 





0 0 0 






  p(H) 

{z
}
|
0 0 0 

X0 (H)


0 0 0
{z
}
|
(2.8)
[Ω0H ]
X0 (H) and X(0) are solution solution of:



 X0 (H)
0
[T][ΩH ]| − [Ω0 ]


 X(0)




= S.
(2.9)



Depending of the problem, linear systems (2.7) and (2.9) can be solved which provides
reflexion and transmission coefficients.
55
2.1.4
Expression of the Transfer Matrix of a transversely isotropic poroelastic material [A8]
In this section, the transfer matrix for a transversely isotropic porous material is established by the way of the classical technique. It has been established in Ref. [A8].
Each component of the state vector can be expressed as a sum of the contributions
of the six waves. For example, the solid phase velocity in z direction at z = 0 and at
z = H,are linked to the amplitudes fk of the waves by
vzs (0) =
6
X
r1 (k)fk , vzs (H) =
k=1
6
X
r1 (k)ek fk ,
(2.10)
k=1
where
r1 (k) = iωµz,s (k), ek = exp(iωqz (k)H), k = 1, 2, 3, ek+3 =
1
,
ek
(2.11)
and fk denotes the amplitude of the waves. Expressions of the µ quasi plane waves factors
are given section 1.11. Similar relations can be obtained from the 5 remaining components
of the state vector. They involve functions ri , i = 2, 3, 4, 5, 6
r2 (k) = iωµx,s (k) , r3 (k) = iω,
(2.12)
r4 (k) = λp (k) , r5 (k) = λx (k) , r6 (k) = λz (k),
(2.13)
where the λ are given by
λx (k) = −iLω[qz (k)µx,s + qx µz,s ], λx (k + 3) = λx (k),
(2.14)
λz (k) = −iω[F̂ qx µx,s + Ĉqz (k)µz,s ], λz (k + 3) = −λz (k),
(2.15)
λp (k) = iω K̃eq [qx µx,t + qz (k)], λp (k + 3) = −λp (k).
(2.16)
From expressions (2.10), it is first possible to express the coefficients mij of the [M(0)]
matrix:
mij = ri (j).
(2.17)
In a second time, expression of [M(H)] can be deduced from [M(0)] and the ek :
[M(H)] = [M(0)][E],
(2.18)
where [E] is the (6 × 6) diagonal matrix of the ek . Hence, expression (2.3) of the transfer
matrix reads:
[T] = [M(0)][E][M(0)]−1 .
(2.19)
Its elements Tij are given by the following expression expressed in [A8]:
3 X
(−1)i+j
ri (k)cj (k),
Tij =
ek +
ek
(2.20)
k=1
where
λx (k + ) − λx (k ++ )
,
iω∆2
(2.21)
λp (k + )λz (k ++ ) − λp (k ++ )λz (k + )
,
iω∆/∆1
(2.22)
c1 (k) =
c2 (k) =
56
c3 (k) =
µz,s (k + )λx (k ++ ) − µz,s (k ++ )λx (k + )
,
iω∆2
(2.23)
c4 (k) =
µx,s (k ++ )λz (k + ) − µx,s (k + )λz (k ++ )
,
∆/∆1
(2.24)
µz,s (k ++ ) − µz,s (k + )
,
∆2
(2.25)
λp (k ++ )µx,s (k + ) − λp (k + )µx,s (k ++ )
,
∆/∆1
(2.26)
c5 (k) =
c6 (k) =
P3
k=1 µz,s (k)(λx (k
∆1 = 4
e1 e2 e3
∆2 = 2e1 e2 e3
3
X
k=1
∆ = −8
" 3
X
− λx (k ++ ))
,
(2.27)
µx,s (k)(λp (k + ) − λp (k ++ )),
(2.28)
#
+
+
λp (k )(µx,s (k )λz (k
k=1
"
×
+)
3
X
k=1
++
) − µx,s (k
++
+
)λz (k ))
#
λx (k)(µz,s (k ++ ) − µz,s (k + )) .
(2.29)
In all the preceding expressions the + (respectively ++) superscript corresponds to a first
(respectively second) following index in circular permutation of the {1,2,3} set (for instance
2+ = 3, 2++ = 1).
Figure 2.2 presents the variation of real and imaginary part of the surface impedance
of a mineral wool and the comparison transverse isotropic and equivalent isotropic model.
It exhibits the need to take anisotropy into account.
2.1.5
Stroh formalism for materials with gradient of properties [A16]
In all the preceding section, the property of the material are supposed to be constant in
the layer. It is possible to extend the technique for inhomogeneous material for which
the properties are not constant in the z direction. The case of an isotropic material is
considered. It is based on the Stroh formalism and the Peano series expansion. Stroh
formalism consists in expressing equation of motion and constitutive laws as a differential
problem of order 1. One then has:
∂
W(z) = [α(z)]W(z),
∂z
57
(2.30)
Figure 2.2: Surface impedance of a transverse isotropic mineral wool; red: experiments;
Solid: Transverse isotropic model; Dashed: Isotropic model
with












[α(z)] = 










0
0
0
0
0
0
0
0
0
jkx
Â
−
γ
ejk1
P̂
ω2
0
P̂
−
0
ω2
k2
+ 1
e eq
ρeeq
K
jk1
−e
ρs
−e
ρeq γ
e
0
0
0
ρeeq γ
e
ρeeq
0
0
ω2
N
jk1
0
0
0
Â2
−
P̂ 2

k12 − ρe 



Â

jk1

P̂



−jk1 γ
e





0





0



0
P̂ ω 2
(2.31)
The expression of the transfer matrix [M] is given by the way of Peano series:
Z z
Z H
Z H
[M] = [I] +
[α(z)]dz +
[α(z)]
[α(ζ)]dζ dz + ....
(2.32)
0
0
0
The matrix can then be evaluated with an iterative scheme:
Z z
0
n
[M (z)] = [I] , [M (z)] = [I] +
[α(ζ)][Mn−1 (ζ)] dζ
(2.33)
0
The transfer matrix is then defined by a limit:
[M] = lim [Mn (H)].
n→∞
(2.34)
As most of advanced numerical methods, this procedure cannot be validated with existing methods only in simple cases. Fig. 2.3 presents the results for a bilayer porous
58
Figure 2.3: Reflection coefficient of a bilayered structure for two incident angle
material. Real and Imaginary part of the reflection coefficient are represented as function
of frequency for two incident angle. The method perfectly fits with the TMM and Fig. 2.4
presents the validation with experimental results.
2.1.6
Divergence of TMM and alternative method [C16]
Even if the TMM is exact from a mathematical point of view, it can diverge for some
configurations. It is the case for sample of large thickness or in the high frequency range.
It can be observed Fig. 2.5 adapted from the PhD of Stephan Griffiths. Methods (2.7) and
(2.9) are not the only ones that can be used for the resolution (Brouard used for example determinants) but all the methods based on Transfer matrices presents this drawback.
This drawback is due to increasing exponentials and can be overcome with some adequate
algebra manipulations. This is the objective of the alternative method proposed in this
subsection. This method is equivalent from a mathematical point of view and is thereby
exact. It is presented for a poroelastic layer but the method can be easily extended to any
type of medium.
The main idea of the method is that among the 6 parameters of the state vector, only 3
of them are really independent. It can be highlighted for example by relations (2.4), (2.5)
and (2.8) in which the state vector is expressed with only 3 unknowns.
Let [T] be the transfer matrix of the layer and let Wr = [Ωr ]Xr be the state vector
(of length 6) at the right interface; Xr is the vector (of length 3) of the physical unknowns
and [Ωr ] is a 6 × 3 correspondence matrix. This last matrix is supposed to be known (It is
the case for transmission (2.5) and rigid backing problems (2.8) and it can be generalized
59
Figure 2.4: Modulus of the reflection coefficient of a bilayered structure
Reflexion coefficient
1
0.5
0
−0.5
0
1
2
3
Frequency
4
5
4
x 10
Figure 2.5: Reflection coefficient for a highly absorbing material; Blue: TMM - Imaginary
part; Red: TMM - Imaginary part; Black: Proposed alternative method
60
for any type of problem). At the left interface similar relations can be written
Wl = [Ωl ]Xl + S,
(2.35)
where Xl are the physical unknowns and S is the vector associated to the excitation.
The idea of the method is that the transfer matrix can be diagonalized. This can be
obtained by two means: the first one is linked to expression (2.19) and the second one
can be obtained through Stroh formalism in the case of an homogeneous porous layer
([α(z)] = [α]):
Z H
[T] = exp
[α] dz = exp([α]H) = [Φ]exp([λ]H)[Ψ],
(2.36)
0
where [Φ] is the matrix of the eigenvectors of [α], [Ψ] = [Φ]−1 and [λ] is the diagonal matrix
of the eigenvalue of [α]. A short comparison with (2.19) indicates that [Φ] = [M(0)]
and that the eigenvalues are the z−projection of the wave vector. The eigenvalues and
eigenvectors can be order by decreasing real part:


 Ψ1



[Φ] = [Φ1 |Φ2 |[Φr ]] , [Ψ] =  Ψ2



[Ψr ]




.



(2.37)
The transfer matrix now reads:
[T] = eλ1 H Φ1 Ψ1 + eλ2 H Φ2 Ψ2 + eλ3 L [α0 ],
(2.38)
with

 1





0
[α ] = [Φr ] 






e(λ4 −λ3 )H
e(λ5 −λ3 )H
e(λ6 −λ3 )H






 [Ψr ].





(2.39)
On then has:
Wl = [T]Wr = (eλ1 H Φ1 Ψ1 [Ωr ] + eλ2 H Φ2 Ψ2 [Ωr ] + eλ3 H [α0 ][Ωr ])Xr .
(2.40)
Let now define the projection of [Ωr ] matrix on the first two left eigenvectors:
[C|D|E] = Ψ1 [Ωr ] , [F |G|H] = Ψ2 [Ωr ].
(2.41)
The propagation of Xr through the transfer matrix leads to the definition of an new
unknown vector X0l :



 eλ1 L



X0l = 
e λ2 L



eλ3 L
61
 C D E 






  F G H  Xr .






0 0 1
(2.42)
Figure 2.6: Porous layer with corrugations
This relation can be analytically inverted and one has:





Xr = 



Ge−λ1 L
GC − DF
F e−λ1 L
−
GC − DF
De−λ2 L
−
GC − DF
Ce−λ2 L
GC − DF
(DH − EG)e−λ3 L
(GC − DF )
(F E − HC)e−λ3 L
(GC − DF )
0
0
e−λ3 L
{z
|




 0
 Xl



(2.43)
}
[T0 ]
State vector Wl can now be expressed from the propagated unknowns:
Wl = [Ω0l ]X0l
(2.44)
[Ω0l ] = [φ1 |φ2 |0] + [α0 ][Ωr ][T0 ].
(2.45)
with
There is now two ways to express the state vector Wl : (2.35) and (2.44). It is then
straightforward to write the linear system in Xl and X0l to solve:
0
[Ωl ]| − [Ωl ]



 0 

 Xl 
= S.
(2.46)



 Xl 

This last system does not exhibit numerical divergence. The result of the proposed method
are presented in black in Fig. 2.5.
2.1.7
Coupled Mode Model (CMM) for corrugated porous materials
[A14]
The last example of plane wave technique is the coupled Mode Method. It has been proposed by Allard et al. [A14]. This technique does not consider multilayered structures but
corrugated layers. The problem is still considered to be invariant along the y direction and
it is assumed to be periodic (of period D) in the x direction. In addition, the corrugation
is assumed to be even (i.e. with a profile symmetric of x = 0 axis). The origin of the
62
z axis is associated to the top of the corrugation. The incident plane plane wave is still
associated to a kx wave-number component on x axis creates in air. The periodicity implies
the existence of Bloch waves with x-wave numbers kx,n+1,l given by:
kx,n+1,l = kx +
2πl
, l ∈ [−2L; 2L]N .
D
(2.47)
A spatial period of the porous surface is represented in Fig. 2.6. The origin of the
x axis is at the middle of the period. Under the corrugation there is a layer of porous
medium which can be infinite (corrugations over a semi-infinite layer (half space), finite,
or equal to 0 for corrugations on a rigid impervious surface). The corrugation is replaced
by a superposition of n elementary volumes separated by horizontal plane surfaces, where
the limits between the porous medium and the air are vertical plane surfaces. The elementary volumes are numbered with j. For j = 1, the elementary volume material is the
porous medium, and for j > 1 volumes are subdivided in three sub-volumes, one with
porous medium (for |x| < D/2 − dj , where dj is the length of the air sub-volume) and two
with air. The geometrical difference between two adjacent elementary volumes should be
limited otherwise a large number of modes are needed to satisfy the continuity of pressure
and z velocity component. Hence the increase of the length dj is constant from j = 1 to
j = n and dj = (j − 1)D/(2n). Let ej be the thickness of layer j. ej can be equal to zero
for elementary volumes called "virtual" elementary volumes. These volumes are added in
the case of strong discontinuities to avoid numerical drawbacks. The different corrugation
shapes are related to different dependences of ej on j.
In the calculations, the number of the refracted modes is finite and the truncation is
controlled by integer N so that |l| < N . The subscript n + 1 is used because the air at
z < 0 is above the elementary volume n. The related z-wave number vector components
are given by:
q
±
2
= ± k02 − kx,n+1,l
kz,n+1,l
.
(2.48)
±
z)).
The spatial dependence of a mode above the corrugations is exp(j(kx,n+1,l x+kz,n+1,l
The pressure presents 2N + 1 different dependences on x and 4N + 2 modes are retained.
−
+
is related to the specular
is related to the incident wave and kz,n+1,0
If kx is real kz,n+1,0
+
+
corresponds to an
reflected wave. For l 6= 0, with the convention =(kz,n+1,l ) > 0, kz,n+1,l
−
unphysical mode which increases when z → −∞ and kz,n+1,l to a physical mode. For j = 1
in the lower elementary volume, and above in the case of a semi-infinite layer, the x-wave
numbers kx,1,l are equal to kx,n+1,l and the z-wave number vector components, given by:
q
±
2
kz,1,l
= ± k12 − kx,1,l
(2.49)
are complex. The acoustic field in the elementary volume j is approximated by a superposition of 4N + 2 modes. These modes correspond to 2N + 1 x−dependences and are
related to two opposite z−wave number vector component ±kz,j,l . These z wave numbers
are common for the three sub volumes. Therefore a mode can propagate vertically in an
elementary volume as a plane wave.
For layer j and wave l, mode pj,l is defined as the solution of the problem:
2
Lj pj,l = −αj,l
pj,l ,
63
(2.50)
with Lj a differential operator defined by:
Lj = −
ρj (x)
d2
+ (k02 − kj2 (x)) with kj2 (x) = ω 2
.
2
dx
Kj (x)
(2.51)
e eq )
Density ρj (x) is ρ0 (resp. Kj (x) = χ0 ) in subvolumes 1, 3 and ρj is ρeeq (resp. Kj (x) = K
in subvolume 2. Let αj,l and βj,l be the x wave number components in the air sub-volumes
and in the porous sub-volume respectively. The x dependences of mode pj,l can be written
in the form:
pj,l (x) = A1,j,l exp(−iαj,l x) + B1,j,l exp(iαj,l x) , D/2 > x > D/2 − dj ,
(2.52a)
pj,l (x) = A3,j,l exp(−iαj,l x) + B3,j,l exp(iαj,l x) , D/2 > −x > D/2 − dj ,
(2.52b)
pj,l (x) = A2,j,l exp(−iβj,l x) + B2,j,l exp(iβj,l x) , |x| > dj .
(2.52c)
The following conditions should also be satisfied:
• Periodicity condition for pressure pj,l and x−velocity component vx (Floquet theorem)
pj,l (x = D/2) = pj,l (x = −D/2)δ , vj,l (x = D/2) = vj,l (x = −D/2)δ,
(2.53)
with
δ = eikx D .
(2.54)
• Common z−wave number vector component for the three sub-volumes:
2
2
2
= keq
− βj,l
k02 − αj,l
(2.55)
• Pressure continuity and x−velocity component continuity at the boundaries between
adjacent sub-volumes.
All these conditions lead to the following dispersion relation:
w(αj,l , βj,l ) = 0,
αe
ρeq
)(1 +
βρ0
αe
ρeq
+ cos(β(D − 2dj ) − 2αdj )(1 −
)(1 −
βρ0
w(α, β) = 4cos(kx D) − cos(β(D − 2dj ) + 2αdj )(1 +
(2.56)
βρ0
)
αe
ρeq
βρ0
).
αe
ρeq
It can be shown that x pressure dependences in any elementary volume are orthogonal:
D/2
pj,l (x)pj,m (−x)
dx = 0 , l 6= m.
ρj (x)
−D/2
Z
(2.57)
After normalization by a factor γj,l :
p̂j,l
pj,l
=√
, γj,l =
γj,l
D/2
pj,l (x)pj,l (−x)
dx.
ρ(x)
−D/2
Z
64
(2.58)
The x dependences satisfy the orthonormalization relation:
D/2
p̂j,l (x)p̂j,m (−x)
dx = δlm .
ρ(x)
−D/2
Z
(2.59)
The coupling equations between modes in elementary volumes in contact are obtained
from the equations related to the continuity of pressure and z−velocity components at the
interface between the elementary volumes j and j + 1.
The x−dependence of the total pressure in the elementary volume j and j + 1 can be
written:
N
X
+
−
Pj (x) =
(Cj,m
+ Cj,m
)p̂j,m (x),
(2.60a)
m=−N
Pj+1 (x) =
N
X
+
−
(Cj+1,l
+ Cj+1,l
)p̂j,l (x).
(2.60b)
l=−N
+
−
Cj,m
is the amplitude of the mode with kz = kz,j,m and Cj,m
is the amplitude of the
mode with kz = −kz,j,m at the interface. Pressure and velocity continuity equations at the
interface can be written:
N
X
+
(Cj,m
+
−
Cj,m
)p̂j,m (x)
=
m=−N
−
+
)p̂j,m (x),
+ Cj+1,l
(Cj+1,l
(2.61a)
l=−N
j
+
−
N
X
(Cj,m
− Cj,m
)p̂j,m (x)
kz,m
ρj (x)
m=−N
N
X
j+1
+
−
N
X
(Cj+1,l
− Cj+1,l
)p̂j+1,l (x)
kz,l
=
.
(2.61b)
−
Cj,m
,
(2.62)
ρj+1 (x)
l=−N
Using the orthogonality relations, Eqs. (2.61) lead to:
+
Cj+1,l
N
1 X
=
2
"
N
1 X
+
2
"
N
1 X
=
2
"
N
1 X
+
2
"
DRL (j, m, j + 1, l) + DLR (j, m, j + 1, l)
m=−N
m=−N
−
Cj+1,l
m=−N
DRL (j, m, j + 1, l) − DLR (j, m, j + 1, l)
DRL (j, m, j + 1, l) − DLR (j, m, j + 1, l)
DRL (j, m, j + 1, l) + DLR (j, m, j + 1, l)
m=−N
j
kz,m
#
+
Cj,m
j+1
kz,m
j
kz,m
#
j+1
kz,m
j
kz,m
#
+
Cj,m
j+1
kz,m
j
kz,m
j+1
kz,m
#
−
Cj,m
.
(2.63)
DLR and DRL are defined by:
D/2
Z
DLR (j, m, j + 1, l) =
0
D/2
Z
DRL (j, m, j + 1, l) =
0
65
p̂j,m (x)p̂j+1,l (−x)
dx,
ρj (x)
(2.64a)
p̂j,m (x)p̂j+1,l (−x)
dx.
ρj+1 (x)
(2.64b)
0.6
0.5
|R|
0.4
0.3
0.2
0.1
0
0.2
0.4
sin(θ)
0.6
0.8
1
Figure 2.7: Reflexion coefficient versus incident angle for a corrugated semi-infinite porous
layer. Solid: No corrugation; Dashed: CM; Dashed-dotted: BTHT; Dot: ET
The amplitudes of the modes at the lower face of the elementary volume j + 1 are modified by the propagation in the z direction on a length equal to the thickness ej+1 of the
±
elementary volume. The coefficients Cj+1,l
must be multiplied by exp(±ikz,j+1,l ej+1 ), the
axis z being directed toward the rigid backing. For each interface between two elementary
volumes, a (4N + 2) × (4N + 2) transfer matrix [Mj+1,j ] links amplitudes in the elementary
volume j at the interface j, j + 1 to amplitudes in the elementary volume j + 1 at the interface j + 1, j + 2. The product [Mt ] = [Mn+1,n ][Mn,n−1 ]...[M2,1 ] links the amplitudes in
the elementary volume 1 at the interface with the elementary volume 2 to the amplitudes
in air at z=0.
For the rigid termination, parameters for the calculation of the CM are the same than in
the previous subsection. For the semi-infinite medium the 2N + 1 modes with <(kz,1,l ) < 0
have an amplitude equal to 0. The amplitude of the 2N retrograde modes in air is equal
to 0 and the amplitude of the incident wave is arbitrarily taken equal to 1. Matrix [Mt ]
is used to relate these amplitudes to the amplitude of the 2N + 1 unknown amplitudes in
the first elementary volume. It is then possible to evaluate these latters. Knowing all the
amplitudes in the first elementary volume, it is possible to evaluate the amplitude of the
specular reflected mode. The modulus of the reflexion coefficient is shown in Fig. 2.7 for
the corrugated semi-infinite layer and in Fig. 2.8 for the corrugations over a rigid backing.
In Fig. 2.7, predictions of the reflexion coefficient modulus obtained by the CM and
heuristic methods are shown as a function of the sinus of the incident angle θ. Prediction
for a single semi-infinite layer without corrugations are also given. All the models predict
a limit equal to 1 when sin(θ) tends to 1. In order to get a clear representation of the
different predictions with a small vertical scale, the chosen superior limit of sin(θ) is 0.97.
The CM model and the FEM predictions are represented by only one curve. The admittance of the semi-infinite layer without corrugations is close to the one of the BTHT and
the ET models. Predictions with the BTHT and the ET model have the same order of
magnitude as predictions with the CM model, the largest discrepancies appearing around
the minimum of the modulus. The modulus of the reflexion coefficient of the semi-infinite
layer is not strongly modified by the corrugations. In Fig. 2.8, predictions of the reflexion
66
1
0.9
|R|
0.8
0.7
0.6
0.5
0
0.2
0.4
sin(θ)
0.6
0.8
1
Figure 2.8: Reflexion coefficient versus incident angle for a corrugated porous layer on a
rigid-backing. Solid: Constant thickness Layer; Dashed: CM; Dashed-dotted: BTHT; Dot:
ET
coefficient modulus obtained with the CM model and the FEM, the BTHT and the ET
models, and also for a layer of constant thickness l = πr2 /(2D) having the same volume
of porous medium per unit area as the corrugations, are shown as a function of sin(θi ).
It appears that the modulus is much smaller with the corrugations than for the layer of
constant thickness for all models. The predictions with the ET model and both CM model
and FEM are very close at all angles of incidence, but at small angles of incidence the
BTHT model provides a modulus significantly lower than the other models.
2.2
2.2.1
Finite-element models
Outline of the finite-element method
The Finite Element Method (or FEM) is nowadays the most common numerical method
for the resolution of static or dynamic mechanical problems. The general principle of this
method is to approach the continuous problem by a discretized description with a finite
number of degrees of freedom. This method derives its popularity from its flexibility, its
generality which allows its implementation for several types of problems. Moreover, it is
based on mathematical convergence theorems from functional analysis, allowing the obtention of validity criteria. On the other hand, it may be directly interpreted from a physical
point of view either as the solution of a minimization problem or as an analogy with the
classical principle of virtual powers.
Unlike other numerical methods such as the finite difference method, the discretization
is not directly processed on the partial differential equations themselves, but on a equivalent integral formulation called the variational formulation. The first step of the method is
then to obtain this variational formulation. The domain occupied by the structure is then
divided into a finite number of subdomains in which a number of points called nodes are
selected. A set of interpolation functions are used to calculate the fields parameters at each
element of the subdomain according to the values at the nodes. The discretized problem is
67
then obtained by writing the relations between the unknown nodal parameters. A linear
system of equations is then obtained. Once this system is solved, it is then possible to
determine, represent or use the approximated solution. The FEM is described in detail in
many books [?, ?]. We will focus here on its application to porous materials.
The FEM began to be used to predict the response of structures involving porous materials in the 1990’s. The first formulations proposed involved variables corresponding to
the displacements (solid and fluid phase). We can mention the work of Kang and Bolton
[?, ?, ?, ?] or the work of Panneton and Atalla [?, ?, ?]. Other formulations in which the
displacement of the fluid phase was replaced by a relative displacement between the two
phases have then appeared [?, ?, ?, ?]. It then appeared that the most intuitive variable
to describe a fluid environment was not its displacement, but the pressure. Atalla et al.
proposed a mixed formulation[?] (displacement of the solid phase, intersticial pressure) for
harmonic motion. Several variations were then derived from the original formulation, each
of which is best suited to manage the differrent boundary and coupling conditions. The
method based on this formulation are today the most widely used and can be considered
as mature. Nevertheless, although the number of degrees of freedom per node is 4 instead
of 6, the systems obtained by this method are still large, and frequency-dependent.
The resolution of a vibroacoustic problem involving an elastic structure, a fluid medium
and a poroelastic medium will require a discretization of each of these domains. The domain
that will determine the size of the system to solve is the poroelastic domain, for several
reasons :
• Wavelengths of the Biot’s waves are generally smaller than those of acoustic waves.
To meet the mesh criterion,a larger nodal density in the porous domain than in the
fluid domain is required.
• The poroelastic area is meshed by volume elements, requiring a discretization in the
3 spatial directions, unlike the elements of structures - plates or shells - requiring
only a mesh in 2 directions.
• The number of degrees of freedom per node of the poroelastic elements is 4 to 6
depending on whether it uses a mixed or displacement formulation, 4 to 6 times
higher than for a component of the fluid.
Thus, it appears that the presence of the porous material will impose some constraints for
the determination of an adequate mesh. This therefore induces a huge numerical cost for
such methods. Nevertheless, they are now the only way to give satisfactory results for this
type of problem.
The comparison between the displacement formulation and the pressure formulation
indicates that the number of unknowns per node is more limited for the mixed formulations for an equivalent mesh. It is important to note that although the mixed formulation
has fewer nodes, it requires the calculation of an additional matrix to take into account
the coupling per unit of volume between the two phases. The time of construction of the
systems to resolve is therefore more important, however, for problems of large scale the
most expensive step, in terms of computational time, is the resolution of the linear system
which is directly linked to the number of degrees of freedom of the system.
68
Concerning the discretization, the usual linear elements provide satisfactory results
for a fluid, but may not correctly converge for a poroelastic material. These elements
do not correctly take into account the shear and bending deformations. To overcome
this problem, the order of the functions of interpolation can be increased using quadratic
elements, or hierarchical finite elements. These elements can converge without altering the
mesh. However, the elementary matrices require a greater computation time depending on
the order of these matrices [?, ?]. Due to the size of systems to be solved, the use of finite
poroelastic elements remained until recently limited to reduce-sized configurations that can
give a good representation of real structures. Nevertheless, the arrival of powerful means
of calculation at affordable prices now permits relatively easily to treat complex problems.
2.2.2
Implementation of the FEM for poroealstic materials: example of
the {us , ut } formulation [A10]
As title of illustration, we will present the implementation of the finite element formulation to the {us , ut } of Biot’s equations. The method to discretize a poroelastic variational
formulation has been proposed by various authors[?, ?, ?]. Let Ω be the porous material
domain and ∂Ω its boundary. This boundary can be divided in two parts ∂ΩD (Dirichlet
conditions) and ∂ΩN (Neumann conditions). The methodology to obtain the variational
formulation is in three steps. The first one is the multiplication of equations (??) and (??)
by variation fields δus and δut . These fields are chosen in order to be kinematically admissible (i.e. to satisfy the Dirichlet boundary conditions on ∂ΩD ). The two equations are
then integrated over Ω and a Green formula is applied to the stress terms. The variational
forms associated to solid and fluid phase are then:
Z
s
Ω
s
2
s
t
s
σ̂(u ) : (δu ) − ω (e
ρs u + γ
eρeeq u )δu dΩ =
Z
Ω
t
t
2
s
t
t
Keq ∇.u ∇.δu − ω ρeeq (u + γ
eu )δu dΩ =
I
σ̂(us ).nδus dΓ
(2.65a)
e eq ∇.ut .nδut dΓ
K
(2.65b)
∂ΩN
I
∂ΩN
n is the outwarding normal vector at the boundary. The boundary integral on ∂ΩD is null
due to Dirichlet conditions. These formulations can then be discretized by finite element.
The porous domain Ω is first partitioned in n elementary sub-domains Ωe . On each subdomain, a finite number of nodes are chosen as well as interpolation functions. The solid
and total displacement fields can then be approached on element e in terms of the nodal
displacements approximations use and ute .











s
s
 ue 
 ue 

≈ [Nues |Nuet ]
(2.66)







 ut 

 ut 
e
e
where [Nues ] (resp. [Nuet ]) corresponds to the shape function associated to the solid (resp.
total) displacement. In all the following, the interpolation functions of the solid and total
displacement are assumed equals (and denoted by [Nu ]). Three elementary matrices are
defined on element e:
Z
Z
e
e t
e
e
[M0 ] =
[Nu ] [Nu ]dΩ , [Ki ] =
[Bue ]t [Di ][Bue ]dΩ , i = 0, 1
(2.67)
Ωe
Ωe
[Me0 ]
is associated with the u.δu terms of the variational formulations. [Ke0 ] is associated
with the ∇.u ∇.δu products and [Ke1 ] corresponds to the shear terms. Two elementary
69
vectors are also defined. On ∂ΩD , the normal stresses are imposed and thereby known.
The elementary force vectors are:
[Fes ]
I
[Nue ]t σ̂(us )dΓ ,
=
[Fet ]
I
=
e eq ∇.ut .ndΓ.
[Nue ]t K
(2.68)
∂Ωe
∂Ωe
∂Ωe corresponds to the surface of the element included in the boundary ∂ΩN . If an element
is not linked to ∂ΩN , the elementary forces are null. The global mass and stiffness matrices
and forcing vector are obtained by the summation of elementary matrices. Let ui (i = s, t)
be the global displacement vector corresponding to the global degree of freedom of the
problem. The displacement unknowns in the element and in the global vector are linked
through a boolean matrix [Le ] so that uie = [Le ]ui . The global pseudo-mass and pseudo
stiffness matrices and forces then reads:
[M0 ] =
X
[Le ]t [Me0 ][Le ] , [Ki ] =
e∈Ω
X
[Le ]t [Kei ][Le ] , Fs,t =
e∈Ω
X
[Le ]t [Fes,t ].
(2.69)
e∈Ω
They are called pseudo as they are not respectively homogeneous to mass and stiffness
but it can be shown that they just differ by a multiplicative constant. The final linear
system then reads:
 

 




 us 
 
 Fs 

[0]
eρeeq [M0 ]  

 ρes [M0 ] γ
 P̂ [K0 ] + N [K1 ]
2
−ω 


=
.


 







t 
t




e
[0]
Keq [K0 ]
γ
eρeeq [M0 ] ρeeq [M0 ]
u
F



(2.70)
and
are both of size n. This system is of size 2n and mathematically equivalent to
the one obtained by the discretization of either {us , uf } or {us , w} formulations of Biot’s
theory. By comparison to these two formulations, this system has the advantage to provide
a diagonal by block stiffness matrix.
Finite-element implementation of {us , p} formulations can be obtained through similar
procedures. They are presented in references [?, ?] and reads in discretized form:
us
ut

 P̂ [K]



[0]


 ρe[M]
−e
γ [C] 

2
−ω 


1
γ
e[C]t
[H]
ρeeq
 
 





 us 
[0]
 
 
 Fs 


=
,

 

 
1

 P 
 
 FP 

[Q]
e eq
K
(2.71)
with [K] = [K0 ]+(N/P̂ )[K1 ]. us and P respectively represent the nodal unknown vector of
solid displacement and intersticial pressure. First vector is of length nu and the second is of
length nP . [K], [M], [C], [H] and [Q] are real matrices associated to the finite-element discretization of the spatial operators. They can respectively be associated to the stiffness of
the solid, compression of the solid, mass of the solid phase, to the coupling between the displacement and pressure and to the kinetic and compression energy of the fluid phase. Note
that these real matrices are not frequency dependent and only depend on the nature of the
differential operators. Fs is the in-vacuo solid force. FP denotes the excitation of the fluid
phase and is homogeneous to a displacement. The different variational forms (from which
the discretized systems can be obtained by a similar procedures) are given in Appendix 2.A
70
< Ks >
Kinetic energy of the solid phase
< Kf >
Kinetic energy of the fluid phase
< Wdef >
Strain energy of the porous material
s >
< Ŵdef
Partial strain energy of the solid phase
f
< Ŵdef
>
Partial strain energy of the fluid phase
s >
< Wdef
In-vacuo strain energy of the solid phase
f
< Wdef
>
Compression energy of the fluid
Wvis
Wstruct
Wth
Viscous dissipated power
Structural dissipated power
Thermal dissipated power
Table 2.4: Definition of the stored energies and dissipated powers
2.2.3
Expression of energies and dissipated powers [A7]
Due to the biphasic nature of porous materials and also to the fact that in mixed pressuredisplacement formulations the fields are of different nature, the derivation of the expressions
of energies and powers in these media is not obvious; in particular they can not be obtained
through separating the real and imaginary parts of the variational formulations. This identification, well known for classical vibration problem (as well as the {us , uf } formulation),
is not valid in this case.
The method to obtain the expressions of stored energies is as follows. The first step is
to express in the time domain the kinetic energy theorem and the thermodynamics first
principle. Both of these theorems are expressed in the case of the {us , uf } formulation as
this one is fitted for a clear explanation and separation of the different physical mechanisms.
Once the temporal expressions of stored energies and dissipated powers are written, their
mean value over a vibrating cycle can easily be obtained. The expressions of these indicators
in the case of the {us , p} formulation are directly derived from these last one through a
variable change. Hence, no energetic demonstration is done in the case of the mixed
formulation, but the expressions are derived from the displacement ones.
The convergence of the finite element method with the energetic indicators is now
studied in order to obtain quantitative estimation of the validity of the discretization by
a comparison to a theoretical convergence model. A finite-element scheme is said to be of
order d if:
||X − Xh ||I ≈ C(I, ω)hd ,
(2.72)
71
< Ks >
< Kf >
< Wdef >
s >
< Ŵdef
f
>
< Ŵdef
s >
< Wdef
f
>
< Wdef
Wvis
Wstruct
Wth
Z
ω2
ρ1 |u0 |2 dΩ
4 Ω
Z
ω2
ρ2 |U0 |2 − ρ012 |u0 − U0 |2 dΩ
4 Ω
Z
1
σ̂ s (u0 ):εs (u∗0 ) + Rr |ξ ∇.u0 + ∇.U0 |2 dΩ.
4 Ω r
Z
1
σ̂ s (u0 ):ε(u∗0 ) dΩ.
4 Ω r
Z
Rr
|ξ ∇.u0 + ∇.U0 |2 dΩ.
4 Ω
Z
1
σ̂ s (u0 ):ε(u∗0 ) + Qr ξ | ∇.u0 |2 + Qr <( ∇.U0 ∇.u∗0 ) dΩ.
4 Ω r
Z
Rr
ξ<( ∇.u0 ∇.U∗0 ) + | ∇.U0 |2 dΩ.
4 Ω
Z
br πω |u0 − U0 |2 dS
Ω
Z
π σ̂ si (u0 ):ε(u∗0 ) dΩ
Ω
Z
Ri π |ξ ∇.u0 + ∇.U0 |2 dΩ.
Ω
Table 2.5: Energy expressions in the case of {us , uf } formulation
72
< Ks >
< Kf >
< Wdef >
s >
< Ŵdef
f
< Ŵdef
>
s >
< Wdef
f
>
< Wdef
Wvis
Wstruct
Wth
Z
ω2
ρ1 |u0 |2 dΩ
4 Ω
Z
ω2
φ2
2
φ
<(e
ρf ) |u0 |2 + <( ∗ 4 ) | ∇P 0 |2 − 2 =( )=(u∗0 ∇P 0 ) dΩ
4 Ω
ω
α
e
ρf
22 ω
Z
2
1
φ Rr
σ̂ sr (u0 ):εs (u∗0 ) + 2 |P0 |2 dΩ
e
4 Ω
R
Z
1
σ̂ s (u0 ):ε(u∗0 ) dΩ.
4 Ω r
Z
φRr
2
2 |P0 | dΩ
e Ω
4 R Z
1
φ
σ̂ sr (u0 ):ε(u∗0 ) − Qr <(
∇.u0 P0∗ ) dΩ
∗
e
4 Ω
R
Z
∇.u0 .P0∗
Rr
φ2
ξφ<(
) + 2 |P0 |2 dΩ
∗
e
e
4 Ω
R
R Z
φ2
2
φ
−πω 2 =(e
ρ) |u0 |2 − =(
) | ∇P 0 |2 + 2 =( )<(u∗0 ∇P 0 ) dΩ
4
ρ
f
ω
ω
α
e
22
Ω
Z
π σ̂ si (u0 ):ε(u∗0 ) dΩ
Ω
Z
2
φ π Ri
|P0 |2 dΩ
2
e
Ω
R
Table 2.6: Energy expressions in the case of modified {us , p} formulation
73
Figure 2.9: A two layer structure
where X and Xh are the exact and discretized solution, ||.||I is the norm associated with
indicator I. h is the spatial discretization step. C(I, ω) is a quantitative parameter of
the convergence and d corresponds to the order of the interpolation set. Relation (2.72) is
more tractable in logarithm representation:
log(||X − Xh ||I ) ≈ log[C(I, ω)] + d × log(h).
(2.73)
Hence the logarithm of the relative error is an affine function of the logarithm of the discretization step whose slope (resp. y-intercept) is d (resp. log[C(I, ω)]).
The configuration of interest is a mono-dimensional two layers problem depicted in Fig.
2.9. The second layer is bonded on a rigid wall and the first one is excited by a normalized
pressure plane wave. The properties of the two porous media are given in Table 2.7. This
problem is numerically interesting as the acoustical behavior of these two porous materials
is more complex than the case for a single porous structure. The second advantage is that
an exact analytical solution can be obtained. Hence, the accuracy of the discretization
scheme can be estimated by a comparison to the exact analytical solution of the problem
considered as reference.
The problem is discretized with both {us , uf } ({us , ut } formulation provides exactly se
same results) and {us , p} linear elements with a regular spatial mesh. The numerical simulations have been done in a frequency range from 100 Hz to 3500 Hz with a 25 Hz step. For
each frequency the substructures are meshed with the same number of nodes. This number
of nodes goes from 5 to 300. Hence, more than 80 000 numerical simulations (137×296×2)
were performed. 300 nodes for each layer are of course not necessary to obtain an adequate
solution in terms of industrial classical approximation. It is nevertheless interesting from
a numerical point of view as it confirms for very refined meshes the results obtained for
standard ones.
Figure 2.10 (resp. 2.11) presents the convergence of the Mean total energy (resp. Mean
total dissipated energy) of the whole structure at 1 kHz in a log-log representation. The
error is plotted as a function of the discretization step.
It can be noticed that both relative errors tend to zero with the discretization step. This
means that both {us , uf } and {us , p} discretization converge toward the exact analytical
value. Hence, even if there is an energetic ambiguity with the mixed formulationit is underlined underline that there is no doubt on the results of the finite-element discretization. In
74
Material 1
Material 2
φ
[1]
0.952
0.937
σ
[N sm−4 ]
21300
50485
α∞
[1]
1.9
2.57
Λ
[µm]
100
57.41
Λ0
[µm]
300
61.62
ρ1
[kgm−3 ]
38.4
95.66
E
[kP a]
30
66
ηs
[1]
0.04
0.105
d
[cm]
5
5
Table 2.7: Material 1 and 2 properties
Norm of relative error, logarithm scale
−2
−3
−4
−5
−6
−7
−8
−9
−8
−7
−6
−5
Discretization step, logarithm scale
−4
Figure 2.10: Convergence of the Mean Total stored energy. solid :{us , uf } formulation;
dashed :{us , p} formulation
75
−2
Norm of relative error, logarithm scale
−2.5
−3
−3.5
−4
−4.5
−5
−5.5
−6
−6.5
−7
−9
−8
−7
−6
−5
Discretization step, logarithm scale
−4
Figure 2.11: Convergence of the Mean Total dissipated power. solid :{us , uf } formulation;
dashed :{us , p} formulation
addition, it is also interesting to add a comment on the order of the convergence. The two
proposed results (Fig. 2.10 (resp. 2.11) are representative of the whole set of simulations
and one can identify an affine function of slope around 1. This result means that the order
d of the convergence of linear finite-element for these two energetic indicators is equal to
unity. These figures enable one to identify the value of the convergence parameters C(I, ω)
as the exponential of the y-intercept of the linear interpolation line. This value is a function of ω as the physical parameters of the model are themselves functions of the pulsation.
A fitting process has been done on the whole set of simulations. It is as follows: for
each frequency, relations (2.73) are fitted and d and C(I, ω) are obtained. Figure 2.12
shows the evolution of these two parameters C(I, ω) and d versus the frequency for the
kinetic energy of the fluid phase of the second substructure. The value of C(I, ω) is divided
by 60 in order to plot the two evolution on the same graph. It can be noticed that for
both formulation the order d is really close to 1 for each frequency thereby validating that
the energetic convergence of finite-element scheme for linear poroelastic elements is unity.
This has been observed for all the other indicators. In addition, function C(I, ω) is always
a crossing function of ω inducing that the spatial step must be shortened with increasing
frequencies. It can also be noticed that the frequency evolution of C(I, ω) can be fitted
through a linear interpolation in this case. It is unfortunately not possible to obtain a
general interpolation for C(I, ω) law available for all indicators.
This numerical study proposed a methodology to obtain the quantitative parameters
of the convergence of linear finite elements for the proposed indicators. This result shows
that the energetic indicators may be used to evaluate the convergence of finite element
schemes and that the order of convergence is equal to 1 for all of them.
2.2.4
Periodic finite-elements
The acoustic properties of a periodic rigid frame porous layer with multiple irregularities
or inclusions is of great interest as the acoustic response of this structure can exhibit high
absorption peaks at low frequencies which are below the frequency of the quarter-wave76
2
1.8
1.6
Fitted coefficients
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
500
1000
1500
2000
Frequency
2500
3000
3500
Figure 2.12: Evolution of the fitted coefficients for Kf , second substructure. o :{u, U}
formulation d; + :{u, P } formulation d;. :{u, U} formulation log(C(I, ω))/60; ∇ :{u, P }
formulation log(C(I, ω))/60
length resonance typical for a flat homogeneous porous layer backed by a rigid plate. It is
explained by excitation of additional modes in the porous layer and by a complex interaction between various acoustic modes. These modes relate to the resonances associated
with the presence of a profiled rigid backing and rigid inclusions in the porous layer.
In order to model these structures, periodic finite-element can be implemented. It is
presented here for the problem associated to the rugous porous material of section 2.1.7.
The problem is presented in Fig. 2.13. A rectangular domain Ω of length D and
thickness h is considered. Note that, a different coordinate system is defined in this section
compared to the previous one as a sake a simplification: the origin of the system is at the
bottom-left corner. Ω is subdivided in two subdomains: Ωp (resp. Ω0 ) corresponds to the
porous material (resp. air). The porous material is modeled by an equivalent fluid model
so that the propagation equation on the whole Ω domain can be written in the following
form:
1
p
∇2 p +
= 0.
(2.74)
2
ω ρ(x)
K(x)
e eq in Ωp .
ρ(x) is equal to ρ0 in Ω0 and to ρeeq in Ωp . K(x) is equal to K0 in Ω0 and to K
The boundary of the Ω domain, denoted by Γ, is divided in four interfaces as depicted in
Fig. 2.13. Γt correspond to a rigid wall so that the normal displacement is equal to zero.
Γl and Γr correspond to the periodicity interfaces so that both displacement and pressure
should satisfy the Floquet theorem (2.53). Γb is the interface with a semi-infinite air layer.
On this boundary, both pressure and normal displacement can be expressed as function of
the amplitude of the incident and reflected waves:
p|Γb (x) = e−jkx x +
X
Rl e−jkx,n+1,l x ,
(2.75)
l∈ Z
un|Γb (x) =
X
j
(
kz,n+1,l Rl e−jkx,n+1,l x − kz,n+1,0 e−jkx x ).
ρ0 ω 2
l∈ Z
77
(2.76)
Figure 2.13: Porous corrugated layer: FE mesh
The incident wave is assumed of unity amplitude so that Rl are the reflexion coefficient of
the plane waves denoted by subscript l. These coefficients are unknowns of the problem.
The variational formulation of the problem reads:
∀q ∈ Vf ,
Z
Ω
∇p ∇q
pq
−
dΩ =
ω 2 ρ(x)
K(x)
I
un q dS.
(2.77)
Γ
Ω is discretized by a Delaunay triangulation with the Ωp − Ω0 interface as internal
boundary so that the density ρ(x) and compressibility K(x) are constant on each triangle
of the mesh. The boundary term can be rewritten as:
!
I
X
j
l −jkx,n+1,l x
−jkx x
un q dS =
kz,n+1,l R e
− kz,n+1,0 e
q dS
ρ 0 ω 2 Γb
Γ
l∈ Z
I
I
+
un q dS + δ un q dS.
I
Γl
(2.78)
Γl
The pressure is approximated by quadratic finite element thereby leading to a discretized problem of ne element and n nodes. So as to ensure the periodicity this mesh has
also similar nl nodes on Γl and Γr . Let nb be the number of nodes on Γb . The finite-element
system is of the following form:
!
e
[H]
e
− [Q]
P = un .
ω2
(2.79)
e and [Q]
e represent the kinetic
P is the unknown vector of the n pressures at the nodes. [H]
78
and compression energy matrices:
ne Z
X
[Le ]t [ ∇ Ne ]t [ ∇ Ne ][Le ]
dΩ,
ρe
ie =1 Ωe
ne Z
X
[Le ]t [Ne ]t [Ne ][Le ]
e =
[Q]
dΩ.
Ke
Ωe
e =
[H]
(2.80)
(2.81)
ie =1
(2.82)
[Ne (x)] is the 1 × 6 matrix of the shape function on Ωe . [Le ] is ne × n element e correspondence matrix between the local and global degrees of freedom. ρe and Ke represents
respectively the values of ρ(x) and K(x) on element e. un is the discretized vector of the
displacement at the boundary. Its values are null on Γt , verify the periodicity relations on
Γl and Γr . In order to model continuity of displacement and pressure on Γb , sums (2.75)
and (2.76) are truncated to the nb first terms (order: 0, 1, −1, 2 ....). The corresponding
nb reflexion coefficients are merged in the R vector which is an unknown of the problem.
Note that this number of reflected waves is not equal to N of the CM method. nb additional equations are then added, associated to the pressure continuity relations (2.75).
The displacement continuity relations are formally written so that the displacements are
expressed as function of R and system (2.79) is transformed to:
 



e



[H]




0
e [S]   P   u 
 ω 2 − [Q]
n 


=
.
(2.83)









[T]
[U]  R   Pb 
Algebraic manipulations are now processed to account for the periodicity and to impose
the plane-wave loading on the system. Equations and degrees of freedom of (2.83) are first
reordered. Let Pl (resp. Pr , Pi ) be the degrees of freedom associated to the left boundary
(resp. right boundary, internal nodes). System (2.83) is then reordered:
 














P
u

 [A] [B] [C] [D]  



l
l







 

















 P 
 
 u 

 [E] [F] [G] [H]  

i
i


=
.
(2.84)











 [I] [J] [K] [L]  





 Pr 
 
 ur 






























R
Pb 
[M] [N] [O] [P]
The periodicity induces that Pr = δ Pl . On the other hand ul and ur can be written:
ul = u0l + up , ur = u0r − δ up .
(2.85)
up is the displacement at the boundary which remains an unknown of our problem. It can
be eliminated by linear combination of the rows of system (2.84). Finally, one has:









 




 


[A]δ + [I] + δ([C]δ + [K]) [B]δ + [J] [D]δ + [L]  



Pl 
u0l + u0r









 



[E] + δ[G]
[F]
[H]
Pi  = 
ui













 


 Pb
 R 
 
[M] + [O]δ
[N]
[P]
79

















(2.86)
1
0.9
|R|
0.8
0.7
0.6
0.5
0
0.2
0.4
sin(θ)
0.6
0.8
1
Figure 2.14: Specular reflection coefficient versus angle of incidence for different meshes.
Solid: CM method; Dashed: λ/2; Dash-dotted: λ/4; ’+’: λ/10.
The resolution of this system provides the reflexion coefficients R of the plane waves.
The Coupled Mode Method of section 2.1.7 and the periodic FEM are first compared.
The results are presented for the rigid backing configuration. The predictions with the
CM model are performed with 2N + 1 = 31 and n = 380 elementary volumes. For the
comparizon, different meshes are considered for the FEM. All of them should satisfy the
similarity of the nodes on Γl and Γr . All these meshes differ from each other by the size of
the elements (triangles) used for the discretization. This size is compared to the wavelength
in the porous medium (20.4 cm at 1 kHz), which is lower than that in air. The meshes are
characterized by a λ/n criterion where n is the number of elements per wavelength. Fig.
2.14 presents the modulus of the specular reflection coefficient for n = 2, 4, 10 discretizations. The convergence can be observed and there is no visual difference between the CM
and the λ/10 FEM result.
In order to check the convergence of the two methods, the error in the euclidian norm
is calculated for each discretization. This error is defined by:
ni q
X
RCM (sin(θi )) − REF (λ/n) (sin(θi ))2
ε(n) =
i=1
ni
.
(2.87)
ni is the number of incident angles. In the present case, ni = 174. The angles are chosen
so that the sines sin(θi ) are linearly spaced between 0.1 and 0.97. The different models
predict a limit equal to 1 when sin(θ) tends to 1. In order to get a clear representation of
the different predictions with a small vertical scale, the chosen upper limit of sin(θ) is 0.97
with a lower limit of 0.1, the implemented version of the coupled mode model cannot be
used at normal incidence. RCM (sin(θi )) and RF EM (λ/n) (sin(θi )) are the specular reflection
coefficient obtained respectively by the CM and FEM for a λ/n discretization. Figure 2.15
presents the evolution of ε(n) for several discretizations. The asymptotic error is around
3.2 × 10−5 , which indicates that both methods agree for the first four digits. This precision
correspond to a −36dB error expressed in pressure level.
80
−2
10
Error ε(n)
−3
10
−4
10
−5
10
0
20
40
60
80
Number of elements per wavelength
Figure 2.15: Evolution of error ε(n) between CMM and FEM versus the discretization size
Z1
Z1,1
x=0
Config K1
Tested sample
Plane waves
Config K2
Z2,1
Z2
d
Z2,2
d
0
x=0
d
Figure 2.16: Configurations considered in the inversion procedure of fluid parameters
2.2.5
Inversion method for recovering equivalent fluid parameters [A19]
Finite-element can also be used to simulate experiments. A numerical method to find
the equivalent fluid properties of a double porosity material is now presented. the porous
frame is assumed motionless. Kundt tube measurements ar smulated through axisymmetric
finite-element in two different configurations (denoted by K1 and K2 in Fig. 2.16). K1
corresponds to a layer of thickness d of double porosity material bonded onto a rigid wall.
K2 is quite similar to K1 but a porous layer of thickness d0 is added between the double
porosity material and the rigid backing. Note that it is preferred to consider a porous media
instead of an air cavity (plenum) to avoid numerical discrepancies due to singularities of
the cotangent function.
Impedances Z1 and Z2 associated to configurations K1 and K2 are first calculated with
the Finite-Element Method. Assuming that the layers can be homogenized, the porous layer
(resp. double porosity, air) is modeled by the way of characteristic impedance Zeq (resp.
Zdp , Z0 ) and wavenumber keq (resp. kdp , k0 ).
81
1
0.9
0.8
Abs. coefficient
0.7
Abs. coeff.
0.6
0.5
0.4
0.3
Difference
0.2
0.1
0
0
500
1000
Frequency [Hz]
1500
Figure 2.17: Sound absorption coefficient (for plane waves at normal incidence) for material
W1. 0 − −0 Experiments, ’o’ Inversion proposed model, ’+’ No diffusion double porosity
model, ’O’ Diffusion double porosity model
For both configurations (i = 1, 2), impedances at the surface of the double porosity
layer Zi,1 can be deduced from Zi :
Zi,1 = Z0
jZ0 − Zi cot(k0 d0 )
, i = 1, 2
jZi − Z0 cot(k0 d0 )
(2.88)
Impedances Z1,1 and Z2,2 are classical:
Z1,1 = −jZdp cot(kdp d) , Z2,2 = −jZeq cot(keq d0 )
(2.89)
For configuration 2, surface impedances are linked by:
Z2,2 = Zdp
jZdp − Z2,1 cot(kdp d)
.
jZ2,1 − Zdp cot(kdp d)
(2.90)
From the two above equations, we get :
2
Zdp
= Z2,1 Z1,1 − Z2,2 (Z1,1 − Z2,1 ).
Zdp is the square root with the positive real part and
jZdp
atan −
Z1,1
kdp =
d
(2.91)
(2.92)
The two quantities are computed at each frequency and the surface impedance (and then
the absorption) is obtained by the way of (2.89).
In order to validate the accuracy of the inversion procedure proposed above for double
porosity materials, the procedure has been tested on several experimental double porosity
82
1
0.9
0.8
Abs. coefficient
0.7
Abs. coeff.
0.6
0.5
0.4
0.3
Difference
0.2
0.1
0
0
500
1000
Frequency [Hz]
1500
2000
Figure 2.18: Normal incidence sound absorption coefficient (for plane waves at normal
incidence) for material W2. ’- -’ Experiments, ’o’ Inversion proposed model, ’+’ No diffusion
double porosity model, ’O’ Diffusion double porosity model
configurations. It is validated on the basis of comparisons with sound absorption data
measured using the recommended standard ISO 10534-2 [?] and with direct computations
of sound absorption coefficients obtained using the theory of Olny & Boutin [?] adapted
to the present axisymmetric problem (the initial theory was devoted to circular mesopores
in rectangular samples). For these latter calculations, denoted hereinafter semi-analytical
models, pressure diffusion effects may also be accounted for. Two examples are given in
Fig. 2.17 and 2.18 respectively for the two fibrous materials W1 and W2.
To ease the analysis, differences (|αmodel − αexp |) with the experimental data are also
given for the three models. Note that the effect of the solid frame deformation can be
observed between 500 and 800 Hz and cannot be captured by the models considered in this
section which all assume a rigid and motionless skeleton. The absorption peak observed
on the measured data around 400 Hz is associated to double porosity phenomena. The
diffusion frequency [?] is estimated at 361 Hz for material W1 and at 498 Hz for material
W2. Under this transition frequency, pressure diffusion effects shall be dominant and
the semi-analytical model accounting for these latter effects should be more suited. As
expected, for the two materials tested here, and for the entire frequency range observed,
this model allows to capture the overall behavior of the measured sound absorption data.
Accordingly, the analytical model without diffusion effects fails to give accurate predictions.
2.3
Resolution methods for PE FEM
The resolution of Finite-Element Models including poroelastic materials is the key point
of the numerical process as it is the most time-consuming. This section presents some
techniques based on normal modes. The principle of modal technique is first presented so
as to help the understanding of the following sections. A semi-analytical method is then
presented for problem in which there is no shear waves as well as an iterative technique for
the {us , ut } formulation. Component mode synthesis is then explained on monodimensional
83
bilayer structures. Finally, a resolution technique based on normal modes is presented for
the {us , p} formulation.
2.3.1
Principle of modal decomposition techniques [A13]
This section is dedicated to the presentation of modal decomposition techniques. The
physical nature of the substructures is not considered for the moment. The structure of
interest is composed of only 2 substructures. Notations relative to the second substructure
correspond to the primed of the ones of the first substructure; they thereby won’t be defined
explicitly. If more than 2 substructures are involved, generalization of this methodology
can be proceeded. The discrete problem in the frequency domain can be written in the
following form:
 














 [A] [0] [λ]  




u
F

 












 



0
0
=
(2.93)
 [0] [A0 ] [λ0 ] 
u   F .



























0 t
t

[λ] [λ ] [0]  µ   0 
u is the discretized field vector (or physical coordinates vector) of length n, [A] is the
discrete matrix of the first substructure and F is the external force acting on the first
substructure. In equation (2.93), Lagrange multiplier are involved to ensure continuity
relations (force and displacement) at the interface between the two substructures. The
number of these relations correspond to the number of degrees of freedom at the interface
and is denoted by nj . This number is rather small compared to n and n0 . µ is an additional
unknown vector of length nj associated to the coupling conditions and −[λ][µ] can be
interpreted as the force from substructure 2 on substructure 1. More generally the last
block-column of the matrix is associated to the interaction force between substructures
as the last block-row consists in linear relations imposing the displacement continuity
conditions. Solution of this problem of size n + n0 + nj can be obtained through direct or
iterative methods which leads to tremendous calculations if the dimensions of the problem
become important as it is the case for structures involving porous materials. It should
be noticed that if a direct solver is used an imposition of the continuity relations by an
adequate assembling of the discrete problem is generally preferred and it is straightforward
to obtain the expression of this problem from (2.93):

 

















 [Â] [0]
F̂
û
[0]  














 




(2.94)
=
 [0] [Âb ] [0] 
ûb   F̂b 









 











0 
0 



0

[0] [0] [Â ]  û   F̂ 
û correspond to the displacement vector (of length n − nj ) of the first substructure of dof
which do not belong to the boundary. The common displacement vector is denoted by ûb
(which is of length nj ). Size of problem (2.94) is n + n0 − nj and is quite similar to these
of problem (2.93) as nj is small. The advantage of Lagrange multiplier is mainly formal as
they prevent from mixing degrees of freedom of substructures 1 and 2 in ub . In the case
of modal techniques, the eigen-modes of each substructure should be computed separately
and it is not direct to project problem (2.94) on the modal basis. In this case expression
84
(2.93) is preferred and the Lagrange multiplier are eliminated by the procedure of synthesis.
This paragraph presents the formal modal decomposition of substructure 1. The discrete modal decomposition in the frequency domain of the response to an excitation E can
be written in the following form:
u(E, ω) =
m
X
Φi qi (E, ω) +
i=1
|
n
X
Φi qi (E, ω),
(2.95)
i=m+1
{z
}
S(E,ω)
|
{z
H(E,ω)
}
Φi (i = 1..n) are the eigenmodes and qi (i = 1..n) are the modal coordinates. From a purely
mathematical point of view, there is equivalence between the problem in physical or modal
coordinates as far as the modal basis [Φ] = [Φi ]i (i=1..n) is complete. The resolution in
modal coordinates, which needs the prior calculation of the eigenvectors Φi has two main
advantages; the first one is that the modal coordinates are decoupled and the second one is
that only a few modes generally contributes to the response of the system thereby leading
to a reduction of the complexity of the problem. In most cases, these modes are the first
one (i.e. the one with the lowest eigen-frequencies). In eq. (2.95), the modal decomposition
is split in two terms: S(E, ω) is associated to the first m modes and H(E, ω) is concerned
with higher modes. The reduction is said to be significant if m << n. In this case there
is no need to compute the n − m higher modes of the problem (only the first modes can
be calculated by the way of subspace methods thereby reducing the numerical cost of the
method).
In order to reduce the complexity of the problem with the minimum loss of precision, an
approximation of the contribution H(E, ω) is needed. As higher modes are corresponding
to eigen-modes with frequencies higher to the maximum frequency of the spectrum of
excitation, their response to the excitation is below their resonance and stiffness controlled.
Hence, each contribution of the higher modes can be approximated by the static one (i.e.
n
X
the contribution at null frequency.) and H(E, ω) ≈ H(E, 0) =
Φi qi (E, 0). This
i=m+1
approximation can be simply calculated by knowing only the first m modes. The static
response of the structure u(E, 0) should previously be computed. The static contributions
of the first m modes is then calculated, S(E, 0) is deduced and H(E, 0) is obtained by
difference. One then has:
m
X
u(E, ω) ≈
Φi qi (E, ω) + H(E, 0).
(2.96)
i=1
H(E, 0) then acts as a corrective term corresponding to higher modes. For the case of single
structure, this term is generally omitted. In the case of aggregated structures composed of
several substructures there is a need to consider this type of correction.
Problem (2.93) can be rewritten in the following form:








 

 [A] [0]  


F
−
F̂












 u  

0


0
0
, F̂ = [λ][µ] , F̂ = [λ0 ][µ].
=
 [0] [A0 ] 
F
−
F̂



 



 u0 
 












0 t
t


0
[λ] [λ ]
85
(2.97)
0
F̂ and F̂ are the interaction forces between the substructures and have non null values only at the degree of freedom corresponding to the interface (and although they are
in the right hand side of equation (2.97), they remain unknowns of the problem). Let
[B] be the nj × n boolean matrix of the dof of the interface in which the ith line has a
unit at the index associated to the ith dof of the first substructure. Let ξ = [B]F̂ be the
vector of length nj of the forces at the interface. Action-reaction principle induces ξ = −ξ 0 .
Due to linearity, one has:
u(F − F̂, ω) ≈ [Φ]q(F − F̂, ω) + H(F, 0) − H(F̂, 0).
(2.98a)
u0 (F0 − F̂ , ω) ≈ [Φ0 ]q0 (F0 − F̂ , ω) + H0 (F0 , 0) − H0 (F̂ , 0).
(2.98b)
0
0
0
[Φ] and q correspond to the matrix of the first m modes of substructure 1 and the
contribution vector of these modes. As the interaction forces are unknowns of the problem, elimination should be done. For each one of the nj dof s of the interface, the static
contribution of higher modes H(1j , 0) are first calculated. 1j corresponds to a boolean
vector of length n in which the only non null value is associated to dof j of the interface.
One has, due to linearity,
H(F̂, 0) =
nj
X
H(1j , 0)ξj = [H(1j , 0)]ξ
(2.99)
j=1
The continuity of the displacements corresponds to the last block-row of problem (2.97) and
by the way of relations (2.98) the interaction force ξ is solution of the following problem:
[Hb (1j , 0)] − [H0b (10j , 0)] ξ = [Φ0b ]q0 (F0 − F0 ,ˆω) + [Φb ]q(F − F̂, ω) + Hb (F, 0) + H0b (F0 , 0)
|
{z
}
[Rjun ]
(2.100)
with
[Hb (1j , 0)] = [λ]t [H(1j , 0)] , [Φb ] = [λ]t [Φ] , Hb (F, 0) = [λ]t H(F, 0).
(2.101)
[Rjun ] is a nj × nj matrix which can be inverted to express ξ as a function of the modal
contribution of the preserved modes and one has:








 q(F − F̂, ω) 


0
0
0
.
ξ = [Rjun ]−1 
[Φ
|Φ
]
(F
,
0)
(2.102)
+
H
(F,
0)
+
H
b
b
b
b




0


 q0 (F0 − F̂ , ω) 
This relation can allow to express the displacements as a function of only the modal
contributions of the preserved mode while introduicing it in (2.98) with relation (2.99):










 u 

 q 

+ Γ,
(2.103)
≈ [Ψ]






 u0 

 q0 

with


 [Φ] −
[Ψ] = 

0
[H
[H(1j , 0)][Rjun ]−1 [Φb ]
−[H(1j , 0)][Rjun ]−1 [Φ0b ]
(10j , 0)][Rjun ]−1 [Φb ]
0
0
[Φ ] + [H
86
(10j , 0)][Rjun ]−1 [Φ0b ]



(2.104)
and
Γ=



 H(F, 0) − [H(1j , 0)][Rjun ]−1 (Hb (F, 0) − H0b (F0 , 0))




.
(2.105)



 H0 (F0 , 0) + [H0 (10 , 0)][Rjun ]−1 (Hb (F, 0) − H0 (F0 , 0)) 

b
j
For any contributions q and q0 , the methodology ensures the continuity relations on
displacement and force for the unknown nodal vector in (2.103). The projection of problem
(2.93) in which Lagrange multiplier,now useless, are omitted on the [Ψ] family leads to the
reduced problem:


 



 








 q 
 [A] [0] 
 F   [A] [0]  
t
 [Ψ]
 Γ
[Ψ]t 
=
[Ψ]
−
(2.106)




 







0
0 
0
0



[0] [A ]
q
F
[0] [A ]
The unknown vector of modal contribution is of length m + m0 and then a reduction is
obtained for the resolution of the problem. This problem can be solved by a direct procedure and the physical unknowns of the problem can be deduced from the modal ones with
relation (2.103).
2.3.2
Automatic selection of the modes [A13]
One key point of modal techniques is to find the adequate number of modes in the selection
of each modal basis. Some empirical criterion are often used (as for example to preserve
modes with eigenfrequencies lower to twice the frequency of excitation). In this section
an automatic selection procedure is proposed. The idea of the method is to compare the
0
accuracy of the modal solution in terms of residual. Let Û, Ûb and Û be the solution
of the modal problem obtained with a selection of m and m0 modes and decomposed as
proposed for problem (2.94). One residual vector can be computed for each substructure:
 

 














 


 

 [Â] [0]   Û   F̂ 
 [Âb ] [0]   ûb   F̂b 
0




R=
−
,R =
−
.





 
0
0 







0 








[0] [Âb ]
Ûb
F̂b
[0] [Â ]
û
F̂
(2.107)
These residuals (which should be null if the displacements correspond to the exact ones)
allow to control the number of modes for each substructure. There is a need of two scalar
parameters ε and ε0 that must be chosen. If kRk > ε, the number of modes for the first
substructure is not sufficient and m is incremented. Similar procedure can be done for
second substructure. Hence, the modal families can be selected separately for the two
substructures. In the case of a bandwidth frequency resolution, the method is as follows.
1 mode is selected for both substructures and the modal resolution is undertaken If the
criterion condition are verified the following frequency is considered. In the other case
a mode is added to the substructure having the maximum kRk /ε ratio and the modal
resolution is done another time. The procedure is continued until the criterion are reached
for both substructures; when it is the case the following frequency is considered.
Choice of ε and ε0 is crucial and is the key point of this automatic selection procedure.
Examples are given in the application section but two general consideration can be done.
87
First, they shouldn’t be too small otherwise too many modes will be selected, mainly
of them adding correction to the solution which can be below the range of the desired
precision for the final results. The second remark is that it shouldn’t be to high otherwise
the precision on the solution is not sufficient and contributed modes are not selected. The
adequate choice consists in keeping the modes that are significantly excited.
2.3.3
Semi-analytical method for compression problems [A10]
This section is concerned with the presentation of an analytical solver for numerical system
(2.70) in configurations for which only purely compressional motions are considered. It is
the case or example for 1D configurations. The validity of this assumption in 3D cases and
its applications has been enlightened by Chazot an Guyader[?]. For this type of problems,
the N [K1 ] term can be neglected in (2.70).
Let [Φ0 ] and [k20 ] be the eigenvectors and the diagonal matrix of eigenvalues of the
generalized problem [K0 ] − ω 2 [M0 ]. It can be shown that the eigenvalues are homogeneous
to a square of wave number. The basis can be chosen so that the modes are normalized
with respect to the [M0 ] matrix:
[Φ0 ]t [K0 ][Φ0 ] = [k20 ] , [Φ0 ]t [M0 ][Φ0 ] = [I]
(2.108)
I is the identity matrix. Orthogonality relations (2.108) involves the decoupling of modes.
Projection of (2.70) on the modal basis leads to the system (2.116):
 



 

 qs 
 
 F0s
2


[0]
eρeeq [I]  
 P̂ [k0 ]

 ρes [I] γ
2

−ω 

=



 
 

 qt 
 
 F0t
e eq [k2 ]
[0]
K
γ
eρeeq [I] ρeeq [I]
0




.
(2.109)



It can be noticed that this system exhibits a decoupling of the unknowns {qis , qit } which
are solution of the two equations:
P̂ ki2 qis − ω 2 (e
ρs qis + γ
eρeeq qit ) = Fi0s
e eq k 2 q s − ω 2 (e
K
γ ρeeq qis + ρeeq qit ) = Fi0s .
i i
(2.110)
(2.111)
Hence qis and qit can be obtained analytically by solving this equations:
qis = pi,1 Fi1 + pi,2 Fi2 , qit = µ1 pi,1 Fi1 + µ2 pi,2 Fi2 .
with
pi,j =
Fi0s + µj Fi0t
1
j
,
F
=
.
i
e eq µ2
ki2 − δj2
P̂ + K
j
(2.112a)
(2.112b)
ki and µi are wave numbers and ratio of the total displacement over the solid one of wave
i defined subsection 1.7. In these last expressions pi,j is the polar form associated to mode
i and relative to wave j. Even if the notion of resonance is not obvious as δj is a complex
and frequency dependent function, these functions are linked to the mechanical resonances
of the medium. The modal forces Fij involves two contributions relative to both phases
and the contribution of each phase is linked to the ration µi of the two displacements.
The denominator of Fij is a normalization factor linked to the stiffness. Thereby the solid
and total contributions are expressed in (2.112a) as the sum of two similar parts, each one
88
Mat
Porosity
Flow resistivity
Tortuosity
VCL
TCL
φ
σ (N m−4 s)
α∞
Λ (m)
Λ0 (m)
A
0, 97
87000
1, 52
3.7 10−5
1.2 10−4
B
0, 97
40000
1, 06
0, 56 10−4
0, 112 10−3
Density
Young modulus
Loss factor
Poisson
ρ1 (Kg m3 )
E (Pa)
ηs
ν
A
31
1, 43 107
0,055
0,3
B
130
0, 44 107
0,
0,1
Mat
Table 2.8: Parameters of porous material A and B
associated to the two Biot waves.
A single porous layer of thickness 5 cm and infinite lateral dimensions backed by a
rigid impervious wall is considered. An harmonic unit force is applied on the free face
of the porous layer. Absorption coefficient is plotted in Fig. 2.19. As the analytical and
{us , ut } formulation results perfectly match, this example provides a validation of the formulation. Two main resonances can be observed (around 500 Hz and 1300 Hz), the second
one being more damped than the first one. In both cases, the resonance is associated with a
particular phenomenon that can be called "decrease-increase". While increasing frequency
there is first a decrease of the absorption coefficient followed by a brutal increase. It has
been observed in several cases when Biot’s effects are involved. In the case of motionless
solid frame (equivalent fluid) the resonance just corresponds to a maximum of absorption.
It is interesting to analyze this effect by the way of a modal approach.
The polar functions p1 and p2 associated to mode 1 and mode 2 are presented in Fig.
2.20 (module) and 2.21 (phase). For mode 1, a resonance of wave 2 is observed at 485
Hz. At this frequency, one has δ1 = 49.5 − 38.6i and δ2 = 31.52 − 2.78i and the value of
the first eigenvalue of the problem k1 = 31.4. Wave 2 is mainly associated with the solid
and is not as damped as the fluid one. That is the reason why no resonance is observed
for pole 1 which is over damped. For the second mode, the resonance is at 1185 Hz and
δ1 = 90.9 − 40.6i and δ2 = 75.3 − 18.0i with k2 = 94.2. Hence, the two observed resonances
at 500 and 1300 Hz are linked to the resonances of the modes. Nevertheless it does not
explain the decrease-increase phenomenon. In order to better understand it is necessary to
remind that the main dissipation mechanism in porous material is viscous effects when the
relative motion of the fluid to the solid is important; hence there won’t be viscous effects if
the solid and fluid displacement are equal even if they are important. The solid and total
contribution of mode 1 and 2 are shown in figure 2.22 (module) and 2.23 (phase). One
can see that at the resonance of mode 1 the complex contribution of the solid and total
are quite equal (in module and phase). Consequently at this frequency the motion of the
89
1
0.9
Absortpion coefficient [1]
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
500
1000
1500
Frequency [Hz]
2000
2500
Figure 2.19: Validation of the {us , ut } formulation. (continuous) analytical, o finite element
{us , ut } formulation
−2
10
−3
2
−2
polar contribution [s m ]
10
−4
10
−5
10
−6
10
−7
10
−8
10
1
10
2
10
3
10
Frequency [Hz]
4
10
5
10
Figure 2.20: Amplitude of polar functions. solid: pole associated to wave 1 and mode 1;
dash-dot: pole associated to wave 2 and mode 1;dot: pole associated to wave 1 and mode
2; dash: pole associated to wave 2 and mode 2
90
0
−0.5
Phase [rad]
−1
−1.5
−2
−2.5
−3
0
500
1000
1500
Frequency [Hz]
2000
2500
Figure 2.21: Phase of polar functions. solid: pole associated to wave 1 and mode 1; dashdot: pole associated to wave 2 and mode 1;dot: pole associated to wave 1 and mode 2;
dash: pole associated to wave 2 and mode 2
fluid relative to the solid is nearly null thereby reducing viscous effects. The important
variation of phase few Hz after the resonance induces an important amplitude variation
and a different phase, there is then a significant relative motion and this corresponds to
the maximum of absorption.
2.3.4
Direct and iterative resolution methods [A10]
Two methods are presented which involves all the modes of the problem. In fact two modal
basis can be considered. One associated to the compressional modes defined in the preceding section and one associated to the solid modes associated to the full stiffness matrix.
Let now [Φ1 ] and[k21 ] be the eigenvectors
and the diagonal matrix of eigenvalues of the
N
generalized problem K0 + K1 − ω 2 [M0 ], normalized with respect to [M0 ] matrix:
P̂
N
t
[Φ1 ] K0 + K1 [Φ1 ] = [k21 ] , [Φ1 ]t [M0 ][Φ1 ] = [I].
(2.113)
P̂
One then defined:
[C1 ] = [Φ0 ]t [K1 ][Φ0 ] and [C2 ] = [Φ0 ]t [M0 ][Φ1 ].
(2.114)
In general, these two matrices are not diagonal nor even sparse.
The first numerical method only involves the compressional normal modes: one express
the total and fluid displacement by the way of their modal contributions qs and qt on the
[Φ0 ] basis. One then has:

 
 
 


t 









 us 
  [Φ0 ] [0]  
 
  [Φ0 ] [0]  

 qs 
 F0s 
 Fs 



=
,
=
(2.115)

 

 


 






0t 
t 
t
 ut 







[0] [Φ0 ]
q
F
[0] [Φ0 ]
F
91
−6
10
−7
10
−8
Amplitude [m]
10
−9
10
−10
10
−11
10
−12
10
−13
10
1
2
10
10
3
10
Frequency [Hz]
4
10
5
10
Figure 2.22: Amplitude of modal contributions. solid: solid contribution of mode 1; dashdot: total contribution of mode 1;dot: solid contribution of mode 2; dash: total contribution
of mode 2
2
1
Phase [rad]
0
−1
−2
−3
−4
0
500
1000
1500
Frequency [Hz]
2000
2500
Figure 2.23: Phase of modal contributions. solid: solid contribution of mode 1; dash-dot:
total contribution of mode 1;dot: solid contribution of mode 2; dash: total contribution of
mode 2
92
The projection of (2.70) then reads:



 
 

 
2
 qs 
 
 F0s
eρeeq [I]  
[0]
 ρes [I] γ

 P̂ [k0 ] + N [C1 ]
2

−ω 

=
 








2
t


 F0t
e
γ
eρeeq [I] ρeeq [I]
q
[0]
Keq [k0 ]







(2.116)
It is now straightforward that the second half of the linear system only involves diagonal
matrices. It can then be easily invertible, thereby allowing a simple relation between qs
and qt
F0t
(2.117)
qt = γ
e[Req ]qs + [Req ] 2
ω ρeeq
with the diagonal matrix [Req ] defined by
[Req ] = diag
2
δeq
2
ki2 − δeq
!
2
, δeq
=
ω 2 ρeeq
e eq
K
(2.118)
ki denotes the elements of [k0 ] and δeq is the wave number of the equivalent fluid model
wave. The expression of qt can then be reintroduced in the first half of the system and
one has:
P̂ [k20 ] + N [C1 ] − ω 2 ρes + γ
e2 ρeeq [Req ] qs = F0s + γ
e[Req ]F0t
(2.119)
This system is of size n. The modal decoupling property was then used to divide by 2 the
size of the system. The problem (2.119) can now be solved by a direct solver to find the
value of qs .
To summarize, this technique is performed in two steps. The first step is the modal
calculation and the projection of the matrices to obtain [C1 ]. These two operations are
performed on real matrices of size n, are just done once and do not need to be done at
each frequency. The second step is the resolution of the complex system (2.119) of size
n at each frequency, the post-processing to obtain qt through (2.117) and the unknown
displacements with (2.115). This post-processing is very low consuming compared to the
resolution of the linear system. The classical resolution methods involves the resolution of a
complex linear system of size 2n at each frequency. The comparison of computational time
of both methods is presented below but it is straightforward that the proposed approach
is better fitted to large systems which need to be solved at many frequency.
A second numerical method is now proposed to solve system (2.70). Unlike the previous
one, it is not a direct method but an iterative one. The nodal displacements are expressed
in the {[Φ1 ], [Φ0 ]} basis as:
 


 
 
t 
















s
0s
s
s

 u   [Φ1 ] [0]   q   F   [Φ1 ] [0]   F 



,
=
(2.120)
=

 

 


 






0t 
t 
t




 ut 



F
q
F
[0] [Φ0 ]
[0] [Φ0 ]
The projection of (2.70) then reads:



2
[0]
ρes [I]
 P̂ [k1 ]



 − ω2 



e eq [k2 ]
[0]
K
γ
eρeeq [C2 ]t
0
 
 

 
 qs 
 
 F0s
γ
eρeeq [C2 ]  

=
 





t


 F0t
ρeeq [I]
q
93







(2.121)
The matrix of this problem can be written as the difference of two matrices
defined by:
 
 




 

2
 
 F0s
ρes [I]
γ
eρeeq [C2 ]   qs 
[0]


 P̂ [k1 ]
2

−ω 

=
 



 

2
t
 qt 
 
 F0t
e
γ
eρeeq [C2 ]
ρeeq [I]
[0]
Keq [k0 ]
[M] and [N]




(2.122)



[M] is a diagonal matrix which is easy to invert. This iterative "porous-Jacobi" method
consists in choosing an initial vector X0 and to iterate the relation:






 F0s 



Xi+1 = [M]−1 
[N]X
+
(2.123)
i






 F0t 
until a tolerance condition is obtained. This tolerance condition is:






0s


F
([M] − [N])Xi −
6 ε.




 F0t 
(2.124)
The value of Xi when this tolerance is reached is the approached solution of the problem.
The first step of this technique is the modal calculation (now two modal basis have to
be found) and the projection of the matrices to obtain [C2 ]. As noted previously these
operations are performed on real frequency independent matrices of size n and are executed
only once. The second step is the resolution of the complex system of size 2n (2.121) at
each frequency. The numerical cost of this technique is linked to the number of iterations
at each frequency. This number cannot be predicted a-priori but the closer from the solution of the problem the initial vector is, the smaller the number of iteration is. Hence for
multiple frequency calculations, solution vector at the previous frequency can be used as
the initial vector for the current frequency. In the case of frequencies close to each other,
solution is not greatly modified and this method do not need a lot of iterations at each
frequency and will allow a great reduction of calculation time.
These two methods are now compared. A block of porous material of material B of
section 10 cm by 10 cm and infinite lateral extent is bonded onto a impervious rigid wall. It
is excited by a line force of 1 N/m in the middle of his top face. As this problem is infinite
in one direction, it can be modelled through a 2D approximation. Triangular isoparametric
quadratic elements are used for the discretization and a Delaunay mesh triangulation is
first performed for several mesh refinements. The more refined is the mesh and of better
quality the result is, higher the numerical cost is.
Fig. 2.24 presents the transfer function of the excitation node for different formulations.
This transfer function is defined as the total displacement of the porous material at the
node over the force for different frequencies. The considered formulations are {us , uf } ,
{us , ut } and the classical {us , p} ones. The mesh is composed of 1680 triangular quadratic
elements and the approximate size of the elements is 2.6 mm and corresponding approxλ
imatively to a
discretization of the shear wave which is the most discriminative one.
15
94
Method
Time in seconds
{us , uf }
3595
{us , p}
1896
{us , ut } (direct)
3563
{us , ut } (modal direct)
944
{us , ut } (modal iterative)
809
Table 2.9: Computational time for the example of Fig. 2.24 for the different methods
It is mentioned that this mesh is only considered as a convergence example. A perfect
agreement is shown between these 3 formulations thereby showing an additional validation
of the proposed formulation compared to {us , uf } and {us , p} ones.
A comparison of computational time is presented in Fig. 2.25. The transfer function is
computed for 400 frequencies linearly spaced between 1 and 1000 Hz by the way of {us , uf } ,
{us , p} as well as {us , ut } formulation with a direct and iterative resolution method. Programs are written in Fortran and sparse algorithms are used to calculate eigenpairs and the
direct solutions of the systems. As the computational time mainly depends on the calculator and its configuration, computation times won’t be directly presented. It is preferred
to compare each one of them to the time of the {us , uf } formulation chosen as a reference.
Hence, for each number of elements, the computational time of the {us , uf } formulation
is divided by the one of the other method. The results are presented in Fig. 2.25. For
example, for 2200 elements, the computation with the Porous-Jacobi method is 5.5 times
faster than the {us , uf } method. For each method, the time are calculated through the
cpu_time() function. Matrices are first allocated and matrices [M0 ], [K0 ] and [K1 ] are
calculated. This first step is of course common to all the solver procedures. Hence, the
considered computational times only considers the resolution methods. For modal method
the initial mode computation(s) is of course taken into account. As an illustration, the
computational times for the result of Fig. 2.24 are given in table 2.9.
It can be observed that {us , uf } and {us , ut } with a direct solver methods are equivalent
in terms of computational time. Even if it is not significative, {us , ut } is a bit faster (a
few seconds for the example of table 2.9). Those seconds corresponds to the time of the
assembling of stiffness coupling for the {us , uf } formulation. {us , p} formulation is around
2 times quicker than the {us , uf } method due to the reduction of dof per node. This reduction is more significant on 3D problems. The proposed direct modal resolution exhibits
a calculation 4 times faster and the porous-Jacobi methods is shown to be more and more
efficient as the number of elements increases.
95
−5
1.8
x 10
Transfer Function [N/m]
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
200
400
600
Frequency [Hz]
800
1000
Figure 2.24: Transfer function of the displacement over the force at the node of excitation.
solid: {us , uf } formulation; o {us , ut }; * {us , P } formulation.
7
Ratio of computational time
6
5
4
3
2
1
0
0
500
1000
1500
Number of element
2000
2500
Figure 2.25: Ratio of computational times. (∇) {us , uf } formulation with direct solver; (+)
{us , P } formulation with direct solver; (o) {us , ut } formulation with the proposed direct
modal resolution; (∗) {us , ut } formulation with the proposed iterative modal resolution
96
2.4
Component Mode Synthesis (CMS) with normal modes
[A13]
The methods of the preceding section involves all the normal modes of the structure. A
technique with a modal truncation and correction is now presented on a simplified monodimensional case. The problem is concerned is a two layered porous structure bonded on a
rigid wall. The modal results are compared with a purely-analytical solution of the problem
that can be found for such a problem by the way of plane waves techniques.
In the general presentation of modal techniques section 2.3.1, it was shown that the
corrections concerns both the interface between substructures as well as the excitation on
substructures. This technique is called FICMT (Force Interface Corrected Modal Technique). If Γ is replaced by a null vector, only the correction at the interface is considered;
it corresponds to Craig and Chang technique and this technique is denoted called ICMT
(Interface Corrected Modal Technique). In the case when no correction is considered the
technique is denoted by Direct Modal Technique (or DMT).
The nodal problem reads:


[0]
[0]
[ν] [0]
[0]
 P̂ [K0 ]



e eq [K0 ]
 [0]
[0]
[0]
[0] [ν]
K



 [0]
P̂ 0 [K00 ]
[0]
[0]
[ν 0 ] [0]




e 0 [K0 ] [0] [ν 0 ]
 [0]
[0]
[0]
K
eq
0



 [λ]t
[0]
[λ0 ]t
[0]
[0] [0]



[0]
[λ0 ]t
[0]
[ν 0 ]t
[0] [0]






















−ω 2
eρeeq [M0 ]
[0]
[0]
 ρes [M0 ] γ



 γ
[0]
[0]
 eρeeq [M0 ] ρeeq [M0 ]



[0]
[0]
ρe0s [M00 ]
γ
e0 ρe0eq [M00 ]





[0]
[0]
γ
e0 ρe0eq [M00 ] ρe0eq [M00 ]




[0]
[0]
[0]
[0]



[0]
[0]
[0]
[0]
[0] [0]
[0] [0]
[0] [0]
[0] [0]
[0] [0]
[0] [0]
 



 
 


 
 


 
 


 
 


 
 

 
 
 


 
 


 

 

 

 

 







us 







ut 







0

us 











Fs 








Ft 
u0t
=









Fs 
















F


t













 F0s 




0 



F
t 
















0















 0 

(2.125)
For each porous structure solid and total displacement are discretized with linear finite
element and the same mesh is used. [K0 ] and [M0 ] are the shape form of stiffness and
mass matrix. us and ut correspond to the solid and total displacement nodal vector which
97
are both of length n + 1 (resp. n0 ) for the first (resp. second) substructure. Fs and Ft
correspond to the interaction forces (for the solid in-vacuo and the pressure) and are both





 Fs 

scalar unknowns of the problem (and ξ =
). [ν] (resp. [ν 0 ]) is a vector of length



 Ft 

n + 1 (resp. n) whose only non-null component is 1 (resp. −1) at the last (resp. the
first) index. Hence, the last two lines correspond to the continuity of the solid and total
displacements. Concerning forces in the right hand side, the only non-null value is a unit
for Ft (1).
The eigen-modes [Φ] of the first substructure are obtained by solving the generalized
eigenvalue problem associated to matrix [K0 ] and [M0 ]. The eigenvectors are normalized
with respect to [M0 ] so that [Φ]t [M0 ][Φ] is the identity matrix and [Φ]t [K0 ][Φ] is the
diagonal matrix [k2 ] of the eigenvalues ki2 . Note that for this first substructure a rigid
body mode exists whose displacements are all equal. For each one of the porous, the solid
and total displacements should be approached by a modal decomposition. If no correction
is applied for non preserved modes (DMT), the displacements are expressed in their modal
form and Lagrange multiplier are kept and one has:

 













 [Φ] [0] [0] [0] [0]  
q
u



s
s 
































 [0] [Φ] [0] [0] [0]  




u
q
t





t 










 





0
0
0
=
(2.126)
[0] [0] [Φ ] [0] [0]   qs  .
 
 us 


























0
0
0






[0]
[0]
[0]
[Φ
]
[0]
q
u




t
t






























 ξ 
ξ 
[0] [0] [0] [0] [I2 ]
[Φ] is the (n + 1 × m) matrix of the first m modes of substructure 1. The same number of
modes is preserved for the solid and total displacement and the modal matrix of (2.126) is
of size (2(n + n0 + 1) × 2(m + m0 + 1)).
Techniques accounting for static correction are now considered. Their first step is to
calculate the attachment modes associated to the static contributions of non preserved
modes. Frequency dependance of the elastic coefficients should be also considered. Second
substructure is first considered. Let now define:
0
u00
=
[K00 ]−1 101 ,
S00
=
n
X
Φ0 (1)
i
i=1
ki0 2
Φ0i , H00 = u00 − S00 .
(2.127)
[K00 ] is real symmetric and positive-definite so that there is no problem of existence in the
preceding equations. H00 is not exactly the static contribution of higher modes at frequency
e eq is frequency dependent, this should
ω but it has to divided by a compressibility. As K
be done with care. As mentioned in the introduction H(E, ω) ≈ H(E, 0) and this division
should be handled with care. The modal contributions qis and qit at frequency ω to modal
forces Fis and Fit are solution of the following equations:
e eq ki2 qit − ω 2 (e
P̂ ki2 qis − ω 2 (e
ρs qis + γ
eρeeq qit ) = Fis , K
γ ρeeq qis + ρeeq qit ) = Fit .
98
(2.128)
Fit
. These two
e eq k 2
P̂ ki2
K
i
relations indicates that the elastic properties that should be considered are the one of the
e0 .
current frequency and not the one at frequency ω. Hence, H0 should be divided by K
eq
The neglecting of inertial effects induces that qis =
Fis
and qit =
Concerning substructure 1 two attachment modes should be computed, one associated
to the excitation and the other one associated to the interaction force. [K0 ] is not invertible
as the first mode Φ1 is not elastic but a constant displacement rigid body motion. The
attachment mode associated to the force is now investigated. Let consider the following
matrices:
[P] = [In+1 ] − Φ1 Φt1 [M0 ] , [KP0 ] = [P]t [K0 ][P] , FPt = [P]t Ft .
(2.129)
[P] is a projection matrix which filter the rigid mode Φ1 ; [KP0 ] and FPt are the projections
through [P]. [KP0 ] is not invertible but let uP be the displacement vector of length n
obtained while solving the problem obtained by removing its last line and column as well
as the last line of FPt . One then defines:



 u 

n0

X
P 
Ftt
uF = [P]
, SF =
Φi , HF = uF − SF .
(2.130)


k0 2

i=2 i
 0 

e eq corresponds to the static response of the structure to excitation Ft from which the
uF /K
e eq is the contribution of elastic modes and HF /K
e eq is the contriburigid is filtered. SF /K
tion of non preserved modes.
While applying symmetry and linearity properties, it is straightforward to calculate
e eq associated to a unit force at the interface between porous 1
the attachment mode H0 /K
and 2. Hence, it is now possible to build the matrices and vectors of eq. (2.103) and then
obtain the solution of the problem with ICMT and FICMT. They are:















0
 0n+1 



H(F, 0) =
, Hb (F, 0) =
(2.131)


HF 
HF (n + 1) 








 e



e eq
Keq
K
(2n+2×1)
(2×1)




H00
H0
0 
0
 0


 P̂


 P̂
0 0
, [H (1j , 0)] = 
(2.132)
[H(1j , 0)] = 

0 


H0 
H0 
0
0
0
e eq
e eq
K
K
(2n+2×2)



[Rjun ] = 

H0 (n + 1)
P̂
(2n+2×2)

H00 (1)
+
P̂ 0
0
0



H0 (n + 1) H00 (1) 
+
0
e eq
e eq
K
K
(2.133)
(2×2))


 Φ(n + 1)
[Φb ] = 

0
0
Φ(n + 1)


0



 −Φ (1)
, [Φ0b ] = 

0
(2×2m)
99
0
0
−Φ (1)



(2×2m0 )
(2.134)
Porous total displacement [m]
−7
1
x 10
PEM 2
0.8
0.6
0.4
PEM 1
0.2
0
−0.04
−0.03
−0.02
−0.01
Position [m]
0
Figure 2.26: Total displacement of the two porous structures. Solid: analytical solution;
Dash-dot: DMT; Dash; ICMT ; Circles: FICMT
In these equations, the index in the parenthesis corresponds to the dimension of the vectors
and matrices.
The proposed example considers two porous structures of 2 cm thickness each. Figure
2.27 represents the absorption coefficient as a function of frequency. The first resonance is
at 600 Hz and the second one at 1800 Hz. The three modal techniques are compared and
for each one of them only 1 mode in each substructure is considered. It can be noticed
that the DMT does not provide accurate results in this case. The ICMT agrees till the
first resonance, and then diverges from the analytical solution and the FICMT is in good
agreement till the second resonance. Hence the static correction for the contribution of
higher modes to the excitation allows to maintain the performance of the technique in
a additional 700 Hz frequency range. After the second resonance there is a need for an
additional mode and there is no doubt that the resonance of a mode cannot be replaced
by a static correction. As intermediate conclusion, it appears that the FICMT is the most
accurate techniques among these three and that it is able to limit the number of kept
modes to the adequate ones.
The automatic selection procedure is now studied and this method is only studied for
the case of FICMT technique as the two other techniques are less efficient. Figure 2.28
(resp. 2.29 represents the evolution of the residual error ε1 and ε2 (resp. the number
of preserved modes for the first and second substructure) as a function of frequency. For
frequencies lower to 555 Hz, only one mode were retained for both structures. In this range
the residual error increases with frequency until the residual error on the eft substructure
reach the tolerance. A mode is the added to the left part. This induces a brutal decrease
of the residual error of the first substructure. Even if the range of the figure does make it
noticeable, the error on the second substructure also decrease (from 0.0198 to 0.0168). In
the second frequency part, 2 modes are considered for the left structure and one for the
right one. It can also be noticed that the error is not always increasing with the frequency.
Around 2300 Hz several modes are added; one to the left at 2270 Hz and one to the right
at 2420 Hz. The total displacement shape is plot at 2300 Hz in figure 2.30 for different
100
Absorption coefficient
1
0.8
0.6
0.4
0.2
0
0
1000
2000
Frequency [Hz]
3000
Figure 2.27: Absorption versus frequency for 1 mode in each porous substructure. Solid:
analytical solution; Dash-dot: DMT ; Dash; ICMT ; Circles: FICMT
0.1
Error
0.08
0.06
0.04
0.02
0
0
1000
2000 3000 4000
Frequency [Hz]
5000
Figure 2.28: Error with a truncation criterion on the residual ε1 = ε2 = 0.1. Solid: Error
relative to the left substructure; Dash-dot: Error relative to the right substructure
number of modes to understand the influence of added modes.
Figure 2.31 proposes the error on the absorption coefficient as a function of frequency
for different values of the residual error criterion. It can be noticed that this error do
not coincide with the residual one (there is no mathematical or physical reason for this).
Nevertheless, the more the residual criterion is weak and the lower is the error on the
absorption coefficient. Nevetheless, for this problem, errors are very weak (it was shown
that 1 mode in each substructure is sufficient until 1800 Hz.).
2.5
CMS with normal modes for the {us , p} formulation
This technique has not still been published, only its principle is exposed
in this section.
101
Number of selected modes
4
3
2
1
0
0
1000
2000 3000 4000
Frequency [Hz]
5000
Figure 2.29: Number of selected modes versus frequency. Solid: Modes of the left substructure ε = 0.1; Dash-dot: Modes of the right substructure ε = 0.1
−8
Porous total displacement [m]
x 10
1
0.5
PEM 1
0
−0.04
−0.03
PEM 2
−0.02
−0.01
Position [m]
0
Figure 2.30: Total displacement at 2300 Hz. Solid: Analytical solution; Dash-dot: Solution
with (2;1) modes; Dash: Solution with (2;2) modes; Circles: Solution with (3;2) modes.
102
Error on the absorption coefficient
0.04
0.03
0.02
0.01
0
0
1000
2000 3000 4000
Frequency [Hz]
5000
Figure 2.31: Error on the absorption coefficient versus frequency. Dash; ε = 0.1; Solid:
ε = 0.5; Dot: ε = 2
A last technique is now presented. It is associated with the mixed displacement pressure formulation. It is applied to system (2.71) whose frequency dependance is due to the
porous inertial and stiffness coefficients and not to shape matrices.
Instead of computing generalized complex modes, the proposed approach is based on
normal modes. Two different generalized eigenproblems are considered: The first one is associated to {[K], [M]} and the second one is associated to {[H], [Q]}. These four matrices
are real and symmetric and positive. [M] and [Q] are also definite. It is then ensured that
the eigenvectors and eigenvalues associated to these spectral problems are all real and that
the eigenvalues are positive.
A truncation of each modal family need to be performed to warranty the performance
the resolution process. Hence, contribution of non selected modes should be accounted for
in order to minimize truncation errors. Following a classical results of modal synthesis,
the selected modes correspond to the one associated to the lowest eigenvalues. Even if
non selected modes should not be computed, they will be considered as formally known to
let the reader understand where are the approximations of this reduced model. In all the
following, matrices indices denote their size. If there is no ambiguities, this size is omitted.
Let [Φs ]nu ×ms (resp. [ΦP ]nP ×mP ) be the matrix of the first (including rigid-body) ms
(resp. mP ) modes of the generalized eigenvalue problem associated to matrices {[K], [M]}
(resp. {[H], [Q]}). Let [Φ0s ]nu ×(nu −ms ) (resp. [Φ0P ]nP ×(nP −mP ) ) be the matrix of the
higher modes. Let [k2s ] and [r2s ] (resp. [k2P ] and [r2P ]) be the diagonal matrices of the
preserved and higher eigenvalues. System (2.71) is now projected on this decoupled modal
103
basis:


 



2



− ω ρe[I]
[0]
−e
γ [Γkk ]
−e
γ [Γkh ]



 Φts Fs
q


s


 
















2
2



0




[0]
P̂ [rs ] − ω ρe[I]
−e
γ [Γhk ]
−e
γ [Γhh ]


 Φst Fs
hs




=


ω2
1 2


2
t
2
t

 −e


[k
]
−
[I]
[0]
γ
ω
[Γ
]
−e
γ
ω
[Γ
]



kk
hk
P





q
ΦtP FP
e
P


ρeeq
Keq



















1 2
ω2



2
t
2
t


 Φ0Pt FP
[I]
[rP ] −
−e
γ ω [Γkh ]
−e
γ ω [Γhh ]
[0]
hP
e eq
ρeeq
K
(2.135)
qs and hs (resp. qP and hP ) are the contributions of the preserved and higher modes
and one has:
us = [Φs ]qs + [Φ0s ]hs , P = [ΦP ]qP + [Φ0P ]hP .
(2.136)
P̂ [k2s ]
Projection of the coupling matrix [C] reads:


 [Γkk ] [Γkh ]  
 = Φs |Φ0s t [C] ΦP |Φ0P .


[Γhk ] [Γhh ]
(2.137)
Systems (2.71) and (2.135) are equivalent as they are the representation of the problem
in nodal and modal coordinates.
Approximations are now made on modal system (2.135). First one is to neglect inertial
terms for solid and pressure higher modes. (e.g. it means that the excitation pulsation is
ω2
1 2
[kP ] >>
[I].). The second approximasufficiently low to have P̂ [r2s ] >> ω 2 ρe[I] and
e eq
ρeeq
K
tion is to neglect the higher modes inter coupling term (−e
γ ω 2 [Γhh ]t ) in the block-equation
associated to pressure higher modes. System (2.135) now reads:














P̂ [k2s ]
− ω 2 ρe[I]
[0]
[0]
P̂ [r2s ]
−e
γ ω 2 [Γkk ]t
−e
γ ω 2 [Γhk ]t
−e
γ ω 2 [Γkh ]t
[0]t


−e
γ [Γkk ]
−e
γ [Γkh ]  









−e
γ [Γhk ]
−e
γ [Γhh ]  



1 2
ω2

[kP ] −
[I]
[0]


e eq
ρeeq


K





1 2

[0]
[rP ] 
ρeeq



qs 








hs 
qP
hP












=




Φts Fs







0

 Φst Fs

















ΦtP FP







 Φ0Pt FP












.
(2.138)
The following step of the proposed method is to condense contributions hs and hP . It
should be enlightened that there is no additional approximations in the following: all the
following developments will leave to systems equivalent to (2.138). By the way of the last
equation-block , it is possible to find the pressure Ph associated to non-preserved modes:
0
0
γ ω 2 [Φ0P ][r2P ]−1 ΦPt [C]t [Φs ] qs ).
Ph = [Φ0P ]hP ≈ ρeeq ([Φ0P ][r2P ]−1 ΦPt FP +e
{z
}
|
{z
}
|
ΨP
[Ξs ]
104
(2.139)

























.
With respect to the frequency dependent coefficient ρeeq , ΨP is the response of higher modes
to excitation FP . This vector is real and frequency independent and can be computed
by the way of the preceding section and without knowing non-preserved eigenvalues and
eigenvectors. [Ξs ]nP ×ms is the matrix of the response of higher modes to the ms unitary
excitations [C]t [Φs ]. Hence, the coupling terms associated to non-preserved solid modes are
interpreted as forces on the fluid part. Ph can be introduced in the second block-equation
(2.138):
0
0
0
P̂ [r2s ]hs ≈ Φst Fs + γ
eΦst [C]ΦP qP + γ
eΦst [C]Ph .
(2.140)
With equation (2.139), displacement uh of non preserved modes can be expressed as a
function of the contribution of solid and pressure preserved modes:
uh = [Φ0s ]hs ≈
1
P̂
(Ψs + γ
e[ΞP ] qP + ρeeq Ψ0s + ρeeq γ
eω 2 [Ξ0s ]qs )
(2.141)
with
0
0
Ψs = [Φ0s ][r2s ]−1 Φst Fs , [ΞP ] = [Φ0s ][r2s ]−1 Φst [C]ΦP
(2.142)
and
0
0
Ψ0s = [Φ0s ][r2s ]−1 Φst [C]ΨP , [Ξ0s ] = [Φ0s ][r2s ]−1 Φst [C][Ξs ].
(2.143)
Hence, system (2.138) is equivalent to:





P̂ [k2s ]
−
ω 2 ρe[I]
−e
γ ω 2 [Γ
kk
]t

 

 

−e
γ [Γkk ]
  qs 
 
 Φts Fs + γ
e[Φs ]t [C]Ph

=

 

1 2
ω2
 q 
 
 Φt FP + γ
[kP ] −
[I] 
eω 2 [ΦP ]t [C]t uh
P
P
e
ρeeq
Keq




.



(2.144)
and the solid and pressure modes contributions qs and qP are solution of the following
system:




−
−
−e
γ [Γkk ]

 qs 






2
2
2
γ
eω ρeeq ω



1 2
γ
e2 ω 2
ω2
 q 

[ΦP ]t [C]t [Ξ0s ]
[ΦP ]t [C]t [ΞP ] 
[I] −
[kP ] −
−e
γ ω 2 [Γkk ]t −
P
e eq
ρeeq
P̂
P̂
K
(2.145)










Φts Fs + γ
e[Φs ]t [C]e
ρeq ΨP
=
.
(2.146)
2


γ
e
ω


t
0

[ΦP ]t [C]t (Ψs + ρeeq Ψs ) 

 ΦP FP +
P̂

P̂ [k2s ]
2.A
ω 2 ρe[I]
ρeeq γ
e2 ω 2 [Φs ]t [C][Ξs ]
Variational forms for poroelastic problems
Equivalent fluid formulation
Z
I
1
1
1 ∂p
∀q ∈ Vf ,
∇p ∇q −
pq dΩ = −
q dΓ.
2
e
eeq ω
eeq ∂n
Keq
Ω ρ
∂Ω ρ
105
(2.147)
{us , ut } Formulation
Z
s
ZΩ
s
2
s
t
s
σ̂(u ) : (δu ) − ω (e
ρs u + γ
eρeeq u )δu dΩ =
Ω
Keq ∇.ut ∇.δut − ω 2 ρeeq (us + γ
eut )δut dΩ =
I
σ̂(us ).nδus dΓ
I∂ΩN
(2.148a)
e eq ∇.ut .nδut dΓ
K
∂ΩN
(2.148b)
{us , p} original formulation
∀v ∈ Vu ,
Z
∀q ∈ Vf ,
s
Ω1
2
σ̂ (u) : ε(v) − ω ρeu.v − γ
e ∇p .v dΩ =
"
Z
Ω1
∇p ∇q
pq
−
−γ
eu. ∇q
2
e eq
ρeeq ω
K
#
I
[σ̂ s (u).n]v dΓ
(2.149)
∂Ω1
1 ∂p
q dΓ
γ
eun −
dΩ =
ρeeq ∂n
∂Ω1
(2.150)
I
{us , p} modified formulation
φ
∀v ∈ Vu , σ̂ s (u) : ε(v) − ω 2 ρeu.v − ∇p .v − φ 1 +
α
e
Ω
I
= [σ̂ t (u, p).n]v dΓ
Z
e
Q
e
R
!
p ∇.v dΩ
(2.151)
Γ
∀q ∈ Vf ,
Z "
Ω
I
=
∂Γ
∇p ∇q
φ
pq
− u. ∇q − φ 1 +
−
2
e
ρeeq ω
α
e
Keq
[Un − un ] q dΓ
e
Q
e
R
!
#
∇.u q
dΩ
(2.152)
106