Lucie Gallerand

Mathias Legrand

Thomas Dupont

Raymond Panneton

Philippe Leclaire

Damping performance of finite microperforated plates using multi-size and spatial distributions of perforations

Keywords:

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Introduction

Microperforated plates (MPP) are usually employed as lightweight acoustic absorbers for noise reduction and are regarded as an alternative to conventional porous materials or conventional acoustic resonators. MPP can be used in many fields to reduce noise, such as meeting room [START_REF] Adams | Sound materials: A compendium of sound absorbing materials for architecture and design[END_REF][START_REF] Hoshi | Implementation experiment of a honeycomb-backed MPP sound absorber in a meeting room[END_REF], acoustic coatings in flow ducts [START_REF] Wu | Micro-perforated panels for duct silencing[END_REF] or nuclear engines and reactors [START_REF] Wang | Investigation of the effect of perforated sheath on thermal-flow characteristics over a gas turbine reverse-flow combustor-Part 2: Computational analysis[END_REF] for example. These simple structures can be safe, environment-friendly and can be made of different materials. They can be designed to be resistant to harsh environments, translucent or biodegradable. MPP transform acoustic energy into heat by exchanges in the viscous and thermal boundary layers near the fluid-solid interface of the microperforations.

The acoustic properties of microperforated plates were investigated through various models, including models based on the Kirchhoff equations [START_REF] Maa | Microperforated-panel wideband absorbers[END_REF] and equivalent fluid models such as the Johnson-Allard approach [START_REF] Champoux | On acoustical models for sound propagation in rigid frame porous materials and the influence of shape factors[END_REF][START_REF] Atalla | Modeling of perforated plates and screens using rigid frame porous models[END_REF]. Vibrations of MPP were also considered with the aim of investigating their influence on the acoustic properties of structures [START_REF] Takahashi | Flexural vibration of perforated plates and porous elastic materials under acoustic loading[END_REF][START_REF] Dupont | Acoustic properties of lightweight micro-perforated plate systems[END_REF][START_REF] Bravo | Vibroacoustic properties of thin micro-perforated panel absorbers[END_REF] 1 . Research was also conducted to improve and extend MPP' acoustic absorption by using different partitioned cavity depths [START_REF] Ruiz | Optimization of multiple-layer microperforated panels by simulated annealing[END_REF] or multi-size perforation [START_REF] Miasa | An Experimental Study of a Multi-Size Microperforated Panel Absorber[END_REF][START_REF] Kim | Absorption performance optimization of perforated plate using multiple-sized holes and a porous separating partition[END_REF], i.e., an MPP whose perforations have various diameters. In fact, Kim and Yoon [START_REF] Kim | Absorption performance optimization of perforated plate using multiple-sized holes and a porous separating partition[END_REF] proposed a multi-size microperforation configuration to improve the sound absorption of MPP in a wide frequency band. An equivalent electro-acoustic circuit method was used to explore the sound absorption properties of a perforated panel with microperforated partitions [START_REF] Carbajo | Assessment of methods to study the acoustic properties of heterogeneous perforated panel absorbers[END_REF][START_REF] Carbajo | Perforated panel absorbers with micro-perforated partitions[END_REF]. Qian et al. [START_REF] Qian | Optimization of multi-size micro-perforated panel absorbers using multi-population genetic algorithm[END_REF] also used an equivalent electroacoustic circuit method to model multi-size microperforations. They proposed a multi-population genetic algorithm to optimize the design of multi-size MPP absorbers. Theoretical results showed that only a multi-size MPP absorber with grouped perforations can improve the sound absorption capability of MPP. Mosa et al. [START_REF] Mosa | Theoretical model of absorption coefficient of an inhomogeneous MPP absorber with multi-cavity depths[END_REF] explored the absorption coefficient of an inhomogeneous MPP with multi-cavity depths. Results showed that introducing an inhomogeneous perforation improved the absorption capability of an MPP absorber compared to a homogeneous one. Experimental and numerical studies have also been performed in an acoustic MPP context with multi-size perforation diameters. Miasa et al. [START_REF] Miasa | An Experimental Study of a Multi-Size Microperforated Panel Absorber[END_REF] explored experimentally the sound absorption performance of a microperforated plate with multi-size perforation diameters. The results showed that multi-size MPP absorbers can exhibit high sound absorption over a wider frequency range than uniformly sized MPP. The authors also concluded that in the case of an MPP with multiple perforation diameters, the benefits of each MPP with a single perforation size were combined. Qian and Zhang [START_REF] Qian | Influence of arranged patterns on the absorption performance of parallel MPP absorbers[END_REF] used finite element analysis to investigate the influence of an MPP with multi-size perforation diameters on the normal incidence sound absorption performance of a parallel MPP absorber. Some authors have also conducted research on the effect of perforation distribution on the sound absorption of MPP. Temiz et al. [START_REF] Temiz | Modelling vibro-acoustic coupling in flexible micro-perforated plates by a patch-impedance approach[END_REF] proposed to numerically explore the effect of perforation distribution on sound absorption by assuming that perforations are discrete impedance patches, as proposed in [START_REF] Ouisse | Patch transfer functions as a tool to couple linear acoustic problems[END_REF][START_REF] Maxit | Modeling of micro-perforated panels in a complex vibroacoustic environment using patch transfer function approach[END_REF]. Although their works focused on MPP sound absorption, they observed that the distribution of perforations on the MPP could have a significant effect on the viscous damping mechanism. Different diameters and perforation arrangements were modeled using the finite element method and then through an experimental study [START_REF] Putra | The effect of perforation on the dynamics of a flexible panel[END_REF]. They observed that the perforation ratio and the perforation position have an influence on the dynamics of the plate.

In addition to acoustics, the structural dynamics of MPP was also explored. A recent work by the authors [START_REF] Gallerand | Vibration and damping analysis of a thin finite-size microperforated plate[END_REF], and based on [START_REF] Leclaire | Transverse vibration of a thin rectangular porous plate saturated by a fluid[END_REF] and [START_REF] Atalla | Modeling of perforated plates and screens using rigid frame porous models[END_REF], has shown, both theoretically and experimentally, that energy dissipation at the fluid-solid interface in the microperforations drives substantial added damping for the structure. Parametric studies have demonstrated the existence of a single characteristic frequency, which depends on the constant diameter of the perforation and the perforation ratio, at which the added damping reaches a maximum. If this characteristic frequency coincides with one natural frequency of a plate, the vibration in the associated mode will be mitigated by the added damping. However, the added damping will be weaker in the other natural modes. Its impact will be all the more limited as their frequencies are further from the characteristic frequency. An MPP with multiple perforation sizes, and thus with multiple characteristic frequencies (one for each perforation diameter), should theoretically have maximum added damping to its modes having a frequency that coincides with one of the characteristic frequencies. Moreover, additional damping capabilities are due to viscous frictions and thermal exchanges in the boundary layers occurring during the motion of the structure. It is therefore expected that more perforations distributed over the areas where the structural displacement is maximum will induce a magnified added damping. The present work proposes to improve and enhance the additional damping presented by microperforations by using multi-size perforation diameters and a spatial distribution of perforations.

In this paper, each section proposes to study a different type of MPP, as listed below: Section 2 MPP with uniformly distributed perforations of the same diameters corresponding to the reference MPP; Section 3 MPP with uniformly distributed perforations of different diameters; Section 4 MPP with non-uniformly distributed perforations of the same diameters; Section 5 MPP with non-uniformly distributed perforations of different diameters. For each section, a model and experimental validations are proposed. To this end, it is first proposed to recall in Section 2 the governing equations of the dynamic of a thin MPP saturated by a lightweight fluid, already detailed in [START_REF] Gallerand | Vibration and damping analysis of a thin finite-size microperforated plate[END_REF]. The model used to capture the effect of multi-size perforation diameters in Section 3 is based on a homogenization approach. In Section 4, the spatial distribution of perforations is considered by defining a spatial perforation ratio. Section 5 proposes to combine the two models presented in Section 3 and Section 4. Finally, conclusions are given in Section 6.

MPP: Uniform distribution and single diameter

Vibration model

The previous work [START_REF] Gallerand | Vibration and damping analysis of a thin finite-size microperforated plate[END_REF] by the authors developed a vibration model of a finite-size MPP of dimensions L x × L y × h, as illustrated in Figure 1, obtained by identifying the MPP with a porous plate and using an alternative form of the Biot's theory [START_REF] Biot | Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range[END_REF][START_REF] Leclaire | Transverse vibration of a thin rectangular porous plate saturated by a fluid[END_REF]. It is used again in the present work, but multi-size and spatially distributed microperforations are considered in Sections 3 to 5. In this model, in-plane displacements are ignored and only the normal displacement is accounted for. The MPP is regarded as a homogeneous plate excited by an external load of the form f ext (x, y, t). Following a low-frequency assumption, the dynamic response of the MPP saturated by a light fluid is the solution of the two coupled governing equations [START_REF] Gallerand | Vibration and damping analysis of a thin finite-size microperforated plate[END_REF] h(ρ ẅs (x, y, t)

+ ρ f ẅ(x, y, t)) + D + α 2 M f h 3 12 ∇ 4 w s (x, y, t) = f ext (x, y, t), (1a)
ρ f ẅs (x, y, t) + ρ f α ∞ ϕ (0) ẅ(x, y, t) + σ (0) ẇ(x, y, t) + αM f ∇ 2 w s (x, y, t) = 0, ( 1b
)
where ∇ is the vector differential operator so that

∇ 2 (•) = ∂ 2 (•) ∂x 2 + ∂ 2 (•) ∂y 2 and ∇ 4 (•) = ∇ 2 (•) 2 .
The relative fluid-solid motion is w(x, y, t) = ϕ (0) (w f (x, y, t) -w s (x, y, t)) with w f (x, y, t), the fluid displacement, and w s (x, y, t), the solid displacement, and ϕ (0) is the perforation ratio. Note that there was a sign error in [START_REF] Gallerand | Vibration and damping analysis of a thin finite-size microperforated plate[END_REF] but this had no impact on the results presented. The equation system is corrected in this paper. Equation (1a) models the elastic response of the equivalent non-perforated homogeneous solid plate and Equation (1b), the relative fluid-solid motion in the perforations. The coefficient D in Equation (1a) is the bending stiffness defined as a function of the plate thickness h, Young's modulus E and Poisson's ratio of the non-perforated structure ν. To capture the effect of microperforations on plate response, Young's modulus E should be adapted to account for the effect of perforations on plate rigidity [START_REF] Boccaccini | A new approach for the Young's modulus-porosity correlation of ceramic materials[END_REF][START_REF] Gallerand | Vibration and damping analysis of a thin finite-size microperforated plate[END_REF].

The bending stiffness therefore depends on ϕ (0) as follows:

D = EC φ h 3 12(1 -ν 2 ) with C φ = (1 -ϕ (0) ) 2 1 + (2 -3ν)ϕ (0) . (2)
For a vibrating MPP saturated by a lightweight fluid, the elastic modulus of the plate is much greater than that of the fluid, that is, 1) and in the rest of the paper. The density of fluid-solid mixture is captured by ρ = (1 -ϕ (0) )ρ s + ϕ (0) ρ f where ρ s and ρ f are the solid and fluid densities, respectively. In Equation (1b), M f corresponds to the elastic modulus of the equivalent fluid. The dimensionless quantity α characterizes the elastic coupling between the equivalent fluid and the solid. In the context of an MPP saturated by a lightweight fluid, we have αM f ≈ K f [START_REF] Leclaire | Transverse vibration of a thin rectangular porous plate saturated by a fluid[END_REF] where K f is the bulk modulus of the fluid. With the above assumptions, Equation (1) becomes (temporal and spatial dependencies are omitted)

E ≫ M f : accordingly, D + α 2 M f h 3 12 ≈ D in Equation (
h(ρ s (1 -ϕ (0) ) ẅs + ρ f ϕ (0) ẅf ) + D∇ 4 w s = f ext , (3a)
ρ f ((1 -α ∞ ) ẅs + α ∞ ẅf ) + σ (0) ϕ (0) ( ẇf -ẇs ) + K f ∇ 2 w s = 0. ( 3b
)
Some porous parameters used in Biot's theory (airflow resistivity σ (0) and tortuosity α ∞ ) are, for an MPP, functions of the perforation diameter d, perforation ratio ϕ (0) , plate thickness h, and fluid in the perforations. Thus, the airflow resistivity reads [START_REF] Atalla | Modeling of perforated plates and screens using rigid frame porous models[END_REF]

σ (0) = ς ϕ (0) with ς = 32µ f d 2 (4)
where µ f is the dynamic viscosity of the fluid. In order to consider the distortion of airflow in the perforations and the fluid-solid interactions between the perforations, an empirical correction is taken into account via the tortuosity as proposed elsewhere [START_REF] Atalla | Modeling of perforated plates and screens using rigid frame porous models[END_REF]. Consequently, the tortuosity is defined as:

α ∞ = 1 + B(1 -1.14 ϕ (0) ) with B = 0.48 h √ πd 2 . (5)
Here, the term in parentheses contains the edge interaction between neighboring perforations. Note that the analytical vibration model presented in this section was validated by experimental measurements on MPP in [START_REF] Gallerand | Vibration and damping analysis of a thin finite-size microperforated plate[END_REF].

Added damping

An MPP features interactions in viscous and thermal boundary layers associated with fluid-solid interactions, known to induce a non-neglecting additional damping in the plate [START_REF] Gallerand | Vibration and damping analysis of a thin finite-size microperforated plate[END_REF]. This phenomenon reaches a maximum at the characteristic frequency

f c = 16µ f πα ∞ ρ f d 2 (6)
defined from Biot's frequency for porous materials [START_REF] Biot | Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range[END_REF][START_REF] Leclaire | Transverse vibration of a thin rectangular porous plate saturated by a fluid[END_REF]. In Equation ( 6), f c depends only on the perforation diameter d and the fluid properties ρ f and µ f . The perforation diameter d can be adapted to induce maximum added damping at a resonance frequency, that is, in a way where f c coincides with a natural frequency of the plate. The additional damping provided by the microperforations is greatest at the characteristic frequency, but also acts in a frequency range centered on f c . Figure 2 represents the damping capacities of an aluminum MPP of dimension 490 mm×570 mm×1 mm. The MPP is excited by a normal point force F ext (x F , y F ) located at (x F , y F ) = (80, 70) mm on its surface. The level of mobility

L Y = 10 log ẇs (x R , y R ) F ext (x F , y F ) (7)
predicted by the model for the MPP at point (x R , y R ) = (61.25, 71.25) mm is compared to that of a reference plate of the same dimensions without perforations. The perforation diameter of the MPP is set to d ≡ d 1 = 1.6 mm in order to induce maximum added damping on the first MPP mode. The perforation ratio is set to ϕ (0) = 10 %. The reduction in vibration and the loss factor at each MPP ) MPP with a perforation diameter, denoted d1, set to induce maximum added damping in the first MPP mode, and (

) MPP with the same equivalent stiffness as the reference non-perforated plate and with d ≡ d1 maximizing the added damping on the first MPP mode. Perforation parameters are d1 = 1.6 mm and ϕ (0) = 10 %. Isotropic structural loss factor of aluminum ηs = 10 -4 or 0.01 % is also considered.

resonance frequency are listed in Table 1. The loss factor η mn is related to the modal damping ratio ζ mn by the following expression η mn = 2ζ mn at the eigenfrequency. The modal damping ratio ζ mn is obtained for mode (m, n) after solving Equation (1) via a modal analysis as detailed in [START_REF] Gallerand | Vibration and damping analysis of a thin finite-size microperforated plate[END_REF]. The structural loss factor for aluminum is assumed to be frequency-independent and is set to η s = 10 -4 or 0.01%. The mobility reduction ∆Y mn is determined by the difference in mobility level between the MPP and the reference plate in mode (m, n). For the MPP, the shift towards low frequencies of the modes is due to the reduction of the Young's modulus via C φ in Equation ( 2). The perforations increase the loss factor and therefore the added damping relative to the reference plate by a factor of 106 in the first mode, 35 in the second mode, 25 in the third mode and 12 in the fourth mode.

(m, n) ηmn(%) ∆Ymn (dB) (1,
In Figure 2, an additional comparison is made, where the MPP keeps the same specific stiffness as the reference plate. Here, the specific rigidity is the ratio between the rigidity and the surface mass (D/ρh). This comparison aims to confirm that the increase in damping is indeed due to the losses added by the perforations and not to the reduction in the rigidity of the plate.

MPP: Uniform distribution and multiple diameters

Homogenization model

In this section, an MPP with N groups of perforations is considered. All perforations in a same group have the same diameter d k with k = 1, 2, • • • , N . Each group of perforations is homogeneously distributed over the MPP which can be considered homogeneous. To capture the dynamic response of an MPP with multi-size perforation diameters, a homogenization approach is proposed. The principle is to model the heterogeneous MPP structure as an equivalent continuous structure. In this context, 2N equivalent plates, defined in the same mathematical spatial domain, are considered: N equivalent homogeneous solid plates and N equivalent homogeneous fluid plates as shown in Figure 3 for an MPP with two groups of perforations. Each homogeneous solid plate is related to its perforations diameter d k and its corresponding perforation ratio ϕ coupling). The properties of each subdomain (perforation ratio, resistivity) are considered separately. A perforation ratio ϕ

(0)
k is associated with each perforation diameter d k and the total perforation ratio is

ϕ (0) tot = N k=1 ϕ (0) k . ( 8
)
2 Note that the equivalent solid plate of subdomain k is defined by its perforation ratio ϕ

k . The definition of the global equivalent solid plate is achieved by an electroacoustic analogy [START_REF] Miasa | An Experimental Study of a Multi-Size Microperforated Panel Absorber[END_REF][START_REF] Qian | Influence of arranged patterns on the absorption performance of parallel MPP absorbers[END_REF]. According to this approach, Young's modulus and density of the overall structure are defined as a function of the overall perforation ratio given in Equation ( 8).

The overall airflow resistivity of the equivalent plate with N perforation diameters takes the form [START_REF] Kim | Absorption performance optimization of perforated plate using multiple-sized holes and a porous separating partition[END_REF][START_REF] Carbajo | Perforated panel absorbers with micro-perforated partitions[END_REF][START_REF] Zhang | The study of sound absorption characteristic of micro-perforated panel with different diameter holes[END_REF]

1 σ = N k=1 1 σ (0) k (9)
where σ

(0)
k is given by Equation ( 4) with d = d k . If α ∞ ≈ 1, only the resistivity is modified by the addition of perforations of different sizes. For the considered ϕ (0) tot , the tortuosity end correction has limited influence on the results. However, to account for the size of multiple perforations in the empirical formulation of the tortuosity of the length correction from [START_REF] Atalla | Modeling of perforated plates and screens using rigid frame porous models[END_REF] given in Equation ( 5), an average approach is proposed here. The tortuosity is thus rewritten

α ∞ = 1 + B tot (1 -1.14 ϕ (0) tot ) with B tot = 1 ϕ (0) tot N k=1 ϕ (0) k B k ( 10
)
where B k defined in Equation ( 5) is applied to each subdomain k.

The bending stiffness is also modified in order to capture the effect of the multi-size microperforations in the dynamic response of the plate. To this end, the correction coefficient C φ in Equation ( 2) becomes a function of ϕ (0) tot . Equations ( 8) to [START_REF] Champoux | On acoustical models for sound propagation in rigid frame porous materials and the influence of shape factors[END_REF] are then inserted in Equation ( 1) to obtain the structural response for an MPP with homogeneous distribution of multiple microperforations.

Multi-size microperforations are expected to increase the frequency range over which the added damping is effective. As said in Section 2.2, the added damping reaches a maximum at a characteristic frequency that is only a function of the diameter of the perforation, as defined in Equation [START_REF] Boccaccini | A new approach for the Young's modulus-porosity correlation of ceramic materials[END_REF]. Assuming that each added damping phenomenon is independent, the use of N groups of perforations of different diameters induces N characteristic frequencies of maximum damping on the same MPP and thus increases its efficiency. In Figure 4, we vary the thickness of the plate described in the previous section, in the range h ∈ 0.5 -3 mm, in order to vary the resonance frequency of its first two modes. Consequently, the loss factor η i is obtained as a function of the resonance frequency f i in Figure 4(a) for i = 1 and in Figure 4(b) for i = 2. Three configurations are considered in terms of d. In the first, the diameter of the perforation d 1 is chosen to induce a maximum added damping in the first natural mode. In the second, the diameter of the perforation d 2 is set to maximize the added damping in the second natural mode. In the third configuration, a combination of d 1 and d 2 is considered. The corresponding perforation ratios are ϕ (0) 1 = 3 % and ϕ (0) 2 = 7 %. For each configuration, the overall perforation ratio, defined from Equation [START_REF] Carbajo | Assessment of methods to study the acoustic properties of heterogeneous perforated panel absorbers[END_REF], is set to 10 %. For mode i with i = 1, 2, an MPP with two groups of perforations provides similar results to the case of an MPP with a single group of perforations, where the diameter is chosen to maximize the added damping on the considered mode. The damping induced by the MPP with two groups of perforations on mode i is always higher than the one induced on mode k, i ̸ = k, by an MPP with a single group of perforations. As a consequence, a multi-size perforation MPP does not decrease (or slightly decreases) the added damping efficiency on mode i compared to an MPP where d is chosen to maximize damping on mode i only but allows for efficiency on multiple modes. The perforation ratio of each subdomain also influences the added damping. In fact, mode i is all the more damped as the perforation ratio of the subdomain k is large. It is therefore possible to adjust ϕ (0) k to maximize the added damping on a mode or to maximize the effect on the N modes.

In order to maximize the added damping in the frequency range between the natural frequencies p and q, the formula of the perforation diameter of each subdomain, denoted d p and d q respectively, is defined as follows

d p = 16µ f πf p ρ f α ∞ , (11)
where f p is the natural frequency of the pth mode. A similar expression is given for d q from f q . The ω-dependent loss factors η p (ω) and η q (ω) can be calculated from Equation (48) of reference [START_REF] Gallerand | Vibration and damping analysis of a thin finite-size microperforated plate[END_REF]. These loss factors respectively show maximum added damping η max p at f p and η max q at f q . If one defines the absolute difference

|η p (ω) -η max p |/η max p -|η q (ω) -η max q |/η max q |,
it can be shown that an overall maximum added damping occurs at the geometric mean of f p and f q given by

f c = f p f q . ( 12
)
This overall maximum damping would typically be obtained when

ϕ (0) p = ϕ (0) tot 1 + fq fp and ϕ (0) q = ϕ (0) tot 1 + fp fq . ( 13
)
Using the latter equations, with ϕ tot = 10% and the graphs in Figure 4, one obtains ϕ

(0)
p ≈ 3% and ϕ (0) q ≈ 7%. This gives the green curves in Figure 4 for the combination of the two sets of perforations. These curves nearly reach the maximum loss curves for the single-perforation configuration for both modes.

Experimental validation

In this section, the experimental validation of the previous approach is performed. The experimental setup presented is used in Sections 4 and 5. The mechanical parameters of the MPP such as Young's modulus and structural loss factor are determined from the Oberst test bench developed by Mecanum Inc. and presented in Figure 5(a). The measuring method follows the ASTM E756-05(2017) [START_REF]Standard Test Method for Measuring Vibration-Damping Properties of Materials[END_REF] standard. The tested sample is clamped at x = 0 and free at x = L x . It is excited by an external force of amplitude F ext at (x F , y F ) = (L x , L y /2) and the sample vibratory response is measured by a magnetic sensor located at point (x R , y R ) = (R, L y /2).

In order to validate the homogenized model of the multi-size MPP, the three samples presented in Figure 5(b) are considered and noted MPP 1 ○, MPP 2 ○ and MPP 3 ○. They are made of steel and have a length L x = 130 mm, a width L y = 30 mm, and a thickness h = 0.87 mm. For MPP 1 ○, d ≡ d 1 is set to induce the maximum added damping around the first natural frequency f 1 . For MPP 2 ○, d ≡ d 2 is chosen to maximize the added damping around the second natural frequency f 2 . The perforation ratios associated with diameters d 1 and d 2 are ϕ (0) 1 = 10 % and ϕ (0) 2 = 2.3 % respectively. MPP 3 ○ is a combination of MPP 1 ○ and MPP 2 ○. The resulting MPP therefore has two groups of perforation with the two perforation diameters d 1 and d 2 with corresponding perforation ratios ϕ ○ has therefore the double number of perforations compared to MPP 1 ○ and MPP 2 ○. The perforation parameters of each MPP are listed in Table 2. Plate mobility levels, as defined in the previous section, are plotted for the three MPP in Figure 6. The figure compares the experimental measurements made on MPP 1 ○, MPP 2 ○, and MPP 3 ○. It also compares the analytical prediction made on MPP 3 ○. As expected, the levels of mobility at the first resonance are similar for MPP 1 ○ and MPP 3 ○. However, compared to MPP 1 ○, a reduction of about 3.8 dB is observed ○ are given in Table 2.

ϕ (0) 1 (%) d 1 (mm) ϕ (0) 2 (%) d 2 (mm) ϕ (0) tot = ϕ (0) 1 + ϕ (0) 2 (%) (8) MPP 1 ○ 10 1 • • 10 MPP 2 ○ • • 2.3 0.4 2.3 MPP 3 ○ 10 1 2.3 0.4 12.3
on the second resonance for the multiple perforations MPP 3 ○. It can also be noticed that the analytical model gives similar results as the experiments and which validates the model presented in Section 3.1.

From the plate mobility levels shown in Figure 6, the loss factors are measured using the half-power bandwidth method at the first two resonances for the three MPP. The measured loss factors are summarized in Table 3. For MPP 3 ○, the loss factor obtained from the theoretical homogenization model presented in Section 3.1 is also provided. It can be seen that for MPP 3 ○, the global damping is maximized at both resonances, unlike an MPP with single-size perforations, where the damping is only maximized locally on one mode.

The previous results show that using an MPP with a multi-size perforations increases the frequency band over which the added damping is effective. Moreover, it also increases the damping factor of the first modes. Finally, the comparison between the measured and analytical results obtained for MPP 3 ○ in Figure 6 and Table 3 allows one to validate the model proposed in Section 3.1.

MPP: Nonuniform distribution and single diameter

Perforation ratio gradient

The distribution of the perforations can also influence the added damping. For a given mode, the closer the perforations are to the maximum displacement zones, the higher the added damping. In fact, the damping added by the microperforations is due to the viscous friction in the boundary layers of the MPP. This viscous friction is all the more important as the relative velocity between the fluid and the structure Mode 1 Mode 2 9(a). Each loss factor is determined for the ith mode. Perforation parameters are given in Table 2 for each MPP configurations.

f 1 (Hz) η 1 [×10 -3 ] f 2 (Hz) η 2 [×10 -3 ] Measured MPP 1 ○ 44.2 ±
is large. To enhance the added damping effect, it is therefore possible to concentrate the perforations on the antinodes of the modes whose amplitude must be reduced. In this section, only a single perforation diameter is considered, and the perforation ratio is written in terms of a distribution function to explore this effect:

ϕ(x, y) = ϕ (0) Ih(x, y) (14)
where Ih(x, y) is a normalized inhomogeneity function and ϕ (0) the maximum value of the perforation ratio obtained at max Ih(x, y) = 1. The perforation ratio in the previous equation is a local variable. Its integration over the whole plate surface yields the global perforation ratio

ϕ g = ϕ (0) L x L y Lx 0 Ly 0 Ih(x, y)dydx. ( 15
)
In Equation ( 3), D and α ∞ depend on the perforation ratio and become a spatial function by considering the perforation ratio gradient defined in Equation ( 14). The correction function in the bending stiffness of Equation ( 2) is rewritten as

C φ (x, y) = (1 -ϕ(x, y)) 2 1 + (2 -3ν)ϕ(x, y) , (16)
Similarly, the corrected tortuosity given by Equation ( 5) now rewrites α ∞ (x, y) = 1 + B -1.14B ϕ(x, y).

Note that the perforation ratio also has an influence on ν [START_REF] Arnold | Prediction of the poisson's ratio of porous materials[END_REF]. However, for the perforation ratios considered, ν is assumed to be constant [START_REF] Lutz | The effect of pore shape on the poisson ratio of porous materials[END_REF][START_REF] Arnold | Prediction of the poisson's ratio of porous materials[END_REF]. Equations ( 14) and ( 16) are inserted into Equation (3) to capture the effect of the perforation ratio gradient in the structural dynamics of MPP. The system of equations describing the autonomous response of a finite MPP saturated by a lightweight fluid in its perforations then becomes h(ρ s (1 -ϕ(x, y)) ẅs + ρ f ϕ(x, y) ẅf ) + D(x, y)∇ 4 w s = 0, (17a)

ρ f ((1 -α ∞ (x, y)) ẅs + α ∞ (x, y) ẅf ) + ς( ẇf -ẇs ) + K f ∇ 2 w s = 0. ( 17b
)
Classical modal analysis is now performed in the same vein as in [START_REF] Gallerand | Vibration and damping analysis of a thin finite-size microperforated plate[END_REF]. The equations must be space semi-discretized and projected onto the non-perforated plate mode. The plate displacement is assumed to be of the form

w s (x, y, t) = ∞ m=1 ∞ n=1 w s mn (t)Ψ mn (x, y), (18)
where w s mn (t) represent the participation of non-perforated plate eigenmode Ψ mn (x, y) determined from boundary conditions. A similar expression holds for the fluid. Equation ( 18) is rewritten for a finite number of modes and rearranged in lexicographic order according to i. Each pair (m, n) ∈ N × N corresponds to a single index i ∈ N. Equation ( 18) is therefore rewritten in terms of i such that

w s (x, y, t) = N dof i=1 w s i (t)Ψ i (x, y) (19)
where N dof is the number of degrees-of-freedom (dof) in plate discretization. The corresponding vector Ψ(x, y) is of size N dof × 1 and stores the terms Ψ i (x, y). After this reorganization, Equation ( 17) are discretized and projected onto the plate and fluid eigenmode basis. To this end, the terms in Equation ( 17)

are multiplied by Ψ ⊤ (x, y) and integrated on the plate surface S. The resulting system is written in the matrix form

ż(t) = Dz(t), where D = 0 Id -M -1 K -M -1 C , z(t) = x(t) ẋ(t) ( 20
)
where Id is the identity matrix of rank N dof /2 and

x(t) = w s (t) w f (t) ; M = M s1 M f1 M s2 M f2 ; C = 0 0 C s2 C f2 ; K = K s1 0 K s2 0 . ( 21
)
The displacement contributions of the solid are stored in w s and those of the fluid, in w f . Each matrix of Equation ( 21) is written as follows:

M s1 = hρ s S (1 -ϕ(x, y))ΨΨ ⊤ dxdy, M s2 = ρ f S (1 -α ∞ (x, y))ΨΨ ⊤ dxdy, M f1 = hρ f S ϕ(x, y)ΨΨ ⊤ dxdy, M f2 = ρ f S α ∞ (x, y)ΨΨ ⊤ dxdy, C s2 = -ς S ΨΨ ⊤ dxdy, C f2 = ς S ΨΨ ⊤ dxdy, K s1 = S D(x, y)∇ 4 ΨΨ ⊤ dxdy K s2 = K f S ∇ 2 ΨΨ ⊤ dxdy. ( 22
)
An analytical solution of the equation of motion is possible in the state space. Solutions for the ith mode take the form of eigenvalues λ i = β i ± jγ i where β i = -ζ i /ω i is the damping term involved into the exponential decrease of the mode due to viscous friction and ζ i is the modal damping ratio. The imaginary part γ i = ω i 1 -ζ 2 i corresponds to the natural frequency, where ω i is the undamped angular frequency. The loss factor is related to ζ i by η i = 2ζ i .

In order to explore the influence of the spatial distribution of the perforation on the added damping3 , the loss factor of the first mode η 1 is plotted as a function of the global perforation ratio in Figure 7. In this figure, three spatial distributions are considered. The first configuration consists in perforations homogeneously distributed on the MPP. It corresponds to the reference MPP, i.e. Ih(x, y) = 1. In the second configuration, the perforations are distributed according to the shape of the first plate mode without perforation thus Ih(x, y) = sin(πx/L x ) sin(πy/L y ). The normalized inhomogeneity function is equal to 1 at the maximum plate deflection. The spatial perforation ratio is defined via Equation [START_REF] Hoshi | Implementation experiment of a honeycomb-backed MPP sound absorber in a meeting room[END_REF]. Finally, the third configuration consists in perforations distributed according to the step function

Ih(x, y) = Π(x, y) = 1 if 1 4 ≤ x Lx ≤ 3 4 and 1 4 ≤ y Ly ≤ 3 4 , 0 else. ( 23
)
As mentioned previously, the additional damping exhibited by MPP is closely related to the viscous friction in the boundary layers of MPP, which is more important the higher the relative velocity between the fluid and the structure. Adapted from [START_REF] Biot | Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher frequency range[END_REF] for an MPP with inhomogeneous spatial distribution of perforation under low frequency assumptions, the viscous friction force can be recast as

f v (x, y, t) = 32µ f ϕ(x, y) d 2 ( ẇf (x, y, t) -ẇs (x, y, t)). (24)
The larger the velocity difference, the greater the friction force. From the discretization in Equation ( 19), Equation ( 24) reads

f v (x, y, t) = 32µ f ϕ(x, y) d 2 N dof i Ψ i (x, y)( ẇf i (t) -ẇs i (t)). (25)
For a given mode, maximizing Ψ i (x, y) leads to maximizing the friction force and thus the additional damping of the MPP. Values of x and y for which Ψ i (x, y) reaches a maximum can be obtained depend on the size of the plate and on the boundary conditions involved in defining the eigenfunction. For a given d, the viscous force and, thus, the additional damping induced by the microperforations, is maximal for the ith mode when x = x max is the solution of ∂ x Ψ i (x, y) = 0 and when y = y max is the solution of ∂ y Ψ i (x, y) = 0. The added damping increases with the perforation ratio. It is therefore possible to increase the added damping by increasing the perforation ratio around the point (x max , y max ). However, the concentration of perforations on the zones of interest, i.e. presenting maximum deflection amplitudes, will have a lower mechanical resistance, due to the modification of the Young's modulus. In addition, the more abrupt the change, the greater the stress concentrations at the point of failure, which reduces the mechanical strength of the structure.

Moreover, the spatial distribution of the perforation has an influence on the Young modulus of the structural part and thus on the MPP stiffness. For the first MPP mode of the three MPP represented in Figure 7, the MPP stiffness K 11 s1 , where exponent 11 corresponds to the term's position in the stiffness matrix, given in Equation [START_REF] Ouisse | Patch transfer functions as a tool to couple linear acoustic problems[END_REF], is plotted versus ϕ g in Figure 8. For the same ϕ g , the plate stiffness decreases significantly when the concentration of perforations is shifted to the maximum deflection area of the plate.

Experimental validation

In this section, the experimental validation is performed on the Oberst test bench detailed in Section 3.2. The tested MPP samples are presented in Figure 9(a) and measure 131 mm × 30.7 mm × 1.08 mm. The overall perforation ratio and the diameter of the perforations are constant for the three plates and fixed at ϕ g = 10 % and d = 1 mm, respectively. For MPP 4 ○ the perforations are homogeneously distributed over the structure. In this validation section, MPP 4 ○ acts as the reference MPP. For MPP 5 ○, the perforation ratio is chosen as a linear function of x with a maximum at x = L x (maximum plate deflection for the first mode) such that ϕ(x) = ϕ (0) x/L x with ϕ (0) = 20 %. For MPP 6 ○, the perforation ratio is defined by

ϕ(x) = ϕ (0) Π(x) with Π(x) 1 if x Lx ≥ 1 2 , 0 else. ( 26
)
where ϕ (0) = 20 %. All perforation parameters and Ih(x) homogeneity functions for the three MPP of Figure 9(a) previously presented are reported in Figure 9(b). The experimental results are presented in Figure 10 and Table 4. In Figure 10, the plate mobilities for the first mode of MPP 5 ○ and MPP 6 ○ are compared to those obtained for the reference MPP 4 ○ and non-perforated plate. On the one hand, an amplitude reduction of about 8 dB is observed with MPP 5 ○ and MPP 6 ○ compared to the case of the non-perforated plate. On the other hand, the amplitude reduction between MPP with spatial distribution (MPP 5 ○ and MPP 6 ○) and the reference MPP 4 ○ is about 3 dB. The resonance frequency is shifted due to the reduction in stiffness, which is successively higher for MPP 6 ○, MPP 5 ○ and MPP 4 ○. Finally, the good comparisons between the measurements and theoretical predictions shown in Figure 10, for MPP 5 ○ and MPP 6 ○, validate the analytical model presented in Section 4.1.

From the mobilities measured, the loss factor is calculated at the first mode i = 1. The results are given in Table 4 for the studied MPP and reference non-perforated plate. In addition, the analytical model of Section 4.1 was also considered for calculating the loss factor of the first mode for the three MPP. One can observe that the analytical values agree well with the measurements. Also, for the same overall perforation ratio, distributing the perforations over the maximum deflection areas of the MPP increases the added damping by 62 % over the reference MPP 4 ○ and 133 % over the non-perforated plate. To conclude this section, we have used an MPP whose microperforations are distributed along the vibrational antinodes of the modes we wish to damp. The proposed analytical model is validated by experimental measurements and shows that an MPP with distributed perforations maximizes the added damping compared to a uniform MPP. This is because the added damping is linked to the dissipation induced in the boundary layers that support the motion. The greater the deflection of the plate, the more viscous friction is present, thus increasing damping. To achieve damping in a given mode, it is therefore advisable to distribute the perforations over its vibratory antinodes. However, a large concentration of Measured Analytical

f 1 (Hz) η 1 [×10 -3 ] f 1 (Hz) η 1 [×10 -3 ]
Reference non-perforated plate 1 ○ 47.9 ± 0.36 0.9 ± 0.073 ○ and the three MPP presented in Figure 9(a) and perforation parameters are given in Table 2. Theoretical results obtained though the model proposed in Section 4.1 provided for i = 1.

perforations in a small area can generate a stress concentration that could reduce the mechanical strength of the structure. This aspect must be carefully considered when designing the MPP.

MPP: Nonuniform distribution and multiple diameters

The aim of this section is to combine the two effects presented in Sections 3 and 4. Recall that Section 3 proposes to increase the frequency range of effective damping by employing multiple perforation diameters, and Section 4 suggests increasing the maximum added damping by using a perforation distribution according to modal deflections. It is therefore proposed to explore the damping capabilities of the MPP with a spatial distribution of multi-sized perforations. The idea is to design an MPP that improves the frequency range of the effective damping while maximizing the added damping on the frequencies of interest, i.e. the resonance frequencies of the structure.

Here, both analytical models exposed in Sections 3 and 4 are combined to obtain the dynamic response of an MPP involving space-dependent distributions of multi-size perforations. The MPP is assumed to be equivalent to N independent MPP, each with its own perforation diameter d k , perforation ratio ϕ (0) k , and inhomogeneity function Π k , k = 1, 2, . . . , N . Based on a homogenization model similar to the one presented in Section 3, the equations of motion generalize to (time and space dependencies are dropped out for the purpose of readability)

h(ρ s (1 -ϕ(x, y)) ẅs + ρ f ϕ(x, y) ẅf ) + D(x, y)∇ 4 w s = 0 (27a) ρ f ((1 -α ∞ (x, y)) ẅs + α ∞ (x, y) ẅf ) + σ(x, y)ϕ(x, y)( ẇf -ẇs ) + K f ∇ 2 w s = 0 (27b) with 1 σ(x, y) = N k=1 1 σ k (x, y) . (28)
The spatial bending coefficient D(x, y) is expressed in terms of the global spatial perforation ratio ϕ(x, y) defined through Equation ( 8) applied to ϕ k (x, y). The global spatial resistivity σ(x, y) is defined from

f 1 (Hz) η 1 [×10 -3 ] f 2 (Hz) η 2 [×10 -3 ]
Measured reference non-perforated plate ○ presented in Figure 11. Each loss factor is determined for the ith mode. Perforation are described in Equation [START_REF] Wang | Investigation of the effect of perforated sheath on thermal-flow characteristics over a gas turbine reverse-flow combustor-Part 2: Computational analysis[END_REF]. mobility calculated for the MPP. Table 5 summarizes the main comparison data. On the one hand, the measured results show a magnitude reduction of approximately 12 dB for the first resonance frequency, while the magnitude reduction is approximately 10 dB for the second resonance frequency. On the other hand, the loss factor is nearly doubled at both resonances in the case of MPP 7 ○ compared to the case of the non-perforated plate. Finally, the predicted theoretical results for MPP 7 ○ are in close agreement with the measurements for MPP 7 ○. These results seem to show that the implementation of multi-size microperforations combined with inhomogeneous spatial distributions of perforations has two advantages: (1) depending on the diameters chosen, the added damping is effective over a wider frequency band compared to a plate with a single set of perforations; (2) the well-chosen distribution of the perforations over certain areas of interest also maximizes the added damping.

Conclusion

This article extends the theoretical model of reference [START_REF] Gallerand | Vibration and damping analysis of a thin finite-size microperforated plate[END_REF] to study the added damping effect exhibited by microperforated plates (MPP) embedding: (i) multi-size perforation diameters; (ii) spatial distributions of perforation, and (iii) a combination of (i) and (ii). For this purpose, a homogenization approach was proposed in the context of MPP with multi-size perforations. Then, to consider an MPP with a spatial distribution of microperforations, the perforation ratio was introduced as a space-dependent function. Finally, these two models were combined. Experimental measurements were performed to validate the analytical models. The results provide evidence that:

(i) MPP with multi-size perforations can broaden the frequency band of the effective added damping. When the perforation diameters are chosen so that each characteristic frequency coincides with a resonance frequency of the MPP, the frequency band of the effective damping is extended.

(ii) MPP with spatial distribution of perforations can maximize the added damping on a given mode. Indeed, the distribution of perforations around the antinodes of the considered mode maximizes the added damping compared to a homogeneous MPP.

(iii) The damping effects of the multi-size perforations and the spatial distribution of perforations can be cumulated. The combination of multiple perforation sizes with inhomogeneous spatial distribution of perforations had the two main advantages. The first is to expand the range of frequencies in which the additional damping is effective. The second is to optimize the amount of added damping across a carefully chosen set of frequencies.

In practice, the perforations could be distributed around the vibration antinodes of the plate modeshapes that need to be dampened.

CRediT authorship contribution statement

Supplementary material

Python scripts used to perform the modal analysis presented in this article are available in commit 292f7864 of the [Git repository] Inhomogeneous Perforated Plate

Figure 1 :

 1 Figure 1: System of coordinates for an equivalent solid plate excited by an external mechanical force along z-axis.

Figure 2 :

 2 Figure 2: Predicted mobility level of aluminum plates of dimensions 490 mm × 570 mm × 1 mm: ( ) reference non-perforated plate, () MPP with a perforation diameter, denoted d1, set to induce maximum added damping in the first MPP mode, and () MPP with the same equivalent stiffness as the reference non-perforated plate and with d ≡ d1 maximizing the added damping on the first MPP mode. Perforation parameters are d1 = 1.6 mm and ϕ (0) = 10 %. Isotropic structural loss factor of aluminum ηs = 10 -4 or 0.01 % is also considered.

k . 2 Figure 3 :

 23 Figure 3: MPP with two groups of perforations with two perforation diameters d1 and d2. The MPP shown in (a) is decomposed into two equivalent MPP, each associated with a perforation diameter and ratio, as illustrated in (b). The two MPP in (b) are considered as the sum of two pairs of equivalent homogeneous plates shown in (c): an equivalent homogeneous fluid plate and an equivalent homogeneous solid plate. Each pair is related to the corresponding d k and ϕ (0) k with k = 1 or 2.

Figure 4 :

 4 Figure 4: Loss factor for two modes. (a) first mode. (b) second mode. ( ) d1 set to maximize added damping on the first mode with ϕ (0) = 10 %; ( ) d2 set to maximize the added damping on the second mode with ϕ (0) = 10 %; ( ) combination of d1 and d2 with associated perforation ratios ϕ (0) 1 = 3 % and ϕ (0) 2 = 7 % where the total perforation ratio, defined from Equation (8), is ϕ (0) tot = ϕ (0) 1 + ϕ (0) 2 = 10 %.

1 ○

 1 and MPP 2 ○ have an equal number of perforations but d 1 > d 2 and thus ϕ

Figure 5 :Figure 6 :

 56 Figure 5: Test bench in (a) and MPP sample in (b) used in the experimental validation of the model presented in Section 3.1. The zoom (c) corresponds to a focus on the perforations of (b).

[×10 - 3 ]Figure 7 :

 37 Figure 7: Loss factor for the first mode of a 570 mm × 490 mm × 1 mm simply-supported aluminum MPP with d = 1.4 mm: ( ) uniform distribution with Ih(x, y) = 1; ( ) distribution along plate mode shape according x with Ih(x, y) = sin(πx/Lx) sin(πy/Ly); ( ) distribution according to the rectangular function Ih(x, y) = Π(x, y) defined in Equation (23).

11 s1Figure 8 :

 118 Figure 8: Equivalent solid plate stiffness K 11 s1 for the first mode of a 570 mm × 490 mm × 1 mm simply-supported aluminum MPP with d = 1.4 mm as a function of the global perforation ratio ϕg for three different spatial perforation distributions: ( ) uniform distribution with Ih(x, y) = 1; ( ) distribution along plate mode shape according x with Ih(x, y) = sin(πx/Lx) sin(πy/Ly); () distribution according to the rectangular function Ih(x, y) = Π(x, y) defined in Equation[START_REF] Putra | The effect of perforation on the dynamics of a flexible panel[END_REF].

Figure 9 :

 9 Figure 9: Tested MPP samples with various space-dependent distributions of the perforation ratio. (a) -MPP 4○ homogeneous distribution: reference MPP, MPP5 ○ linear distribution, MPP 6 ○ concentrated distribution located at the maximum of the deflection amplitude. (b) -Perforation parameters and homogeneity function Ih(x) for the three MPP presented in Figure9(a). Perforation diameter is set to d = 1 mm in order to induce maximum damping around the first mode (i.e. i = 1) and global perforation ratio ϕg = 10 % identical for all three MPP.

Figure 10 :

 10 Figure 10: Theoretical and experimental mobilities around mode 1 of (a) MPP 5 ○ and (b) MPP 6 ○. Comparisons between ( ) measured and ( ) theoretical. In (a) and (b), the measured mobilities of the reference MPP 4 ○ ( ) and non-perforated plate ( ) are also provided. MPP perforation parameters are given in Figure 9(b).

Lucie Gallerand :

 Gallerand Conceptualization, Methodology, Software, Validation, Writing -original draft, Writing -review & editing. Mathias Legrand: Conceptualization, Methodology, Supervision, Writing -original draft, review & editing. Thomas Dupont: Conceptualization, Methodology, Supervision, Resources, Funding acquisition, Writing -original draft, review & editing. Raymond Panneton: Conceptualization, Methodology, Supervision, Writing -original draft, review & editing. Philippe Leclaire: Conceptualization, Methodology, Supervision, Writing -original draft, review & editing.

Table 1 :

 1 Modal loss factor ηmn and mobility reduction ∆Ymn for mode (m, n) of a simply supported MPP whose mobility is indicated in Figure2. Loss factor of aluminum is set to a constant value of ηs = 0.01%.

	1)	1.06	38
	(1, 2)	0.35	30
	(2, 1)	0.25	27
	(2, 2)	0.12	21

Table 2 :

 2 Measured sample perforation parameters.

Table 3 :

 3 Values of experimental and theoretical loss factor with standard deviation for the three MPP presented in Figure

Table 4 :

 4 Measured with standard deviation and analytical frequencies of the first mode plate and corresponding loss factors for the reference non-perforated plate1

	•	•

Table 5 :

 5 Values of experimental and theoretical loss factor for the MPP7

For a more detailed literature review on the acoustics and vibroacoustics of uniform MPP, the reader is invited to refer to the introduction in[START_REF] Gallerand | Vibration and damping analysis of a thin finite-size microperforated plate[END_REF].

Due to the distribution of perforations along the deformations of the first resonance mode, i.e. i = 1, only the added damping presented by the first mode is taken into account in this section. The same considerations can be made for another mode, i.e. i > 1.

Acknowledgment

This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC: RGPIN-2019-06573) and by Fonds de Recherche du Québec -Nature et Technologies (FRQNT).

the resistivity of each equivalent plate derived from d k and ϕ k (x, y). The global spatial tortuosity is also expressed using Equation [START_REF] Champoux | On acoustical models for sound propagation in rigid frame porous materials and the influence of shape factors[END_REF] applied to the spatial perforation ratio ϕ k (x, k) and the perforation diameter of the subdomain k.

The validation of the generalized model in Equation ( 27) is now exposed. To this aim, two plates of dimension 195 mm × 30.7 mm × 1.17 mm are considered: a non-perforated reference plate 2 ○ and MPP 7 ○ with multiple perforation diameters spatially distributed along a non-homogeneous pattern. To maximize the damping added by the microperforations on modes 1 and 2, two diameters of perforations are chosen: d 1 = 1.3 mm and d 2 = 0.7 mm, corresponding to characteristic frequencies f c1 ≈ 27 Hz and f c2 ≈ 114 Hz, respectively. MPP 7 ○ which is presented in Figure 11 is decomposed into three equivalent plates with respective perforation ratios and inhomogeneity functions defined as:

and ϕ

The ratio ϕ 1 (x) is associated with perforations of diameter, d 1 while ϕ 2 (x) and ϕ 3 (x) are related to perforations of diameter d 2 . ○ is considered as three equivalent MPP inhomogeneous spacial distribution of perforations defined in Equation [START_REF] Wang | Investigation of the effect of perforated sheath on thermal-flow characteristics over a gas turbine reverse-flow combustor-Part 2: Computational analysis[END_REF]. Spacial distribution (29a) associated with d1 = 1.3 mm while spacial distributions (29b) and (29c) are related to d2 = 0.7 mm. Loss factor and Young's modulus used in the theoretical model are obtained experimentally from measurements on the non-perforated plate.

The test bench described in Section 3.2 is used to obtain the MPP mobility. In Figure 12, the mobility measured for MPP 7 ○ is compared to that of the non-perforated reference plate 2 ○ and the analytical


James Costa
email: james.costa@ens-lsh.fr

Patricia Lambert
email: patricia.lambert@ens-lsh.fr

France and Language(s): Old Policies and New Challenges in Education. Towards a Renewed Framework?

This article is concerned with the way the French education system deals with new challenges in terms of language, while referring to the (traditional) general framework of language planning in the country. The same principles and mindset that governed the way regional minority languages were treated in the past are now largely determining the way the languages of immigrants and their speakers are treated and considered.

We thus seek to remind readers how the theoretical and practical background to language planning in education functions in France, showing how historical factors led to considering linguistic diversity as an unnecessary heritage rather than as an asset. We then move on to examine the state of language teaching generally speaking in the education system, in order to provide a general outlook on the subject.

Finally, we show that albeit seemingly monolithic at first, the French system can also accommodate diversity to a certain extent, through looking at two particular contexts.

Introduction

The traditional view of France, in terms of language, is that of a monolingual country. In fact, very few people outside France know that over 70 languages are currently listed as Languages of France [START_REF] Cerquiglini | Les langues de France: Rapport au Ministre de l'Education Nationale, de la Recherche et de la Technologie[END_REF]). Yet, none of them are in any way recognised as official in any part of the French territory. Education was long seen as one of the main instruments to implement the desired monolingualism (Martel, 2007a) in a country where French only became a language spoken by the entire population by the middle of the 20 th century.

Yet, no ideology, however potent and ancient, is monolithic. The official language policy in France has undergone considerable change over the past few years, and this has had repercussions in terms of language education policies, the focus of this article.

The situation is in fact largely one of tension, or stress, between conflicting aspects and demands from different segments of the French population regarding language. While the dominant and officialyet in many ways unspokendogma might still be one asserting the superiority of the French language and the necessity to disregard minority languages, whether indigenous or immigrant languages, plurilingualism is officially valued in the education system, in a recontextualisation of the Council of Europe's discourse, and all pupils are required to study at least two foreign languages. The French position remains equally ambiguous regarding the status and position of English in France and in the education system. This article will thus seek to situate the debate around language education policies in its historical context, and to make more explicit the tensions we referred to above, by identifying the most salient ones, but also by showing that the system itself is not as monolithic as could be thought from the outside. Two examples will be analysed for this purpose, one institutionally supported, the other one on the margins of the Institution.

In this article, the terms regional minority language, regional language, minority language are used interchangeably.

France, language, and language policy

Shohamy has recently defined language policy as "a manipulative tool in the continuous battle between different ideologies" (2006, p. 46). She adds that "these manipulations occur on a number of levels and in a number of directions but especially in relation to the legitimacy of using and learning certain language(s) [...] in given contexts and societies [...]" (p. 46).

Her use of the concept of ideology is of particular relevance here, since France was defined in ideological terms long before it came to be defined in terms of practice. Erasure, one of the mechanisms in ideology formation identified by [START_REF] Gal | The boundaries of languages and disciplines: How ideologies construct difference[END_REF] is of particular importance in France: for the French nation to come into existence, a large part of its history and diversity was to be reinterpreted and redefined as non-important or even non existing.

According to Shohamy (2006, p. 76), "language education policy (LEP) refers to a mechanism used to create de facto language practices in educational institutions, especially in centralized educational systems. LEP is considered a form of imposition and manipulation of language policy as it is used by those in authority to turn ideology into practice through formal education".

These definitions enable us to define our area of investigation for this article. They point to the fact that dealing with language policies is not an innocent exercise: such an object of investigation deals with the founding ideological principles of nation-states.

The questioning of language policies for research purposes must lead to the questioning of national ideologies, i.e. the very beliefs at the core of national policies. Among the many aspects of ideology, we suggest the following formulation as a basis for our presentation here:

"On the one hand, ideology is no mere set of abstract doctrines but the stuff which makes us uniquely what we are, constitutive of our very identities; on the other hand, , it presents itself as an "Everybody knows that", a kind of anonymous universal truth" (Eagleton, 1991, p. 20).

We ought, of course, to be wary of universal truths, especially when questioning them. Universal truths such as "French is the language of France", or "French people are bad at languages", or even "the system cannot be changed, this is the way it is" inevitably point to ideological attitudes. Those views are both influenced by national language policies and influences on those same polices. They can be found in the discourse of media, in textbooks, on the street or, obviously, in schools. Other similar views can be found in official texts and documents. Language policies thus have an official, explicit, aspect, as well as an implicit, unofficial one. We will try to briefly develop an analysis of both those aspects.

Official policy: a historical approach

The Ordinance of Villers-Cotterêts ( 1539) is usually believed to mark the beginning of the making of French as an official language for administrative purposes throughout the kingdom of France [START_REF] Balibar | L'institution du français : Essai sur le colinguisme des Carolingiens à la République[END_REF], although there is some debate around the question. This Ordinance is still referred to, even today, in the media or by politicians who wish to oppose any pro-regional language policy. Yet it is not believed to have altered significantly the daily life of the inhabitants of the kingdom: clerks had already began to use some French in their official documents, even in the Occitan-speaking lands of the South, where Occitan retained prestige for some time [START_REF] Judge | Linguistic Policies and the Survival of Regional Languages in France and Britain[END_REF]. The populations continued speaking their local vernaculars.

The situation was to be altered dramatically at the time of the French Revolution beginning in 1789. If, at first, the new established powers sought to use local languages to communicate with the population [START_REF] Martel | Les « patois » pendant la période révolutionnaire : recherches sur le cas occitan Mots[END_REF], after the Terror in 1793, this was ended and French was to be made the sole official language in France. In 1790, Grégoire's survey had shown that only three million people, out of a total population of fifteen million, could speak French fluently. Grégoire's survey was conducted in order to justify the elimination of the various vernaculars in competition with the central norm, which was to become the only legitimate variety of speech in what was to become the French Republic. In 1794, Barère is famous for having stated, in a report to the revolutionary Comité de Salut Public:

"The voice of federalism and of superstition speaks Breton; the émigrés and those who hate the Republic speak German. The counter-revolution speaks Italian; fanaticism speaks Basque. Let us smash these instruments of damage and error... For our part we owe it to our citizens, we owe it to our republic, in order to strengthen it, that everyone on its territory is made to speak the language of the Declaration of the Rights of Man" (quoted in Judge, 2007, p. 22).

A law was subsequently passed on 20 July 1794 prohibiting the use of any other language but French for official use and official documents [START_REF] Encrevé | La langue de la République[END_REF].

Yet, those measures still did not really affect the population in its daily life. Indeed, in 1835, an 1835 study (quoted in [START_REF] Weber | Peasants into Frenchmen. The Modernization of Rural France, 1870-1914[END_REF] shows that only a handful of départements (the new revolutionary administrative unit) were fully French-speaking (although it is not exactly clear what was meant then by this), all located around Paris and North-western France. In 1863, a survey conducted by Duruy, the Minister for Instruction in Napoleon III's government, and analysed in Weber (1977, pp. 498-501) shows that out of 30 million inhabitants, about 7.5 million were monolingual in a local vernacular. Those were to be found mostly in Brittany, Corsica, Occitan-speaking areas, the Basque Country, Catalonia and Alsace.

The 1870 defeat against Prussia, the advent of the Third Republic, and compulsory schooling were to accelerate the spread of French. The Great War gave regional languages a final blow, and by 1920 most parents would be speaking French, or a regional form of French, to their children. Today, regional language transmission in the homes is a very rare phenomenon, although it seems to have persisted longer in Corsica, Alsace, the Basque Country and some parts of Brittany and Bearn, in South-West France.

It must be noted that bilingualism was never considered a serious option, and French was iconically connected with France and Frenchness as from the 19 th century and Michelet's monumental work on the history of France [START_REF] Encrevé | La langue de la République[END_REF]. The Alsatian case is slightly different, due to the fact that the Province, where a Germanic dialect is spoken, was part of Germany between 1870and 1918, and then again between 1940and 1945(cf. Tabouret-Keller and Luckel, 1981). In today's overseas territories, the situation is also radically different and many languages are still spoken on a daily basis in French Polynesia, French Guyana, the West-Indies and Reunion Island (cf. [START_REF] Cerquiglini | Les langues de France[END_REF].

Language in education policy

In terms of education policy, things were clear from the very beginning. At the time of the Revolution a schoolmaster was to be appointed in every village to teach the French language, but this was in fact never enacted. If it was generally understood that education was to be given in French, masters are known to have used the local vernaculars in various locations and on many occasions [START_REF] Martel | L'école française et l'occitan -Le sourd et le bègue[END_REF]). Yet, in 1870, Gaidoz, Charencey and de Gaulle (the General's great uncle) sent a petition in favour of the acceptation of local idioms in schools to the National Assembly. The arguments used then are still used today: bilingualism was presented as an intellectual asset, citizens were thought to deserve equal respect disregarding what language they spoke, and local languages were presented as bridges towards related languages across national borders.

The 1870 French defeat made it impossible for the petition to even be considered. Times had changed, and revenge on Germany was to become a priority. It was then out of question to promote languages which could be used to communicate with neighbours which could all be seen as potential enemies. Local languages could only be seen as a threat, which in fact comforted the arguments given at the time of the Revolution. Even today, pro-French language discourses frame regional languages as an inside enemy in the struggle against English.

When the famous 1882 Jules Ferry school laws were passed, no mention whatsoever was made of languages other than French. The question was obviously not on the cards, and languages other than French were seen as non-existent. In fact, Colonisation rendered the question obsolete, and politicians had their minds now set upon other questions in terms of language.

Several debates took place in Parliament around the question of regional languages and education [START_REF] Martel | Le "patois à l'école"? Retour sur un débat (XIXème-XXème)[END_REF], to no avail, until 1925 when the minister in charge of education, A. de Monzie, ordered that only French be used in all schools. Monolingualismand monolingualism in the legitimate normwas seen as the only acceptable choice.

It was not until 1951 that a bill was passed in Parliament authorising Occitan, Breton, Catalan and Basque to be taught in schools as an optional subject outside normal school hours by voluntary teachers [START_REF] Martel | L'école française et l'occitan -Le sourd et le bègue[END_REF].

The situation has moved on, and regional languages can now officially be taught as part of the curriculum, and some bilingual primary schools exist in parts of the country, as we will see later in this article. Regional languages can no longer be considered a threat to the supremacy of French, although it can be argued that they never were. In fact, they were used instrumentally to promote a certain vision of France as a homogeneous country. In France, "the search for self-identification led to a reification of France itself as a natural and indivisible entity" (Blommaert and Verschueren, 1998b, p. 197).

Non official discourses

The effects of the dominantmonolingual and centralistideology is reflected in the dominant medias to this day, and in the dominant public discourse. It is common in the discourse of teachers [START_REF] Lambert | Les répertoires plurilectaux de jeunes filles d'un lycée professionnel. Une approche sociolinguistique ethnographique Thèse de doctorat de sciences du langage sous la direction de Jacqueline Billiez[END_REF], and it is also to be found in schoolbooks.

In a study conducted in 2009, we analysed several history and geography books designed for teaching those subject at secondary school level (4 e classes, pupils between 13-14 years old), as well as the official programmes to which they referred, and we found that despite a general discourses in favour of individual plurilingualism, societal multilingualism was neither perceived nor presented as a desirable option.

Language diversity is still presented as a potential source of problems and violence and is constantly presented alongside the religious question in Europe, thus contributing even more to the association of diversity with tension.

Present day manuals still illustrate what Blommaert & Verschueren (1998a, 1998b) call the dogma of homogeneism, i.e.:

A view of society in which differences are seen as dangerous and centrifugal and in which the "best" society is suggested to be one without intergroup differences. In other words, the ideal model of society is monolingual, monoethnic, monoreligious, monoideological (Blommaert and Verschueren, 1998b, p. 195).

Schoolbooks, as well as the official syllabus, are still very much marked by the dominant ideologies which they tend to reproduce, despite claims to the contrary. In fact, the situation seems to have changed very little since the 1980s [START_REF] Martel | Los noveus manuals d'istòria de terminala e la question regionala[END_REF].

Regional languages of France are now recognised in the Constitution as a part of the national heritage, which incidentally does not grant their speakers any specific right to use their language in public. Regional languages are by now almost totally gone as means of everyday communication [START_REF] Héran | La dynamique des langues en France au fil du XXe siècle[END_REF], recognising them symbolically is thus, in practical terms, of limited importance.

Languages in the current education system

So far, we have discussed France and language. The general attitude to language in France does of course determine, to a certain extent, the way in which languages are perceived and conceived of. Many other elements would nevertheless need to be taken into account and the way in which languages are treated in the education system responds to conflicting logics. There is indeed a tension between the imagining of France as a monolingual nation and the necessity to teach foreign languages on the one hand, and the necessity to take immigration languages on the other, particularly in the context of a reframing of the dominant discourse on the French language itself, which now tends to value linguistic diversity as a desirable, yet abstract, state of affairs. Romance languages, English, regional minority languages, "rare languages", as the system calls languages such as Russian, Chinese, Arabic or even Portuguese, are all seen in different ways according to what stakes they convey and to what ideological positions they refer to. English is both the arch-enemy, and also a most desirable language to possess in one's linguistic repertoire. The former Minister for Education even declared in September 2008 that he wanted every pupil to become bilingual in English. The ideologies surrounding languages are clearly becoming more and more complex, and vary according to the situation in which they are expressed and the people who voice them.

School is obviously one of the most potent instruments of language policy, and has been used to redefine legitimacy and authority in terms of language, and to reframe identity in a way that suited the Central government [START_REF] Jaffe | Ideologies in Action: Language Politics on Corsica[END_REF][START_REF] Jaffe | Authority and Authenticity: Corsican Discourse on Bilingual Education[END_REF]. Education thus continues to be the principle medium to reproduce the dominant ideologies, yet at the same time it is a site where contradictory discourses and tensions are to be found.

Assets

As can be read in the 2008 Eurydice Network report, "Since the beginning of the 2007/08 school year, it has in principle become compulsory for all pupils aged 7 to learn a foreign language. At 14, only the pupils who have taken the option "decouverte professionnelle" (initiation to professional life) (6 hours per week), no longer learn two foreign languages as compulsory subject" (Eurydice Network, 2008, p. 30).

Languages are thus at the very core of the French education system, it can be said, and the Common Base for Knowledge and Skills requires that a foreign language must be mastered by all pupils at the end of compulsory education (cf. [START_REF] Coquidé | The Common Base for Knowledge and Skills in France: debates over the concept of "competence" and other obstacles[END_REF]. It must be added that there are a host of optional languages which can also be studied, in addition to the compulsory ones: classical languages such as Latin and Greek, regional minority languages (Basque, Breton, Catalan, Corsican, Creole languages, Tahitian, Occitanin all its regional varieties -, as well as German in Alsace and Flemish in the areas of Northern France where it is traditionally spoken).

Yet, it must be said that in fact the vast majority of pupils study English as a first language, and all must study English at some stage during their compulsory school years. This is both a result of the existence of a utilitarian ideology which views languages primarily as assets in terms of economic success, as well as of the education policy in France which has tended to promote a small number of languages, namely English, German and Spanish. For various reasons, German is declining steadily, except in Alsace, and Spanish seems to be mainly chosen as a second language. In fact, most parents and pupils demand English as the first foreign language, and this trend is more and more obvious.

As far as regional languages are concerned, over 400,000 pupils follow some form of teaching in or of a regional language (cf. [START_REF] Costa | Langues et cultures régionales, langues vivantes étrangères. Quels enjeux pour une convergence didactique?[END_REF]. While this might seem an important figure, in covers a wide range of situations, which might include bilingual classes as well as classes where a song might occasionally be learnt in a regional language. Also, conditions are still far from ideal, and many teachers in secondary education must still teach during lunch hours, as the system is clearly not designed to accommodate such a wide variety of situations.

A plan is currently being implemented to promote German in the education system, which includes an offer in terms of German language in schools in all Académies (the educational administrative divisions in France), and an increase of 20% within five year of the total number of pupils studying German at primary school level. The language should also be offered in secondary education in all locations where it can be studied in primary schools, and it will be increasingly possible to study both German and English in the first year of secondary education1 . The whole education system is undergoing change as regards language. A Plan de Rénovation des Langues was set up in 2005 in order to develop skills in foreign languages and to introduce the Common European Framework of Reference for Languages into the system. This is still taking place as we write, and major changes are being implemented in the way foreign languages are taught in schools.

Tensions

As has by now become apparent, a large number of languages are present in the French education system, illustrating a wide panel of sociolinguistic situations. While English is the dominant foreign language throughout the system and throughout most of the territory, regional languages are still present, although clearly not a priority. For political reasons, German is promoted at all levels and Spanish has a well-established niche as a second foreign language for most pupils (Eurydice Network, 2008).

So far both foreign languages as well as regional minority languages have been mentioned. One (highly heterogeneous) group of languages has been strikingly absent, i.e. languages spoken by immigrants and their descendents. Over 400 languages are spoken in France, as was found in a 1999 survey [START_REF] Héran | La dynamique des langues en France au fil du XXe siècle[END_REF], and among them, Arabic, Portuguese, Spanish, Italian as well as some Bantu languages and other Asian and African languages are being transmitted to the younger generations. They are conspicuously absent from the education system. It is a fact that Arabic is offered as a foreign language in some secondary schools, but in forms most often quite dissimilar to the ones found in the pupils' repertoires [START_REF] Billiez | Contacts de langues à l'école: disjonctions et tentative de raccordements[END_REF]. Similarly, some immigrant language classes are offered in areas where a demand exists, as part of an ELCO2 programme. Such programmes have led to a series of difficulties and have raised many issues [START_REF] Billiez | Plurilinguisme, variations, insertion scolaire et sociale[END_REF].

As Billiez, et al. point out (2003, p. 301), it is only recently that sociolinguists have begun to study the way plurilectal repertoires were valued and used in educational settings in France: what type of language classes could be implemented? In what type of curriculum? What would be the effects of such measures on the children themselves?

A large amount of research is still needed in this field, especially as competing discourses advocating an all-French approach are still dominant in the media as well as among teachers. In fact, the education system still functions with two basic assumptions:

"the integration assumptionthat is, the assumption that multilingualism is an obstacle for societal and national integration into a coherent nation-state. [...] The second assumption could be called the efficiency assumptionthat is, the assumption that efficient government, as well as economic growth and development, are hampered by multilingualism" (Blommaert and Verschueren, 1998b, p. 206).

There is therefore a deep rooted tension between a system which is opening up to European multilingualism, on paper at least, and the plurilingual reality of the country, in conjunction with an underlying assumption which still furthers integration as assimilation.

Challenges: towards a renewed system?

The French education system, like all systems, is however not monolithic, far from it. Many changes have occurred in the past few years as regards language and languages, and despite the fact that old ideological reflexes are still dominant, a large amount of variation does exist, and the dominant model is also being challenged, both from within and as a result of greater European integration. As Shohamy put it, "Yet, at times, LEP [Language Education Policy] is also used as a bottom-up, grassroots mechanism to nogociate, demand and introduce alternative language policies" (2006, p. 76).

We shall now examine two examples of such attempts to establish alternative language policies.

Bilingual education

Although by bilingual education we mean a system which integrates two languages as both object and medium of education, which would include, in France, several types of experimentations (such as European and International Sections, where one academic subject is partially taught in a foreign language), we will concentrate here on a form of bilingualism which gives both languages equal representation in terms of time. This system is only available for some regional languages, namely Breton, Occitan, Corsican, Basque, Catalan as well as German, considered a regional language in Alsace.

As a result of parents' pressure in the 1970s, and, with the opening of private immersion schools in Brittany, the Basque Country, Northern Catalonia, Languedoc and Aquitaine, the state took action and created its own bilingual primary system, where children are educated in French and in a regional minority language for equal numbers of hours.

The system welcomes an ever-increasing number of pupils: in 2008-20093 , there were over 57,000 pupils involved in bilingual education in France at primary level, 70% of which in the public system, 15% in religious private schools and an equal number in private non-profit immersion schools run by parents.

Although systems vary, there is a tendency, in the public sector, to opt for an organisation whereby the same teacher teaches in both French and the minority language, thus enabling cross-subject work in both languages, as well as a more global approach of language as a phenomenon [START_REF] Cortier | Français et langues régionales et/ou minoritaires: une mise en convergence difficultueuse[END_REF].

In terms of the organisation of the system itself at the level of schools, while Corsica has adopted all-bilingual schools, on the Continent there tends to be one bilingual section among otherwise monolingual schools, which is not without begging questions as to the finality of bilingualism. The chosen approach does not focus on language revitalisation but on the children's cognitive development, although both can be compatible. Such an organisation in terms of language repertoires fits with Candelier's definition of pluralistic approaches: "While "singular" approaches address one particular language or culture taken in isolation, pluralistic approaches are teaching approaches in which the learner works on several languages or cultures simultaneously" (Candelier, 2008, p. 225).

In a way, bilingual education thus constitutes a more and more institutionalised integration of pluralistic approaches. Yet, the system is only really operational for primary education. Bilingual sections do exist in secondary schools in the Occitan-speaking regions, in Brittany, Corsica and elsewhere, but they usually consist in a greater number of hours in the regional language and the teaching of one academic discipline, usually history-geography, through the medium of the minority language.

Bilingual education needs to be analysed "as a component of a wider social economic cultural and political framework" [START_REF] Hélot | Language Policy and the Ideology of Bilingual Education in France[END_REF]. It is yet unknown what consequences the development of this system could bear on language policy in the education system as a whole.

Pluralistic approaches

We gave the definition of pluralistic approaches in the former section, and bilingual education as a potential illustration in some cases where languages are thought together, and not as discrete entities.

Other approaches include the integrated teaching and learning of languages taught (building for instance on the learner"s own language to facilitate access to a first foreign language, or on a first foreign language to facilitate access to a second one [...], the intercomprehension between related languages [...], and, of course, the inter-(or cross-) cultural approach [...] (Candelier, 2008, p. 225), and, most saliently, language awareness programmes. Those approaches, even though they might be part of some teacher's everyday class experience, are by no means institutionalised as such. They nevertheless aim at transcending the problems caused by a dichotomous approach in terms of monolingualism vs. bilingualism, and more generally they seek to explore new ways of teaching and approaching languages as well as language as a phenomenon.

In 2007, the European Centre for Modern Languages accepted an international project as part of its 2008-2011 programme ("empowering language professionals") which aimed to combine intercomprehension approaches and language awareness activities to integrate regional minority languages and other languages present in the children's environments. The project, named EBP-ICI 4 , and to which the authors of this article participate, seeks to identify various interlinguistic strategies which guide intercomprehension processes, at primary and secondary school level. Thus, working with teachers of Occitan in Provence and with partners in the Val d'Aoste, Catalonia and Scotland, we have developed partnerships between schools in which several related languages are spoken or taught, as well as, generally, English (see [START_REF] Cortier | Propositions sociodidactiques face à la diversité des contextes. Vers une didactique convergente des langues collatérales et de proximité : éducation bi/plurilingue et projets interlinguistiques[END_REF] for a more detailed presentation).

Conclusion

As can be seen from the above presentation, at the roots of the French education system stands a deep rooted ideological system which promotes monolingualism as well as one specific social norm of French. A homogeneous vision of society is both sought and promoted through education, but this is the case throughout Europe (Blommaert and Verschueren, 1998b). In this respect, France is no exception.

Yet the dominant ideology is itself not homogeneous, and the structure of power relations between languages, i.e. between their speakers, has evolved over the past 50 years. Regional languages may have almost disappeared from the public scene, yet activists have succeeded in establishing them in the public system of education, although at its margins.

Other initiatives involving pluralistic approaches have been successfully developed over the past 15 years, but they are not part of the main curriculum and show no signs of being accepted by mainstream policy-makers. Those include ways of integrating all the languages present in the pupils' environment into the system, bearing in mind that no child should feel downgraded for the languages they hold in their repertoire.

Experimentation is therefore possible in the French education system, and many others are currently taking place. Yet, one may question their ability to ever become generalised, given the vitality of traditional ideologies and the lack of concern for societal multilingualism in France.

http://eduscol.education.fr/D0156/all-plan-langue.htm?rub=101

Enseignement de Langue et de Culture d'Origine, or Teaching of Language and Culture of Origin. Those programmes are set up between France and the countries of origin of pupils.

http://www.flarep.com/

Education Bi-Plurilingue, Intercompréhension et Compétences Interlinguistiques. See the prject website for more information, in French and English : http://ebp-ici.ecml.at.

About the Authors

James Costa is a researcher at the French National Institute for Pedagogical Research (INRP), where he works in research areas connected with regional minority languages in education in France as well as issues linked with language revitalisation. He was part of the team who, in 2009, completed a report on the sociolinguistic situation of regional minority languages in Rhône-Alpes (the Lyon area in France) and worked on the language policy section of that report, which is available online on the Conseil Régional Rhône-Alpes website. He is currently completing a PhD on the discourses and ideologies of language revitalisation in Provence and Scotland. His research interests range from sociolinguistics and language policy and planning to linguistic anthropology.

Patricia Lambert is a university professor (maître de conferences) in sociolinguistics and language didactics, and a researcher at the National Institute for Pedagogical Research (INRP). She was part of the team that worked on the experimentation and implementation of language awareness materials (Evlang) in 1998-2001. She is currently the coordinator of a research project at the INRP concerned with the sociolinguistics of education, language variation and miltilingualism. Her other research interests include linguistic anthropology, the ethnography of communication and plurilingual situations in relation with questions linked to immigration.