F Dumortier
email: florent.dumortier@insa-lyon.fr

V Meyer

L Maxit

Scattering from a partially coated shell immersed in water using a subtractive modelling technique

Keywords: Vibroacoustic, Subtractive modelling, Acoustic scattering, Numerical methods, Cylindrical shells

In this study, we focus on the prediction of the pressure field scattered from an immersed cylindrical shell partially coated by a soft rubber, impacted by an acoustic plane wave. As the coating covers only a partial portion along the circumference of the shell, the considered system is not axisymmetric. As a result, a spectral (Fourier) resolution of the mathematical problem would induce the coupling of the different circumferential orders, which can lead to prohibitive computing times. To circumvent this issue, the reverse Condensed Transfer Function (rCTF) method has recently been developed to decouple vibroacoustic subsystems initially coupled along lines or surfaces. From an analytical model of the fully coated shell impacted by the acoustic plane wave and a finite element (FE) model of the missing coating material, the rCTF approach predicts the vibroacoustic behavior of the coated shell with a voided section instead of the removed part. This voided section can then be filled by a FE model of the water domain replacing the removed coating material, using the direct CTF approach. The principle of the rCTF approach, some numerical validations, and results for the scattering from the partially coated shell are presented in this paper.

Introduction

Studying the vibroacoustic performances of cylindrical shells finds great interest for industrial applications. They are particularly used in the naval context, in order to model hulls of underwater vehicles. Indeed, acoustic performances of underwater vehicles must be predicted in a large frequency range following the increasing capabilities of Sonar antennas. In order to enhance the vibroacoustic performances of underwater vehicles, acoustic coatings can be applied at the surface of the hull. Those coatings are generally made of soft material and can mainly be divided into two categories. Decoupling coatings are used to reduce the radiation by decoupling the hull from the external fluid medium. On the other hand, anechoic coatings are used to limit the scattering from the hull by absorbing incoming waves thanks to their impedance match with water. [START_REF] Skelton | Theoretical acoustics of underwater structures[END_REF] Over the past decades, a substantive amount of work has been produced to study such technologies and how well they contribute to the mitigation of the noise produced by the structure they are mounted on. Several approaches to model the coating have been investigated, among which we can cite the locally reacting material approach convenient for the simpleness of its implementation. Locally reacting materials have been used to study radiation [START_REF] Laulagnet | Sound radiation from a finite cylindrical shell covered with a compliant layer[END_REF] and scattering 3 from cylindrical shells, but it was observed that this theory is limited in a physical point of view, mainly because it is unable to describe the behavior of the coating in its thickness. A second, more robust, theory is to model the coating using the three-dimensional theory of elasticity. [START_REF] Laulagnet | Sound radiation from finite cylindrical coated shells, by means of asymptotic expansion of three-dimensional equations for coating[END_REF][START_REF] Cuschieri | The modeling of the radiation and response Green's function of a fluid-loaded cylindrical shell with an external compliant layer[END_REF] This approach gives better results as it accounts for both longitudinal and shear waves travelling in the material, with the main drawback of being much more time consuming. A third approach consists in modelling the coating as an equivalent fluid, meaning that only the longitudinal waves are accounted for in the coating. [START_REF] Sharma | Acoustic performance of gratings of cylindrical voids in a soft elastic medium with a steel backing[END_REF] This approach, particularly indicated when the coating is made of soft material, is a good compromise between the two aforementioned approaches, as it has proven to better capture the physical phenomena in the coating than the locally reacting approach, while being less computationally intensive than the three-dimensional theory of elasticity.

While only the case of homogeneous media has been discussed here, heterogeneous coatings also have been studied in literature. In particular, cylindrically layered media can show great interest as they could allow for example overlaying anechoic and decoupling coatings.

A mathematical framework to study cylindrically layered media infinite along the axial direction has been presented by Skelton and James. [START_REF] Skelton | Theoretical acoustics of underwater structures[END_REF] It is based on a spectral approach where the equations of motion of the shell are solved in the wavenumber domain, by the means of a Fourier transform along the length of the shell and a Fourier series decomposition along its circumference. A global matrix assembly procedure is then used to account for several kinds of layers, which can either be solid or fluid, [START_REF] Dana | A spectral global matrix method for modelling the response of a fluid-loaded multilayered cylindrical shell excited by an acoustic plane wave[END_REF] allowing extending the classical Transfer Matrix Method for so-called large fd problems (i.e. at high wavenumber-thickness product) where high numerical instabilities occur. Heterogeneous coatings formed with resonant inclusions have also been studied as way to improve the sound absorption by the coating, either in terms of performances or frequency range. In particular, coatings constituted by a matrix of soft rubber with void inclusions, [START_REF] Sharma | Acoustic performance of gratings of cylindrical voids in a soft elastic medium with a steel backing[END_REF] hard inclusions [START_REF] Sharma | Acoustic performance of periodic steel cylinders embedded in a viscoelastic medium[END_REF] or both void and hard inclusions [START_REF] Sharma | Sound absorption by rubber coatings with periodic voids and hard inclusions[END_REF] have been investigated using analytical homogenisation theories. The previously cited studies were focused on the material technology in itself, but its influence when mounted on a cylindrical shell has also been studied. [START_REF] Sharma | Acoustic radiation from a cylindrical shell with a voided soft elastic coating[END_REF] Other theories can be used such as the layer-multiple-scattering method, which allows to consider the double-periodic problem of anechoic coatings with periodic cavities as a 2-D case because of the invariance in axial direction. [START_REF] Ivansson | Sound absorption by viscoelastic coatings with periodically distributed cavities[END_REF][START_REF] Ivansson | Numerical design of Alberich anechoic coatings with superellipsoidal cavities of mixed sizes[END_REF][START_REF] Ivansson | Anechoic coatings obtained from two-and three-dimensional monopole resonance diffraction gratings[END_REF] If those studies focus on fully coated cylindrical shells, less attention has been paid to the study of partially coated cylindrical shells. Studying such coating configurations can find physical or practical interests, for example to study the impact of a missing coating tile on the vibroacoustic behavior of the system, or to account for areas where the shell is covered with anechoic coating, and other areas where the shell is covered with decoupling coating. However, a difficulty arises from the loss of axisymmetry induced by the partial coating. Indeed, the spectral procedure described in 1 is based on the assumption of axisymmetry of the system. Hence, the only studies focused on partial coating are restricted to the locally reacting material approach to describe the coating. [START_REF] Cuschieri | Influence of circumferential partial coating on the acoustic radiation from a fluid-loaded shell[END_REF][START_REF] Laulagnet | Sound radiation from finite cylindrical shells, partially covered with longitudinal strips of compliant layer[END_REF][START_REF] Liu | Vibratory response and acoustic radiation of a finite cylindrical shell partially covered with circumferential compliant layers[END_REF] On the other hand, from a numerical point of view, substructuring methods have emerged during the past decades to overcome the frequency limitations of element-based methods such as the Finite Element Method (FEM). They allow studying rather complex systems by separating them into simpler subsystems which can be studied separately, before coupling the subsystems by the means of frequency transfer functions. In particular, substructuring methods based on admittance and impedance concepts will be of interest here. In the Patch Transfer Function (PTF) method, [START_REF] Ouisse | Patch transfer function as a tool to couple linear acoustic problems[END_REF] the interface between the subsystem is separated into a number of elementary surfaces called patches, and computing the transfer functions between the patches for each subsystem allows coupling them together. This approach has proved to be popular in the automotive industry, [START_REF] Ouisse | Patch transfer function as a tool to couple linear acoustic problems[END_REF][START_REF] Veronesi | Patch transfer function approach for analysis of coupled vibro-acoustic problems involving porous materials[END_REF][START_REF] Rejlek | A combined computationalexperimental approach for modelling of coupled vibro-acoustic problems[END_REF] but has found applications in other kind of industries, such as the naval industry [START_REF] Aucejo | Convergence acceleration using the residual shape technique when solving structure-acoustic coupling with the Patch Transfer Functions method[END_REF][START_REF] Maxit | Improving the Patch Transfer Function approach for fluid-structure modelling in heavy fluid[END_REF] to study the transmission of sound through the ballasts of a submarine vessel. Still initially focused for naval applications, the Circumferential Admittance Approach (CAA) [START_REF] Maxit | Prediction of the vibro-acoustic behavior of a submerged shell non periodically stiffened by internal frames[END_REF] has been developed to couple a model of a cylindrical shell immersed in water to axisymetric internal structures such as stiffeners or bulkheads.

In this method, the fluid-loaded cylindrical shell is studied using the spectral procedure described in. [START_REF] Skelton | Theoretical acoustics of underwater structures[END_REF] This method has then been applied to study the scattering from a stiffened cylindrical shell. [START_REF] Maxit | Scattering model of a cylindrical shell with internal axisymmetric frames by using the Circumferential Admittance Approach[END_REF] Later on, the principle of the method was extended to non-axisymetric internal frames such as floors or engine foundations by the way of the Condensed Transfer Function (CTF) method, to investigate their contribution on radiation and scattering from the shell. [START_REF] Meyer | A condensed transfer function method as a tool for solving vibroacoustic problems[END_REF][START_REF] Meyer | Prediction of the vibroacoustic behavior of a submerged shell with non-axisymmetric internal substructures by a condensed transfer function method[END_REF][START_REF] Meyer | A substructuring approach for modeling the acoustic scattering from stiffened submerged shells coupled to non-axisymmetric internal structures[END_REF] As such, the CTF method can be seen as a generalisation of both the PTF and CAA methods. Following these recent advances, the reverse Condensed Transfer Function (rCTF) has been developed as a reverse formulation of the CTF method to decouple subsystems initially coupled along lines or surfaces. One of the main advantages of these methods is that it allows to study non-axisymetric systems, proving an interesting potential to address the issue of the partially coated cylindrical shell. The mathematical framework of the method was proposed in, [START_REF] Dumortier | Vibroacoustic subtractive modeling using a reverse condensed transfer function approach[END_REF] while the procedure to compute the so-called Condensed Transfer Functions (necessary to apply the method) from numerical models has been described in. [START_REF] Dumortier | Subtractive modelling using the reverse condensed transfer function method: influence of the numerical errors[END_REF] A global approach of subtractive modelling was then proposed to improve the convergence of the method by extending the definition of the decoupling interface to interior boundaries of the initial system. [START_REF] Dumortier | A global decoupling technique for subtractive modelling on acoustic and vibration problems[END_REF] This paper is focused on addressing the problem of the scattering of an incident plane wave by a partially coated cylindrical shell, by using a coupled CTF-rCTF approach. Hence, the rCTF formulation is extended to predict the pressure field scattered by a partially coated cylindrical shell. The theoretical framework is presented in Section 2, whereas the numerical processes used to evaluate the condensed transfer functions are addressed in Section 3. The numerical results of the proposed approach concerning the scattering from the partially coated shell immersed in water are then discussed in Section 4, before concluding this paper.

Principle of the coupled CTF-rCTF method

Ω M 1 M 1 Figure 1.
Studying a partially coated cylindrical shell. The '-' sign illustrates the rCTF method; the '+' sign illustrates the CTF method.

In order to study the scattering from a partially coated cylindrical shell, the coupled CTF-rCTF method, illustrated in Figure 1, is applied. It consists in considering initially a model of the fully coated cylindrical shell impinged by a harmonic acoustic plane wave (system 1+2 in Figure 1). From this system, a model of the missing part of the coating (subsystem 2 in Figure 1) is removed. This first operation, called the decoupling procedure, is performed using the rCTF method and is illustrated in Figure 2. This results in an intermediary subsystem: the partially coated cylindrical shell, for which the missing part of the coated has been replaced by a rigid void (subsystem 1 in Figure 2, where the white area correponds to the void). Then the obtained subsystem is coupled to a model of the water domain replacing the removed coating material (subsystem 2' in Figure 1). This second operation is called the recoupling procedure, is performed using the CTF method and is illustrated in Figure 3. The obtained system is finally the partially coated cylindrical shell immersed in water (system 1+2' in Figure 1) impacted by the harmonic acoustic plane wave. In the following sections, the mathematical frameworks for the decoupling and recoupling procedures will be recalled.

Decoupling procedure using the rCTF method

In order to consider the decoupling problem (Figure 2), the rCTF method is used. The theoretical fundamentals of this procedure have been explained in details in reference, [START_REF] Dumortier | Vibroacoustic subtractive modeling using a reverse condensed transfer function approach[END_REF] but the main steps will be recalled here.

For the sake of conciseness, a 2D model of our system is considered. Let us consider two systems initially coupled along a line Ω. The system 1+2, corresponding to the fully coated cylindrical shell, is excited by an external acoustic plane wave, and the responses are

Ω M 1 M 1 Figure 2.
Decoupling procedure using the rCTF method.

calculated in harmonic regime. The objective of this decoupling procedure is to obtain the response at any point M 1 of subsystem 1. To this end, a set of N orthonormal functions, called condensation functions, is defined on Ω: tφ i u 1¤i¤N . Then, the pressures p α and normal velocities u α on Ω can be approximated, for each (sub)system α, as a linear combination of the condensation functions

p α pxq N i1 P i α φ i pxq and u α pxq N i1 U i α φ i pxq, x Ω (1)
where P i α and U i α are the unknowns. They can be estimated by defining, for each uncoupled subsystem (i.e. subsystems 1 and 2), condensed transfer functions (CTFs) between φ i and φ j . These CTFs are defined by applying a prescribed velocity u α φ j on Ω, meaning that the CTFs will result in condensed impedances

Z ij α xp α , φ i y xu α , φ j y xp α , φ i y (2)
where xf, gy ³ f pxqg ¦ pxq dx (with * denoting the complex conjugate), and pα corresponds to the resulting pressure on Ω when the subsystem is excited by u α φ j . Concerning system 1+2, as the decoupling boundary Ω is a fictitious one, a prescribed velocity jump corresponding to φ j must be considered to define its condensed impedances

Z ij 1 2 P i 1 2 δU j 1 2 d p1 2 , φ i h xφ j , φ j y d p1 2 , φ i h (3)
The condensed impedances are calculated for each couple of condensation functions for each system/subsystem. The condensed impedance matrix for system/subsystem α can subsenquently be defined as

Z α Z ij α % N ¢N
. Then, following some developments of the rCTF approach described in reference, [START_REF] Dumortier | Vibroacoustic subtractive modeling using a reverse condensed transfer function approach[END_REF] the condensed impedance matrix of subsystem 1 can be obtained from the condensed impedance matrices of system 1+2 and subsystem 2:

Z 1 Z 2 pZ 2 ¡ Z 1 2 q ¡1 Z 1 2 (4)
The contribution from the plane wave excitation must be introduced through the definition of the free condensed pressures of the system 1+2

P i 1 2 xp 1 2 , φ i y (5)
where p 1 2 is the pressure induced by the acoustic plane wave on the the decoupling boundary Ω. Then, to estimate the response at any point M 1 of the uncoupled subsystem 1, the quantity Z i 1 2 pM 1 q, corresponding to the response at point M 1 in the system 1+2 when Ω is excited by a velocity jump φ i , must be introduced. This quantity can be inferred from the free condensed pressures induced by a monopole of unit volume velocity located at point M 1 , using a reciprocity principle 29, 30

Z i 1 2 pM 1 q P M 1 i 1 2 ¦ xp M 1 ¦ 1 2 , φ i y ¦ (6)
where p M 1 ¦ 1 2 is the complex conjugate of the pressure induced on the decoupling boundary Ω by a monopole of unit volume velocity located at point M 1 . It must be underlined that the introduction of the complex conjugates in Eq. 6 comes from the use of the reciprocity principle. Indeed, the complex conjugate of the condensation function must appear in the definition of a condensed quantity, which is not the case when using the reciprocity principle, leading to the necessity of adding the complex conjugates to have a correct correlation between the different quantities. The condensed pressure vectors of system 1+2 can subsequently be defined as P 1 2

P i 1 2 $ N ¢1 and P M 1 ¦ 1 2 P M 1 i 1 2 ¦ % N ¢1
. Finally, the pressure at any point M 1 of the uncoupled subsystem 1 can be written as

p 1 pM 1 q p 1 2 pM 1 q ¡ I Z 1 Z 2 ¡1 ¨P M 1 ¦ 1 2 © T Z 2 ¡1 P 1 2 (7)
where p 1 2 pM 1 q is the pressure induced by the acoustic plane wave at point M 1 of system 1+2.

Recoupling procedure using the CTF method

Once the pressure has been obtained in the decoupled subsystem 1, we are interested in recoupling this subsystem to a water tile occupying the missing part of the coating material, to obtain the partially coated cylindrical shell (Figure 3). The recoupling procedure is performed using the CTF method, for which the theoretical fundamentals have been developed

in [START_REF] Dumortier | Vibroacoustic subtractive modeling using a reverse condensed transfer function approach[END_REF] for acoustical systems and in 24 for mechanical systems. We can find the pressure at any point of the coupled system 1+2', for which the expression depends if the considered point initially belonged to subsystem 1 or subsystem 2'

M 2' Ω M 1 Ω M 1 M 2'
5 p 1 2 IpM 1 q p1 pM 1 q Z 1 T pM 1 qU 1 2 I p 1 2 IpM 2 Iq ¡Z 2 I T pM 2 IqU 1 2 I (8a) (8b)
where the point M 1 initially belongs to subsystem 1, while the point M 2 I initially belongs to subsystem 2'. Z 1 pM 1 q is the vector of the point condensed impedances of subsystem 1, and its components are defined similarly as Z i 1 2 pM 1 q, using a reciprocity principle. [START_REF] Fahy | Some applications of the reciprocity principle in experimental vibroacoustics[END_REF][START_REF] Marchetto | Vibroacoustic response of panels under diffuse acoustic field excitation from sensitivity functions and reciprocity principles[END_REF] The vector can then be expressed as 26

Z 1 pM 1 q P M 1 ¦ 1 I Z 1 Z 2 ¡1 ¨PM 1 ¦ 1 2 (9)
Concerning U 1 2 I, it reads

U 1 2 I ¡ pZ 1 Z 2 Iq ¡1 P1 (10)
As Z 1 has already been computed, the necessary quantities here are Z 2 I (which is as- sumed to be known), and P1 . The latter corresponds to the condensed pressure at the surface Ω of the uncoupled subsystem 1 when the excitation is the acoustic plane wave and can then be noted P1 . It is calculated in a similar way as PM 1 1 in Equation 9

P1 I Z 1 Z 2 ¡1 ¨P1 2 (11)
From these information, Eq. 8a can be rewritten

p 1 2 IpM 1 q p 1 2 pM 1 q ¡ I Z 1 Z 2 ¡1 ¨PM 1 ¦ 1 2 © T ¡ Z 2 ¡1 P 1 2 ¡ pZ 1 Z 2 Iq ¡1 P1 © ( 12
)
where Z 1 and P1 are expressed from Eqs. 4 and 11, respectively. On the other hand, Eq. 8b

gives

p 1 2 IpM 2 Iq Z 2 I T pM 2 Iq pZ 1 Z 2 Iq ¡1 P1 (13)
where Z 2 IpM 2 Iq can be computed by evalutating the pressure at point

M 2 I of subsystem 2'
when the excitation is a condensation function.

From Eqs. 12 and 13, the pressure scattered by the partially coated cylindrical shell can be estimated at any point in the fluid domain, from the condensed impedances of the fully coated shell (i.e. system 1+2), of the removed coating patch (i.e. subsystem 2) and of the added water domain (i.e. subsystem 2'). In the next section, we are going to focus on the numerical computation of these condensed impedances (Z 1 2 , Z 2 and Z I

2 ).

Numerical calculation of the condensed impedances

In order to apply the process proposed in Section 2, the condensed impedance matrices Z 1 2 , Z 2 and Z I 2 must be computed. After a preliminary introduction of the considered system and of the condensed transfer functions in subsection 3.1, the procedure to evaluate numerically the condensed impedances will be developed in subsections 3.2 and 3.3 for the system 1+2 and subsystems 2 and 2', respectively.

Description of the system and of the condensation functions

In the following, we consider a cylindrical shell of radius R, thickness h, Young's modulus E s , density ρ s and Poisson's ratio ν s . It is excited by a harmonic acoustic plane wave of oblique incidence θ i compared to the vertical direction (see Figure 4a). The behavior of the shell are described using Flügge equations for thin shells. [START_REF] Karczub | Expressions for direct evaluation of wave number in cylindrical shell vibration studies using the Flügge equations of motion[END_REF] The coating is an anechoic coating of thickness h p and density ρ p , and will be modelled as an equivalent fluid, meaning that the transverse waves travelling in the coating are not taken into account. The celerity of longitudinal waves in the coating is noted c p , while its damping loss factor is η p . The acoustic properties of the coating correspond to the ones used for the BeTSSi II (Benchmarking of Target Strength Simulations) international workshop. [START_REF] Nijhof | BeTSSi II : Submarine target strength modeling workshop[END_REF] The coated shell is surrounded by a fluid medium of density ρ f and speed of sound c f , and a damping loss factor η f is introduced to avoid numerical instabilities. [START_REF] Dumortier | Vibroacoustic subtractive modeling using a reverse condensed transfer function approach[END_REF] Both fluid domains are modelled using the Helmholtz equation, and the Sommerfeld radiation condition is verified at infinity where k f is the acoustic wavenumber. Besides, the continuity of normal velocities and pressures are considered at the shellcoating and coating-exterior fluid interfaces. The characteristics of the shell, the coating and the surrounding fluid domain are presented in Table 3.1. It must be stressed that, in the following, the computations are carried out using 2D models. This choice was made in order to be able to compare the obtained results to a FEM reference calculation. Therefore, the oblique incidence of the acoustic plane wave in the azimuthal plane was not taken into account.

φ i psq 5 1 c Ls if pi ¡ 1qL s ¤ s iL s 0 elsewhere , i 1, N 0 ¤ s ¤ Γ ( 15
)
The size of the segments follows a wavelength-based criterion, and must be smaller than half the minimal wavelength (between the acoustic wavelength in the surrounding fluid, the acoustic wavelength in the coating and the flexural wavelength of the shell) at the highest considered frequency. In the following, the calculations are carried out between 5 Hz and 1500 Hz. At this frequency, the acoustic wavelength in the coating is the smaller one, and is then the one retained to define the size of the segments. Following this criterion, the decoupling boundary Ω is divided into 22 segments as illustrated in Figure 4b.

Condensed impedances of the fully coated cylindrical shell

According to the definition of the condensed impedances given in Eq. 3, a velocity jump must be applied on the decoupling boundary and the resulting pressure must be calculated. Besides, to correspond to a condensation function, the velocity jumps will be applied successively on each segment of the decoupling boundary according to Eq. 15. To emulate these velocity jumps, an integral formulation derived from the indirect BEM formulation is retained. Following this formulation, the pressure field from a radiating surface can be expressed as ppM q » Ω νpP qGpM, P q dΩpP q, M Ω, P Ω

where νpP q is the single layer potential due to a layer of monopole sources and represents a velocity jump at the crossing of Ω. It can be expressed from the velocity jump δU as νpP q jωρ f δU pPq

GpM, P q is the free-field Green function and corresponds to the pressure at point M due to a monopole source located at point P GpM, P q ¡ e jk f p1¡jη f q|M¡P| 4π|M ¡ P |

with |M ¡P| being the Euclidian distance between the two points. According to the definition of the single layer potential, the velocity jump can then be emulated by placing on Ω an array of monopole sources, which allows to evaluate numerically the integral in Eq. 16.

In practice, the formulation to obtain the pressure induced at a single point by a monopole excitation must be derived. This calculation is done via a semi-analytical approach in the wavenumber domain as described in, 1 by the means of a Fourier series decomposition along the circumferential coordinate. It is worth mentioning that, if we were to work with 3D models, a Fourier transform along the axial coordinate would also be performed. Finally, the total pressure at a single point n i due to a monopole excitation located at point n j can be retrieved by adding the contribution of all the circumferential orders

p n i n j tot n V ņ¡V pn i n j tot e jnθn i (19)
where pn i n j tot is the total spectral pressure obtained using the procedure in. [START_REF] Skelton | Theoretical acoustics of underwater structures[END_REF] The sum in Eq.

19 is theoretically infinite, but is in practice truncated to a finite value N max derived from the maximal value of the different wavenumbers of the problem,

N max int rκ n Rmaxpk f , k p , k s qs (20)
with k f the acoustical wavenumber in the surrounding fluid medium, k p the acoustical wavenumber in the anechoic coating, and k s the flexural wavenumber of the shell. κ n is a margin coefficient generally fixed to 1.5 for such problems.

The condensed impedance between the incident segment j and the receiving segment i is then obtained by summing the resulting pressures at all the points N i belonging to the segment i, due to monopole excitations on all the points N j belonging to the segment j

Z ij 1 2 N i ņi 1 N j ņj 1 p n i n j tot L j L i δξ j δξ i (21)
where L i and L j are the length of the segments i and j, respectively, and δξ i and δξ j correspond to the discretization step in the segments (which depends on the location of the segment on the decoupling boundary).

It must be emphasized that the expression of the pressure in Eq. 12 also relies on the calculation of the condensed pressures P 1 2 and P M 1 1 2 . These condensed pressures are numerically evaluated in a similar way as Z 1 2 , in the sense that the expressions of the pressure induced by the acoustic plane wave (for P 1 2 ) and by the unit monopole (for

P M 1 1 2
) are projected on the segments using the discretization points. The details of these calculations will not be addressed here, but they can be found in. [START_REF] Dumortier | Principle of vibroacoustic subtractive modelling and application to the prediction of the acoustic radiation of partially coated submerged cylindrical shells[END_REF]

Condensed impedances of the missing coating material and the water patch

Concerning the condensed impedances of the missing coating material and of the water patch added to fill up the voided section, they must be computed by applying a prescribed velocity on the decoupling boundary, as defined in Eq. 2. To do so, and given the small size of the subsystem, a Finite Element (FE) formulation was used. An example of the FE model of the removed patch can be seen in Figure 4c. As we are working here with acoustical systems, the FEM formulation which must be solved yields rKs ¡ ω 2 rMs p1 ¡ 2jη ¨tPu tQu (22) where rKs and rMs are the acoustic stiffness and mass matrices, respectively, η is the damping loss factor (either in the coating or in the surrounding fluid), tPu is the output pressure vector, and tQu is the input volume velocity vector.

The procedure to compute the condensed impedances of the subsystems is then similar to the one described in subsection 3.2. The nodes of the FE model are firstly associated to the different segments to which they belong, in order to define the condensation functions.

Then, a volume velocity excitation is applied on the nodes belonging to the incident segment j according to the definition of the condensation functions in Eq. 15. Following this, the input volume velocity vector tQ j u associated to the incident segment j will have N j non-null components Q j n , with N j being the number of nodes belonging to the segment j

Q j n δS n L j , n 1, N j (23)
where δS n is the length around the node n, and L j the length of the segment j. The resulting pressure vector tP j u of the FE model is then obtained by inverting directly the dynamic matrix of Eq. 22. This operation is not computationally costly as the system has a low number of degrees of freedom.

tP j u rKs ¡ ω 2 rMs p1 ¡ 2jη ¨¡1 tQ j u

Finally, the condensed impedance between the incident segment j and the receiving segment i is obtained by summing the resulting pressure at all the M i nodes belonging to the receiving segment

Z ij α 1 c L i M i m1 P j m δS m (25)
where P j m is the pressure at node m belonging to the segment i when the system is excited on the segment j, δS m is the length around the node m, L i is the length of the segment i, and α corresponds either to subsystem 2 or subsystem 2'.

Scattering from the partially coated shell immersed in water

In Section 3, the formulations to obtain the necessary quantities have been demonstrated.

We can now proceed with the application of the method to predict the scattering from the partially coated cylindrical shell impacted by a plane wave of unit amplitude, for which the missing coating part spreads from 0 to π{4 as illustrated in Figure 5. Two loading configurations will be considered. In the first one, the acoustic plane wave will be incident on the uncoated part of the shell, while for the second one, the acoustic plane wave will impinge the other side of the shell. A first paragraph will be dedicated to address some theoretical aspects of the scattering problem, in order to understand the quantities that will be studied hereafter. Then, we will be interested in investigating the impact of partial coating on the near-field and far-field scattered pressure.

Theoretical aspects concerning the scattering problem

When dealing with the scattering problem, the total pressure in the domain can be decomposed as three different components: the incident pressure p inc corresponding to the pressure induced by the acoustic plane wave, the reflected pressure p ref l corresponding to the pressure scattered as though the boundary were hard, and the reradiated pressure p rad stemming from the cylinder motion. Generally, when the scattering problem is considered, the contribution of the elastic part (i.e. the reradiated pressure) is studied separately from the contribution of the rigid part (i.e. the sum of the incident and reflected pressure, also called the blocked pressure). This can be summarized in Eq. 26

1 p p inc p ref l l jh n p block p rad (26)
In the following, we will be interested into evaluating the reradiated pressure by the cylindrical shell. Indeed, one must keep in mind that the problem investigated here constitutes a first step towards the study of 3D partially coated, stiffened cylindrical shells.

For such studies, we can be interested in the elastic component of the scattered pressure (i.e. p rad ) as the spectra of monostatic backscattered pressures show patterns of periodic discontinuities (namely Bragg scattering and scattering from Bloch-Floquet waves) arising from the interaction between the incident wave and the elastic stiffeners. [START_REF] Maxit | Scattering model of a cylindrical shell with internal axisymmetric frames by using the Circumferential Admittance Approach[END_REF][START_REF] Liétard | Acoustic scattering from a finite cylindrical shell with evenly spaced stiffeners: Experimental investigation[END_REF] To obtain the pressure reradiated by the partially coated shell, we have seen in Eq. 12 that the first step is to compute the pressure reradiated by the fully coated shell, p 1 2 pM 1 q.

The procedure to obtain this reradiated pressure is not trivial and will be developed here.

Let us consider the problem presented in Figure 4a

The Helmholtz equation can be solved in the wavenumber domain using a linear combination of Bessel functions. As such, the pressure inside the coating, p1 , and outside the coating, pV , accounting for the previously stated conditions, can be expressed as 35

p1 pr, nq A 1 H n pk pr rq H n pk pr Rq B 1 J n pk pr rqH n pk pr R 1 q if r ¤ R 1 (28a) pV pr, nq A V H n pk f r rq H n pk f r R 1 q j n J n pk f r rq if r ¥ R 1 (28b)
where A n and B n are the unknowns, J n and H n are the Bessel function of the first kind and the Hankel function of the second kind, respectively, R is the radius of the shell, R 1 the exterior radius of the coating, and k ir k i cospθ i q (with k i the acoustic wavenumber in the domain i). The unknowns can be obtained by considering first the Neumann condition at the shell surface, the continuity of pressures and velocities at the interface between the fluid and the coating, and then solving the subsequent system of equations

" " " " " " ! ¡1 H n pk pr R 1 q H n pk pr Rq J n pk pr R 1 qH n pk pr R 1 q ¡ k f r r ρ f H I n pk f r R 1 q H n pk f r R 1 q k pr ρ p H I n pk pr R 1 q H n pk pr Rq k pr ρ p J I n pk pr R 1 qH n pk pr R 1 q 0 k pr H I n pk pr Rq H n pk pr Rq k pr J I n pk pr RqH n pk pr R 1 q ( 0 0 0 0 0 0 ) ! A V A 1 B 1 ( ) " " ! j n J n pk f r rq k f r r ρ f j n J I n pk f r rq 0 ( 0 0 ) (29)
The blocked pressure at the surface of the shell can then be expressed as

pbloq pr R, nq A 1 B 1 J n pk pr RqH n pk pr R 1 q (30)
As stated previously, the next step is then to use the expression of Eq. 30 as an excitation term of the cylindrical shell. To do so, the behavior of the shell will be described using the Flügge equations 31, 36

Lpnq ! Ũ pnq Ṽ pnq W pnq ( ) γ ! 0 0 pbloq pr R, nq ( ) (31)
where L is the Flügge operator, γ ¡ p1¡ν 2 s qR 2 Eshs and U , V and W are the axial, tangential and radial displacements, respectively. The resolution of this system of equations allows expressing the radial displacement as W pnq ¡ γ pbloq pr R, nq

¡ ZUU pnq ZV V pnq ¡ Z2 U V pnq © ∆pnq (32)
with Zij being the coefficients of the Flügge operator, and ∆ is the determinant of the Flügge matrix taking into account the loading of the fluid and of the coating. It can be expressed as

∆ ZUW p ZUV ZV W ¡ ZUW ZV V q ZV W p ZUW ZUV ¡ ZV W ZUU q p ZW W γ Zf qp ZUU ZV V ¡ Z2 U V q ( 33
)
where Zf is the effective spectral impedance of the fluid and coating. This quantity can be obtained similarly as the blocked pressure in the system of Eq. 29, with the following conditions homogeneous Helmholtz equation in the exterior fluid domain and in the coating, Sommerfeld radiation condition at infinity.

Euler equation depending on the radial shell velocity at the shell surface, continuity of pressures and velocities at the interface between the fluid and the coating.

The system of equations which must be solved now reads

" " " " " " ! ¡1 H n pk pr R 1 q H n pk pr Rq J n pk pr R 1 qH n pk pr R 1 q ¡ k f r r ρ f H I n pk f r R 1 q H n pk f r R 1 q k pr ρ p H I n pk pr R 1 q H n pk pr Rq k pr ρ p J I n pk pr R 1 qH n pk pr R 1 q 0 k pr H I n pk pr Rq H n pk pr Rq k pr J I n pk pr RqH n pk pr R 1 q ( 0 0 0 0 0 0 ) ! A V A 1 B 1 ( ) ! 0 0 ρ p ω 2 ( ) (34)
and the fluid and coating loading impedance is obtained

Zf A 1 B 1 J n pk pr RqH n pk pr R 1 q (35)
Finally, the shell's radial displacement can be used to obtain the displacement of the external part of the coating, and, subsequently, the pressure reradiated by the shell (using the Euler equation)

Wcoat pnq k pr ρ p ω 2 ¢ A 1 H I n pk pr R 1 q H n pk pr Rq B 1 J I n pk pr R 1 qH n pk pr R 1 q W pnq (36)
where A 1 and B 1 are the unknowns obtained when computing the effective fluid and coating impedance in Eq. 34.

In the next sections, the influence of the partial coating will be evaluated via two quantities: the evaluation of the near-field reradiated pressure at the ring frequency of the shell, and the target strength of the shell in the entire frequency spectrum. The target strength is defined as the ratio between the far-field scattered pressure brought back to 1 m from the object, and the incident pressure of the acoustic plane wave.

Evaluation of the near-field pressure at the ring frequency

At first, we are interested in the near-field reradiated pressure at the ring frequency. The ring frequency corresponds to the frequency at which the longitudinal wavelength in the shell is equal to the circumference and is defined as

f r 1 2πR d E s ρ s p1 ¡ ν 2 s q (37)
Around the ring frequency, the radiation efficiency of the cylindrical shell is particularly high, and this frequency also exhibits a change in behavior of the shell. Indeed, above the ring frequency, it is generally admitted that the cylindrical shell behaves like a plate, [START_REF] Heckl | Vibrations of point-driven cylindrical shells[END_REF] in the sense that the flexural motions are mainly decoupled from the longitudinal and shear motions. It will then be interesting to study the behavior of the partially coated shell at this particular frequency. For the case of interest, the ring frequency yields f r 958 Hz.

The cartography of the near-field reradiated pressure will be presented for two loading cases, as illustrated in Figure 5. The first one corresponds to an acoustic plane wave having a normal incidence on the missing coating patch (θ i π 8 ), while the second one will be directed towards the positive x-axis (θ i π). For each loading case, a comparison is made between the uncoated shell, the fully coated shell, and the partially coated shell. And for the latter, the results of the coupled CTF-rCTF approach are compared to a reference FEM calculation which has been conducted using COMSOL Multiphysics ® . For the FEM calculation, it must be mentioned that the elastic radiated pressure cannot be obtained directly. To obtain this quantity from a FEM calculation, it is necessary to go back to the definition of the radiated pressure given in Eq. 26. As such, the reradiated pressure can be infered by subtracting the blocked pressure from the total pressure. Following this definition, two FEM calculations must be performed: a full calculation accounting for the elastic motion of the shell, and a blocked pressure calculation where the shell is not modelled, in order to be considered as rigid.

The first loading case, corresponding to an incident acoustic plane wave impacting the shell at the location of the missing coating material, is shown in Figure 6. The direction of the incident plane wave is specified in Figure 6a. The first observation that can be made regarding Figures 6a (uncoated shell) and 6b (fully coated shell) is that the anechoic coating has a beneficial impact as the values of the reradiated pressure are globally much lower when the coating is applied on the shell. Then, regarding Figures 6c (FEM reference calculation) and 6d (CTF-rCTF calculation) which show the pressure field for the partially coated shell, we can see that there is a very good agreement between the two calculations, which validates the formulation proposed in Section 2. The different patterns of maxima and minima of pressure are correctly localized, and the amplitudes are well correlated. It can also be said that the absence of coating on the impact area of the plane wave increases the reradiated pressure, especially at the vicinity of the shell, as it could have been expected.

Finally, it is important to stress that the symmetry of the system with respect to the angle of incidence of the plane wave has been well complied with.

Concerning the second loading case, corresponding to an acoustic plane wave arriving towards the positive x-axis, the reradiated pressure is shown in Figure 7 for the four configurations. It can be observed that the fully coated shell and the partially coated shell exhibit a very similar behavior. This shows that the partial coating has a very low influence on the behavior of the shell when the missing coating material is located in the shadow area of the shell. This figure also highlights that the CTF-rCTF and FEM calculations exhibit almost the same results, as it was observed in Figure 6 for the previous loading case. This is associated to the previous observation concerning the low impact of the partial coating for this configuration, hence limiting the influence of the sensitivity of the rCTF method.

To conclude on this part, the results presented in this paragraph show that the partial coating can have an impact on the reradiated pressure, compared to the fully coated shell.

But this change in behavior is restricted to the case where the incident plane wave impinges directly the shell around the uncoated area. When the plane wave is incident on a coated part of the shell, the impact of the partial coating on the reradiated pressure is very limited.

Impact of the partial coating on the target strength of the shell

We are now interested into evaluating the elastic component of the target strength of the shell, and the impact of the partial coating on this quantity, for the two lading configurations.

As stated earlier, the target strength is defined as the ratio between the far-field scattered pressure brought back to 1 m from the object, and the incident pressure of the acoustic plane wave. It can be evaluated at the same angle as the incident angle of the plane wave (monostatic target strength) or at a different angle (bistatic target strength). In this study, we are only interested in the monostatic target strength for the two loading cases.

The elastic component of the target strength corresponds to the evaluation of the target strength when only the reradiated pressure p rad is accounted for. In practice, it can be obtained for a 2-D case by using the asymptotic expression of the Hankel function for large arguments. This allows predicting the far-field reradiated pressure from the radial spectral displacement of the shell for uncoated shells, or of the radial spectral displacement of the external boundary of the coating for coated shells. Its expression for a coated shell yields

p f ar rad pr, θq ρ f ω 2 k 0 2 πrk 0 n V ņ¡V Wcoat pnq H p2q I n pR 1 k 0 q e ¡jrk 0 j pn 1qπ 2 jnθ , r 4 2π k 0 (38)
where W coat n is the spectral radial displacement of the external boundary of the coating obtained in Eq. 36, and H p2q I n is the derivative of the Hankel function of the second kind and of order n. It must be noted that Eq. 38 exhibits an infinite sum. In practice, this sum is truncated to a maximal circumferential order, and the choice of this value is explained in details in. [START_REF] Maxit | Scattering model of a cylindrical shell with internal axisymmetric frames by using the Circumferential Admittance Approach[END_REF] For the partially coated shell, however, we have seen in Section 2 that the final quantity that is obtained using the CTF-rCTF method is the radiated pressure, and there is no information concerning the radial displacement of the shell or of the coating. Hence, the farfield pressure will be evaluated in the same way as the near-field pressure, using Eq. 12. To obtain the reradiated far-field pressure, the process is the same as for the FEM calculation.

Two calculations are performed to obtain the total pressure and the blocked pressure, and the reradiated pressure is obtained by subtracting the latter from the former. When the far-field reradiated pressure has been computed for all of the coating configurations, the monostatic target strength can be obtained from the following expression

T S mono pθq 10 log ¤ ¦ ¥ ¡ c rp f ar rad pr, θq © 2 p 2 0 ( 39
)
with p 0 being the amplitude of the incident plane wave. The monostatic target strength for the three different coating situations (uncoated, fully coated and partially coated) is shown in Figure 8a, for the first loading case (i.e. the acoustic plane wave impinges the missing coating area of the shell). We can observe that the target strength of the uncoated shell is higher than for the fully and partially coated shells, which shows that the anechoic coating also proves its interest when it comes to limiting the scattering in the far-field. It is also interesting to notice that there is not much difference between the fully coated and the partially coated shell up to 800 Hz. Then the target strength tends to be slightly higher for the partially coated shell up to 1200 Hz, and the gap increases for frequencies above 1200 Hz. To analyze this result, one could look for a relation between the size of the opening and the acoustic wavelength in the coating.

Indeed, we could assume that, if the acoustic wavelength is greater than the opening, then the plane wave will be attenuated by the coating nevertheless. With the opening spreading from 0 to π{4, its length is then π{4 m (as the shell's radius is 1 m). From this value, we can define a limit frequency f l above which the acoustic wavelength in the coating will be smaller than the opening. In this case, we obtain f l =1070 Hz. This is hence consistent with the hypothesis stating that the partial coating will mainly have an impact at frequencies where the acoustic wavelength in the coating is smaller than the opening. Besides, it can be observed in Figure 8b that there is an excellent agreement between the rCTF and the FEM calculations, which is consistant with the previous observations made in Section 4.2 and validates the coupled CTF-rCTF approach. Concerning the second loading case, the comparison between the different coating situations is shown in Figure 9a. The results displayed in this figure are very consistent with the analysis made from Figure 7, which state that the absence of coating on a given part of the shell does not have a significant impact on the scattering of the shell when this specific part is not impinged by the plane wave. Hence, as the behavior of the fully coated and partially coated shells are similar for this loading case, the CTF-rCTF exhibits no calculation error compared to the FEM calculation. The conclusions drawn in this study will have to be investigated in more details in further studies, and particularly for 3D stiffened cylindrical shells for which very specific scattering phenomena can be observed.

Conclusion

In this paper, the scattering problem of an acoustic plane wave by a partially coated cylindrical shell has been addressed using the CTF and rCTF methods. The theoretical fundamentals of the method which were previously developed for studying radiation problems have been extended in the present paper to the case of scattering problems. A theoretical framework for the calculation of elastic radiated pressure and elastic target strength has been proposed for the cases of fully coated and partially coated cylindrical shells.

A comparison with FEM simulations on a 2D model validates the proposed coupled CTF-rCTF approach, while it is expected that the latter will require much less computing time to deal with 3D models than with a FEM simulation. The comparison with the FEM calculation exhibits accurate results for both the near-field pressure and the far-field pressure (represented by the elastic target strength). These first results show that the influence of the partial coating is particularly highlighted when the incident acoustic plane wave is heading towards the area where the coating material is missing, and for frequencies where the acoustic wavelength in the coating is smaller than the opening.

It is important to mention that these results, although giving a first insight into the possibilities brought by the rCTF method, are not completely of interest for naval applications, as the calculations were performed on a 2D model of an unstiffened shell. However, this study must be seen as a first step towards the possibility of studying the scattering from a 3D partially coated, stiffened cylindrical shell.

Figure 3 .

 3 Figure 3. Recoupling procedure using the CTF method.

5 ∆ppr, θ, xq k 2 f

 52

Figure 4 .

 4 Figure 4. (a) Fully coated cylindrical shell impacted by an acoustic plane wave. (b) Discretization of the decoupling boundary for the system 1+2. (c) FE model for the subsystem 2.

Figure 5 .

 5 Figure 5. Loading cases and angular spreading of the partial coating.

  . The calculation is performed in two steps. The first one consists in solving the blocked pressure problem, i.e. the response of the system when the shell is considered as rigid. The blocked pressure at the surface of the shell is then used as an excitation term to derive the radial displacement of the shell and ultimately the reradiated pressure. The blocked pressure satisifies the following conditions Helmholtz equation with a source term in the exterior fluid domain, homogeneous Helmholtz equation in the coating, Sommerfeld radiation condition at infinity. The problem is solved in the wavenumber domain by the means of a Fourier series decomposition along the circumference of the shell, allowing to express the pressures as a function of the cylindrical harmonics ppr, θq V ņ¡V pn prqe jnθ

Figure 6 .

 6 Figure 6. Reradiated pressure (in dB ref. 1e-6 Pa) for θ i π 8 . (a) Uncoated shell. (b) Fully coated shell. (c) Partially coated shell -reference calculation. (d) Partially coated shell -CTF-rCTF calculation.

Figure 7 .

 7 Figure 7. Reradiated pressure (in dB ref. 1e-6 Pa) for θ i π. (a) Uncoated shell. (b) Fully coated shell. (c) Partially coated shell -reference calculation. (d) Partially coated shell -CTF-rCTF calculation.

Figure 8 .

 8 Figure 8. Monostatic target strength when the plane wave is incident on the uncoated part of the shell (θ i π{8). (a) Comparison between the coating situations. (b) Comparison between the rCTF and FEM calculations.

Figure 9 .

 9 Figure 9. Monostatic target strength when the plane wave is incident on the coated part of the shell (θ i π).

( a )

 a Comparison between the coating situations. (b) Comparison between the rCTF and FEM calculations.

Table 1 .

 1 Material and fluids characteristics.

	Parameter	Notation	Value	Unit
	Shell radius	R	1	m
	Shell thickness	hs	0.01	m
	Shell Young's modulus	Es	210	GPa
	Shell Poisson's ratio	νs	0.3	-
	Shell density	ρs	7800 kg.m -3
	Shell damping loss factor	µs	0.01	-
	Coating thickness	hp	0.1	m
	Coating density	ρp	1600 kg.m -3
	Coating speed of sound	cp	841.5	m.s -1
	Coating loss factor	µp	0.236	-
	Exterior fluid density Exterior fluid speed of sound	ρ f c f	1000 kg.m -3 1500 m.s -1
	Exterior fluid loss factor	µ f	0.001	-
	In this work, as the decoupling boundary Ω is a 1D one (because we are working with
	2D models), the condensation functions are chosen to be gate functions. This means that

the decoupling boundary Ω, of length Γ, is divided into N segments. The condensation functions are then defined depending on their length L s

Acknowledgements

This work was funded by the French National Research Agency (ANR) within the plan "France Relance", and was performed within the framework of the LabEx CeLyA (ANR-10-LABX-0060) of Université de Lyon, in collaboration with Naval Group Research.

Bibliography


domain: shs shs shs.scipo

text:
The Absurd Injunction Not to Belong. Belonging, Un-belonging and the Biduns in Kuwait.

HAMM: We're not beginning to... to... mean something? CLOV: Mean something! You and I, mean something! (Brief laugh.) Ah that's a good one! Beckett, Samuel (1958) Endgame, A Play in One Act, London: Faber and Faber (p. 26) In a tweet posted on 29 March 2018, a bidun activistwho was later jailed from July 2019 to January 2020 for peacefully protesting against the biduns' inhumane conditionsshared a video. 1 The brief video zooms very closely on an ID card, recognizable as one of those issued to the biduns, or long-term residents of Kuwait whose legal status is an object of contention between them and the state. 2 More precisely, the mobile phone camera focuses, at the back of the bidun ID card, on one line with a special mention added by the administration in charge of the biduns, the Central System. 3 Other magnetic strip cards hide the personal data written above and below it. A male voice can be heard saying that he will read this additional remark, but before even doing so starts bursting into laughter. The faceless voice goes on to read out the label in an unrestrained laugh: "ladih qarib… ladihi qarina…dalat jinsiyya iraqiyya/ he has a relative… who has a presumptive evidence… suggesting an Iraqi nationality." The video shakes as the result of the laugh, a contagious laugh, growing in intensity. In the Kuwaiti dialect, the voice continues commenting: "Aqsum bil-Allah, ga'adt sa'at ufakir sheno ma'anat al hatchi/ I swear by God, it took me an hour to figure out the meaning of this nonsense", before reading the sentence again, stopping and guffawing, and asking if it should "repeat it a third time…" expressing amazement at its absurdity. The tweet, addressed at the head of the Central System mentioned in the hashtag (#scandal Salah al-Fadhala), reads: "Don't bother, write what you want" (in lam tastahi fa'ktub ma sha't).

The scene is comedic. Not only comedic, it is also extremely meaningful. Laughter and humor are to be taken seriously, when it comes to their political meaning. They can signify a refusal to submit to processes of subjugation. 4 They can be understood as transgressive acts because, when out of place or contrary to the reaction of fear and cowering expected in the face of oppression, they destabilize existing social hierarchies and disavow the power they are faced with. Humor lays bare the absurdity of the situation the subjugated are forced to suffer.

In the above clip, the scene poking fun at the bidun's ID delivered by the Central System in Kuwait sheds light on the potentiality of humor to question the rationalizations for injustice, and the absurdity of a life-world produced by a decades-long of policies of rights denial, that sometimes contradict each other.

To capture this feeling of absurdity, I do an intellectual detour outside of my original discipline, political science. I borrow a concept from the literary techniques and critique and draw modes of thought that belong to the social sciences, literary critique and literature itself -beyond a somewhat arbitrary separation. I use the lenses of the absurd to emphasise the human and emotional depth in the face of aberrant and incomprehensible mechanisms of power and injustice, and which have metaphysical dimensions. I understand the term here both in the everyday meaning of nonsensical and in the broader sense of an expansive category of literary strategies and methods that highlight the feelings of incongruity or incongruous realities. 5 This effect is achieved by the following extract from Mahmoud Darwish's Journal of an Ordinary Grief, which was inspired by the author's own experience.

You despair of the passport and ask for a laissez-passer. You find that you are not a resident of Israel because you have no certificate of residence. You think this is a joke and rush to tell to your lawyer friend: "Here I'm not a citizen, and I'm not a resident either. Then where and whom am I? You are surprised to find that the law is on their side, and you must prove you exist. You ask the Ministry of the Interior, "Am I here, or am I absent? Give me an expert in philosophy, so that I can prove to him I exist."

Then you realize that philosophically you exist but legally you do not. 6 What matters in the absurd is less the pessimistic vision of humanity as a whole, struggling vainly to find a purpose. What matters is the nonsensical as experienced by individuals such as, for instance, the main character in the Trial of Franz Kafka, a novel in which the judicial process turns into a farce, mixing tragedy and comedy, emotions of anxiety and derision. The nonsensical in the biduns' lives take the form of not knowing what the rules of the naturalisation or regularisation game are, of having to repeat procedures with no idea of where they stand in the process -having lost sight of the outcome, and above all of being told who they are in contradiction to who they feel they are. Resorting to this literary concept illuminates fundamental elements of the lived experience of the biduns that more conventional approaches to bidun life fail to account for, namely the opacity and contradictions of their administrative situation and temporary condition and the painful dislocation of identity both defined by tragedy and farce. It also takes into account the complexity of the metaphysical anguish and resilience that results from these social processes and takes an invisible toll on bidun individuals. It brings the biduns' experience away from the sole victim narratives towards subjects using derision, and metaphorically transgressing norms and their condition.

Questioning the Rationalizations of Nationality Attribution

Albeit underexplored, the transgressive power of laughter lies in the fact that it calls into question the normative rationalizations that the Central System has developed. In particular, it challenges the norms, narratives and basic reasoning, drawn from principles of nationalism and sovereignty that the Central System takes to extremes. Here, the tenuous conjecture by which nationality is attributed to biduns fosters laughter and a comic tone. Power is disavowed in the sense that helplessness has changed sides. Laughter is not a coping strategy but a response to an oppressive political structure.

The protracted struggle of the biduns has lasted for decades. The first decades of Kuwait's independence (1960s-80s) were marked by uncertain and inconsistent policies toward those who were later known as biduns. During the nineties and early noughties, the government gradually manufactured a category for all individuals of undetermined status called "illegal residents," per its official wording in administrative policy. In a performative way, this policy stripped its members from their rights. It contributed to the pauperisation of the group, by estranging them from the privileged citizens, increasingly defined in classist and materialist terms. In parallel, special committees were mandated with the task to "uncover the true origins" of the bidun claimants. Over the past two decades, the government has tried two administrative approaches to ascertain the alienness of the group: a genealogical conjecturing approach and the resort to genetic testing, deemed scientifically infallible but never practically implemented. Since 2010, the Central System is the entity responsible for the genealogical searches. Yet after decades of policies designed to pressure the biduns into claiming other nationalities as their "true" ones, the tenuous conjecture on which the Central System relies eventually suggests a desperate absence of solution. The spontaneous laughter it triggers, exemplified in the tweet which opened this paper, if not clearly oppositional, is a form of defiance. It lays bare the tacit admission of impotence on the part of the power.

Genealogical Trees and the Impossible Rooting

Since 1993, six years after a secret decree imposed the label of "illegal residents" to all individuals of undetermined status, the Kuwaiti Ministry of Interior has tasked special committees with the registration and follow-up of the biduns' files. These administrative entities have established figures and collected data by centralising scattered information across various state ministries and institutions. Officially, this practice of cross-checking data seeks to "uncover the true origins" of the claimants. The Central System was set up in November 2010 as an independent entity that, contrary to previous committees was distinct from the Ministry of the Interior. It was endowed with a five-year mandate to resolve once and for all the issue of the biduns claiming Kuwaiti nationality. But its mandate was renewed in 2015 for want of reaching its stated goal and the temporary institution seems to become rather permanent.

The Central System works to identify and ascertain the national origin of biduns on the basis of the secret gathering of information, drawn from the institution's own genealogical research or from data held by other government agencies. Biduns are assigned a foreign nationality and foreign identity, based on the Central System's reconstitution of their family history. Yet "there is here a fundamental misunderstanding between nationality, the official state recognition and origin, inferred from stated documentations," recognises Dr. Rashid Hamad Al-Anezi, a Kuwaiti academic and lawyer specialising in nationality. 7 The state's inference of origin is no more than a presumption, obtained by deduction.

In Kuwait, as in most Gulf states, nationality law is premised on the idea that nationals must prove the purity of their lineage (as in being original or pure /asli in Arabic) and their pre-oil presence in the Emirate. 8 Nationality gives entitlement to benefit from the bounty of the soil and share in the hydrocarbon wealth drawn from it, and by extension, forms the basis to the right to own property as opposed to aliens who, until recently, could not. On the contrary, those who are deemed from dubious genealogical lineage and portrayed as post-oil migrants or latecomers are denied nationality. Their foreign origins exclude them from the bounties of the land.

During the last decade, the Central System has issued documents supplemented with "presumptions of nationality." Here are some examples of what these documents include: "presumptive evidence of Iraqi nationality" -the official translation for the Arabic, which literally reads "the father/ the grand-father has a presumptive evidence suggesting an Iraqi nationality/ lada al ab/ ladayhi al jadd qarina dalat ala al jinsiyya al iraqiyya ) or "Saudi/Iraqi due to a father's paternal uncle" (Saudi al jinssiyya taba'an li-'am al ab). The Arabic word used, presumption (qarina), is the same one as presumption of innocence. This is far from providing evidence or proof. During my fieldwork, for instance, I recall one bidun mentioning that his family was considered "of Syrian origin" on the basis of a neighbour's testimony stating that his paternal uncle came to register in Kuwait in 1975.

[add pictures of the sentence "presumptive Iraqi" if found good enough] These attributions of forced identities have been gradually systematised since 2010 but they were practised before. In January 2007, as the government eased access to temporary driving licenses, a great number of biduns refused to register to get the document. They pointed to the fact that the "nationality" field was pre-filled with the mention "illegal resident" (muqım bi sura ghayr qanuniyya). Likewise, other biduns denounced the fact that the Ministry of Health issued birth certificates with the "nationality" field of parents arbitrarily filled in with foreign nationalities. Likewise, when people resorted to the court to get married, the court would detain files under their name with the note 'min usul 'iraqiyya', 'of Iraqi origin'.

The bone of contention is that biduns refuse to formally acknowledge a forced, attributed origin that erases their life-long experience in Kuwait. Moreover, any acknowledgement would tacitly mean a formal drop of their claims to citizenship. In the face of the biduns' rejection of this forced identity, the practice of the Central System is tantamount to blackmail. It forces biduns to accept un-belonging in order to acquire identification papers, without which they cannot interact with the rest of Kuwait's government bodies nor can they register their children in schools. In recent years, biduns were allegedly requested to sign a form that certified the correctness of the information written on their identity card before they were actually handed over to them. They were also asked to sign a blank piece of paper on which the Central System would later write a letter, according to the 2018 Department of State's Human Rights report, "purportedly "confessing" [their] "true" nationality such as Saudi, Iraqi, Syrian, Iranian, or Jordanian, which rendered them ineligible for recognition or benefits as Bidoon."9

The biduns' continued belonging is paradoxically conditioned on their formal un-belonging.

The Central System's main objective is to release a series of figures that support the gradual diminishment of the biduns' number, no matter how. From an already-underestimated official number of around 106,000 in 2010, biduns reportedly numbered only 88,000 eight years later. even though no large batch of naturalisation had occurred. On the contrary biduns have joined other refugees on the deadly routes to seek asylum abroad. The Central System excels at statistics; yet the Kuwaiti state and society have yet to engage in a reflection on access to nationality as a longer-term process of regulating presence and identity.

Biology in the stead of Politics: Genetics as the solution?

Following the same genealogical logic, the government of Kuwait also considered the idea of resorting to genetic tests to help arbitrate its contention with this population. Just like alienness can be inferred from a father's paternal uncle, the genetically-proven affiliation with a Kuwaiti citizen could support cases for naturalisation. Many Kuwaitis believe, out of weariness with the protracted bidun issue, that resorting to DNA could provide an incontestable scientific arbitration. It would establish the truth, where politics and political authority fail to act decisively.

The obsession with ascertaining truth through genetics is linked with the idea of preserving purity. 10 The biduns' existence at the margins of Kuwait's society challenge the myth of the original purity, as a pillar of the citizen's definition. In September 1998, the Kuwaiti government announced that genetic tests would be required from all biduns to prove their claim to Kuwaiti nationality through family lineage or through kinship to a Kuwaiti mother, father or other relative. Yet using biology as a criterion to determine the fate of the biduns rather than the duration of their residence or their feeling of belonging or cultural affinity with Kuwait was never applied, 11 despite being proposed several times. It remained at the stage of functioning as a threat to pressurise biduns into designing their own alternative and often desperate solutions to their administrative conundrum. Using the naïve belief of the infallibility of science, the state's threat to use genetic testing aims at unmasking "fraudulent" claims, i.e. not supported by a genealogy defined as pure.

The Absurdities of the Injunction to not Belong

Those laughing also denounce the absurdity of the policies towards biduns. The extreme obstinacy with which the Central System has tried to fit each biduns' case into a "previous nationality" box as part of the official policy of attributing foreign nationality boggles the mind. It is on this loss of meaning and the emotional implications of it, that I want to stress. The felt reality is that spontaneous processes of rooting and belonging create emotional links between long-term residents on the one hand, and places, persons and memories associated with them, on the other. Yet these processes are set on a collision course with administrative procedures that seek to unravel identity claims. Such procedures use a vertical logic, that of the genealogic and genetic rationale, in order to impose an injunction not to belong, or to 'unbelong'.

The Absurd: between Tragedy and Farce

10 Kuwait has the financial and technical means to carry out mass DNA tests. An August 2015 DNA test law made it mandatory for all citizens and foreign residents to be included in a national DNA database. Initially designed to address security concerns, the law was widely criticised internationally for its privacy implications and subsequently abandoned when the Constitutional court judged it contrary to the Constitution. 11 The Kuwaiti DNA database was not set up before 2002.

The Central System utilizes biological and genealogical approaches to resolve a fundamentally political issue. The stubborn endeavour to attribute nationality is sometimes deceitful and inconsistentwith some biduns, for instance, holding documents in which their presumed nationality is Iranian and others in which it is Iraqi. This reiteration of performative yet contradictory statements reminds one of the Absurd Drama's use of dislocated language, repetitions and incoherent plots to signify the loss of meaning. The characters in Ionesco's Bald Primadonna sit and talk like broken records, repeating the obvious until it becomes nonsense, reflecting the influence of comic tradition using disembodied characters or puppets such as in commedia dell'arte or mimes and pantomimes. The repetition is part of the farce: biduns are expected to repeat the same actions and renew their IDs, whose validity is often now as short as six months, all with the hope of an eventual different outcome. Within the circumvolutions of the administrative process, they never know at which step they stand. The literary concept of the absurd captures the processes which people are subjugated to with more clarity than political science approaches to citizenship can do.

The biduns experience state-made or state-led dislocation of identities that fosters dissonance between their lived and state-controlled experiences. The biduns are not part of the statesanctioned population yet at the same time they cannot escape the state they are excluded from either. What is absurd here is not human existence proper but the frame through which it is moulded and produced. The imposition of an unwanted identity is brutal. The bidun category most probably is cemented partially and paradoxically through this shared experience of being given an identity through administrative violence that they reject. Such policies produce a situation where people feel and think that they are at home where they actually live but are denied both this bond and this belonging. They are told to live transiently in a place without growing roots. It means they can live this rootless life for many generations. This is the "injunction to not belong." However, despite oppressive policies of dislocated identity and denied belonging, the biduns have a strong survival force: they resist, they are resilient and they appropriate places and identities. People prevented from belonging are not passive, as shown by Yasser Elsheshtawi in Temporary Cities: Resisting Transience in Arabia. For instance, migrants appropriate the places they inhabit, like street corners and display a huge and natural ingenuity to assert their belonging. 12 During the Covid-19, biduns in Kuwait displayed their selfless commitment for the country, "as they have stepped forward as healthcare volunteers". 13 This strong civic engagement echoes the campaign of blood donation that the biduns had organised in 2011 in the wake of the mobilisation movement as a demonstration of their natural belonging to and caring for Kuwait.

Existence as Resilience: Political Activism and Literary Unavoidability

Biduns have shown resilience in their ways of belonging. Until the end of the 2000s, the biduns had nurtured individual hopes to be naturalised. Since then efforts to decompartmentalize their claims led to the emergence of a bidun movement as Kuwaiti biduns, emphasizing their national feeling of belonging and their shared experience of discrimination as a social group. The bidun movement became mobilized in 2011 with the Arab popular mobilisation. Since then, the collision of patterns of belonging and un-belonging has become more visible and the so-called protracted "issue of biduns" has garnered attention far beyond the circles of human rights defenders, with the Central System stepping up efforts to formally reduce their numbers.

In February 2011, the biduns mobilised and took to the streets. Emboldened by the Arab uprisings, they broke with their hitherto non-confrontational practice of using patronage links to obtain concessions from the state. The protest movement was met with fierce repression. Yet, in the face of renewed oppression, the biduns experimented with a new creative, dynamic and bold repertoire of contentious action to claim their entitlement to nationality and demand the recognition of their basic rights.

First, biduns appropriated street protest and sit-ins as a new contentious practice. As "illegal residents," the biduns, like foreign residents, are not allowed to demonstrate according to the 1979 Law on Public Gathering. Protest was a departure from the awareness-raising campaigns of the late 2000s. Working with Kuwaiti activists to garner support from citizens, the biduns organised a sustained demonstration campaign that persisted, even in the face of repression, until 2019. This constituted an act of unequivocal political contestation against the government's policies. Importantly, the 2011 protest created an iconic site dedicated to the contestation: the waste parking lot adjacent to the Sha'bi mosque in the bidun popular housing district of Jahra is now known as Freedom Square (sahat al hurriya). The Square belongs to biduns' history and memory.

[picture here of the Freedom Square] Through taking to the street, being arrested, contesting judicial decisions and putting their bodies at risk through hunger-strikes and even committing suicides, the biduns forced the state to recognise them as legal subjects rather than inexistent beings, administered solely by the Central System in an attempt to shield them from any type of legal interaction. Claiming universal values of freedom and dignity, bidun activists also articulate a new understanding of citizenship that emphasises their current presence in Kuwait and embraces cultural difference over the genealogical determinism that justified discrimination and left no room for diversity of origin in the national identity. Genealogical determinism shares a lot of commonalities with racism, as a systemic and systematic exclusion.

Finally, a student solidarity campaign, mobilisation for education and an active literary movement among the biduns have shaped new community and civic engagement. They overcame the dichotomy between biduns and the majority of nationals that the government had actively created, fostered and capitalised on by portraying the biduns as fraudsters interested in the financial worth of nationality only so as to prevent nationals to feel sympathetic to their plight. Kuwaiti writer Abrar Al-Shammari notes the rising numbers of literary works from Kuwaiti citizens and Kuwaiti bidun authors that deal with the situation of the biduns, particularly since 2011. It has become an important sub-genre of Kuwaiti literature." She cites Khaled al Torki, a bidun author who "believes that the circulation of bidun literature contributes to countering dehumanizing state and media narratives around the bidun community." 14 These new forms of public recognition and creative acts of contestation are transforming the terms of political activism in ways that question prevailing norms of citizenship. They contest the idea of living in a state of un-rootedness and lay bare the delusionary character of the absurd injunction to not belong.

Faced with the absurdity of being told they are what they feel they are not, they have resisted the attribution of nationality and the ascribing of origin. Faced with the absurdity of the Central System's far-fetched links to rationalize these nationality attributions, they cannot help but resort to laughter. The paradox of the absurd is that it is often tragic and violent but eventually also becomes comic. While the absurd reflects an extremely deep anxiety, a metaphysical loss of meaning, a painful dislocation of identity, it also holds potentialities for more disruptive or liberating practices. To put it like Martin Esslin: "The shedding of easy solutions, of comforting illusions, may be painful, but it leaves behind it a sense of freedom and relief. And that is why, in the last resort, the Theatre of the Absurd does not provoke tears of despair but the laughter of liberation." 15

https://twitter.com/bobashig/status/979440149854654465

The biduns, literally, the people "without" (nationality), refer to people who have been excluded from nationality since the 1960s as a result of inconsistent state policies and who, since then, have been in contention over their entitlement to nationality, with the state of Kuwait that regards them as illegal on its territory since a 1986 decree.

The Central System is short for the official name, the Central System for the Remedy of the Situations of the Illegal Residents.

See in the Palestinian context of precarity and occupation, the excellent piece of Bhungalia, Lisa, Laughing at power: Humor, transgression, and the politics of refusal in Palestine. Environment and Planning C:Politics and Space, 38(3), 2020, 387-404.

As a reminder, the concept of Absurd was coined originally to refer to a non-organized literary movement and specifically but not exclusively, to playwrights such as Samuel Beckett, Eugene Ionesco and Jean Genet. The literary critic Martin Esslin, who theorised it, noted common points between playwrights that primarily conveyed a sense of incomprehension and even despair, in the face of what was felt as a lack of consistency and cohesion in the world. The plays are characterized by incomprehensible plots, that seem to start and end arbitrarily, unmotivated actions and meaningless dialogues. Esslin, Martin "Introduction" to Penguin Plays-Absurd Drama, Harmondsworth: Penguin, 1965. p. 12. Here the concept of absurd is understood more broadly than the definition of the sole Absurd Drama.

Darwish, Mahmoud, Journal of an Ordinary Grief (transl. by Ibrahim Muhawi), Brooklyn: Archipelago books, 2018. p.66

My emphasis. Beaugrand Stateless in the Gulf: Migration, Nationality and Society inKuwait, IB Tauris, 2018. p. 128

Kuwaitis by origin are legally defined as the persons and their descendants who are able to prove their continued presence since 1920 in "Kuwait," a yet to be defined sovereign territory.

Department of State (DoS), Kuwait 2018 Human Rights Report, p.16 https://www.state.gov/wp-content/uploads/2019/03/KUWAIT-2018.pdf

Elsheshtawi, Yasser Temporary Cities: Resisting Transience in Arabia, London: Routledge, 2019.

McGee,Thomas "The COVID-19 Crisis and New Agency for Stateless Bidoon in Kuwait", LSE Blog Post, 27 April 2020. https://blogs.lse.ac.uk/mec/2020/04/27/the-covid-19-crisis-and-new-agency-for-stateless-bidoon-in-kuwait/

Al-Shammari, Abrar "The Rise of Bidun Literature: Representation and Advocacy in Kuwait", Blog Post, The Arab Gulf States Institute in Washington, 2 December 2019. https://agsiw.org/the-rise-of-bidun-literature-representation-and-advocacy-in-kuwait/