Designing ecient hull coatings for acoustic stealth of submerged vehicles remains a challenge that evolves along with improvements in sonar performance. Low frequency performance is particularly important for military applications, due to the long range propagation properties of low frequency signals in the sea. Acoustic metamaterials are now widely recognised as promising candidates for such underwater coatings. The aim of this research is to study metamaterials for underwater applications, from theoretical analysis to experimental characterisation. This thesis presents analytical or semi-analytical models based on a transfer matrix formalism to homogenise periodic structures. These methods are then used in a computationally ecient optimisation approach to obtain metamaterial designs that meet a goal in terms of their anechoism coecient, hull decoupling coecient, or both. The entirety of this approach is rst carried out for multilayered structures then extended to metamaterials with macro-inclusions to which a degree of complexity is added. Finally, measurement methods in a water tank facility are implemented and validated. They are then used for the experimental characterisation of panels manufactured within the framework of the thesis.

Thank you to all the Frenchies! And my deepest thanks to the amazing Tatjana, Iris and Beeh, for the support and the great times we spent together.

IEMN, Lille

De retour en France pour ma dernière année de thèse, j'ai été chaleureusement accueillie par l'équipe d'acoustique de l'ISEN de Lille, que je remercie ! En particulier, je remercie Charles Croenne, pour les échanges toujours pertinents et intéressants et pour m'avoir tant aidé à prendre du recul sur mon travail. Je tiens aussi à remercier Monique Pouille et sa bonne humeur qui ont fait des journées de mesures des moments intenses mais agréables, où j'ai beaucoup appris. Enn, j'adresse un immense merci à Anne-Christine Hladky, pour son encadrement plein d'énergie, de gentillesse, d'idées pertinentes et de connaissance scientique. Son enthousiasme dans nos discussions m'a aidé à rester motivée et dévouée jusqu'à la n de la thèse. Un grand merci ! Enn, je remercie mes amis de longues dates, Lucie, Audette, Jordane, mes pompoms, qui m'ont toujours apporté leur soutien et bonne humeur malgré la distance. Je remercie aussi mes nouveaux collègues du CEMIS/AC et amis de l'aire Toulonnaise pour leur soutien et encouragement au quotidien lors de la dernière ligne droite. Je tiens aussi tout particulièrement à remercier ma famille, et notamment mes parents et à ma s÷ur pour avoir fait le voyage jusqu'en Australie pour passer Noël à mes côtés. Je peux

Chapter 1 Introduction and Literature Review

In this introduction, the concepts of underwater acoustic discretion and stealth are introduced and some of the current challenges are highlighted. In this context, acoustic metamaterials occur to be promising candidates to improve the discretion and stealth performances of underwater vehicles. A literature review is then conducted, focusing on the concept of metamaterials and some of their specic uses in an underwater environment. Following the literature review, the denition for concepts employed in this thesis are given. Finally, the thesis overview states the main goals of the current research and presents the thesis plan. Underwater passive detection consists in analysing ambient sound in order to recognise characteristic sounds from a submarine or marine vessel. Those sounds mainly originate from vibrations of items of machinery, engines and propellers, which overall result in radiated noise that can be detected by an adverse passive SONAR system. Against such a threat, the discretion of a submarine or underwater vehicle is crucial. A solution, amongst others, is to place hull decoupling coatings on the submarine, which signicantly reduce noise radiated from the hull.

Active detection is the other main acoustic threat, consisting in an acoustic wave sent in the underwater environment and in the detection of the potential reected waves from an obstacle, thus informing on its presence and location. The vehicle stealth is thus also essential. Target strength, dened as the ratio between the back-scattered acoustic energy and the incident acoustic energy, can be reduced by adapting the external shape, by using acoustic deectors or by applying anechoic coatings made of absorbing materials on the submarine outer hull in order to minimise back-scattered echoes.

Anechoic and hull decoupling coatings have been developed and used since World War II, starting with the Alberich-type design which consists of lattices of resonant cavities in an elastomer or rubber matrix, as represented in Figure 1.2. The cavities' diameters are tuned to maximise absorption for several specic frequencies. Nevertheless, there are also some non-acoustic requirements for the design of coatings, such as the weight and the total thickness. In particular, the static compressibility and stability of acoustic performances as a function of depth in the ocean must be taken into account in the design process of a coating. Hence, some coatings such as Alberich coatings can lead to interesting acoustic performances. However, their performance is often limited to specic frequencies, and in addition they may not comply to non-acoustic requirements. metamaterials appear to be a potential solution to such a challenge, opening up new opportunities for the design of submarine coatings, mainly due to their strong acoustical eects at frequencies much lower than the ones corresponding to the typical coating thickness or inclusion sizes. In the following literature review, the concept of acoustic metamaterials is presented.

Thesis Overview 1.2.1 Goals

The aim of this research is to study, develop, implement and apply methods and tools that are useful to analyse and design metamaterials. More specically, these tools and methods are to be used to examine structures for underwater applications, such as anechoic and hull decoupling coatings.

In this work, tools and methods are implemented for a full research and development process, from the theoretical analysis to the experimental characterisation of a structure. As such, analytical models or semi-analytical models need used or developed to theoretically study a design. Such analytical models or semi-analytical models are then to be used in a computationally ecient optimisation approach aiming to obtain ideal metamaterial designs for the anechoism, the hull decoupling, or both coecients. Experimental validation are then carried out, for which measurement methods are developed and validated.

Plan

The manuscript is divided in two main parts. In Part I, multilayered designs are studied.

They are simple but still interesting designs, for which the homogenisation, optimisation and experimental approaches are introduced. Designs of greater complexity with macroinclusions are then considered in Part II. A summary of each Chapter is presented in what follows.

Chapter 2 introduces the notions of the transfer matrix and homogenisation models. In particular, it presents an eective medium theory based on a transfer matrix approach.

This homogenisation process aims to eectively represent any multilayered arrangement as a uid or a Willis-type uid (depending on the symmetry of the medium). Eective properties, such as the eective wavenumber, impedance, density, bulk modulus, can be derived and used to characterise an eective medium of a given length. Such eective media can predict the scattering response of multilayered media of any length.

In Chapter 3, the eective medium method for multilayered media presented in Chapter 2 is used in an optimisation approach. The objective is to improve the anechoism performance or the hull decoupling performance by optimising the layer arrangement. Layers may be either steel or polyurethane with micro-balloons. The algorithm is rst validated on a test case. Optimisation results are then analysed and a feature of performance is found for each coecient. An asymmetric design is selected for its anechoic performance.

Chapter 4 aims at presenting the measurements for the selected conguration of Chapter 3. Beforehand, the principle of underwater measurements is exposed. An original method, referred to as the 3-point method, is also presented. The method's validity and robustness are assessed through measurements on an aluminium panel, for which edge-diraction eects are signicant. The method is then applied to the polyurethane used in Chapter 3 in order to experimentally retrieve the eective speed of sound in the medium. Finally, the layer arrangement selected is measured and the anechoism coecient is compared to the predictions.

Two retrieval methods, referred to as the direct inversion method and the dierential method, are presented in Chapter 5. Both methods lead to the eective properties of a symmetric unit cell containing a macro-inclusion. These eective properties can then be used to characterise eective media, from which predictions of the scattering coecients are made for designs comprising dierent numbers of unit cells. The retrieval methods are validated on a multilayered structure, for which the retrieved eective properties are found to be the same as those obtained using the transfer matrix model in Chapter 2.

Two-dimensional designs comprising periodic cylindrical rigid or voided inclusions in a soft elastic matrix are then studied. Results led to the creation of an hybrid prediction method, which uses both the direct inversion method and the dierential method, to predict the scattering response of a two-dimensional design of any length.

In Chapter 6, the two retrieval methods are extended for asymmetric structures with macro-inclusions. These extended methods are applied to an asymmetric variant of the locally resonant medium comprising steel inclusions, studied in Chapter 5. Eects of the asymmetry are investigated. A second case study considers an Alberich-type medium for which Willis coupling is introduced by combining two asymmetric layers to compare the results of the extended method to another approach that breaks the structure down in symmetric layers. It is shown that with such approach, unexpected results can be obtained.

Chapter 7, the two retrieval methods are used to create a database of eective properties associated to dierent unit cell designs. This database is used as an input into an optimisation algorithm that aims to optimise the unit arrangement of structures for anechoism performance. As such, each unit is seen as an eective layer, so the algorithm approach introduced in Chapter 3 is used. Complex structures with macro-inclusions can be therefore be optimised with short computation times. Moreover, similar performance features as in Chapter 3 are found using the eective impedance along the medium thickness.

In Chapter 8, the Alberich medium studied in Chapter 6 is experimentally analysed.

For this purpose, the new challenges originated from the periodicity of the grating are highlighted. The 3-point method is tested on this non-homogeneous medium and extended for greater accuracy. Experimental results are then compared to predictions.

A general summary of the manuscript is nally proposed, in which of the main novelty and relevant results are summarised. Ideas for further work are also given.

Literature Review 1.3.1 Concept of Acoustic Metamaterial

Etymologically, metamaterial qualies a material that goes beyond materials. The reason for such a term is found in the capacity of these articial materials to exhibit properties that cannot be found for conventional materials. In the context of sinusoidal steadystate analysis for acoustic metamaterials, the dynamic material properties can reach negative values or strongly vary with frequency, contrarily to conventional materials.

Hence, the most common accepted denition of an acoustic metamaterial is : "Articial material with the capacity to exhibit unusual properties, that is to say with values or strong variations with frequency of the constitutive acoustic parameters not found for conventional materials. These unusual aspects arise from the collective manifestation of the internal constituent units in the structure, such as resonant inclusions arranged in a host matrix" [2]. Nevertheless, a clarication has to be added regarding these unusual acoustics properties. Due to the complex structure of acoustic metamaterials, the most common study scheme involves the use of an equivalent medium that reproduces the overall behaviour of the metamaterial. The equivalent medium is referred to as an eective medium, as shown in Figure 1.3. Eective properties of a metamaterial are thus the material properties of its eective medium. Such a homogenisation approach is usually valid if the scale of the constituent unit is much smaller than the wavelength.

Nonetheless, eorts have been made to extend conventional eective medium theories to higher frequency regimes [START_REF] Quan | Maximum willis coupling in acoustic scatterers[END_REF].

Effective medium Figure 1.3 Homogenisation approach, in which a composite material is represented by an eective medium.

Most acoustic metamaterial concepts' features rely on the behaviour of inclusions within a matrix, which may be arranged periodically or randomly distributed. It is then worth mentioning the distinction of a phononic crystal, which is a medium comprising of periodic arrangements of inclusions which produce wave phenomena resulting from the periodicity of the scatterers. If the inclusions are resonant, the medium may also be referred to as a locally resonant phononic crystal, and thus become a periodic metamaterial since the behaviour of the inclusion is involved [START_REF] Croënne | Band gaps in phononic crystals: Generation mechanisms and interaction eects[END_REF].

Most research on metamaterials include the development or application of an homogenisation method to obtain one or several eective material properties, which are commonly the mass density, the bulk modulus or the refractive index. The wavenumber and impedance are also widely used. Acoustic metamaterials have been of great interest to the scientic community as unusual values and strong variations of these dynamic eective properties are a translation of physical phenomena occurring in the composite structure. For ex-ample, a negative eective bulk modulus corresponds to a resonance that results in an expansion of the structure while loaded under a compressive force, as illustrated in Figure 1.4(a).

Figure 1.4 Schematic illustrations of the dynamic behaviors of locally resonant materials with negative-valued eective parameters submitted to harmonic excitation. (a) A material possessing a negative eective bulk modulus (κ < 0) supports a volume expansion (∆V > 0) upon an isotropic compression (∆P > 0) (b) The acceleration a of a material possessing a negative eective mass density (ρ < 0) is opposite to the driving force F . [START_REF] Brunet | Soft acoustic metamaterials[END_REF].

A negative bulk modulus can be achieved by using a Helmholtz resonator, which creates the phenomenon of air resonance in a cavity [START_REF] Fang | Ultrasonic metamaterials with negative modulus[END_REF], as shown in Figure 1. 5(a). It can also be obtained using constituent units comprising side holes with both ends opened [START_REF] Lee | Composite acoustic medium with simultaneously negative density and modulus[END_REF], as in Figure 1.5(b), or with periodic distributions of cylindrical boreholes in a waveguide [START_REF] Graciá-Chocano | Quasi-two-dimensional acoustic metamaterial with negative bulk modulus[END_REF],

with balloon-like soft resonators [START_REF] Jing | Soft resonator of omnidirectional resonance for acoustic metamaterials with a negative bulk modulus[END_REF], or even with some specic unit shapes resulting from an optimisation algorithm [START_REF] Lu | Topology optimization of an acoustic metamaterial with negative bulk modulus using local resonance[END_REF]. Broadly, the negative bulk modulus results from a monopolar resonance of the component units [START_REF] Li | Double-negative acoustic metamaterial[END_REF].

Membrane

Side hole The inset illustrates the analogy between a Helmholtz resonator and an inductorcapacitor circuit, showing the uidic inductance due to the neck, and the acoustic capacitance due to the cavity [START_REF] Fang | Ultrasonic metamaterials with negative modulus[END_REF]. (b) Composite structure consisting of interspaced membranes and side holes [START_REF] Lee | Composite acoustic medium with simultaneously negative density and modulus[END_REF].

Another typical metamaterial feature is an eective density that becomes negative. This is the result of a dipolar resonance of the building units [START_REF] Li | Double-negative acoustic metamaterial[END_REF]. The dipolar resonance corresponds to a resonance that leads to a motion of the centre of mass of the particle which is out of phase with an incident directional pressure eld, as illustrated in Figure 1. 4(b).

With the appropriate polarisation, dipolar resonance can lead to signicant transmission attenuation. The blocking eect capacity due to negative eective density can be easily demonstrated using a mass-spring structure [START_REF] Zhou | Elastic metamaterials with local resonances: An overview[END_REF], as shown in Figure 1. 6(a). Nonetheless, the unusual low-frequency bandgaps observed in such negative density metamaterials are generally only present in a narrow bandwidth around resonant frequencies. Practically, dipolar resonances that lead to negative values of the eective density can appear using constituent units consisting of a heavy mass core surrounded by a soft polymer, both being arranged in a relatively rigid matrix [START_REF] Liu | Locally resonant sonic materials[END_REF] (Figure 1.6(b)). A similar eect can be achieved using, amongst others, membrane-type metamaterials [START_REF] Yang | Membrane-type acoustic metamaterial with negative dynamic mass[END_REF], mass-spring systems [START_REF] Yao | Experimental study on negative eective mass in a 1D mass-spring system[END_REF] or anisotropic cylindrical scatterers penetrating in one of the rigid walls dening a 2D

acoustic waveguide [START_REF] Graciá-Salgado | Negative mass density and ρ-near-zero quasi-two-dimensional metamaterials: design and applications[END_REF], as shown in Figure 1.6(c).

Figure 1.6 (a) Mass-spring structure with negative eective mass [START_REF] Zhou | Elastic metamaterials with local resonances: An overview[END_REF]. (b) Cross section of a coated lead sphere that forms the basic structure unit [START_REF] Liu | Locally resonant sonic materials[END_REF]. (c) Schematic view of an array of anisotropic scatterers embedded in a 2D waveguide [START_REF] Graciá-Salgado | Negative mass density and ρ-near-zero quasi-two-dimensional metamaterials: design and applications[END_REF].

Therefore, local resonance of a constituent unit cell can lead to negative values and/or strong variations of either the bulk modulus or the eective density, which then add an imaginary part to the speed of sound. This in turn means that the acoustic wave amplitude dissipates with increasing distance. A considerable amount of research on acoustic metamaterials also aims at combining constitutive units leading to a monopolar resonance (negative bulk modulus) with resonant inclusions creating a dipolar resonance (negative mass density). As such, the phase speed is again a real-valued number, implying wave propagation, although with a twist: the energy and phase velocities are in opposite directions. This response provides metamaterials with the capability to produce unnatural eects like negative refraction. Some examples are metamaterials composed of membranes and side holes [START_REF] Lee | Composite acoustic medium with simultaneously negative density and modulus[END_REF] or of Helmholtz and rod-spring resonators [START_REF] Fok | Negative acoustic index metamaterial[END_REF]. Another way to achieve the double negativity is to design hybrid resonators, producing a hybridisation between monopolar and dipolar resonances. Such resonators could be chiral microstructures (Figure 1.7(a) [START_REF] Liu | An elastic metamaterial with simultaneously negative mass density and bulk modulus[END_REF]) or a composite of soft rubber spheres suspended in water [START_REF] Li | Double-negative acoustic metamaterial[END_REF]. Other resonators can also create anti-resonances [START_REF] Zhou | Broadband lowfrequency membrane-type acoustic metamaterials with multi-state anti-resonances[END_REF], quadrupolar resonances (Figure 1.7(b) [START_REF] Zou | An ultra-thin acoustic metasurface with multiply resonant units[END_REF]), or multiple resonances, for example the multi-mass resonating unit cell of Lai et al. [START_REF] Lai | Hybrid elastic solids[END_REF] shown in Figure 1.7(c) that leads to monopolar, dipolar and quadrupolar resonances, and results in novel elastic properties.

Figure 1.7 (a) Unit cell of the 2D metamaterial made from solid media. The unit cell is composed of three-component continuum media by a chirally soft-coated heavy cylinder core embedded in a matrix [START_REF] Liu | An elastic metamaterial with simultaneously negative mass density and bulk modulus[END_REF]. (b) Inclusion composed of a soft rubber cylinder embedded with a hard rubber cylinder, surrounded by four rectangular steel rods in the shape of an X [START_REF] Zou | An ultra-thin acoustic metasurface with multiply resonant units[END_REF]. (c) Elastic metamaterial unit showing multiple resonances [START_REF] Lai | Hybrid elastic solids[END_REF].

Determination of Eective Properties

Eective properties are therefore important for an in-depth analysis of a metamaterial design and its specic features. In the following section, a non-exhaustive overview of techniques to obtain eective properties is given.

Analytical Models

A range of analytical techniques have been employed to derive the eective properties of composite structures. For example, eective properties of multilayered structures are often derived using fully analytical models for one-dimensional wave propagation [2426]. If the multilayered medium is periodic and symmetric, the derivation of the overall eective parameters can be reduced to the calculation of the eective properties for the constitutive unit [START_REF] Yang | Full-band exact homogenization of one-dimensional elastic metamaterials[END_REF]. For such cases, as presented in Figure 1.8(a), the transfer matrix method is found to be a suitable tool to characterise a multilayered structure [START_REF] Xu | Double-negative dynamic properties in onedimensional multi-phase metamaterial based on the symmetrical equivalent layer[END_REF]. However, most studies based on the the transfer matrix method are limited to the analysis of symmetric unit cells [START_REF] Song | A transfer-matrix approach for estimating the characteristic impedance and wave numbers of limp and rigid porous materials[END_REF]. Multilayered media with non-symmetric units can still be using such an approach, but the investigation is limited to the dispersion curves [START_REF] Wang | One-dimensional phononic crystals with locally resonant structures[END_REF][START_REF] Zhao | The band gap of 1D viscoelastic phononic crystal[END_REF] or using an average density [START_REF] Rytov | Acoustical properties of a thinly laminated medium[END_REF]. In recent works, several methods have been extended to take into account the asymmetry of a structure which may result in Willis coupling, as presented in the next subsection. refers to a unit cell [START_REF] Xu | Double-negative dynamic properties in onedimensional multi-phase metamaterial based on the symmetrical equivalent layer[END_REF]. (b) Layer homogenisation applied to a PDMS medium with single layer of periodic voids [START_REF] Sharma | Sound transmission through a periodically voided soft elastic medium submerged in water[END_REF].

Analytical models are also used to homogenise two-dimensional designs with macroinclusions, such as periodically voided soft elastic media [3234], as shown in Figure 1. 8(b).

Whilst analytical methods provide insight into physical mechanisms for wave propagation and are generally computationally ecient, they are limited to simple geometric designs.

Methods for the Study of Willis Structures

In recent literature, there has been a considerable eort to extend methods to determine the eective parameters to non-symmetric structures. The scope of possibilities for the design of metamaterials has thus been opened up with the consideration of the Williscoupling, also referred to as acoustic bianisotropy [START_REF] Quan | Maximum willis coupling in acoustic scatterers[END_REF]. In acoustics, a Willis material shows a pressure-velocity coupling [START_REF] Su | Retrieval method for the bianisotropic polarizability tensor of willis acoustic scatterers[END_REF], which arises from inherent asymmetry in the microstructure. More precisely, the asymmetry appears when the element center's of mass is misaligned with the centroid [START_REF] Muhlestein | Willis Coupling in Acoustic and Elastic Metamaterials[END_REF], as illustrated in Figure 1.9.

Figure 1.9 Schematic description of the reaction of standard and Willis materials to compression and to a uniform force in one direction. The standard material under compression becomes smaller with the center unshifted, and under a uniform force translates without changing size. A Willis material, by contrast, both translates and changes size under both compression and a uniform force [START_REF] Muhlestein | Analysis of one-dimensional wave phenomena in willis materials[END_REF].

The acoustic bianisotropy can be optimised in order to be maximal at a subwavelength scale, which can lead to interesting properties resulting from the fact that both the dipole and monopole resonances can be excited by pressure and velocity elds [START_REF] Quan | Maximum willis coupling in acoustic scatterers[END_REF]. However, neglecting bianisotropy in the eective material properties may result in parameters that may violate the principles of causality and passivity for an asymmetric unit cell [3941].

To study such Willis metamaterials, extra parameters have to be determined compared to conventional symmetric materials. Muhlestein [START_REF] Muhlestein | Experimental evidence of willis coupling in a one-dimensional eective material element[END_REF] and Amirkhizi [START_REF] Amirkhizi | Homogenization of layered media based on scattering response and eld integration[END_REF] published methods to derive the eective properties for uniaxial Willis materials, which can be used to design unidirectional perfect absorbers [START_REF] Merkel | Unidirectional zero sonic reection in passive PT -symmetric willis media[END_REF] or bi-layer plate-type acoustic metamaterials [START_REF] Ma | Bi-layer plate-type acoustic metamaterials with willis coupling[END_REF].

Retrieval Methods

Retrieval methods are an alternative approach to derive eective properties that utilise scattering coecients obtained on the incidence and transmission sides of the medium, as illustrated in Figure 1. 10 [44, 45]. For such methods, the scattering coecients have been obtained analytically for a periodic multilayered medium [START_REF] Zhu | Eective dynamic properties and multiresonant design of acoustic metamaterials[END_REF], numerically for periodic metamaterials [START_REF] Croënne | Eective parameter retrieval of phononic crystal slabs[END_REF][START_REF] Méresse | Acoustic coatings for maritime systems applications using resonant phenomena[END_REF] or experimentally for acoustic metamaterials composed of split hollow spheres [START_REF] Ding | Multi-band and broadband acoustic metamaterial with resonant structures[END_REF] or periodically arranged hollow tubes [START_REF] Chen | Acoustic metamaterial with negative mass density in water[END_REF]. A retrieval method has also been extended for an asymmetric design [START_REF] Aghighi | Low-frequency micro-structured mechanical metamaterials[END_REF]. Park et al. [52] also proposed a method to homogenise anisotropic metamaterial slabs by determining eective properties of the constitutive unit cell. Within the slab, the eective properties dier whether the unit cell is an edge cell or an inner cell. An advantage of retrieval methods is that they are not constrained to simple geometries.

Figure 1.10 Technique used for obtaining the eective properties of a metamaterial, with the structure as well as with incident, reected, and transmitted plane waves. The eective material is also shown, along with the technique used for obtaining eective properties of a metamaterial [START_REF] Fokin | Method for retrieving eective properties of locally resonant acoustic metamaterials[END_REF].

Materials for Underwater Applications

Metamaterials have been developed for various underwater applications, such as carpet cloak [START_REF] Zou | An ultra-thin acoustic metasurface with multiply resonant units[END_REF][START_REF] Bi | Design and demonstration of an underwater acoustic carpet cloak[END_REF] or a screen converting underwater cylindrical waves to plane waves [START_REF] Chen | Broadband and high-transmission metasurface for converting underwater cylindrical waves to plane waves[END_REF]. This section reviews metamaterials used as coatings for underwater vehicles.

A simple way to improve wave attenuation is to use a large amount of absorbent material of a total thickness corresponding to several wavelengths. However, this is not suitable for the design of underwater coatings, which in practice can not exceed a certain thickness due to limitations related to naval architecture requirements for the vessel. In this context, metamaterials bring a new and promising perspective for thin coatings with good acoustic performance [START_REF] Bai | From local structure to overall performance: An overview on the design of an acoustic coating[END_REF]. Some concepts applicable to underwater applications are reviewed here.

Elastomers as Matrix

For underwater applications, elastomers are widely used, providing good vibration energy dissipation due to their viscoelastic properties. In particular, they are very dispersive and dissipative for shear waves but not signicantly for longitudinal waves. Elastomers are therefore widely used as a matrix material, in which elements are included to take advantage of these material properties. For example, a resonating inclusion may convert the longitudinal waves to shear waves, thus strongly dissipating the acoustic energy due to the shear material properties of the elastomer [START_REF] Hinders | Particle-loaded composites for acoustic anechoic coatings[END_REF]. However, if the dierence between the speed of sound in a resonating element and the surrounding elastomer is too high, the energy conversion may not occur [START_REF] Zhao | Tri-component phononic crystals for underwater anechoic coatings[END_REF].

In addition, the longitudinal sound velocity and density of an elastomer are generally close to the properties of water, resulting in impedance matching. An adequate impedance matching between the material interface and the surrounding water enhances coupling of the incident acoustic energy with the absorbers [START_REF] Ouahabi | Gradient metamaterial layers as impedance matching devices for ecient sound absorption[END_REF]. Hence, without good impedance matching at the interface, only a small proportion of the acoustic wave energy would be dissipated in the material. More generally, structures with gradient properties are a promising way to manipulate acoustic waves [START_REF] Feng | Acoustic behavior of composites with gradient impedance[END_REF]. For example, Naify et al. [START_REF] Naify | Underwater acoustic omnidirectional absorber[END_REF] developed an underwater acoustic omnidirectional absorber based on a gradient index medium.

However, that concept leads to thicknesses comparable to the wavelength, which may not comply with non-acoustic design criteria.

Micro-inclusion Materials

Micro-inclusion materials are elastomers comprising micro-cavities or soft-wall microballoons. There is usually a few percent volume fraction of these micro-inclusions, sometimes among other micro-inclusions such as carbon black or minerals in order to adjust the overall density. These inclusions of air bubbles or voids lower the eective speed of sound and provide better absorption. For high frequencies, micro-inclusion media can be regarded as metamaterials [START_REF] Lepert | Determination of the eective mechanical properties of inclusionary materials using bulk elastic waves[END_REF][START_REF] Brunet | Soft 3d acoustic metamaterial with negative index[END_REF]. However, for low frequencies, the size of the micro-inclusions, generally of a few tens of microns, remains very small compared to the wavelength in water. Thus, a micro-inclusion medium can be considered as a homogeneous material. Further, the air cavities can deform under pressure, which could aect the performance of the coating [START_REF] Beretti | Réponse acoustique d'élastomères micro-inclusionnaires soumis à la pression d'immersion[END_REF].

Alberich Coatings

Alberich coatings, introduced in Figure 1.2, were developed by Erwin Meyer and his team, who were working on new absorbers of underwater sound in Berlin since 1945. At least one submarine of the German Navy, the U-480, was coated with this new technology as shown in Figure 1.11, making it the rst stealth submarine [START_REF] Meyer | Sound absorption and sound absorbers in water[END_REF]. Moreover, such a coating has the advantage of being signicantly lighter than its counterpart with hard inclusions.

Figure 1.11 Alberich acoustic absorption coating used on the German submarine U-480 [START_REF] Meyer | Sound absorption and sound absorbers in water[END_REF].

In early work based on the nite element method (FEM), Hladky-Hennion and Decarpigny [START_REF] Hladky-Hennion | Analysis of the scattering of a plane acoustic wave by a doubly periodic structure using the nite element method: Application to Alberich anechoic coatings[END_REF] developed a code to compute the sound transmission in a rubber medium submerged in water and embedded with periodic cylindrical cavities of nite height. A unit cell of the coating comprising a single cavity in rubber was simulated and periodicity of the geometry was implemented by applying classical Bloch conditions on the boundaries of the unit cell. Using the same nite element code, Langlet et al. [START_REF] Langlet | Analysis of the propagation of plane acoustic waves in passive periodic materials using the nite element method[END_REF] calculated the eective wavenumber of a periodically voided elastic medium under plane wave excitation.

Alberich coatings have latter been the subject of many studies to better understand their physical phenomena [START_REF] Gaunaurd | One-dimensional model for acoustic absorption in a viscoelastic medium containing short cylindrical cavities[END_REF][START_REF] Lane | Absorption mechanisms for waterborne sound in Alberich anechoic layers[END_REF]. Analytical [START_REF] Leroy | Superabsorption of acoustic waves with bubble meta-screens[END_REF][START_REF] Thieury | Phenomenological law for the acoustic reection by an array of cylindrical cavities in a soft elastic medium[END_REF] and semi-analytical [START_REF] Sharma | Sound transmission through a periodically voided soft elastic medium submerged in water[END_REF][START_REF] Sharma | Acoustic performance of gratings of cylindrical voids in a soft elastic medium with a steel backing[END_REF] models have been developed to examine the acoustic performance of a periodically voided soft elastic medium. This type of coating can be derived in many ways, by modifying the shape of the cavities in other classical forms [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 diraction gratings[END_REF] or in unusual new forms [7377]. The use of pyramidal inclusions, as shown in Figure 1.12, helps with the impedance adaptation.

It can also be achieved by cavities with gradient changes of radii and distances between cavities in a soft elastic medium [START_REF] Wang | Broadband underwater sound absorbing structure with gradient cavity shaped polyurethane composite array supported by carbon ber honeycomb[END_REF]. However, the resonance frequency of a cavity depends mainly on the volume of the cavity. It is therefore dicult to improve the absorption at low frequencies because the thickness of the anechoic layer is limited and the coating must also withstand hydrostatic pressure.

Figure 1.12 Schematic diagrams of the coating with optimised cavities, from (a) [START_REF] Zhong | On the accuracy and optimization application of an axisymmetric simplied model for underwater sound absorption of anechoic coatings[END_REF] and (b) [START_REF] Zhang | Sound absorption performance of underwater anechoic coating in plane wave normal incidence condition[END_REF].

The steel support corresponding to the hull of the submarine also aects the performance of the coating as constructive interference can occur [START_REF] Sharma | Acoustic performance of gratings of cylindrical voids in a soft elastic medium with a steel backing[END_REF]. In addition, the mass of the steel support inuences the low frequency absorption. An increase in the mass can decrease the frequency of the absorption peak [START_REF] Zhao | Backing eects on the underwater acoustic absorption of a viscoelastic slab with locally resonant scatterers[END_REF].

Hard Inclusions

A metamaterial comprising hard inclusions embedded in a viscoelastic matrix is an ideal candidate as an external coating on the hull of an underwater vehicle, due to impedance matching of the viscoelastic medium with water and robustness of the hard scatterers under hydrostatic pressure [START_REF] Hinders | Particle-loaded composites for acoustic anechoic coatings[END_REF]7981]. Scattering of sound waves by inclusions results in the conversion of longitudinal waves into shear waves, the latter being easily dissipated in a viscoelastic medium [START_REF] Hinders | Particle-loaded composites for acoustic anechoic coatings[END_REF][START_REF] Wen | Eects of locally resonant modes on underwater sound absorption in viscoelastic materials[END_REF]. An adaptation of the core-shell concept by Liu et al. [START_REF] Liu | Locally resonant sonic materials[END_REF] is shown in Figure 1.6(b) and consists of hard inclusions coated by a viscous rubber layer. This design has been shown to improve the acoustic performance of an underwater anechoic coating [START_REF] Zhao | Tri-component phononic crystals for underwater anechoic coatings[END_REF][START_REF] Méresse | Matériaux absorbants à structure périodique et inclusions résonantes pour l'acoustique sous-marine (Absorbing materials with periodic arrangements and resonant inclusions for underwater acoustics[END_REF][START_REF] Shi | Underwater sound absorption performance of acoustic metamaterials with multilayered locally resonant scatterers[END_REF]. It has also been shown that an o-centred core leads to higher absorption [START_REF] Zhong | Eect of core position of locally resonant scatterers on acoustic absorption[END_REF][START_REF] Duan | A new eccentric resonance matching layer with anti-reection characteristics for underwater acoustic scattering suppression[END_REF], as shown in Figure 1.13. Multiple layers of scatterers of dierent resonant frequency can also lead to broadband performance [START_REF] Meng | Optimization of locally resonant acoustic metamaterials on underwater sound absorption characteristics[END_REF], which can also be achieved by the simultaneous use of voided and hard inclusions [START_REF] Sharma | Sound absorption by rubber coatings with periodic voids and hard inclusions[END_REF][START_REF] Jin | Sound absorption behaviors of metamaterials with periodic multi-resonators and voids in water[END_REF]. core position [START_REF] Zhong | Eect of core position of locally resonant scatterers on acoustic absorption[END_REF].

Strong absorption can be obtained with the integration of resonant elements into a lossy viscoelastic material. The foremost simultaneous for energy dissipation is the presence of viscoelastic damping in at least one of the constituents [START_REF] Krushynska | Visco-elastice eects on wave dispersion in three-phase acoustic metamaterials[END_REF]. It is nonetheless noted that the use of hard inclusions is sometimes limited by restrictions on the coating weight.

Multilayered Media

Despite their simple geometry, multilayered media may also lead to interesting acoustic performance. For example, sub-wavelength sound wave isolation has been achieved by utilising strong Gibbs-type oscillation in a nite periodic multilayered medium [START_REF] Lu | Low-frequency gibbs-type oscillation in nite soliduid sonic crystals and its application in sub-wavelength wave isolation for waterborne sound[END_REF]. In addition, multilayered designs are easier to study and to homogenise than two-or threedimensional structures. Multilayered media may also lead to negative parameters, such as eective density or eective bulk modulus, for a symmetric unit cell [START_REF] Xu | Single-negative properties based on the bandgaps of onedimensional phononic crystal[END_REF][START_REF] Xu | Multi-objective optimization of layered elastic metamaterials with multiphase microstructures[END_REF]. However, examples of these structures for underwater applications are rare in the literature.

Other Designs

As mentioned previously, rubbers and polyurethane are the most commonly used matrix materials for underwater applications because of their ability to absorb incident sound wave energy. However, the elastic modulus of these materials increases with pressure. To cope with this constraint, Jiang and Wand [START_REF] Jiang | Phononic glass: A robust acoustic-absorption material[END_REF] created the phononic glass for underwater applications, which includes a metal skeleton and polyurethane llings with an interpenetrating network structure, as shown in Figure 1. 14(a). The physical connection between the locally resonant structure units is useful for exciting more modes, leading to a high sound absorption coecient. In addition, the metal skeleton lled with polyurethane oers high mechanical strength and therefore has good resistance to hydrostatic pressure.

However, with a metal skeleton, the total weight poses a diculty for a coating application.

Experimental Techniques

Experimental characterisation of a panel requires the insonication of the sample by a known acoustic eld. The reection loss of a sample is obtained from the acoustic eld reected by the sample, while the transmission loss is calculated with the acoustic eld passing through the sample [START_REF] Bobber | Underwater Electroacoustic Measurements[END_REF]. The reection and transmission coecients can be measured in an impedance tube, a pressurised water tank or an open water tank [START_REF] Zeqiri | Measurement and testing of the acoustic properties of materials: a review[END_REF].

The present review focuses on normal incidence characterisation techniques for panels in open water tanks.

One of the main diculties of experimental measurements is the reproduction of the set of assumptions used for the theoretical or numerical calculations, which are dicult to implement in practice. For example, theoretical models or nite element calculations on periodic structures often consider innite lateral dimensions, which is obviously impossible to achieve in practice. The innite panel assumption becomes particularly problematic for low frequency measurements. One of the main parasitic eects resulting from the nite lateral dimensions of the panel is the diraction of acoustic waves by its edges. There exist techniques [START_REF] Piquette | Technique for detecting the presence of nite sample-size eects in transmitted wave measurements made on multilayer underwater acoustic panels[END_REF] and models [START_REF] Audoly | Scattering from thin elastic screens[END_REF][START_REF] Audoly | Improvement of the measurement of the transmission coecient of panels at normal incidence using surface receivers[END_REF] to obtain direct evidence of the edge-diraction phenomenon. It has even been shown that the importance of the edge diraction perturbations is largely inuenced by the material properties of the sample [START_REF] Smith | Diraction from the edge of a thin elastic half plane and implications for panel measurements[END_REF]. Edge diraction may complicate measurements of the transmission loss for panels made of acoustically soft materials, since the amplitude of the transmitted waves can be very small compared to that of the diracted waves. Moreover, if the speed of sound in the sample is rather low, the diracted waves can overlap the reected and transmitted signals in the time domain, which makes their dierentiation dicult [START_REF] Piquette | An analytical technique for reducing the inuence of edge diraction in reection measurements made on thin acoustical panels[END_REF][START_REF] Piquette | An extrapolation procedure for transient reection measurements made on thick acoustical panels composed of lossy, dispersive materials[END_REF]. Edge diraction also signicantly contaminates measurements on acoustically hard panels, for which the diracted pattern shows strong variations in space within measurements planes [START_REF] Audoly | Scattering from thin elastic screens[END_REF].

Several methods have been developed to estimate the amplitude of diracted waves. One method requires an air box to be placed in front of the panel (Figure 1.15(a)), for which the transmission loss is more than 60 dB. Thus, the signal received on the transmitted side is assumed to result from the sole contribution of the edge-diracted waves [START_REF] Piquette | Direct measurements of edge diraction from soft underwater acoustic panels[END_REF]. An alternative technique has recently been presented which involves surrounding the panel in a reective bae so that its edges are less excited, thus reducing the contamination [START_REF] Szabo | Reduction in edge eects for small panels characterized by a parametric array source[END_REF] (Figure 1.15(c)). (c) Panel mounted in bae [START_REF] Szabo | Reduction in edge eects for small panels characterized by a parametric array source[END_REF].

Another experimental means of reducing the parasitic eects of edge diraction is the use of a parametric array as the insonication is maximised at the centre of the panel and reduced at the edges [START_REF] Humphrey | The measurement of acoustic properties of limited size panels by use of a parametric source[END_REF]. The directivity of a parametric array for low-frequency waves is obtained by the interaction of high-frequency waves, which implies a non-linear mixing of the sound [START_REF] Westervelt | Parametric acoustic array[END_REF]. Nevertheless, it must be ensured that non-linear eects do not complicate or invalidate the measurement process [START_REF] Humphrey | Acoustic characterization of panel materials under simulated ocean conditions using a parametric array source[END_REF], so a low-pass acoustic lter is required in the measurement zone to attenuate the higher frequency primary waves.

A surface receiver may also be used to reduce the scattered eld perturbations on the transmission coecient measurements (Figure 1. 15(b)). This approach helps to reduce the eects of the edge-diracted pressure assuming that the optimal size and position of the receiver have been found [START_REF] Audoly | Improvement of the measurement of the transmission coecient of panels at normal incidence using surface receivers[END_REF]. Perturbations caused by edge-diracted waves can also This process, known as the ONION method, has given good results for thick multilayered panels made of materials whose properties are known [START_REF] Piquette | An extrapolation procedure for transient reection measurements made on thick acoustical panels composed of lossy, dispersive materials[END_REF][START_REF] Piquette | The ONION method: A reection coecient measurement technique for thick underwater acoustic panels[END_REF][START_REF] Piquette | Transmission coecient measurement and improved sublayer material property determination for thick underwater acoustic panels: A generalization and improvement of the ONION method[END_REF]. These techniques that have been implemented to reduce the edge diraction eects. However, often these techniques have their own limitations, as they often require the manufacture or the use of some specic equipment or they may impose constraints on the panel design (thickness, material, etc. ). Even with these techniques, a good compromise must still be found for the position of the hydrophones with respect to the panel. The hydrophones must be positioned close enough to reduce the detection of diracted waves and far enough to avoid the overlap of the incident and reected signals.

Experimental Study of Metamaterials

Experimental evidence of metamaterial features is not common in the literature, especially in an underwater environment. Bi et al. [START_REF] Bi | Design and demonstration of an underwater acoustic carpet cloak[END_REF] designed and experimentally performed an underwater acoustic carpet cloak, comprising multiple layers of brass plates separated by layers of water. A soft 3D acoustic metamaterial with negative index has been developed by Brunet et al. [START_REF] Brunet | Negative-index metamaterials: is double negativity a real issue for dissipative media?[END_REF]. Hladky et al. [START_REF] Hladky-Hennion | Negative refraction of acoustic waves using a foam-like metallic structure[END_REF] experimentally demonstrated the negative refraction in water for medical imaging. These underwater achievements are interesting but such designs cannot be practically used as a coating as they do not consider all the related constraints such as thickness, hydrodynamics and weight.

There are only a handful of papers reporting the achievement of metamaterials with negative eective mass density in the water medium. Fok and Zhang [START_REF] Fok | Negative acoustic index metamaterial[END_REF] developed a negative acoustic index metamaterial in water that combines Helmholtz and rod-spring resonators to control eective bulk modulus and mass density, respectively. Eective properties from numerical simulations of their metamaterial design showed that negative real components of bulk modulus and density occur simultaneously, resulting in a negative real component of the acoustic index. Experimental measurements on a sample of their metamaterial design conrmed that the real components of the acoustic index and bulk modulus reach negative values, but the density does not become negative, attributed to the material loss. Popa et al. [START_REF] Popa | Anisotropic acoustic metauid for underwater operation[END_REF] also retrieved a positive eective mass density of an anisotropic acoustic metauid, shown in Figure 1. 16(a). This was achieved using an inverse method by measuring the reected and transmitted elds on planes situated in front and behind the samples, and calculating the sample reection and transmission coecients based on the elds averaged along these planes. The coecients were then inverted using a standard method proven to be very eective in acoustics. Chen et al. [START_REF] Chen | Acoustic metamaterial with negative mass density in water[END_REF] designed a 2D metamaterial with a simple design consisting of tubes on two pieces of epoxy.

The tube was open and contained the water of the surrounding medium as shown in Figure 1. 16(b). Eective properties extracted from experimental transmission and reection coecients showed that the eective density exhibits negative values for a narrowband frequency range. Plates are connected through solid spacers (purple) [START_REF] Popa | Anisotropic acoustic metauid for underwater operation[END_REF]. (b) Acoustic metamaterial with negative eective mass density in water, designed by periodically arranging hollow tube "meta-atoms" [START_REF] Chen | Acoustic metamaterial with negative mass density in water[END_REF].

Notes on Concepts and Notations 1.4.1 Time Convention

In this manuscript, an incident wave refers to a harmonic plane wave excitation in a uid domain that arrives under normal incidence on a medium. Using the notations of Figure 1.17, the sound propagates along the x-axis.

Figure 1.17 Incident plane wave on a medium of thickness h.

The phase convention e i(ωt-kx) is used, where i = √ -1, ω is the angular frequency, k is the wavenumber, t is time and x denes the position on the x-axis.

Longitudinal and Shear Waves

In liquids or gases, only pressure waves can propagate. The propagation of a longitudinal wave is characterised by oscillations occurring in the direction of wave propagation. It thus corresponds to compression and expansion movements. In solids, longitudinal waves and shear waves can propagate. In the case of shear wave propagation, particles oscillate transversely to the direction of propagation.

For the study of a solid homogeneous medium at normal incidence, only the longitudinal waves are excited in the medium, provided that it is uniform in the lateral dimensions.

Since there are only compression / expansion movements, as in a uid, a uid formalism is employed to study acoustic propagation in solids under these conditions. It is then applied for both a single layer of material and a multilayered structure.

Material Damping Representation

In acoustics, an isotropic material is commonly characterised by three parameters: mass density ρ, Young modulus E and Poisson ratio ν; or equivalently, mass density and longitudinal and shear speeds of sound denoted by c l and c s , respectively. Those parameters may depend on frequency or on environmental conditions, such as pressure and temperature.

For elastomeric materials that have the ability to dissipate acoustic energy, an imaginary part can be added to given material properties. In such complex notations for damped material properties, the imaginary part represents the damping. It has to be noted that damping is physically represented by the imaginary part of the wavenumber. Nevertheless, in this manuscript a complex speed of sound notation is used as c l = c l (1 + iη), with c l corresponding to the real part of the longitudinal speed of sound and η the loss tangent.

Homogenisation

In this manuscript, the homogenisation process consists of applying a theoretical model to an eective medium so that its eective properties can be used to describe the fareld scattering response for normal incidence of a composite structure. Two models are considered. The rst one, which happens to be the most commonly used in the literature, describes the eective medium as a uid. The second model aims to describe asymmetric structures for which Willis-coupling eects may appear. The eective medium is then assumed to be a Willis uid-like medium for which properties depend on the direction of wave propagation.

Passivity Constraint

Extreme eective properties can be obtained for a metamaterial. However, eective properties must comply with fundamental physical laws. For example, standard elastic materials and structures made of elastic materials are assumed to be passive. Implications of these restrictions were fully explored by Muhlestein [START_REF] Muhlestein | Willis Coupling in Acoustic and Elastic Metamaterials[END_REF], where a passive material is dened as a material that can not create mechanical energy.

In the present manuscript using the time convention e iωt , passivity constraints set the sign of the wavenumber imaginary part to be negative, as shown in Section B.1 of Appendix B. Passivity also constrains the real part of the impedance to be positive, as described in [START_REF] Rossing | Springer handbook of acoustics[END_REF].

Acoustic Performance

The acoustic performance of a structure is generally evaluated using the reection and transmission coecients for the structure placed between two semi-innite uid media.

The reection coecient describes the amount of acoustic wave reected by an impedance discontinuity created by the introduction of the structure in-between the surrounding media. It is equal to the ratio of the amplitude of the reected wave to the incident wave.

The transmission coecient is a measure of the amount of acoustic energy passing through a structure. It is calculated from the ratio of the amplitude of the transmitted wave to the amplitude of the incident wave. For experimental determination of the reection and transmission coecients, free-eld measurements on a test panel are used.

For underwater applications, the eciency of an coating in reducing acoustic radiation from a structure is assessed using the hull decoupling coecient C D , while the stealth performance is evaluated with the anechoism coecient C A . For determination of the anechoism coecient, the sample may be xed on a steel plate. However, this approach is not practical at low frequencies as the steel backing plate might not be thick enough. The hull decoupling coecient may be obtained by placing the sample on a vibrating support and comparing results with and without the sample. This approach gives a qualitative comparison, but post-processing is complex [START_REF] Audoly | Global characterization of multilayer panels for underwater applications[END_REF].

In order to simplify the determination of the anechoism and hull decoupling coecients, the common approach consists in conducting free-eld measurements of the panel and then deriving these coecients numerically using the reection and transmission coecients of the structure, as demonstrated in Appendix C.

Chapter 2 Homogenisation

In this chapter, an analytical model based on a transfer matrix formalism is presented to characterise an eective medium that can eectively replace a multilayered medium and from which the reection and transmission coecients can be calculated. Two variants of the homogenisation method are proposed. The rst method is specic to symmetric multilayered media whereas the second method can be applied to both asymmetric and symmetric designs. A quasi-static approach is also introduced to verify the homogenisation method at low frequencies. Numerical results are then presented for a case study consisting of a multilayered structure comprising alternating layers of acoustically rigid and soft materials.

Contents

Transfer Matrix

Denition

In the context of one-dimensional propagation of waves in a uid-like structure, the transfer matrix M relates the acoustic pressure p and the normal particle velocity v of the medium on the two sides of the layer. Based on the notations in Figure 2.1, the transfer matrix M is dened by:

p left v left = M p right v right .
(2.1) Figure 2.1 Pressures and normal particle velocities at the inlet/outlet of a medium of thickness h, where the subscript left refers to the face on the left hand-side and the subscript right to the face on the right hand-side of the medium.

Expression

Expressions of the transfer matrix elements depend on the nature of the medium. Two cases are considered in what follows: classical uid or Willis uid. Details on how the expressions are derived are given in Appendix A.

Case 1: Fluid Medium

The simplest case considers a medium that can be assimilated to a uid. With k the wavenumber and Z the acoustic impedance, the expression of the transfer matrix is given by:

M = cos(kh) iZsin(kh) iZ -1 sin(kh) cos(kh) , (2.2)
where h is the thickness of the medium in the direction of the sound propagation. Since only harmonic plane waves at normal incidence on multilayered structures are considered in Part I, any homogeneous material can be approximated as a uid medium and can thus be described by Equation (2.2).

Case 2: Willis Fluid Medium

The second case considers a uid-like medium that is asymmetric, also referred to as a Willis uid. Material properties thus depend on the wave propagation direction. For the one-dimensional wave propagation considered here, two directions are possible: waves are referred to as forward or backward propagating waves. The distinction is reected on the eective impedance, since the wavenumber is the same for both directions [START_REF] Amirkhizi | Homogenization of layered media based on scattering response and eld integration[END_REF]. The acoustic eld can therefore be decomposed along these two directions, which leads to the transfer matrix form given by:

M = 1 Z + -Z - Z + e ikh -Z -e -ikh Z + Z -(e -ikh -e ikh ) e ikh -e -ikh Z + e -ikh -Z -e ikh , (2.3)
where k is the wavenumber and Z + and Z -are the impedances for forward and backward propagating waves, respectively. It can be noted that for a symmetric medium, for which

Z + = -Z -= Z, the simplication of Equation (2.
3) gives the transfer matrix given by Equation (2.2).

Calculation

When a plane wave arrives at normal incidence on a multilayered medium of innite lateral dimensions, the acoustic elds at the interfaces are uniform. They may thus be described by only a scalar for the pressure and a scalar for the normal particle velocity, as shown in Figure 2.1. The use of transfer matrices is particularly suitable for the analysis of multilayered media as they can easily be connected to each other like a chain to represent a succession of layers. The technique that uses transfer matrices to study chained acoustic structures is also known as the transmission line method. As such, given the continuity of acoustic pressure and velocity at the interfaces between two layers, the total transfer matrix M n of a multilayered medium comprising n layers is obtained by:

M n = m 1 m 2 ...m n , (2.4)
where m i denotes the transfer matrix of layer i. The transfer matrix m i can be calculated using the transfer matrix given by Equation (2.2), assuming that each layer is a uid-like medium with known material properties.

Homogenisation

The homogenisation approach described herein aims to replace a nite periodic multilayered structure by a uid-like homogeneous medium that would have the same scattering response. For this purpose, the characteristic unit cell of the multilayered medium is dened as the smallest recurring unit. This unit cell is used to derive eective properties, such as the eective wavenumber and eective impedance, which are the material properties of an equivalent homogeneous unit cell, as illustrated in Figure 2.2. for which a uid eective medium is dened by an eective wavenumber and an eective impedance obtained using the transfer matrix M Lu of the periodic unit.

Unit cell

In this manuscript, the notion of eective medium is therefore referring to a medium dened by an homogenisation model and which is intended to replace a given nonhomogeneous medium. The simplest case is treated rst, considering that the eective unit is a uid that can replace a symmetric unit cell. Then, the case of non-symmetric unit cells is detailed, where eective properties are obtained using a Willis uid homogenisation model.

Fluid Homogenisation Model

The transfer matrix of a unit cell of length L u , denoted as M Lu , can be numerically computed since the properties for longitudinal wave propagation of the materials composing each layer of a multilayered unit are known. For this purpose, the transfer matrix for each layer is rst calculated using Equation (2.2), and they are then multiplied according to Equation (2.4). For example, in the case presented in Figure 2.2, the transfer matrix of the unit M Lu is m 1 m 2 m 3 , where m i (i = 1, 2, 3), are the transfer matrices for layers 1 to 3.

Another method of expressing the transfer matrix of the eective medium is as follows:

M e = cos(k e L u ) iZ e sin(k e L u ) iZ -1 e sin(k e L u ) cos(k e L u ) , (2.5)
where L u is the length of the medium, k e is the eective wavenumber and Z e is the eective impedance. As the eective medium aims to represent the multilayered structure, both transfer matrices M Lu and M e are the same, that is:

M Lu = M e .
(2.6)

Eective parameters can then be extracted from the elements of the transfer matrix M e .

The eective wavenumber expression, derived in Appendix B.1, is given by:

k e = ln(λ) iL u + 2πm L u , (2.7)
where m is an integer whose value is determined in such way that the eective wavenumber as a function of frequency is continuous, and λ is the eigenvalue of the transfer matrix M Lu , chosen so that the imaginary part of k e is negative to comply with material passivity requirements (as mentioned in section 1.4.5 and in Appendix B).

The eective impedance Z e can be directly extracted from the o-diagonal elements of the transfer matrix, using:

Z e = ± M e (1, 2) M e (2 , 1) . (2.8)
Material passivity requirements constrain the real part of the eective impedance Z e to be positive.

With the eective wavenumber k e and the eective impedance Z e , the eective uidmedium representing a given symmetric multilayered unit cell is fully characterised, given that the length of the eective medium in the direction of sound propagation is the same as the length of the unit cell, in this case, L u . The eective medium approach described here is valid independently of the number of layers comprised in the unit cell, as long as it remains symmetric.

Willis Fluid Homogenisation Model

For a non-symmetric multilayered medium, the behaviour is dierent for a wave travelling towards x = +∞ with respect to a wave travelling towards x = -∞. In order to eectively represent non-symmetric media, the eective medium is now assumed to be an asymmetric uid-like medium, also referred to as a Willis uid. The eective medium approach follows the same scheme as before but it uses the more general form of the transfer matrix given by Equation (2.3).

One way to calculate the total transfer matrix for the unit cell of length L u , dened as M Lu , is to take the product of the transfer matrices for each layer as per Equation (2.4).

The second method is to use the transfer matrix of the eective Willis uid given as:

M e = 1 Z + e -Z - e Z + e e ikeLu -Z - e e -ikeLu Z + e Z - e (e -ikeLu -e ikeLu ) e ikeLu -e -ikeLu Z + e e -ikeLu -Z - e e ikeLu , (2.9)
where k e is eective wavenumber and Z ± e are the eective impedances for forward and backward propagating waves. The eective wavenumber is also given by Equation (2.7).

Eective impedances can be obtained by the direct identication (M Lu = M e ), which yields:

Z ± e = -(M e2,2 -M e1,1 ) ± (M e2,2 -M e1,1 ) 2 + 4M e1,2 M e2,1 2M e2,1 , (2.10)
where M ei,j are the elements of the total transfer matrix M e . Due to the passivity material constraint, solutions are chosen in a way that the impedance for forward propagating waves

Z + e has a positive real part. The eective impedance Z - e then corresponds to the other branch.

The eective medium for a non-symmetric unit cell is now fully characterised with the eective wavenumber k e and the eective impedances Z ± e . It is worth noting that the eec- tive medium approach for non-symmetric structures is also valid for symmetric structures, as the symmetry only implies some simplications, such as the equality M e2,2 = M e1,1 , from Equation (2.2), which results in the simplication of Equation (2.10) into Equation (2.8).

Willis Parameters

As introduced by Mulhestein et al. [START_REF] Muhlestein | Experimental evidence of willis coupling in a one-dimensional eective material element[END_REF], a Willis uid may also be characterised using Willis parameters instead of impedances for forward and backward propagating waves.

These parameters correspond to the asymmetry coecient W and the characteristic impedance Z W . The eective impedances Z ± e are linked to W and Z W as per:

Z ± e = Z W (±1 + iW ).

(2.11)

When the asymmetry coecient W is equal to 0, the eective impedance Z e is the same as the Willis characteristic impedance Z W .

The asymmetry coecient W and the characteristic impedance Z W can be obtained from the transfer matrix form given by:

M e = cos(k e L u ) -W sin(k e L u ) iZ W sin(k e L u )(1 + W 2 ) i sin(k e L u )/Z W cos(k e L u ) + W sin(k e L u ) .
(2.12)

This expression is derived in Appendix A.4. Willis parameters are therefore given by:

W = -M e1,1 + cos(k e L u ) sin(k e L u ) , (2.13)
Z W = M e1,2 M e2,1 (1 + W 2 )
.

(2.14)

Eective material properties for a Willis uid are then derived using the expressions:

κ = ωZ W k e , (2.15)
ρ = k e Z W ω .

(2.16)

Scattering Coecients

The scattering response of a multilayered medium here refers to the reection and transmission coecients dened as in a two-port network, where the ports correspond to the interfaces of the medium with the surrounding uid (see Appendix A for the notion of two-port network and the denition of S-parameters). For a given multilayered structure, scattering coecients can be calculated using (1) the total transfer matrix of the entire multilayered medium, or (2) the transfer matrix of an equivalent homogeneous medium.

Both approaches are presented in what follows, based on Figure 2.3.

Effective medium

Figure 2.3 Finite periodic multilayered medium of length L and eective medium of same length whose eective properties have been derived from the constituent unit cell.

Multilayered Medium

The transfer matrix M L is analytically calculated by taking the product of the transfer matrix of each layer comprised in the multilayered medium, as per Equation (2.4). It is worth noting that this approach is still valid for non-periodic multilayered media.

Elements M L i,j of the total transfer matrix are used to obtain the S-parameters as per:

S 11 = M L 1,2 + Z f (M L 1,1 -M L 2,2 ) -Z 2 f M L 2,1 ∆ , (2.17)
S 21 = 2Z f ∆ , (2.18)
S 22 = M L 1,2 -Z f (M L 1,1 -M L 2,2 ) -Z 2 f M L 2,1 ∆ , (2.19)
S 12 = 2Z f Det(M L ) ∆ , (2.20)
where Z f is the impedance of the surronding uid and ∆ is:

∆ = M L 1,2 + Z f (M L 1,1 + M L 2,2 ) + Z 2 f M L 2,1 .
(2.21)

Those expressions are derived in Appendix C.

Eective Medium

When a homogeneous eective medium is used to replace a multilayered medium, scattering coecients can be obtained from its transfer matrix M e L . This is achieved using Equations (2.17) to (2.20) by replacing the matrix elements M L i,j with the elements M e L i,j .

For an eective medium replacing a symmetric multilayered design of length L, S-parameters are obtained using the eective properties k e and Z e that have been derived for the symmetric unit cell, by calculating the transfer matrix M e L given by:

M e L = cos(k e L) iZ e sin(k e L) iZ -1
e sin(k e L) cos(k e L) .

(2.22)

For a non-symmetric conguration, the transfer matrix of the eective medium of length L is:

M e L = 1 Z + e -Z - e Z + e e ikeL -Z - e e -ikeL
Z + e Z - e (e -ikeL -e ikeL ) e ikeL -e -ikeL

Z + e e -ikeL -Z - e e ikeL .

(2.23)

Quasi-static Approach

At very low frequencies, values of the longitudinal wavenumber are very small. A quasistatic approach can be used to approximate eective parameters of a periodic unit cell.

The quasi-static eective density is given by the average density for the various layers in the equivalent homogeneous medium as follows:

ρ QS = N i=1 (h i ρ i ) h u , (2.24)
where h i and ρ i respectively are the thickness and density of the layer i and h u = N i=1 h i is the total thickness of the unit.

The eective wavenumber can be obtained using the dispersion relation given for a two layer unit [START_REF] Langlet | Analysis of the propagation of plane acoustic waves in passive periodic materials using the nite element method[END_REF], as follows:

cos(k(h 1 + h 2 )) = cos(k 1 h 1 ) cos(k 2 h 2 ) - 1 2 (γ + 1 γ ) sin(k 1 h 1 ) sin(k 2 h 2 ), (2.25)
where h 1 , k 1 , Z 1 and h 2 , k 2 , Z 2 are the thickness, wavenumber and impedance respectively, of material 1 and material 2 in the periodic unit, and γ = Z 1 /Z 2 . For low values of the wavenumber, a polynomial approximation of Equation (2.25) leads to:

k 2 QS (h 1 + h 2 ) 2 (k 1 h 1 ) 2 + (k 2 h 2 ) 2 + (γ + 1 γ )k 1 h 1 k 2 h 2 .
(2.26)

The quasi-static eective sound speed c QS and the quasi-static eective impedance Z QS are then given by:

c QS h 1 + h 2 ( h 1 c 1 ) 2 + ( h 2 c 2 ) 2 + (γ + 1 γ ) h 1 h 2 c 1 c 2 , (2.27)
Z QS = ρ QS c QS .

(2.28)

2.5 Numerical Results

Designs

Numerical results for periodic multilayered structures will now be presented. Media are formed by alternating layers of silicone rubber and aluminium. Material properties are given in Table 2.1. reaches a local maximum. This shows that there is a greater attenuation of acoustic waves in the band gaps than in the pass bands. Moreover, it can be seen that this maximum increases with higher loss factor η. For the real part of the reduced wavenumber, an increase of the damping η results in a more pronounced "smoothing" of the curve, which no longer meets the edges of the band gaps. and aluminium, considering 0%, 2% and 8% loss factor in the silicone layer. For the case η = 2%, the eective wavenumber is also unwrapped (k un e,2% ), as well as compared to the quasi-static value k QS . The shaded areas represent band gaps dened using the wavenumber k e,0%

ρ c l η (kg.m -3 ) (m.s -1 ) (%)
The real part of the wavenumber for η = 2% is unwrapped, represented by k un e,2% , in order to illustrate the unwrapping process using the integer m introduced in Equation (2.7). This manipulation is only required if the eective wavenumber is then used to derive other eective parameters, such as the density and the bulk modulus, in order to get a continuous evolution with frequency.

Finally, the quasi-static wavenumber k QS , obtained with Equation (2.26) and represented by the red crosses, is consistent with the eective wavenumber k e,2% at low frequencies, thus validating the latter.

Eective Impedance

In contrast to the eective wavenumber, the eective impedance depends on the design of the unit cell. Considering η = 2% for the silicone, Figure 2.6 presents the real and imaginary parts of the eective impedances for units (a) to (d). Band gaps for zero damping are also shown by the shaded areas. The rst line of graphics represents the real part of the eective impedances, while the second line corresponds to the imaginary part. The shaded areas represent the band gaps obtained using zero damping for silicone. The red crosses show the quasi-static eective impedance.

For the symmetric versions (a) and (b), the eective impedance is derived using Equation This observation can also be mathematically proven by inverting the transfer matrix. For these two asymmetric units, the characteristic impedance Z W , given by Equation (2.11), diers from the eective impedance Z ± e as the asymmetry coecient W is non-null.

The variations of the eective impedance for unit (a) greatly diers from the three others and even reaches negative values. In order to further study this dierence, the eective density and bulk modulus are calculated in what follows.

Eective Density and Bulk Modulus

For the 4 unit cells, the eective density is plotted in Figure 2.7 using Equation (2.16).

The eective bulk modulus is calculated using Equation (2.15) and is shown in Figure 2.8.

The real part of the eective density of version (a) reaches negative values in the rst band gap. This specic metamaterial feature is usually the translation of a mass-spring resonance in most metamaterial studies. This observation can be made only for unit (a),

where the heaviest layer (aluminium -mass) is placed in-between two soft and damped layers (silicone -spring). For all the other band gaps, the bulk modulus becomes negative, which is the translation of a compression-extensional motion. Hence, either a negative mass density or a negative bulk modulus is obtained for each band gap produced by the four unit versions. The version where the heavy layer is surrounded by the damped layer is the only one that can lead to a negative mass density.

Scattering Coecients

As presented in Section 2.3, two approaches are used to calculate the acoustic performance in terms of S-parameters for a medium of length L = n × L u . The rst method to nd the total transfer matrix M L of the n-unit medium is calculated using the transfer matrix of each layer. Equations (2.17) to (2.20) are then used to derive the scattering coecients.

For the second method, an eective homogeneous medium is also dened with the eective wavenumbers and impedances previously calculated. As illustrated in For all versions, the scattering coecients of the eective medium (black lines) lead to the exact same reection and transmission coecients of the multilayered medium (red lines). The same observation has been made for larger structures, although results

are not presented here to avoid repetition. As such, for these multilayered designs, the homogenisation approach introduced here is exact and multilayered media can therefore be accurately replaced by an eective medium to obtain the scattering coecients using either a uid or Willis uid homogenisation model. In Figure 2.9, it can also be seen that the reection and transmission coecients dier depending on the unit cell, since the rst layer is dierent. For better comparison, the coecients are superimposed in Figure 2.10.

The transmission coecient obtained for unit (c) is the same as that of unit (d), which is consistent with the equality S 12 = S 21 as unit (c) is the reverse design of unit (d), and vice-versa. It is also observed that the reection coecient for the medium made of unit (b) is very low within the pass bands, even though this medium starts with a thin layer of aluminium. On the contrary, the transmission coecient is signicantly dierent depending on the number of unit cells, especially within the pass bands. The band gaps, associated to low transmission, are visible from 5 unit cells and are clearly dened for 8 unit cells and above.

Summary

In this chapter, a method has been presented to replace a given periodic multilayered medium by an equivalent homogeneous medium. The eective parameters that characterise the eective medium are obtained from the transfer matrix of the periodic unit.

Two cases have been distinguished depending on the symmetry of the unit. If the unit is symmetric, the eective medium is assumed to be a uid. For an asymmetric unit cell, the eective medium is a Willis uid medium, characterised by impedances specic to forward and backward propagating waves, or by a characteristic impedance and an asymmetry coecient.

The homogenisation approach has been applied to four variant designs of an unit cell. The eective medium approach leads to the eective wavenumber, identical for all variants, which gives the frequency ranges where wave propagation is strongly attenuated. The eective impedance highlights the necessity to adopt an homogenisation model that can describe an asymmetric structure. Eective density and eective bulk modulus have also been obtained and exhibit some of the specic features of metamaterials, that is, negative values of their real parts in some frequency bands. Eective parameters have been validated at low frequencies using a quasi-static approach.

It was also shown that the eective medium accurately predicts the reection and transmission coecients of a multilayered medium of any length multiple of the unit cell thickness. It is found that the scattering coecients are noticeably dierent depending on the variant design chosen for the unit cell. Acoustic performance for increasing number of units has also been shown. As expected, it was observed that the reection coecient is mostly dened by the rst unit.

Chapter 3

Optimisation

In this chapter, layer arrangements of one-dimensional multilayered structures are optimised to improve their broadband acoustic performance. The acoustic performance of any multilayered medium can be rapidly obtained using transfer matrices as presented in Chapter 2. As such, it is worth using this approach to optimise the layer arrangement of multilayered media. The gures of merit are the anechoism coecient (Section 3.2), the hull decoupling coecient (Section 3.3) and both coecients combined (Section 3.4). The main objective of the optimisation study is to broaden the frequency range of eective acoustic performance through the layer arrangement. It may be found that non-periodic media, multi-periodic media or gradual media help reduce these coecients better than an usual periodic medium.

For the layer optimisation in this Chapter, only two materials are considered : stainless steel or polyurethane with micro-balloons. The mass density for stainless steel is 7700 kg.m -3 and the longitudinal speed of sound is 5800 m.s -1 . Micro-balloons are introduced into a polyurethane matrix in order to tune the longitudinal speed of sound as needed. In particular, it is possible to obtain a lower speed of sound and a greater damping loss factor in matrices where the damping loss factor of longitudinal waves is usually very small. For the polyurethane with micro-balloons considered here, with 6% of micro-balloons, the estimated mass density is ρ = 984 kg.m -3 . The estimated speed of sound is given by c = c L × (1 + iη), with the real part c L and the damping η both plotted in Figure 3.1. These values have been experimentally derived as described in the following Chapter 4, for a temperature of 15 o C. Simple optimisation algorithms, such as a greedy algorithm, could thus be used for this layer optimisation. Nevertheless, the optimisation approach implemented here for multilayered media will also be applied to metamaterials with macro-inclusions using their eective properties in Chapter 7. The number of "material" possibilities will then be considerably higher than two. A controlled elitist genetic algorithm is thus selected.

In what follows, the methodology to implement the optimisation problem is detailed.

Methodology

A genetic algorithm works on a population of individuals that are candidate solutions to an optimisation problem. These individuals are vectors made of several genes chosen amongst a set of given genes. For the present optimisation problem, the layer arrangement of multilayered media is optimised for a bi-objective function over a given frequency range.

For this purpose, an individual is dened as a multilayered medium of n l layers of thickness l, for a total length called L. For example, the multilayered conguration shown in Figure 3.2 comprises n l = 15 layers, each being l = 4 mm-thick, for a total length of L = 60 mm.

The algorithm selects a material from the available materials (genes) for each layer. In what follows, the various steps of the selected optimisation algorithm are presented. Thicknesses are displayed in meters. The acoustic wave is incident on the face x = 0 whereas x = 0.06 corresponds to the transmission side. For the calculation of both the anechoism and hull decoupling coecients, the hull is considered at on the right-hand side.

Algorithm

A controlled elitist genetic algorithm, variant of NSGA-II [START_REF] Deb | Controlled elitist non-dominated sorting genetic algorithm for better convergence[END_REF], is used through the MATLAB Global Optimization Toolbox. The search method operates on a population of potential solutions applying the principle of survival of the ttest to produce better individuals. At each generation, a new population is created by the process of selecting individuals according to their gures of merit and they are bred together using operators borrowed from natural genetics, such as selection, recombination and mutation. The biobjective optimisation problem usually has no unique solution but a set of non-dominated solutions known as the Pareto-optimal set. The later is the result of a non-dominated ranking approach as illustrated in Figure 3.3, in which an individual i dominates an individual j if i is strictly better than j for at least one objective function and i is no worse than j for both objective functions. It is also possible to say that j is dominated by i or that i has a lower rank than j. As such, two individuals have the same rank if neither dominates the other. The Pareto-front is dened as the group of best t solutions, with the lowest rank. An elitist genetic algorithm always favors individuals with better tness value (rank), whereas a controlled elitist genetic algorithm also favors individuals that can help increase the diversity of the population even if they have a lower tness value. It is very important to maintain the diversity of the population in order to converge to an optimal Pareto front. Hence, to compute the next generation of the population, the non-dominated ranking system is used in the current generation. Individuals with equal rank are then compared using a crowding distance measure of individuals, that calculates the distance between the neighboring solution in order to help maintain diversity on a front by favoring individuals that are relatively far away from each other. As such, solutions are sorted according to the denition [START_REF] Deb | Controlled elitist non-dominated sorting genetic algorithm for better convergence[END_REF] : Solution i is better than solution j if (i rank < j rank ) or ((i rank = j rank ) and (i distance > j distance )). 1 The algorithm randomly generates individuals to create the initial population, whose size N is xed.

2 The optimisation process starts with the parents selection for the next genera- tion. The set of parents is then transformed by means of two genetic transformations: crossover or mutation. A percentage of C % of children is created by crossover and (100 -C % ) is the result of mutation. Hence, in order to create the same amount of children than the number of individuals in the current population, the set of parents must contain N × (100 + C % ) individuals. For the parents' selection, a binary tournament is used, which chooses each parent by choosing two individuals at random and then choosing the best individual out of that set to be a parent.

3 Crossover operates on a pair of parents to produce new children by exchanging segments from the parents' vectors. In this work, the two-point crossover approach is used.

The latter approach randomly generates two integers that will dene segments within the pair of parents. The algorithm then concatenates these genes to form a single individual. In the case of mutation, a process comprised of two steps is applied. First, the algorithm selects a fraction of the vector entries of an individual for mutation, where each entry has a probability m of being mutated.

In the second step, the algorithm replaces each selected entry by a random number selected uniformly from the range for that entry.

4 Those children created by crossover and mutation are then combined with the set of parents, resulting in a recombination of N p individuals.

5 Each individual is then evaluated for both objective functions in order to estab- lish the non-dominating ranking. 6 The choice of the individuals transferred into the next generation follows a controlled selection. The latter consists in restricting the number of individuals in the current best non-dominated rank and in maintaining a predened distribution of individual numbers in each front. From the rst rank, a ratio r 1 of the population N p is kept and transferred into the next generation. For higher fronts, the maximum number of individuals n i that can be retained from the front i is:

n i = N p 1 -r 1 -r K r i-1 , (3.1)
with K being the number of non-dominated fronts in the combined population of N p individuals (K = 4 in the ow chart in Figure 3.4) and r is the geometric progression ratio, whose value is kept constant and equal 0.8. Therefore, the number of solutions kept in each front is exponentially reduced, so that solutions from all non-dominated fronts are forced to co-exist in the population.

7 The new generation is nally evaluated and repeats the entire optimisation process until the stopping criterion is met. Herein, the optimisation process stops when the maximum number of generations G max is reached.

Results for Anechoism

Objective Functions

The optimisation process is rst applied to minimise the anechoism coecient C A , given by:

C A = S 11 - S 21 S 12 S 22 -1 . (3.2)
This expression is derived in Appendix C. The S-parameters of the entire multilayered medium are calculated using the transfer matrix method considering a water environment.

A bi-objective function is implemented in order to minimise the anechoism coecient C A over a nite number of frequencies n F , each frequency f of the frequency range being indexed by i F . The rst component of the objective function aims to minimise the Root Mean Square (RMS) of the anechoism coecient, referred to as |C A | and given by:

|C A | = 1 n F n F i F =1 |C A (f (i F ))| 2 . (3.3)
This rst objective function aims to reduce the values of the anechoism coecient. The anechoism coecient must also remain low over the entire frequency range of study and not only for narrow frequency ranges. The anechoism coecient may indeed be very low for some specic frequencies, which would give a satisfactory |C A |, but that does not prevent the anechoism coecient to be relatively high elsewhere, as illustrated in Figure

t C A (f (i F )) = |C A | if |C A (f (i F ))| < |C A | |C A (f (i F ))| if |C A (f (i F ))| > |C A | . (3.4)
Using the intermediate function t C A , the normalised standard deviation (NSD), referred to as σ C A , is then given by:

σ C A = RMS(t C A -|C A |) |C A | . (3.5)

Validation

The algorithm is rst validated for a case where all possible solutions can be calculated in order to check if the optimisation algorithm converges towards the best set of solutions.

The parameters for the algorithm and for the multilayered medium to be optimised are given in Table 3.1. Six sets of parameters have been dened, where Set 1 is used for reference. In Sets 2 to 6, one parameter diers compared to Set 1 in order to assess its inuence on the results. As such, using the notations in Figure 3 for Set 3 show that it is important to maintain a sucient proportion of children created by crossover, since the rest of the children are created by mutation with a probability of 0.02. Results for Set 5 show that, when the algorithm is too elitist with r 1 = 0.5, then best solutions may be missing.

Around |C A | = 0.8, Set 4 and Set 6 do not lead to the absolute set of best solutions. For Set 4, mutation probability is too low, which hinders the creation of suciently diverse individuals. For Set 6, the maximum number of generations is relatively low, which does not guarantee the best set of solutions but still gives good results. Finally, only Set 1 and Set 2 lead to the exact set of best solutions, which shows that the parameters for Set 1 are adapted to the problem and there is no need to increase the population size.

In addition, the scores for a medium made entirely of polyurethane and for a medium entirely made of stainless steel are also visible, respectively shown by the orange and black squares. It shows that better acoustic performances are obtained by using both materials in a multilayered conguration.

Results

The optimisation process is now applied using parameters presented in Table 3.2. The objective here is to obtain layer arrangements that would be suitable for manufacturing and testing in a water tank. The maximum length of the medium is set to 4 cm, consisting In Figure 3.8, it is observed that some patterns are recurrent. For example, for the lowest values of |C A | (i.e. arrangements at the top of the graph), there are about 11 layers, beginning and ending with a polyurethane layer. Moreover, for these arrangements, the rst layer thickness is always the same, whereas the thicknesses of the other layers between x = 0.015 m and x = 0.035 m are more subject to variations. Overall, it seems that layers tend to get thicker as a function of their depth inside the arrangement, creating an impedance gradient. By going down along the Pareto front, it is then observed that the middle layers tend to disappear to form thicker layers. It can also be seen that nonperiodic solutions are suitable here to minimise the anechoism coecient on the wide frequency range, as none of these solutions is perfectly periodic.

Four of these arrangements are now selected for further investigation. After analysis of the solutions of the Pareto front, low values of |C A | were preferred since these solutions do not show signicant peaks. Their scores are shown by the white squares in Figure 3 12 kHz, but also have the highest values of |C A | for the lower frequencies. Selection 2 thus appears as a good compromise for low |C A | for most frequencies and it is thus selected for measurements in a water tank. Moreover, this selection also has the advantage of being made of "thick layers", that is to say it does not include a polyurethane layer which is less than 2 millimetre thick which would be dicult to manufacture. Selection 2 is now further analysed in order to determine features that make a medium satisfy both objective functions. As previously mentioned, most arrangements of the Pareto front are made of layers with gradually increasing average density as a function of depth inside the arrangement, just as a medium with gradient impedance. This feature can be observed with the moving average µ mov of the individual layers' impedances, calculated along the medium's thickness. The average is realised on a maximum of 10 successive layers of minimal thickness l. It is represented as a function of the thickness in Figure 3.10, for the frequency 22.5 kHz since it corresponds to the frequency of lowest |C A |. To summarise this optimisation for the anechoism coecient, the optimisation algorithm was rst validated, then applied to a case with more variables. The goal was to lower the mean of the anechoism coecient (using |C A |) and keep its values low (using σ C A ) over the frequency range of study. A design has been selected (Selection 2) for manufacturing and it will be measured in the next Chapter. A performance feature has also been found, which links the low anechoism coecient to a specic impedance pattern.

Results for Hull Decoupling

The hull decoupling performance is optimised for a multilayered medium considering the exact same objective functions as for the anechoic performance, dened in Section 3.

|C D | = S 12 -S 22 + 1 . (3.6)
This expression is derived in Appendix C. The optimisation process is applied taking into account the same parameters as previously, displayed in Table 3.2, aiming to dene a layer arrangement that would give a low hull decoupling coecient over the frequency range from 3 kHz to 25 kHz. In Figure 3.12, it can be seen that the rst solutions are mostly made of two layers : one layer of stainless steel rst, followed by one layer of polyurethane. Overall, most of the layer arrangements present a thick polyurethane layer to be placed on the vibrating hull. Four of these arrangements are now selected for further investigations. The scores of this selection are highlighted by the white squares in Figure 3.11.

For the four selected congurations, the hull decoupling coecient as well as the layer arrangement are presented in Figure 3 The impedance of water is shown by the blue dotted line

The moving mean µ mov of the impedance simply shows an impedance break almost centred within the medium. Consequently, an ecient hull decoupling medium can be made of a thick layer of absorbing material followed by a thick layer of reecting material. This design ensures that the waves remain trapped within the rst layer, which dissipates the energy.

The optimisation study for the hull decoupling coecient assisted in nding the impedance pattern leading to interesting performance. It is then observed that the ideal eective impedance for the hull decoupling coecient is very dierent from that for the anechoism coecient. In the next subsection, layer arrangements are optimised for both coecients in order to analyse the evolution of the impedance patterns.

Results for Anechoism and Hull Decoupling

Objective Functions

The goal is herein to minimise both the anechoism and the hull decoupling coecients so that they comply with the given required performance. For this purpose, the standard deviation functions are modied and performance levels are introduced. It is expected that the coecients are less than 0.1, corresponding to -20dB. It should be noted that this level is arbitrary and is only given to develop and illustrate the optimisation approach.

With this level requirement, the goal is now to minimise the deviation of the anechoism and hull decoupling coecients according to the level, but only for higher values since lower values imply a better performance. As such, the rst objective function is based on the normal standard deviation dened in Equation (3.5) and in Figure 3.5, but using the given level L instead of the root mean square of the coecient. Therefore, an intermediary function l C A is considered for each frequency f (i F ) in the frequency range of optimisation, dened as:

l C A (f (i F )) = L if |C A (f (i F ))| < L |C A (f (i F ))| if |C A (f (i F ))| > L . (3.7)
The level value is L = 0.1. The normalised standard deviation (NSD) according to the level requirements, referred to as σ L C A , is then obtained with:

σ L C A = RMS(l C A -L) L . (3.8)
Hence, with this objective function, the compliance for a given acoustic performance is optimised. A similar function is considered for the hull decoupling coecient C D and the second objective function is therefore expressed as:

σ L C D = RMS(l C D -L) L .
(3.9) All the layer arrangements of the Pareto front are given in Figure 3.16, noting that x = 0 corresponds to the water/coating interface, whereas x = 0.04 m would be the coating/hull interface. The main observations made in the two last sections are conrmed here: good anechoism performance is obtained with gradually increasing average density as a function of depth inside the arrangement, while the hull decoupling performance is better for a bi-layer design, whereby the layer at the coating/hull interface is polyurethane. As such, the layer at the interface x = 0.04 m tends to be thicker as the hull decoupling coecient diminishes, so that the media comprising multiple layers converge to a 2-layer medium. Among these solutions, four congurations, presented in Figure 3.17, are selected, highlighted by the white squares in Figure 3 It is observed that the anechoism coecient at low frequency is higher than it was for its optimisation on its own, whereas the hull decoupling is higher for the entire frequency range, but remains relatively low. As such, improving the overall hull decoupling coecient by increasing the last layer of polyurethane comes at the price of a poor anechoism performance at low frequency. The reverse observation is also veried here.

Results

Nonetheless, it is observed that both impedance patterns, associated with the anechoism coecient and the hull decoupling coecient are compatible.

Summary

In this chapter, an optimisation approach has been presented aiming to optimise the layer arrangement of multilayered media for the anechoic and hull decoupling performance. The optimisation algorithm has been validated on a test case.

Results were presented for the anechoic performance, which led mainly to media made of layers with gradually increasing average density as a function of depth inside the arrangement. A performance pattern has been drawn based on the eective impedance of the multilayered selection. In terms of a moving mean, a gradually increasing impedance along the medium thickness is obtained, which then drops close to the end of the medium (theoretically corresponding to the rigid end in an anechoic set up).

Results for the hull decoupling performance are simpler. It was shown that a simple bi-layer medium made of two thick layers with the soft layer placed on the hull performs best. The performance pattern for such a design shows a sudden and signicant impedance change that creates strong reection. Waves entering the coating from the soft layer are therefore trapped within this layer.

Finally, the optimisation algorithm was applied to simultaneously improve anechoism and hull decoupling gures of merit. The implemented optimisation approach thus appears as an ecient tool to optimise the layer arrangement of multilayered media for a variety of objective functions, materials and congurations.

Chapter 4

Experiments

The multilayered medium selected in Chapter 3 has been manufactured to be measured. In this chapter, a measurement method, referred to as the 3-point method, is presented and analysed. The 3-point method is rst used to characterise the polyurethane comprised in the selected multilayered medium. Its eective speed of sound is experimentally retrieved from the underwater measurements. Finally, the multilayered medium is measured and experimental results are compared to the theoretical ones. These pressures are used to calculate the reection and transmission coecients of a panel at positions A and B, as per:

R(A) = p r (A) p i (A) , (4.1)
T (B) = p t (B) p i (A) . (4.2)
A phase shift is then applied to these equations so that R corresponds to the reection coecient at the front face of the panel and T corresponds to the transmission coecient between the front and the back faces, as per the two-port network conguration.

At position A, in front of the panel, it is often dicult to temporally distinguish the reected pressure p r from the incident pressure p i , especially for low frequencies. The incident pressure can be obtained by measurement without the panel, or by calculation using several measurements as in the 3-point method described in this Chapter.

Methodology

In this section, a method is presented for acoustic panel measurements in water tanks.

This method is based on a conventional measurement conguration consisting of the panel, an acoustic source and two hydrophones, one on either side of the panel. In the experimental set-up, only three dierent positions for the hydrophones are considered.

This method, called the 3-point method, provides an estimation of the amplitude of the edge-diracted waves as well as the amplitudes of the transmitted, reected and incident waves. The method is detailed below, using Points A 1 , A 2 and A 3 correspond to the three positions taken by the hydrophone on the reection side, while points B 1 , B 2 and B 3 are the positions of the hydrophone on the transmission side. Pressure contributions and geometric parameters are also presented.

On both sides of the panel, the total acoustic pressure is measured at three points. The 3-point method then treats the reection and transmission sides in slightly dierent ways.

For the determination of the reection coecient, the three points A 1 , A 2 and A 3 are considered. The central position A 2 is taken as a reference, with the distances

d A12 = A 1 A 2 and d A23 = A 2 A 3 .
At each point, the total pressure p can be expressed as a function of the contributions of the incident pressure p i and the reected pressure p r at the central position as well as the diracted pressure on the reection side p diff R :

     p(A 1 ) = p i (A 2 )e -ikd A12 + p r (A 2 )e ikd A12 + p diff R (A 1 ) p(A 2 ) = p i (A 2 ) + p r (A 2 ) + p diff R (A 2 ) p(A 3 ) = p i (A 2 )e ikd A23 + p r (A 2 )e -ikd A23 + p diff R (A 3 ) . (4.3)
It is then assumed that the total contribution of the edge-diracted pressure is equal at these three positions when the three points are placed close to each other on the x-axis.

To legitimise this hypothesis, the analytical model presented in Appendix D is used by comparing the edge-diracted pressure in dierent positions. This hypothesis then leads to:

p diff R (A 1 ) = p diff R (A 2 ) = p diff R (A 3 ) = p diff R . (4.4)
The system of equations given by Equation (4.3) can be simplied and written in a matrix form as follows:

  p(A 1 ) p(A 2 ) p(A 3 )   =   e -ikd A12 e ikd A12 1 1 1 1 e ikd A23 e -ikd A23 1     p i (A 2 ) p r (A 2 ) p diff R   . (4.5)
By inverting Equation (4.5), the pressures p i (A 2 ), p r (A 2 ) and p diff R are obtained:

  p i (A 2 ) p r (A 2 ) p diff R   =   e -ikd A12 e ikd A12 1 1 1 1 e ikd A23 e -ikd A23 1   -1   p(A 1 ) p(A 2 ) p(A 3 )   . (4.6)
Once the total pressures at points A 1 , A 2 and A 3 are measured, the contribution of the incident, reected and edge-diracted pressures can be deduced using the previous system.

The reection coecient at point A 2 is then obtained using Equation (4.1).

On the transmission side, the same approach is adopted, considering the three points B 1 , B 2 and B 3 and the distances d B12 = B 1 B 2 and d B23 = B 2 B 3 , where position B 2 is taken as the reference. In this case, the pressure eld is decomposed into contributions of transmitted pressure p t , diracted pressure p diff T as well as pressure p ∞ called innite pressure. The latter describes a wave coming from innity in the direction of decreasing

x. On the reection side, this pressure would correspond to the contribution of reected waves, but on the transmission side, only waves reected on the water tank wall could be travelling in this direction. However, such waves are not recorded within the acquisition time window considered for the measurements. Thus, the innite pressure is assumed to be zero. On the transmission side, the same hypothesis on the edge-diracted pressure is used, so p diff T is also assumed to be equal for each of the three points. Total pressures at points B 1 , B 2 and B 3 are therefore expressed by:

     p(B 1 ) = p t (B 2 )e ikd B12 + p ∞ (B 2 )e -ikd B12 + p diff T p(B 2 ) = p t (B 2 ) + p ∞ (B 2 ) + p diff T p(B 3 ) = p t (B 2 )e -ikd B23 + p ∞ (B 2 )e ikd B23 + p diff T . (4.7)
The contributions of the transmitted, innite and diracted pressures are then calculated using:

  p t (B 2 ) p ∞ (B 2 ) p diff T   =   e ikd B12 e -ikd B12 1 1 1 1 e -ikd B23 e ikd B23 1   -1   p(B 1 ) p(B 2 ) p(B 3 )   . (4.8)
With the measurements of the total pressure at points B 1 , B 2 and B 3 , the contributions of the transmitted, innite and edge-diracted pressures can be deduced using the previous system. The transmission coecient between A 2 and B 2 is then obtained using Equation(4.2). The experimental bench is fully controlled via a LabView interface, which has been coded in such a way that the user simply needs to indicate the frequency range as well as the dierent positions of the hydrophones along the x-axis. It is also possible to automatically change the hydrophone positions for each frequency, in order to dene the distance between two positions in terms of wavelength λ instead of absolute distance.

Experimental Facilities

The water temperature of 15.5 • C remains stable for all measurements. Under such conditions, the speed of sound in water is estimated at 1489 m.s -1 .

The source emits a packet of sine waves at the frequency of interest for a duration of 2.2 ms. The signal received by each hydrophone is synchronised with this emitted signal.

The signal received on each hydrophone is initially observed globally over a large time window using a conventional oscilloscope. This makes it possible to dene the acquisition window to consider only the desired signals simultaneously (which will include the dierent pressures of interest) on each hydrophone using a picoscope. The choice of this window is made on several criteria: (1) it must be located before the reception of the own echo signals linked to the walls of the water tank, (2) it must start a little after the reception of the signal by each hydrophone once the signal is well established (without start-up bounce), and (3) it must be relatively homogeneous over the duration of the acquisition.

A start of the window at 2.7 ms over a period of 1 ms is often a satisfactory setting for measurements carried out in the water tank given the classic positioning of the dierent elements and the size of the tank. As a result, measurements for frequencies below 3 kHz can be dicult because there is not enough level on the source side and not enough number of waves observed on the time side, before the arrival of the echoes.

Once this acquisition window is dened, the automatic acquisition program is launched and the picoscope records in this acquisition window for each hydrophone, the frequency, the modulus and the phase of the received signal. The quality of the measurement can be evaluated using the dierence between the frequency measured on the hydrophones and that of the reference (imposed on the transmitter). A pre-processing step is thus always performed to check the quality of the recorded signals: pairs of positions X and frequencies f of unsteady-state signal are detected using the frequencies extracted from signals recorded in the acquisition window. When a retrieved frequency is signicantly dierent from the reference frequency imposed on the transmitter, the pair (X , f ) is not taken into account in the post-processing step. 4.1 thus complies with the hypothesis that the edge-diracted pressure is equal in the three measurement positions, as per Equation (4.4). Moreover, with this distance, the total pressures at the three positions are suciently dierent (cf. Test 3 in 4.2.2).

Method Validation

d X 2 (cm) d X 1,2 (λ) d X 2,3 (λ) θ ( • ) 8 λ/15
λ/15

9.1 Table 4.1 Geometric parameters for validation of the 3-point method.

The total pressures recorded at the three positions on each side of the panel are plotted in On the reection side, the total pressures are used to derive the incident pressure p i , the reected pressure p r and the diracted pressure on the reection side p diff R . On the transmission side, the 3-point method leads to the transmitted pressure p t as well as the diracted pressure p diff T . Pressure p ∞ coming from innity and travelling towards the negative x is also obtained for the transmission side. The fact that the innite pressure must be zero can help to detect an invalid application of the method. All these deduced pressures are shown in The incident pressure is compared to that obtained by measurements without panel, referred to as p WP i . It can be observed that both measurements of the incident pressure with and without the panel are consistent over the whole frequency range, validating the estimation of the incident pressure provided by the 3-point method. In addition, the innite pressure p ∞ is almost zero over the entire frequency range, which is expected when the 3-point method is valid and correctly implemented. It can also be noted that the diracted pressure on both sides oscillates approximately between 2% and 70% of the total pressure, highlighting that the edge-diraction eects cannot be neglected.

For comparison, the reection and transmission coecients can be obtained analytically for a medium of innite lateral dimensions [START_REF] Hennion | Analysis of the scattering of a plane acoustic wave by a periodic elastic structure using the nite element method: Application to compliant tube gratings[END_REF]. The material properties for aluminium are 2700 kg.m -3 for the mass density and 6183 m.s -1 for the longitudinal speed of sound. Since the 3-point method eliminates edge-diraction contributions, the experimental coecients can be directly compared to those of an innite panel, as in Figure 4.7.

Experimental coecients have also been tted to a quadratic curve. This t weakens the few slight discrepancies which are mainly explained by the structural resonant modes of the nite panel, such as the plate bending modes [START_REF] Audoly | Improvement of the measurement of the transmission coecient of panels at normal incidence using surface receivers[END_REF]. Therefore, the 3-point method leads to reection and transmission coecients that t well with the ones calculated analytically for an innite panel. Data recorded at the central position X 2 are also used to calculate the scattering coecients, as per the direct conventional method using a single point.

Results are dierent from the theoretical calculation, which highlights the benet of using the 3-point method instead of direct measurements to remove the contribution of the edge-diraction, since its eects are also strongly dependent on the hydrophone position, as shown latter in Section 4.2.3. analytically for a panel of innite lateral dimensions and measured with the direct single point method and with the 3-point method. Positions for the 3-point method are dened in Table 4.1 and the central position X 2 is used for the direct single point method. Results from the 3-point method are tted to a quadratic curve.

The reection and transmission coecients given by the 3-point method are complex values, so their phase can be calculated. In Figure 4.8, phases of the coecients derived from the 3-point method (dotted lines) are compared to the phases calculated theoretically. Phases obtained with the 3-point method were modied considering the distances hydrophone/panel d X 2 + 2.2cm for the reection side and d X 2 + 2.5cm for the transmission side.. It can be seen that their variations are similar but they dier from one to another.

This dierence can be explain mainly by the uncertainty on the absolute position of the hydrophone relative to the panel. Indeed, as a hydrophone is xed to a 3-meter-long section, it is unlikely that the position on the surface and in the water will be exactly the same. For instance, with a 3-meter-long section xed at a right angle on a support, an oset of 1 • of the support at the right angle would create a 5 cm displacement of the hydrophone in water. When the phase has to be shifted to the panel interface (for the calculation of the anechoism coecient for instance) there is then a strong uncertainty due to the lack of precision on the hydrophone/panel distance. However, it should be understood that this does not prevent the 3-point method from providing precise results (in modulus), as it only requires a relative position from one hydrophone to another and this displacement is precisely controlled by the motors. A distance correction is applied on the latter to retrieve the variation of the theoretical phase: Phases obtained with the 3-point method were modied considering the distances hydrophone/panel d X 2 + 2.2cm for the reection side and d X 2 + 2.5cm for the transmission side.

Validity Limits

It has been shown with the previous example that when the hypothesis on the diracted pressure is satised, the 3-point method leads to reection and transmission coecients that t well with those of an innite panel. The current section studies the limits of validity of the method based on three tests dened in Table 4.2. Measurements for these tests have been conducted between 1 kHz and 15 kHz. The rst test considers three points close to each other (separated by 1 cm so that the hypothesis on the edge-diracted pressure being equal in the 3 points is assumed to be satised) but far away from the panel, where the central position forms an angle θ = 25.6 • with the panel face. In the second test, a signicantly greater distance between hydrophone positions is considered using d X 1,2 = d X 2,3 = 10 cm, thus challenging the hypothesis on the edge-diracted pressure equal in the three points. Finally, Test 3 considers three positions so that the angle θ is now equal to 5.7 For these three tests, the reection and transmission coecients given by the 3-point method are plotted in Figure 4.9. It is rst observed that these experimental scattering coecients are very dierent from those of an innite panel obtained theoretically (lines without markers). Moreover, all of them reach values higher than one, which makes no sense physically for a passive material. Results for each test are further analysed. 4.2. Reection and transmission coecients obtained theoretically for a panel of innite lateral dimensions are also presented (same as in Figure 4.7).

In Test 1, the relative dierence between the edge-diracted pressure levels in each point remains less than 1% according to the model in Appendix D. Even though the hypothesis on the edge-diracted pressure can be considered as valid here, the experimental coecients are very dierent from the theoretical ones because when the hydrophones are this far away from the panel, the signal may be polluted by waves other than those diracted by the panel. The angle θ shall therefore remain less than 15 • . Test 2 considers a greater distance between hydrophone positions. This conguration gives acceptable results at very low frequencies, but leads to poor results at higher frequencies as the distances d X become too important in terms of wavelength. The relative dierence between the edge-diracted pressure levels is oscillating between 5% and 10% according to the analytical model in Appendix D. Poor quality results are thus obtained when the hydrophone set-up does not comply with the hypothesis on the hydrophone positions being close enough to each other so that the diracted pressure is assumed identical for the three positions. Ideally, the distance between hydrophone positions should be lower than λ/5.

Finally, for Test 3, the distance d X = 0.5 cm between hydrophone positions is equivalent to λ/300 at 1kHz and λ/20 at 15kHz. For this conguration, the hypothesis is satised and the relative dierence is less than 0.1% according to the analytical model. Moreover, these three points are very close to the panel. However, the points are so close to each other that the total pressures measured are almost identical at the three points, which leads to results strongly dependent on measurement and numerical errors. Therefore, a compromise must be found to dene three positions suciently close to each other and close to the panel, while being suciently spaced in terms of wavelength. It is thus preferable to maintain a minimum distance of λ/20 between two hydrophone positions.

The aforementioned criteria apply for the current aluminium panel and therefore will also generally apply to all panels of input impedance higher than the impedance of water.

Nevertheless, it is worth noting that the behavior may be dierent in terms of the edgediracted signal, for a panel having an input impedance lower than the impedance of water [START_REF] Audoly | Scattering from thin elastic screens[END_REF]. For such a case, one has to make sure that measurements are made within an area where a diracted signal exists at the 3 chosen positions so that it is accurately extracted from the total signal with the 3-point method.

Pressure Mapping

Acoustically hard panels are known for having a pronounced diraction pattern. In order to evaluate and visualise the variations of the edge-diracted pressure in a plane parallel to the panel, a pressure mapping is proposed. Both the analytical model in Appendix D and the 3-point method are applied for a pressure mapping along the y-axis.

Hydrophone positions along the x-axis dened in Table 4.1 are considered hereafter. Along the y-axis, 31 positions are considered, from y = -15 cm to y = 15 cm, with y = 0 cm corresponding to the panel's center. The distance between each y-position is then 1 cm. For frequencies 3.5 kHz, 5.5 kHz and 7.5 kHz, the calculated pressures on each side of the panel are plotted in Figure 4.10. For frequencies 14 kHz, 14.5 kHz and 15 kHz, the pressure mapping is given in Figure 4.11. Finally, variations of the edge-diracted pressure given by the model in Appendix D are plotted in Figure 4.12. It should be noted that Figure 4.12 does not provide the amplitude of the edge-diracted pressure, but only its variations. Experimental coecients are also given and compared to the theoretical coecients for a panel of innite lateral dimensions. On these pressure maps, the innite pressure p ∞ is almost zero everywhere, which shows that the 3-point method is valid and correctly implemented. Reected, transmitted and incident pressures are almost identical along the y-axis, which reinforces the assumption of a plane wave condition. However, on both sides of the panel, the diracted pressure strongly varies along the y-axis. This last observation is consistent with the results

given by the analytical model in Figure 4.12. The pattern of the diracted pressure eld then depends on frequency, since for low frequencies the edge-diracted pressure slowly varies along the y-axis, but as frequency increases, variations become steeper and closer. In addition, the center of the panel (y = 0) is always a maximum for the edge-diracted pressure. This pressure mapping highlights the strong variations of the diracted pressure along the y-axis and shows that a means of either reducing the edge-diraction or calculating and removing its contribution is crucial for accurate measurements.

Experimental Speed of Sound Determination 4.3.1 Measurements

In this section, the longitudinal speed of sound in the polyurethane medium comprising 6% of micro-balloons is experimentally determined. This experimental determination ensures that the material properties used in the optimisation study of Chapter 3 are really those of the material that can be manufactured and not purely theoretical properties. The polyurethane panel comprising of 6% of micro-balloons is of dimensions 900 mm×900 mm and of thickness 40.9 mm (on average), as shown in Figure 4.13. The theoretical mass density is 969 kg.m -3 . In practice, the measured mass density is 984 kg.m -3 . The case of a panel that has an input impedance lower than that of water is dierent from the case of an aluminium panel. The diraction pattern for such panel is indeed not as signicant as for a panel with higher impedance [START_REF] Audoly | Scattering from thin elastic screens[END_REF]. In addition, the amplitude of the diracted waves is so low that their eects on the reection and transmission coecients may be negligible. The 3-point method, with θ = 13

Retrieval Approach 1

Reection and transmission coecients of the panel at normal incidence are used to retrieve the complex speed of sound in the panel with the retrieval method of Fokin et al. [START_REF] Fokin | Method for retrieving eective properties of locally resonant acoustic metamaterials[END_REF]. This method is based on the transfer matrix method, which can be fully calculated for the medium using the scattering coecients (Appendix A). From the theoretical expression for the transfer matrix, the eective speed of sound can then be retrieved assuming that the mass density is known. The retrieved eective parameters are given in

Retrieval Approach 2

The unknown values of c L and η are obtained by numerically minimising over these two parameters the error between the measured values of the coecients R and T and their calculation using given values of c L and η. Based on the estimation given by the retrieval approach 1, the two parameters to retrieve are bounded. The range of possible values for the real part of the speed of sound is [800 m.s -1 -1100 m.s -1 ] and the loss factor taking its values within the range [2% -12%]. The minimisation criterion C is given by:

C(c L , η) = |R calc (c L , η) -R meas | + |T calc (c L , η) -T meas |. (4.9)
This criterion should provide a robust solution as it is over-determined [START_REF] Audoly | Evaluation of sound velocity inside underwater acoustic materials using test panel acoustic measurements[END_REF]. For each frequency, the criterion is calculated in the 2D-space (c L , η). For example, for the frequency 15 kHz, the criterion map is given in Figure 4.16. It is observed that for the region between 15 kHz and 17 kHz, there is no solution that satises the criterion C < 0.13. It is also noted that for frequencies less than 10 kHz, there is greater uncertainty, as shown by the blue bars. Overall, the real part of the speed of sound is growing slightly from 900 ± 10 m.s -1 between 5 kHz to 10 kHz to 950 ± 10 m.s -1 , whereas the loss factor η is about 5 ± 0.5%. These values are consistent with those obtained using the retrieval method in

Verication

The retrieved complex speed of sound has been tted, as presented in The thickness of each layer has been measured and is consistent with the required design.

The main diculty with this panel was to ensure that the center of mass, which is dierent from the geometric center since the panel is strongly asymmetric, was correctly aligned with the mounting post. Misalignment can in fact lead to the creation of an unknown angle for the panel which is then not characterised for normal incidence. The theoretical center of mass was thus calculated in order to align it with the mounting center using specially designed spacers to be placed on either side of the panel holding points (Figure 4.21(c)). However, it is important to note that, despite all the attention paid to the correct placement of the panel, there is still uncertainty since the center of mass calculation was performed on theoretical mass densities and thicknesses of the layers, which cannot be checked independently for this assembled panel.

Measurements

The 3-point method was used to measure the panel. The central position was placed 11 cm away from the face of the panel. A spacing of λ/10 was dened between the 3 positions of the hydrophone. Both sides of the panel are measured since the multilayered medium is not symmetrical. To do so, the panel was rotated but the acoustic source position remained unchanged.

Pressure magnitudes calculated by the 3-point method are presented in Figure 4. 22. This shows that the retrieved incident pressure matches with the pressure obtained by measurements without the panel. In addition, pressure p ∞ is almost zero over the entire frequency range, which is consistent with the theory. These two observations show that the 3-point method is applied under valid conditions. Moreover, it can be seen that from 20 kHz, the transmitted pressure is very low and reaches the same values as the edgediracted pressure. There is good overall consistency between the experimental and analytical results for the modulus. The transmission coecients measured on both sides is almost identical and also in agreement with the predictions for the modulus. However, it can be seen for both sides the panel is less reective at low frequency than what has been predicted. On the contrary, from 10 kHz it becomes slightly more reective, especially for Face 1. The phases of the two reection coecients are notably dierent from the prediction. This can be explained by signicant uncertainty on the exact distance between the hydrophone and the panel. In fact, this distance is considered twice in the calculation of the phase of the reection coecient: once to phase-shift the pressure reected on the face of the panel and once to phase-shift the pressure incident on this same face.

Phase corrections are applied by modifying the hydrophone/panel distances.

The 3 corrections are as follows :

Correction 1 corresponds to the distance corrections determined on the aluminium panel (Figure 4.8). The hydrophone on the reection side is thus assumed to be placed at d X 2 + 2.2cm from the panel and the transmission hydrophone is estimated to be at d X 2 + 2.5cm away from the panel.

Correction 2 corresponds to the distance corrections determined by attempting to t the theoretical curves for the insonication of Face 1. The hydrophone on the reection side is thus assumed to be placed at d X 2 + 1.5cm from the panel and the transmission hydrophone is estimated to be at d X 2 + 3.3cm away from the panel.

Correction 3 corresponds to the distance corrections determined by attempting to t the theoretical curves for the insonication of Face 2. The hydrophone on the reection side is thus assumed to be placed at d X 2 + 2.4cm from the panel and the transmission hydrophone is estimated to be at d X 2 + 2.3cm away from the panel.

These corrected phases are plotted in Figure 4.25 for the insonication of Face 1 and in Figure 4.26 for the insonication of Face 2. As such, it can be seen that Correction 1, determined on the aluminium panel, is a correction that also applies well for this panel, except for the reection coecient obtained by insonication of Face 2.

Calculation of the Anechoism Coecient

The anechoism coecient, dened in Appendix C, can now be derived using the experimental reection and transmission coecients. In addition to the calculation using coecients derived directly from the 3-point method, the anechoism coecient is also calculated using the coecients with corrected phases. For the rst calculation with phase correction, Correction 1 is applied to all coecients, as per :

C A,Correction1 = R Face1,Correction1 - T Face1,Correction1 T Face2,Correction1 R Face2,Correction1 -1 . (4.10)
For the second calculation, Correction 2 is applied on the coecients obtained for insonication of Face 1 and Correction 3 for that of Face 2, as given by:

C A,Mixed Correction = R Face1,Correction2 - T Face1,Correction2 T Face2,Correction3 R Face2,Correction3 -1 (4.11)
The resulting tted anechoism coecients are given in Figure 4.27 and compared with the theoretical prediction. This shows that the anechoism coecient obtained experimentally diers from the prediction in a non-negligible way. Since phase corrections are obtained from position corrections, this highlights that it is crucial to know the panel-hydrophone distance exactly if the reection and transmission coecients are to be used to calculate the anechoism coecient, especially at low frequencies. However, the mixed correction seems to provide an anechoism coecient close to the theoretical value. The slight dierences visible for this anechoism coecient appear to be due to the modulus variations. As mentioned previously, the panel is less reective than expected for low frequencies and for both sides of the panel, which yields a low anechoism coecient for this frequency range.

On the contrary, the panel becomes more reective at higher frequencies, thus leading to an higher value of the anechoism coecient. Coecients are also calculated considering the phase correction 1 obtained for the aluminium panel and considering mixed phase correction by the use of correction 2 and 3.

To summarise, the multilayered panel optimised in Chapter 3 has been measured in the water tank for both sides, in order to calculate the anechoism coecient. Modulus of both coecients, for both sides, are in good agreement with theoretical calculations. Slight dierences between the predictions and the experimental modulus can be explained by the use of tted material properties (obtained from the measurements) in the analytical calculation and by the possible local dierences between the theoretical and actual thicknesses and material properties of each layer. However, as for the aluminium panel, phases of the two coecients are dierent from the theoretical results since the panel/hydrophone distance is not well known. It is recalled that phase shifts are applied to these experimental coecients. However, this phase shift requires precise knowledge of the distance between the hydrophone and the panel. As such, the calculation of the anechoism coecient requires phase corrections to overcome this uncertainty. When the phase of the coecients are corrected independently for each side and then used together in the calculation of the anechoism coecient, the agreement between experimental and theoretical anechoism coecients is better.

Summary

In this Chapter, the conventional set-up for measurements in open water tank have been described. An experimental method, namely the 3-point method, has also been presented. Using measurements at three dierent positions, the 3-point method yields estimations of the incident, reected and transmitted pressures, from which the reection and transmission coecients can be derived. The method also provides an estimation of the total contribution of the edge-diracted waves. The method has been applied to the challenging case of a homogeneous panel made of aluminum, for which edgediracted eects usually strongly corrupt the measurements. Experimental reection and transmission were close to the theoretical calculation. The validity limits of the method have then been studied. It has been shown that it is important to choose the right positions for the hydrophones, so that the assumption on the identical edge-diracted pressure at the three points is satised. A mapping of the dierent pressures along the y-axis was presented, where the strong variations of the edge-diracted pressure are clearly visible.

The 3-point method has the great advantage of being easy to implement and it does not require any additional device other than the conventional experimental bench for open water tank measurements. This work has been published in [START_REF] Roux | Panel measurement method for the determination of scattering coecients and edge diraction in an open water tank[END_REF].

The 3-point method has then been applied to an acoustically soft panel to retrieve its eective speed of sound. This panel is made of polyurethane with the inclusion of 6% of micro-balloons, which is the material used in the multilayered medium. From the scattering coecients given by the 3-point method, the eective speed of sound has been retrieved using two dierent approaches, for which results are in agreement.

Finally, the multilayered design selected in Chapter 3 was measured using the 3-point method. Moduli of the reection and transmission coecients are consistent with analytical predictions. For Face 1, which is intended to be at the interface with water in an anechoic conguration, the reection coecient is rather low over the entire frequency range, while the transmission coecient is decreasing which is a characteristic of good anechoic media. Dierences were however noted on the anechoism coecient, which proves to be extremely sensitive to the distance between the panel and the hydrophones.

Nonetheless, the use of phase corrections seems to overcome this lack of accuracy and yields an anechoism coecient close to the theoretical prediction.

Part I introduced a homogenisation method for symmetric or asymmetric multilayered media (Chapter 2), which then was used for optimising layer arrangements. This study has shown patterns of performance for the anechoism and hull decoupling coecients (Chapter 3). The acoustic performance of an optimised multilayered medium was then veried experimentally using the 3-point method. The experimental method was initially validated using an reference aluminum panel and then applied to the polyurethane panel used in the multilayered medium for retrieving its eective speed of sound (Chapter 4).

In Part II, the emphasis is on designs with macro-inclusions, for which homogenisation methods, optimisation approaches and experimental performances are presented and are more complex than for their 1D multilayered counterparts.

Part II

DESIGNS WITH MACRO-INCLUSIONS 103

Chapter 5

Homogenisation for Symmetric Designs

In structures with macro-inclusions, additional physical phenomena may occur compared to one-dimensional periodic nite structures, such as local resonances and boundary eects. It therefore becomes more dicult to apply a simple homogenisation model. In this chapter, the eective properties of acoustic metamaterials are obtained by applying two retrieval methods, referred to as the direct inversion method and the dierential method. They employ the scattering coecients on the reection and transmission sides of structures immersed in a uid (in this work, water). As with Chapter 2, a uid homogenisation model is considered and a transfer matrix approach is used to describe the propagation in the constituent unit cell. A validation case study on a multilayered periodic design is rst presented. Two case studies consisting of designs with periodic cylindrical steel or void macro-inclusions in a soft elastic matrix are then examined.

Homogenisation issues related to dierent underlying assumptions in the two retrieval methods are discussed. A hybrid method is also introduced to provide an accurate eective representation of a structure with macro-inclusions. The uid homogenisation model is evidently simplistic when applied to complex periodic media with macro-inclusions. Several homogenisation diculties will be underlined in this Chapter and are rst mentioned here. First of all, only longitudinal waves can propagate in a uid whereas shear wave propagation occurs in solid media. As structures are studied under normal incidence, this assumption was valid for one-dimensional multilayered structures where only longitudinal modes were excited. However, for structures made of a elastic matrix with macro-inclusions, local resonances may appear and convert longitudinal waves into shear waves, which cannot be described by the present homogenisation model. Shear wave propagation is therefore not accurately modelled but is implicitly taken into account. The uid model also cannot consider the eects of the near-elds that may be created by media with macro-inclusions, as in practice, uid media of innite lateral extent do not produce near-eld eects. Aware of the approximations involved by the choice of a uid model, it is worth emphasising that the retrieval methods are implemented to give approximate predictions of a medium's acoustic performance so that it can be used in an optimisation process. As such, it is not an exact homogenisation model that is sought, but a simple method to get a reasonably accurate representation of the behavior of metamaterials with macro-inclusions.

It is worth remembering that propagation in a uid medium may be represented by a transfer matrix under the form given by Equation (2.2), recalled below : M = cos(kh) iZsin(kh) iZ -1 sin(kh) cos(kh) .

(5.1)

Eective properties k and Z are derived using two retrieval methods, that are techniques based on the scattering response of a structure, such as the reection and transmission coecients. These methods are presented in what follows and are referred to as the direct inversion (dir) method and the dierential (di ) method.

Direct Inversion Method

The direct inversion method uses the reection and transmission coecients of a structure to retrieve its eective parameters, similar to the method proposed by Fokin et al. [START_REF] Fokin | Method for retrieving eective properties of locally resonant acoustic metamaterials[END_REF].

The medium is modelled as a segmented structure comprising of periodic repetitions of a constituent unit cell, as shown in Figure 5.1. The segmented medium comprises n unit cells of length L u . Coecients R and T , respectively, correspond to the coecients of the reected and transmitted pressures calculated at the interfaces between the segmented medium and the surrounding uid medium. In this method, all unit cells are identical. The direct inversion method follows the same approach as the homogenisation method presented in Chapter 2 for multilayered structures, using the fact that transfer matrices can be expressed and calculated in several ways. One way to obtain the global transfer matrix M n of the periodic medium comprising n unit cells is to use its reection and transmission coecients in the surrounding uid of impedance Z f as follows:

M n = 1 2T 1 -R 2 + T 2 ((1 + R) 2 -T 2 ) Z f ((1 -R) 2 -T 2 ) Z -1 f 1 -R 2 + T 2 .
(

This expression is derived in Appendix A. Since all unit cells are identical and the homogenisation model is applied to each cell, the global transfer matrix of the segmented medium may also be written as the product of the transfer matrix of each cell. This leads to M n = (M u ) n , where M u is the transfer matrix of a constituent unit cell and L u is its thickness. It is assumed that the transfer matrix M u has the same form as the transfer matrix given in Equation (5.1). It is then easy to show that for any transfer matrix under the form given by Equation (5.1), the calculation of (M u ) n for any n integer is equivalent to only replacing the thickness L u by nL u without changing the wavenumber or the impedance associated with the matrix M u . Therefore, if the eective parameters may be found for a single unit cell, the same parameters apply for a segmented medium comprising n unit cells. The global transfer matrix expression is thus given by :

(M u ) n = cos(nk dir L u ) iZ dir sin(nk dir L u ) iZ -1 dir sin(nk dir L u ) cos(nk dir L u ) , (5.3)
where Z dir is the eective impedance and k dir is the eective wavenumber obtained for the direct inversion method. The global transfer matrix expression given by Equation (5.3) is equal to the global transfer matrix calculated using the scattering response of the medium given by Equation (5.2). The eective parameters are derived from this equality.

We herein introduce the notation M n i,j to express the elements of the global transfer matrix M n , with 1 ≤ i, j ≤ 2. The eective impedance is given by:

Z dir = ± M n 1,2 M n 2,1 .
(5.4)

Material passivity condition requires the real part of the eective impedance to be positive (See section 1.4.5). The eective wavenumber is obtained using either M n 1,1 or M n 2,2 , as per:

k dir = ± cos -1 (M n 1,1 ) nL u + 2πm nL u .
(5.5)

The passive material condition is satised when the imaginary part of k dir is negative. Integer m is chosen to ensure continuity of the real part of k dir as a function of frequency (see Section 1.4.5).

Dierential Method

The second retrieval method, referred to as the dierential method, is an extension of the Bianco and Parodi method [START_REF] Bianco | Determination of the propagation constant of uniform microstrip lines[END_REF] and is detailed below. Two versions of the dierential method are introduced. In the rst version, the transfer matrix of the unit cell is assumed to be of the form given by Equation (5.1), whereas in the second version of the dierential method, eective parameters are obtained independently of this transfer matrix form.

Bianco and Parodi Method

The Bianco and Parodi method (BP) is a retrieval method which was originally developed for electromagnetism, in order to derive the propagation constant in a medium of innite length by examining wave propagation within a section of a nite medium [START_REF] Bianco | Determination of the propagation constant of uniform microstrip lines[END_REF]. Figure 5.2 presents the conguration required to apply the Bianco and Parodi method. Two media are considered, diering only by their length, where the second medium is greater in length by ∆ L compared to the rst medium. The propagation constant is calculated for this extra length. R i and T i (i = 1, 2) respectively correspond to the coecients of the reected and transmitted pressures calculated at the interfaces between the two media and the surrounding water. The Bianco and Parodi method does not require the application of a descriptive model to the entire media. The homogenisation model is only applied to the extra portion ∆ L in the second medium, assuming that the medium is long enough to create a region where wave propagation is monomode, thus described by the wavenumber k BP in the medium core. Firstly, the rst medium (m1) is virtually segmented in two sections, referred to as section 1 (s 1 ) and section 2 (s 2 ), for which the transfer matrices are respectively M s 1 and M s 2 . The global transfer matrix for the shorter medium is then given by:

M m1 = M s1 M s2 .
(5.6)

The global transfer matrix for the second medium (m2) only diers by the introduction of the section ∆ L in-between the sections 1 and 2, which yields:

M m2 = M s1 M ∆ L M s2 .
(5.7) Equation (5.7) is multiplied by the inverse of the global transfer matrix of the rst medium given by Equation (5.6), which leads to:

M m2 (M m1 ) -1 = M s1 M ∆ L M s2 (M -1 s2 M -1 s1 ) = M s1 M ∆ L M -1 s1 .
(

By denition, the matrix M s1 M ∆ L M -1 s1 and M ∆ L are similar. As such, these two matrices have the same trace, which is also the same as the product M m2 (M m1 ) -1 according to Equation (5.8). Moreover, the transfer matrix M ∆ L , describing the core of the second medium, is assumed to be under the form of Equation (2.22). Consequently, the eective wavenumber k BP is obtained using:

Tr M m2 (M m1 ) -1 = Tr (M ∆ L ) = 2cos(k BP ∆ L ).
(5.9)

The eective wavenumber k BP is thus derived using the trace of the transfer matrix of the unit cell, which is

Tr (M n+1 (M n ) -1 ) = Tr (M u ) = M u 1,1 + M u 2,2 .
The eective wavenumber expression is then:

k BP = ± cos -1 ( Tr(Mu) 2 ) L u + 2πm L u .
(5.10)

The Bianco and Parodi method gives the eective wavenumber in the medium of theoretical innite length. The method is valid even when the wave coupling process at the inlet/outlet cannot be modelled by a transfer matrix under the form of Equation (5.1), as it is unnecessary to apply a homogenisation model for these sections. The Bianco and Parodi method thus accurately removes physical phenomena that may arise from the nite length of the medium and from the coupling at the interfaces with the surrounding uid on the incidence and transmission sides of the medium [START_REF] Croënne | Eective parameter retrieval of phononic crystal slabs[END_REF]. However, it is worth noting that the Bianco and Parodi method, as well as the following dierential method, requires the calculation of the scattering coecients for two media, unlike the direct inversion method which only needs one medium.

Dierential Method

The Bianco and Parodi method is extended here to calculate more eective properties than only the eective wavenumber. This extended method is referred to as the Dierential Method. The two media are modelled as segmented structures comprising periodic repetitions of a constituent unit cell, as shown in Figure 5.3. The extra portion added to the second segmented medium is one unit cell of length L u . The shorter segmented medium comprises n unit cells and the longer medium comprises n + 1 unit cells. Coecients R i and T i (i = 1, 2) respectively correspond to the coecients of the reected and transmitted pressures calculated at the interfaces between the segmented media and the surrounding uid medium. In this extended method, all unit cells are identical. Transfer matrices of the two media are expressed and calculated using their scattering coecients. The global transfer matrix M n of the shorter periodic medium comprising n unit cells can be calculated using its refection R 1 and transmission T 1 coecients as follows:

M n = 1 2T 1 1 -R 2 1 + T 2 1 ((1 + R 1 ) 2 -T 2 1 ) Z f ((1 -R 1 ) 2 -T 2 1 ) Z -1 f 1 -R 2 1 + T 2 1 .
(5.11)

Similarly, the global transfer matrix M n+1 of the longer segmented medium can be calculated using its scattering coecients R 2 and T 2 as given by (Appendix A):

M n+1 = 1 2T 2 1 -R 2 2 + T 2 2 ((1 + R 2 ) 2 -T 2 2 ) Z f ((1 -R 2 ) 2 -T 2 2 ) Z -1 f 1 -R 2 2 + T 2 2 .
(5.12)

Two versions are from now on distinguished.

-Version 1

Since all unit cells are identical, the global transfer matrix of a segmented medium may also be written as the product of the transfer matrix of each unit cell. This leads to M n = (M u ) n for the shorter segmented medium and M n+1 = (M u ) n+1 for the longer medium, where M u is the transfer matrix of a constituent unit cell. The transfer matrix M u of a unit cell can be now deduced from the product of the global transfer matrix of the longer segmented medium with the inverse of that of the shorter medium, that is:

M n+1 (M n ) -1 = (M u ) n+1-n = M u .
(5.13)

The left hand side of Equation (5.13) corresponding to the product M n+1 (M n ) -1 can be calculated in terms of the scattering responses of the two segmented media, using Equations (5.11) and (5.12).

The transfer matrix M u on the right hand side of Equation (5.13) is assumed to be of the form of Equation (5.1), similar to the direct inversion method. Therefore all unit cells, including the extra unit in the second segmented medium, can be modelled by a transfer matrix in the form of Equation (5.1). This hypothesis was not present in the original Bianco and Parodi retrieval method, where a descriptive model was only applied to the core of the second medium. In this version of the dierential method, it is assumed that all the segments (units) can be described by the transfer matrix M u . As later shown through case studies, this assumption may aect the retrieved parameters particularly if the transfer matrix in the form of Equation (5.1) fails to describe the wave coupling at the medium interfaces. The right hand side of Equation (5.13) is thus expressed as follows:

M u = cos(k diff 1 L u ) iZ diff 1 sin(k diff 1 L u ) iZ -1 diff 1 sin(k diff 1 L u ) cos(k diff 1 L u ) , (5.14)
where k diff 1 is the longitudinal eective wavenumber and Z diff 1 is the eective impedance of the eective unit cell of length L u . Using Equations (5.11) to (5.14), the eective parameters Z diff 1 and k diff 1 of the unit cell can be extracted as follows. Elements of the unit cell M u are referred to as M u i,j , 1 ≤ i, j ≤ 2. The eective impedance is given by:

Z diff 1 = ± M u 1,2 M u 2,1 .
(5.15)

As for the direct inversion method, the real part of the eective impedance is required to be positive according to the material passivity constraint. The eective wavenumber is obtained using either M u 1,1 or M u 2,2 as given by:

k diff 1 = ± cos -1 (M u 1,1 ) L u + 2πm L u , (5.16)
where m is an integer chosen for continuity of the eective wavenumber as a function of frequency. The imaginary part of the wavenumber must be negative with respect to the material passivity constraint.

Since M u 2,2 = M u 1,1 according to the transfer matrix form of the unit cell given by Equation (5.14), the eective wavenumber from the original Bianco and Parodi method (Equation (5.10)) is supposed to be the same as the wavenumber derived with this version of the dierential method (Equation (5.16)). However, it will be shown later with case studies that M u 2,2 may dier from M u 1,1 when the transfer matrix in the form of Equation (5.1) fails to describe the unit cell. A second version of the dierential method is therefore implemented to overcome this diculty which disregards the form of the transfer matrix for the constituent unit cell.

-Version 2

For the second version of the dierential method, the eective wavenumber is obtained in a similar way to the Bianco and Parodi original method. Assuming that wave propagation within the second medium extra portion L u is monomode and described by the eective wavenumber k diff 2 , the uid homogenisation model is only applied to this core portion L u which is then described by a transfer matrix in the form given by Equation (5.1), yielding

Tr (M u ) = 2cos(k diff 2 L u ).
The expression for the eective wavenumber can be obtained as:

k diff 2 = ± cos -1 ( Tr(M n+1 (Mn) -1 ) 2 ) L u + 2πm L u , (5.17)
where m is an integer whose value is determined such that the eective wavenumber as a function of frequency is continuous.

For the determination of the eective impedance, Equation (5.13) is considered for which the left hand side is calculated using Equations (5.11) and (5.12). Contrarily to the rst version however, the eective impedance is obtained independently of the theoretical expressions for the elements of the transfer matrix M u . For this purpose, the denition of the transfer matrix of a medium is written as:

p left = M u 1,1 p right + M u 1,2 v right v left = M u 2,1 p right + M u 2,2 v right .
(5.18)

The impedance Z right is dened as the ratio of p right and v right and the impedance Z left is dened as the ratio of p left and v left . Equation (5.18) then becomes:

Z left = p left v left = M u 1,1 Z right + M u 1,2 M u 2,1 Z right + M u 2,2 .
(5.19)

Impedances Z left and Z right are dened locally. Assuming that this denition is valid and using the denition of a uid, the eective impedance at the inlet is equal to that of the outlet. The impedance Z of the medium is thus dened as Z = Z right = Z left and it can be obtained by solving the following quadratic equation:

Z 2 M u 2,1 + Z(M u 2,2 -M u 1,1 ) -M u 1,2 = 0.
(5.20)

There are then two solutions for Z = Z diff 2 , given by:

Z diff 2 = -(M u 2,2 -M u 1,1 ) ± (M u 2,2 -M u 1,1 ) 2 + 4M u 1,2 M u 2,1 2M u 2,1 .
(5.21)

It is reminded that elements M u i,j are calculated using Equation (5.13). However, it is not assumed that their form comply with the element forms given by Equation (5.1) as it was the case in Version 1. Nevertheless, it can be noted that if M u 2,2 = M u 1,1 , then the expression for the eective impedance can be simplied into the expression for the eective impedance in Version 1. Such an equality would also mean that the transfer matrix in the form given by Equation (5.1) for the uid model accurately represents the unit cell.

Consequently, this equality between the transfer matrix elements M u 1,1 and M u 2,2 can be used as an indicator to identify if the transfer matrix of the uid homogenisation model fails to describe the unit cell. The dierence ∆ Mu is introduced and dened by:

∆ Mu = M u 2,2 -M u 1,1 . (5.22)
When this dierence is equal to zero, it does not necessarily imply that the transfer matrix of the homogenisation model accurately describes the unit cell. However the reciprocal implication is valid: if this dierence is not equal to zero, then the uid homogenisation model is not suitable for the unit cell, otherwise it would have been possible to describe the unit cell with a transfer matrix in the form of Equation (5.1). In such cases, Version 2 of the dierential method would be more suitable as it disregards the constraints of the transfer matrix form, even thought it still considers some hypotheses of the uid homogenisation model.

In the following case studies, the dierence ∆ Mu is evaluated as a function of frequency in the form of a relative dierence ∆ r Mu , given by:

∆ r Mu = 2(M u 2,2 -M u 1,1 ) (M u 2,2 + M u 1,1
) .

(5.23)

Implementation

The two retrieval methods require the reection and transmission coecients of a periodic structure to retrieve the eective properties of a constituent unit cell. Those scattering coecients may be obtained from an analytical model for simple geometries such as multilayered media, from experimental measurements or from nite element models.

The scattering coecients are here numerically calculated using the FEM code ATILA [START_REF]ATILA Finite Element Software Package for the analysis of 2D & 3D structures based on smart material[END_REF]. Water is modelled on the incidence and transmission sides of the media, with the uid speed of sound set as c f = 1500 m/s and uid density ρ f = 1000 kg/m 3 . Each periodic medium is subject to harmonic plane wave excitation at normal incidence from the uid domain. A periodic boundary condition is applied on the lateral boundaries of the medium to simulate an innite array of unit cells in the direction transverse to the direction of sound propagation. Reection and transmission coecients are calculated from the pressure at the interface between water and the incidence and transmission sides of the media, respectively.

It has to be noted that in practice, for media with macro-inclusions, the scattering coecients are not directly calculated at the interfaces but are taken further away and then phase-shifted to be brought back at the interfaces. The reason is that the near eld of a locally resonant medium is not uniform, unlike one-dimensional multilayered media, as shown in Figure 5.4. This procedure eectively ensures that, in the presence of a nonuniform near-eld, the scattering coecients describe the behaviour of the nite structure in the far eld. It should be pointed out that by denition, the uid homogenisation model used here cannot describe any complex near eld at the medium inlet/outlet.

Validation

The objective of the current section is to assess the validity of the retrieval methods by comparing the retrieved eective parameters to those obtained with the transfer matrix method of Chapter 2. The multilayered medium examined here corresponds to arrangement version (a) of the periodic cell in Chapter 2. This design is recalled in Dispersion curves in terms of the dimensionless (reduced) wavenumber are shown in Figure 5.6, for a frequency range up to 40 kHz. Variations of the eective impedance are presented in Figure 5.7. It can be seen that the direct inversion method, the Bianco and Parodi method and versions 1 and 2 of the dierential method give the same wavenumber and the same eective impedance, which is also identical to those calculated with the transfer matrix model presented in Chapter 2. Therefore, both retrieval methods are valid for the description of the propagation in the multilayered medium.

The multilayered structure is not analysed in detail here, as it has already been studied in Chapter 2. As in Chapter 2, the eective properties obtained with the retrieval methods can be used to characterise an eective medium that can accurately predict the reection and transmission coecients of a symmetric periodic multilayered medium of arbitrary length. The uid homogenisation model is thus suitable for the analysis of such structures.

Eective Parameters

Dispersion curves in terms of dimensionless wavenumber are plotted in Figure 5.9 up to 30 kHz. Firstly, it can be seen that only the wavenumber from the Bianco and Parodi method (k BP ) and the wavenumber from Version 2 of the dierential method (k diff 2 )

are identical, thus validating the fact that the dierential method accurately describes propagation in the core of the medium, as does the Bianco and Parodi method. However, the eective wavenumber k diff 1 diers signicantly, in particular at some frequencies, even though it should also be the same. To explain this dierence, the relative dierence between the diagonal elements of the transfer matrix M u calculated with the dierential method, as per Equation (5.23), is plotted in Figure 5.10. This dierence reaches a maximum around 13.5 kHz, which exactly corresponds to the frequency where the wavenumber k diff 1 diers the most from the others. This signicant dierence implies that the transfer matrix M u of the unit cell does not exist or cannot be in the form of Equation (5.1), as diagonal terms are too dierent. This explains why Version 1 of the dierential method, which is based on the assumption that the transfer matrix M u has the form of Equation (5.1) with equal diagonal terms, fails. Version 1 of the dierential method is from now on put aside, since by construction it does not provide a good approximation of the unit cell. Version 2 of the dierential method is now referred to as the dierential method only. As a reminder, Version 2 gives eective parameters without relying on a transfer matrix form, which is why it still gives good results even though the unit cannot be exactly modelled as a uid. For the sake of clarity, dispersion curves have also been plotted in Figure 5.11 with only the wavenumber from the direct inversion method and from the dierential method (Version 2). It can be observed that both retrieval methods lead to the same eective wavenumber up to approximately 10 kHz. Dierences appear for frequencies around points A, B and C.

These frequencies correspond to resonant modes for which the displacement elds are given in Figure 5.12. Figure 5.12A presents the displacement eld in the four-unit medium for frequency A (13.6 kHz). The displacement eld exhibits a mass-spring (dipole) resonance where the inclusion represents the rigid mass, translating without undergoing any shape changes, while the host matrix represents the spring that retains the oscillations of the masses. For point C (25.7 kHz), Figure 5.12C shows that the rubber matrix is moving, whereas the steel inclusions are mostly stationary. The waves scattered by the rigid cylinders are trapped in-between the cylinders, leading to resonance of the host rubber.

This mode is therefore only existing for the matrix portions which are not at the interfaces in contact with the surrounding water. In Figure 5.11, as the imaginary part of the wavenumber represents the attenuation per distance, the wavenumber derived with the dierential method for resonances A and C leads to slightly greater attenuation than the one from the direct inversion method, but the variations remain similar. Another specic displacement eld of the rubber can be observed in Figure 5.12B for frequency B (19.8 kHz), but very localised at the input uid/medium interface. Point B thus also corresponds to a resonant mode in the matrix which only exists between the interface and the rst layer of rigid inclusions. As such, unlike the two other resonances, the boundary interference eects of point B do not vary with the number of unit cells placed after the rst unit cell at the interface. In Figure 5.11, the interface mode at point B is associated with an attenuation peak for the direct inversion method. Such attenuation is not apparent using the dierential method, which aims to describe the interior of the medium away from the boundary interfaces with the uid. In contrast, the direct inversion method describes the nite medium including boundary interface eects.

The eective impedance obtained with the direct inversion method and the dierential method are plotted in Figure 5.13. It can be seen that the two impedances are fairly similar, except at frequencies around the interface resonance, highlighted by the orange vertical line. For this resonance, there is a local maximum compared to water impedance, which means the medium is a good reector for these frequencies. This is consistent with the analysis of the displacement elds in Figure 5.12B corresponding to a coupling between the incident wave and a surface mode only. Nevertheless, it can be noted that the eective impedance from the dierential method Z diff is lower than the eective impedance from the direct inversion method Z dir . with the dierential method (Z diff ). The impedance of water is also represented.

The eective properties for the locally resonant medium dier depending on the retrieval method used, which was not the case for the multilayered medium. This is due to the fact that the uid homogenisation model does not take into consideration complex near eld eects and shear wave propagation, as the eective medium is assumed to be a uid medium in which only longitudinal waves can propagate. This assumption is valid for multilayered structures under normal incidence where only longitudinal modes were excited. However, for structures made of a solid matrix comprising of inclusions, the conversion of longitudinal waves into shear waves due to local resonances cannot be fully described by the present homogenisation model. However, both retrieval methods capture the behaviour of the locally resonant metamaterial with reasonable accuracy. The inuence of the number of unit cells used to apply the retrieval methods is examined in the following subsection.

Inuence of the Number of Unit Cells

In order to assess the robustness of the two retrieval methods, they are both applied on media of dierent length. The direct inversion method is applied to four media comprising of designs with n = 2, n = 4, n = 8 and n = 18 unit cells. The dierential method is applied to the four designs, corresponding to (2, 3), (4, 5), (8, 9) and (18, [START_REF] Fok | Negative acoustic index metamaterial[END_REF], where the shorter medium is the design used for the direct inversion method, comprising n unit cells. As previously, the wavenumber and eective impedance are calculated for all congurations to visualise the eects of the number of unit cells on the eective parameters.

Figure 5.14 shows the eective wavenumbers obtained by the four applications of the direct inversion method. It can be seen that the eective wavenumbers from the direct inversion method vary depending on the number of units n in the medium. In particular, the attenuation peaks associated to resonances (blue lines) tend to be stronger with increasing n, as indicated by the blue arrows. In contrast, the attenuation peak due to the interface resonance tends to be lower for higher values of n, as indicated by the orange arrow. For the dierential method, in Figure 5.15, there is nearly no dierence between the wavenumbers corresponding to the four combinations. As such, the eective wavenumber from the dierential method seems independent of the number of unit cells contained in the two segmented media. the dierential method for the unit combinations (2,3), (4,5), [START_REF] Fang | Ultrasonic metamaterials with negative modulus[END_REF][START_REF] Lee | Composite acoustic medium with simultaneously negative density and modulus[END_REF] and [START_REF] Graciá-Salgado | Negative mass density and ρ-near-zero quasi-two-dimensional metamaterials: design and applications[END_REF][START_REF] Fok | Negative acoustic index metamaterial[END_REF].

Similar observations to the eective wavenumber can be made for the variations of the eective impedance. It can therefore be concluded that the eective wavenumber from the direct inversion method is dependent on the number of unit cells n. Further, with increasing number of unit cells, the eective wavenumber converges towards the result obtained using the dierential method, corresponding to the eective wavenumber in the medium interior away from boundary interface eects. More specically, at the interface resonance around 20 kHz, the direct inversion method is strongly inuenced by the number of unit cells n whereas the dierential method removes the interface eects.

Scattering Predictions

Eective properties from both methods are now used to characterise eective media in order to predict the reection and transmission coecients of a segmented metamaterial comprising N unit cells. The length of eective media is thus L = N × L u . The eective parameters used were obtained for n = 4 and are given in characterised by the eective wavenumber and eective impedance obtained using the two retrieval methods. A hybrid eective medium is also introduced in order to take advantages of both methods, by using the eective properties of the direct inversion method at the interfaces and the eective properties from the dierential method for the interior of the eective medium. The hybrid medium is therefore made of three domains as shown in Figure 5.16(d). the reection and transmission coecients are predicted using an eective medium characterised by the eective wavenumber and impedance obtained using (b) the dierential method (diff), (c) the direct inversion method (dir), (d) a hybrid eective medium comprising the direct inversion method at the boundary interfaces and the dierential method for the medium away from the interfaces.

Results are presented in Figure 5.17 the resulting reection coecient will be close to that for the medium used to derive Z dir , which here is the n = 4-unit medium. Both the direct inversion method and the hybrid method are therefore less suitable for the prediction of the reection coecient for various N -layer media than the dierential method. For the transmission coecient, each property characterising the medium is relevant. Accurate predictions are obtained using the dierential method, except for the transmission dip around 20 kHz arising from the interface mode. Therefore, the transmission coecient is better predicted using the hybrid eective medium, as it physically restricts the eective properties of the direct inversion method to the interfaces and the eective properties from the dierential method to the core of the eective medium. obtained with the direct inversion method (k dir ) and with the dierential method (k diff ).

The dispersion curves are given in Figure 5. [START_REF] Fok | Negative acoustic index metamaterial[END_REF]. We observe that eective wavenumbers dier around 13.8 kHz, which corresponds to a resonant mode in the matrix at the interface similar to the resonant modes observed with the presence of a hard inclusion. The eective wavenumbers using the two retrieval methods signicantly dier above 23 kHz.

The displacement elds corresponding to point A (23.75 kHz) and B (28.5 kHz) exhibit exural motion of the matrix layer at the interface, similar to a drum-like up and down motion [START_REF] Hladky-Hennion | Analysis of the scattering of a plane acoustic wave by a doubly periodic structure using the nite element method: Application to Alberich anechoic coatings[END_REF]120]. This results in signicant reection of the acoustic energy by the rst unit cell. As there is very little propagation within the medium beyond the rst unit cell, the dierential method cannot accurately dene the wavenumber in the interior of the medium. The accuracy of the predictions using the eective parameters is studied in what follows.

Scattering Predictions

The reection and transmission coecients are presented to check the validity of the homogenisation approach for the voided medium design. Results are presented in Figure 5.21 for N = 5, 9, 19 unit cells using the various eective designs shown in Figure 5.16. The reection coecients for all eective designs are in close agreement with each other. Since the reection coecient is mostly dependent on the medium impedance at the interface on the incident side and since Z dir ≈ Z diff , all eective media give similar results. The accuracy of the predictions then validates the eective impedance given by both retrieval methods. characterised by eective parameters of the dierential method (dashed lines), the direct inversion method (dotted lines) that represent an Alberich medium comprising N = 5, N = 9 or N = 19 unit cells. Scattering coecients from the hybrid eective medium (circle markers) and from FEM (full lines) are also presented.

We observe that the interface mode around 13.8 kHz does not lead to a signicant dip in the transmission coecient. The hybrid method nonetheless predicts the local minimum related to the interface resonance with greater accuracy than the dierential method, which removes interface eects, and the direct inversion method which accentuates the eects of the interface resonance. At higher frequencies, the transmission coecient becomes extremely low, which is accurately predicted by the homogenised media. There are however a few discrepancies above 23 kHz for the prediction using the dierential method only, for N = 5, which do not occur using the hybrid method or the direct inversion method. It thus conrms that when most of the acoustic energy does not propagate further than the rst unit cell, the dierential method cannot properly dene the eective wavenumber in the interior medium. However, the direct inversion method accurately captures the strong reection by the rst unit cell. As such, the hybrid method provides an accurate prediction as it physically restricts the eective properties of the direct inversion method at the eective medium input.

Summary

Two retrieval methods have been introduced, corresponding to the direct inversion method and the dierential method. The direct inversion method uses the reection and transmission coecients of a medium comprising a number of identical unit cells, immersed in a uid, while the dierential method requires two segmented media comprising n and n + 1 unit cells. Both retrieval methods lead to an eective wavenumber and an eective impedance of a homogeneous uid medium. The eective wavenumber from the dierential method was able to describe the interior of the medium whereas the direct inversion method captures the eects related to the medium's nite size and interfaces with the surrounding uid media on the incidence and transmission sides. As such, to predict the scattering coecients, a hybrid eective medium was developed, whereby the rst and last layers are dened with the eective properties from the direct inversion method and the core layer is dened by the dierential method. Reection and transmission coecients have been predicted using eective media characterised by the direct inversion method, by the dierential method and by the hybrid medium. The reection coecient is accurately predicted by the dierential method. The hybrid medium provides accurate predictions of the transmission coecient as it physically restricts the eective properties of the direct inversion method to the interface and uses the dierential method to describe the core of the medium. This work has been published in [START_REF] Roux | Homogenization of acoustic metamaterials using retrieval methods[END_REF].

Three case studies have been analysed corresponding to a multilayered medium and locally resonant designs with hard or voided cylindrical inclusions. For the multilayered medium, both retrieval methods lead to the same eective parameters, also identical to those obtained with the transfer matrix-based model presented in Chapter 2. However, the two retrieval methods dier for a locally resonant media with either steel or voided cylinders in a viscoelatic matrix. For the design with hard scatterers, variations of the eective parameters have been analysed for each physical phenomena observed, namely, dipole resonance, a trapped mode and an interface mode. The dierential method accurately removes the interface eects and gives the eective properties describing the interior of the medium, independent of the number of unit cells initially considered. In contrast, using the direct inversion method, variations in the eective properties at frequencies associated with an interface resonant mode can be observed. Similar observations have been made for the design with voided scatterers. For this design, the dierential method cannot provide the eective wavenumber due to very little wave propagation in the medium interior.

The retrieval methods described here may be benecial to get a better understanding of wave propagation in a metamaterial coating. Moreover, the hybrid method is an ecient tool to predict and optimise the acoustic performance of coating designs that may contain less classical inclusions. In Chapter 7, the retrieval methods will be applied to several metamaterial designs in order to create a database of eective properties associated with various inclusion shapes, sizes and materials. Eective properties will then be used in an optimisation process in order to obtain structures with a good acoustic performance over a wide frequency range. The work in the present chapter is therefore useful to quickly and easily make a topological optimisation of acoustic coatings for underwater applications using uid properties.

In general, the goal of these methods is to build homogeneous equivalent media which are a precise approximation of their micro-structured counterparts. Eective media can be used to study metamaterials with complex or arbitrary geometries for which analytical models do not apply. The proposed homogenisation approach may be especially useful in the study of large metamaterial-based systems including structural gradients, such as the ones considered in transformation acoustic designs. Similarly, the proposed homogenisation approach could prove useful in the study of multi-scale metamaterials to alleviate the simulation issues related to the presence of complex geometries at dierent scales, by allowing the replacement of the smaller structures by equivalent media.

Chapter 6

Homogenisation for Non-symmetric Designs

In this chapter, the direct inversion method and the dierential method are extended in order to obtain the eective parameters of nite periodic and asymmetric structures with macro-inclusions. It is assumed that such a structure can be described by an eective Willis uid similar to the asymmetric multilayered unit cells in Chapter

M = cos(kh) -W sin(kh) iZ W sin(kh)(1 + W 2 ) i sin(kh) Z W cos(kh) + W sin(kh) . (6.1)
Three eective parameters are thus required (k, Z W , W ), whereas there were only two eective parameters for the symmetric case (k, Z). The extensions presented in this Chapter can also be applied to symmetric congurations, since it implies that the asymmetry coecient W is equal to 0, which reduces the number of undetermined parameters to two.

Direct Inversion Method

As in Chapter 5, the medium is modelled as a segmented structure comprising n repetitions of an identical unit cell of length L u . Coecients S are calculated at the interfaces between the segmented medium and the surrounding uid medium of impedance Z f . It is worth remembering that S 11 is dierent from S 22 for asymmetric structures. Two calculations are thus required, considering rst an incident wave-front towards the x positive and then towards x negative, as shown in Figure 6.1. The global transfer matrix M n of the periodic medium containing n unit cells can be calculated using its S-parameters: This expression is given in Appendix A. The homogenisation model is applied to each unit cell, so the global transfer matrix of the segmented medium may also be written as the product of the transfer matrix of each unit cell. This leads to M n = (M u ) n , where M u is the transfer matrix of a constituent unit cell of the form given by Equation (6.1). The calculation of (M u ) n for any n integer is equivalent to replacing only the thickness L u by nL u without changing the wavenumber or the impedance associated with the matrix M u . The global transfer matrix expression is thus given by :

M n = S 11 -
M n u = cos(nk dir L u ) -W dir sin(nk dir L u ) iZ dir sin(nk dir L u )(1 + W 2 dir ) i sin(nk dir Lu) Z dir cos(nk dir L u ) + W dir sin(nk dir L u ) . (6.
3)

The global transfer matrix expression given by Equation ( 6.3) is equal to the global transfer matrix calculated using the scattering response of the medium given by Equation (6.2). The eective parameters are derived from this equality as detailed in what follows.

We herein introduce the notation M n i,j to express the elements of the global transfer matrix M n , with 1 ≤ i, j ≤ 2. The eective wavenumber k dir is rst obtained with:

k dir = ln(Λ 1,2 ) inL u + 2πm nL u , (6.4)
where m is an integer whose value is determined in such a way that the eective wavenumber as a function of frequency is continuous, and where Λ 1 or Λ 2 are the eigenvalues of the transfer matrix M n . The material passivity condition imposes the choice of the eigenvalue so that the imaginary part of k dir is negative. The asymmetry coecient W dir and the Willis characteristic impedance Z dir are then given by:

W dir = -M n 1,1 + cos(k dir nL u ) sin(k dir nL u ) , (6.5
)

Z dir = M n 1,2 M n 2,1 (1 + W 2 dir )
. The transfer matrices of the two media are expressed and calculated using their S coecients. The global transfer matrix M n of the shorter periodic medium with n unit cells can be calculated as follows:

M n = S 11n -S 22n -S 11n S 22n +S 12n S 21n +1 2S 21n Z f S 11n +S 22n +S 11n S 22n -S 12n S 21n +1 2S 21n 1 Z f -S 11n -S 22n +S 11n S 22n -S 12n S 21n +1 2S 21n -S 11n +S 22n -S 11n S 22n +S 12n S 21n +1 2S 21n . (6.7)
The global transfer matrix M n+1 of the longer segmented medium can be calculated using this same equation but replacing n by n + 1.

As all unit cells are identical, the global transfer matrix of a segmented medium may also be written as the product of the transfer matrix of each unit cell. This leads to M n = (M u ) n for the shorter segmented medium and M n+1 = (M u ) n+1 for the longer medium, where M u is the transfer matrix of a constituent unit cell. The transfer matrix M u of a unit cell can now be deduced from the product of the global transfer matrix of the longer segmented medium with the inverse of that of the shorter medium, that is:

M n+1 (M n ) -1 = (M u ) n+1-n = M u . (6.8) Using M + u i,j
where 1 ≤ i, j ≤ 2, the elements of this transfer matrix, the eective impedance for forward propagating waves Z + diff is given by:

Z + diff = -(M + u 2,2 -M + u 1,1 ) ± (M + u 2,2 -M + u 1,1 ) 2 + 4M + u 1,2 M + u 2,1 2M + u 2,1 . (6.10)
Similarly, the eective impedance for backward propagating waves Z - diff is given by:

Z - diff = -(M u - 2,2 -M - u 1,1 ) ± (M - u 2,2 -M - u 1,1 ) 2 + 4M - u 1,2 M - u 2,1 2M - u 2,1 . (6.11)
The real part of the eective impedances must be positive to satisfy the material passivity constraint. It is now possible to calculate the Willis characteristic impedance Z diff and the asymmetry coecient W diff as follows:

Z diff = Z + diff + Z - diff 2 , (6.12) W diff = i(Z + diff -Z - diff ) -Z - diff -Z + diff . (6.13)
6.2 Case study: Hard Inclusions

Design

Both retrieval methods have been validated by considering the asymmetric multilayered design presented in Chapter 2, for which the retrieved properties are found to be identical to those obtained using the transfer matrix method for the unit cell. A locally resonant unit is now examined, formed by an o-centered cylindrical steel inclusion of 4 mm diameter in a square polyurethane matrix with a side length of 1 cm. Properties of steel and polyurethane are given in Table 5.1 and the design is shown in Figure 6.4. Figure 6.4 Representation of the asymmetric locally resonant unit cell. Dimensions are in millimetres. The dierential method is applied to the segmented media comprising of 4 and 5 unit cells for application of the dierential method. The direct inversion method is applied to the rst medium, containing 4 unit cells.

Homogenisation

The eective wavenumber for the asymmetric unit cell is shown in Figure 6.5. It is rst observed that the dierential method leads to the same wavenumber as for the symmetric unit cell considered in Chapter 5. In an innite periodic medium, the eective wavenumber is indeed independent from the choice of the unit cell, as shown in Chapter 2 with the 4 versions of a multilayered unit cell. Since the dierential method provides the wavenumber within the interior of the medium by removing the interface eects, the same wavenumber is obtained for the symmetric unit as for any non-symmetric version of this unit when this method is used. Slight dierences between the two retrieval methods are observed amongst the variations of the asymmetry coecient W , given in Figure 6.7. It is rst noted that for the massspring resonance corresponding to the rst blue dotted line, both methods give almost the same asymmetry coecient, which is very close to 0, corresponding to the value that W takes for a symmetric medium. This implies that for this resonance, the asymmetry of the unit has very little inuence and the behaviour of the asymmetric medium is similar to that of the symmetric medium. This is also conrmed by the comparison between the scattering coecients of the asymmetric medium and those of the symmetric medium, respectively shown by the red and green colored dotted lines in Figure 6.8. They are indeed dierent over the entire frequency range except just after the rst resonant frequency, where the asymmetry coecient of the asymmetric unit is almost null.

In addition, Figure 6.8 leads to the same conclusion as for the symmetric case regarding the accuracy of the predictions, which is that S 11 is better predicted by the dierential method and T = S 12 = S 21 by the direct inversion method since the medium is made of only 5 unit cells. Ultimately, the hybrid method best predicts both. method (FEM) and eective medium approach with the eective properties from the direct inversion, dierential and hybrid method.

Consequently, the extended retrieval methods both lead to the eective properties of the asymmetric unit cell. The eective properties are then used to predict the scattering coecients as per Chapter 5 and identical conclusions are drawn regarding the accuracy of these predictions using the hybrid method. In comparison with the symmetric version, the major change is related to the interface resonance. The evolution of this resonance depending on the position of the resonant element is investigated in the following subsection.

Parametric Study

A parametric study is conducted in order to assess the inuence of the asymmetry on both the acoustic performances and the eective properties. The quantity d int , introduced in It is then observed that the transmission loss from the interface resonance is the greatest for the symmetric unit than for any other conguration. For this same frequency, a local maximum occurs for the absorption coecient in Figure 6.11. The interface resonance therefore appears to be more benecial in terms of acoustic performance if the inclusion is centered within the unit. However, for the anechoism coecient, given in Figure 6.12, the symmetric conguration is not better than any other asymmetric version at the frequency of interface resonance (around 20 kHz). For other frequencies, the absorption coecient is greater for higher values of d int , especially under 15 kHz. For these frequencies, the transmission coecient is mostly unchanged for the various distances d int , meaning that this gain in performance is related to the reection coecient, as shown in Figure 6.10. Moving the hard inclusion away from the interface with water helps to reduce the reection coecient. As such, the anechoism coecient is also better for higher values of d int .

It appears to be benecial to place the inclusion as far as possible from the medium/water interface, except if the objective is to broaden the frequency range of high absorption. In that case, placing the inclusion at the center of the unit (d int = 3 mm) helps to maximise the interface resonance, which here is well located in-between the dipole resonance A and the matrix resonance C.

The inuence of d int on the eective properties, and in particular on the eective density and the eective bulk modulus, is now further investigated. These two parameters are indeed commonly studied given the fact that they may reach negative values that can be linked to resonance phenomena, as mentioned in Chapter 1. Eective density and eective bulk modulus are thus given in Figure 6.13. It should be noted that only half of the lines (units (f ) to (k)) are visible, since a unit design and its reverse counterpart have the same eective properties k, Z W and W , and as such yield identical ρ and κ. The asymmetry of the unit cell appears to have a very small impact on the eective density, for both its real and imaginary parts, especially under 15 kHz. Upon closer inspection, it can be seen that the dierences above 15 kHz are not negligible, but they seem small due to the scale. Nevertheless, there are no signicant changes in the overall variations, and the real part of the eective density becomes negative only for the mass-spring (dipole) resonance A. Similarly, the asymmetry does not signicantly change the bulk modulus under 15 kHz. However, above 15 kHz, the bulk modulus strongly varies. The values of its real part also remain positive over the entire frequency range independently of the asymmetry of the unit cell.

To summarise this case study, the position of the resonant element is found to impact the acoustic performance. It is preferable to move the hard inclusion away from the interface. It is also found that for this design, the interface resonance has a maximum impact when the inclusion is centred within the unit. It is also observed that when the asymmetry coecient W is close to 0, the asymmetric medium behaves like its symmetric counterpart. As such, for the dipole resonance around 13.5 kHz, all unit cells are found to have the same eective bulk modulus and eective density, the latter being negative which is characteristic of the dipole resonance in such media.

Case study: Voided Inclusions

In this section, an asymmetric Alberich-type medium is homogenised. As such media usually perform well as hull decoupling coatings, the acoustic performance is evaluated in terms of transmission coecient, which is a good indicator of the hull decoupling performance. In Chapter 5, it was observed that the acoustic performance shows very little variation for increasing numbers of unit cells. Alberich coatings are in practice not made of a large number of inclusions in the direction of sound propagation. Here, a single-layer voided medium and a bi-layer voided medium are considered.

Homogenisation

In this subsection, asymmetry is introduced as previously by o-centering the resonant element.

The design is presented in Figure 6.14. The host matrix is made out of polyurethane, for which the mass density is 1150 kg.m -3 . The longitudinal speed of sound is around 1500 m.s -1 , whereas the shear speed of sound strongly varies with frequency between 50 m.s -1 and 200 m.s -1 with a loss factor of roughly 30%. All three units lead to similar variation in the eective bulk modulus, which also shows strong variations around 12.5 kHz. It is also observed that the real part of the bulk modulus is negative below this frequency, which is the translation of a monopole resonance of the void. Such voided media for underwater applications are in fact well known for their ability to block acoustic waves by using monopole resonances of the voided inclusions [START_REF] Sharma | Sound transmission through a periodically voided soft elastic medium submerged in water[END_REF].

As for the case study with hard inclusions, it is observed that the Willis asymmetry introduced by moving the resonant inclusion out of the geometric center of a unit does not signicantly change the specic metamaterial feature, which here is the negative bulk modulus associated to the monopole resonance. An alternative approach to introducing Willis coupling is presented in the next subsection.

Variant Approach

Another approach to introduce Willis coupling is to create an asymmetric combination of symmetric components having dierent behaviours. With such an approach, novel properties may be obtained resulting from the interaction between each component. This may result in better broadband performance. A bi-layer unit illustrated in Figure 6.17 Acoustic performance is rst studied. The transmission coecients for the rst layer, the second layer and for the bi-layer are shown in Figure 6.18. The onset of the rst diraction orders for the periodical array, dened by f c = 1489/(2d 2 ) is highlighted by the orange dotted line. From this frequency, the acoustic response of the panel is no longer specular, so the approach used to calculate the reection and transmission coecients needs to be adapted. This aspect will be further studied in Chapter 8. It is observed that the transmission coecient of the bi-layer medium remains very low for the entire frequency range of study, attributed to the combination of the two layers, since each layer resonates at a specic frequency. The second layer creates a low transmission peak just above 5 kHz, whereas the rst layer of voids leads to a transmission dip just below 15 kHz, creating broadband low transmission.

The eective properties of the medium are now derived. The direct inversion method introduced in previously in the current chapter is applied. Since the unit is a combination of two symmetric layers, another homogenisation approach can be introduced here : each symmetric part is individually homogenised. The transfer matrix method is then used to manipulate the homogenised layers. This option is labeled by Option 2 in Figure 6.19.

The rst option is the direct homogenisation of the asymmetric medium considering Willis coupling, using the given extended retrieval methods. Both options are applied to investigate whether or not they are equivalent in terms of the nal eective medium. The direct inversion is applied to n = 1 unit cell, since for such Alberich this method was shown to yield an accurate eective medium (see Chapter 5).

For the rst option, the bi-layer unit is therefore considered and all four S-parameters are required. For the second option, each layer is modelled individually and only the coecients R and T are required. A bi-layer unit is then homogenised using the transfer matrix approach of Chapter 3.

The dispersion curves are rst obtained and are shown in Figure 6.20. It is observed that both options lead to very similar eective wavenumbers. Moreover, the imaginary parts of the wavenumber of layers 1 and 2 show the attenuation peaks associated with low transmission. There are therefore small dierences between the two homogenisation approaches presented herein. It is dicult to assess which one is more suitable or representative of the medium, since dierences are negligible here. They may however dier more for other structures where the nature of the 2 layers is dierent and for which the interface eects captured by the method are signicant. In terms of computation time, both approaches are equivalent since they both require two nite element simulations.

Eective density and bulk modulus are now derived for the bi-layer voided medium and are given in Figure 6.22. Independently, both layers lead to negative values for the real part of the eective bulk modulus when their voids create a monopole resonance. The bilayer unit also provides a bulk modulus that has negative values. The unexpected result to highlight here is the eective density of the bi-layer medium that becomes negative, whereas the eective density of both layers individually remains positive. Such negative values of the eective density may seem surprising for a voided medium, since it is known that the eective density is linked to the translational vibration of an inclusion, whereas the bulk modulus relates the compressibility (bulk oscillations) behaviour of an inclusion. Negative bulk modulus is therefore expected for voided media, but the eective negative density is not expected since there is no mass inclusion. It has been theoretically demonstrated that negative mass density can be obtained for spherical voided inclusions when the motion of the center of mass of the particle is out of phase with an incident directional pressure eld [START_REF] Li | Double-negative acoustic metamaterial[END_REF][START_REF] Lee | Origin of negative density and modulus in acoustic metamaterials[END_REF][START_REF] Fedotovskii | A porous medium as an acoustic metamaterial with negative inertial and elastic properties[END_REF]. The phase dierence between the displacement of the mass center, located at 32.83 mm from the input interface, and the incident eld is plotted in Figure 6.23. In this gure, frequencies where the eective density is negative are highlighted by the blue shaded area (Option 1) and by the orange shaded area (Option 2). It is noted that these areas exactly correspond to a phase opposition between the phase of the incident signal and the phase of the center of mass displacement. They also exactly correspond to the maximum of the asymmetry coecient W . This study case may be one of the rst evidence of a design with 2D macro-inclusions that exhibits negative density appears when the displacement of the center of mass is in phase opposition with the incident eld. To summarise this case study, Willis coupling has been introduced by combining two symmetric layers. It has been shown that this method of introducing asymmetry considerably helps to improve the broadband performance compared to o-centering the resonant element. Moreover, negative values for both the eective density and eective bulk modulus have been obtained. The origin of the negative mass density, which is uncommon for such voided media, is explained. The bi-layer Alberich design analysed in this section is selected for fabrication, and its characterisation will be presented in Chapter 8.

Summary

In this Chapter, the direct inversion method and the dierential method were extended to consider asymmetric media. These methods were then employed for two case studies, in which the introduction of Willis coupling through the asymmetry is investigated, including the impact on the acoustic performance and on the eective properties.

The rst case study is used as an example of application of the retrieval methods, in which a locally resonant medium containing an o-centred hard inclusion is homogeneised. It was found that the acoustic performance is slightly improved by moving the hard inclusion away from the input interface. However, the position of the inclusion does not particularly aect the eective parameters, including the eective density at the frequency of dipole resonance where it reaches negative values. It was found that this absence of dierence between the unit versions at this dipole frequency is also linked to low values of the asymmetry coecient W .

For the second case study, an asymmetric voided medium is considered. The asymmetry was introduced so that the acoustic performance was improved over a broadband frequency range. The asymmetry coecient for this asymmetric unit was also found to be maximised when the eective density of the unit was negative, attributed to a phase opposition between the center of mass displacement and the incident eld. Consequently, these two case studies are examples of the application of the extended retrieval methods. Further, these case studies show how the introduction of Willis coupling may aect both the acoustic performance and the eective properties.

Chapter 7

Optimisation

In this chapter, a topological-like optimisation of metamaterials with macro-inclusions is conducted. Media to be optimised are made of several units with inclusions of dierent shape and size. In order for the optimisation process to be computationally ecient, each unit is represented by its eective unit counterpart that has been obtained with the retrieval methods presented in Chapter 5. For this purpose, a database of eective properties associated with unit cell designs is created. The inuence of the inclusion shape and inclusion size on the eective properties are also investigated. The database is then used as input for an optimisation process, based on the layer optimisation approach presented in Chapter 3 for multilayered media, with the goal of minimising the anechoism coecient at low frequencies. The work in the current chapter is therefore useful to make a topological-like optimisation of acoustic coatings for underwater applications using uid properties. The optimisation approach for two dimensional designs is detailed in what follows and is illustrated by Figure 7.1. Both the direct inversion method and the dierential method are applied to dierent periodic designs in order to get the eective properties of the constituent unit cells. The direct inversion method is applied to n = 3 unit cells. Since it has been shown in Chapter 5 that the dierential method is almost independent to the parameter n, the method is applied to (n, n+1) = (2, 3) unit cells. As a result, a database is created with short computation times.

Contents

This database comprises a range of eective wavenumbers and eective impedances associated with an inclusion design. For the sake of simplicity, only symmetric unit cells are considered. This database of eective properties is used to optimise the layer arrangement of a segmented medium. With this optimisation approach, the optimisation tools for multilayered media optimisation are used for structures with two-dimensional macroinclusions through the use of eective media. That is, for a given number of unit cells (characterising the minimal layer thickness and the total medium thickness), each layer is aected with a set of eective properties (wavenumber and impedance) from the database.

As for the optimisation for multilayered media, the objective functions are then calculated and the layer arrangement is optimised until the genetic algorithm gives a set of best solutions.

Database of effective properties

Optimisation of the layer arrangement Figure 7.1 Diagram of the optimisation approach, which uses a database of eective properties derived with the retrieval methods for various inclusion shapes.

For the optimisation of the anechoism coecient, the same objective functions used in Chapter 3 are used : the rst objective function is the root mean square of the anechoism coecient, which aims at reducing its overall value, while the second objective function is the normalised standard deviation to this value of the root mean square, which aims at avoiding high peaks.

In order to calculate the anechoism coecient, the scattering coecients of the multilayered medium of eective layers are calculated. Nevertheless, the segmented media obtained are more likely to be asymmetric. It is thus crucial to calculate the anechoism coecient using all S-parameters, as per (Appendix C):

C A = S 11 - S 2 21 S 22 -1 (7.1)
7.2 Database

Presentation

The database of unit cells is presented in Figure 7. These dimensions are then given in Table 7.1 considering three dierent surface areas for each one of the inclusion shapes. These surface areas are A = 4 mm 2 , A = 16 mm 2 and A = 25 mm 2 . The inclusions are centered within a 1 cm 2 polyurethane unit, with L u = 10 mm. Using this database, the inuence of the inclusion shape for a same surface area as well as the inuence of the inclusion size for a given shape can be studied. Table 7.1 Dimensions dening the size of the inclusion depending on their surface area A.

Inuence of the Inclusion Shape

The inuence of the inclusion shape on the variations of the eective properties is investigated. For this purpose, the four inclusions of surface area A = 16 mm 2 are considered.

The eective wavenumber for each of these units is obtained using both the direct inversion method and the dierential method, respectively given in Figure 7.4 and Figure 7.5. It is observed that even if these dierent inclusions have the same surface area, their eective wavenumber diers depending on the shape, for both inversion methods. In Figure 7.5, it appears that the eective wavenumbers for the square and the circle inclusions are very similar up to 20 kHz, which includes the mass-spring resonance. The same observation is made for the eective wavenumber for the diamond and rotated rectangular inclusions. Moreover, the mass-spring resonance for these last two inclusion shapes appears at higher frequencies than for the square and circle inclusions.

The variations for the second resonance, corresponding to the matrix resonance around 25 kHz, are studied using the eective wavenumber obtained using the dierential method such that there is no confusion with the interface resonance since the dierential method erases the interface eects. It is observed that the rotated rectangular inclusions correspond to a matrix resonance that appears at lower frequencies than the other shape, but also corresponds to a low wave attenuation as shown by the small amplitude peak just under 25 kHz. The wavenumber for the diamond inclusions and the circle inclusions are similar and lead to a matrix resonance at roughly 26 kHz. The matrix resonance for the square inclusion appears at higher frequencies, almost at the end of the frequency range of study.

As such, the eective wavenumber for two inclusions may have similar variations for one type of resonance, but quite dierent variations for another type of resonance. Each inclusion shape thus leads to its own variations for the wavenumber although the surface area is the same.

The frequency of the interface resonance, around 20 kHz, also diers depending on the inclusion shape, as visible in Figure 7.4. The diamond, the circle, the rotated rectangular and the square inclusions respectively correspond to an interface resonance at the frequencies 20.2 kHz, 20.3 kHz, 21.3 kHz and 21.6 kHz.

Therefore, the change of inclusion shape does not modify the general behaviour of the medium since it is still possible to observe the mass-spring resonance, the interface resonance as well as the matrix resonance. However, these phenomena are shifted in frequency depending on the shape. Moreover, it appears that the attenuation per distance, represented by the amplitude of the imaginary part of the wavenumber, may also be slightly modied depending on the inclusion shape. The same observations as for the eective wavenumber can be made for the eective impedance obtained with the dierential method, shown in Figure 7.6.

Inuence of the Inclusion Size

The inuence of the inclusion's surface area is further investigated in this section. The diamond inclusion in Figure 7.3(c) is selected for the three surface areas of 4 mm , 16 mm 2 and 25 mm 2 . The dispersion curves are given in Figure 7.8 for the dierential method, and in Figure 7.9 for the direct inversion method. In these two gures, the dierences are signicant amongst the curves for the 3 surface areas in both amplitude and frequency of resonances. It is therefore observed that a small inclusion leads to small amplitude variations at resonance frequencies, and that these resonances occur at low frequencies.

A large inclusion leads to signicant amplitude variation of the eective wavenumber and resonance phenomena that are shifted to higher frequencies. For the eective wavenumber in Figure 7.9 obtained using the direct inversion method it is also observed that the interface resonance shifts to higher frequencies with increasing surface area.

The eective impedance is also strongly aected by the inclusion surface area, especially for such heavy inclusions made of steel, as shown in Frequency ( The aim of this rst optimisation is simply to check the validity and accuracy of the approach that uses eective properties obtained using the hybrid prediction method in such an optimisation scheme. The layer arrangement is optimised for a 10-unit medium, considering the database of Figure 7.2 without the homogeneous steel unit.

The Pareto front is plotted for all generations in Figure 7.12. For the last generation, the 10 leftmost arrangements, having the lowest |C A |, as well as the 10 rightmost arrangements, with the minimum σ C A , are also represented. The rst arrangement of the Pareto front for which |C A | is minimum, as well as the last arrangement with the minimum σ C A , are selected to assess the validity of the optimisation approach. Their scattering coecients obtained using the eective medium approach (full lines) are compared to those obtained using a nite element model (dashed lines) in Figures The optimisation approach is therefore judged satisfactory and suitable to predict the anechoism coecient of such complex media with macro-inclusions.

Low Frequencies

The optimisation process is now conducted aiming to lower the anechoism coecient for the subwavelength domain, for a frequency range up to 10 kHz. Optimisation parameters are given in Table 7. For these four selections, it is observed that the anechoism coecient remains low only in the frequency range of optimisation, and then increases above 10 kHz, especially for Selections 1 and 2.

The rst arrangement is further studied for a frequency range up to 10 kHz. The scattering coecients obtained using the eective medium approach are given in Figure 7.17 and compared to those obtained using FEM. Predictions and FEM results are very close to each other. It is also observed that the optimised arrangement improves the anechoic performance from 2 kHz compared to the matrix alone, as shown by the dierence between the blue and orange lines in Figure 7.17 (right). Nevertheless, since the performance is optimised below 10 kHz, the origin for this improved performance is not a resonance eect.

According to the results in sections 7.2.2 and 7.2.3, the only main dierence between the units of the database is the eective impedance within this frequency range. coecient C A for the rst arrangement of the Pareto front, obtained using the eective medium approach (full lines) and nite element simulation (dashed lines). (Left) Anechoism coecient, in decibels, of the optimised medium and of the matrix free of inclusions.

The eective impedance along the medium is thus calculated and is given in This result therefore reinforces that such an impedance pattern leads to low anechoism, as it has been concluded for multilayered media, even though the gradual feature is less visible here. Resonances phenomena associated with the presence of inclusions are then not necessary to improve anechoic performance on the subwavelength domain. As such, it is interesting to verify if better performance could be obtained without inclusions, considering a multilayered medium only. The same optimisation algorithm has been run

for multilayered media, considering the same material properties and total length as for the media-counterparts with macro-inclusions (which are dierent than those used in Chapter 3). The Pareto front is given in Figure 7.19. It can be seen that the dierence between the Pareto front for multilayered media (grey squares) and the Pareto front for media with macro-inclusions (white dots) is not signicant. The former provides slightly better solutions for |C A |, whereas the latter gives solutions of lower σ C A . It is also observed that the arrangements are similar : thick layers of steel are found where homogeneous steel units are located in the arrangements with macro-inclusions, and thinner layers of steel are found at similar locations as small inclusions. One of the multilayered media has a score very close to that of the selected medium with macro-inclusions. They are both compared in Figure 7.20. It is observed that the multilayered solution leads to the same -even slightly betteranechoism coecient as the solution with macro-inclusions under 10 kHz. However, the dierence is signicant at higher frequencies, where the solution with 2D-macro-inclusions provides a much lower anechoism coecient, even though this solution was the one with the worse performance at higher frequencies according to Figure 7.16.

Therefore, inclusions and their resonances are not essential to optimise the anechoism coecient at low frequencies, or within a relatively narrow frequency range. However, they may help generate strong and sudden variations of the eective impedance at the resonance frequencies, as shown in Figure 7.10, which may help lower the anechoism coecient for these resonance frequencies. They are also crucial to broaden the frequency range of performance using their various resonances that can be tuned to be nicely spread in the frequency range of study. Consequently, an ecient approach for an anechoic coating of broadband performance may be to combine uniform layers and macro-inclusions, whereas steel layers lower the anechoism coecient at low frequencies and inclusions help to dissipate the energy at higher frequencies through their resonances. This optimisation approach could also be improved by adding multilayered unit cells to the database, with various steel thicknesses.

Performance Levels

The layer arrangement is now optimised for both the anechoism and the hull decoupling coecient, using the objective functions dened in Chapter 3, Section 3.4.

The Pareto front along generations is given in

Summary

In this section, a new approach for optimising structures with macro-inclusions has been introduced. It uses the eective properties of dierent polyurethane units comprising inclusions of dierent sizes and shapes. The inuence of the shape and size of the inclusion on the variations of the eective properties was investigated. The database was then used to optimise a 10 cm-thick media for subwavelength anechoic performance. Similar to what was found for multilayered media, an impedance pattern leading to a low coecient of anechoism arose. This pattern transcribes the fact that, in the optimised medium, acoustic waves propagate in the medium and are trapped in the damped layer at the end of the medium, resulting in a strong dissipation of energy. This study has also shown that the design of an anechoic coating need not be limited to multilayered media or media with macro-inclusions, as the combination of the two (ie. media with macro-inclusions and uniform layers) can greatly help improve the broadband acoustic performance.

Such an optimisation approach has also proved to be very satisfactory in terms of prediction accuracy and calculation time. In a short time, a complex optimisation problem of 1.3786 × 10 11 potential solutions was treated analytically. In addition, this new optimisation concept can be used, adapted and applied to thousands of other problems, for example by considering other criteria of acoustic performance (reection, transmission, absorption, eective properties, etc.), other materials, other shapes, other total unit lengths and thicknesses, other frequency ranges, and so on. As such, this optimisation tool is easily adaptable.

Chapter 8 Experiments

In this Chapter, the bi-layer Alberich panel designed in Chapter 6 is measured in the acoustic water tank presented in Chapter 4. Since the panel contains periodically embedded void inclusions, the experimental characterisation becomes more complex. The grating diraction phenomenon is rst presented to highlight these diculties. The 3point method, presented in Chapter 4, as well as an extension of the method are then applied to measure the acoustic response of the panel. The scattering coecients of the voided medium are also compared to those obtained using a FE model. For the homogenisation and optimisation of periodic metamaterials with macro-inclusions, the diraction grating phenomenon was ignored since its inuence can easily be neglected for the frequency range of study and the numerical method used. However, the measurement of the Alberich panel poses new challenges due to its periodic arrangement of macroscopic inclusions. In order to apply the 3-point method under valid conditions, the phenomenon of diraction induced by gratings, which are periodically structured media, is rst described. The implications, in terms of the additional challenges for the experimental characterisation, are then discussed.

Grating Diraction Phenomenon

For an acoustic panel containing a diraction grating, the scattered eld consists of a superposition of an innite number of diracted waves. This phenomenon is also well known in optics, using optical diraction gratings. Each diracted wave is associated with a diracted mode number n, where n is an integer. In the case of plane wave excitation of a panel at normal incidence, the 0 th -order corresponds to the specular reection (transmission). As such, it has the same propagation direction as the incident plane wave while the other modes |n| > 0 propagate in other directions, as illustrated in In what follows, the reected and transmitted pressures of a panel containing a periodic array are expressed based on [1,116] but using the coordinate system presented in Chapter 4 for the water tank. The incident plane wave is therefore travelling in the direction x, while the panel is in an (y, z)-plane. For the theoretical approach, the inclusions are considered to be innite along the z-axis and the grating is periodic in the y-direction from -∞ to +∞.

p t (x, y) = +∞ n=-∞ c t,n e -iqny e -ikx,nx (8.2)
where q n and k x,n are given as follows:

q n = nπ d , (8.3) k 2 x,n = k 2 -q 2 n , (8.4)
where k is the wavenumber in water and n is the mode number (or mode order) of the diracted wave whose contribution coecient is c r,n or c t,n . When k 2 x,n < 0, the corresponding diracted wave propagates parallel to the y-axis and decays exponentially along the x-axis. In this case, it only contributes to the near-eld and it is referred to as an evanescent wave.

On the contrary, if k 2

x,n > 0, the diracted wave propagates away from the panel in the x-direction. The cut-o frequency f c for diracted wave n is therefore given by:

f c (n) = c w n 2d , (8.5)
where c w is the speed of sound in water.

Consequently, there is a cut-o frequency for each diracted mode n for which a diracted wave converts from a evanescent wave to a propagating wave. The diracted waves that do not propagate along the x-axis decay exponentially with distance to the plane which contains the centers of the inclusions. For example, for a frequency f such that f < c w /(2d), only the 0 th mode exists in the far-eld since all the other diracted waves are evanescent. As the frequency increases, a greater number of diracted waves convert to propagating waves.

Challenges

In the past, the grating diraction phenomenon appearing in Alberich-type panels has been studied analytically [1,[START_REF] Hennion | Analysis of the scattering of a plane acoustic wave by a periodic elastic structure using the nite element method: Application to compliant tube gratings[END_REF][START_REF] Achenbach | Reection and transmission of an obliquely incident wave by an array of spherical cavities[END_REF][START_REF] Achenbach | 3-d reection and transmission of sound by an array of rods[END_REF]. Additionally, Ivansson [START_REF] Ivansson | Reection from steel plates with doubly periodic anechoic coatings[END_REF] highlighted some of the array diraction features through experiments.

For an Alberich-type coating comprising spherical cavities of 2 mm diameter periodically spaced every 10 mm, ultrasonic pulse measurements showed that above the rst cut-o frequency (of 148kHz), the specular reected wave is closely followed by diracted waves of higher modes. The author also presented a pulse computation for an Alberich coating below the rst cuto frequency, showing that evanescent waves that exponentially decay in the normal direction are no longer discernible 1 meter away from the panel but are non negligible at the interface [START_REF] Ivansson | Sound absorption by viscoelastic coatings with periodically distributed cavities[END_REF]. This phenomenon has also been mentioned in the study of materials other than Alberich-type designs, but results above the rst cut-o frequency are usually not investigated [START_REF] Robert | Propagation of elastic waves through twodimensional lattices of cylindrical empty or water-lled inclusions in an aluminum matrix[END_REF][START_REF] Zong | Eects of periodicity perturbations on the transmission by underwater phononic crystals[END_REF].

Therefore, the experimental characterisation of a metamaterial with a periodic arrangement of macroscopic inclusions such as the Alberich coating is not common in the literature since lattice-diracted non-specular waves make the experimental study of such media even more complex than for a homogeneous panel. Diculties in panel measurements

Below the Cut-o Frequency Applicability

The panel design presented in Figure 8.4 is used. The 3-point method set-up is also shown. For this panel, the rst cut-o frequency for the grating is f c = cw 2d = 18.75 kHz. Below the rst cut-o frequency, only the 0 th -order mode propagates. The main challenge here is to ensure that the evanescent eld does not pollute measurements near the panel, where the hydrophones are placed. For the 3-point method, the hydrophones must be placed suciently close to the panel so that the edge-diracted pressure can be assumed to be the same at the 3 positions. In order to verify if the experimental method can be correctly applied to measure the Alberich grating, the near-eld evanescence is analysed. Using Equation (8.4), the term e ikx,nx is evaluated, since it weights the contribution coecients c n of the diracted mode n. For the panel studied here, only the n = ±1 modes are considered since there are no other cut-o frequencies included in the measurement frequency range up to 25 kHz for the water tank used. In Figure 8.5, the position x at which e ik x,±1 x becomes equal to 0.01, 0.05 and 0.1 are plotted for each frequency. Theses curves show that at the corresponding position x, the amplitude of the evanescent diracted wave ±1 has decreased by 99%, 95% or 90%. In addition, the 3 positions of the hydrophone on the reection side are plotted for each frequency, considering a central position X 2 at 11cm from the interface and a spacing of λ/10 between the 3 positions. from the large inclusion array is less than 1%, 5% and 10% of its initial amplitude. The positions of the hydrophones for the 3-point method set-up are also shown.

Below 15 kHz, the diracted modes ±1 should not disturb the 3-point method measurements since their amplitude should be reduced by at least 99% of their initial amplitude. It can nevertheless be observed that the evanescent feature of the diracted waves decreases as the cut-o frequency approaches. Between 15 kHz and the rst cut-o frequency, some discrepancies may consequently appear due to a potential contribution of the diracted modes ±1.

Results

Hydrophone positions along the x-axis dened previously (X 2 = 11cm, X 1 = X 2 -λ/10 and X 3 = X 2 + λ/10) are considered hereafter. Along the y-axis, In both gures, it can be observed that the measured pressures are slightly dierent depending on the position along the y-axis, due to the contribution of the edge-diracted pressure. For higher frequencies, the evanescent eld may also contribute slightly to the measured pressures as shown in Figure 8.5.

It is crucial here to draw attention to the fact that the acoustic source resonates around 16 kHz which sometimes leads to an overload of the received signal, thus invalidating the measurement. Usually, the power of the source is reduced to avoid such an overload.

However, a strong acoustic emission was required here in order to be able to detect a signal on the transmission side that stood out from the noise, since the transmission through the panel is extremely low (as expected for such panels). Therefore, results from 15 kHz should be taken with caution.

On the reection side, the total pressures measured are used to derive the incident pressure p i , the reected pressure p r and the edge-diracted pressure on the reection side p diff R .

These calculated pressures are shown in Figure 8.8. The incident pressure is compared to that obtained by measurements without the panel, referred to as p WP i . This incident pressure p WP i is maximum at the resonance frequency of the acoustic source. It can be seen that the calculated incident pressure p i is consistent with p WP i up to 15 kHz, thus validating the estimation of the incident pressure using the 3-point method. However, there is a signicant dierence around 16 kHz, which invalidates the results from 15 kHz for the reection side. It is nonetheless dicult to identify whether this divergence originates from the overload of the received signal due to the source resonance or from the higher contribution of the evanescent diracted waves close to the cut-o frequency. ) is also plotted. The graduation of grey represents the y-positions along the unit cell.

On the transmission side, the 3-point method leads to the transmitted pressure p t and the edge-diracted pressure p diff T . Pressure p ∞ coming from innity and travelling towards the negative x is also obtained for the transmission side. These pressures are plotted in Figure 8.9, where it can be seen that the innite pressure is almost null, thus validating that the 3-point method is correctly applied. The transmitted pressure p t is also signicantly low. It can nally be noticed that the edge-diracted pressure varies signicantly as a function of the y-position and most often appears as the main component of the pressure eld behind the panel. Finally, since the 3-point method eliminates edge-diraction contributions, the experimental coecients could, in practice, be directly compared to those of an innite panel.

Reection and transmission coecients are thus calculated using the pressures from the 3-point method given in Figures 8.8 It is rst noted that the coecients of the calculation without the 3-point method are signicantly dierent from the numerical results. Even by averaging these coecients over the unit cell, the acoustic response obtained would be very dierent from the numerical prediction. On the contrary, the coecients derived from the 3-point method are consistent with the predictions. There are some slight discrepancies which may be attributed to the fact that the panel is not innite. Consequently, the 3-point method leads to reection and transmission coecients which correspond well to those calculated numerically for an innite panel up to 15 kHz. This again underlines the importance of removing the contribution of the edge-diracted pressure.

To summarise, the 3-point method provides accurate and robust measurements of the Alberich medium without being inuenced by the evanescent eld created by the grating.

Nevertheless, it must be ensured that the distance between the grating and the hydrophone is large enough but also not too large so that the assumption on the edge-diracted pressure can be applied. This can be achieved by using criteria established in Chapter 4

for homogeneous panels. However, between 15 kHz and the rst cut-o frequency, there are signicant discrepancies, which may be due to the resonance of the acoustic source or to the evanescent eld of higher amplitude close to the cut-o frequency.

where c m n is the elementary contribution obtained using the pressures p m and p m+1 measured/calculated at the points y m and y m+1 as per:

c n,m = α (n,m) p m + α (n,m+1) p m+1 . (8.11) The expression for the coecients α (n,m) and α (n,m+1) are:

α (n,m) = e -iqnym ( 1 iqn -1 (iqn) 2 e -iqn(y m+1 -ym) -1 -y m+1 +ym ) α (n,m+1) = e -iqny m+1 ( 1 -iqn -1 (iqn) 2
1-e iqn(y m+1 -ym) y m+1 -ym

) (8.12)
Thus, the use of measurements/calculations on a meshed line covering a complete unit cell should, in theory, lead to the reected and transmitted pressures of the grating diracted modes 0, +1 and -1 for the Alberich panel. This approach is used in a FEM calculation to obtain the reection and transmission coecients for orders 0 and ±1. It is also used to perform a raw calculation of the coecients using the pressures measured along an elementary cell. Such a calculation therefore does not take into account an eventual contribution of the panel's edge-diraction and it can thus be seen as an equivalent to the results without the 3-point method presented in the preceding section below the cut-o frequency. These coecients will be compared later in this Chapter. An extension of the 3-point method is rst proposed to calculate both the grating diracted modes and the edge-diracted pressure.

The 5-point method

Above the rst cut-o frequency, diracted modes ±1 are no longer evanescent and begin to propagate. Diracted modes ±2 are still present but remain evanescent up to 37.5 kHz. For measurements, this means that the associated pressures, p -1 and p +1 , must be calculated in addition to the other unknowns of the 3-point method. It is therefore necessary to add two unknowns, resulting in two additional inputs which are measurement points. Since the 3-point method is applied in several positions y m along the y-axis to cover a complete unit cell, a fourth and a fth point, X 4 and X 5 , are chosen and correspond to the central point at the previous and next y positions where the 3-point method is applied, so X 4 = X 2 (y m-1 ) and X 5 = X 2 (y m+1 ). This adaptation is illustrated in Figure 8.12. It should be noted that the elementary cell is not represented as centered since, in practice, it may not be possible to know where the hydrophones are located in relation to the position of inclusion. However, if the inclusions were centered with respect to the measurement line, the 5-point method could be reduced to the 4-point method since p -1 = p +1 . The system of equations to implement a ve-point method can now be written by adding two unknowns to the 3-point method system as follows:

               p(A 1 ) = p i (A 2 )e -ikd A12 + p 0 r (A 2 )e ikd A12 + p diff R (A 1 ) + p +1 r (A 2 )e ikd A12 cos(θ) + p -1 r (A 2 )e ikd A12 cos(θ) p(A 2 ) = p i (A 2 ) + p 0 r (A 2 ) + p diff R (A 2 ) + p +1 r (A 2 ) + p -1 r (A 2 ) p(A 3 ) = p i (A 2 )e ikd A23 + p 0 r (A 2 )e -ikd A23 + p diff R (A 3 ) + p +1 r (A 2 )e -ikd A23 cos(θ) + p -1 r (A 2 )e -ikd A23 cos(θ) p(A 4 ) = p i (A 4 ) + p 0 r (A 4 ) + p diff R (A 4 ) + p +1 r (A 2 )e -ikd A24 sin(θ) + p -1 r (A 2 )e ikd A24 sin(θ) p(A 5 ) = p i (A 5 ) + p 0 r (A 5 ) + p diff R (A 5 ) + p +1 r (A 2 )e ikd A25 sin(θ) + p -1 r (A 2 )e -ikd A25 sin(θ)
. (8.13) Since the incident wave travels along the x-axis and it is a plane wave, p i (A 2 ) = p i (A 4 ) = p i (A 5 ). The same observation applies for the specular reection p 0 r . It is also assumed that the 5 positions are close enough so that

p diff R (A 1 ) = p diff R (A 2 ) = p diff R (A 3 ) = p diff R (A 4 ) = p diff R (A 5
). Equation (8.13) can thus be simplied and written in a matrix form as follows:

      p(A 1 ) p(A 2 ) p(A 3 ) p(A 4 ) p(A 5 )       =      
e -ikd A12 e ikd A12 1 e ikd A12 cos(θ) e ikd A12 cos(θ) 1 1 1 1 1 e ikd A23 e -ikd A23 1 e ikd A23 cos(θ) e ikd A23 cos(θ) 1 1 1 e -ikd A24 sin(θ) e ikd A24 sin(θ) 1 1 1 e ikd A25 sin(θ) e -ikd A25 sin(θ)

            p i (A 2 ) p 0 r (A 2 ) p diff R (A 2 ) p +1 r (A 2 ) p -1 r (A 2 )      
. (8.14) By inverting Equation (8.14), the pressures p i (A 2 ), p 0 r (A 2 ), p +1 r (A 2 ), p -1 r (A 2 ) and p diff R (A 2 )

are obtained.

On the transmission side, the same approach is adopted, considering the ve points B 1 , B 2 , B 3 , B 4 and B 5 and the distances

d B12 = B 1 B 2 , d B23 = B 2 B 3 , d B24 = B 2 B 4 and d B25 = B 2 B 5
, where position B 2 is taken as the reference. Assuming that the edgediracted pressure is the same at the ve positions, the total pressures at these points are expressed by:

               p(B 1 ) = p 0 t (B 2 )e ikd B12 + p ∞ (B 2 )e -ikd B12 + p diff T (B 2 ) + p +1 t (B 2 )e ikd B12 cos(θ) + p -1 t (B 2 )e ikd B12 cos(θ) p(B 2 ) = p 0 t (B 2 ) + p ∞ (B 2 ) + p diff T (B 2 ) + p +1 t (B 2 ) + p -1 t (B 2 ) p(B 3 ) = p 0 t (B 2 )e -ikd B23 + p ∞ (B 2 )e ikd B23 + p diff T (B 2 ) + p +1 t (B 2 )e -ikd B23 cos(θ) + p -1 t (B 2 )e -ikd B23 cos(θ) p(B 4 ) = p 0 t (B 4 ) + p ∞ (B 4 ) + p diff T (B 2 ) + p +1 t (B 2 )e -ikd B24 sin(θ) + p -1 t (B 2 )e -ikd B24 sin(θ) p(B 5 ) = p 0 t (B 5 ) + p ∞ (B 5 ) + p diff T (B 2 ) + p -1 t (B 2 )e -ikd B25 sin(θ) + p -1 t (B 2 )e ikd B25 sin(θ) . (8.15)
Since the innite pressure is theoretically a non-existing wave travelling along the x-axis and is a plane wave, p ∞ (B 2 ) = p ∞ (B 4 ) = p ∞ (B 5 ). The same observation applies for the specular transmission p 0 t . The contributions of the transmitted, innite and edge- diracted pressures are then be calculated by solving the following system of equations:

      p(B 1 ) p(B 2 ) p(B 3 ) p(B 4 ) p(B 5 )       =       e ikd B12 e -ikd B12 1 e ikd B12 cos(θ) e ikd B12 cos(θ) 1 1 1 1 1 e -ikd B23 e ikd B23 1 e -ikd B23 cos(θ) e -ikd B23 cos(θ) 1 1 1 e -ikd B24 sin(θ) e ikd B24 sin(θ) 1 1 1 e ikd B25 sin(θ) e -ikd B25 sin(θ)             p 0 t (B 2 ) p ∞ (B 2 ) p diff T (B 2 ) p +1 t (B 2 ) p -1 t (B 2 )       . (8.16)

Numerical Results

In order to verify the equations implemented in the 5-point method, a nite element model is carried out in which the pressures are recorded at the same positions as for the measurements, as illustrated in Figure 8.13. The FEM model includes only one unit cell of the bi-layer Alberich. Periodic Bloch-Floquet conditions are applied on the lateral sides. In the nite element model, the total pressures are calculated at 16 lateral positions covering one spatial period of the medium, 11 cm away from its interface with water. For such a numerical model with theoretically innite lateral dimensions, there is no edgediracted pressure. To verify the amplitudes of the reected and transmitted pressures given by the 5-point method, a direct calculation using Equations (8.8) to (8.12) is also applied to the nite element model. From the total pressure obtained from the FEM calculation, Equations (8.8) to (8.12) are used to calculate the contribution of the 0 th mode and the ±1 st modes.

By also calculating the incident pressure in the numerical model, it is possible to derive the reection and transmission coecients from the 0 th mode using p 0 /p i and from the ±1 st modes with p ±1 /p i . Reection and transmission coecients for the 0 th and ±1 st modes are given in Figure 8. [START_REF] Liu | Locally resonant sonic materials[END_REF]. This shows that the direct calculation and the 5-point method lead to the same scattering coecients, thus validating the approach implemented in the 5-point method. ±1 st order obtained using a direct calculation as per section 8.2.3 and the 5-point method as per section 8.2.3, both using the total pressures calculated with a nite element model of a the constituent unit cell.

Experimental Results

The 5-point method is now applied on measured pressures. It is rst observed that the incident pressure obtained with the 5-point method is consistent with the one measured without a panel. It is also seen that the 0 th mode is predominant on the reection side over the ±1 st modes. Moreover, discrepancies are visible along the y-axis for all these calculated pressures where theoretically only the edge-diracted pressure should vary from one y-position to another. It can also be noted that at such "high frequencies" (in the context of coatings for underwater acoustics), the results may be more sensitive to the material properties and geometry irregularities. These discrepancies may therefore be representative of the panel and its local irregularities. On the transmission side, the innite pressure is almost null for the whole frequency range, as it is expected when the simplifying assumptions of the method are acceptable. Therefore, the results seems overall consistent and might, as for the reection side, be inuenced by local irregularities having more impact since the wavelength is getting smaller. The reection and transmission coecients are now derived from these calculated pressures and compared to numerical results presented in Figure 8.15. In addition to the coecients obtained by the 5-point method and those calculated with the FEM model, a direct calculation from measurements using Equations (8.8) to (8.12) is made. The latter therefore ignores the existence of the edge-diracted pressures and only requires, as for the FE model, the total pressure over a linear mesh covering the unit cell and the incident pressure obtained separately. Using the measurements made, this direct calculation can be applied 3 times : on the lines of A 1 , A 2 and A 3 for the reection side and on the lines of B 1 , B 1 and B 3 for the transmission side. All results are compared in Figure 8.20. Coecients R +1 and R -1 are averaged under the label R ±1 in the gure (the same is done for the transmission coecients). For each side, three direct calculations are also performed as per Equations (8.8) to (8.12) using the total pressures measured on each y-line, thus not taking into account the edge-diracted pressure.

For the reection coecient, it is rst observed that the result of the 5-point method for the 0 th mode is lower to the one calculated numerically, whereas they are identical for the ±1 modes. The level dierence for R 0 is non negligible and is attributed to the fact that the panel is not innitely periodic. Further, the panel material properties and geometries may slightly dier from those in the FE model. The same observations apply for the transmission coecients.

The coecients R 0 from the direct calculations over the 3 y-lines are signicantly dierent from both other results. This can be explained by the fact that the edge-diracted pressure is not taken into account in these calculation and, even if its amplitude is quite low (Figure 8.18), it contributes to the pressure eld. The direct calculation is based on the exact knowledge of the reected and transmitted elds. It is therefore not applicable in the presence of non-modeled acoustic sources, such as the edge-diraction phenomenon. This shows that the 5-point method is more suitable for measurements of an Alberich-type coating since it takes into account the main contributions, which are edge-diraction and the rst modes diracted by the diraction grating . This thesis studied the concept of metamaterials for underwater applications, from theoretical analysis to experimental characterisation. For this purpose, methods were developed to homogenise a range of metamaterial designs. Further, this thesis presented ways to optimise complex structures with low computation cost. This thesis also studied experimental methods to obtain the acoustic performance of metamaterials.

In the rst chapter of the manuscript, the concept of hull coatings used for the discretion or stealth of submerged vehicles was introduced. Metamaterials are seen as good candidates for subwavelength coatings with good acoustic performances at low frequencies.

A literature review has been conducted to present the concept of metamaterials and some methods to study them. A non exhaustive state of the art of metamaterial designs optimised for underwater applications was presented, followed by a review of measurement techniques.

In Part I of this thesis, tools for the study and design of multilayered media were introduced. Chapter 2 describes a homogenisation method based on transfer matrices.

Either a uid homogenisation model or a Willis uid homogenisation model was adopted depending on the symmetry of the layer arrangement. The eective medium was found to be an exact representation of a multilayered medium studied under normal incidence.

The homogenisation theory was then used to optimise multilayered media in Chapter 3. The optimisation approach is computationally ecient and it was validated on a case study. Objective functions aiming to reduce the anechoism coecient for a wide frequency range have then been dened. It was found that the best performing multilayered media show an impedance adaptation pattern. For the hull decoupling coecient, it was found that simple bi-layer media give the best results. The optimisation approach was then applied for performance levels of both the anechoism and hull decoupling coecients, which conrmed the observations made for the separate optimisations of each coecient. Following the optimisation studies, a multilayered arrangement was selected for fabrication.

In Chapter 4, a measurement method in an open water tank, namely the 3-point method, has been introduced. The main advantage of this method is that it determines the contribution of the edge-diracted waves. The method has been validated on an aluminium panel and the validity limits of the method have been studied. The method was then applied to a polyurethane panel containing micro-balloons in order to retrieve the eective speed of sound in the medium. Two approaches were used, for which results are comparable. Finally, the multilayered medium presented in Chapter 3 has been measured and it was found that moduli of the reection and transmission coecients are consistent with analytical predictions. Signicant dierences were however noted on the anechoism coecient, which proves to be extremely sensitive to the distance between the panel and the hydrophones. By applying phase shifts, it was found that the anechoism coecient converged to the predicted value.

In Part II, metamaterials with macro-inclusions were homogenised, optimised and measured. The homogenisation of such designs was achieved using two retrieval methods, the direct inversion method and the dierential method. These two retrieval methods were presented in Chapter 5. Both retrieval methods lead to an eective wavenumber and an eective impedance of a homogeneous uid medium. The eective wavenumber from the dierential method was able to describe the interior of the medium whereas the direct inversion method captured the eects related to the medium's nite size and interfaces with the surrounding uid media on the incidence and transmission sides. Three case studies were examined corresponding to a 1D multilayered medium and 2D locally resonant designs with hard or voided cylindrical inclusions. Through these case studies, applicability and limitations of the methods have been discussed. It was found that the proposed hybrid medium provides accurate predictions of the transmission coecient as it physically restricts the eective properties of the direct inversion method to the interface and uses the dierential method to describe the core of the medium.

Chapter 6 presented extensions of both retrieval methods in order to obtain the eective properties of asymmetric unit cells with inclusions. These methods were then employed for two case studies, in which the introduction of Willis coupling through asymmetry was investigated. The impact of asymmetry on the acoustic performance and on the eective properties was examined. In the rst case study, o-centred hard inclusions were considered and the inuence of the inclusion position was investigated. For the second case study, an asymmetric voided medium was analysed. The asymmetry coecient was found to be at its maximum when the eective density of the unit cell was negative. This physical mechanism was attributed to an anti-phase dierence between the center of mass displacement and the incident eld.

In Chapter 7, the retrieval methods for symmetric media were applied to several metamaterial designs in order to create a database of eective properties associated with various inclusion shapes, sizes and materials. Eective properties were then used in an optimisation process in order to obtain structures with a low anechoism coecient over a wide frequency range. This work is therefore useful to make a topological-like optimisation of acoustic coatings for underwater applications using uid properties. For the example illustrating this new approach for optimising structures with macro-inclusions, the same impedance pattern leading to a low coecient of anechoism as in Chapter 3 was found.

This study also showed that it is worth not limiting the design of an anechoic coating to multilayered solutions or designs with inclusions, as the combination of the two, by alternating homogeneous layers with layers comprising macro-inclusions, can greatly help improve the broadband acoustic performance.

In Chapter 8, the bi-layer Alberich panel designed in Chapter 6 was measured in the acoustic water tank presented in Chapter 4. Since the panel contains periodically embedded void inclusions, the experimental characterisation becomes more complex because of the grating diraction phenomenon. The experimental characterisation of the panel was divided into two dierent approaches: below the rst cut-o frequency and above the rst cut-o frequency. Below the rst cut-o frequency, the 3-point method was found to be robust up to 15 kHz, after which the source resonance and the ±1 modes invalidated the results. Above the cut-o frequency, the ±1 modes were no longer evanescent and needed to be calculated to fully characterise the panel. For this purpose, an extension of the 3point method was implemented and results were consistent with numerical predictions. In particular, it was highlighted that it is crucial to take into account edge diraction, which makes the 5-point method suitable for measurements of such Alberich coating designs.

Therefore, the 3-point method and its extension to the 5-point method were shown to be robust and unique approaches to characterise panels embedded with periodic inclusions.

Perspectives for Future Work

As the experiments of this thesis have shown, it is sometimes dicult to accurately know the dynamic properties of polyurethane. Moreover, the manufacturing processes can introduce uncertainties to both the material and geometric properties of acoustic coatings. Studying the inuence of these uncertainties on the acoustic performance could help explain some experimental results.

It would also be interesting to look at dierent solutions to accurately measure the distance between the hydrophone and the panel in the water tank, since it was shown that the anechoism coecient is very sensitive to this distance. A laser solution could be implemented. Ultrasound patches may be placed on the panel to determine the panel/hydrophone distance.

The optimisation approach for metamaterials with resonant inclusions could be used with a larger database, in which eective media could be obtained quickly with analytical models for simple geometries (using models by Sharma et al. for instance [START_REF] Sharma | Sound transmission through a periodically voided soft elastic medium submerged in water[END_REF], [START_REF] Sharma | Acoustic performance of gratings of cylindrical voids in a soft elastic medium with a steel backing[END_REF]) or with the retrieval methods for complex geometries. The retrieval methods implemented in this thesis as well as the optimisation approach can also be applied to coatings containing three-dimensional inclusions of dierent shapes, such as spheres, elliptical and conical shapes. Further, the analytical and semi-analytical models presented in this thesis could also be extended to consider oblique incidence.

The homogenisation methods for metamaterials with inclusions could be modied to predict the vibro-acoustic behavior of a submerged hull in order to assess the noise radiated by a cylindrical hull coated with a metamaterial. Indeed, since the homogenisation models provide eective properties of a uid, the coating could be integrated using a uid-loaded shell.

Usually the radiated noise of a submerged vehicle is assessed using large-scale models of uncoated hulls. The dierence between the desired performances and these predictions then leads to the denition of anechoic or hull decoupling performances that a coating must satisfy. Assuming that coatings can be modeled easily by uid eective media in large scale models, a future study could be to derive a metamaterial design from a given set of eective uid properties. For this purpose, it is interesting to investigate the link between eective uid properties and acoustic performance. This last point was the subject of a preliminary study which includes performance mapping in the eective property space, the rst elements of which are presented in Appendix E.

notions can also be suitable for acoustic analyses using an analogy between the physical quantities involved. The scattering parameters (or S-parameters) of a two-port network are herein introduced and linked to the elements of the transfer matrix (also called the The transfer matrix elements A, B, C and D are dened as [START_REF] Frickey | and T parameters which are valid for complex source and load impedances[END_REF]:

p 1 u 1 = A B C D p 2 u 2 . (A.2)
The relationship between the reected, incident power waves and the S-parameter matrix is given by: b

1 b 2 = S 11 S 12 S 21 S 22 a 1 b 2 , (A.3)
where S 11 is the reection coecient on the left-hand side and S 21 the transmission coecient and where S 12 is the transmission coecient for an excitation on the righthand side for which the reection coecient is S 22 , as per the expressions: The transfer matrix and S-parameters are linked together by:

S 11 = b 1 a 1 a 2 =0 , (A.4) S 12 = b 1 a 2 a 1 =0 , (A.5) S 21 = b 2 a 1 a 2 =0 , (A.6) S 22 = b 2 a 2 a 1 =0 . (A.
1 + S 11 1-S 11 Z f = A B C D S 21
S 21 Z f , (A.8) S 12 -S 12 Z f = A B C D 1 + S 22 -1+S 22 Z f . (A.9)
By combining the two previous equations, elements of the ABCD-matrix are expressed with the S-parameters as given by:

A = -R 2 + T 2 + 1 2T , (A.14)
e -ikLu = e -i(k +ik )Lu = e k Lu e -ik Lu .

(B.5)

The modication of the wave amplitude, under the eect of losses, is represented in the term e k Lu . This term is required to be inferior to unity if the material is passive. Thus, the imaginary part of wavenumber has to be negative.

Other Expression

Eigenvalues of the transfer matrix M u are given by:

λ ± = 1 2 (Tr(M u ) ± Tr(M u ) 2 -4Det(M u )). (B.6)
Thus, λ + + λ -= Tr(M u ). However, λ + + λ -can also be expressed with the wavenumber k using Bloch-Floquet conditions, yielding:

λ + + λ -= e ikLu + e -ikLu = 2 cos kL u . (B.7)
Consequently, cos kL u = Tr(M u )/2.

B.2 Representation

At normal incidence, the wavenumber k derived for a unit cell of thickness h can be represented as a function of frequency f , which is commonly referred to as the dispersion To express the anechoism coecient C A , the following boundary conditions are thus introduced :

     p left = C A + 1 v left = 1-C A Z f v right = 0 . (C.5)
Considering these boundary conditions, the transfer matrix denition is given by:

1 + C A 1-C A Z f = A B C D p right 0 , (C.6)
where A, B, C and D are the elements of the transfer matrix. The anechoism coecient C A can then be obtained by expressing the ABCD-elements in terms of S-parameters as per Appendix A, which yields:

C A = S 11 - S 21 S 12 S 22 -1 . (C.7)
If the medium is symmetric, then S 11 = S 22 = R and S 12 = S 21 = T , which simplies the anechoism coecient expression into: It is assumed that without the material, this support radiates into the uid a plane wave of complex amplitude H 0 in the plane of the vibrating surface. When the support is coated with the material of thickness h, then the plane wave complex amplitude calculated in the same reference plane becomes H. The hull decoupling coecient in the reference plane is then dened by C D = He -ik f h /H 0 .

C A = R - T 2 R -1 . (C.
To express the hull decoupling coecient C D , the following boundary conditions are thus introduced :

     p left = H v left = -H Z f v right = v 0 = -H 0 Z f . (C.9)
Considering these boundary conditions, the transfer matrix denition is given by:

H -H Z f = A B C D p right -H 0 Z f . (C.10)
where A, B, C and D are the elements of the transfer matrix. The hull decoupling coecient C D can then be obtained by expressing the ABCD-elements in terms of Sparameters as per Appendix A, which yields:

C D = He -ik f h H 0 = S 12 -S 22 + 1 e -ik f h . (C.11)
The hull decoupling coecient can thus be obtained without the use of a vibrating support and by only measuring the reection and transmission coecients R and T : It is considered that each edge portion of the square behaves as a point source for edgediracted waves. It is assumed that each of these portions produces the same response to a plane wave excitation, so that the elementary pressure of edge-diracted waves, called P e diff , is identical along edges of the square. With these simplifying considerations, the total contribution of the edge-diracted pressure at point 0, referred to as P diff (0), can be expressed using the elementary edge-diracted pressure P e diff and a source term that is summed over the square edges:

C D = T -R + 1 e -ik f h .
P diff (0) = P e diff Γ e -ikr r dl, (D.1)
where r is the distance between point 0 and a point on the edge. Distance r is expressed as r = a 2 + y 2 for horizontal edges or r = √ a 2 + z 2 for vertical edges. As the panel is square, Equation D.1 is simplied into:

P diff (0) = 4P e diff a -a e -ik √ a 2 +l 2 √ a 2 + l 2 dl. (D.2)
Edge diracted pressure P diff (M ) at a point M away from the panel face, of coordinates (x M ,y M ,z M ), can then be expressed using the same approach, with projections of the distance r in the 3D space.

Appendix E Performance Maps and Frequency Paths

For the scientic community interested in metamaterials, a topic of interest is the search for designs presenting single or double negative values for the eective properties. Eective properties can also be used to simplify an optimisation process by using homogenised media instead of nite element models of a complex structure, as in Chapter 7. However, there are very few studies on the link between eective properties and acoustic performance. It is of course known, and it has also been exhibited in the preceding chapters, that a dipole resonance can result in a negative eective density and that a negative bulk modulus appears during a monopolar resonance. But what is the link between the values of these eective properties and how much the structure will absorb, dissipate or transmit the acoustic energy? For example, is it better to use metamaterials with eective density or negative bulk modulus to improve anechoism?

This appendix presents a general study on the relation between the eective properties and the anechoic performances of a structure. The aim of this appendix is not to cover the full extent of such a study, but simply to open up the subject for possible future studies or discussions. In the rst section, a visual concept is introduced to investigate this link for the coecient of anechoism. A multilayered structure is then dened in the second section, for which the eective properties are used to follow a frequency path.

221

E.1 Mapping Approach

E.1.1 Presentation

The link between absorption and eective properties has already been investigated by the scientic community. For example, the concept of double-negative [START_REF] Kumar | Double negative acoustic metastructure for attenuation of acoustic emissions[END_REF], or doublezero [START_REF] Wang | An experimental study on the radar absorbing characteristics of folded core structures[END_REF] eective parameters has been exploited for the absorption of sound waves. Duan et al. [START_REF] Duan | Theoretical requirements for broadband perfect absorption of acoustic waves by ultra-thin elastic meta-lms[END_REF] have theoretically studied perfect absorption of elastic waves with either (1) imaginary large mass density and a free space boundary, or (2) imaginary small bulk modulus and hard wall boundary. Total absorption could also be realised by an acoustic coherent perfect absorber (CPA) in which the scattered waves at the resonant frequency can be cancelled through interference with another coherent incident acoustic wave with a suitable phase and amplitude [START_REF] Song | Acoustic coherent perfect absorbers[END_REF][START_REF] Wei | Symmetrical and anti-symmetrical coherent perfect absorption for acoustic waves[END_REF]. These links between eective properties and absorption performance are very specic to a given design, set-up and are rather frequencylocalised.

In this section, a general approach is proposed to visually study the link between the anechoism coecient and the eective properties of a structure. These eective properties are the properties of a uid. Such a uid equivalent model may be accurate for the description of a multilayered medium under normal incidence, as in Chapter 2. These eective properties could also be used for the approximate representation of a metamaterials with macro-inclusions above the homogenisation limit, as in Chapter 5. It is worth noting that the performance mapping approach introduced here can also be applied to other performance coecients, such as the hull decoupling coecient or the absorption coecient. For the anechoism coecient, a visual approach is particularly useful as it is mathematically dicult to guess how the eective properties should vary to improve anechoic performance. For a symmetric medium, the anechoism coecient is given by:

C A = R - T 2 R -1 , (E.1)
where R and T are respectively the reection and transmission coecients. These coecients are themselves expressed using the elements A, B, C and D from the transfer matrix, as per Appendix C. Finally, the elements of the transfer matrix of the medium are expressed using its eective properties, as in Appendix A. The expression of the anechoism coecient as a function of the eective properties is therefore quite convoluted, especially since the eective properties are complex numbers. A performance mapping approach is thus introduced in order to visually exhibit the link between eective properties and anechoism performance.

Building the anechoism map is however not simple, since the anechoism coecient C A is a complex-valued function F of 6 real variables :

C A = F( (ρ), (κ), (ρ), (κ), f, L), (E.2)
where (ρ) and (ρ) are respectively the real and imaginary parts of the eective density, (κ) and (κ) are respectively the real and imaginary parts of the eective bulk modulus, f is the frequency and L is the length of the medium. At rst, frequency f and medium length L are xed. These two parameters can also be expressed in terms of a reduced wavenumber in water, k w L = ωL/1500, where 1500 m.s -1 is the usual value for the speed of sound in water, commonly used in underwater acoustics.

For these given parameters, the mapping approach to be developed needs to show the anechoism coecient according to the four other variables, which means that a 4D-space has to be represented in 2D using projections. Additionally, it is worth noting that the anechoism coecient C A can also be expressed as a complex-valued function with only 4 real variables ( (z), (z), (kL), (kL)), where z is the reduced impedance and kL is the reduced wavenumber.

Two types of representation are introduced. The rst representation is presented in Figure E.1. The main performance map (on the left) is a representation in the space of the real parts, where the x-axis corresponds to the real part of the bulk modulus (κ) and the y-axis corresponds to the real part of the density (ρ). For a given value of (κ) and (ρ), the anechoism coecient is calculated for all values of the imaginary parts (κ) and (ρ). This leads to a performance map of |C A | in the -coordinate system ( (κ), (ρ)), In each map, four quadrants are identied and referred to as Quadrants 1 to 4 as per A condition is added in order to satisfy the passivity constraint, based on Section 1.4.5: if the imaginary part of the wavenumber is positive or if the real part of the impedance is negative, then passivity constraint is not satised.

E.1.2 Analysis

Pre-Analysis

The mapping approach is applied for the frequency f = 5 kHz and a medium length L = 6 cm surrounded by water. This corresponds to k w L = 1.25. The main map is rst pre-analysed to dene areas that are then further investigated. The denition of the bounds of the map was based on the values of the eective density ρ e and eective bulk modulus κ e obtained in Chapters 2 and 5. The ranges for the real part of the eective density and bulk modulus are [-10000 10000] kg.m -3 and [-10 10] × 10 9 Pa, respectively. Ranges for the imaginary parts (ρ) and (κ) are the same as for their real part counterparts. These bounds may seem extreme, but such values can be reached by the eective properties at dipole resonance frequencies for example (see Chapter 5). Duan et al. [START_REF] Duan | Theoretical requirements for broadband perfect absorption of acoustic waves by ultra-thin elastic meta-lms[END_REF] have also reported that perfect absorption can be obtained for large values of (ρ).

Central Symmetry

In all 3 maps of Figure E.4, it is observed that Quadrant 3, corresponding to ( (κ) < 0, (ρ) < 0), is the central symmetric about the coordinate system origin of Quadrant 1 ( (κ) > 0, (ρ) > 0). The same observation is made for Quadrants 2 and 4. Further investigation remains to be done to demonstrate if (ρ,κ) leads to the same anechoism coecient as (-ρ,-κ). A way to do so could be to write the anechoism coecient in terms of the eective wavenumber k 2 e = ω 2 ρ/κ and the eective impedance Z e = √ ρκ. In an anechoic conguration, that is to say for a medium applied on a rigid backing, the anechoism coecient corresponds to the total reection coecient R, resulting from the multiple reections within the medium. The expression of the total reected wave is obtained using Figure E.5 and is given by: C A = R = r + (1 -r)t(1)t(1 + r) + (1 -r)t(1)t(-r)t(1)t(1 + r) + ...

(E.5)
This last equation can be also written as:

C A = r + (1 -r)(1 + r)t 2 ∞ n=0 (-rt 2 ) n , (E.6)
in which there is a geometric series for | -rt 2 | < 1 converging toward 1/(1 + rt 2 ). The nal expression of the anechoism coecient is thus given by:

C A = r + t 2 1 + rt 2 .
(E.7)

This last Equation as well as Equations (E.3) and (E.4) can be used to show that the anechoism coecient is the same for (ρ,κ) as for (-ρ,-κ), which leads to the central symmetry.

Quadrant 1

In Quadrant 1 ( (κ) > 0, (ρ) > 0), several blue branches stand out, corresponding to very low values of |C A |. The origins and conditions of these low |C A |-branches are discussed in more details here. Since they seem to be related to specic values of the wavenumber, the anechoism coecient expression in terms of the eective wavenumber k e and the eective impedance Z e given in Equation (E.7) is used in order to determine the conditions on these parameters which make it possible to obtain exactly |C A | = 0.

The only non-trivial solution that can exactly cancel the anechoism coecient is r = -t 2 .

In terms of eective wavenumber and eective impedance, this equality becomes:

z -1 z + 1 = -e -2ikeL .

(E.8)

The reduced wavenumber is thus given by:

k e L = - 1 2i log(| 1 -z 1 + z |) - 1 2 arg( 1 -z 1 + z ) -mπ, (E.9)
where m is an integer, since the complex logarithm is multi-branched function.

Type 2 Representation is now used for the wavenumber and eective impedance to visually evaluate Equation (E.9).

In (ρ) and (κ), which conrms that on the branches, and only precisely on the branches, numerical errors are present due to the discontinuity so that there is no consistency for the values of the imaginary parts. In order to understand why minimal values of the anechoism coecient seem to be obtained when the imaginary and real parts of ρ and κ are equal, the eective properties are written as:

ρ = ρ (1 + i) κ = κ (1 + i) , (E.12)
with ρ = (ρ) = (ρ) and κ = (κ) = (κ). These expressions are used to express the wavenumber k = ω ρ/κ, which yields: k = ω ρ κ .

(E.13)

E.2 Frequency Path E.2.1 Introduction

In this section, performance maps are generated for several frequencies in order to follow the "frequency path" of a multilayered structure. The notion of frequency path is explained/illustrated later in this section. Two case studies are conducted, described below :

1. A layer arrangement is dened aiming to maximise the number of frequencies for which the eective properties remain in Quadrant 4 ( (κ) > 0, (ρ) < 0) since it has been observed that this area can always provide low values of the anechoism coecient independently of the reduced frequency. The objective of this approach is thus to determine if, by containing the dispersion on the waves' properties in the medium to this area, it is possible to obtain broadband low anechoism.

2. The second layer arrangement is dened so that the anechoism coecient is minimised over the entire frequency range, as per Chapter 3. As such, it would be possible to observe the behaviour of performance-driven dispersion within the maps.

In both case studies, it is the layer arrangement of a unit cell of thickness L u that is dened. The transfer matrix of the unit cell is rst calculated, from which eective properties and scattering coecients are derived as per Chapter 2. The same materials as in Chapter 3 are considered, that is to say a polyurethane containing 6% of micro-balloons as well as stainless steel. The thickness of the unit is L u = 16 mm and the minimal layer thickness is l = 2 mm. There are thus 2 8 = 256 possible designs for the unit cell. Even though the performance mapping can be applied to asymmetric media by adding extra parameters, a symmetric unit is herein selected for the sake of simplicity. This reduces the number of possible designs to 2 4 = 16.

E.2.2 Quadrant Driven Design

In this section, a multilayered medium is selected based on its eective properties so that they remain in the quadrant ( (κ)>0, (ρ) < 0) for a large number of frequencies within the frequency range of interest which is [3 kHz -20 kHz]. As a reminder, the objective here is to contain the dispersion in this area, which seems unchanged with frequency, in order to obtain broadband low anechoism.

The layer arrangement shown in Figure E.14 is selected. Amongst all possible symmetric units, it corresponds to the largest range of frequencies with negative values of (ρ). For each of the frequencies previously selected, the ( (κ), (ρ))-map is plotted. The eective properties obtained for the unit cell are then spotted on the -map. For this given set of values for ( (κ) and (ρ)), the -map is generated, in which the values of ( (κ) and (ρ)) obtained for the unit cell are highlighted, along with the value of the anechoism coecient. With this approach, applied for successive frequencies, the frequency path of a design is created and is given in For frequency (g) 20 kHz, a interesting location is reached in Quadrant 4 of the -map, but, as previously, the imaginary parts are such that the anechoism coecient remains very high. It can nonetheless be noted that for frequencies (f ) and (g), the -maps tend to be split into two spaces: very low anechoism for (κ) > 0 and very high anechoism for (κ) < 0.

Consequently, it was observed that the imaginary parts of the eective properties play a signicant role in the acoustic performance. Even though a specic metamaterial feature (being a negative eective density) is obtained, it is not associated here to a good anechoism coecient. A better anechoism performance may therefore be obtained with a better control of the dispersion and more particularly of the imaginary parts of the eective properties. In what follows, a design is chosen for its broadband anechoism performance in order to observe the "natural" evolution of the dispersion for a performance driven design.

E.2.3 Performance Driven Design

In this section, a multilayered medium is selected based on its broadband anechoism At frequency (e) to (g), the area z < 1 is reached, where extremely low values of the anechoism coecient could be obtained. For both (e) and (f ), it can be seen however that |C A | is quite high here since the imaginary parts are not suitable. Similarly to the design previously studied, it would be benecial here to have (κ) > 0. In fact, it is observed that very low values of |C A | are obtained at 24 kHz (frequency (g)) in the the area z < 1 thanks to values of (κ) going towards positive values.

To summarise, low values of the anechoism coecients have been obtained as a result of slab resonances associated with the branches. Once on the branch, it does not seem to be dicult to obtain the imaginary parts that lead to a good anechoism coecient. Slightly o the branches, it was observed that it is benecial to aim for positive values of (κ) or alternatively for near-zero values of (κ).

This conrms what has been observed for the 2D optimisation (Chapter 7), that is that multilayered media can lead to low anechoism in a given frequency range, but fail to provide a broadband performance since they do not provide local resonances that could help tuning the imaginary parts. Multilayered media are therefore found to provide low anechoism coecient due to slab resonances, thus depending on the length of the medium.

The fact that (ρ) < 0 is therefore not an indicator that the anechoism coecient is low, but simply signies that the center of mass displacement is in phase opposition to the incident elds (Chapter 6), which for multilayered media seems to only happen when a heavy layer is located in-between two damped layers.

Although the presented results are not always intuitive, this concept of performance maps and frequency path helped understanding the feature that lead to low anechoism and it gave keys on how to tune the dispersion on waves' properties to achieve the desired goal.

Contents 1 . 1 19 1. 3 . 3 22 1. 3 . 4 26 1. 4 30 1. 4 . 2 30 1. 4 . 3 31 1. 4 . 5 11 Figure 1 . 1

 11193322342643042304331451111 Figure 1.1 Concepts of passive and active detection. To avoid detection by passive sonar, a hull decoupling coating can be used. To avoid detection by active sonar, anechoic coating to reduce the back-scattered energy can be used.

Figure 1

 1 Figure 1.2 (a) Section view, (b) top view and (c) sample picture [1] of a typical design of an acoustic coating comprising soft rubber embedded with voided inclusions.

Figure 1 . 5 (

 15 Figure 1.5 (a) Schematic cross-sectional view of a Helmholtz resonator. The sample is made of aluminium, consisting of a rectangular cavity and a cylindrical neck. The cavity and neck are lled with water, and are connected at the same side to a square water duct. The inset illustrates the analogy between a Helmholtz resonator and an inductorcapacitor circuit, showing the uidic inductance due to the neck, and the acoustic capacitance due to the cavity [8]. (b) Composite structure consisting of interspaced membranes and side holes [9].

Figure 1

 1 Figure 1.8 (a) Schematic diagram of a multi-phase metamaterial. The dashed rectangle

Figure 1 .

 1 Figure 1.13 (a) Cross section of one locally resonant core-shell unit, and (b) optimal

Figure 1 .

 1 Figure 1.14 (a) A schematic diagram of a synthesis and structure of phononic glass. [93] (b) Schematic diagram of the sandwich anechoic coating embedded with a micro-perforated panel in high-viscosity condition for underwater sound absorption [94].

Figure 1 .

 1 Figure 1.15 (a) Measurement rig, sample and airbox [103]. (b) Surface hydrophone [99].

  be minimised as part of the post-processing by considering the transient-state reected and transmitted signals, which are less contaminated. The transient-state signals are then extrapolated to obtain the steady-states used to derive the scattering coecients.

Figure 1 .

 1 Figure 1.16 (a) Fluid-like structure made of perforated steel plates suspended in water.

Figure 2 . 2

 22 Figure 2.2 Finite periodic multilayered medium characterised by a symmetric unit cell,

Figure 2 . 5

 25 Figure 2.5 Dispersion curve for the multilayered units comprising layers of silicone

Figure 2 . 6

 26 Figure 2.6 Eective impedances for forward propagating waves (dashed blue lines) and backward propagating waves (dashed-dotted blue lines) as well as characteristic impedance Z W (black lines) obtained for the unit designs shown in Figure 2.4(a) (rst column), Figure 2.4(b) (second column), Figure 2.4(c) (third column) and Figure 2.4(d) (fourth column).The rst line of graphics represents the real part of the eective impedances, while the second line corresponds to the imaginary part. The shaded areas represent the band gaps obtained using zero damping for silicone. The red crosses show the quasi-static eective impedance.

( 2 . 8 )

 28 whereas for the two asymmetric units (c) and (d), Equation (2.10) is used, leading to the eective impedances for forward and backward propagating waves, respectively represented by the dashed and dashed-dotted blue lines. The black lines represent the characteristic impedance Z W obtained using Equation (6.6). These plots rst conrm that Z e = Z W for the symmetric units (a) and (b). The quasi-static impedance, represented by the red crosses, is also consistent with the eective impedance obtained at low frequencies.Dierences in the eective impedances amongst the dierent unit cells are more visible for the imaginary part. For the two asymmetric versions (c) and (d), the real parts of the eective impedances are quite similar but the imaginary parts dier signicantly. It is also important to note that the eective impedance for forward propagating waves Z + e for one unit is the opposite of the eective impedance for backward propagating waves Z - e for the other (reverse) unit. In other words, Z + e,(c) = -Z e,(d) and Z + e,(d) = -Z e,(c) .

Figure 2 . 7

 27 Figure 2.7 Eective density obtained for the unit designs represented by Figure 2.4(a) (rst column), Figure 2.4(b) (second column), Figure 2.4(c) (third column) and Figure 2.4(d) (fourth column). The top row shows the real part while the bottom row corresponds to the imaginary part. The shaded areas represent the band gaps obtained using zero damping for silicone.

Figure 2 . 8

 28 Figure 2.8 Bulk modulus obtained for the unit designs represented by Figure 2.4(a) (rst column), Figure 2.4(b) (second column), Figure 2.4(c) (third column) and Figure 2.4(d) (fourth column). The top row shows the real part while the bottom row corresponds to the imaginary part. The shaded areas represent the band gaps obtained using zero damping for silicone.

Figure 2 .

 2 Figure 2.9.

Figure 2 . 9

 29 Figure 2.9 Coecients R = S 11 and T = S 12 = S 21 of the four unit designs in Figure 2.4, obtained from the total transfer matrix for 5 unit cells (red lines) and from the eective media (black lines). The shaded areas represent the band gaps obtained with zero damping for silicone.

Figure 2 . 4 .

 24 Figure 2.10 Reection and transmission coecients of the forward scattering coecients R = S 11 and T = S 12 = S 21 obtained for the four unit designs of Figure 2.4.

Figure 2 .

 2 Figure 2.11 Reection coecients R = S 11 for media comprising 5, 8, 15 or 30 units.

Figure 2 .

 2 Figure 2.12 Transmission coecients T for media comprising 5, 8, 15 or 30 units.

Figure 3 . 1

 31 Figure 3.1 Estimated complex speed of sound in the selected polyurethane matrix comprising 6% of micro-balloons.

Figure 3 . 2

 32 Figure 3.2 Multilayered medium before the aectation of materials by the optimisation algorithm.Thicknesses are displayed in meters. The acoustic wave is incident on the face x = 0 whereas x = 0.06 corresponds to the transmission side. For the calculation of both the anechoism and hull decoupling coecients, the hull is considered at on the right-hand side.

Figure 3 .

 3 Figure 3.3 (a) Schematic diagram showing how a solution divides the objective space into four quadrants for the non-dominated ranking approach. (b) Illustration of the Pareto front according to the non-dominated ranking system.

  The optimisation process is detailed in what follows and illustrated on Figure 3.4 and described in what follows.

Figure 3 . 4 Flow

 34 Figure 3.4 Flow Chart for the Controlled Elitist Genetic Algorithm.

3. 5 .

 5

Figure 3 . 5

 35 Figure 3.5 Anechoism coecient as function of frequency f , for which the root mean square, being the rst objective function, is given by |C A |. The second objective function, σ C A , aims at lowering the values of the coecient that are higher than |C A |, as shown by the downwards arrows. Values lower than |C A |, highlighted by the red crosses, are thus not considered.

For this 15 -

 15 layer conguration, there are 32768 possibilities. Scores for each one of these individuals are plotted in Figure3.6, creating the cloud of grey dots. Scores of the Pareto front solutions are also given and are highlighted by the colored dots for sets 1 to 6.

Figure 3 . 6

 36 Figure 3.6 Scores for all the possible layer arrangements for the conguration dened in Table3.1. The colored dots represent the Pareto front for Sets 1 to 6. The orange square shows the score of a medium comprising only polyurethane layers, while the black square corresponds to the score of a stainless steel medium.

N

  = n l × 4 = 160 Proportion of children created by Crossover C % = 80% Proportion of children created by Mutation 100 -C % = 20% Mutation probability m = 0.02 First rank ratio r 1 = 0.4 Maximum number of Generations G max = 300Table 3.2 Parameters to optimise the layer arrangement of a medium to lower the anechoism coecient.

Figure 3 .Figure 3 . 7 Figure 3 . 8

 33738 Figure 3.7 presents the population's scores over the generations, represented by the colored dots. The Pareto front obtained with the last generation is highlighted by the grey squares.

Figure 3 . 9

 39 Figure 3.9 Anechoism coecient (left) and layer arrangement (right) for four selected solutions of the Pareto front.

Figure 3 .

 3 Figure 3.10 Layer arrangement selected moving average µ mov of the individual layers' impedances, calculated along the medium's thickness. The impedance of water is shown by the blue dotted line.

  2.1. Equations (3.3) and (3.5) are thus used replacing C A by C D . The root mean square of the hull decoupling coecient and the normalised standard deviation to this value are calculated. The hull decoupling coecient is given by :

Figure 3 .Figure 3 .Figure 3 .

 333 Figure 3.11 presents the population's scores over the generations, represented by the colored dots. The Pareto front obtained with this last generation is highlighted by the grey squares. The layer arrangement for each solution of the Pareto front is shown in Figure 3.12, where the orange color corresponds to polyurethane and the dark grey color to stainless steel. The solutions are sorted from top to bottom in ascending |C D | order. In all cases x = 0 corresponds to the water/metamaterial interface, and x = 0.04 to the metamaterial/rigid hull interface.

Figure 3 .Figure 3 .

 33 Figure 3.13 Hull decoupling coecient (left) and layer arrangement (right) for four selected solutions of the Pareto front.

Figure 3 .

 3 Figure 3.15 presents the scores of the population over generations, represented by the colored dots. The Pareto front of the last generation is highlighted by the grey squares. It is observed that some of the solutions are close to level requirements for the hull decoupling coecients since σ L C D reaches low values compared to σ L C A. This observation is consistent with the two previous sections, that showed that the input parameters (materials and total length) are more suitable to the design of a hull decoupling coating than an anechoic coating.

Figure 3 .

 3 Figure 3.15 Population's scores along generations. The grey squares represent the Pareto front obtained at the last generation, whereas the white squares are the scores of selected congurations.

Figure 3 .

 3 Figure 3.16 Layer arrangements for all the solutions of the Pareto front, arranged according to decreasing σ L C D , and thus increasing σ L C A . The orange color represents the polyurethane while the dark grey color corresponds to the stainless steel.

Figure 3 .

 3 Figure 3.17 Layer arrangement for four selected solutions of the Pareto front.

Figure 3 .

 3 Figure 3.18 Anechoism coecient and hull decoupling coecient for four selected solutions of the Pareto front.

Figure 4 . 1

 41 Figure 4.1 Conventional set-up for panel measurements in an open water tank.

Figure 4 .

 4 2 to illustrate the experimental setup specic to the method.

Figure 4 . 2

 42 Figure 4.2 Experimental set-up specic to the application of the 3-point method.

  Measurements are conducted in a large open water tank, shown in Figure4.3, which is 8 meters long, 6 meters wide and 7 meters deep. This facility is located in the academic laboratory ISEN, in Lille (France). An omnidirectional source is placed at a distance of approximately 3 m from the panel perpendicular to the panel surface and in its median plane. Two hydrophones are placed on either side of the panel, in symmetrical positions with respect to the central plane of the panel (parallel to its large surface). These hydrophones are xed at the end of aluminium sections, which other end is xed to a motor on the surface. The position of the receivers on the x-axis and y-axis can be easily modied using Rexroth motors able to move the sections, on which hydrophones are xed with an accuracy of the order of one tenth of a millimetre. It is worth emphasising that such precision is valid on the surface. However, given that the hydrophones are placed about 3 meters deep, there is certainly a loss of accuracy in the position of the hydrophones in the water. Displacement along the z-axis must be done manually by changing the immersion of the panel.

Figure 4 . 3

 43 Figure 4.3 Open Water Tank Facility of ISEN, Lille (France).

4. 2 . 1

 21 Application on a Test PanelThe 3-point-method is applied on a reference panel made of 5083 aluminum alloy of dimensions 1 m × 1 m and of thickness 4 cm, as shown in Figure4.4. In the water tank, the hydrophones and the projector are completely immersed to a depth of 2.73 m, corresponding to the position of the panel's center. The spherical projector is placed 2.87 m away from the panel. It is assumed that the incident wavefront is plane when it reaches the panel.

Figure 4 . 4

 44 Figure 4.4 Aluminium reference panel.

Figure 4 . 5 .

 45 Figure 4.5. As previously mentioned, the total pressure on the reection side may reach local minima resulting from destructive interference. Pairs of positions / frequencies (X , f ) that have been removed using the pre-processing step are highlighted with circles.

Figure 4 . 5

 45 Figure 4.5 Pressure amplitudes measured at the 3 positions A 1 , A 2 and A 3 for the reection side and at the 3 positions B 1 , B 2 and B 3 for the transmission side. The circles represent (X, f ) pairs of destructive interference.

  Figure 4.6.

Figure 4 .

 4 Figure 4.6 (a) Incident pressure p i , reected pressure p r and diracted pressure p diff R at the central position A 2 on the reection side. The incident pressure measured without panel (p WP i ) is also plotted. (b) Transmitted pressure p t , innite pressure p ∞ and diracted pressure p diff T at the central position B 2 on the transmission side.

Figure 4 . 7

 47 Figure 4.7 Modulus of the reection coecient and the transmission coecient obtained

Figure 4 . 8

 48 Figure 4.8 Phase of the reection coecient and the transmission coecient obtained analytically for a panel of innite lateral dimensions and measured with the 3-point method.A distance correction is applied on the latter to retrieve the variation of the theoretical phase: Phases obtained with the 3-point method were modied considering the distances hydrophone/panel d X 2 + 2.2cm for the reection side and d X 2 + 2.5cm for the transmission side.

Figure 4 . 9

 49 Figure 4.9 Amplitude of (a) the reection coecient and (b) the transmission coecient obtained with the 3-point method for the tests 1 to 3 dened in Table 4.2. Reection and transmission coecients obtained theoretically for a panel of innite lateral dimensions are also presented (same as in Figure 4.7).

Figure 4 .

 4 Figure 4.10 Pressure mapping along the y-axis for 3.5 kHz, 5.5 kHz and 7.5 kHz.

Figure 4 .

 4 Figure 4.11 Pressure mapping along the y-axis for 14 kHz, 14.5 kHz and 15 kHz.Experimental coecients are also given and compared to the theoretical coecients for a panel of innite lateral dimensions.

Figure 4 .

 4 Figure 4.12 Variations of the diracted pressure along the y-axis obtained with the analytical model in Appendix D.

Figure 4 .

 4 Figure 4.13 Polyurethane panel comprising 6% of micro-balloons. The panel is hold by a Plexiglas frame.

  • and d X = λ/5 , has yielded excellent results and has highlighted the existence of a diracted signal with low amplitude and very little variation. The coecients shown in Figure4.14 are then averaged and used to retrieve the eective speed of sound in the panel.

Figure 4 .

 4 Figure 4.14 Modulus and unwrapped phase of the reection and transmission coecients obtained with the 3-point method at three dierent positions along the yaxis.

Figure 4 . 15 .

 415 Figure 4.15. The real part of the speed of sound rises slightly from about 900 m.s -1 up to 950 m.s -1 at 25 kHz. The loss factor for the eective speed of sound slightly decreases from roughly 10% to 5%.

Figure 4 .

 4 Figure 4.15 Complex speed of sound c L (1 + iη) obtained with the retrieval method of Fokin et al. [45].

Figure 4 .

 4 Figure 4.16 Values of the criterion C for the frequency 15 kHz.

  Figure 4.15. These curves have been tted to be used in Chapter 3, as shown in Figure 4.18.

Figure 4 .

 4 Figure 4.17 Speed of sound obtained using the minimisation criterion C given in Equation (4.9) for C < , with = 0.13. The grey dots are the values of the complex speed of sound that give the global minimum of C whereas the error bars are the sets of solutions that satisfy C < .

Figure 4 .

 4 Figure 4.18 Estimated complex speed of sound in the selected polyurethane matrix comprising 6% of micro-balloons.

Figure 4 .

 4 [START_REF] Graciá-Salgado | Negative mass density and ρ-near-zero quasi-two-dimensional metamaterials: design and applications[END_REF]. Reection and transmission coecients calculated with theses values are given in Figure4.19 and compared with coecients obtained experimentally. Close agreement between calculated and experimental values for both the modulus and unwrapped phase can be observed.

Figure 4 .

 4 Figure 4.19 Modulus and unwrapped phase of the reection and transmission coecients obtained with the 3-point method (black lines) and calculated with the t of the retrieved speed of sound in the panel (red lines).

4. 4

 4 Multilayered Medium Measurements 4.4.1 Panel and Experimental Set-up In this section, the selected multilayered medium selected in Chapter 3 (Figure 3.10) is experimentally characterised. Its design is reminded in Figure 4.20 and photographs of the panel are presented in Figure 4.21.

Figure 4 .

 4 Figure 4.20 Theoretical layer arrangement and thicknesses (in millimetres) of the manufactured panel. In the anechoic conguration, Face 2 would be on the hull and the incident plane wave arrives on Face 1.

Figure 4 .

 4 Figure 4.21 Photo of (a) the panel before being mounted in the water tank, (b) the layer arrangement as per Figure 4.20, and (c) panel holding points that contain all steel layers.

Figure 4 .

 4 Figure 4.22 Pressure levels calculated with the 3-point method on the reection and transmission side, for an incident plane wave on Face 1 (left) and on Face 2 (right). The incident pressure measured without the panel is also given (p WP i (A 2 )).

FromFigure 4 .Figure 4 .

 44 Figure 4.23 Reection and transmission coecients of the multilayered medium obtained by insonication of Face 1.

Figure 4 .

 4 Figure 4.25 Corrected phases of the reection and transmission coecients of the multilayered medium obtained by insonication of Face 1.

Figure 4 .

 4 Figure 4.26 Corrected phases of the reection and transmission coecients of the multilayered medium obtained by insonication of Face 2.

Figure 4 .

 4 Figure 4.27 Anechoism coecient for the multilayered medium obtained theoretically in Chapter 3 and using the measured reection and transmission coecients (blue).Coecients are also calculated considering the phase correction 1 obtained for the aluminium panel and considering mixed phase correction by the use of correction 2 and 3.

Contents 5 .

 5 1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.1.1 Fluid Homogenisation Model . . . . . . . . . . . . . . . . . . . 107 5.1.2 Direct Inversion Method . . . . . . . . . . . . . . . . . . . . . . 107 5.1.3 Dierential Method . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.3 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.4 Case Study: Hard Inclusions . . . . . . . . . . . . . . . . . . . . 118 5.4.1 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.4.2 Eective Parameters . . . . . . . . . . . . . . . . . . . . . . . . 118 5.4.3 Inuence of the Number of Unit Cells . . . . . . . . . . . . . . 123 5.4.4 Scattering Predictions . . . . . . . . . . . . . . . . . . . . . . . 124 5.5 Case Study: Voided Inclusions . . . . . . . . . . . . . . . . . . 127 5.5.1 Eective Parameters . . . . . . . . . . . . . . . . . . . . . . . . 127 105 5.1 Methodology 5.1.1 Fluid Homogenisation Model This chapter is limited to the study of symmetric and periodic structures with macroinclusions, characterised by a unit cell. Similar to the homogenisation theory for multilayered structures presented in Chapter 2, the objective is to dene the eective medium that would replace the unit cell. A uid model is selected and is thus characterised by an eective wavenumber k and an eective impedance Z.

Figure 5 . 1

 51 Figure 5.1 Schematic diagram of a segmented medium comprising n identical unit cells. Reection and transmission coecients on the incidence and transmission sides of the medium are also shown.

Figure 5 . 2

 52 Figure 5.2 Schematic diagram illustrating the Bianco and Parodi method in which two nite media dier only by length ∆L in the propagation direction. Reection and transmission coecients of the two media are dened.

Figure 5 . 3

 53 Figure 5.3 Schematic diagram illustrating the dierential method in which two nite segmented media comprising identical unit cells dier only by a single unit cell. Reection and transmission coecients of the two segmented media are dened.

Figure 5 . 4

 54 Figure 5.4 Fluid pressure at the interface with (a) the locally resonant medium and (b) with the multilayered medium.

Figure 5 . 5 .Figure 5

 555 Figure 5.5. The thickness of silicone and aluminium are selected as d s = 15 mm and d a = 10 mm, hence the length of a unit cell becomes L u = 40 mm. The direct inversion method is applied to the multilayered medium comprising n = 4 unit cells, as shown in Figure 5.5(b). The dierential method is applied for (n, n + 1) = (4, 5).

Figure 5 . 6

 56 Figure 5.6 Dispersion curves for the reduced wavenumber obtained with the direct inversion method (k dir ), with the Bianco and Parodi method (k BP ), with the dierential method Version 1 (k diff 1 ), with the dierential method Version 2 (k diff 2 ) and with the analytical model for multilayered media presented in Chapter 2 (k cell ).

Figure 5 . 7

 57 Figure 5.7 Eective impedance obtained with the direct inversion method (Z dir ), with the dierential method Version 1 (Z diff 1 ), with the dierential method Version 2 (Z diff 2 ) and with the analytical model for multilayered media presented in Chapter 2 (Z cell ).

Figure 5 .

 5 Figure 5.8 (a) Representation of the locally resonant unit cell. (b) Two segmented media comprising n = 4 and n + 1 = 5 unit cells. Each cell has length L u .

Figure 5 . 9

 59 Figure 5.9 Dispersion curves for the reduced wavenumber obtained with the direct inversion method (k dir ), with the Bianco and Parodi method (k BP ), with the dierential method Version 1 (k diff 1 ) and with the dierential method Version 2 (k diff 2 ).

Figure 5 .

 5 Figure 5.10 Relative dierence between the diagonal elements of the unit transfer matrix calculated with the dierential method as per Equation (5.23). The dashed lines highlight the three selected frequencies A, B and C dened in what follows.

Figure 5 .

 5 Figure 5.11 Dispersion curves for the real part (a) and imaginary part (b) of the reduced eective wavenumber for a locally resonant unit cell obtained with the direct inversion method (k dir ) and with the dierential method (k diff ). The dashed lines highlight the three selected frequencies A, B and C.

Figure 5 .

 5 Figure 5.12 Displacement elds for the four unit medium at given frequencies A, B and C. The arrows indicate the direction of displacement and their length is proportional to the displacement modulus. The shades of grey also reect the amplitude of the displacement.

Figure 5 .

 5 Figure 5.13 Eective impedance obtained with the direct inversion method (Z dir ) and

Figure 5 .

 5 Figure 5.14 Eective wavenumbers as a function of angular frequency obtained with the direct inversion method for media comprising 2, 4, 8 and 18 unit cells.

Figure 5 .

 5 Figure 5.15 Eective wavenumbers as a function of angular frequency obtained with

Figures 5.11 and 5 . 13 .

 513 The scattering coecients of the eective medium are also compared to coecients derived from a FE model of the metamaterial comprising N unit cells, as shown in Figure5.16(a). Figures 5.16(b) and 5.16(c) respectively show the corresponding eective medium

Figure 5 .

 5 Figure 5.16 (a) Finite periodic medium of length comprising N unit cells, for which

  for N = 5, 9, 19 unit cells. All three eective media represented in Figures 5.16(b) to 5.16(d) accurately predict the reection andtransmission coecients up to 15 kHz. Above this frequency, the eective parameters from the direct inversion method are strongly dependent on the parameter n, particularly for the interface mode according to Section 5.4.3. As such, using the direct inversion method, the scattering coecients for a high number N tend to present overestimated interface eects since the eective properties have been obtained for a smaller number n. For the reection coecient, the impedance of the rst layer plays a major part in the results, as it has been mentioned in Chapter 2 for variant designs of a unit cell. This is because the reection coecient is roughly R ≈ (Z in -Z f )/(Z left + Z f ), with Z in the eective impedance of the input layer and Z f the impedance of the uid. Since Z in = Z dir for both eective media represented in Figures 5.[START_REF] Yang | Membrane-type acoustic metamaterial with negative dynamic mass[END_REF](c) and 5.16(d),

Figure 5 .Figure 5 .

 55 Figure 5.17 Reection and transmission coecients calculated for the eective media characterised by eective parameters of the dierential method (dashed lines) or the direct inversion method (dotted lines) that represent a periodic media comprising N = 5, N = 9 or N = 19 unit cells. Scattering coecients from the hybrid eective medium (circle markers) and from FEM (full lines) are also presented. The dotted orange line highlights the interface resonance frequency.

Figure 5 .

 5 Figure 5.19 Dispersion curves for the real part (a) and imaginary part (b) of the reduced eective wavenumber for the unit cell obtained with the direct inversion method (k dir ) and with the dierential method (k diff ). Two frequencies, referred to as A and B, are selected and highlighted by the orange dotted lines.

Figure 5 .

 5 Figure 5.20 Displacement elds for the four unit cell medium at frequencies corresponding to points A and B in Figure 5.19. The arrows indicate the direction of displacement and their size is proportional to the displacement modulus. The shades of grey also reect the amplitude of the displacement.

Figure 5 .

 5 Figure 5.21 Reection and transmission coecients calculated for the eective media

Figure 6 . 1

 61 Figure 6.1 Schematic diagram illustrating the direct inversion method, applied to a segmented medium comprising n identical asymmetric unit cells. Coecients S of the medium are also shown.

  Two media are modelled as segmented structures comprising n and n + 1 periodic repetitions of an identical constituent unit cell, as shown in Figure6.2. The extra portion added to the second segmented medium is one unit cell of length L u .

Figure 6 . 2

 62 Figure 6.2 Schematic diagram illustrating the dierential method, applied to segmented media comprising n and n + 1 identical asymmetric unit cells. Coecients S n and S n+1 of these media are also dened.

Figure 6 . 5 Figure 6 . 6

 6566 Figure 6.5 Eective wavenumbers obtained using the direct inversion method and the dierential method for the asymmetric unit with macro-inclusions.

Figure 6 . 7

 67 Figure 6.7 Non-dimensional asymmetry coecient obtained using the direct inversion method and the dierential method for the asymmetric unit with macro-inclusions.

Figure 6 . 8

 68 Figure 6.8 Scattering coecients for a 5-unit medium, obtained by nite element

Figure 6 . 4 ,

 64 Figure 6.4, denes the distance between the resonant inclusion and the input face of the unit cell. It is given dierent values, which are : 0.5 mm, 1 mm, 1.5 mm, 2 mm, 2.5 mm, 3 mm, 3.5 mm, 4 mm, 4.5 mm, 5 mm and 5.5 mm, as respectively shown in Figures 6.9(a) to 6.9(k).

Figure 6 . 9

 69 Figure 6.9 Schematic diagram of the asymmetric inclusions studied, in which the distance interface-inclusion varies.

Figure 6 .

 6 Figure 6.10 Scattering coecients S 11 and T = S 12 = S 21 for various distances d int (in millimetres).

Figure 6 .

 6 Figure 6.11 Absorption coecient A = 1 -|S 11 | 2 -|T | 2 for various distances d int (in millimetres).

Figure 6 .

 6 Figure 6.12 Anechoism coecient for various distances d int (in millimetres).

Figure 6 .

 6 Figure 6.13 Eective density and bulk modulus of asymmetric unit cells derived with the dierential method for various distances d int (in millimetres).

  Medium thickness is h = 35 mm and periodicity is dened by 2d = 80 mm. The medium is studied below the cuto frequency of the periodical array dened by f c = 1489/(2d). Inclusions are voided cylinders of radius r = 10 mm. Three versions of the units are considered : (a) and (c) are asymmetric, with the inclusion placed at a distance d int = 2 mm from the input and output interfaces, respectively. Unit (a) is the reverse design of unit (c).

Figure 6 .

 6 Figure 6.14 Design and dimensions of the voided medium unit.

Figure 6 .

 6 Figure 6.15 Reection and transmission coecients for the units (a), (b) and (c) shown in Figure 6.14.

Figure 6 .

 6 Figure 6.16 Eective density and bulk modulus for the asymmetric layers presented in Figure 6.14.

  is considered. The following dimensions are selected : h 1 = 22 mm, h 2 = 35 mm, d 1 = 20 mm, d 2 = 40 mm, r 1 = 5 mm and r 2 = 10 mm.

Figure 6 .

 6 Figure 6.17 Schematic diagram of the bi-layer Alberich.

Figure 6 .

 6 Figure 6.18 Transmission coecients for the bi-layer Alberich and for each layer separately.

2 Figure 6 .

 26 Figure 6.19 Transmission coecients for the bi-layer Alberich and for each layer separately.

Figure 6 .Figure 6 . 21 .Figure 6 .

 66216 Figure 6.20 Dispersion curve for the eective bi-layer voided unit derived using the homogenisation Options 1 and 2. The eective wavenumber for each individual unit is also shown.

Figure 6 .

 6 Figure 6.22 Eective bulk Modulus and mass density for the bi-layer Alberich design.

Figure 6 .

 6 Figure 6.23 Phase dierence (in degree) between the displacement of the center of mass and the incident eld.

  2 and contains 14 possibilities for the unit cell. Two of them are homogeneous, made of either polyurethane or steel, whereas the others comprise a polyurethane matrix containing a steel inclusion. Material properties are given in Chapter 5.

Figure 7 . 2

 72 Figure 7.2 Schematic diagram of the unit cells composing the database. Inclusions are not drawn to scale.

Figure 7 .

 7 Figure 7.3 introduces the dimensions of (a) the circle inclusion, (b) the square inclusion, (c) the diamond inclusion, (d) the rotated rectangular inclusion and (e) the unit cell.

Figure 7 . 3

 73 Figure 7.3 Dimensions for the four inclusions, consisting of (a) a circle, (b) a square, (c) a diamond and (d) a rotated rectangle. The dimensions of the unit cell are also shown (e).

Figure 7 . 4

 74 Figure 7.4 Dispersion curves obtained using the direct inversion method for the four dierent units of surface area A = 16 mm 2 .

Figure 7 . 5

 75 Figure 7.5 Dispersion curves obtained using the dierential method for the four dierent units of surface area A = 16 mm 2 .

Figure 7 . 6

 76 Figure 7.6 Eective impedance obtained using the dierential method for the four dierent units of surface area A = 16 mm 2 .

Figure 7 . 7

 77 Figure 7.7 Reection (R), transmission (T ) and absorption coecients (A = 1 -|R| 2 -|T | 2 ) for four 3-unit media, each one made of repetitions of one of the four dierent units of surface area A = 16 mm 2 .

5 Figure 7 . 8

 578 Figure 7.8 Dispersion curves obtained using the dierential method for the diamond inclusion of surface area A = 4 mm 2 (small markers), A = 16 mm 2 (medium markers) and A = 25 mm 2 (large markers).

5 Figure 7 . 9

 579 Figure 7.9 Dispersion curves obtained using the direct inversion method for the diamond inclusion of surface areas A = 4 mm 2 (small markers), A = 16 mm 2 (medium markers) and A = 25 mm 2 (large markers).

Figure 7 . 10 .

 710 Contrary to the wavenumber, dierences between inclusions are herein especially signicant at low frequencies. Such signicant dierences in the impedances are of interest as it has been shown in Chapter 3 that strong variation in impedance may be benecial to improve anechoism performance.

Figure 7 .Figure 7 .

 77 Figure 7.10 Eective impedance obtained using the dierential method for the diamond inclusion of surface areas A = 4 mm 2 (small markers), A = 16 mm 2 (medium markers) and A = 25 mm 2 (large markers).

7. 3

 3 Results for Anechoism 7.3.1 Approach Validation The optimisation is now conducted for a broadband frequency range [200 Hz -30000 Hz].

Figure 7 .

 7 Figure 7.12 Scores of the Pareto front for all generations (left) and layer arrangements for the 10 rst and last solutions of the Pareto front (right), from the lowest |C A | to the highest, considering the left hand-side as the incidence side and the right hand-side as the hull side.

Figure 7 .Figure 7 .

 77 Figure 7.13 Scattering coecients, S 11 , S 22 and T = S 12 = S 21 , as well as the anechoism coecient C A for the rst arrangement of the Pareto front, obtained using the eective medium approach (full lines) and nite element simulation (dashed-dotted lines).

  Proportion of children created by Crossover C % = 80% Proportion of children created by Mutation 100 -C % = 20% Mutation probability m = 0.02 First rank ratio r 1 = 0.35 Maximum number of Generations G max = 1000Table 7.2 Parameters to optimise the layer arrangement of a medium to lower the anechoism coecient up to 10 kHz.

Figure 7 .

 7 Figure 7.15 presents the evolution of the Pareto front for all generations as well as the 30 leftmost layer arrangements of the last generation. It is observed that all the arrangementshave their last unit cell free of inclusion, and all comprise a small inclusion in their rst unit cell. Heavy steel layers close to the back end of the medium also seems to be a pattern that stands out for low anechoism in the selected frequency range.

Figure 7 .

 7 Figure 7.15 Scores of the Pareto front along generations (left) and 30 rst layer arrangements of the Pareto front (right), from the lowest |C A | to the highest, considering the left hand-side as the incidence side and the right hand-side as the hull side. White squares highlight the scores of four selected arrangements.

Figure 7 .

 7 Figure 7.16 Layer arrangements and anechoism coecients of four selected congurations chosen for their low anechoism coecient under 10 kHz.

Figure 7 . 17 (

 717 Figure 7.17 (Right) Scattering coecients, S 11 , S 22 and T , as well as the anechoism

Figure 7 . 18 at 9 .Figure 7 .

 71897 Figure 7.18 Eective impedance for each unit at a frequency of 9.8 kHz.

Figure 7 .

 7 Figure 7.19 Scores for all generations for the optimisation of the multilayer (1D) arrangement (colored dots). The Pareto front for the last generation is represented by the grey squares. Layer arrangements are also presented on the right hand-side. The white circles show the Pareto front obtained previously when optimising the arrangement of units with macro-inclusions, shown in Figure 7.15.

Figure 7 .

 7 Figure 7.20 Comparison of the design and anechoism coecient of two solutions having the same scores.

Figure 7 .

 7 21, which also shows a selection of 35 solutions. It appears that with the parameters given in the optimisation (materials, frequency range, medium thickness), a better anechoic performance than hull decoupling performance is obtained, since σ L C A is mostly lower than σ L C D . It is observed that the portion of polyurethane on the hull side is increased compared to the optimisation for the anechoism coecient alone. Inclusions are then moved to the water/medium interface as the hull decoupling performance improves. As such, results here are similar to those obtained with the optimisation for multilayered media, that is, better hull decoupling performance is obtained with media close to a simple bi-layer, with the softer absorbing material being in contact with the hull. It is also worth highlighting that the "steps" in the Pareto front are linked to the position of the unit cell made of steel.

Figure 7 .

 7 Figure 7.21 Scores of the Pareto front along generations (left) and selection of 35 layer arrangements of the Pareto front (right), from the lowest σ L C A to the highest, considering the left hand-side as the incidence side and the right hand-side as the hull side.

Contents 8 . 1

 81 Experimental Characterisation of Diraction Gratings . . . . 8.1.1 Grating Diraction Phenomenon . . . . . . . . . . . . . . . . . 8.1.2 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Measurements of the Alberich . . . . . . . . . . . . . . . . . . . 8.2.1 Presentation of the panel . . . . . . . . . . . . . . . . . . . . . 8.2.2 Below the Cut-o Frequency . . . . . . . . . . . . . . . . . . . 8.2.3 Above the Cut-o Frequency . . . . . . . . . . . . . . . . . . . 8.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 8.1 Experimental Characterisation of Diraction Gratings

Figure 8

 8 Figure 8.1.

Figure 8 . 1

 81 Figure 8.1 Schematic diagram illustrating the scattered eld from a grating. Only the 0 th , 1 st and 2 nd orders on both the reection and transmission sides are shown.

Figure 8 .

 8 2 introduces the projections of the n mode propagation vector in this coordinate system.

Figure 8 . 2

 82 Figure 8.2 Projections of the n mode propagation vector in the (x,y)-plane.

highlighted in Chapter 4 ,

 4 such as panel edge-diraction, are still present and these challenges are increased by the presence of diracted waves traveling in directions other than the incident plane wave and further, may or may not be evanescent depending on the frequency. Therefore, in what follows, the applicability and robustness of the 3-point method for measurements of the Alberich panel is studied. The physical phenomena that occurs below and above the rst cut-o frequency are studied separately.8.2 Measurements of the Alberich8.2.1 Presentation of the panel The panel is shown in Figure 8.3. Its lateral dimensions are 980 × 980 mm and is maintained by a Plexiglas frame. Due to the important concentration of air in the panel, a weight has been placed at the bottom to prevent it from oating.

Figure 8 . 3

 83 Figure 8.3 Photography of the Alberich panel before immersion (left) and during the experiments (right).

Figure 8 . 4

 84 Figure 8.4 Alberich panel measured with the 3-point method below the rst cut-o frequency.

Figure 8 . 5

 85 Figure 8.5 Distances at which the amplitude of the evanescent diracted waves n = ±1

  Figures 8.6 and 8.7 respectively for the incident side and transmission side.

Figure 8 . 6

 86 Figure 8.6 Pressure amplitudes measured at the 3 positions A 1 , A 2 and A 3 for the reection side. The graduation of grey represents the y-positions along the unit cell.

Figure 8 . 7

 87 Figure 8.7 Pressure amplitudes measured at the 3 positions B 1 , B 2 and B 3 for the transmission side. The graduation of grey represents the y-positions along the unit cell.

Figure 8 . 8

 88 Figure 8.8 Incident pressure p i , reected pressure p r and edge-diracted pressure p diff R at the central position A 2 on the reection side. The incident pressure measured without the panel (p WP i

Figure 8 . 9

 89 Figure 8.9 Transmitted pressure p t , innite pressure p ∞ and edge-diracted pressure p diff T at the central position B 2 on the transmission side. The graduation of grey represents the y-position along the unit cell.

and 8. 9 .Figure 8 .

 98 Figure 8.10. Additionally, a raw calculation of the coecient is performed, using the total pressure measured at the central position X 2 and the incident pressure measured without a panel p WP i , according to :

Figure 8 .

 8 Figure 8.12 Experimental set-up specic to the application of the 5-point method along one unit cell. Pressure contributions and geometric parameters are also presented.

Figure 8 .Figure 8 .

 88 Figure 8.13 Set of measurements points on the reection side.

Figure 8 .

 8 Figure 8.15 Reection and transmission coecients for the specular response and the

  Contrary to the previous results based on the nite element model, the measured pressures are now inuenced by the edge-diracted pressure. The total pressures recorded at the three positions on each side of the panel are plotted in Figures 8.16 and 8.17 for the incident side and transmission side, respectively. It can be seen that the pressure levels are very low on the transmission side in comparison with the reection side.

Figure 8 .

 8 Figure 8.16 Pressure amplitudes measured at the 3 positions A 1 , A 2 and A 3 on the reection side. The graduation of grey represents the y-positions along the unit cell.

Figure 8 .

 8 Figure 8.17 Pressure amplitudes measured at the 3 positions B 1 , B 2 and B 3 on the transmission side. The graduation of grey represents the y-positions along the unit cell.

Figure 8 .

 8 Figure 8.18 Incident pressure (p i ), reected pressure of the 0 th mode (p 0 r ) and of the diracted modes ±1 (p ±1 r ) and edge-diracted pressure (p diff R ) at the central position A 2 on the reection side. The incident pressure measured without the panel (p WP i ) is also plotted. The graduation of grey represents the y-positions along the unit cell.

Figure 8 .

 8 Figure 8.19 Transmitted pressure for the 0 th mode (p 0 t ) and for the diracted modes ±1 (p ±1 r ), the innite pressure (p ∞ ) and the edge-diracted pressure (p diff T ) at the central position B 2 on the transmission side. The graduation of grey represents the y-positions along the unit cell.

Figure 8 .

 8 Figure 8.20 Magnitudes of the reection coecient (top) and the transmission coecient (bottom) obtained numerically for a panel of innite lateral dimensions and measured with the 5-point method. For each side, three direct calculations are also performed as per Equations (8.8) to (8.12) using the total pressures measured on each y-line, thus not taking into account the edge-diracted pressure.

Figure A. 2

 2 Figure A.2 Two-port network with (a) pressure/ow notations and (b) power waves notations.

7 )

 7 Note : As one may not be familiar with S-parameters, equivalents in terms of the usual reection coecient and transmission coecient notations are given. R and T are the coecients for an incident wave arriving on one face of the medium. When the medium is reversed, R b and T b are the coecients for the same incident wave arriving on the other (backward) face. As such, S-parameters are : S 11 = R, S 12 = T b , S 21 = T and S 22 = R b .

Figure A. 3 S

 3 Figure A.3 S-parameters depending on the incident side, represented by the red arrow.

  curve. An example of such graphical representation is given in FigureB.1 for the reduced wavenumber kh. Band gaps are frequency ranges where the acoustic waves do not propagate in the given direction, assuming the periodic medium to be innite. Band gaps are obtained when there is no mode with k and ω real.

Figure B. 1

 1 Figure B.1 Example of dispersion curve on the irreducible Brillouin zone for a normal incidence angle.

8 )C. 3

 83 Figure C.2 Conguration used to calculate the hull decoupling coecient, showing the radiation from the vibrating support with and without the decoupling material.

  model is presented here to express the diracted pressure at one point.A square at panel of side 2a is considered and represented in FigureD.1, where the point of coordinates (0,0,0) corresponds to the center 0 of the panel .

Figure D. 1

 1 Figure D.1 Schematic diagram of a square panel of side 2a.

  shown on the right hand side in Figure E.1, where the color of each pixel corresponds to the value of |C A |. Within this |C A |-map in the -coordinate system, the minimum value of the anechoism coecient is spotted and reported in the main |C A |-map in the -coordinate system ( (κ), (ρ)). Therefore, the main map shows the minimal values of the anechoism coecient calculated for all (κ) and (ρ).

Figure E. 1

 1 Figure E.1 Representation 1 of the performance map for a given frequency and medium length.

Figure E. 2

 2 Figure E.2 Representation 2 of the performance map for a given frequency and medium length.

Figure E. 3 .

 3 Figure E.3.

Figure E. 3

 3 Figure E.3 Denition of Quadrants 1 to 4.

  The performance map as per Representation 2 (Figure E.2) is given in Figure E.4. A few observations are made: (1) There seems to be a central symmetry around the coordinate system origin, (2) in the main map (on the left hand side) branches with low |C A | stand out in Quadrants 1 and 3 and are associated to strong discontinuities of the imaginary parts, and (3) low |C A | values are obtained in Quadrants 2 and 4 for very low values of (κ) and/or (ρ). These three observations are further investigated in what follows.

Figure E. 4

 4 Figure E.4 Anechoism performance mapping based on Representation Type 2, showing |C A |-maps in the ( (ρ), (κ)) coordinate system and values of (ρ) and (κ) that led to these |C A |, for the frequency f = 5 kHz and the medium length L = 6 cm.

Figure E. 5

 5 Figure E.5 is used to obtain an expression of the anechoism coecient based on the partial reection at an interface and on the transmission coecient through one layer of medium of thickness L, respectively r and t, which themselves are expressed using k e and Z e , as per:

Figure E. 5

 5 Figure E.5 Multiple reections and transmission through a layer in an anechoic conguration.

  One branch is very close to the diagonal of the map, while the other blue branches are condensed close to the (κ) = 0 axis. Maps have been drawn in Figure E.6 for other values of k w L, around 2 and 3.5, to observe how these branches evolve. They dier greatly as a function of this parameter, therefore as a function of the frequency and of the thickness. Their variations for dierent k w L can be explained by the fact that these low |C A | are related to the resonances of the slab and therefore depend on the thickness and the frequency.

Figure E. 6

 6 Figure E.6 |C A |-map for k w L = 2.09 (left) and k w L = 3.44 (right).

Figure E. 7 ,

 7 minima of |C A | have been plotted in the ( (kL/pi), (z))-plan. The horizontal grey line shows the impedance of water, corresponding to a real normalised impedance (z) = 1. In the |C A | map, the branches of low anechoism can be seen. There is a clear distinction between the areas with higher and lower impedances than that of water.

Figure E. 7 2 π 2 4f 2 L

 7222 Figure E.7 Type 2 Representation for the |C A |-map in the ( (kL/pi), (z)) coordinate system where the grey dashed lines show the impedance of water.

Figure E. 8 Figure E. 9

 89 Figure E.8 |C A |-map in the ( (ρ), (κ)) coordinate system highlighting four couples ( (κ), (ρ)) by white circles. The grey line shows the delimitation between areas z < 1 and z > 1. The red lines are plotted according to Equation (E.11) for m = 0, m = 1 and m = 2.

Figure E. 10 1

 10 Figure E.10 Type 2 representation for the Quadrant 1, showing |C A |-maps in the ( (ρ), (κ)) coordinate system and the values of (ρ) and (κ) that led to these |C A |.

Figure E. 12

 12 Figure E.12 Type 1 representation, showing |C A |-maps in the ( (ρ), (κ)) coordinate system for the four couples ( (κ), (ρ)) represented in Figure E.11 by the white circles and highlighted here as well. The white areas break the passivity constraint.

Figure E. 13

 13 Figure E.13 Type 2 representation of the Quadrant 4, showing |C A |-maps in the ( (ρ), (κ)) coordinate system and the values of (ρ) and (κ) that led to these |C A |.

Figure E. 14

 14 Figure E.14 Layer arrangement for the selected unit cell.

Figure E. 15

 15 Figure E.15 Eective bulk modulus and eective mass density for the selected unit cell. Seven frequencies are selected and highlighted by the red dotted lines.

1

  Figure E.16 Anechoism, reection and transmission coecients for a 3-unit medium.

Figure E. 17

 17 Figure E.17Frequency paths for a medium of length L = 3 × L u = 4.8 cm at the frequencies (a) 6 kHz, (b) 9.7 kHz, (c) 12.3 kHz, (d) 13.7 kHz, (e) 14.7 kHz, (f) 16.5 kHz and (g) 20 kHz. On the -maps (left hand side), the white circle shows the values of (κ) and (ρ) obtained for the unit cell in Figure E.14. The red line represents the complete path. In the -map obtained for these values (right hand side), the white circle shows the values of (κ) and (ρ) obtained for the unit cell and the value of the anechoism coecient.

Figure E. 18

 18 Figure E.18 Scores of the 16 possible designs and layer arrangements of congurations in the Pareto front.

Figure E. 19

 19 Figure E.19 Layer arrangement for the selected unit cell.

1

  Figure E.20 Anechoism, reection and transmission coecients for a 3-unit medium.

Figure E. 21

 21 Figure E.21 Eective bulk modulus and eective mass density for the selected unit cell. Seven frequencies are selected and highlighted by the red dotted lines.

Figure E. 22

 22 Figure E.22 Frequency path for a medium of length L = 3 × L u = 4.8 cm at the frequencies (a) 4 kHz, (b) 10 kHz, (c) 13 kHz, (d) 15.7 kHz, (e) 18 kHz, (f) 20 kHz and (g) 24 kHz. On the -maps (left hand side), the white circle shows the values of (κ) and (ρ) obtained for the unit cell in Figure E.19. The red line represents the complete path. In the -map obtained for these values (right hand side), the white circle shows the values of (κ) and (ρ) obtained for the unit cell and the value of the anechoism coecient.

  For frequencies (a) to (d), the real parts of the eective properties are in Quadrant 1 in the area where z > 1. Amongst these frequencies, (a), (b) and (d) are located on a slab resonance branch. For frequencies (b) and (d), the imaginary parts at these frequencies are also well located, close to the isoline of null |C A |. As a result, a very low anechoism coecient is obtained for these two frequencies, especially for the frequency (d) with |C A | = 0.15. For the frequency (c), the real part is slightly o the branches and very low values for the imaginary parts are obtained. The anechoism coecient is relatively good, but could be improved with higher values of the imaginary parts (κ).

Table 2 .

 2 1 Density ρ and longitudinal speed of sound c l for the silicone and aluminium.Damping is expressed using the loss factor η as a percentage of the speed of sound.

	Silicone rubber	1250	1000	2
	Aluminium	2700	6200
	Four dierent versions of the unit cell are considered, represented in Figure 2.4 and for
	which layer thicknesses are given in Table 2.2. Two of these designs are symmetric and
	the two other are non-symmetric. The total thickness of silicone is always 30mm and the
	total thickness of aluminium is 10mm.
	(a)	(b)		(c)	(d)
	Figure 2.4 Schematic diagram of the two symmetric unit cells (a) and (b) and the two
	asymmetric unit cells (c) and (d). Silicone rubber is represented by the orange layers and
	aluminium is represented by the grey layers.
		Layer 1		Layer 2	Layer 3
	Version (a)	Silicone		Aluminium	Silicone
	(mm)	15		10	15
	Version (b)	Aluminium	Silicone	Aluminium
	(mm)	5		30	5
	Version (c)	Silicone		Aluminium
	(mm)	30		10
	Version (d)	Aluminium	Silicone
	(mm)	10		30
	Table 2.2 Layer arrangements and thicknesses for the four versions of the unit cell, for
	which the total thickness is L u

Table 3 .

 3 1 Optimisation Parameters for the algorithm validation.

	Optimisation Parameter	Set 1	Set 2	Set 3	Set 4	Set 5	Set 6
	Total thickness L (m)			0.06
	Layer thickness l (m)			0.004
	Number of layers n l			15
	Frequency range (kHz)			[3 -20]
	Population size N	150	300	150	150	150	150
	Children created by Crossover C % (%)	80	80	60	80	80	80
	Children created by Mutation 100 -C % (%)	20	20	40	20	20	20
	Mutation probability m	0.02 0.02 0.02 0.005 0.02 0.02
	First rank ratio r 1	0.3	0.3	0.3	0.3	0.5	0.3
	Maximum number of Generations G max	300	300	300	300	300	150

.4, Set 2 is used to assess the inuence of parameter N , Set 3 for C % , Set 4 for m, Set 5 for r 1 , Set 6 for G max .

  .7.

	1					1
	0.8						0	0.005	0.01	0.015	0.02	0.025	0.03	0.035	0.04
						2
	0.6
							0	0.005	0.01	0.015	0.02	0.025	0.03	0.035	0.04
	0.4					3
	0.2						0	0.005	0.01	0.015	0.02	0.025	0.03	0.035	0.04
						4
	0 0	5	10	15	20	25	0	0.005	0.01	0.015	0.02	0.025	0.03	0.035	0.04
			Frequency (kHz)							Thickness (m)

  .13.

	1					1
	0.8						0	0.005	0.01	0.015	0.02	0.025	0.03	0.035	0.04
						2
	0.6
							0	0.005	0.01	0.015	0.02	0.025	0.03	0.035	0.04
	0.4					3
	0.2						0	0.005	0.01	0.015	0.02	0.025	0.03	0.035	0.04
						4
	0 0	5	10	15	20	25	0	0.005	0.01	0.015	0.02	0.025	0.03	0.035	0.04
			Frequency (kHz)							Thickness (m)

  .15. Anechoism and hull decoupling coecients are presented in Figure 3.18.

	1
	0	0.005	0.01	0.015	0.02	0.025	0.03	0.035	0.04
	2
	0	0.005	0.01	0.015	0.02	0.025	0.03	0.035	0.04
	3
	0	0.005	0.01	0.015	0.02	0.025	0.03	0.035	0.04
	4
	0	0.005	0.01	0.015	0.02	0.025	0.03	0.035	0.04
					Thickness (m)

Table 4 .

 4 X 2 (cm) d X 1,2 (cm) d X 2,3 (cm) θ ( • ) 2Conguration parameters tested to assess limitations of the 3-point method.

	Test 1	24	1	1	25.6
	Test 2	15	10	10	16.7
	Test 3	5	0.5	0.5	5.7

•

, where hydrophone positions are separated from each other by only 0.5 cm. Test 3 then satises the hypothesis on the edge-diracted waves. d

Table 5

 5

	.1.
		ρ	E	ν
		(kg.m -3 )	(Pa)
	Steel	7800	2.15 × 10 11	0.31
	Polyurethane	1100	9.9045 × 10 7 0.49333
	(loss factors)		12 %	0.17 %

Table 5

 5

.1 Density ρ, Young modulus E and Poisson ratio ν for steel and polyurethane. Loss factors for the polyurethane are expressed as a percentage of the Young modulus and Poisson ratio.

  Case study: Hard Inclusions . . . . . . . . . . . . . . . . . . . . 139 6.2.1 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6.2.2 Homogenisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 6.2.3 Parametric Study . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.3 Case study: Voided Inclusions . . . . . . . . . . . . . . . . . . . 147 6.3.1 Homogenisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 6.3.2 Variant Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 149 6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 , the objective here is to dene the eective medium that would replace an asymmetric unit cell. A Willis uid model is selected to describe the eective medium, thus characterised by an eective wavenumber k, a Wills characteristic impedance Z W and an asymmetry coecient W , as previously presented in Chapter 2. Propagation in a Willis uid can be represented by a transfer matrix of the form:

	6.1 Methodology
	6.1.1 Willis Fluid Homogenisation Model
	Similar to the homogenisation theory for one-dimensional multilayered structures pre-
	sented in Chapter 2
	133

2. A rst case study is presented, consisting of an asymmetric version of the locally resonant medium with hard inclusions studied in Chapter 5. This case study is completed by a parametric study that shows the inuence of o-centering the resonant inclusion on the variations of the eective parameters and acoustic performances. A second case study focuses on an asymmetric voided medium in which Willis coupling is not introduced by o-centering the inclusion as in the rst case study, but by combining two symmetric layers. The eects on the eective properties and acoustic performance are examined. Contents 6.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 6.1.1 Willis Fluid Homogenisation Model . . . . . . . . . . . . . . . . 134 6.1.2 Direct Inversion Method . . . . . . . . . . . . . . . . . . . . . . 134 6.1.3 Dierential Method . . . . . . . . . . . . . . . . . . . . . . . . . 136 6.2

  S 22 -S 11 S 22 +S 12 S 21 +1 2S 21 Z f S 11 +S 22 +S 11 S 22 -S 12 S 21 +1 2S 21 1 Z f -S 11 -S 22 +S 11 S 22 -S 12 S 21 +1 2S 21 -S 11 +S 22 -S 11 S 22 +S 12 S 21 +1 2S 21 .

	(6.2)

  Results for Anechoism . . . . . . . . . . . . . . . . . . . . . . . 167 7.3.1 Approach Validation . . . . . . . . . . . . . . . . . . . . . . . . 167 7.3.2 Low Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . 169 7.4 Performance Levels . . . . . . . . . . . . . . . . . . . . . . . . . 174 7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

	7.1 Optimisation Approach
	157

7.1 Optimisation Approach . . . . . . . . . . . . . . . . . . . . . . . 158 7.2 Database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 7.2.1 Presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 7.2.2 Inuence of the Inclusion Shape . . . . . . . . . . . . . . . . . . 161 7.2.3 Inuence of the Inclusion Size . . . . . . . . . . . . . . . . . . . 164 7.3

  Thesis Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 9.2 Perspectives for Future Work . . . . . . . . . . . . . . . . . . . 203

	9.1 Thesis Summary
	Chapter 9
	Conclusion and Perspectives
	Contents
	9.1 199

  S 11 + S 22 + S 11 S 22 -S 12 S 21 + 1 2S 21 , -S 11 -S 22 + S 11 S 22 -S 12 S 21 + 1 2S 21 , 11 + S 22 -S 11 S 22 + S 12 S 21 + 1 2S 21. face or on the other, which means that S 11 = S 22 = R. Reciprocity implies that S 21 = S 12 = T . Elements of the transfer matrix can thus be simplied as follows:

				(A.11)
	C =	1 Z f	(A.12)
	D =	-S (A.13)
	For a symmetric medium, scattering coecients are identical whether the acoustic wave is
	incident on a

A = S 11 -S 22 -S 11 S 22 + S 12 S 21 + 1 2S 21 , (A.10) B = Z f

premiers mois de ma thèse dans les locaux de Naval Group, à Ollioules. Je tiens particulièrement à remercier Christian Audoly, mon superviseur industriel, pour avoir mis en place ce projet pour m'avoir donné la chance de réaliser cette thèse. Son investissement et son regard avisé ont été d'une aide précieuse tout au long de la thèse. Je souhaite aussi remercier mes collègues de Naval Group qui étaient là dès mes débuts et qui m'ont chaleureusement accueilli, Rania, Valentin, Ygaäl, Patrick, Gilles. Merci encore pour ce poster de Lorie qui m'a suivi jusqu'en Australie, et qui est désormais l'egie de l'open space du CEMIS/AC. Je remercie aussi Pierre et Arthur, du CESMAN, pour leur aide à la réalisation des panneaux acoustiques. UNSW, Sydney I then ew to Sydney, where I was warmly welcomed by my supervisor Nicole Kessissoglou.I am deeply grateful for her support and comments, which have helped me a lot with my editing style. I would also like to thank the group of Nicole: Daniel, Gyani, David, Mahmoud and Jerey, with whom I enjoyed chatting from time to time around the open space. Among those I met at UNSW, I especially want to thank Sean, for the enjoyable discussions around a coee in the morning. Above all, thanks to Jeorey, who was friendly from day one and who made my integration in Sydney so much easier and more fun.My stay in Sydney wouldn't have been the same without the good friends I have made.

Acknowledgements

To begin with, I would like to express my gratitude to the members of my defense committee, Valentin Leroy, Thomas Brunet, Bertrand Habert, Alex Skvortsov, Jean Dassé, Charles Croënne, Olivier Bou Matar and my 3 supervisors, for their presence at the defense and for the interesting exchanges that followed. In particular, I would like to thank the two reviewers, Valentin Leroy and Thomas Brunet, for their careful reading of the manuscript and for their valuable feedback. Since I spent half the time of my doctorate in France and the other half in Australia, I must continue my thanks in both languages.

Naval Group, Ollioules

Je commence avec la France, et plus particulièrement dans le sud, où j'ai passé les

The left hand side of Equation (6.8) corresponding to the product M n+1 (M n ) -1 can be calculated in terms of the scattering responses of the two segmented media, using Equation (6.7). The transfer matrix M u is then used to derived the eective wavenumber k diff as per:

where m and Λ are chosen as in the direct inversion method. As for the symmetric case, this expression is independent of the expression of the transfer matrix elements.

The determination of the Willis characteristic impedance and the asymmetry coecient is less straightforward than the determination of the eective wavenumber. The diculty here comes from the fact that, unlike the direct inversion method, the eective parameters for the dierential method must be obtained without considering the transfer matrix form given by Equation (6.1). This was relatively easy when the unit cell was symmetric since

Nevertheless, the asymmetry of the unit breaks all these equalities. Therefore, eective impedances for forward propagating waves and for backward propagating waves are rst obtained, in dissociated ways, as developed below.

The transfer matrix M u calculated as per Equation (6.8) is used to obtain the transfer matrix from which the eective impedance for forward propagating waves is derived, as per Figure 6.3(a). The transfer matrix M u derived using Equation (6.8) is thus now referred as M + u . A distinction is indeed necessary as it is found that the transfer matrix of the unit changes depending on the orientation of the unit cell. As such, the transfer matrix of the reversed-orientation unit shown in Figure 6 The matrix M + u is then used to calculate the eective impedance for forward propagating waves Z + , while the transfer matrix M - u leads to the eective impedance Z -for backward propagating waves.

Above the Cut-o Frequency Direct Calculation

Above the rst cut-o frequency, the acoustic response is no longer specular and diracted waves convert from evanescent to propagating. Since the reected and transmitted pressures are written as an innite sum of waves diracted by the grating (Equations (8.1) and (8.2)), the approach used to calculated the reection and transmission coecients is The expression of the contribution coecient for the 0 th mode is rst given by :

where c m 0 is an elementary contribution calculated on each segment [y m ,y m+1 ] using the pressures p m and p m+1 measured/calculated at these points as per:

(8.9)

The contribution coecient of the n th mode for n = 0 is:

Summary

The Alberich panel containing a grating of voided cylinders has been characterised experimentally. The post-processing of the experimental data has been divided into two dierent approaches: below the rst cut-o frequency and above the rst cut-o frequency. Below the rst cut-o frequency, the 3-point method is applied as for a homogeneous panel.

A preliminary analysis was carried out to ensure that the hydrophones were located far enough from the panel so that the evanescent waves were no longer discernible. Results

were consistent with predictions, except around the rst cut-o frequency, where the source resonance and the ±1 modes invalidated the results. Above the cut-o frequency, the ±1 modes are no longer evanescent and must be calculated to fully characterise the panel. For this purpose, the 5-point method was implemented and led to consistent results.

Nevertheless, the frequency range above the cut-o frequency is such that geometric irregularities and local uncertainties in material properties further inuence the results, thus leading to dierences with the nite element model of exact geometric and material properties. In addition, the panel is not innitely periodic as in the FE model. Finally, it was shown by the signicant divergence of the direct calculations from the FE predictions that it is crucial to take into account the edge-diracted pressure as in the 3-and 5-point methods. Therefore, both the 3-and 5-point methods appear as a robust and unique way to characterise panels embedded with periodic inclusions above the rst cut-o frequency.

Appendix A Transfer Matrix

A.1 Denition

Transfer matrix M for one-dimensional acoustic wave propagation through a layer relates acoustic pressure p and normal particle velocity v of the uid on either side of the medium, as follows: In the eld of Acoustics, physical quantities (p,v) are commonly used, rather than (p,u), u being the displacement. Nevertheless, the transition from the quantity u to v is possible with a derivative, equivalent to multiplying by iω for harmonic problems.

A.2 Expression with S-parameters

Elements of the transfer matrix can be expressed using S-parameters. S-parameters and two-port network notions are widely used in electrical and microwave engineering. These 205

(A.17)

A.3 Expression with Material Properties

Considering the time convention e iωt , expressions of the transfer matrix elements are found by decomposing the acoustic eld into forward and backward-propagating waves, as shown in Figure A.1. Pressures and particle velocities on both sides of the medium of length h are written at x = 0 and x = h as:

where k + and k -are the wavenumbers for forward and backward-propagating waves respectively and where Z + and Z -are the impedances for forward and backward-propagating waves respectively.

Wave amplitudes A + and A -are extracted from system (A.18) using the rst two lines, which leads to:

Those expressions for wave amplitudes A + and A -are next injected in the last two lines of system (A.18) so that p right and v right are expressed as a function of p left and v left . These two lines can be rewritten in a matrix form as follows:

This system can be reversed to obtain the transfer matrix expression according to Equation (A.1), which nally leads to:

For a uid medium, the material properties are meant to be the same whether the acoustic wave is forward or backward propagating. However, the distinction is voluntarily kept to in order to apply to an asymmetric uid-like medium, which is one of the homogenisation models considered in the manuscript. However, even if the medium is asymmetric, it will always be reciprocal (ie S 12 = S 21 ) and consequently k + = -k -= k. This equality can be proved using the ABCD transfer matrix expressed with Equations (A.10) to (A. [START_REF] Li | Double-negative acoustic metamaterial[END_REF]), for which the determinant is equal to 1 when S 12 = S 21 as well as using the fact that the determinant of the transfer matrix form given by Equation (A.21) is also equal to 1 if and only if k + = -k -. Hence, there is no need to keep the distinction between k + and k -.

The transfer matrix for an asymmetric uid-like medium is then given by:

Z + e -ikh -Z -e ikh .

(A.22)

Moreover, if the medium is symmetric, then Z -= -Z + = -Z. The transfer matrix expression of Equation (A.21) can then be simplied as follows:

M = cos(kh) iZsin(kh) iZ -1 sin(kh) cos(kh) .

(A.23)

A.4 Expression for Determination of Willis-parameters

For Willis-type systems, the eective properties may depend on the direction of wave propagation. For example, the eective impedance Z + , mentioned in the previous section, is the eective impedance of a medium for forward-propagating waves. In the publication of Muhlestein et al. [START_REF] Muhlestein | Experimental evidence of willis coupling in a one-dimensional eective material element[END_REF], the eective impedances Z + and Z -are referred to as specic impedances for a Willis-type system. Considering the time convention e iωt , the specic acoustic impedances in the ± directions may be written as:

where W = ωψ/Z W the complex asymmetry coecient, where ψ is the Willis-coupling coecient and where Z W is the complex characteristic impedance. Therefore, the transfer matrix expression given by Equation (A.22) can be changed considering the parameters Z W and W instead of the specic impedances Z ± . For this purpose, pressures and particle velocities on both sides of the medium of length h are written at x = 0 and x = h as:

.

(A.25)

Wave amplitudes A + and A -are extracted from system (A.25) using the rst two lines, which leads to:

.

(A.26)

The last two equations are then injected into the last two lines of the system (A.25) to obtain expressions of p right and v right according to p left and v left , as given by:

(A.27)

The latter system can be reversed to obtain the transfer matrix expression according to Equation (A.1), which nally leads to:

i sin(kh) Z W cos(kh) + W sin(kh) .

(A.28)

Appendix B

Eective Wavenumber

B.1 Expression

Wave propagation in an innite periodic structure can be studied by using the constitutive unit cell and periodic boundary conditions. The unit cell is dened as the smallest pattern that leads to the innite structure when it is endlessly repeated. Its transfer matrix for longitudinal modes M u is dened by:

The longitudinal eective wavenumber k is derived by applying the Bloch-Floquet relation on the unit cell of length L u . Such condition can be written as:

Equation B.2 is subtracted to Equation B.1, which yields:

Solving Equation (B.3) comes down to the calculation of the eigenvalues λ ± of the transfer matrix M u . The eective wavenumber is thus given by:

where m is an integer whose value is determined in such way that the eective wavenumber as a function of frequency is continuous. Moreover, for a forward propagating wave, eigenvalue λ is chosen so that the imaginary part of k is negative in order to comply with the material passivity requirement considering the phasor convention e i(ωt-kx) . In fact, the decomposition into real and imaginary parts of the wavenumber, respectively k and k for a forward propagating wave leads to: Consequently, expressions for the S-parameters are now given by:

where ∆ is:

215

Since ρ < 0 and κ > 0, the wavenumber is shown to be purely imaginary, which means that waves do not propagate and are purely evanescent.

Consequently, minima of the anechoism coecient are obtained for Quadrant 4 has then be studied. This Quadrant seems promising since there is a small area of low |C A | which seems to remain at this location independently of the frequency and medium thickness. This area is mainly concentrated near the (κ) = 0 axis. Consequently, to obtain a broadband performance, it would be interesting to tune the dispersion in order to be either next to a branch or below |z| < 1 in Quadrant 1 or close to the (κ) = 0 axis in Quadrant 4. In both cases, it has been observed that the values of the imaginary parts have a signicant impact. In the following section, two multilayered media are homogenised and their eective properties are derived. They are then used to create a "frequency path", so that one can follow the evolution of the anechoic performance based on the maps.

The frequency path is analysed along with the variations of the anechoism coecient for (d), it appears that, in terms of anechoic performance, being out of a branch with suitable imaginary parts may be as benecial as being on a slab resonance branch with unsuitable imaginary parts.

In This thesis presents analytical or semi-analytical models based on a transfer matrix formalism to homogenise periodic structures. These methods are then used in a computationally ecient optimisation approach to obtain metamaterial designs that meet a goal in terms of their anechoism coecient, hull decoupling coecient, or both. The entirety of this approach is rst carried out for multilayered structures then extended to metamaterials with macro-inclusions to which a degree of complexity is added. Finally, measurement methods in a water tank facility are implemented and validated. They are then used for the experimental characterisation of panels manufactured within the framework of the thesis.

Résumé

Concevoir des revêtements de coque pour la furtivité acoustique des véhicules submergés deumeure un dé évoluant constament avec l'amélioration des performances sonar. Les performances à basses fréquences sont particulièrement importantes pour les applications militaires, en raison des propriétés de propagation à longue distance dans l'eau des signaux à basses fréquences. Les métamatériaux acoustiques sont désormais largement reconnus comme des candidats prometteurs pour de tels revêtements. L'objectif de cette recherche est d'étudier les métamatériaux pour des applications sous-marines, de l'analyse théorique à la caractérisation expérimentale.

Cette thèse présente des modèles analytiques et semi-analytiques basés sur un formalisme de matrices de transfer pour homogénéiser des structures périodiques. Ces méthodes sont ensuite utilisées dans une approche d'optimisation peu couteuse en temps de calculs pour obtenir des designs de métamatériaux qui répondent à des objectifs en termes de coecient d'anéchoïsme, de masquage, ou les deux. L'ensemble de cette approche est d'abord réalisée pour des structures multicouches puis étendue aux métamatériaux à macro-inclusions auxquels s'ajoute un degré de complexité. Enn, des méthodes de mesure pour bassin acoustique sont mises en ÷uvre et validées. Elles sont ensuite utilisées pour la caractérisation expérimentale de panneaux fabriqués dans le cadre de la thèse.