N Guenfoud

C Droz

M Ichchou

O Bareille

B Pluymers

E Deckers

Modeling and experimental measurements of the sound transmission loss for multi-layer core topology systems

In this paper, we focus on multi-layer core topology systems which consists in piling up layers made by different kind of cores and offers innovative solutions to periodic structures design. The main objective of this study is to analyze the phenomena occurring in these multilayer core topologies, and then compare the numerical models with experimental measurements while the effects of the phase shift on the STL will be investigated. The model is implemented using MATLAB and ANSYS apdl. The mass of the unit cell is taken as constraints to eliminate its influence on the STL and the compression modulus is calculated to verify the mechanical properties. To perform the experimental measurements, the samples are designed with CATIA and then made by an industrial 3D printer using the fused deposed model technic. An impedance tube is used to measure the STL and compare with the vibroacoustic phenomena.

Introduction

A great challenge of these last decades is to make lightweight structures with efficient acoustical and mechanical performances for many industrial applications (aeronautics, aerospace, building construction, automotive, …). Sandwich panels with laminates, composite structures as well as porous media have been developed to propose new solutions. Alternatively, periodic structures are widely used in aerospace industry since the stiffness to weight ratio is low. In terms of mechanical properties, the rigidity of the core enhances the tensile and compression modulus of the whole structure. The core geometry creates an anisotropic material with several main directions. Under wave excitations and at some frequencies it will make occur specific characteristics of the structure. Periodic structures can be easily characterized by the unit cell and many models can be applied to investigate their vibroacoustic properties. The principle of the wave method firstly explained in [START_REF] Brillouin | Wave propagation in periodic structures[END_REF] has been implemented using Finite Element Method and became the Wave Finite Element Method (WFEM) described in [START_REF] Manconi | Modelling wave propagation in two-dimensional structures using a wave/finite element technique[END_REF] for 1D and 2D periodicity. Analytical formulation combined with homogenized formulation can be used to determine the Sound Transmission Loss as it is done in [START_REF] Zergoune | Assessments of shear core effects on sound transmission loss through sandwich panels using two-scale approach[END_REF] using the k-space obtained with the WFEM. Nevertheless, it is limited in terms of structure complexity due to the use of homogenized method. Besides, another approach using the Transfer Matrix Method ( [START_REF] Allard | Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials[END_REF]) is proposed in [START_REF] Parinello | Transfer Matrix representation for periodic planar media[END_REF] and reduce drastically the computational cost. In [START_REF] Christen | Acoustic transmission through composite shells with noise treatment: Modelling and sensitivity[END_REF], the method is based on nodal surfaces and uses a simple formulation to obtain the acoustic properties. Similarly, [START_REF] Yang | Prediction of sound transmission through, and radiation from, panels using a wave and finite element method[END_REF] manipulates the shape functions associated with the displacement of the panel to develop the formulation to obtain the acoustic properties. Recently, the wave-based method has been used in [START_REF] Deckers | Prediction of transmission, reflection and absorption coeeficients of periodic structures using a hybrid Wave Based -Finite Element unit cell method[END_REF] allowing to model any kind of acoustic excitations. By exploiting these models, the study of design space is done by changing the geometry of the core and by analyzing the influence on the STL. In [START_REF] Zergoune | Meso-macro approach for modeling the acoustic transmission through sandwich panels[END_REF][START_REF] Baho | Conception de structures sandwiches à fort pouvoir d'atténuation acoustique : analyse de sensibilité et optimisation[END_REF], by computing a parametric survey as well as an optimization, they showed that modifying the shape of the core can improve the sound properties or by altering some parameters of the core and keeping the same geometry, leading to the possibility to optimize the structure. These effects have been investigated in [START_REF] Griese | The effect of honeycomb core geometry on the sound transmission performance of sandwich panels[END_REF][START_REF] Mazloomi | Vibroacoustic of 2D gradient auxetic hexagonal honeycomb sandwich panels[END_REF] as well. In [START_REF] Droz | Vibro-acoustic optimisation of sandwich panels using the wave/finite element method[END_REF], several core geometries (rectangular, triangular, hexagonal, …) are compared to optimize the transition frequency, corresponding to a vibroacoustic indicators. Concerning multi-layer systems, [START_REF] Magnucka-Blandzi | Stability and vibrations of a metal seven-layer rectangular plate with trapezoidal corrugated cores[END_REF][START_REF] Lewinski | Determination of shear modulus of elasticity for thinwalled trapezoidal corrugated cores of seven-layers sandwich panels[END_REF][START_REF] Paczos | Experimental and numerical investigations of fivelayered trapezoidal beams[END_REF][START_REF] Magnucka-Blandzi | Strength of a metal seven-layer rectangular plate with trapezoidal corrugated cores[END_REF][START_REF] Magnucka-Blandzi | Buckling and vibrations of metal sandwich beams with trapezoidal corrugated coresthe lengthwise corrugated main core[END_REF][START_REF] Magnucka-Blandzi | Three-point bending of seven layers beamstheoretical and experimental studies[END_REF] have exploited the way to pill up several layers made of corrugated cores. These studies were mainly restricted for vibration and mechanical properties. In [START_REF] Wen-Chao | Sound insulation improvement using honeycomb sandwich panels[END_REF], the experimental measurement indicated the efficiency of a double-layer honeycomb panel compared to a single-layer especially in low frequency.

In this paper, a sandwich panel will be studied. By shifting the unit cell, it will offer the possibility to obtain a multi-layer that will be investigated in terms of mechanical and acoustic properties. Results will be validated by experimental measurements to conclude on the efficiency of making this type of structure.

Model

A sandwich panel with a standard rectangular core is used as a reference result. The principle is to make the new geometry by shifting a third of the unit cell in two directions x and y as illustrated in the Fig. 1. This layout allows to keep the mass of the unit cell constant. The total thickness of the unit cell is 24 mm (ydirection), 15 mm in the x-direction and 15 mm in the z-direction. The thickness of the skins is 2 mm and the thickness of the core is 0,7 mm. The following sandwich panels are made of ABS with E = 1,6 x 10 9 Pa, ρ = 985 kg/m 3 and ν = 0,33. The damping η is evaluated at 2 %. Theses values were obtained by characterizing the material.

A finite element analysis is made with the commercial software Ansys apdl. SHELL181 with a size of 1 mm is used to build the finite element model shown in Fig. 1.

a) b) c)

Sound Transmission Loss

The studied structure is excited by normal incident waves considered as plane waves. It is then divided in three kind of acoustic plane waves: the reflected, the absorbed and the transmitted waves. The sandwich panel separates two semi-infinite fluids with ρ0 = 1,25 kg/m 3 and c0 = 343,6 m/s. Finally, the Sound Transmission Loss (STL) is obtained using the Eq.1:

= 10 log10 i t W STL W    (1)
The method developed in [START_REF] Christen | Acoustic transmission through composite shells with noise treatment: Modelling and sensitivity[END_REF] uses the following assumptions:

• The structure is assumed as an infinite panel.

• Only real wavenumbers are considered.

• The structure is excited by plane waves.

• The angles of incidence and reflection are equal.

A modal analysis is performed with Ansys apdl with free boundary conditions. The stiffness and mass matrix are extracted from the unit cell and allow to calculate the dynamic stiffness. The WFEM is then used to apply the Bloch-Floquet theory. The displacement on the incident and transmitted side and the external forces applied on the structure are linked with the dynamic stiffness and yields to Eq. 2:

ii it i i ti tt t t b b u e b b u e       =            (2)
with ui and ut the node displacement on the incident and transmitted side whereas ei and et are the external forces applied on the structure, respectively. Considering the nodal surfaces of the incident and transmitted side (called Si and St) of the unit cell, the external forces applied on the nodes can be rewritten as the product of the nodal surfaces multiply by the pressure evaluated on the node.

The continuity of normal particle velocity at the interfaces gives: ² ( )

² i y I R t y T u ik p p u ik p   = - -    =-   (3)
ky propagative constant in z-direction, pi, pr and pt the incident pressure, the reflected and the transmitted wave, respectively.

Then, a matrix equation between the acoustic pressures of each side is obtained and becomes:

0 0 0 i ii it i ii R I T t ti tt ti S bb S b Y p Y p p S bb b Y  +-   -    =      --    (4)
Consequently, by solving the Eq. 4 the reflected and transmitted pressures are known and allow to calculate the STL. It is easy to implement the methodology with MATLAB since it consists on solving a matrix equation. This method is applied for the three geometries and compared on the following Fig. 2. The frequency range is chosen from 100 Hz to 1600 Hz corresponding to the valid frequency range of the impedance tube. It is occurred that shifting the core improves the sound transmission loss and shifts the coincidence frequency to higher frequencies. The low mass is kept for a larger frequency band. The x-shift sample turns out being the best layout with the best acoustic performances. Indeed, the more the coincidence frequency will be in higher frequencies the more the structure will be efficient and reliable in terms of sound transmission loss. Finally, it seems that the shift in y-direction decrease the acoustic efficiency of the sandwich panel.

Compression Modulus

Since the geometry of the core changes, it has been chosen to calculate the compression modulus as a relevant indicator to verify the mechanical properties of the sandwich panel. The same boundary conditions as a compression machine are applied: 1. only the displacement in y-direction is free 2. A unit force is applied on the top skin 3. All nodes of the top skin are constraint to have the same displacement.

The commercial software Ansys apdl and CATIA are used to model the unit cell. SHELL181 is still used with Ansys apdl whereas Solid Elements are chosen for CATIA. The finite element models are illustrated in Fig. 3 and the result obtained shown in Table 1. Table 1 : Calculation of the compression modulus

The differences between the two software could be mainly due to the element used since with solid elements the physics is better captured and closer to the reality. Nevertheless, the conclusion remains the same for both simulations: the x-Shift sample occurs to have the highest compression modulus. Indeed, adding the phase shift lead to well distribute the load on the skins and increase the rigidity of the whole sandwich panel. Shifting the unit cell in the y-direction creates nodes contact at some locations and drastically decrease the compression modulus. Finally, the x-Shift sample appears to be the optimize structure to obtain the best acoustic mechanical performances and keeping the mass constant. These results still need to be validated by measurements.

Experimental Measurements

Impedance tube

The experimental measurements were performed to verify the efficiency of the optimize structure as expected from the modeling part. A four microphones B&K standard wave tube (Fig. 4) with the two loads method was used. The loudspeaker is located at the end of the tube generating a random noise signal in the frequency range from 200 Hz to 1600 Hz. The B&K 4206 large tube with a diameter of 100 mm was chosen.

The frequency response functions are measured with the four microphones and all transfer functions are computed. Two measurements are needed to calculate the sound transmission loss: 1.

Measurement with an open tube termination 2. Measurement with an anechoic termination made using 4 standards sound absorbing samples with a 100 mm depth in total.

Manufacturing process using the 3D printing technic

Both samples were made using a the FORTUS 250, a 3D printer using the fused deposed model technic making the structure layer by layer. The 3D printer can add two kinds of matter: the model material, corresponding to the material of the structure, and the support material helping to build the structure, but which will be removed after immerging the structure inside a solution.

The samples were firstly designed with CATIA and then preprocessed using INSIGHT as illustrated in Fig. 4. To improve the quality of the structure it is needed to add the support material. The thickness of the deposed rob is 0.3556 mm while the thickness of each layer is 0.1778 mm. The final samples are shown in Fig. 6. Since the structure is rigid and it is difficult to accurately reach 100 mm to fit inside the tube, it has been chosen to manufacture the sample with a diameter of 97 mm to overcome the uncertainty of the manufacturing process. Then, to set up the sample inside the tube, a circular joint made of rubber and scotch tape (Fig. 7) was made to avoid air leakage. The weight of both samples was measured and listed in Table 2. Without the joint the difference between both samples is 1 % while adding the joint it reaches 1,7 %. These differences are neglectable and will not affect the mass low corresponding to low frequencies.

Without rubber joint

Results

The obtained result with the impedance tube are shown in Fig. 8. Three measurements were done with the standard case to verify the repeatability. As it was expected the x-Shift sample reveals the best acoustic performance. Before 1000 Hz, the difference is mainly due to the boundary conditions that are not perfectly respected and experimental conditions including measuring errors by microphones. After 1000 Hz, the x-Shift shows a great efficiency and reaches more than 7 dB better than the standard sandwich panel.

Conclusion

This work proposes a new way to design sandwich panels creating multi-layer core topology systems. Three samples were designed by shifting a third of the unit cell in two directions. They were modeled under normal incident waves using the Wave Finite Element Method and then compared in terms of acoustic and mechanical properties. The sound transmission loss and the compression modulus were chosen as acoustic and mechanical relevant indicators.

The model revealed the possibility to increase the compression modulus of the sandwich panel as well as the sound transmission loss by shifting the unit cell in the x-direction and keeping the mass constant. A 3D printer was used to manufacture the samples and then, a joint made of rubber was added to ensure the samples to fit inside the impedance tube without air leakage. The measurements were done using a Tube de Kundt from B&K with a diameter of 100 mm and shown the efficiency of shifting a part of the unit cell. Further studies can be carried out to predict the efficiency of any type of unit cells and an optimize structure could be obtained.

Figure 1 :

 1 Figure 1 : a) Standard; b) x-Shift c) xy-Shift

Figure 2 :

 2 Figure 2 : Comparison of the Sound Transmission Loss of three sandwich panels made of three different unit cell

Figure 3 :

 3 Figure 3 : Finite element model a) Ansys apdl b) CATIA

Figure 4 :

 4 Figure 4 : Impedance tube (B&K)

Figure 5 :

 5 Figure 5 : (a) Samples designed using CATIA (b) Preprocessing using INSIGHT

Figure 6 :Figure 7 :

 67 Figure 6 : Printed samples

Figure 8 :

 8 Figure 8 : Comparison of the sound transmission loss with the impedance tube (B&K)

Table 2 :

 2 Samples weigth

	With rubber joint

Acknowledgments

This project has received funding from the European Union Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No. 675441. The author would like to acknowledge all the Institutions and Partners involved in the VIPER project.


N Guenfoud

C Droz

M Ichchou

O Bareille

E Deckers

W Desmet

PARAMETRIC ANALYSIS OF A TRIPLE CORE PERIODIC UNIT CELL INCLUDING A MIDDLE RANDOM CORE -SMART 2019

Keywords: multi-layer core, sandwich panels, transmission loss, periodic structures

In the last decades, the main issue concerning structures is related to weight constraint. Many solutions are available as composites or sandwich panels. Nevertheless, it seems that metamaterials such periodic structures focus a specific attention nowadays. In this field, honeycomb structures remain wide used solutions especially in the aerospace industry, but they exhibit a high stiffness-to-weight ratio leading to poor acoustic properties. Consequently, some new designs are created to obtain better acoustic indicators. In this context, this paper proposes to study a new kind of periodic structures using multi-layer core topology systems consisting on stacking layers made of different geometry of cores (auxetic, hexagonal, rectangular…). Several new parameters should be considered to fully understand the dynamic and acoustic behaviour. In this paper we will limit the study to rectangular core combined with a random core located in between. Consequently, we will stack 3 kind of cores: Rect-Random-Rect. The main objective of this study is to analyse the results of the parametric survey. All configurations are constraint to have the same surface density. The chosen acoustic indicator is the Sound Transmission Loss (STL). The transition frequency will be also investigated as well as the compression and shear modulus and the bending waves. Modelling these structures is possible using either Wave Finite Element Method (WFEM) only or combining WFEM and the Transfer Matrix Method (TMM). We assume an infinite panel, real wavenumbers and that the structure is excited by plane waves with the angles of incidence and reflection equal. The model is implemented using MATLAB and ANSYS apdl. Because such structures are made using a 3D printer, the ABS is used as the material for the study. From the parametric study, it turned out that multi-layer core systems correspond to relevant solutions to highly improve the STL. Moreover, it is possible to find out an optimized design giving the best STL. However, as it will be shown, it goes along with less efficiency in terms of mechanical properties and thus should be relevant for some specific applications.

INTRODUCTION

Lightweight structures including composite panels, sandwich panels or porous media are still widely used in the industry due their low mass. Besides, sandwich panels made of periodic cores exhibit a great interest having a low stiffness to weight ratio but leading to unsatisfactory acoustical properties. Many researches are carried out to provide them better mechanical and acoustical performances according to the industrial application. Proposed designs in the literature begin from simple cases as composite panels, studying the stacking sequences between layers to improve their mechanical efficiency [START_REF] Roque | Maximization of fundamental frequency of layered composites using di_erential evolution optimization[END_REF][START_REF] Reddy | Mechanics of laminated composite plates and shells: theory and analysis[END_REF], up to more complex structures using more exotic shapes and/or adding add-ons [START_REF] Droz | Vibro-acoustic optimisation of sandwich panels using the wave/finite element method[END_REF][START_REF] Han | Performance enhancement of sandwich panels with honeycomb-corrugation hybrid core[END_REF][START_REF] Mazloomi | Vibroacoustics of 2d gradient auxetic hexagonal honeycomb sandwich panels[END_REF][START_REF] De Melo Filho | Dynamic massbased sound transmission loss prediction of vibro-acoustic metamaterial double panels applied to the mass-air-mass resonance[END_REF], especially within the periodic media field. More recently, multi-layer core topology systems have been developed by stacking different layers of cores (auxetic, hexagonal, rectangular, random geometry) to operate with new parameters. A double-layer honeycomb panel has been studied in [START_REF] Chung-Fai | Sound insulation improvement using honeycomb sandwich panels[END_REF] to drastically reduce the effect of the critical frequency of the Sound Transmission Loss (STL). Besides, corrugated and truss cores are stacked for building applications [START_REF] Hou | Experimental and numerical studies on multi-layered corrugated sandwich panels under crushing loading[END_REF][START_REF] Magnucka-Blandzi | Strength of a metal seven-layer rectangular plate with trapezoidal corrugated cores[END_REF][START_REF] Xiong | Compression and impact testing of two-layer composite pyramidal-core sandwich panels[END_REF] trying to reach a more desirable impact resistance. Most of time, authors add damping layers separating the cores. These last designs are mainly considered for their capacity to absorb energy for specific mechanical applications but not employed as concerns acoustics. It is then clearly expected that more complex structures will continue to be investigated in the future to obtain improved properties. Consequently, this paper intends to analyze multi-layer core topology systems within the framework of periodic structures and to perform a parametric survey with the involvement of new parameters. Such approaches have been already completed for vibroacoustic properties in standard cases using honeycomb cores [START_REF] Baho | On global bending-shear core transition e_ects for the vibroacoustic of sandwich structures: Analytical and numerical investigations[END_REF][START_REF] Zergoune | Assessments of shear core e_ects on sound transmission loss through sandwich panels using a two-scale approach[END_REF] or by changing gradually the core from a honeycomb to an auxetic unit cell [START_REF] Griese | The effect of honeycomb core geometry on the sound transmission performance of sandwich panels[END_REF][START_REF] Galgalikar | Design optimization of honeycomb core sandwich panels for maximum sound transmission loss[END_REF]. They have shown up the influence of geometrical parameters on vibroacoustic indicators (transition frequency, the modal density, the group velocity) and the STL. All these indicators reveal a better comprehension of the phenomenon leading to the improvement of the acoustical performances. In this paper, the parametric survey will be characterized by a study of the geometrical parameters of one layer over three opening new configurations and new perspectives not limited as one-core unit cell structures and leading to a high improvement of the STL.

MULTI-LAYER CORE TOPOLOGY SYSTEMS

Description

Multi-layer core topology systems are made of several layers of periodic structures (hexagonal, auxetic, rectangular, …), Fig. 1. Such structures involve new parameters to be considered to figure out the dynamic and the acoustic behavior. These parameters are listed as following:

1. The phase shift between layers 2. The rotation between layers 3. The interaction between geometrical parameters of each layer giving the possibility to modify the shape of the unit cell for each layer. 4. The size of the unit cell for each layer. 5. The periodicity of each layer.

The phase shift and the rotation make occur lower rigid contacts between layers leading to new phenomena which tend to affect the transition frequency and decrease the compression modulus of the structure. Nevertheless, their impact on the STL still need to be verified. The phase shift is possible along the direction x and y. In addition, the rotation between layers produces a nonperiodic structure. The existing methods using the Wave Finite Element Method (WFEM) for the calculation of vibroacoustic indicators are not applicable. Finally, they offer a great added value by keeping the mass constant. Changing the geometrical parameters for each layer turns out the possibility to have different shape for each layer, and then, combining a hexagonal core with an auxetic or a rectangular core, or more generally random cores and increasing the number of possibilities and the complexity of the unit cell. Ultimately, the size of the unit cell and the periodicity for each layer are linked. Indeed, the only way to extract a unit cell in this configuration comes out when each layer has a unit cell size multiple to each other and without rotation. Resulting from these parameters, it is then not always possible to extract the unit cell through the depth of the structure. The entire sandwich panel must be modeled and a high computational is expected. Therefore, some listed parameters will be constrained to have the opportunity to use the WFEM and to apply the model allowing to obtain the STL.

Parametric model

The illustrated parametric model (Fig. 2) will be used for the study. Each layer has the same unit cell size and no rotation is considered. The main purpose in this study is to identify the geometrical influence of the shape of the core. The thickness of the core and the skins as well as the height of each layer will be not altered since it is shown in [START_REF] Han | Performance enhancement of sandwich panels with honeycomb-corrugation hybrid core[END_REF] that they have a strong influence on the mechanical and acoustical properties. Thereby, by keeping the mass constant, only parameters defined in the parametric model influence the STL. An algorithm, developed in the lab, gives the possibility to obtain all configurations having the same surface density for a given size of unit cell. Since the size of the unit cell is maintained and the depth of each layer are equal and the parameter n is set to 1/2, only 5 parameters are modified: a1, a2, a3, α and β. However, to represent correctly the result coming from the parametric survey it is necessary to reduce the number of parameters. Consequently, it is decided that a1 = a2. The size of the unit cell is fixed to Lx = 15 mm and Ly = 15 mm. A sandwich panel with 3 layers is considered. The top and bottom layers correspond to a rectangular core while the middle core is a random core defined by the set of parameters. The depth of each layer is 5 mm. The thickness of the skins and the core is 0,6 mm. The sandwich panel is made of ABS since the 3D printing technic is commonly used to manufacture these structures, with E = 1,8 x 10 9 Pa, ρ = 985 kg/m 3 and υ = 0,33. The damping η is evaluated at 2 %. These values were acquired using a DMA test. From the algorithm, the number of configurations is obtained (Fig. 3) given a certain range of surface density with a3 = Lx /4. They will be compared to a standard periodic structure with a rectangular core whose surface density is equal to 0,57 kg/m². Thereby, 216 configurations are possible. The parameter a3 is then used to be shifted in the along x with 6 different values between 0 and Lx. Consequently, 1296 configurations in total will be computed. For each configuration the STL is calculated. A wise choice for the model must be done to avoid high computational cost. By taking this range of surface density and having all the possibilities for each parameter, it is possible to plot each parameter as a function of others. It occurs that the angle α and β are linked by a logarithmic function as shown in Fig. 4. Consequently, the set of parameters could be reduced to a1, a3 and α since α = f(β).

MODELLING

During the last decade many models were developed to calculate vibroacoustic indicators. This paper is mainly focused on the STL as the acoustic indicator. Considering an incident plane wave impinging a structure with a specific angle, this will create 3 other waves corresponding to a reflected, transmitted and absorbed wave as illustrated in Fig 5 . The STL is defined with the ratio τ = | Wt / Wi | which is the ratio between the acoustic power of the transmitted and incident wave and be obtained for specific angles (Eq. 1).

𝑆𝑇𝐿(𝜃, 𝜙, 𝜔) = -10 log 10 𝑊 𝑡 𝑊 𝑖 (1)

The STL is also calculated for a diffuse field by the integration over all angles θ ∈ [0°, 90°] and φ ∈ [0°, 360°] and yield to Eq. 2.

𝑆𝑇𝐿 𝑑 (ω) = ∫ ∫ τ(ω, θ, ϕ) sin(θ) cos(θ) θ 𝑚𝑎𝑥 0 2π 0 𝑑θ 𝑑ϕ ∫ ∫ sin(𝜃) θ 𝑚𝑎𝑥 0 cos(θ) 2π 0 𝑑θ 𝑑ϕ (2)
In 2016, two methods released [START_REF] Christen | Wave finite element formulation of the acoustic transmission through complex infinite plates[END_REF][START_REF] Parrinello | matrix representation for periodic planar media[END_REF] using the nodal surfaces and the Transfer Matrix Method (TMM) concept combined with the WFEM respectively. They allow to model complex structures without homogenization. More recently, [START_REF] Deckers | Prediction of transmission, reflection and absorption coe_cients of periodic structures using a hybrid wave based-finite element unit cell method[END_REF] added their contribution by developing a hybrid wave-based method with finite element to add more complexity to the structure as resonators. It is possible to model other type of fluid on both sides of the structure such as strong fluid-structure interaction. In this paper, the method developed in [START_REF] Parrinello | matrix representation for periodic planar media[END_REF] will be used. Indeed, the computational cost is drastically reduced since the skins are modeled using the analytical formulation for simple plate. The core is modeled with Ansys apdl using SHELL elements and MATLAB for post processing. It is assumed an infinite panel, real wavenumbers and that the structure is excited by plane waves with the angles of incidence and reflection equal. The principle of this method is illustrated in Fig. 6. Figure 6 : Procedure for applying the method developed in [START_REF] Parrinello | matrix representation for periodic planar media[END_REF] Finally, to perform the parametric analysis two options are chosen for simplification. Firstly, due to the orthotropy of the structure resulting from the honeycomb core, it is not necessary to calculate the diffuse field. From the dispersion curves of the standard case corresponding to the rectangular core, bending waves reveals the direction x as critical to determine the acoustic efficiency of the structure. It is shown that the intersection with the acoustic wavenumber and bending waves occurs in the direction x. Thus, the STL will be calculated for an angle φ = 0° and θ = 45° exciting the structure in its x-direction only. In addition, the target indicator is defined as the integration of the STL over the frequencies as it is done in [START_REF] Galgalikar | Design automation and optimization of honeycomb structures for maximum sound transmission loss[END_REF]. The more the value is high the more the structure is considered acoustically efficient.

PARAMETRIC SURVEY

The first result was obtained having all configurations with different values of a3. It is shown in Fig. 8 that a quasi-symmetry occurred related to the chosen parametric model. As it can be noticed, the parameter a3 has a strong influence on the STL and the highest values of STL is obtained when a3 = Lx/4. The next steps of the parametric analysis are carried out with a3 = Lx/4. 9 shows the effect on the STL of the angles α and β for different values of a1. The influence of the angles is highly dependent on the value of the parameter a1. The angles α and β become critical parameters when a1 = Ly/4. Consequently, for such configuration it is necessary to make a wise choice on the angles. The highest values of the STL are obtained with a1 = 1,25 mm and a1 = 6,25 mm. Finally, the effect of the pair (α, β) is illustrated in Fig. 10. It is shown that depending on the value of the STL, the pairs (α, β) correspond to a linear function in which the value of the intercept has a critical influence. The highest value of STL occurs when the intercept is the lowest and the pairs (α, β) are highlighted in Fig. 10.

From this parametric study the optimized configuration can be listed in the following Tab. 1. with a1 and a2 equal to 1,25 mm or 6,25 mm and a3 = Lx/4. Since the unit cell size is 15 mm by 15 mm, the angles are almost neglectable and could be considered as 0°. Nevertheless, the optimized unit cell with the highest STL value correspond to (1,4°, 0°).

OPTMIZED VS STANDARD STRUCTURE

Thanks to the parametric survey, it is then possible to compare the optimized structure with the standard rectangular core. The Fig. 11. shows the great improvement obtained with the multilayer core system in all the frequency range. The coincident frequency is shifted to higher frequencies. This result is confirmed with the comparison of bending waves in Fig. 12. The multi-layer core has lower equivalent mechanical properties in both direction x and y and lead to an intersection with the acoustic wavenumber to a higher frequency. The acoustic wavenumber is calculated for a diffuse field and the intersection correspond to the critical frequency. Consequently, the optimized structure should be still acoustically efficient in a diffuse field. Finally, the mechanical and acoustical properties for both structures are summarized in the following table (Tab. 2). The compression modulus is calculated along z-axis. As it is noticed, the improvement of the STL goes along with a reduction of the compression modulus and the transition frequency in both directions. Multi-layer core systems show a great interest in terms of acoustic properties while mechanical properties seem to be lower than a standard one core unit cell. These results were obtained without altering the mass which is a critical parameter especially in the aerospace industry.

CONCLUSIONS

This paper has been focused on the study of multi-layer core topology systems made of 3 layers with a top and bottom rectangular core and a random middle core. It has been shown that this new kind of structure involves new geometrical parameters to be considered. A parametric survey has been suggested to study the influence of this geometrical parameters on the random core. The parameter a1 and a2 were considered equal. The STL has been chosen as the target acoustic indicator and a WFEM combined with the TMM method has been used to model the unit cell. It occurs through the analysis that all parameters seem to have an interaction with the other and could become critical in some configurations. The parameter a3 have a strong effect on the STL and is maximum when a3 = Lx/4. In addition, the pair (α, β) have an impact on the STL when a1 = a2 = Ly/4 and their influence decrease when a1 get close to 0 or Ly/2. Moreover, after fixing a1 = 1,25 mm or 6,25 mm, it was shown that the pair (α, β) turned out a linear relationship according to the value of STL. Thereby, the optimized pair (α, β) corresponded to the equation with the lowest intercept. Consequently, an optimized structure with the same surface density of the standard rectangular core was obtained as well as the comprehension of the influence of each parameter of the parametric model. Ultimately, the parametric analysis may introduce new target indicators as mechanical and vibroacoustic indicators. In addition, this parametric model proposes some tools to verify the influence of each parameter and could be applied to other standard structure as hexagonal and auxetic core widely spread in the industry. Finally, some vibroacoustic indicators as the transition frequency seems to be strongly influenced by the geometry of the multi-layer core and should be studied.

Figure 1 :

 1 Figure 1 : Multi-layer core topology systems

Figure 2 :Figure 3 :

 23 Figure 2 : Parametric model

Figure 4 :

 4 Figure 4 : α = f(β)

Figure 5 :

 5 Figure 5 : Plane wave incidence, reflection, transmission and absorbed at the interface of a sandwich panel.

Figure 7 :

 7 Figure 7 : Bending waves of the standard case (rectangular core)

Figure 8 :

 8 Figure 8 : Effect of the parameter a3 on all configurations

Figure 9 :

 9 Figure 9 : Effect of the angles of the core α and β with specific values of the parameter a1.

Figure 10 :

 10 Figure 10 : Influence of the pair (α, β) given with a1 = 1,25 mm and a1 = 6,25 mm.

Figure 11 :

 11 Figure 11 : Comparison of the STL (θ = 45 ° and φ = 0°) of the optimized structure with the standard case.

Figure 12 :

 12 Figure 12 : Bending waves comparison of the optimized and standard structure.

Table 1 :

 1 Pairs of (α, β) for the optimized unit cell.

	α (°) -4,9	-3,3	-1,6	0	1,4	3	4,8
	β (°) -6,5	-4,9	-3,2	-1,6	0	1,4	3

Table 2 :

 2 Mechanical and acoustical properties.

		standard	optimized
	Compression	227	92
	modulus (MPa)
	Transition	(2910, 1694)	(1388, 251)
	Frequency (Hz)
	(ftx, fty)
	STL integration	26,71	32,88
	(dB)

ACKNOWLEDGEMENT

This project has received funding from the European Union Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No. 675441. The author would like to acknowledge all the Institutions and Partners involved in the VIPER project. The research work of Elke Deckers is financed by a post-doctoral grant of the Research Foundation-Flanders (FWO)