Numerical modelling of acoustic metamaterial made of periodic Helmholtz resonator containing a damping material in the cavity Zacharie Laly 1 CRASH, Centre de Recherche Acoustique-Signal-Humain, Université de Sherbrooke, 2500 Boul. de l’Université, Sherbrooke, Québec, J1K 2R1, Canada. Department of Mechanical and Materials Engineering, Queen's University, 99 University Ave, Kingston , ON, K7L 3N6, Canada. Christopher Mechefske 2 Department of Mechanical and Materials Engineering, Queen's University, 99 University Ave, Kingston , ON, K7L 3N6, Canada. Sebastian Ghinet 3 National Research Council Canada, Aerospace, 1200 Montreal Road, Ottawa ON, K1A 0R6, Canada Charly T. Kone 4 National Research Council Canada, Aerospace, 1200 Montreal Road, Ottawa ON, K1A 0R6, Canada ABSTRACT

Acoustic metamaterials are frequently used in many fields such as aerospace, buildings and ground transportation industries for low frequency noise control applications. Different solutions based on membrane or Helmholtz resonators have been investigated in the past few years. In the present study, a numerical design of an acoustic metamaterial made of a Helmholtz resonator with a membrane in its cavity is presented. The resonator with a neck protruding into the cavity is periodically embedded within a porous material. The membrane inside the resonator cavity is modelled as a linear isotropic elastic material with free and fixed boundary conditions. The transmission loss (TL) of the proposed metamaterial design, predicted using a finite element method presents multiple resonant peaks while only one peak is obtained with a conventional

1 zacharie.laly@usherbrooke.ca

2 chris.mechefske@queensu.ca

3 sebastian.ghinet@nrc-cnrc.gc.ca

4 tenoncharly.kone@nrc-cnrc.gc.ca

resonator. Two TL resonance peaks are observed when the membrane circumferential boundary is free inside the resonator cavity. For fixed boundary conditions, more than two resonance peaks are obtained. The proposed metamaterial design can therefore be used in many industrial applications for low frequency noise attenuation. It can be useful in limited volume space condition. The membrane mounting can be easier, and the transmission loss is improved than the TL of existing resonator design.

1. INTRODUCTION

The reduction of low frequency noise is a challenge in many industrial fields and acoustic metamaterials are widely used for this purpose. The design of aircraft cabin panels can incorporate acoustic metamaterial in order to reduce the low frequency noise inside the cabin. Kuntz et al. [1] studied the acoustic response of the cabin sidewalls of a trimmed Gulfstream II aircraft with embedded periodic Helmholtz resonators and demonstrated that the transmission loss (TL) of the double wall structure was improved at the resonance frequencies of the resonators. Magliacano et al. [2] presented numerical analysis of a fuselage acoustic panel made of a meta-core inside an acoustic package. The acoustic performance of porous material with embedded periodic Helmholtz resonators was investigated by Doutres et al. [3] using the parallel transfer matrix method and a homogenization based model. They compared the theoretical results with experimental measurements and finite element method (FEM) results and showed the improvement of the transmission loss at the resonance of the Helmholtz resonator. Abbad et al. [4] investigated the transmission loss and sound absorption performance of a front membrane-cavity Helmholtz resonator embedded in a porous matrix. They observed the transmission loss improvement at the Helmholtz resonance with a degradation of the sound absorption. Ghinet et al. [5] studied a design of glass wool layers with embedded periodic Helmholtz resonators and presented the experimental measurements results on single and double wall configurations. Langfeldt et al. [6] studied theoretically and experimentally the vibro-acoustic response of a double wall with embedded Helmholtz resonators in the cavity. Hu et al. [7] studied membrane-coupled Helmholtz resonators using theoretical approach and illustrated multiple resonances. Groby et al. [8] demonstrated the sound absorption coefficient improvement of porous material with embedded periodic air-filled Helmholtz resonators. Li et al. [9] presented a design of multilayer honeycomb membrane-type acoustic metamaterial and showed its sound transmission loss performance. Laly et al. [10,11] proposed a transfer matrix method to characterize multilayer acoustic metamaterial including inhomogeneous materials with complex inclusions. This method was used to predict the acoustic transmission loss of porous material with periodic embedded Helmholtz resonators in single and double wall configuration and the results were in good agreement with finite element method. Acoustic metamaterial with complex neck shape was investigated by Kone et al. [12,13] for low frequency noise control. Yamamoto et al. [14] studied acoustic metamaterial constituted by a plate of plastic foam with embedded periodic Helmholtz resonators and illustrated good transmission loss performance.

In this study, an acoustic metamaterial made of a porous material with embedded periodic Helmholtz resonators is proposed and investigated numerically. A damping material, in the form of a membrane is integrated into the resonator cavity. The acoustic transmission loss of the metamaterial is obtained using finite element method of the commercial software COMSOL Multiphysics and the effect of the membrane boundary condition inside the resonator cavity is illustrated. It is shown that the membrane in the resonator cavity induces multiple resonances for the transmission loss while only one resonance peak

is obtained with a conventional resonator. The influence of the membrane position within the resonator cavity is illustrated. The proposed resonator design can be used in limited available space and its manufacturing process with the membrane mounting within the resonator cavity can be easier than other existing design [4] while the transmission loss is improved. 2. FINITE ELEMENT DESIGN AND ANALYSIS OF THE PROPOSED METAMATERIAL

In the following, a design of acoustic metamaterial based on Helmholtz resonator periodically embedded within a porous material is presented using finite element method. The resonator contains in its cavity a damping material in the form of a membrane. The commercial code COMSOL Multiphysics is used and the simulations are conducted on the Periodic Unit Cell (PUC), which consists of the structure that is connected to an incident fluid and a transmission fluid as illustrated in Figure 1. The structure is periodic along the x and y directions and periodic boundary conditions are applied on a pair of parallel planes along x and y directions. A normal incidence plane wave with pressure amplitude of 1 Pa is applied on the inlet plane while plane wave radiation condition is applied on the inlet and outlet planes to minimize the reflection of the acoustic waves. The effects of evanescent waves are accounted for by using in the numerical simulations suitable lengths for the incident and transmission fluids.

Figure 1: Periodic Unit Cell of the system.

Figure 2 illustrates the design of the resonator where a damping material in the form of a membrane is inserted into the resonator cavity. The neck of the resonator with a cylindrical shape is extended into the cavity in order to use the design in limited available space. The radius and the length of the neck are denoted respectively by R and H. Figure 2(b) shows the resonator embedded within a porous material. The length of the resonator cavity and the thickness of the porous material are identical. The wall of the resonator is considered rigid.

Incident fluid Transmission fluid Inlet plane. Outlet plane x . z Structure

Figure 2: Helmholtz resonator with a membrane in the cavity: (a) Helmholtz resonator with neck extended into the cavity (b) Helmholtz resonator embedded in porous layer.

The proposed acoustic metamaterial design can be used in many industrial applications such as aerospace. It can be incorporated into the aircraft cabin panels in order to reduce the low frequency noise levels inside the cabin.

Figure 3 shows the single wall configuration which consists of Helmholtz resonator embedded periodically within the porous material that is separated from the panel by an air gap. The Helmholtz resonator contains a membrane in its cavity. The membrane within the resonator cavity induces multiple transmission loss peaks and its mounting within the cavity can be easier than existing design with a front wall membrane [4]. The present design can be useful in limited space condition where more space can be required with other design [4].

a) Neck Cavity b) ——_-Helmhottz resonator Neck Porous material yembrane

Figure 3: Periodic Unit Cell of single wall configuration.

The membrane within the resonator cavity is modeled as a linear isotropic elastic material using the solid mechanics interface of COMSOL Multiphysics where the circumferential boundary conditions are free or fixed. For the free boundary, the membrane can move inside the cavity of the resonator while for the fixed boundary, the velocity is set to zero. The equivalent fluid model proposed by Johnson- Champoux-Allard is used to characterize the porous material as well as the air inside the resonator neck in order to account for the viscous and thermal dissipation effects. The equivalent density  eq   and

bulk modulus  eq K  are expressed by [15,16]

‘Air gap Neck Porous material Panel eee: wena

           

      

2 0 0 2 2 2 0

4 1 1 eq

j j



 

, (1)



                     

1

   

1



, (2)

K P

0 '2 0 '2 0

eq

   

j P j P

8 1 1 16

r

r

where  is the porosity,  the static airflow resistivity,   the tortuosity, 0  is the density ,  the

viscous characteristic length, '  the thermal characteristic length,  is the dynamic viscosity, r P the

Prandtl number,  is the specific heat ratio, and 0 P is the atmospheric static pressure. The static airflow

resistivity of the air within the resonator neck is calculated using the following relation [16-20]

    . (3)

2 8

R

Note that Equation 3 represents the linear term of the airflow resistivity of a micro perforated panel [17-20]. The porosity and the tortuosity of the air in the resonator neck are equal to 1 while the viscous characteristic length  is equal to the radius of the neck R . The sound transmission loss is determined by the following relation

W TL W       

in 10 10log

, (4)

out

where in W and out W represent respectively the incoming power at the inlet plane and the outgoing power

at the outlet plane.

Figure 4 presents the cut-out view of the mesh which is a physics-controlled mesh created by COMSOL Multiphysics with 17 025 domain elements and 4 360 boundary elements. The number of degrees of freedom is 40 186. The length of the incident fluid is 150 mm and is equal to the one of the transmission fluid. The lateral dimensions of the PUC are 100 mm x 100 mm. For all the numerical simulations, the membrane is Ethylene-vinyl acetate rubber material with Young’s modulus of 5 MPa, a density of 660 kg/m 3 and a Poisson’s ratio of 0.45. The static airflow resistivity of the porous material is 26 000 N s m -4 , the open porosity is 99%, the tortuosity is 1.02, and the characteristic viscous and thermal lengths are respectively 150 μm and 300 μm. The resonator has a cylindrical cavity with diameter of 70 mm and a length of 40 mm, which is equal to the thickness of the porous material.

Figure 4 : Bisected view of the mesh for numerical simulation.

Figure 5 shows the influence of the boundary condition of the membrane inside the resonator cavity on the transmission loss. The membrane with a thickness of 1 mm is inserted into the cavity of the resonator at 10 mm from the bottom wall of the cavity. The radius of the neck is 15 mm with a length of 20 mm. The transmission loss of the porous layer with embedded periodic resonator obtained using Equation 4 is illustrated in Figure 5 for the case with and without a membrane inside the cavity of the resonator.

Figure 5: Transmission loss of metamaterial.

In Figure 5, the transmission loss of the metamaterial without the membrane in the resonator cavity presents only one resonant peak at 670 Hz where the TL peak value is 38.8 dB. When the membrane is integrated within the resonator cavity with free boundary, two transmission loss resonant peaks with amplitudes of 38 dB and 36 dB are observed at 580 Hz and 1024 Hz, respectively. For membrane fixed boundary conditions, the transmission loss presents five peaks with amplitudes of 43 dB, 36 dB, 39 dB, 38 dB and 25 dB at resonant frequencies of 334 Hz, 540 Hz, 648 Hz, 924 Hz and 1076 Hz respectively. The interaction between the Helmholtz resonance and the membrane vibration, which depends on its boundary condition results in multiple TL peaks. The membrane fixed boundary conditions should be

Incident fluid Porous material Transmission fluid T ‘Neck HEWWKOETOSCRENOr ‘dkinirane:

used in the resonator cavity for TL improvement. During the membrane mounting process, one should ensure that all the entire edge of the membrane is fixed.

Figure 6 illustrates the cut-out view of the single wall configuration mesh. It is a physics-controlled mesh created by COMSOL Multiphysics with 45 676 domain elements and 8 884 boundary elements. The number of degrees of freedom is 77 464 . The Helmholtz resonator with a membrane in its cavity is periodically embedded into a porous material, which is separated from the panel by an air gap. The length of the incident and transmission fluid are identical, it is 150 mm and the lateral dimensions of the PUC are 100 mm x 100 mm. The panel is aluminum layer with a thickness of 2 mm and the thickness of the air gap is set to 30 mm. The membrane with a thickness of 1 mm is located at 10 mm from the bottom wall of the resonator cavity. The length and the radius of the neck are equal to 15 mm.

Figure 6: Cut-out view of the single wall configuration mesh. Figure 7 shows the transmission loss of the single wall configuration in the case with and without a membrane in the resonator cavity.

Figure 7: Comparison of the transmission loss for single wall configuration.

Incident fluid Porous material Transmission fluid Pane!“ gieday Helmholtz Membrane Ai a i? ee

The transmission loss of single wall configuration in Figure 7 shows only one peak with an amplitude of 75 dB at 716 Hz in the case of resonator without membrane. When a membrane is inserted with a free circumferential boundary in the resonator cavity, two resonant transmission loss peaks are observed at 606 Hz and 1058 Hz with amplitudes of 74.15 dB and 73.25 dB respectively and for membrane fixed boundary conditions; four resonant transmission loss peaks are obtained. Thus, the use of the membrane in the cavity of the resonator and its boundary condition result in additional resonances by comparison to the case when no membrane is used.

Figure 8 illustrates the effects of the membrane thickness on the transmission loss. The resonator whose neck has a radius of 15 mm and a length of 20 mm is periodically embedded within the porous material. The membrane with fixed boundary conditions is located at 10 mm from the bottom inner wall of the resonator cavity and its thickness is varied from 0.5 mm to 2 mm.

Figure 8: Effect of the membrane thickness on the transmission loss: (a) t = 0.5 mm, (b) t = 1 mm, (c) t

= 1.5 mm, (d) t = 2 mm.

In Figure 8, one observes that the resonant frequencies and the number of TL peaks change when the membrane thickness changes. For the membrane thickness of 1 mm, there are five resonant TL peaks in Figure 8(b) and four resonant peaks in Figure 8(c) for the membrane thickness of 1.5 mm.

The influence of the membrane position inside the cavity of the resonator is illustrated in Figure 9. The length and the radius of the neck are set to 15 mm. The distance between the membrane and the bottom wall of the resonator cavity is denoted by p D . In Figure 9, the transmission loss is shown for p D

varying between 5 mm and 20 mm. When p D increases, the membrane gets close to the neck of the

resonator.

Figure 9: Effect of the membrane position on the transmission loss.

In Figure 9, it can be observed that the values of the resonant frequencies change when the position of the membrane in the cavity of the resonator changes. For p D equal to 15 mm there are five TL resonant

peaks and for p D values of 10 mm and 20 mm, there are six resonance peaks and four resonance peaks

for p D equal to 5 mm. The position of the membrane in the resonator cavity affects therefore the resonant

frequencies and the number of TL peaks.

The present resonator design can be manufactured easily with the membrane mounting. It can be used in limited space compared to other existing designs [4], which can require more space, and its transmission loss is improved. 3. CONCLUSION

An acoustic metamaterial consisting of Helmholtz resonators periodically distributed within a porous material was studied using the finite element method. The resonator contains a damping material in the form of a membrane in its cavity, which is modeled as a linear isotropic solid material while the porous material was characterized by the equivalent fluid model of Johnson-Champoux-Allard. The transmission loss of the proposed acoustic metamaterial presented two resonant peaks when the membrane circumferential boundary is free and more than two resonant peaks when the boundary is fixed while only one peak was observed with a conventional resonator. The proposed acoustic metamaterial design can therefore be used in many industrial applications such as aerospace for low frequency noise attenuation.

4. ACKNOWLEDGEMENT

The authors would like to acknowledge the National Research Council Integrated Aerial Mobility Program for the financial support.

5. REFERENCES

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