Numerical design of acoustic metamaterial based on parallel Helmholtz resonators for multi-tonal noise control 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

The reduction of multi-tonal noise at multiple frequencies simultaneously is a challenge in many industrial fields. Different solutions such as metamaterials consisting of periodic Helmholtz resonators embedded into a porous layer have been studied in the literature. Generally, a classical resonator made of a cavity connected to a neck provides only one resonant transmission loss peak. In this study, a design of acoustic metamaterials is proposed numerically using the finite element method for multi-tonal noise reduction. Parallel assemblies of Helmholtz-resonators are periodically distributed within a porous material. The cylindrical global cavity is partitioned into several sub-cavities, which are separated from one another by a rigid wall, and each sub-cavity is

1 zacharie.laly@usherbrooke.ca

2 chris.mechefske@queensu.ca

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

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

connected to one neck. The presented designs with 2, 3, 4, and 5 necks exhibit multiple resonance transmission loss peaks which correspond respectively to the number of the necks. Depending on the target frequencies and available volume space, one can use a suitable design by properly choosing the dimensions of each neck and each sub-cavity. The proposed acoustic metamaterial designs can be used for multi-total noise reduction at several frequencies simultaneously. 1. INTRODUCTION

The noise pollution has serious impact on people’s health, comfort and well-being. The reduction of multi-tonal noise at low frequency is a challenging topic due to the weak energy dissipation with traditional sound absorbing materials. Guo et al. [1] used a transfer matrix method and the equivalent medium model to characterize analytically a thin absorber based on a Helmholtz resonator with an extended neck, which can exhibit effective low frequency sound absorption. They observed that the resonant frequency could be flexibly tuned by adjusting the geometry of the extended neck. In order to increase the absorption frequency band, they proposed a checkerboard absorber that is made of parallel assemblies of Helmholtz resonators. They showed that the parallel resonators operate almost independently when the adjacent resonators units are largely dissimilar. Mahesh and Mini [2] investigated analytically, numerically and experimentally the sound absorption properties of series and parallel configurations of Helmholtz resonators. They used parallel transfer matrix method to model analytically parallel assembly of Helmholtz resonators and the analytical results agree well with finite element method results. They observed that parallel arrangement of dissimilar Helmholtz resonators induced an increase of the absorption bandwidth. Selamet et al. [3] studied theoretically, numerically, and experimentally the acoustic response of a concentric circular Helmholtz resonator with an extended neck. They proposed a two-dimensional analytical method for an extended neck with constant cross- sectional area and illustrated the influences of the parameters of the neck on the resonance frequency and the transmission loss. Guo et al. [4] investigated a compact sound-absorbing structure, which is made of 11-coupled parallelly arranged double layer Helmholtz resonators with extended necks for broadband sound absorption. They validated the numerical and theoretical results with experiments. Mercier et al. [5] used a homogenization procedure to study the resonance of a Helmholtz resonator with arbitrary neck shape. Langfeldt et al. [6] proposed a theoretical model to estimate the resonance frequency and the input impedance of a Helmholtz resonator with multiple necks. They showed that even small holes in the Helmholtz resonator can induce significant increase of the resonance frequency and reduce considerably the sound absorption performance. Guo et al. [7] proposed a compact metasurface consisting of Helmholtz resonator with an embedded spiral neck using analytical, numerical and experimental investigations. They showed that perfect absorption at tunable resonance frequency can be achieved by the proposed design. Duan et al. [8] presented a multi-layer Helmholtz resonance metamaterial, which can achieve multiple sound absorption peaks. Laly et al. [9,10] used a transfer matrix method which combines with finite element calculations to characterize acoustic metamaterial made of Helmholtz resonators periodically distributed into a porous material. The results of the proposed method for single and double wall configurations agreed well with finite element results. Doutres et al. [11] observed that the transmission loss of porous material with embedded periodic Helmholtz resonators was improved at the resonance of the Helmholtz resonator. A front membrane-cavity Helmholtz resonator embedded in a porous matrix was studied by Abbad et al. [12]. Glass wool layers with embedded periodic Helmholtz resonators in single and double wall configuration were investigated by Ghinet et al. [13]. They carried out experimental measurements under diffuse field and presented the transmission loss characteristics of the metamaterial designs. Selamet et al. [14] used analytical, numerical and experimental investigations

to study Helmholtz resonators with the cavity lined with fibrous material. Kone et al. [15,16] presented acoustic metamaterial with complex neck shape which achieves multiple sound absorption peaks.

This study presents an acoustic metamaterial design using the finite element method (FEM) for multi- tonal noise reduction. The design comprises porous material with periodic parallel assembly of Helmholtz resonators. The cylindrical global cavity is partitioned into several sub-cavities, and each sub- cavity is connected to one neck. The results of the transmission loss obtained using the finite element method show multiple resonant peaks, which correspond respectively to the number of the necks. This study illustrates different designs of parallel assembly of Helmholtz resonators periodically embedded within a porous layer. Depending on the targeted sound attenuation frequencies and the available volume space, one can therefore choose a suitable design with appropriate dimensions of the neck and the sub- cavities.

2. DESIGN OF ACOUSTIC METAMATERIAL BASED ON PARALLEL ASSEMBLIES OF

HELMHOLTZ RESONATORS

In this paper, acoustic metamaterial designs based of parallel assemblies of Helmholtz resonators are proposed for multi-tonal noise control. Each of this parallel assembly of Helmholtz resonators is periodically embedded into a porous material. These metamaterial designs can be used in several industrial applications such as aerospace to attenuate multi-tonal noise. The number of parallel Helmholtz resonators depends on the target frequencies for acoustic sound attenuation as well as the available volume space. To attenuate the noise simultaneously at N frequencies, one can make a metamaterial design by assembling N resonators in parallel and the dimensions of the necks and the sub-cavities should be well chosen to fit the target frequencies. 2.1 Design of a metamaterial based on two parallel Helmholtz resonators

Figure 1 shows a global cylindrical cavity that is partitioned into two sub-cavities, which are separated by a wall. Each sub-cavity has the same volume and is connected to a neck. Each neck with the associated sub-cavity represents a Helmholtz resonator. This system is therefore a parallel assembly of two Helmholtz resonators with necks that are extended into the sub-cavity to make the design useful in limited spaces conditions. The length of the i th neck is denoted by i H while the radius is denoted by i R .

Figure 1: Parallel assembly of two Helmholtz resonators.

Figure 2 illustrates the Periodic Unit Cell (PUC) consisting of the incident and transmission fluids and

the parallel assembly of two Helmholtz resonators embedded within a porous material. The geometry is

shown in Figure 2(a) and the mesh in Figure 2(b), which is a physics-controlled mesh created by

COMSOL Multiphysics with 15 164 domains elements and 3 719 boundaries elements. The number of

degrees of freedom is 37 595. The length of the incident and transmission fluid are equal to 150 mm and

the lateral dimensions of the PUC are 100 mm x 100 mm. Periodic boundary conditions are applied on a

pair of all parallel planes and plane wave radiation condition is applied on the inlet and outlet planes of

the PUC for all the numerical simulations.

Figure 2: PUC of the metamaterial made of a parallel assembly of two Helmholtz resonators: (a) geometry (b) mesh.

The porous layer is modeled using the equivalent fluid model proposed by Johnson-Champoux-Allard where t he equivalent density  eq   and bulk modulus  eq K  are given by [17,18]

           

      

2 0 0 2 2 2 0

4 1 1 eq

j j



 

, (1)



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

1

   

1



K P

, (2)

0 '2 0 '2 0

eq

   

j P j P

8 1 1 16

r

r

a) Incident fluid Porous material Transmission fluid

with  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 thermo-viscous

acoustic interface of COMSOL Multiphysics is used to characterize the air inside each neck in order to account for the viscous and thermal dissipations effects. The air inside each neck can also be modeled as equivalent fluid using Johnson-Champoux-Allard model where the static airflow resistivity of the air within the resonator necks, similar to the linear airflow resistivity of a micro perforated plate is given by

i R     . A normal incidence plane wave with pressure amplitude of 1 Pa is applied on the

[18-22] 2 8

i

inlet plane of the PUC and the sound transmission loss is determined by the following relation

      

W TL W

in 10 10log

, (3)

out

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

at the outlet plane. Acoustic and thermo-viscous acoustic boundaries in the numerical model represent the circular top and bottom surfaces of each neck. For all the numerical simulations, 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. Figure 3 shows the transmission loss of the porous material with embedded periodic parallel assembly of two Helmholtz resonators. The diameter of the global cavity is 80 mm and neck 1 has a length of 20 mm and a radius of 7 mm while neck 2 has a length of 15 mm and a radius of 10 mm. The length of the global cavity and the thickness of the porous material are equal to 30 mm. The wall of the resonators is considered rigid.

Figure 3: Transmission loss of acoustic metamaterial based on a parallel assembly of two Helmholtz resonators.

The transmission loss in Figure 3 shows two resonance peaks at 474 Hz and 718 Hz where the TL amplitudes are 32.7 dB and 36 dB respectively. Each resonator operates like independently resulting in two TL peaks.

2.2 Design of a metamaterial based on three parallel Helmholtz resonators

Figure 4 shows a parallel assembly of three resonators. The global cylindrical cavity is partitioned into three sub-cavities of equal volume and each sub-cavity is connected to one neck. A wall is created between the sub-cavities. The parallel assembly of the three resonators is periodically distributed within a porous material and the length of its cavity is equal to the thickness of the porous material. The geometry of the PUC and the mesh are shown in Figures 4(b) and (c). The length i H and radius i R for

each neck are given by H 1 = 15 mm, H 2 = 20 mm, H 3 = 25 mm and R 1 = 8 mm, R 2 = 11 mm, R 3 = 13 mm. The diameter of the global cavity is 80 mm with a length of 30 mm, which is equal to the thickness of the porous layer. The incident and transmission fluid have a length of 150 mm while the lateral dimensions of the PUC are 100 mm x 100 mm. The transmission loss is obtained using Equation 3.

Figure 4: PUC of acoustic metamaterial: (a) parallel assembly of three Helmholtz resonators (b) geometry of the PUC (c) mesh of the PUC.

The transmission loss of the porous layer with embedded periodic parallel Helmholtz resonators is shown in Figure 5.

‘b) incident fluid Porous Resonator

Figure 5: Transmission loss of acoustic metamaterial based a parallel assembly of three Helmholtz resonators.

In Figure 5, the transmission loss presents three resonant frequencies, which are 668 Hz, 866 Hz and 986 Hz where the TL amplitudes are 35 dB, 37.6 dB and 35.6 dB respectively. The number of TL peaks is equal to the number of the necks. 2.3 Design of a metamaterial based on four parallel Helmholtz resonators

In Figure 6, a parallel assembly of four Helmholtz resonators is illustrated. The cylindrical cavity is partitioned into four sub-cavities that have the same volume and are separated from one another by a wall, which is considered rigid. Each sub-cavity is connected to one neck. The air inside each neck is characterized using the thermo-viscous acoustic interface of COMSOL Multiphysics.

Figure 6: Parallel assembly of four Helmholtz resonators.

Figure 7 shown the PUC, which comprises incident and transmission fluids and the porous material with embedded periodic parallel assembly of four Helmholtz resonators. The lateral dimensions of the PUC

are 100 mm x 100 mm and the incident fluid as well as the transmission fluid has a length of 150 mm. The diameter of the global cavity is 80 mm with a length of 30 mm. The mesh illustrated in figure 7(b) is a physics-controlled mesh with 24 682 domain elements and 6 443 boundary elements and the number of degrees of freedom is 65 669. The length of each neck is equal to 20 mm while the radii are R 1 = 6 mm, R 2 = 8 mm, R 3 = 9 mm and R 4 = 10 mm respectively. A normal incidence plane wave with pressure amplitude of 1 Pa is applied on the inlet plane and the sound transmission loss is obtained using Equation 3.

Figure 7: PUC of the metamaterial made of a parallel assembly of four Helmholtz resonators: (a) geometry (b) mesh.

Figure 8 presents the transmission loss of the porous layer with embedded periodic parallel assembly of four Helmholtz resonators.

2) Incident fluid Porous material transmission fluid ) Incident fluid Porous material Transmission fluid

Figure 8: Transmission loss of acoustic metamaterial based on a parallel assembly of four Helmholtz resonators.

The transmission loss in Figure 8 presents four resonant frequencies of 584 Hz, 776 Hz, 878 Hz and 978 Hz with peak amplitudes of 31.8 dB, 33.6 dB, 35.2 dB and 34 dB respectively. These four resonant frequencies depend on the geometrical dimensions of each neck and each sub-cavity. 2.4 Design of a metamaterial based on five parallel Helmholtz resonators

In Figure 9, a design of a parallel assembly of five Helmholtz resonators is shown. There are five sub- cavities with one sub-cavity in the center and four others around it. The sub-cavities are separated by a wall, which is considered rigid. One neck is connected to each sub-cavity. The air inside each neck is modeled using the thermo-viscous acoustic interface to account for the viscous and thermal dissipations effects.

Figure 9: Parallel assembly of five Helmholtz resonators.

Figure 10 shows the PUC of the structure consisting of the parallel assembly of five Helmholtz resonators, which is periodically distributed within a porous material. The structure is connected to an incident fluid and a transmission fluid with have the same length of 150 mm. The lateral dimensions of the PUC remain 100 mm x 100 mm and periodic conditions are applied on each pair of parallel planes. Figure 10(a) illustrates the geometry and the mesh is shown in Figure 10(b). The length of each sub- cavity as well as the thickness of the porous layer is equal to 40 mm. The global cavity has a diameter of 85 mm and the diameter of the sub-cavity, which is in the center is 35 mm. The length of each neck is 30 mm and the radii are respectively given by R 1 = 5 mm, R 2 = 6 mm, R 3 = 7 mm, R 4 = 8 mm and R 5 = 9 mm. The transmission loss of the porous layer with the embedded periodic parallel assembly of five Helmholtz resonators is shown in Figure 11.

Figure 10: PUC of the metamaterial made of a parallel assembly of five Helmholtz resonators: (a) geometry (b) mesh.

Figure 11: Transmission loss of acoustic metamaterial based on a parallel assembly of five Helmholtz resonators.

In Figure 11, one observes five resonant frequencies, which are 374 Hz, 446 Hz, 522 Hz, 606 Hz and 734 Hz where the transmission loss peaks values are 26.2 dB, 29.4 dB, 30 dB, 33 dB and 37.3 dB respectively. The acoustic behavior of each resonator seems independent.

For specific industrial applications, the design of the proposed metamaterial will depend on the available volume space and the target resonant frequencies. A given number of parallel Helmholtz

a) incident fluid Porous material Transmission fluid Resonator b) Incident fluid Porous material Transmission fluid

resonators can be periodically distributed within the porous layer to attenuate the noise simultaneously at multiple specific frequencies.

3. CONCLUSION

A design of acoustic metamaterials constituted by porous material with embedded periodic parallel assembly of Helmholtz resonators was proposed. The transmission loss of the metamaterial was predicted using finite element method. The global cavity is partitioned into several sub-cavities, which are separated from one another by a wall. The results of the transmission loss showed 2, 3, 4, and 5 resonant peaks which correspond respectively to the number of the necks. The proposed metamaterials design can be used in many applications for multi-total noise reduction.

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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