Ventilated noise-insulating metamaterials inspired by sonic black holes
Farid Bikmukhametov Lana Glazko Yaroslav Muravev Dmitrii Pozdeev Evgeni Vasiliev Sergey Krasikov Mariia Krasikova






    These authors contributed equally.
    School of Physics and Engineering, ITMO University, Saint Petersburg 197101, Russia


    These authors contributed equally.
    School of Physics and Engineering, ITMO University, Saint Petersburg 197101, Russia


    These authors contributed equally.
    School of Physics and Engineering, ITMO University, Saint Petersburg 197101, Russia


    These authors contributed equally.
    School of Physics and Engineering, ITMO University, Saint Petersburg 197101, Russia
    

    These authors contributed equally.
    School of Physics and Engineering, ITMO University, Saint Petersburg 197101, Russia


    s.krasikov@metalab.ifmo.ru
    School of Physics and Engineering, ITMO University, Saint Petersburg 197101, Russia


    School of Physics and Engineering, ITMO University, Saint Petersburg 197101, Russia


             




§ ABSTRACT 


Acoustic black holes represent a special class of metastructures allowing efficient absorption based on the slow sound principle. The decrease of the wave speed is associated with the spatial variation of acoustic impedance, while the absorption properties are linked to thermoviscous losses induced by the local resonances of the structure. 
While most of the developments in the field of sonic black holes are dedicated to one-dimensional structures, the current study is concerned with their two-dimensional counterparts. It is shown that the change of the dimensionality results in the change of noise insulation mechanism, which relies on the opening of band-gaps rather then thermoviscous losses. The formation of band-gaps is associated with the strong coupling between the resonators constituting the considered structures. Numerically and experimentally it is shown than the structure is characterized by broad stop-bands in transmission spectra, while the air flow propagation is still allowed. In particular, a realistic application scenario is considered, in which the acoustic noise and the air flow are generated by a fan embedded into a ventilation duct.
The obtained results pave the way towards the development of next-level ventilated metamaterials for efficient noise control.


Ventilated noise-insulating metamaterials inspired by sonic black holes
    Mariia Krasikova
    September 9, 2024
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§ INTRODUCTION

The development of the low-frequency passive sound insulation systems still represent an unsolved challenge. Recent studies in the field of acoustic metamaterials <cit.> open a rout towards the development of low-weight sub-wavelength structures allowing not only efficient noise suppression <cit.>, but also air ventilation <cit.>. A special class of metastructures is represented by the so-called acoustic black holes (ABH) in which the phase velocity reduces until zero due to spatial variation of acoustic impedance <cit.>. In this sense, the incoming wave is trapped inside the structure, in analogy with the corresponding cosmological objects. 
Successfully implemented for suppression of vibrations <cit.>, the concept of ABH was successfully translated for airborne sound, where the structures typically represent a set of plates characterized by the gradually changing sizes and distances between them <cit.>.
The absorption mechanism of ABH in this case relies on resonant states inducing the increase of thermoviscous losses <cit.>. Though, in realistic systems the fulfillment of critical coupling condition is rather difficult to achieve and additional porous inserts might be required <cit.>. 
In order to distinguish between two different types of structures, ABHs for sound propagating fluids are usually referred to as sonic black holes (SBH), while the term vibrational black holes (VBH) is utilized for bending waves <cit.>.



Typically, SBH are utilized as absorbing termination ends of circular waveguides, utilizing the ring-based geometry developed in Ref. <cit.>, or ventilated duct silencers <cit.>. More complicated geometries may also include arrays of discs <cit.> or combination of rings with other structures, such as lattices <cit.>.
The circular geometry implies that the structure effectively is one-dimensional. The increase of the dimensionality might be associated with the periodicity of the structure in the direction perpindicular to the direction of the incident wave propagation. Wile periodic arranges of VBH and SBH were suggested for suppression of vibrations <cit.>, two-dimensional versions of SBH are still under-represented (with the rare exceptions like in Ref. <cit.>).
However, it might be speculated that the change of dimensionality might might allow to improve performance of the noise-insulating systems via the use of band-gaps.


Hence, the work is dedicated to the development of ventilated meta-atoms for efficient noise mitigation in rectangular ducts. Considered meta-atoms represent a set of rectangular plates forming a set of strongly coupled resonators. Noise-suppression properties of the meta-atoms are described in terms of band-structures of the associated infinitely periodic 2D systems. It is shown that wide stop-bands in the transmission spectra originate from the coupling between the resonances formed by the plates. As the consequence, the properties of the considered meta-atoms are not directly defined by the geometric profile of the structure. The obtained experimental results indicate that the broad stop-band covers nearly the whole range from 1000 to 2850 Hz. At the same time, the structure is ventilated and the meta-atom reduces the air flow speed by only 25%. These findings might be useful for the further development of metastructures for passive noise insulation in ventilated ducts.




§ MATERIALS AND METHODS



 §.§ System description

Considered meta-atoms consist of rectangular plates having different widths and arranged at unequal distances with respect to each other (see Fig. <ref>). Similar structures frequently investigated as sonic black holes are usually characterized by axial symmetry meaning that effectively they are one-dimensional, contrary to the considered case. When the meta-atom is placed inside a rectangular duct with sound hard walls, it can be considered as an effectively two-dimensional structure, which is periodic along the directions perpendicular to the direction of the incident wave propagation. The approximation is valid as long as the field distribution is characterized by the mirror symmetry with respect to the direction of the wave propagation, i.e. x-axis. Similarly, the system is effectively two-dimensional until the field distribution along the z-axis is uniform. Mirror symmetry also implies that instead of the whole structure, only its half can be utilized without the loss of generality. The verification of this statement, as well as the detailed description of the design procedure are given in Supplementary Information.






 §.§ Numerical Calculations

Numerical calculations are provided in COMSOL Multiphysics. Transmission spectra calculations presented in the main text are obtained using the “thermoviscous acoustics, frequency domain” physics. The incident wave is implemented as the background pressure field with the amplitude p_0 = 1 Pa. Both ends of the waveguide are supplemented by the perfectly matched layers, imitating absorbing termination ends of the experimental setup. All walls of the resonators are considered to be the sound hard ones. Still, viscous boundary layers are taken into account, such that the thickness of the boundary layers are considered to be 

    d_visc = d_visc, 0√(f_0/f),

where d_visc, 0 = 0.22 mm is the boundary layer thickness at the frequency f_0 = 100 Hz. Correspondingly, the mesh is supplemented by the “boundary layers” feature.

Transmission coefficients given in decibels are obtained as

    T = 20 log_10 P_t/P_b,

where P_t and P_b are the amplitudes of the total and background pressure fields, respectively, calculated via the integration of the pressure over the area behind the structure [see Fig. <ref>(a)]:

    P_t,b = 1/A∫_A |p_t,b|^2 dA.

Note that transmission spectra are calculated for the halved unit cell, contrary to the case of the eigenmodes analysis. 



 §.§ Optimization

The optimization procedure is based on a simple genetic algorithm, similar to the one implemented in Ref. <cit.>. In particular, the goal of the algorithm is to find geometric parameters of the structure for which the value of the cost function 𝒞 is minimized. The considered cost function represents a linear combination of the transmission coefficient averaged over the whole spectra ⟨ T ⟩, the transmission coefficient averaged over the specified part of the spectra ⟨ T ⟩_[f_1,f_2], and its standard deviation σ_[f_1,f_2]:

    𝒞 = a_1 ⟨ T ⟩ + a_2 ( ⟨ T ⟩_[f_1,f_2] + σ_[f_1,f_2]).

All calculations are performed for the spectral range 100 - 3000 Hz, while ⟨ T ⟩_[f_1,f_2] is defined within the target range 300 - 2000 Hz. Such a combination of parameters allows to ensure that the stop-band occurs in the desired range, and the transmission spectra in this case is close to a flat one. At the same time, the consideration of ⟨ T ⟩ allows to ensure that there are no undesired resonances outside of the target range, which might be crucial for practical applications. The procedure is performed for different values of the coefficients a_1,2. Namely, the structure discussed in the main text is obtained with a_1 = 0.1 and a_2 = 0.9. 

For the optimization procedure, the size of the structure is limited by 10 plates. Each plate is characterized by the width, which can take a value between 0 and 55 mm (the width of the waveguide is fixed at 60 mm). Similarly, the thickness of each plate and the distance between the plates can take values from the range from 1 to 10 mm. 

The population size is considered to be 20, such that the initial population is generated randomly. The selection procedure is implemented using the roulette wheel approach with the selection pressure 0.1. For the crossover operation, single point, double point, or uniform crossover operators are utilized, such that the particular operator can be selected with the probability 0.2, 0.2, and 0.6, respectively. The mutation is based on the normal distribution with the standard deviation 0.1, and the mutation rate is fixed at 0.1.



 §.§ Measurements

Experimental measurements are performed in the rectangular waveguide with the height 60 mm and the adjustable width, while the length of the waveguide is 1350 mm. The walls and the bottom of the waveguide are made of 15 mm thick aluminium and the lid is made of plexiglass with the thickness 6 mm. Sound waves are generated by the loudspeaker (Visaton BF 45/4) located at one end of the waveguide, and the recording of the transmitted field is done with the shotgun microphone (RODE NTG4) placed at the other end of the waveguide. To avoid reflections at the end of the waveguide, both the loudspeaker and the microphone are embedded into the porous inserts made of melamine foam. The length of the inserts is 150 mm. The schematics of the setup and its photo are shown in Figs. <ref>(b) and <ref>(d).

Generation and recording of the signals is controlled via the custom software based on the sounddevice python module <cit.>. For that both the loudspeaker and the microphone are connected to the USB audio interface Roland Rubix22, such that the loudspeaker is connected via the amplifier based on the Yamaha YDA138-E microchip. Generated signals are the chirped signals with the duration 10 s and the sampling frequency 44100 Hz. For each measurement the signal was generated and recorded 6 times and then averaged in the frequency domain. The transmission spectra in this case is defined as

    T = 20 log_10 p_tr/p_ref,

where p_tr is a pressure amplitude of the transmitted wave and p_ref is the reference pressure amplitude of the wave in the absence of the structure. The same procedure is performed for the measurements with the noise generated by a fan, i.e. the loudspeaker is substituted by the hair dryer (STINGRay ST-HD801A). The measured signal-to-noise ratio for both the loudspeaker and the hair dryer are presented in the Supplementary Information.

Ventilation measurements are performed in the same waveguide. The airflow is generated by the hair dryer, the same as for the noise measurements [see Fig. <ref>(c)]. The speed of the flow is measured with the anemometer (ADA AeroTemp 30) located at the distance 55 cm from the structure, such that the distance between the hairdryer and the structure is also about 55 cm. 

The samples are manufactured by 3D printing using a PLA filament. To avoid gaps between the lid and the structure, the tops of the plates were supplemented by a thin layers of a porous material [see Fig. <ref>(e)].





§ RESULTS



 §.§ “Spruce” Meta-Atom




The considerations start from the structure in which parameters of the plates change gradually, such that the coordinates of the plates along the x-axis are defined as 

    x_n = 5(n+1) + n(n-1)/2,

where n is the number of a plate, and the corresponding semi-widths are 

    w_n = 115 x_n/255.

The thickness of all plates is fixed at d = 2 mm. In the consequent text, such meta-atom will be refereed to as a “spruce”, just due to associations with some schematic pictures of the corresponding tree.
One-dimensional structures with the similar parameters were considered in Ref. <cit.>, where it was shown that attenuation inside the structure is associated with the resonances of the cavities formed by the plates. As in the considered two-dimensional case the structure is periodic, it is reasonable to start from the consideration of its band structure. The corresponding unit cell [see Figs. <ref>(a) and <ref>(b)] is characterized by the width a_x and the height a_y, which are considered to be 250 and 240 mm, respectively. According to the band structure shown in Fig. <ref>(d), the system is characterized by a set of band-gaps, which are practically merged in the regions 750 – 1450 Hz and 2200 – 2800 Hz, as the modes within this ranges are flat. The field distributions shown in Fig. <ref>(f) indicate that these modes correspond to the resonances in the cavities formed by the plates. Practically, the meta-atom can be considered as a set of resonators strongly coupled to each other. This coupling is the origin of the flat-band formation and the opening of band-gaps as discussed in the Supplementary Material. 

It should be expected, that when the equivalent finite-size structure is considered, the transmission spectra should be characterized by the stop-bands corresponding to the band-gaps of the infinite structure. Indeed, as shown in Fig. <ref>(e), the experimentally measured transmission drops below -40 dB in the corresponding regions 900 – 1450 Hz and 2450 – 2850 Hz. At the same time, not all band-gaps are manifested as the stop-bands. The reason for that is the small thickness of the structure, which is just a single unit cell. When the thickness increases, more stop-bands can be observed (see Supplementary Information). In particular, experimentally measured transmission spectra of a system having a thickness of two meta-atoms [see Fig. <ref>(e)] is characterized by a small dip within the range 300 – 450 Hz, where a band-gap is located. In addition, the two other stop-bands are a bit wider and deeper than for the case of the structures consisting of a single meta-atom. It should be also noted that the experimental results do not quite match the numerical ones, which is associated with the fact that in the real system the finite height of the samples and the presence of the lid and bottom of the waveguide result in the occurrence of additional thermoviscous losses, which are not taken into account in the calculations. In addition, geometric parameters of the samples might differ from the ones used in the numerical calculations due to the limited accuracy of the utilized 3D printer. Still, the qualitative behavior of the measured and calculated spectra is the same. In addition, an inverted structure is considered, such that the meta-atom is flipped inside the waveguide, i.e. the largest plate is located closer to loudspeaker and the the smallest one – closer to the microphone. The resulting transmission spectra [see Fig. <ref>(e)] is practically the same as for the initial orientation of the meta-atom, additionally indicating that the properties of the structure are determined by the resonances, rather than the gradual change of the admittance. Hence, the structure works as a reflector, not an absorber.



Still, an effect similar to the acoustic black hole can be observed for the case when the size width of the plates is equalized [see Fig. <ref>(a)]. In this case the band structure is characterized by two band-gaps, but the most interesting observation is the fact that the tilt of the low-frequency eigenmodes decreases with the increase of frequency until the modes become flat [see Fig. <ref>(b)]. These flat bands correspond to the lower boundary of the first band-gap. The same situation repeats for the modes between the first and the second band-gaps. As the inclination of the modes indicate the group velocity, defined as v_g = ∂ω/∂ k, it can be stated that the group velocity decreases with the increase of the frequency until reaching zero, following by the opening of a band-gap. This is a direct analogy of a conventional ABH effect in which the phase velocity decreases with the propagation coordinate. In this sense, the considered structure can be regarded as a reciprocal ABH. The “spruce” meta-atom in this case is practically the deformed version of such a structure allowing the formation of larger number of band-gaps, which is the aim of the work. While the further exploration of the effect lies beyond the scope of the work, some additional details are provided in Supplementary Information.







 §.§ Optimized meta-atom

While the “spruce” meta-atom demonstrate noticeable reduction of the transmission coefficient in the expected spectral regions, it might be argued that the geometric parameters of the structure are not optimized and the performance might be improved. The particular aim is to merge the stop-bands and simultaneously decrease the size of the structure. Following the analysis of coupled resonators and eigenmodes of periodic systems (see Supplementary Information), it might be expected that the optimized structure should consist of several nearly equivalent cavities characterized by a large width, as it allows to increase the width of band-gaps and shift them towards the low frequencies.
Using the optimization procedure described in Materials and Methods, the meta-atom shown in Fig. <ref>(a) is obtained. It consists of just 5 plates forming four cavities inside a unit cell and two additional cavities formed by the plates from the neighboring unit cells (along the x-axis). The geometric profile in this case is flat, meaning that all plates have the same width, contrary to the “spruce” meta-atom and typical geometries of SBH. There is also no gradual change of the cavity sizes, as it not an important requirement for the adjustment of the coupling between the resonators. The band structure of the infinitely periodic system consisting of such meta-atoms is characterized by broad band-gaps covering almost the whole range from 600 to 2800 Hz [see Fig. <ref>(c)]. Again, the modes correspond to the resonances in-between the cavities [see Fig. <ref>(e)], which can be characterized by both symmetric and anti-symmetric distributions. Here it also should be mentioned, that resonators also interact with their neighbors along the y-axis, which also affects the results (see Supplementary Information). Notably, such effect is unavailable in one-dimensional SBHs, which make a distinguish between these two types of systems. 
The corresponding transmission spectra of the finite-size structure is characterized by the broad stop-band spectrally coinciding with the band-gaps of the infinitely periodic structure [see Figs. <ref>(b) and  <ref>(d)]. Again, there qualitative differences between the numerical and experimental results caused by the limitations of the experimental setup, but still the transmission within the stop-band is mostly below -40 dB, while the stop-band itself is much larger than for the case of the “spruce” meta-atom.






 §.§ Ventilation

Geometry of the structure implies the presence of the gaps between the meta-atom and one of the waveguide walls. Therefore, it might be expected that an airflow propagating through the waveguide will not be fully blocked by the meta-atom. In particular, the specific case of the noise control in the ventilation ducts in which both acoustic noise and air flow propagate simultaneously. As discussed in the Materials and Methods, to test the structure under such conditions, a fan is integrated into one the waveguide termination ends. Obviously, both transmission and ventilation depend on the size of the gap between the meta-atom and the wall of the waveguide [see Figs. <ref>(a) and  <ref>(c)]. It might be expected that the increase of the gap will result in the corresponding increase of the air flow speed behind the structure and, at the same time, in the increase of the transmission coefficient.

Indeed, as shown in Fig. <ref>(b), the averaged transmission coefficient increases  when the gap becomes larger. For the case of the “spruce” meta-atom, the averaged transmission coefficient changes almost linearly, such that the difference between the lowest and the largest value is just -5 dB. For the optimized meta-atom the situation is a bit more complicated, and the dependence of the transmission coefficient on the gap size resembles a logarithmic function. The change in the values is rather drastic, about 15 dB. Notably, the optimized structure demonstrate much better noise-insulating performance at low values of Δ w, than the “spruce”, which was actually the aim of the optimization procedure. When the gap size increases, and becomes more than 10 mm, the “spruce” structure demonstrates better performance, indicating the fact that the optimization procedure should be actually performed for each value of Δ w.

The ventilation properties of both meta-atoms is quite similar to each other [see Fig. <ref>(d)]. Expectantly, the ratio of the air flow speed U and the reference flow speed  U_0 (i.e. the speed without the structure) increases with the increase of the gap between a structure and the waveguide. The performance of the “spruce” meta-atom is slightly better, than for the optimized one, which might be associated with its geometry implying the gradual decrease of the effective channel width. In any case, the ventilation above 50% is achieved when the value of Δ w exceeds 15 mm. For Δ w = 25 mm the ratio U/U_0 is nearly 0.75, meaning that only 25% of the air flow is blocked. The transmission spectra in this case is still characterized by stop-bands [see Fig. <ref>(d)] occurring within the range 1000–2800 Hz. Contrary to the previous measurements with the loudspeaker, the transmission spectra for the case of the hair dryer is noisy, but distinguishable gaps are still clearly visible. Therefore, a compromise between noise-insulation and ventilation properties can be found for the realistic application scenario.







§ CONCLUSION

To conclude, the work is dedicated to the investigation of 2D structures having the geometries similar to those of 1D sonic black holes. It is shown that the change of the dimensionality practically ruins the black hole effect and the decrease in transmission is associated with the opening of band-gaps, such that the structure works as a reflector rather than absorber. The formation of band-gaps is associated with the coupling between the resonators constituting the structure. As a consequence, the geometric profile in the 2D case is not the key factor, and only the size of the cavities matters. Still, an analog of a sound black hole effect is numerically observed, such that the group velocity gradually decreases with frequency until reaching zero, though this effect is not utilized in the work. As the main result, it is shown that fro the realistic scenario of a ventilation duct in which both the airflow and acoustic noise propagate simultaneously, it is possible to find a compromise between ventilation properties and noise insulation.
In particular, it is shown that the reduction of the air flow speed might be about 25% while the broad stop-band is observed within the spectral range 1000–2850 Hz.
These results might be useful for further development of ventilated acoustic metamaterials and noise insulating systems for passive noise insulation, especially in ventilation ducts.



§ AUTHOR CONTRIBUTIONS

FB, LG, YM, DP, and EV contributed equally to this work. 
LG and SK provided the numerical calculations. FB, LG, YM, DP, EV and MK performed the experimental measurements.
SK proposed the idea and supervised the project. MK acquired the funding and co-supervised the project.



§ DECLARATION OF COMPETING INTEREST

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.



§ DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.



§ ACKNOWLEDGEMENTS

The authors thank Aleksandra Pavliuk for the help with the experimental measurements and Mikhail Kuzmin for fabrication of the samples. The authors also thank Anton Melnikov for fruitful discussions and useful comments. 
This work is supported by the Russian Science Foundation (project 24-21-00275). 





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