Continuous near-perfect sound absorption of a slit-resonator acoustic metastructure

With the rapid development of modern industry, global environmental pollution has also occurred, and noise pollution, as a kind of environmental pollution, has become a major hazard to human beings. The effective attenuation of noise for modern people's daily life, machinery, equipment, and transportation is a crucial topic. Sound-absorbing structures or materials can convert sound energy into heat energy for consumption, which is widely applied in various scenarios. Traditional sound absorption structures, including the Helmholtz resonator,^1–4 microperforated plates (MPPs),^5–7 and membrane structure,^8–10 can absorb low-frequency sound waves efficiently but often require larger geometries. Porous materials are ideal for sound absorption, almost exclusively at medium and high frequencies. Thus, it is imperative to propose metamaterials with low-frequency broadband sound absorption performance.

To achieve efficient sound absorption with compact dimensions, an effective strategy is to optimize the cavity configuration. Chen et al.¹¹ designed an acoustic metasurface with an average absorption coefficient of 0.931 at 350–1000 Hz with a thickness of 55 mm, which consists of MPPs and folded cavities. Huang et al.¹² proposed a metasurface formed by embedded apertures combined with convoluted channels. Wang et al.¹³ designed a low-frequency sound-absorbing metamaterial consisting of a perforated plane and a coiled-up cavity, which achieved lower frequency sound absorption by adjusting the number of folds. Applying spiral necks and coiled cavities, Guo et al.¹⁴ presented a metasurface that achieved 98% acoustic energy absorption at 180 Hz with a thickness of 13 mm. Wu et al.¹⁵ proposed an improved finger winding method, and the obtained metastructure achieved broadband absorption at 375–855 Hz. Unlike rigid structures, flexible structures also show good potential for broadband sound absorption. Gao et al.¹⁶ integrated the soft hyperelastic material into the Helmholtz resonator, and the sound absorption was adjustable by mechanical loading. Benouhiba et al.¹⁷ proposed an adjustable Helmholtz resonator based on origami, which can adjust the resonant effect by changing the cavity size. Sun et al.¹⁸ proposed a soft-wall space-coiled metamaterial, in which the deformation of the soft wall helps to improve the dissipation of sound energy.

In addition, synergistic coupling of the multi-cavity combination can effectively widen the vocal absorbing cords. The first method is to combine cells with quasi-perfect absorption, and a sound absorber with broadband absorption capacity is pieced together by adjusting the number of cells, the parameters of each cell, and the combination form of cells.^19–21 Huang et al.²² proposed a metasurface that achieved an average absorption coefficient of 0.957 at 870–3224 Hz. Utilizing the excellent acoustic and mechanical properties of honeycomb sandwich panels, Peng et al.²³ presented a composite honeycomb metasurface that achieved 90% acoustic energy absorption with a thickness of no more than 30 mm. Liu et al.²⁴ proposed a multi-element metasurface based on resonators and perforated plates, with a sound absorption coefficient greater than 0.8 in the range of 450–1360 Hz. The second method is to form a collaborative coupling of the resonator with other elements, such as microperforated plates, resonant membranes, and slit type sound absorbers.^24–27 Ma et al.²⁵ combined the slits with the resonators to obtain an average absorption coefficient of 0.8 at 50 Hz. Liu et al.²⁶ designed a smart MPP consisting of an MPP and a piezoelectric ceramic, and improved sound absorption through different multi-mode shunt design methods. Zhu et al.²⁷ designed a composite metastructure consisting of a central hole and multiple cylindrical layers, and applied it to the noise control of the fairing, reducing the maximum average sound pressure level by 17.5 dB.

To broaden the absorption band, porous materials are introduced into acoustic metamaterials.^28–30 Prasetiyo et al.³¹ proposed a labyrinthine metastructure embedded into a nonwoven material, which reaches a sound absorption coefficient of 0.97 at 1380–2400 Hz. Yuan et al.³² presented a tunable composite metasurface consisting of a micro­perforated panel and porous material, and that reaches an absorption of 0.6 at 55–3000 Hz and 0.8 at 1500–3000 Hz. Dong et al.³³ processed a folded sound absorber with woven prepreg and ventilated felt, and achieved a sound absorption over 0.4 at 400–6300 Hz. Efficient absorption of acoustic waves can also be achieved with metallic porous materials. Ding et al.³⁴ proposed a metastructure that consists of an MPP and neck-embedded Helmholtz resonators that reaches efficient sound absorption in the range of 800–3200 Hz with a thickness of 40 mm. Zhang et al.³⁵ proposed a metastructure composed of folded slits and a porous matrix, which achieves remarkable sound absorption at 200–600 Hz. In summary, previous studies have shown that an organic combination of resonant structures and porous materials help to improve sound absorption performance with a compact structure.

Inspired by the above works, a novel slit-resonator acoustic metastructure (SRAM) composed of a series of parallel-coupled Helmholtz resonators, air slits, and porous materials is presented, as shown as Fig. 1. Various types of porous materials with significant differences in static flow resistance are introduced to enhance the tunability of the structure. In addition, a combination of different porous materials, including parallel and vertical combinations, is adjusted to improve the absorption performance over a wider frequency range. The metastructure is investigated by theoretical analysis, finite element study, and experimental verification. The sound absorption mechanism is revealed on the basis of theoretical and numerical models. In addition, genetic algorithms are used to optimize the parameters to achieve near-perfect sound absorption over a wider range within 200–3000 Hz.

The slit-resonator acoustic metastructure (SRAM) consists of a Helmholtz resonator and porous materials, as shown in Fig. 1(a), where the neck of the Helmholtz resonator is surrounded by porous materials and can be seen as a slit when its diameter is small enough. To simplify the theoretical derivation, the 3D model is simplified to a 2D model, as shown in Fig. 1(b). Geometrical parameters affecting the acoustic performance of SRAM are thickness H, cavity thickness L, porous material thickness L_p, total diameter D_t, and slit diameter D. To achieve effective sound absorption over a wider frequency range, a SRAM with more unit cells is designed, as shown in Fig. 1(c), with the number of unit cells denoted by N.

To facilitate theoretical modeling, the slit-resonator acoustic metastructure (SRAM) is divided into two layers, as shown in Fig. 1(d). Layer 1 is the air cavity layer with thickness L_l1 = L. Layer 2 is composed of an air slit and a porous material domain of the same thickness L_l2 = L_p.

In order to show the sound absorption performance of SRAM more intuitively, the concept of half-absorption bandwidth is introduced, which is the width of the absorption band when the absorption coefficient of SRAM reaches 0.5. Assuming that the absorption coefficient of SRAM is not less than 0.5 from $f 1$ to $f 2$⁠, the half-absorption bandwidth is expressed as $f 2− f 1$⁠, and if there are more than one such frequency range, the total bandwidth is the sum of the individual bandwidths.

The numerical model of SRAM is created in comsol Multiphysics 6.0, as shown in Fig. 2(a). The air domain of the impedance tube and the cavity is set to air, the material parameters are defined, and the solid domain is set as the hard boundary. The speed of sound c₀ is 340 m/s, the temperature is 25 °C, and the pressure is 1 atm. The rigid walls are set as sound hard boundaries without deformation. For the porous material portion of the metastructure, add settings for poroacoustics to the pressure acoustics, frequency domain interface in comsol Multiphysics 6.0. A free tetrahedral meshing model is used, as shown in Fig. 2(b).

In numerical simulations, the incident background pressure field is expressed as $P in= e − j k 0 l$⁠, where $k 0=2πf/ c 0$⁠, representing the number of waves in the pressure field, and l is the distance between the incident surface of the SRAM and the sound waves. When the simulation is performed, the incident plane wave is incident on the upper surface of the metastructure and enters the cavities and porous materials. Moreover, a probe is set to detect the value of incident pressure $P in$ and scattered acoustic pressure $P sc$⁠. The process of calculating the sound absorption coefficient is as follows:

The experimental devices are shown in Fig. 3(a). The experimental devices are composed of a data collector, an impedance tube, a high-precision microphone, monitoring software, a power amplifier, a digitizer, two sensors, and a calibrator. The sample should be installed in the test tube with the right size, without excessive pressure, and not too tight to bulge. The microphone is calibrated using a standard sound level calibrator to calibrate the sensitivity of the microphone to be tested. After the microphone for the test is connected to the system, it is calibrated with a standard sound source (1 kHz, 94 dB).

The experimental principle is shown in Fig. 3(b). The sound source and the sample are at opposite ends of the impedance tube. When sound waves travel through the tube, a part of the energy is absorbed or reflected by the sample, and the other part is reflected by the rigid end of the impedance tube. During this process, the signals of the incident and reflected waves are received by the two sensors, and the absorption coefficient curves are calculated by the monitoring software based on these two signals.

Before the experiment begins, set the hardware parameters, experimental temperature and atmospheric pressure conditions, and test tube type in the software interface. Through the process of testing, calculation, and analysis, the sound absorption coefficient curve of the sound absorber is obtained. The experimental samples are 3D printed using stereo lithography apparatus (SLA) stereoscopic light-curing molding technology, made of a photosensitive resin material. The total thickness H of both samples is 50 mm, with a thickness of 40 mm for the air slit-porous material domain and 10 mm for the backing cavity, and the total diameter D_t is 100 mm. For the SRAM with N = 1, the radius of the central air slit of the slit-porous material domain is 10 mm, the thickness of rigid wall is 0.5 mm, the annular width of the porous material area is 39 mm, and the radius of the backing cavity is 49.5 mm. The SRAM with N = 4 is divided into four units, with unit 1 consisting of an air slit and an annular porous material domain, and units 2–4 consisting of two annular porous material domains and an air slit in the sandwich in the slit-porous material domain. The radius of the air slit in unit 1 is 0.5 mm, the width of the porous material area is 9.5 mm, and the radius of the backing cavity is 10.5 mm. The width of the air slit in units 2, 3, and 4 is 1 mm, the width of the porous material domain near the center is 5.5 mm, and that of the other porous material domain is 5 mm, and all wall thickness is 0.5 mm. Comparisons of the results obtained by the three methods for samples with unit numbers 1 and 4 are shown in Figs. 3(c) and 3(d). The results of the two samples are in good agreement, and small differences may be caused by manufacturing errors. Due to the limitations of the impedance tube experiment, the effective frequency is only 200–1600 Hz. However, because of the consistency of theoretical absorption coefficient, numerical absorption coefficient, and experimental absorption coefficient in the range of 200–1600 Hz, it can be considered that the experimental method is correct and the numerical results at higher frequencies are reliable.

The effect of parameters is discussed in order to investigate the sound absorption performance of the slit-resonator acoustic metastructure (SRAM). The Johnson–Champoux–Allard (JCA) model is used for finite element analysis of porous materials. The porous material applied in the first and second sections is melamine foam (MF), which has JCA parameters of viscous feature length L_v of 240 μm, thermal feature length L_th of 470 μm, flow resistivity R_f of 10 500 Pa s/m², porosity ɛ_p of 0.995, viscous feature length parameter s of 0.49, and curvature τ of 1.0059.³⁷ The sound absorption coefficients mentioned in the revision are obtained from simulations unless specifically mentioned as being obtained from theoretical models or from conducting experiments. In the figures, the theoretical results and experimental results are marked with Ana or Exp in brackets. If there is no special annotation on the curve, it is the theoretical result.

According to the analytical derivation, the sound absorption coefficient of the SRAM is correlated with the total thickness H, cavity thickness L, air space diameter D, and the relative position of the air slit and the porous material to the air space, denoted by L₁. The control structure W0 is set to a total thickness H of 50 mm, a cavity thickness L of 10 mm, an air slit diameter D of 15 mm, and L₁ of 0.

The effect of the total thickness H is shown as Fig. 4(a). With H increasing, the absorption peak of SRAM increases and the frequency decreases. It can be seen from the absorption curves that when H = 50 mm, the perfect absorption band (α > 0.9) of SRAM is 391–501 Hz. When H = 100 mm, it is 211–286 Hz. These two SRAMs show absorption peaks at 441 and 241 Hz with peaks of 0.963 and 0.996, respectively. It can be assumed that the increase of H can cause the perfect sound absorption band to move to low frequency, but the bandwidth will be narrowed.

As shown in Fig. 4(b), by maintaining H at 50 mm, the absorption coefficient of SRAM reaches 0.5 at 421 Hz when L is 5 mm, and when this parameter is increased to 45 mm, the half-absorption frequency of SRAM moves to the lower 268 Hz. The increase in the cavity thickness L in the range of [5,45] mm will significantly cause the decrease of SRAM resonance frequency and the increase of peak value, and also the decrease of the half-absorption bandwidth.

It can be seen from Fig. 4(c) that when L₁ is 0 and 40 mm, the absorption peaks of SRAM are 0.783 at 1000 Hz and 0.963 at 441 Hz, respectively, and effective sound absorption (α > 0.5) can be achieved within 460–1000 and 286–1000 Hz, respectively. Overall, as L₁ increases, i.e., the pure cavity gradually moves away from the sound wave incident surface, the effective absorption frequency shows a decreasing trend, and the peak value shows an increasing trend.

Figure 4(d) shows the influence of the air slit diameter D. The SRAM shows an absorption peak of 0.932 at 441 Hz and 0.920 at 601 Hz with half-absorption bandwidths of 2734 and 2639 Hz when D is 15 and 20 mm, respectively. The decrease in the air slit diameter in the range of [8,40] mm improves the absorption capacity of SRAM at low frequencies and causes an increase in the peak absorption. In addition, two SRAMs (D = 15 and 20 mm) are selected and their sound pressure and thermal viscous dissipation distributions at 201, 601, and 1000 Hz are shown in Figs. 4(e)–4(p). At the resonance frequency, strong air particle vibration occurs in the air slit, so the sound pressure is much higher than the other parts. In the low-frequency range (201 Hz), the sound pressure in the cavity inside the HR is higher, where the absorption performance is mainly carried out by resonance. At high frequencies (1000 Hz), the sound pressure at the top of the metastructure is higher, and the absorption of sound energy mainly relies on the porous material.

Figure 4(q) reveals the absorption mechanism and coupling effect of SRAM. The absorption coefficient curve of SRAM shows one absorption peak at 400 Hz, close to the absorption peak of HR with a rigid boundary at 561 Hz. In the high-frequency range, the absorption coefficient of SRAM has a similar upward trend to the curve of melamine foam (MF) with a microperforated plate (MPP). The results show that HR improves the sound absorption performance in the low-frequency range, while the porous material contributes more in the high-frequency range. It is worth noting that compared with the rigid boundary, the first peak frequency of HR with a porous boundary moves to 541 Hz, and the peak is increased by 0.014, while the second peak moves from 2101 Hz to a lower 2001 Hz with a decrease in peak by 0.047. This is because when local resonance occurs, the porous boundary also absorbs a portion of the acoustic energy, which affects the sound absorption performance of the metastructure.

In addition to the variety in geometric parameters that affect the absorption coefficient of the slit-resonator acoustic metastructure (SRAM), the structural form of SRAM also has an impact on the absorption performance. The influence of the number of unit cells N, the form of the upper surface, and the form of the Helmholtz resonator and porous material on the sound absorption coefficient is discussed in this section in order to explore the effects of structural form.

According to the previous studies on resonant structures, the variation of the unit cell number will not only cause a difference in the effective sound absorption frequency, but also cause an increase or decrease of the absorption peak. The absorption performance of SRAMs with unit numbers N of 1, 2, 4, 7, and 10 is studied as shown in Fig. 5(a) separately. It can be seen that when N = 1, the sound absorption coefficient of SRAM in 0–3000 Hz is below 0.7. When the unit number N increases to 4, the sound absorption coefficient increases, reaching 0.5 at 691 Hz and 0.8 at 1161 Hz, continuing until 3000 Hz. However, it is not that the larger the N, the better the sound absorption. When N increases to 10, the absorption coefficient of SRAM reaches 0.5 at 761 Hz and 0.8 at 1231 Hz. The half-absorption bandwidths of SRAMs with N of 1, 2, 4, 7, and 10 are 1936, 2326, 2309, 2268, and 2239 Hz, respectively. The absorption coefficient curves of these five SRAMs show the same trend, while the half-absorption bandwidth increases and then decreases with the increase of N, indicating that the absorption performance of SRAMs is enhanced and then weakened when N is gradually increased from 1 to 10. However, by comparing the two curves of N = 4 and N = 10, it can be seen that the attenuating effect is not significant.

Figures 5(b)–5(i) show the total thermoviscous dissipation and sound velocity corresponding to the peak frequencies of three SRAMs with N = 1, 4, and 10. When N = 4, the sound velocity in the air slit is very high, and the sound wave propagates the deepest in the SRAM, and promotes the friction between the sound wave and the wall of the air slit, causing energy dissipation. Moreover, the propagation of sound waves allows them to come into full contact with the porous material, causing additional energy dissipation. When N is greater than or equal to 2, SRAMs with different N show similar results.

The thermoviscous dissipation and acoustic velocity of SRAMs with N = 1, 4, 7, 10 at resonance frequencies are analyzed to reveal the absorption mechanism of SRAMs with different unit cell numbers N, as shown in Figs. 5(b)–5(i). f 1–f 4 are 3000, 2001, 1938, and 1921 Hz, respectively. It is worth noting that since there is no significant absorption peak in the range of 200–3000 Hz when N = 1, the frequency corresponding to the maximum absorption coefficient in this range is selected. The results show that the common feature of the four SRAMs is that the thermal viscosity dissipation at the incident surface is significantly higher than that at other locations, and the thermal viscosity dissipation gradually weakens as the sound wave propagates. This is because, on the one hand, a part of the sound waves is absorbed by porous materials during the propagation process, and on the other hand, porous materials hinder the propagation of sound waves due to their high flow resistance. In addition, acoustic velocity in the air slit is significantly higher than that of the porous material, indicating that the sound wave entering the SRAM through the air slit has sufficient friction with the wall of the slit. However, the thickness corresponding to the thermoviscous dissipation and the red region of acoustic velocity is the largest when N = 4, followed by N = 7, N = 10, and the thermoviscous dissipation and acoustic velocity are the weakest when N = 1. This explains that the sound absorption coefficient is relatively large when N = 4 and the worst when N = 1.

The second structural factor that affects the absorption performance analyzed in this section is the form of the upper surface. The SRAM with N = 4 is selected for analysis and two upper surface forms are considered, the air slit and the microperforated plane, as shown in Fig. 6(a). Cases 1 and 2 represent the SRAMs when the upper surfaces of the four cells are all air slits and microperforated planes, respectively. The two SRAMs achieve half-absorption at 688 Hz (f 1) and 381 Hz (f 3), corresponding to the half-absorption bandwidths of 2312 and 1034 Hz, with peak frequencies of 2001 Hz (f 2) and 781 Hz (f 4), respectively. The introduction of microperforated planes narrows the half-absorption bandwidth, but moves the peak frequency toward low frequency, that is, the low-frequency absorption performance of SRAM is improved, but the high-frequency absorption performance is significantly attenuated.

In addition, three forms of cavity thickness of the Helmholtz resonator are analyzed, including a conventional structure and two gradient structures. When the cell near the center of the structure has the thinnest air slit and porous material thickness, denoted as Case 4, the peak of SRAM is lower than the other two structures. However, there is no significant difference in the absorption coefficients of the three SRAMs. The frequencies at which half-absorption is reached are 742 Hz and 938 Hz for Case3 and Case4, respectively, corresponding to half-absorption bandwidths of 2258 Hz and 2062 Hz in the frequency range 200–3000 Hz, with peak frequencies of 2001 Hz (f 5) and 1961 Hz (f 6), respectively. Figures 6(b)–6(m) show the total thermoviscous dissipation and acoustic velocity of Cases 1–4. Comparing the results of Case1 at the two frequencies, the thermoviscous dissipation at 689 Hz is mainly provided by the central unit cell, while all four cells show strong thermoviscous dissipation at 2001 Hz, and the dissipation value is significantly greater than the former. Similarly, Case 2 dissipates thermal viscosity significantly higher at 881 Hz than that at 296 Hz. For the velocity of sound, the law of air slits higher than other locations is shown. Comparing the results of Cases 1, 3, and 4, it can be seen that the thermal viscosity dissipation distribution is positively correlated with the thickness of the porous material. The thicker the cell of the porous material, the greater the thermal viscosity dissipation. The velocity of sound shows the opposite result, the unit with a smaller thickness of the porous material has a larger acoustic velocity. Combined with the coupling effect analysis in Sec. III A, it is shown that porous materials play a major role in sound absorption at the absorption peak, and the location with less porous materials achieves better sound absorption by enhancing the friction between the sound wave and the wall.

Previous studies³⁸ have shown that a combination of microperforated plates, cavities, and porous materials (MPP-AG and MPP-AG-MF) enables effective sound absorption over a wide frequency band. The sound absorption coefficients of SRAM, MPP-AG, and MPP-AG-MF with the same MPP and cavity parameters are shown in Fig. 6(a). The porosity of the four units is 2%, 3%, 4%, and 5%, the aperture is 0.9, 1.0 mm, 1.1 mm, and 1.2 mm, and the microperforated plane thickness is 0.5 mm, 1.0 mm, 1.5 mm, and 2 mm, respectively. The total thickness of the structure is 50 mm, and the air space thickness of the SRAM is 10 mm. As shown by the black curve, a SRAM with MPP reaches half-absorption at 361 Hz and peaks at 0.89 at 1221 Hz, completely covering the sound absorption coefficient curves of the other two metastructures. This suggests that SRAM is better for sound absorption than the other two metastructures.

The analysis above shows that porous materials are critical to the sound absorption performance of SRAM, and the effect of porous material parameters is next investigated. The JCA parameters of three porous materials are shown in Table I, and the sound absorption of SRAMs with porous materials 1, 2, and 3 are shown in Fig. 7(a). The geometric parameters of the three samples are identical: H = 50 mm, L = 10 mm, N = 4, D = 1 mm, and L₁= 0. The SRAM containing material 1, material 2, and material 3 achieve half-absorption of 390–3000, 691–3000, and 337–3000 Hz, respectively, with bandwidths of 2610, 2309, and 2663 Hz, and absorption peak frequencies of 1201, 2001, and 541 Hz, respectively. Compared to the SRAM with fiberglass, the SRAM with rock wool can not only shift the peak absorption frequency to a lower frequency, but also increase the sound absorption bandwidth. However, in terms of smoothness of the sound absorption coefficients, SRAMs with fiberglass and melamine foam are superior to SRAMs with rock wool.

In addition, the total thermoviscous dissipation and acoustic velocity of the three samples are shown in Figs. 7(b)–7(e). In the range of 200–3000 Hz, there is only one peak in the curves of samples 1 and 2, at 1201 and 2001 Hz, respectively, and two absorption peaks at 541 and 2401 Hz in the absorption coefficient curves of sample 3. For all three SRAMs, the maximum thermoviscous dissipation is located at the acoustic incidence surface and the direction of acoustic velocity in SRAMs is mainly from the upper surface to the end of the metastructure. However, there exist significant differences in the distribution of thermoviscous dissipation along the direction of acoustic wave propagation for the three SRAMs. The thermoviscous dissipation in the four cells of samples 1 and 2 gradually decreases with the propagation of the sound wave and the thicknesses required to fall to the minimum value are similar, but the maximum value of the dissipation is larger in sample 1, which implies that the thermoviscous dissipation in sample 1 decays more significantly with propagation. The direction of propagation of sound waves in both sample 1 and sample 2 is from the incident surface to the bottom of the structure. In contrast, the thickness of the thermoviscous dissipative distribution is larger for sample 3, especially at the absorption peak at 541 Hz. At 541 Hz, the thermo-adhesive dissipation of sample 3 along the direction of acoustic wave propagation first diminishes to close to zero, and then increases gradually, with still high values of dissipation at the end of the porous materials. The maximum thermoviscous dissipation of sample 3 at 2401 Hz is larger than that at 541 Hz, but decays more rapidly, albeit slower than samples 1 and 2. This is due to the higher static flow resistance of material 3, which makes the propagation of sound waves more difficult. In addition, unlike samples 1 and 2, the speed of sound in the slit at the two absorption peaks in sample 3 is clearly higher than that at the other locations, and the first absorption peak of sample 3 is higher due to the stronger resonance effect. This is mainly because of the relatively high static flow resistivity of material 3, and the propagation of sound waves in porous materials is impeded more significantly.

In conclusion, a reasonable design of SRAM geometric parameters, number of cells, upper surface form, and selection of appropriate porous materials can help to improve the sound absorption performance of SRAM.

In order to further understand the influence of porous materials on the sound absorption performance of slit-porous material composite metastructures, three materials are combined, and the combination forms are divided into two types, namely, layered combination and interval combination. Wherein, the layered combination is that the porous material in each unit is divided into two layers of the same thickness, and there are six types of layered combinations. The upper material in the layered combination 1 is fiberglass, and the lower material is melamine foam. The upper material in the layered combination 2 is fiberglass, and the lower material is rock wool. Layered combination 3 and combination 4 are composed of melamine foam and fiberglass, and melamine foam and rock wool, respectively. The upper material of combinations 5 and 6 is rock wool, and the lower material is fiberglass and melamine foam, respectively. The interval combination refers to the filling of elements 1 and 3 with the same porous material, and the filling of elements 2 and 4 with another material in the slit-porous material composite structure, with a total of six interval combinations. The interval combinations 1–6 are composed of fiberglass and melamine foam, fiberglass and rock wool, melamine foam and fiberglass, melamine foam and rock wool, rock wool and fiberglass, and rock wool and melamine foam, respectively. The sound absorption coefficients of the slit-porous material combination of the two combined forms are compared with sample 1 in Fig. 7(a), as shown in Fig. 8.

As can be seen in Fig. 8(a), the half-absorption frequencies of the layered combination metastructures 1–6 are 443–3000, 366–3000, 538–3000, 462–3000, 348–3000, and 378–3000 Hz, and the half-absorption bandwidths are 2557, 2634, 2642, 2538, 2652, and 2622 Hz, respectively. The first absorption peaks of the layered combination metastructures 5 and 6 have lower frequencies of 580 and 682 Hz, respectively, and higher peaks of 0.9901 and 0.9949, respectively. Compared with sample 1, although the layered combination metastructures 5 and 6 showed superior sound absorption performance at 200–800 Hz, the absorption coefficient curves of the two metastructures show valleys at 1763 and 1986 Hz, with trough values of 0.6587 and 0.5884, respectively. In addition, the sound absorption performance in the range of 1000–3000 Hz is significantly different from that of sample 1. Compared with sample 1, the peak frequency of the layered combination structure 2 is lower than that of sample 1, the peak frequency is lower than that of sample 1, the peak sound absorption is 0.9346, and the sound absorption coefficient curve in the range of 1000–3000 Hz is higher than that of metastructures 5 and 6 but still lower than that of sample 1. The layered combination structures 1 and 3 have higher peak frequencies of 1421 and 1783 Hz, respectively, but larger peaks of 0.9898 and 0.9984, respectively. In addition, the layered combination metastructures 1 and 3 achieve superior sound absorption at high frequencies. Among them, metastructure 3 still reaches a sound absorption coefficient of 0.9527 at 3000 Hz. The peak frequency of the layered combination metastructure 4 is 1021 Hz and the peak is 0.9715. It can be concluded that the combination of fiberglass and melamine foam is better able to achieve high-frequency sound absorption. When the upper material is rock wool, the absorption effect of the layered composite structure in the low frequency range is improved, but the sound absorption performance in the high frequency range is not so good.

As shown in Fig. 8(b), the half-absorption frequencies of interval combination metastructures 1–6 are 498–3000, 347–3000, 430–3000, 348–3000, 350–3000, and 391–3000 Hz, and the half-absorption bandwidths are 2502, 2653, 2570, 2652, 2650, and 2609 Hz, respectively. The interval combinations 1 and 3 have similar peak frequencies of 1861 and 1701 Hz, respectively, and similar peaks of 0.9948 and 0.9990, respectively. The growth trend of the two curves is the same, and with the increase of frequency, the sound absorption coefficient gradually reaches the peak, and then, the sound absorption coefficient decreases with the frequency increasing. The absorption coefficient curves of interval combination metastructures 2, 4, and 6 show the same trend, with a sound absorption peak first appearing in the low frequency range below 600 Hz, a trough with a lower sound absorption coefficient as the frequency increases, and then, the sound absorption coefficient increases with frequency, and finally, it decreases. The first absorption peaks of structures 2, 4, and 6 are 563, 541 and 542 Hz, respectively, and the peaks of the first absorption peaks are 0.9534, 0.9148, and 0.7394, respectively. It can be seen that the sound absorption coefficient of the interval combination metastructure 2 is high in the frequency range below 1000 Hz, the sound absorption effect of metastructure 6 is excellent in the range above 1000 Hz, and the interval combination metastructure 4 has the advantages of structures 2 and 6, that is, the frequency of the first sound absorption peak is low, the first sound absorption peak is high, and it has high sound absorption performance at high frequency. The sound absorption coefficient curve of the interval combination metastructure 5 is relatively stable, reaching a sound absorption coefficient of 0.5 at 361 Hz, and an average sound absorption coefficient of about 0.9 in the range of 800–3000 Hz. It can be seen that the combination of fiberglass and melamine foam can achieve an absorption coefficient of 0.8 of high-frequency sound waves, while the frequency of effective sound absorption (α > 0.5) is high, and it is difficult to form sound absorption peaks in the low-frequency range. Melamine foam is filled with units 1 and 3, and fiberglass is filled with units 2 and 4 to obtain better sound absorption. The combination of fiberglass and rock wool can achieve effective sound absorption at a lower frequency (<400 Hz), and the sound absorption effect is relatively stable after reaching the sound absorption peak, and the sound absorption coefficient is greater than 0.8 in 500–3000 Hz. Filling the rock wool in units 1 and 3 and fiberglass in units 2 and 4 can make the sound absorption coefficient curve more stable. It can be deduced that the combination of melamine foam and rock wool results in a lower frequency first absorption peak, with unit cells 1 and 3 filled with melamine foam, and unit cells 2 and 4 filled with rock wool to improve the absorption of the first absorption peak.

Based on the parametric analysis of Sec. III, SRAMs with both low frequency and ultra-wideband sound absorption are proposed, as shown in Fig. 9(a). The three SRAMs have the same structural parameters: total thickness H = 50 mm, cavity thickness L = 10 mm, unit cell number N = 4, air space diameter D = 1 mm, and the relative position of the porous material and the air slit to the air space L₁ = 0. The difference is that the unit cells are filled with different combinations of porous materials. In SRAM 1, units 1 and 2 are filled with M2, and units 3 and 4 are filled with M1. Compared with SRAM 1, SRAM 2 replaces the porous material of units 3 and 4 with M3, and SRAM 3 replaces the porous material of units 1 and 2 with M3.

The sound absorption curves of the three designed SRAMs are shown in Figs. 9(b)–9(d). By adjusting the distribution of the porous materials, the three coupled SRAMs can achieve excellent sound absorption in the frequency ranges of 381–3000, 331–3000, and 346–3000 Hz, respectively, and the half-absorption bandwidths are 2619, 2669, and 2654 Hz, respectively.

The sound adsorption of the SRAM can be improved based on the above parameter effect analysis. To achieve a lower frequency of the absorption peak and a wider absorption frequency band, the genetic algorithm is adopted to optimize the SRAM parameters.

In the genetic algorithm, the population size is 200, the number of iterations is 500, the mutation rate is 0.05, the crossover probability is 0.8, and the initial probability of variation is 0.05. Random initialization and real number encoding are adopted. The optimization model and fitness functions of the above geometric variables are calculated.

The optimization process and convergence plot of the genetic algorithm are shown in Figs. 10(a) and 10(b), and the 3D printed samples and acoustic properties of the optimized structure are shown in Figs. 10(c) and 10(d), respectively. The total thickness H of the optimal SRAM is 50 mm, the total diameter D_t is 100 mm, and the wall thickness is 0.5 mm. The radius of the air slit in unit 1 is 0.7 mm, the width of the porous material area is 9.3 mm, and the thickness of the backing cavity is 13 mm. For unit 2, the radius of the air slit is 0.6 mm, the width of the inner porous material domain and the outer porous material domain is 4.6 and 4.8 mm, respectively, and the cavity thickness is 16 mm. In unit 3, the radius of the air slit is 0.9 mm, the width of the inner porous material domain is 3.8 mm and that of the outer porous material domain is 5.3 mm, and the cavity thickness is 17 mm. For unit 4, the radius of the air slit is 0.4 mm, the width of the inner porous material domain and the outer porous material domain is 3.5 mm and 6.1 mm, respectively, and the cavity thickness is 12 mm. The porous material in units 1 and 3 is melamine foam, and that in units 2 and 4 is fiberglass. It can be seen that the analytical and numerical experimental results agree well, which are greater than 0.5 in the range of 281–3000 Hz, and greater than 0.8 at 400–3000 Hz. In addition, the real part of impedance of the optimized SRAM is close to 1 and that of the imaginary part is close to 0, which meets the requirements of a perfect sound absorber.

In this study, a novel slit-resonator acoustic metastructure (SRAM) is proposed to achieve efficient sound absorption at 200–3000 Hz. The metastructure SRAM, which is composed of Helmholtz resonators and porous materials, is investigated attempting to achieve a balance between a high absorption coefficient and a broadband. The SRAM with melamine foam and rock wool achieves an absorption coefficient of more than 0.5 at 331–3000 Hz, and reaches a peak of 0.946 at 501 Hz with a thickness of 50 mm. The sound absorption coefficient of SRAM containing melamine foam and fiberglass is greater than 0.5 in the frequency range of 381–3000 Hz with a thickness of 50 mm, and reaches a higher peak of 0.987 at 1561 Hz. Furthermore, the parameters of SRAM are optimized using the genetic algorithm. The optimized SRAM obtains a sound absorption coefficient greater than 0.5 at a low frequency of 281 Hz and greater than 0.8 at 400 Hz, achieving an effective sound absorption of 200–3000 Hz with a thickness of 50 mm. The SRAM is adjustable to achieve low-frequency and ultra-broadband sound absorption performance, with potential applications in sound absorber’s design and noise reduction in industrial fields such as transportation and architectural construction.

The authors would like to acknowledge the financial support from the National Natural Science Foundation of China (No. 12172383), the National Key R&D Program of China (No. 2022YFB3806101-2), and the science and technology innovation program of Hunan Province (Nos. 2023RC3036 and 2023JJ30644). We are grateful for resources from the High Performance Computing Center of Central South University.

The authors have no conflicts to disclose.

Yingli Li: Conceptualization (equal); Funding acquisition (supporting); Methodology (equal); Resources (equal); Writing – review & editing (equal). Yu Yan: Data curation (equal); Methodology (equal); Validation (equal); Writing – original draft (lead). Jiahui Yan: Investigation (equal); Methodology (equal); Software (equal). Suchao Xie: Supervision (equal); Visualization (supporting). Yong Peng: Investigation (equal); Supervision (equal).

The data that support the findings of this study are available within the article.

Since sound-absorbing structures are usually required to achieve efficient sound absorption under the premise of compact size, the ratio of effective sound absorption bandwidth-to-structure thickness is used to measure the performance of acoustic metastructures.³⁹  Table II lists some metastructures designed in previous studies. These results show that the metastructure designed in this study has a better bandwidth-to-thickness ratio, indicating that the thickness sound absorption capacity of SRAM has been effectively improved and better sound absorption can be achieved.

The optimization variables are the cavity thickness L, porous material width W, and air space diameter D in each unit cell of the SRAM. These parameters have a significant impact on the peak value and the effective sound absorption frequency of the sound absorption coefficient curve.

As shown in Fig. 11(a), the increase or decrease of L would not affect the trend of the sound absorption coefficient curve but affects the peak value. When L increases from 5 to 10 mm, the sound absorption coefficient hardly shifts. When L increases from 10 to 25 mm, the sound absorption coefficient curve decreases significantly, and the peak value decreases from 0.967 with a L of 10 mm to 0.914 with a L of 25 mm, which is a decrease of 5.5%. In addition, the frequency at which the sound absorption coefficient reaches 0.5 is shifted back from 681 to 783 Hz when L is increased from 5 to 25 mm. Therefore, within the appropriate range, the smaller the L is, the more favorable the sound absorption coefficient is.

The innermost cell of the SRAM is denoted as unit 1, and the number of cells away from the center is increased by 1. When analyzing the porous material width W, the width of the porous material domain in unit 1 is set to a constant value of 9.5 mm since there is only one porous material domain in the innermost unit cell. The width of the porous material near the center of each unit is recorded as W₁, and that for the one away from the center is recorded as W₂. The effect of W on the sound absorption coefficient is shown in Fig. 11(b). It can be seen that when the values of W₁ and W₂ are close, the sound absorption coefficient of SRAM in the low frequency is significantly improved. When the difference between the values of W₁ and W₂ is large, changes in the two values have little impact on the sound absorption coefficient.

The sound absorption coefficient changes significantly when the value of D changes, as shown in Fig. 11(c). When D increases between 1 and 5 mm, it not only makes the peak of the absorption coefficient curve decrease significantly but also makes the acoustic performance of the metastructure at low frequencies attenuate evidently.

In summary, the smaller the D and L and the closer the widths of the two porous material domains in the same unit cell (except for unit 1) within a suitable range, then the more favorable it is for the improvement of the sound absorption performance of SRAMs.