Ultra-broadband sound absorption in a compact multi-chamber micro-perforated panel absorber with varying depths

This study presents an enhanced multi-chamber micro-perforated panel absorber (MC-MPPA) with varying sub-chamber depths, offering ultra-broadband low-frequency sound absorption. Traditional micro-perforated panel absorbers are constrained by a limited bandwidth, necessitating impossibly small perforations for optimal low-frequency absorption. Our innovative design addresses these constraints with a lightweight, compact panel structure that uses varied chamber depths and unique porosities. Using the two-point impedance method from graph theory, an MC-MPPA was modeled and optimized. Notably, our MC-MPPA test pieces achieved impressive sound absorption coefficients experimentally of over 0.8 in the whole frequency ranges of [397–1000] and [698–1895] Hz. The absorber’s thickness is a mere 47 mm, equivalent to 1/18.2 and 1/10.5 of the sound wavelength at the minimum operational frequency, respectively. Theoretically, with a maximum sub-chamber depth of just 20 mm, average absorption coefficient values of 0.6780 and 0.6483 were observed in [200–3000] and [200–4000] Hz ranges, respectively. Our optimization algorithm permits the definition of practical geometric parameters, promising substantial industrial benefits. The results have been validated theoretically, numerically


I. INTRODUCTION
Micro-perforated panel absorbers (MPPAs) [1][2][3] have become an increasingly popular solution for sound absorption in recent years.5][6] The combination of micro-perforations with an air-backing cavity creates a compressibility effect resonant system that is highly efficient at absorbing acoustic energy.However, the conventional design of MPPAs exhibits inadequate absorption capability in the low-frequency range, as well as a typically narrow absorption bandwidth, particularly in shallow applications where the space for acoustic treatment is constrained.Therefore, the development of a light, compact micro-perforated panel structure that can effectively absorb lowfrequency and broadband sound continues to present an intractable challenge.
To date, extensive efforts have been expended in the pursuit of the development of a lightweight micro-perforated panel absorber with exceptional acoustic performance.Authors in one research laboratory [7][8][9] introduced an innovative sound absorber that exploits the viscothermal dissipative traits of the micro-perforated plate (MPP) and effectively reduces its frequency response by integrating decorated membrane resonators.Experimental results demonstrate that this compact absorber can achieve low-frequency broadband sound absorption at deep, subwavelength thicknesses.Carbajo et al. 5 and Wang and Bennett 6 explored how micro-perforated partitions can effectively broaden the sound absorption bandwidth of MPPAs.Kim and Yoon 10 discovered that an MPP with multiplesized holes and a porous separating partition in a MPPA can result in a wider sound absorption bandwidth when compared to a conventional rigid partition MPPA.Wang et al. 4 proposed a novel MPPA incorporating a trapezoidal cavity that facilitates superior vibroacoustic coupling between the micro-perforated panel and ARTICLE pubs.aip.org/aip/adv the cavity, ultimately broadening the sound absorption bandwidth of the MPPA.Yang, Choy, and Li 11 conducted a study on the sound absorption capabilities of a wavy MPPA, ultimately devising a compact wavy MPPA that achieves a high random sound absorption coefficient (SAC) through an optimization methodology.Bravo et al. [12][13][14] conducted a study on the effect of panel vibrations on the acoustic properties of thin MPPAs and subsequently improved the sound absorption performance of these MPPAs using the impedance boundary optimization method.Wang and Huang 15 examined the sound absorption properties of a compound MPPA array, comprised of three MPPAs arranged in parallel with varying cavity depths.Their results indicate that the MPPA array requires a lower acoustic resistance for effective sound absorption compared to a single MPPA.Drawing inspiration from the traditional perforated panel concept, Li, Yu, and Zhai 16 introduced a novel sound absorber based on a microlattice, exhibiting an average experimental SAC of 0.77 across a broad frequency range spanning from 1000 to 6300 Hz.Furthermore, this innovative design offers additional multifunctionality in terms of impact resistance.
Research progress in micro-perforated sound absorbers is greatly influenced by the evolution of process and manufacturing technologies. 17,18][21] Despite the substantial advancements achieved in the past two decades, there remain limitations in their scope and nature.As early as 1998, Maa 3 postulated that the attainment of broadband sound absorption in a single-layer micro-perforated sound absorber could be realized through a reduction in the perforation diameter (below 0.1 mm) and an increase in the perforation ratio. 6,22In a prior investigation, 6 the impact of pore size and perforation rate on the sound absorption characteristics of an MC-MPPA was investigated.The analytical results indicated that when the noise of interest was white noise, there was a strong agreement between the theory put forth by Maa 3 and the observed increase in the sound absorption bandwidth with decreasing perforation size.However, the creation of perforations of such diminutive size, particularly in instances where the perforation ratio is substantial, presents a huge challenge and results in a subpar economic efficiency for traditional processing techniques, such as machine punching, and for additive manufacturing technologies, such as 3D printing.Therefore, the design and fabrication of an MPPA with an excellent acoustic performance within the confines of current common processing and manufacturing technologies constitutes a noteworthy undertaking, with far-reaching practical applications in industry and a significant economic value.
In this research, a novel and robust absorber with subchambers of varying depth and porosity is presented.Its design and optimization is facilitated by the introduction and original application of the TpIM, which enables the modeling of a technology of such complexity.The introduction of the TpIM to acoustic metamaterial modeling is a milestone as the ubiquitous equivalent circuit model (ECM) method based on electro-acoustic analogies can only represent very basic circuits and so has been an obstacle to innovative development.The current work has the following objectives: (1) To develop and optimize a multi-chamber MPPA where each sub-chamber can have unique porosity both on the faceplate and on internal partitions as well as unique depth.
(2) To achieve ultra-broadband sound absorption with this novel MC-MPPA.(3) To identify which parameters are most critical to control during production to maintain optimal absorption characteristics through a sensitivity analysis.(4) To design and fabricate a compact MC-MPPA whose geometric details are achievable with reasonable current processing and manufacturing technologies.(5) To validate the compact MC-MPPA design numerically and experimentally.

II. DESIGN AND THE TWO-POINT IMPEDANCE METHOD (TPIM)
As illustrated in Fig. 1, a rectangular MC-MPPA is characterized by adjacent panels with differing porosities and has varying depths of its sub-chambers.The investigated MC-MPPA structure is divided into nine chambers in a 3 × 3 configuration, where each chamber can possess unique depths and is separated by microperforated adjacent panels.The dimensions of the MC-MPPA are denoted by length L and width W. The thickness of the top MPPs in the MC-MPPA is symbolized as τt, whereas the thickness of the internal micro-perforated adjacent panels is represented by τi.The total height, H, of the MC-MPPA is obtained by summing τt and the maximum sub-chamber depth, Dmax.Throughout this paper, the overall MC-MPPA depth specifically refers to the deepest sub-chamber's depth, Dmax.The key parameters of the investigated MC-MPPA are outlined in Table I.We keep the perforation number n as a variable as its control is necessary to prevent an unreasonably high value inhibiting manufacture.In addition, both τt and τi are fixed with predefined thicknesses corresponding to off-theshelf availability.While both of these parameters could be allowed to vary relative to each other and as a function of sub-chamber in the optimization routine, it was decided that a constant plate thickness from chamber to chamber would be much more practical for manufacture/assembly.
In this study, the two-point impedance method (TpIM) is introduced and applied to determine the impedance of the MC-MPPA.The typically used equivalent circuit model (ECM) method uses electrical circuit analogies and can be used to characterize the operational parameters of an acoustic system but only for very simple circuits.The ECM usually uses Δ-Y and Y-Δ transforms to simplify the circuit to estimate acoustic impedance. 5However, as the complexity of the acoustic system increases, the ECM becomes unmanageable. 8For this reason, we have developed a graph-theory approach, based on the two-point impedance theory proposed by ARTICLE pubs.aip.org/aip/adv Internal walls MPP i( 1) Tzeng and Wu 23 previously used in the domain of electronics, but used here instead to estimate acoustic impedance.Figure S1 of the supplementary material shows the investigated MC-MPPA from Figs. 1(a) and 1(b) represented schematically as a bi-pyramidal molecular structure.The objective here is to calculate the impedance of the overall system, i.e., between nodes 1 and 11 using a matrix approach.The primary objective of employing the TpIM in our MC-MPPA technology is to develop an analytical model that allows the performance of the MC-MPPA to be optimized to calculate the maximum SAC possible within a specified set of parameters.In our case, we can establish an optimization cost function that characterizes the total SAC within a specified frequency range: Here, cα represents the comprehensive SAC, proportionally balanced by the sound spectrum of the intended noise source, Here, ω 1 is equivalent to 2πf 1 and ω 2 corresponds to 2πf 2 , where f 1 and f 2 represent the lower and upper bounds of the frequency range under consideration.The SAC, denoted as α(ω), is determined following the TpIM procedure in the supplementary material with the intended sound spectrum characterized by an auto-spectral density function, Gxx(ω).In this research, the target noise is white noise, Gxx(ω) ≡ 1.We employed the Sequential Quadratic Programming (SQP) optimization technique to minimize the cost function.
Full details of the acoustic modeling and optimization procedures for the MC-MPPA employing the two-point impedance method can be found in the supplementary material.Detailed accounts of the experimental and numerical sound absorption procedures conducted on the MC-MPPA and used for validation are also provided in the supplementary material.
Two MC-MPPA experimental samples, which were optimized for frequencies of [20-1000] and [20-2000] Hz, were fabricated, of which the latter is shown in detail in Fig. 2. Both the core (air cavities and perforated adjoining panels) and the top MPPs were 3D printed with a Masked Stereolithography Apparatus (MSLA).The maximum depth of the sub-chamber air-cavity in both samples was 45 mm, and the total thickness of each technology was 47 mm, including the thickness of τt.Masked stereolithography is a 3D printing process, which uses photopolymer resins that can be cured by UV light at a wavelength of 405 nm.The optimized geometric parameters of the experimental Test Pieces 1000 and 2000 can be found in Tables S6-S7 of the supplementary material.II provides the specific values of cα for Fig. 3 in the four different frequency ranges as a function of Dmax.It is clear that the cα value is dependent on the frequency range chosen for evaluation and that values of above 0.8 and even 0.9 over a wide frequency range can be attained through a judicious choice of the frequency limits of optimization.

III. RESULTS
Figure 4 shows a comparison of the analytical results to numerical results, which were calculated using COMSOL, in the same frequency ranges, of the MC-MPPAs with varying overall absorber

TABLE II.
Overall SACs of the MC-MPPA in four different frequency ranges (data correspond to the SACs curves in Fig. 3).
( thickness across different frequency ranges.These figures were generated to investigate the relationship between the absorber thickness and sound absorption performance of the MC-MPPA for various frequencies ranges.The numerical results presented in these figures serve to validate the analytical results depicted in Fig. 3 and increase confidence in the novel TpIM approach to analytical modeling and the optimization algorithm.
Investigating the sensitivity of broadband absorption performance to various types of parameters becomes crucial when considering uncertainties in manufacturing parameters.Through the analysis of the theoretical model, it is possible to determine which parameters significantly impact the broadband absorption performance and which parameters have a relatively minor effect.Figure 5 presents an analysis of the sensitivity of the MC-MPPA's geometric parameters.This analysis aims to derive general principles from the theoretical model that can inform optimization and practical manufacturing processes, thus ensuring the stability and reliability of the broadband absorption performance.
In order to provide experimental validation, Figs.To emphasize the compactness of the MC-MPPA design, illustrated by the ratio of the structure's thickness to its minimum operational wavelength, Fig. 7 presents a comparative study.This study underscores the MC-MPPA's effective performance, especially regarding its absorption bandwidth and efficiency at deep-subwavelength scales.

IV. DISCUSSIONS A. Absorption response of the MC-MPPA
Regarding the acoustic response of the MC-MPPA, the results demonstrate ultra-broadband sound absorption characteristics when all parameters of each chamber are allowed to vary independently.Figure 6(a) shows that an overall SAC over 0.8 can be achieved experimentally by the MC-MPPA at a maximum sub-chamber depth of only 45 mm in the frequency range of 276-1000 Hz, while Fig. 6(c) demonstrates that the same SAC value can be achieved at a maximum sub-chamber depth of 45 mm in the frequency range of 325-2000 Hz.Additionally, for higher frequencies, as presented in Fig. 3 and Table II, the overall SAC of MC-MPPA at a maximum sub-chamber depth of 45 mm is 0.7424 and 0.6984 at frequency ranges of [200-3000] and [200-4000] Hz, respectively.Notably, even at a maximum sub-chamber depth of only 20 mm, the MC-MPPA achieves an overall SAC of 0. Regarding the "compact" acoustic performance of the MC-MPPA, as illustrated in Figs.6(a) and 6(c), the experimental SACs of Test Piece 1000 exceed 0.8 within the frequency range of 397-1000 Hz.Remarkably, the absorber's thickness is a mere 47 mm, equivalent to 1/18.2 of the sound wavelength at the minimum operational frequency.Similarly, the SACs of Test Piece 2000 surpass 0.8 in the 698-2000 Hz frequency range, with the absorber maintaining the same thickness, equating to 1/10.5 of the sound wavelength at its minimum operational frequency.
The MC-MPPA's acoustic absorption capabilities can also be explained by its acoustic reactance and resistance, as shown in Figs.6(b) and 6(c).An ideal impedance match requires the acoustic reactance to be zero and the acoustic resistance to be one, achieved simultaneously.Analysis of the numerical and experimental acoustic reactance and resistance in Figs.reactance approaches zero and the acoustic resistance nears one.This pattern is also observed in Figs.6(c) and 6(d).Specifically, for Test Piece 2000, the experimental data at ∼1400 Hz show acoustic reactance approximately zero and acoustic resistance close to one, aligning with its experimental absorption curve at the same frequency, where SAC nearly reaches one.

B. Impact of the air cavity depth in sub-chambers
As can be seen from Fig. 3 and Table II, an increase in the maximum sub-chamber depth, Dmax, corresponds to an enhancement in the SAC of the MC-MPPA.In this research, it has been observed that fixing the sub-chamber depths to a common value limits the operating frequency of the MC-MPPA.However, as illustrated in Fig. 3 and Table II, allowing the depths to vary from one subchamber to another effectively broadens the operating frequency range from [200-2 000] to [200-4 000] Hz.Upon further investigation, as detailed in the supplementary material, it is observed that when the optimized frequency is confined to 20-1000 Hz, the depths of individual sub-cavities tend to remain consistent.However, as the upper limit of the optimized frequency increases, the difference in depth among sub-cavities within the same MC-MPPA becomes more pronounced.This observation further corroborates the conclusion from an alternative perspective.

C. Sensitivity analysis of MC-MPPA's geometric parameters
Given the precision required in MC-MPPA manufacturing, even slight deviations in geometric parameters can significantly impact performance.A sensitivity analysis is essential to identifying  performance of the MC-MPPA.In this analysis, the ranges of parameters have been chosen under the current common processing and manufacturing technologies.Details of the sensitivity analysis of MC-MPPA's geometric parameters can be found in Sec.S5 of the supplementary material.
The analysis demonstrates that the broadband absorption performance exhibits significant sensitivity to the chamber depth and the porosity of the top and internal MPPs, while showing moderate sensitivity to the panel thickness.Conversely, alterations in the diameters of the top and internal MPPs' perforations have a negligible impact.This evidence indicates that stricter regulation of chamber depth, perforation porosity, and MPP thickness during production can lead to substantial enhancements in performance consistency.These findings offer critical guidance for optimizing MC-MPPA and its manufacturing process in research and can explain the differences between the experimental and analytical/numerical results.

D. Numerical and experimental validation
Regarding the numerical validation, Fig. 4 presents a comparison between the analytical and numerical SACs of MC-MPPAs for varying overall absorber thicknesses across the four distinct frequency ranges.Generally, the numerical simulations agree well with the analytical results.Upon closer examination, minor discrepancies are observed with increasing depth and frequency.These marginal discrepancies may be explained by a violation of the lumped element assumption and the possibility that the spacing between the perforations is not negligible.A more exhaustive exploration of ARTICLE pubs.aip.org/aip/advthese phenomena will ensue in subsequent studies.Regarding the experimental validation, as illustrated in Fig. 6, the experimental results demonstrate a high degree of agreement with both the analytical values and the numerical simulations and demonstrate that the design is more than just conceptual but is realizable.
E. Analysis of the processing and manufacturing challenges associated with the MC-MPPA under investigation Generating numerous micro-holes (with diameters smaller than 0.3 mm) in a thin sheet presents a substantial challenge for both 3D printers and Computer Numerical Control (CNC) milling machines.Even when successfully produced, the process requires significant time and specialized equipment cost.Consequently, large-scale engineering applications employing microperforated plates with diameters less than 0.3 mm are limited by their economic efficiency.In principle, an increase in diameter and a decrease in the number of micro-holes machined in a thin plate result in reduced requirements for the machining and manufacturing process, decreased production costs, and enhanced potential for large-scale industrial applications.
In this study, we took into account not only the SAC performance of the MC-MPPA but also the processing and manufacturing challenges associated with the designed MC-MPPA.Consequently, we controlled the diameter of the micro-holes to be within the range of 0.5-1.0mm and the porosity to be between 0% and 5%.In addition, the plate thicknesses were fixed at 1 and 2 mm.This design can undoubtedly be easily printed or machined by the vast majority of 3D printers and CNC machines.which combines a MPP and coiled-up Fabry-Pérot channels.The yellow bar depicts the MPPA with an integrated Helmholtz resonator, as explored in Li et al. 25 The magenta bar represents the single-layer MPPA from Li et al., 25 while the cyan bar corresponds to the double-layer MPPA, also from Li et al. 25 It is important to note that all the data compared are experimental.The sound acoustic performance of the MC-MPPA is notably high, particularly from the aspects of absorption bandwidth and efficiency at deep-subwavelength levels.This efficiency is distinctly marked by the proportion of the structure's thickness relative to its minimum operational wavelength.

V. CONCLUSION
This research concentrates on exploring the acoustic properties of an MC-MPPA while varying the depths of sub-chambers.The acoustic properties of our designed MC-MPPA have been evaluated, taking into account the limitations of the current processing and manufacturing technologies.The investigation led to the following findings: MC-MPPA is predominantly influenced by several key factors: the chamber depth, the porosity of the top and internal micro-perforated panels (MPPs), and the thickness of their perforations.Consequently, these parameters warrant meticulous regulation throughout the design and fabrication process to guarantee uniform performance.5.The design and fabrication of a compact MC-MPPA exhibiting an exceptional acoustic performance is achievable through the current common processing and manufacturing technologies, providing substantial practical applications and an economic value within the industry.6.While a 3 × 3 chamber MC-MPPA was examined, other geometries can be modeled.Similarly, the depths and frequency ranges were chosen as examples only of what is possible.7.As every optimized curve has a frequency dependent ramp-up and ramp-down in the absorption coefficient, designs can be made such that the high level absorption plateau spans the frequency range of interest.This will result in even high average absorption values.

SUPPLEMENTARY MATERIAL
For detailed information on the acoustic modeling, optimization procedures, sensitivity analyses, and sound absorption measurements of the MC-MPPA, along with its geometric parameters, refer to the supplementary material.

Figure 3
Figure 3 displays the analytical SAC curves of MC-MPPAs that have been optimized for four different frequency ranges: [20-1000],[20-2000],[20-3000], and[20-4000] Hz.The geometric parameters of the MC-MPPA that correspond to each of the absorption curves in Fig.3are provided in Tables S2-S5 of the supplementary material.Furthermore, TableIIprovides the specific values of cα for Fig.3in the four different frequency ranges as a function of Dmax.It is clear that the cα value is dependent on the frequency range chosen for evaluation and that values of above 0.8 and even 0.9 over a wide frequency range can be attained through a judicious choice of the frequency limits of optimization.Figure4shows a comparison of the analytical results to numerical results, which were calculated using COMSOL, in the same frequency ranges, of the MC-MPPAs with varying overall absorber

FIG. 2 .FIG. 3 .
FIG. 2. Photograph of the MC-MPPA test samples.(a) The top MPP and the core are both 3D printed.(b) The sub-chambers of the core have perforated adjacent panels of differing parameters, and the sub-chamber depths differ from each other.
6(a) and 6(c) depict the analytical, numerical, and experimental SACs of the Test Piece 1000 and Test Piece 2000 MC-MPPA samples.These figures provide a comprehensive comparison between the expected and actual SACs of the MC-MPPA in these frequency ranges.The acoustic resistance and reactance of Test Piece 1000 and Test Piece 2000 MC-MPPA samples, both numerical and experimental, are plotted in Figs.6(b) and 6(d).

FIG. 5 .
FIG. 5.Sensitivity analysis of MC-MPPA's geometric parameters.(a) Curve's color gets darker as the depth increases from 10 to 50 mm.[(b) and (e)] Curve's color gets darker as the perforation diameter increases from 0.5 to 1.0 mm.[(c) and (f)] Curve's color gets darker as the perforation porosity increases from 0.5% to 5.0%.[(d) and (g)] Curve's color gets darker as the thickness increases from 0.5 to 2.5 mm.

FIG. 6 .
FIG. 6.(a) Analytical, numerical, and experimental SACs of Test Piece 1000.(b) Numerical and experimental acoustic resistance and reactance of Test Piece 1000.(c) Analytical, numerical, and experimental SACs of Test Piece 2000.(d) Numerical and experimental acoustic resistance and reactance of Test Piece 2000.

Figure 7
Figure 7 presents a comparison of the sound absorption bandwidth (SAC >0.8) between the MC-MPPA and other compact broadband MPP-type sound absorbers.In this figure, the red bars represent the MC-MPPAs under investigation, namely Test Piece 1000 (TP1000) and Test Piece 2000 (TP2000), with data sourced from Figs. 6(a) and The green bar illustrates the hybrid sound absorber by Wu et al.,24 which combines a MPP and coiled-up Fabry-Pérot channels.The yellow bar depicts the MPPA with an integrated Helmholtz resonator, as explored in Li et al.25 The magenta bar represents the single-layer MPPA from Li et al.,25 while the cyan bar corresponds to the double-layer MPPA, also from Li et al.25 It is important to note that all the data compared are experimental.The sound acoustic performance of the MC-MPPA is notably high, particularly from the aspects of absorption bandwidth and efficiency at deep-subwavelength levels.This efficiency is distinctly marked by the proportion of the structure's thickness relative to its minimum operational wavelength.

1 .
A compact MC-MPPA can demonstrate ultra-broadband sound absorption capabilities.The experimental results from MC-MPPAs, with a thickness of only 47 mm, exhibit an overall average SAC exceeding 0.8 within the frequency ranges of [276-1000] and[325-2000] Hz.Theoretically, even with a maximum sub-chamber depth of just 20 mm, the MC-MPPA attains overall SACs of 0.6780 and 0.6483 in the frequency ranges of[200-3000] and [200-4000] Hz, respectively.2. The experimental SACs of Test Piece 1000 exceed 0.8 within the whole frequency range of 397-1000 Hz.Remarkably, the absorber's thickness is a mere 47 mm, equivalent to 1/18.2 of the sound wavelength at the minimum operational frequency.3. The variable sub-chamber depth of the MC-MPPA considerably broadens the sound absorption bandwidth.Varying subchamber depths effectively extends the operating frequency range from [200-2000] to [200-4000] Hz and simultaneously improves performance.4. Sensitivity analyses show that the absorption performance of

TABLE I .
Parameters that define the acoustic behavior of the MC-MPPA under study.