Issue 
Acta Acust.
Volume 8, 2024



Article Number  37  
Number of page(s)  11  
Section  Acoustic Materials and Metamaterials  
DOI  https://doi.org/10.1051/aacus/2024039  
Published online  16 September 2024 
Technical & Applied Article
The potential of additively manufactured porous absorbers in the design of multilayer microperforated absorbers
FriedrichAlexanderUniversity (FAU) ErlangenNuremberg, Institute of Fluid Mechanics, Cauerstr. 4, 91058 Erlangen, Germany
^{*} Corresponding author: benedikt.berchtenbreiter@fau.de
Received:
18
April
2024
Accepted:
18
July
2024
Microperforated absorbers (MPA) are a wellestablished technology for attenuating sound in flow carrying ducts. MPAs usually consist of a microperforated panel (MPP) in combination with a cavity as back volume. The damping maxima of MPAs occur in the range of their resonant frequencies and the effect is narrowband compared to porous or fibrous absorbers, which damp broadband at high frequencies. The resonant frequencies of MPAs decrease with increasing back volume. This provides a challenge, especially in applications with limited installation space when the damping of low frequencies is required. In the literature, the combination of MPPs and porous or fibrous absorbers is reported to reduce the required back volume. Therefore, doublelayer MPAs with an additively manufactured porous absorber underneath the MPP are introduced in this work. The advantage of using additively manufactured porous absorbers as an acoustic metamaterial over conventional absorbers is that the acoustic properties can be specifically adapted to the required impedance boundary conditions. The results of this work show that the additively manufactured absorbers reduce the required back volume by up to 13% compared with liners without absorber underneath the MPP. Experimental validation underscores the robustness of the chosen design approach for doublelayer MPAs.
Key words: Microperforated absorber (MPA) / Acoustic liner / Acoustic metamaterial / Additively manufactured porous absorber / Microperforated panel (MPP)
© The Author(s), Published by EDP Sciences, 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
The design of soundabsorbing liners in flow carrying ducts typically involves the application of different operating principles of sound attenuation: Dissipative through friction in fibrous and porous materials or in submillimeter slots and pores, reactive by reflection or by interference at, for example, Λ/4resonances [1–4]. These two principles can also occur together when the effect of the springmass oscillator is combined with the Λ/4resonance in a Helmholtz resonator. As an alternative to the methods described above, socalled HerschelQuincke liners can be used, which are characterized by destructive interference via an Ushaped bypass [4].
Dissipative sound absorbers in HVAC and ventilation systems or automotive sound absorbers often use fibrous materials due to their effectiveness and costeffectiveness. However, a disadvantage of these materials is that the acoustic effect can diminish if they become contaminated or damp. A lack of mechanical stability can also lead to dents in the sound attenuation, for example due to slippage of the material. Furthermore, flow can cause fibers to dissolve which can lead to health hazards, especially in HVAC systems [5–7]. An alternative proposed by Maa [8–10] in 1975 is the use of microperforated panels (MPP). MPPs are commercially available and typically made of plastic or metal and the cost is comparable to that of conventional porous or fibrous materials [11, 12]. Moreover, metallic MPPs are fire resistant and can be welded [11]. Furthermore, MPPs exhibit a flowguiding effect with low pressure drop serving as soundabsorbing liners and baffles or as diffusors [6, 13]. The submillimeter openings in MPPs are usually circular or slot shaped. An alternative to panels with a large number of symmetrically arranged pores is the use of single slots extending over the entire cavity as back volume [14, 15].
MPPs are often paired with an airfilled cavity to create microperforated absorbers (MPA). These MPAs are characterized by frequencyspecific damping maxima at the Helmholtz and/or Λ/4resonance. Using a single MPP results in narrowband damping maxima at the respective resonant frequencies, giving the silencer one degree of freedom. Alternatively, variants with several degrees of freedom can be used, resulting in more broadband attenuation [4]. For this purpose, the cavity is subdivided in depth direction with several perforated layers. Hence, these variants are also referred to as multilayer MPAs [16]. Multilayer MPAs can consist not only of MPPs, but additionally of fibrous or porous absorbers such as rock wool. The use of porous layers also leads to a more broadband attenuation and, additionally, to a shift of the resonant frequencies towards lower frequencies with the same total back volume [2, 17, 18].
A literature review focusing on the fabrication and materials used with respect to multilayer MPAs with and without porous absorbers reveals that MPPs are typically fabricated from metallic materials [4, 19–22]. However, the production of MPPs also employs additive manufacturing with plastics [16, 23–25]. A wide variety of materials is used as porous absorbers: Glass and rock wool [17, 18, 26], vegetable fibers [16, 23, 27], foams [20, 22, 23], organic materials [28]. The authors of this paper are not aware of any studies in which the porous absorbers are additively manufactured.
Regarding additively manufactured porous absorbers, three structural types are typically distinguished in the literature: tubes [24, 29–31], lattice structures [32–42], cell structures [25, 43, 44]. In much of the research, the absorbers are built as lattice structures and are typically fabricated using fused deposition modelling (FDM). The advantage of the FDM process is that it is costeffective, there is no need to remove excess polymer or powder after the printing process, and the fineness of the lattices is limited only by the diameter of the extrusion nozzle. Therefore, it is possible to produce very fine and closemeshed lattice structures. In addition, very high and low porosities can be realized with only one process. At this point, it is not possible to state which approach mentioned in the various papers yields the best results for sound dissipation, as the absorbers investigated differ in depth and, consequently, in effective volume. According to Cai et al. [33], however, the absorption becomes maximum when half the pore size or half the distance between adjacent lattice bars corresponds to the viscous boundary layer thickness.
When designing multilayer MPAs, additively manufactured porous absorbers as acoustic metamaterial offer the advantage of specifically tailoring and adapting the acoustic properties to meet desired impedance boundary conditions. This opens up additional degrees of freedom in the design of multilayer MPAs and is thus a decisive advantage of additively manufactured porous absorbers. Furthermore, the dimensional stability in combination with the geometric freedoms enables manufacturing according to the subsequent installation situation, for example when a multilayer MPA is installed in a channel bend.
The objective of the present work was to design and validate doublelayer MPA acoustic liners using additively manufactured porous absorbers. These liners combine a MPP, an additively manufactured porous absorber and a cavity and are tuned to a target frequency at which the damping maximum occurs. The absorber is positioned directly below the MPP inside the cavity so as not to lose the MPP’s flow guiding. The back volume was divided both longitudinally and transversely, allowing for the assumption of a locally reacting liner in the modelling. The porous absorber was designed as a lattice structure due to its lower complexity compared to the cellular structure. Typically, latticetype absorbers are manufactured using FDM. However, the FDM process is slow, limiting its applicability for largescale industrial production. In the context of this work, the use in an industrial environment should already be considered. Therefore, the lattices were produced by means of binder jetting. With modern material systems, the process is characterized by a high level of detail, fast production times and low costs.
The focus of this work was to investigate the use and the advantages of additively manufactured porous absorbers. Therefore, the MPPs were fabricated by laser cutting and not additively, as this would have led to additional challenges and uncertainties, especially in the modelling of the impedance. To reduce the complexity of the MPPs, they were slotted, with each cavity being assigned one slot extending along its entire length. The primary question in this work was whether the acoustic effect of the additively manufactured absorbers is sufficient to achieve the described advantage of reducing the total back volume. Furthermore, for the design of the liners, it was necessary to succeed in modelling the acoustic properties of the porous absorbers as a function of geometry.
2 Acoustic modelling of MPAs
When a plane wave hits both a MPA or a multilayer MPA perpendicularly according to Figure 1, the acoustic modelling is based on the surface impedance. The surface impedance Z_{MPA} of a conventional MPA with one MPP and an airfilled cavity is determined as follows:
Figure 1 Schematic illustration of a doublelayer MPA at perpendicular sound incidence. 
In equation (1), Z_{MPP} denotes the impedance of the MPP, while Z_{0} is the characterisic impedance of air. The freefield wavenumber is k_{0} and the depth of the cavity is t_{C}.
2.1 Acoustic modelling of doublelayer MPAs
The impedance of a multilayer MPA with n layers in general is computed using equation (2), which is derived from the total transfer matrix T_{tot} of the individual layers. T_{tot} is obtained by multiplying the transfer matrices of each layer. For a doublelayer MPA this denots multiplying T_{MPP} of the MPP, T_{A} of the porous absorber and T_{C} of the airfilled cavity. [16, 19, 45, 46]
In equation (2), the transfer matrices of the layers are
In equation (3), the propagation of sound within the porous absorber is characterized by the complex equivalent wavenumber and impedance . The parameter t is used to describe the depth of the respective layer. Equation (2) simplifies to equation (1) for a conventional MPA with MPP and cavity.
2.2 Modelling of slotted MPPs
In this work, the modelling of the slotted MPPs is done using the Johnson, Champoux and Allard (JCA) semiphenomenological model [46–48] according to equation (4). Here ρ_{0} is the density and υ is the kinematic viscosity of air and ϕ_{MPP} is the degree of perforation of the MPP. The viscous characteristic length Λ corresponds to the hydraulic radius r_{hyd} of the slots. The static flow resistance with σ = 12νρ_{0}/r_{hyd} is also calculated as a function of the hydraulic radius. The tortuosity α_{∞} = 1 + 2ϵ/t_{MPP} is a function of both the correction length ϵ and the depth of the MPP.
In the literature, different approaches are considered to calculate the correction length ϵ. Attala and Sgard [49] propose ϵ = 0.85r_{hyd}. Zieliński et al. [50] choose the following approach for parallel slots using the slot width w_{S} and the distance w_{0} of adjacent slots:
Investigations in [51] have shown that for the geometrical characteristics of the perforated panels of this work, the approach by Attala and Sgard [49] is advantageous for slot widths in the range of 0.4 mm, while the approach by Zieliński et al. [50] is preferable for slot widths larger than 1 mm. The range in between thus represents a transition. When designing the liners in Section 4, it should be possible to model the impedance of the MPP as a continuous function of the slot width. Therefore, a resulting correction length that accounts for the transition region was formed from both approaches. The course of the correction length over the slot width is shown in Figure 2.
Figure 2 Correction lengths ϵ according to Zieliński et al. [50] and Attala and Sgard [49] as well as the spline fitted from them as a continuous function of w_{S}. 
The influence of the porous absorber underneath the MPP on the fluid motion in the orifices was considered using the dynamic approach proposed in [49]. With an additional reactance term, the dynamic approach takes into account the resistive and inertial effects due to the distortion of the fluid movement in the porous absorber at the interface to the MPP.
2.3 Modelling of additively manufactured porous absorbers
In the model, additively manufactured porous absorbers are treated as an equivalent fluid and the sound propagation in the fluid is characterized by complex quantities, namely equivalent density and equivalent bulk modulus . The modelling of and of the additively manufactured porous absorbers was performed using the Johnson, Champoux, Allard, and Lafarge (JCAL) semi phenomenological model [47, 48, 52, 53] corresponding to the equations (6) and (7) [46]:
The equivalent density and the equivalent bulk modulus depend on the following fluid properties of air: density ρ_{0}, kinematic viscosity υ, isentropic exponent γ, Prandtl number Pr with v′ = v/Pr. In addition, the modelling is based on the socalled JCAL parameters resulting from the micro geometry of the porous absorber: Porosity ϕ, tortuosity α_{∞}, viscous characteristic length Λ, thermal characteristic length Λ′, static viscous permeability q_{0} static thermal permeability. While the fluid properties of air can be measured and calculated directly, the corresponding JCAL parameters are determined using an inverse scheme (see Sect. 3.1).
3 Experimental characterization of additively manufactured porous absorbers
The additively manufactured porous absorbers represent a periodically arranged lattice structure according to Figure 3 and were manufactured from polymethyl acrylate by means of binder jetting. According to Table 1, the lattice constant l_{yz} in yzdirection was systematically varied. This led to lattice structures with different porosities ϕ_{CAD}, which were determined using the CAD (Computer Aided Design) data. The diameter of the bars d_{b} is 0.5 mm.
Figure 3 Geometric structure of the investigated additively manufactured porous absorbers including the elementary cell (bottom left). 
Characteristic geometric parameters of the additively manufactured porous absorbers.
The acoustic properties of the porous absorbers were determined in the twoport test rig TPPK (TwoPortPrüfstand Kreisförmiger Querschnitt) [51] at FAU ErlangenNuremberg. Transmission, reflection and dissipation were determined as well as the equivalent fluid properties and via the transfer matrix method. The procedure and results are described in detail in [51].
3.1 Characterization of the JCAL parameters via inverse scheme
In order to model the additively manufactured porous absorbers as a continuous function of the lattice constant, the initial step involved determining the JCAL parameters x_{JCAL} for each absorber in Table 1. One sample was manufactured for each absorber and its equivalent density and equivalent bulk modulus were determined using the measured transfer matrix. Based on a constrained minimization problem, an inverse scheme was employed to determine the JCAL parameters. The discrepancy between the modeled and experimentally determined values for and forms the cost function, to be iteratively minimized. In this way, it is possible to fit the JCAL parameters of the model to the real parameters of the absorbers. Using the Euclidean norm ∥•∥, the cost function to be minimized g_{min} is as follows [54]:
The weighting factors and are computed using the length N of the frequency vector:
The solution of the minimization problem is thus
The minimization was solved using an interiorpoint algorithm adhering to the constraints proposed by Niskanen et al. [54]: 0 ≤ ϕ ≤ 1, ≤ α_{∞} ≤ 10, 10 · 10^{−6} ≤ Λ, Λ′ ≤ 2000·10^{−6} m, . To increase the probability of finding a solution for each lattice structure that is as close as possible to the global minimum and thus to the putative optimal parameters, the minimization was solved 10,000 times. The initial values were chosen randomly within the above limits. Subsequent the parameters associated with the lowest function values of the 10,000 minima of g_{min} were examined. It was observed that there are differences between the parameters, despite the associated function values being remarkably similar. This means that there is a set of parameter combinations with which and can be modeled in a very good approximation. As a result, the parameter combinations of the 50 lowest function values were selected and the mean value for each parameter was calculated, rather than selecting the parameters associated with the global of the 10,000 minima. With this approach, the best results were obtained in the validation of the inverse scheme.
Finally, by fitting, the individual parameters were determined as a continuous function of the lattice constant l_{yz}. It should be pointed out that when fitting the functions, the main focus was to be able to model the equivalent fluid properties between l_{yz,min} and l_{yz,max} as accurately as possible. Therefore, the functions should not be extrapolated. The best fitting results were obtained with a quadratic polynomial curve. Figure 4 shows the JCAL parameters as a function of the lattice constant.
Figure 4 Mean values of the inverse determined JCAL parameters for each lattice constant l_{yz} (×) and the fitted functions based on it. 
As expected, the porosity ϕ decreases with decreasing lattice constant, thereby increasing filling ratio. Upon comparing the values with the design data for each lattice constant, it becomes evident that the values inversely determined surpass those computed using CAD data. Therefore, the structures were viewed under a microscope and the dimensions of the lattice bars were measured. Assuming rotational symmetry of the lattice bars, the diameters were measured at several positions and an average diameter of 0.48 mm was calculated. This reduction in diameter would correspond to a reduction in the volume of the lattice structure by 8.2%, thereby an increase in porosity. This consideration is a very idealized and simplified calculation, which is intended to illustrate the enormous influence geometric deviations have on the porosity. This serves as an explanatory approach for the discrepancy observed between theoretical and inversely determined porosity.
The tortuosity decreases with increasing lattice constant and porosity. This characteristic aligns with the results obtained by Boulvert et al. [38] in similar studies. In the range considered, both the viscous and thermal characteristic lengths Λ and Λ′ increase approximately linearly with the lattice constant l_{yz}. This is consistent with expectations given that the lattice constant represents the characteristic geometric dimension of the lattice structures. The characteristic lengths of the different lattice constants are of the order of half the length of the free space of adjacent lattice bars. The curves for the viscous and thermal characteristic length are almost identical. This can be attributed to the fact that the lattice structures are not pores in the classical sense with constrictions and expansions. This results in the characteristic lengths being attributed to the same geometric dimension. This result is also confirmed by Boulvert et al. [38]. Thus, the determined characteristic lengths are plausible both qualitatively and quantitatively.
Finally, the static viscous and thermal permeabilities q_{0} and are considered. Both parameters increase with the lattice constant and porosity, as expected. Once again, this behavior aligns with the research of Boulvert et al. [38]. The values obtained are also quantitatively similar to their results. The static viscous permeability is directly related to the flow resistance σ of the lattice structures as expressed by the equation σ = νρ_{0}/q_{0}, and it decreases with increasing permeability. A decreasing flow resistance with increasing lattice constant corresponds to the fluid mechanical expectations. In conclusion, the static permeabilities, both quantitatively and qualitatively, appear to be plausible.
3.2 Validation of the inverse scheme
The final step in the inverse scheme is its validation. To achieve this, the reflection, dissipation and transmission are reconstructed using the function values of the JCAL parameters and these are then compared with the corresponding measurement values of each absorber listed Table 1. The steps in this process are listed below:
Determination of the JCAL parameters by reading the corresponding functions depending on the lattice constant l_{yz}.
Calculation of and with the JCAL model according to the equations (6) and (7).
Calculation of the equivalent wave number and equivalent impedance .
Calculation of the transfer matrix T(ω) with equation (3).
Calculation of reflection and transmission as energy quantities R(ω) und T(ω) [55]:

Calculation of the dissipation D(ω) = 1 − (R + T).
Figure 5 shows the modeled and measured curves for transmission, reflection and dissipation of the absorbers according to Table 1. Model and measurement agree very well in the considered frequency range. Upon considering the upper frequency range, it becomes apparent that for all configurations with the JCAL model the dissipation is slightly underestimated when compared to the measurements. This difference is due to the higher dissipation error when the frequency is close to the upper cutoff frequency of the TPPK [51]. The significance of the dissipation error increases as the measured dissipation decreases. In contrast, this implies that modelling succeeds in eliminating or reducing the systematic dissipation error of the test rig at high frequencies. The same is true for dissipation due to structural vibrations, which depend only on the geometry of the overall lattice structure with d_{tot} and t_{tot} (see Fig. 3) and do not represent any acoustic property of the lattice as an equivalent fluid per se. For instance, in the measured data, an increased dissipation occurs at 2900 Hz for l_{yz} = 1.25 mm, attributed to a structural vibration of the overall structure, but this is not evident in the modeled curve. Thus, the inverse scheme’s ability to reduce or eliminate the influence of measurement uncertainties and structural oscillations, which are dependent on the overall geometry of the measured absorbers, is a significant advantage. This enhancement increases the anticipated accuracy of the results when the equivalent fluid properties are utilized for subsequent calculations, such as numerical simulations or analytical models.
Figure 5 Transmission, reflection, and dissipation: measured data (, , ) and modeled data (T_{JCAL}, R_{JCAL}, D_{JCAL}). 
4 Design of doublelayer MPA liners
The doublelayer MPA liners (DLMPA) were designed according to Section 2 by modelling the surface impedance. This methodology is equally applicable for liners if the boundary condition of a locally reacting wall is fulfilled. Locally reacting implies that the acoustic behaviour does not depend on the spatial geometry of the sound field, and thus not depend on the angle of incidence. Figure 6 shows the structure of the liners based on CAD data. The entire liner consists of 36 doublelayer MPAs. The absorber elements are positioned within the cavities and are directly connected to the MPP. The characteristic dimensions l_{char} of the cavities are 18 mm (width) and 28 mm (length). With resonant frequencies up to 1500 Hz, the boundary condition of a locally reacting liner with k_{0}l_{char} ≪ 1 [56] is thus fulfilled. The wall thickness of the cavities is 2 mm. Thus, the acoustically effective length of the liners is 360 mm. The MPP was fabricated of 1 mm aluminum sheet by laser cutting. According to Figure 7, the liners are designed for the twoport test rig TPPR (TwoPortPrüfstand Rechteckiger Querschnitt) [51]. The crosssection of the flowcarrying channel is 60 mm × 80 mm.
Figure 6 Liner consisting of 36 DLMPA: general view (left), detailed view (top right), porous absorber (bottom right). 
Figure 7 CAD model of the symmetrically designed TPPR. 
The design of the liner targeted the optimum wall impedance according to Cremer [57], at which the maximum damping occurs. Taking into account the losses in the wall boundary layer, the optimal wall impedance of a singlesided liner in a rectangular duct is [56]
The distance to the opposite wall is h. Two liners were specifically designed with the target frequencies f_{opt} = 800 Hz and 1200 Hz. The surface impedance of the liners Z_{W} was calculated using the equations (2)–(7). The impedance was related to the entire channel width and the wall thickness of the cavities was considered (see Fig. 6):
Using a genetic algorithm, Z_{W} was iteratively fitted to Z_{opt} for the two target frequencies. Four geometric degrees of freedom x_{geo} were available: 0.4 mm ≤ w_{S} ≤ 1.2 mm, 0 ≤ t_{C}, 1.25 mm ≤ l_{yz} ≤ 2.0 mm, 3.75 mm ≤ t_{A} ≤ 10 mm. Thus, the function to be minimized was
The minimization problem was solved 200 times with a randomly varying set of initial values using the Matlab function ga with the recommended default settings [58]. Table 2 gives the parameter combinations with the lowest function values for g_{min}. Furthermore, it shows the deviations and from Z_{opt} for the respective real and imaginary parts. The total depth is t_{tot} = t_{C} + t_{A}.
Parameter sets with the lowest function values for g_{min}.
For both target frequencies, f_{opt} = 800 Hz and 1200Hz, the genetic algorithm succeeded in determining parameter sets which approximate Z_{opt} quite accurately. It is evident that the optimal wall impedance is well approximated across different parameter sets. This therefore means that the optimum wall impedance is achieved with different parameter sets. When designing the liners, it is thus possible to choose from various options. For example, the variant that is most advantageous in terms of production can be selected.
To check if the required back volume is reduced with the additively manufactured porous absorbers beneath the MPP, corresponding variants without absorbers were designed for the two target frequencies. To distinguish clearly, these are referred to as singlelayer MPA liners (SLMPA). The basic procedure was identical to the doublelayer variants, with the distinction that only two degrees of freedom were available in the depth of the cavity and the slot width. The following result was achieved:
Singlelayer MPA with f_{opt} = 800 Hz: t_{C} = t_{tot} = 47.32 mm and w_{S} = 0.56 mm.
Singlelayer MPA with f_{opt} = 1200 Hz: t_{C} = t_{tot} = 22.41 mm and w_{S} = 0.48 mm.
For the singlelayer MPAs, Z_{opt} is obtained with only one parameter set. Table 3 compares the total depths of the different parameter sets of the doublelayer MPAs with the total depth of the singlelayer MPAs. It has been confirmed that the additively manufactured porous absorbers significantly reduce the necessary total back volume by up to 13.4%.
Differences in total depth of the DLMPAs and SLMPAs for f_{opt} = 800 Hz and 1200 Hz.
5 Experimental validation
The twoport test rig TPPR [51] was utilized for experimental validation. For each of the two target frequencies, one doublelayer MPA and one singlelayer MPA liner was manufactured. Due to manufacturing considerations, variant 2 (see Tab. 2) was selected for the doublelayer MPAs in each case. Variant 2 combines two advantages for both target frequencies. Firstly, the lattice constant is moderate and not too finemeshed. This allows the excess powder to be removed after additive manufacturing without high energy input. This reduces the risk of destroying the fine lattice bars and the production time. Secondly, with a total depth of 5.6 mm and 5.4 mm respectively, it was possible to design the lattice structures in such a way that the bars end with their full diameter. This prevented the bars in the edge layers from becoming too thin and being damaged when the excess powder was removed. Figure 8 shows the measured transmission loss as a function of frequency. The measurements were conducted without channel flow. A distinction is made between the total transmission loss TL, that due to dissipation TL_{D} and that due to reflection TL_{R}.
Figure 8 Transmission loss: total TL (left), by dissipation TL_{D} (middle), by reflection TL_{R} (right). 
Table 4 presents a comparison between the frequencies , where the damping maxima occur, and f_{opt} according to the design in the previous section. For all four liners there is a very good agreement with a maximum of 2.8% deviation between measurement and design.
Comparison of f_{opt} (design) and (measurement) for DL800, DL1200, SL800 and SL1200.
For the DL800 and SL800, f_{opt} is marginally overestimated, while for the DL1200 and SL1200, it is underestimated. Consequently, there is not a clear dependency whether the modelling approach fundamentally under or overestimates f_{opt}. However, it is evident that the agreement is better for f_{opt} = 800 Hz than for 1200 Hz. One possible reason for this may be that the impedance models for low frequencies are more resilient to geometric deviations due to tolerances in manufacturing. An alternative cause is inaccuracies in the modelling of the impedance. Since the characteristics of the DL1200 and SL1200, with and without absorber elements, are qualitatively very similar, this is an indication that the model for the impedance of the slotted MPP is less accurate with increasing frequency.
Figure 8 illustrates the transmission loss due to dissipation dissipation and reflection . indicates that dissipation plays a significant role in the soundabsorbing effect of the liners. For all liners considered, is significantly below 1. This behavior corresponds to the characteristics of a liner with optimal wall impedance and confirms the modelling approaches with and without porous absorber.
According to the literature [2], an additional advantage of multilayer MPAs is the more broadband damping characteristic. This feature does not occur with the liners investigated here. The transmission loss for frequencies larger and smaller than is similar. The dissipative effect of the porous absorbers is too small to have a noticeable effect due to the relatively large lattice constants. Therefore, future investigations should focus on producing and integrating additively manufactured porous absorbers with even lower lattice constants to increase dissipation. In this context, it is important to consider additive manufacturing processes that make it possible to reproducibly create very fine and closemeshed lattice structures, such as FDM.
6 Conclusion
The objective of this work was the design of doublelayer MPA liners, which are a combination of MPP, additively manufactured porous absorber and cavity, and can be adapted to a target frequency where the maximum damping is expected to occur. For this purpose, it was essential to model the acoustic behavior of the porous absorbers as a function of the lattice constant. Consequently, the equivalent fluid properties were modeled using the JCAL model, based on experimental data. Continuous functions were fitted over the lattice constant for each JCAL parameter. The Validation of the JCAL model demonstrated a strong correlation between measurement and model in terms of transmission, reflection and dissipation of the absorbers studied.
The design of the doublelayer MPA liners targeted the optimum wall impedance, according to Cremer [57]. Using a genetic algorithm, the impedance of the liners was iteratively adjusted to the optimum wall impedance. The degrees of freedom available in the design were the slot width of the MPP, the lattice constant and the depth of the absorber and of the airfilled cavity. It was determined that different parameter sets closely approximate the optimum impedance boundary condition. This is particularly advantageous, since the parameter set can be selected to fit to the requirement of the production technology. Alongside to the doublelayer MPAs, comparable variants without porous absorber were designed. It was found that the required total back volume can be reduced by up to 13.4% by using the porous absorber. Finally, the liners underwent experimental validation. For both the single and doublelayer MPA liners, the target frequencies are met to a very good approximation. In addition, the transmission loss is primarily result of dissipation. This corresponds to the expectations for liners with optimal impedance boundary condition according to Cremer [57]. Contrary to what is described in the literature, a more broadband damping was not achieved with the doublelayer MPAs compared to the singlelayer MPAs.
Thus, the primary questions of this work can be answered. The additively manufactured porous absorbers can be effectively modeled as a function of geometry and this model has been successfully validated. The acoustic effect of the absorbers is substantial enough to achieve a reduction in the necessary back volume.
Based on the results of this work, future research should aim to enhance the complexity of doublelayer MPAs by using MPPs with multiple circular or slotshaped micro perforations. Additionally, it would be intriguing to observe the behaviour of doublelayer MPAs when the absorber elements are positioned above the MPP, as well as the impact of the position of the absorber elements within the cavity on sound damping.
Conflicts of interest
The authors declared no conflicts of interests.
Data availability statement
Data are available on request from the authors.
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Cite this article as: Berchtenbreiter B. Renz A. & Becker S. 2024. The potential of additively manufactured porous absorbers in the design of multilayer microperforated absorbers. Acta Acustica, 8, 37.
All Tables
Characteristic geometric parameters of the additively manufactured porous absorbers.
Differences in total depth of the DLMPAs and SLMPAs for f_{opt} = 800 Hz and 1200 Hz.
Comparison of f_{opt} (design) and (measurement) for DL800, DL1200, SL800 and SL1200.
All Figures
Figure 1 Schematic illustration of a doublelayer MPA at perpendicular sound incidence. 

In the text 
Figure 2 Correction lengths ϵ according to Zieliński et al. [50] and Attala and Sgard [49] as well as the spline fitted from them as a continuous function of w_{S}. 

In the text 
Figure 3 Geometric structure of the investigated additively manufactured porous absorbers including the elementary cell (bottom left). 

In the text 
Figure 4 Mean values of the inverse determined JCAL parameters for each lattice constant l_{yz} (×) and the fitted functions based on it. 

In the text 
Figure 5 Transmission, reflection, and dissipation: measured data (, , ) and modeled data (T_{JCAL}, R_{JCAL}, D_{JCAL}). 

In the text 
Figure 6 Liner consisting of 36 DLMPA: general view (left), detailed view (top right), porous absorber (bottom right). 

In the text 
Figure 7 CAD model of the symmetrically designed TPPR. 

In the text 
Figure 8 Transmission loss: total TL (left), by dissipation TL_{D} (middle), by reflection TL_{R} (right). 

In the text 
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