The main purpose of this section is to design the primary structural configuration and its corresponding acoustic characteristics with the help of topology optimization [23,25,48]. For convenience, the following assumptions are made: (1) the acoustic structure dimensions are much smaller with respect to the acoustic wavelength for the subwavelength structures, which are more prone to reflection and refraction; (2) the reference points and integration boundaries in acoustic analysis remain unchanged in topological optimization iterative design [49]. Then, the sound pressure level (SPL) minimization of solid structures in the ASI system is taken as the objective function in this study. A corresponding topology optimization model can be established based on the SIMP method and combined with a projection basis filter [50]: (15)Find:X=[x1,x2,…,xn]TMin:{p}S.T.:f=Auφp=Gu−Hp∑i=1nxivi−αV≤00≤xmin≤xi≤1,(i=1,2,3,…,n)where x_i is an optimization variable in the optimization model, representing the finite element relative density. v_i represents the finite element relative volume. α represents the maximum volume fraction. V denotes the overall space volume of the structural design space. x_min is the minimum material density taken as 0.001 (non-zero to avoid singularity).

The sensitivities of the first two constraint functions in Eq. (15) can be directly calculated by taking the derivative of the design variable X: (16)∂f∂X=∂A∂Xu+A∂u∂X (17)φ∂p∂X+∂φ∂Xp=∂G∂Xu+G∂u∂X−∂H∂Xp−H∂p∂X

Given that the size of the solid structure is much smaller than the wavelength of the acoustic wave, we have ∂f/∂X = 0. Thus, Eq. (16) can be transformed into the following expression: (18)∂A∂Xu=−A∂u∂X

By substituting Eq. (3) into Eq. (18), it yields: (19)∂A∂X=∂K∂X−ω2∂M∂X

Eq. (19) represents the sensitivity of the dynamic frequency response. Then, by substituting Eq. (17) into Eq. (18), we can have: (20)(G+H)∂P∂X+(∂H∂X+∂G∂X)p=G∂u∂X−A−1∂A∂X∂G∂Xu

Since the reference points and integration boundaries remain unchanged during optimization, ∂G/∂X and ∂H/∂X are equal to zero. Thus, Eq. (20) can be converted into the following expression. (21)∂p∂X=GG+H∂u∂X

Eq. (21) denotes the sensitivity of the acoustic pressure of this optimization model. In this study, the derivative of u with respect to X is computed by the so-called adjoint variable method [49]. After obtaining the sensitivity information, the topology optimization formulation can be solved by the optimality criterion algorithm [13,51,52], which is widely accepted in the community of topology optimization.

The ASI system is assumed to be a bounded and closed system, its relative position and dimensions are shown in Fig. 1. Fig. 1a,b is the 3D and plan schematic diagrams, respectively. The solid part represents the PMAM in the ASI system. Surfaces A and B are two opposite faces of the PMAM. Herein, A is the surface where the acoustic wave impacts. Its structural parameters primarily affect the noise consumption capacity. Meanwhile, B is the structural function surface, and its structural parameters affect the noise reduction performance. In this study, the specific optimization objective is to obtain PMAM with the lowest sound pressure level (SPL) in the ASI system. According to the acoustic pressure expression in Eq. (14), the objective function of topology optimization is minimizing the SPL of surface B, which can reflect the noise filtering level of the PMAM. The primary parameters of the ASI system are shown in Table 1.

In this study, the solid region in Fig. 1 is assumed to be filled with uniform metamaterial microstructures. Thus, in the conceptual design phase using topology optimization, we simplify the solid structure in the ASI system as a planar structure. The 2D structure design domain is discretized by 100 × 100 quadrangle FEs. The volume fraction constraint is defined as 60%. The stimulated SPL is maintained at 70 dB. The initial design is completely solid, and the topology optimization iteration takes 82 steps to converge to an optimized design. The iterative history and several typical designs during the optimization process are plotted in Fig. 2.

As can be seen from Fig. 2, the topologically optimized design (82th step) has several holes compared with the initial solid design (1st step). This characteristic facilitates the diffraction of acoustic waves during the propagation process and the friction of the cavity wall to consume energy. Hence, the sound absorption and noise reduction abilities of the PMAM can be improved. As shown in Fig. 2, the SPL of the initial structure is 66.31 dB. The SPL of the optimized structure is 60.83 dB, with a drop of 8.26%. Furthermore, the material usage of the optimized design is 40% less than that of the initial design. Therefore, a 2D conceptual design of lightweight PMAM can be obtained using the topology optimization design. To strengthen the sound absorption ability of the PMAM, it is necessary to optimize the structure parameters to achieve a detailed design with the manufacturable features.

Parameter optimization of the 3D PMAM based on the conceptual design of topology optimization is performed to enhance its sound absorption and noise reduction performance. First, the 2D configuration obtained through topology optimization is remodelled, as shown in Fig. 3a. Then, the remodelled structure in Fig. 3a is taken as the cross-sectional shape of a 3D porous PMAM with the 10 × 10 periodicity, as shown in Fig. 3b. The porous structure has many repeated unit cells, which exhibit excellent noise reduction and sound absorption properties. Finally, the 3D porous PMAM in Fig. 3c is obtained by stretching the 2D structure in Fig. 3b. As shown in Fig. 4, the parameter settings for the structured unit cell reflect the dimensions of the entire 3D PMAM structure. Table 2 shows the range of values for the main design parameters of the optimized model. Additionally, setting a1 > a2, h1 > h2 is necessary due to the application requirement. In both literature and engineering [53–56], the tapered acoustic surficial slope of the main channel is one of the key parameters to control the sound pressure along the Z direction. Therefore, we considered these parameters when constructing the parametric design model. Referring to the above literature and relevant values in engineering, we gave a range of values for the slope of the conical acoustic surface of the main channel in this study.

Parameter optimization is a key element for the proposed optimization design scheme. The accuracy and efficiency of the parameter optimization directly determine the effectiveness of the proposed scheme. In this study, the parameter optimization of acoustic metamaterials is realized based on the SM. The deformation of a solid structure in the ASI system is a complex elastoplastic process with strong geometric, physical, and boundary nonlinearity. Hence, it is extremely difficult to obtain the strains that are essential for the optimization. SM, based on statistical theory, can construct the explicit function relationship of the complex objective function in a multidimensional design space. The explicit function can significantly reduce the difficulty of structural strain calculation, and thus help to efficiently solve the concerned nonlinear optimization problem.

In the actual design process, if all the microstructures are subjected to finite element analysis, not only will a lot of time be wasted repeating on the setting of boundary conditions, but also each analysis may take several minutes. This will undoubtedly bring a large amount of computational cost. In contrast, by using the SM, the response of the positional points can be predicted by performing only the finite element analysis of the sample points. By training the SM model in advance, we can get the required structural performance results in a few seconds. Compared to repetitive finite element analyses, SM models provide an exponential increase in efficiency.

The core of SM is to use as few initial model solutions as possible to build an alternative model that is as accurate as possible. The essence of SM is to use the response value of sample points in the experimental design to predict the response value of unknown points in the design space. Therefore, we can construct an approximate relationship (i.e., the so-called SM) of input-output parameters with only a little precomputation. The accuracy and efficiency of the optimization can be guaranteed. Fig. 5 shows a typical flowchart of SM for the parametric design of PMAM.

This study combines the adaptive weight method with the Kriging interpolation model, which can effectively avoid SM being stuck in a local optimum solution. In doing so, the global and local search capabilities of the optimization algorithm are also improved. Based on the Kriging model, the association equation between the design variables (X = [x₁, x₂, …, x_m]^T, x_i ∈ Rⁿ) and the response dependent variables (Y = [y₁, y₂, …, y_m]^T, y_i ∈ R^q) in the system could be represented by the equation below:

(22)Y(X)=Z(X)+g(X)among these, Z(X) follows an independent and identically distributed distribution and represents a systematic bias. It has the following statistical properties: Var[Z(X)] = σ², E[Z(X)] = 0, cov[Z(x_i), Z(x_j)] = σ²R(c, x_i, x_j), where x_i and x_j are any two sample points. Meanwhile, g(X) = [g₁(X), g₂(X), …, g_k(X)]^T is the deterministic component. It could be represented by using k known regression functions. Here, σ² denotes the process variance. R(c, x_i, x_j) denotes a spatial correlation function containing the parameter c, such as the Gaussian function R(X) = exp(−X²/c²), etc. Moreover, the parameter c is solved with the following equation:

(23)min{φ(c)=−(ln⁡|R|+mln⁡σ2)/2}where R is the relevance matrix. By linearly weighting and interpolating the response dependent variable y_i based on the independent variable x_i of the sample points, the response dependent variable of the measurement point X can be calculated as follows:

(24)Y^(X)=w(X)TYwhere w(X) = [w₁, w₂, …, w_n] is the weighted coefficient of the solution. Based on conditions of minimum variance and unbiasedness, the final weights vector can be determined by integration using the Lagrange multiplier method:

(25)w(X)=R−1(G(GTR−1G)−1+r(X))⋅(GTR−1r(X)−g(X))where G = [g^T(x₁), g^T(x₂), …, g^T(x_n)]^T is an expanded matrix with n rows and k columns. Among them, g^T(x_i) is obtained by transposing g(X). Inserting Eq. (25) into Eq. (24) yields the corresponding Kriging model, which is expressed as follows:

(26)Y^(X)=g(X)β∗+r(X)TR−1(Y−Gβ∗)where β^∗ = (G^TR⁻¹G)⁻¹G^TR⁻¹Y is the corresponding estimated value of β obtained by the least squares method. Among them, r(X) = (R(c, x₁, X), (R(c, x₂, X), …, (R(c, x_n, X))^T. In addition, R = [R_ij]_n×n = [R(c, x_i, x_j)]_n×n. Here, an adaptive weighting method based on a particle swarm optimization (PSO) algorithm is presented to automatically adjust the weight coefficients in Eq. (24). For more details of the standard PSO, the readers can refer to [57,58]. In the optimization of SM, the fitness value of the sample p_i is set as the response value y_i of the sample point x_i. The optimal sample fitness value is the initial response value y_m, and the average fitness value of the particle swarm is yavg=1n∑i=1nyi. The sample fitness value is better than the average fitness value, which is defined as Δ = |y_m – y^′_avg|.

Herein, the particles in the PSO are divided into three subgroups according to y_i, y_m, and y_avg. Different adaptive procedures are performed in each subgroup. Theoretically, in the standard PSO algorithm, increasing the weight coefficient can improve the global search ability, whereas reducing the weight coefficient can improve the local search ability. The method for adjusting the inertial weight coefficient w is shown below. (1) If y_i > y′_avg, w_i = w_i − (w_i − w_min)| (y_i − y′_avg)/(y_m − y′_avg)|. (2) If y′_avg ≤ y_i ≤ y_m, w_i remains unchanged. (3) If y_i ≤ y′_avg, wi = 1.5 − 1/(1 + k₁·exp(−k₂·Δ)). Herein, k₁ and k₂ are corresponding control parameters. Among them, k₁ is responsible for the upper limit control of w, while k₂ is responsible for the adjustment control of w_i = 1.5 − 1/(1 + k₁·exp(−k₂·Δ)). When the algorithm finishes running, if the sample distribution results are scattered at this point, Δ increases and w_i decreases. Then, the localized searching capability of this algorithm will be enhanced, and this group will show a convergence trend. If the sample distribution results of the cluster are obtained, Δ decreases and w_i increases. In this fashion, the algorithm has a stronger detection capability for deviating from local optimal solutions.

In this study, the PMAM is optimized by using the SM to minimize the sound acoustic pressure. Combinations of different parameters of the PMAM are regarded as the initial sample points. At the same time, these sampling points are also input variables in the SM. Herein, P is the output variable associated with the sampled points, employed to assess the noise attenuation capability of the optimized configuration. Structures with 10 × 10 unit cells are adopted in the simulation process for the SM. These optimized configurations more accurately reflect practical implementation scenarios like stress states, which encompass forces between adjacent unit cells.

In order to get an optimal configuration of a PMAM, the input variable of SM is set to the structural sizes. The simulated SPL on surface B is the output variable P_out for the SM. In the final phase, both input and output variables are synchronously integrated to establish the SM that creates a deterministic linkage across the parameter space:

(27)minPout=SPLs.tPii=1,2⋯11∈Ψwhere P_i is the input variable of the SM, and its value range is Ψ.

The widely adopted Latin hypercube approach generates a maximally effective and representative sampling distribution space within the SM [59,60]. Consequently, the parametric coordinate allocation across the design matrix achieves maximal equidistribution to guarantee SM fidelity throughout the parameter domain [61,62]. Scholars have established consensus through systematic benchmarking of viable methodologies. Specifically, the initial SM input domain originates from 5-dimensional (5D) functional analysis, achieving convergence through sequential refinement to yield optimized solutions satisfying precision thresholds upon completion of 11-dimensional (11D) high-fidelity simulations [63–65]. In this study, D = 11. There are 55 points in the initial sample space, and the SM must calculate 66 points. In total, there are 121 points after the optimization. Here, all the sample points (e.g., PMAM with different combinations of parameters) are analyzed in the environment of software COMSOL Multiphysics.

The simulation analysis diagram of the acoustic wave intensity for the optimized PMAM is plotted in Fig. 6. Fig. 6a,b shows the x-y and y-z cross-sectional views of the obtained PMAM, respectively. Compared with the initial structure in Fig. 3c, it is noted that the typical structure size of the optimized PMAM has changed. It is obvious that the acoustic wave intensity on surface B is significantly reduced when compared to that of surface A. The optimized parameters of the PMAM are shown in Table 3. To examine parameter evolution across structural variables at identical scales, all parameters undergo standardization during optimization. The main design variables’ progression within the SM-assisted optimization framework is depicted in Fig. 7.

As shown in Fig. 7a, d1, d2, and H exhibit fast convergence speed and small steady-state fluctuations. This phenomenon illustrates that d1, d2, and H are not sensitive to the change of output (SPL). The other parameters fluctuated significantly during the convergence process. These parameters are more sensitive to the change in SPL. As shown in Fig. 7b, the SPL approached a constant value of 48 dB after the PMAM was optimized by the SM-based method. Note that the SPL of the initial structure is 60.83 dB, and the SPL in the initial sample space is within [53, 61 dB]. It is seen that the SPL of devised PMAM is significantly reduced. Therefore, parameter optimization based on the SM is effective for the detailed design of PMAM.

During the parameter optimization of SPL for the PMAM, PMAM structures with different parameter combinations are obtained. As previously discussed, there are 121 sample points to be considered in the parameter optimization, which means that the total optimization step as well as the parameter set of PMAM structures during the optimization are all equal to 121. To further verify the effectiveness of the optimized structures in terms of sound insulation and noise reduction, we select typical structures during the parameter optimization to implement simulation analysis.

In a PMAM, surface A is an acoustically consuming surface that consumes the energy of an acoustic wave through the friction, reflection, and refraction of the PMAM. Surface B is an acoustic impacting surface, and the acoustic wave transmittance or reflectance on this surface directly reflects the primary noise consumption capacity. Therefore, the reduction ratio of the acoustic wave for different structures at different locations during the optimization process is analyzed, as shown in Fig. 8a. The reduction ratio of acoustic waves is a critical indicator that measures the absorption of acoustic waves for a PMAM. It represents the quotient between sound pressure amplitudes at a given boundary and the reference interface (i.e., surface A). The range of the reduction ratio is [0, 1], and the initial ratio is set to 1. Note that the location in Fig. 8a denotes the distance from the current surface to surface A. In addition, the SPL change at different locations (starting from surface A) of the optimized structure under different excitation frequencies is analyzed, as shown in Fig. 8b. The typical excitation frequencies of acoustic sources are set to 500, 1500, and 2500 Hz, which represent the noise in low-, medium-, and high-frequency bands, respectively.

As shown in Fig. 8a, noise reduction ratios of various PMAM structures are very large at the locations near the start surface. It again confirms that d1, d2, and H in Fig. 7a are insensitive to the SPL. Moreover, by comparing the acoustic wave reduction curves corresponding to various structures during the optimization, it can be observed that parameter optimization based on SM can effectively improve the noise consumption capacity of the PMAM. As shown in Fig. 8b, the excitation frequency of the acoustic source has a significant influence on the SPL of the PMAM. A higher excitation frequency will always cause a higher SPL at different locations in the PMAM and lead to a slower SPL reduction rate. It implies that the devised PMAM by SM-based parameter optimization can have the potential to absorb the acoustic wave with a lower frequency, which is important and challenging in real-world applications.

The change of maximum amplitude under different excitation frequencies of acoustic waves is also studied. As shown in Fig. 9a, the maximum amplitude on the surface B of optimized PMAM is smaller than that of the initial structure. It indicates that the acoustic wave attenuation ability of the optimized structure is enhanced. Furthermore, attenuation magnitude in the optimized configuration exhibits significantly reduced levels relative to the baseline design under increasing excitation frequency. The widening inter-amplitude divergence demonstrates superior vibration acoustic attenuation characteristics in the optimized configuration relative to its baseline counterpart. Fig. 9b shows the influence of excitation acoustic waves on the hole and solid areas of the devised PMAM. The overall SPL of the hole area is smaller than that of the solid area, which indicates that the porous structure helps improve the noise reduction performance of the PMAM. An explanation of the principle of the formula similar to our experimental results can be found in reference [66].

Due to extreme miniaturization and convoluted internal features in the PMAM design, traditional production techniques cannot achieve required tolerances. Therefore, additive manufacturing is employed in this study [67]. This makes it possible to manufacture complex structures that are previously restricted by traditional manufacturing methods. High-molecular-weight polymers have excellent viscoelasticity and damping characteristics, which are beneficial for integrating their damping properties with other acoustic absorption mechanisms to improve the acoustic absorption properties of a structure. Therefore, the polymer is used as the base material for 3D printing in this work, with a Poisson’s ratio of 0.394 and an elastic modulus of 2.2 GPa. According to the structural unit cell obtained by the above parameter optimization, the particular-shaped PMAMs with repeated unit cells are fabricated by additive manufacturing. Fig. 10a shows the preprocessing models in two sizes, and Fig. 10b presents the printed samples. Particularly, the round silhouette of the PMAM samples is determined by the impedance tube experiment. Table 4 lists the specific shape dimensions of two different sizes of PMAM samples. Table 5 presents a list of typical samples used for 3D printing.

Herein, printed PMAM samples are used for experimental verification to analyze their acoustic characteristics of noise reduction. The sound absorption coefficient is an important index for evaluating the noise reduction performance of the PMAMs. In this paper, the impedance tube test is implemented, which only requires a small sample without utilizing a standard reverberation room. These advantages make the impedance tube suitable for the acoustic test of the devised PMAM. The test system of the impedance tube consists of free-field speakers, impedance tubes, acoustic acquisition cards, signal processing equipment, and signal generators. Fig. 11 shows a schematic of the acoustic impedance tube, wherein ①–⑥ are the acoustic source, acoustic source tube, PMAM sample, sample tube, acoustic collection tube, and microphone, respectively.

The structure of the impedance tube is shown in Fig. 11. Tests with different frequency ranges refer to different tube diameters. The diameter of the large tube is 100 mm, and the test frequency range is [63, 1600 Hz]. The lengths of the large-impedance tube and specimen tube are 520 and 100 mm, respectively. In addition, the diameter of the small tube is 30 mm. The test frequency range is [1600, 6300 Hz]. The lengths of the small-impedance tube and sample tube are 380 and 45 mm, respectively. Therefore, to ensure that the PMAM sample meets the size requirement of the impedance tube, it is necessary to select an appropriate impedance tube according to the test requirements before conducting test experiments.

Two points must be considered when installing the PMAM samples on the impedance tube. Firstly, the PMAM sample should not be excessively compressed when it is installed on the test piece cylinder, and the surface of the sample should be flat to coincide with the reference surface. Secondly, the inner surface of the sample should be leaned on the bottom plate behind the impedance tube to avoid generating an air layer. In this study, the diameters of impedance pipes are 30 and 100 mm, respectively. The test frequency range is [50, 2500 Hz], which means the acoustic wavelength range is approximately [136, 6800 mm]. The unit cells in each printed sample or topology optimization model have a uniform length of 10 mm (i.e., X or Y direction), much smaller than the acoustic wavelength. Acoustic frequencies are discriminated and collected by an experimental device through frequency bands rather than a single frequency. Therefore, the frequency band division method (i.e., the 1/3 octave) for acoustic wave analysis is used in this study.

Through acoustic experiments, the acoustic characteristics of PMAM are analyzed, such as the sound absorption coefficient measured by an impedance tube. Subsequently, the acoustic characteristic change curve of the PMAM is obtained. Finally, the corresponding acoustic absorption coefficient of the 1/3 octave frequency is calculated. Fig. 12 shows the test sites of impedance tubes, including the acoustic absorption test equipment, acoustic insulation test equipment, low-frequency impedance tube, and high-frequency impedance tube.

There are two types of acoustic parameters tested with the impedance tube, including the acoustic absorption coefficient and acoustic insulation coefficient. The acoustic absorption coefficient represents the noise absorption capacity of a structure. The acoustic absorption coefficient equals the proportion of dissipated acoustic energy (including transmitted acoustic energy) assimilated within the PMAM relative to the total incident acoustic energy impinging upon the PMAM surface. Thus, the value of the acoustic absorption coefficient ranges from 0 and 1. The larger the acoustic absorption coefficient is, the better the acoustic absorption performance of the PMAM will be. The acoustic insulation coefficient is also referred to as the acoustic transmission loss. Its value is equal to the incident SPL minus the transmitted SPL. Both the acoustic absorption and insulation coefficients are closely related to the frequency of acoustic waves. Fig. 13 shows the acoustic curves of typical PMAM structures (including No. 1, No. 56, and No. 121) generated during the parameter optimization. Fig. 13a shows the curve of the acoustic absorption coefficient, and Fig. 13b plots the curve of the acoustic insulation coefficient.

The acoustic absorption coefficient shown in Fig. 13a continues to increase with an increase in the excitation frequency. Then, it is stabilized after reaching a peak. Comparing the curves of PMAM No. 1, No. 56, and No. 121 (No. 121 is the final optimized design), it shows that the parameter optimization design improves the acoustic absorption capability of PMAM especially in the moderate and high frequency ranges. Fig. 13b shows that as the excitation frequency of acoustic waves increases, the acoustic transmission loss of the PMAM first decreases and then increases. In particular, the increase rate is larger than the decrease rate. Moreover, the acoustic transmission loss is lower at low frequencies and higher at high frequencies.

Fig. 14 shows the comparison of experimental and simulated acoustic transmission losses in regards to the optimized PMAM at five typical excitation frequencies, e.g., 500, 1000, 1500, 2000, and 2500 Hz. As indicated in Fig. 14, the numerical error between the simulation results and the experimental results is less than 10% at each excitation frequency, which is within the acceptable range [68]. The experimental results of the additively manufactured sample validate the feasibility of the suggested method and devised PMAM [69].