An Extensive Parametric Analysis and Optimization to Design Unidimensional Periodic Acoustic Metamaterials for Noise Attenuation

Abstract

The presented research delineates an extensive study aimed at obtaining and comparing optimal designs and geometries for one-dimensional periodic acoustic metamaterials to attenuate noise within the audible frequency range of 20 Hz to 20 kHz. Various periodic designs, encompassing diverse geometric parameters and shapes—from Basic-Periodic to Semi-Periodic, Tapered-Diverging, and Tapered-Converging unit cells of repeated patterns—are examined to identify the most effective configurations for this application. A thorough parametric analysis is executed employing FE-Bloch’s theorem across these four configurations to determine their bandgaps and to identify the most effective geometry. A normalization process is utilized to extend the domain of the analysis and the range of the system parameters studied in this work, totaling 202,505 design cases. Finally, the optimal design is identified based on achieving the best bandgaps coverage. The study concludes with the presentation of frequency domain acoustic pressure responses at multiple sensing points along the filters, validating the performance and the obtained bandgaps through these optimal geometries.

1. Introduction

Over the past two decades, there has been a growing concern regarding the noise generated by mechanical systems [1,2,3]. These reliable and commonly used systems are found extensively in sectors such as automotive and aerospace. For instance, delivery carts in the automotive industry, and cabin pressurization and air conditioning systems in aviation, contribute to substantial levels of noise, which can have adverse effects on human health and the environment [4,5]. For instance, exposure to high levels of noise has been associated with significant human health consequences, including hearing impairment, chronic stress, and cardiovascular diseases [3]. The presence of vibrating components within these mechanical systems is responsible for generating continuous high levels of noise.

Conventionally, this issue was addressed by proposing and developing noise—reducing devices known as acoustic filters. These filters are designed to decrease the level of noise to a permissible threshold for human hearing. Acoustic filters function by absorbing sound energy, resulting in a decreased level of noise. In industries such as automotive and aviation, traditional acoustic filters like glass fibers and mineral wools have been widely implemented over the last 50 years [4]. However, the production of these conventional filters has negative impacts on both human health and the environment. From a health perspective, manufacturing processes for glass fibers and mineral wool release harmful particles into the air, which can be inhaled leading to respiratory problems [6,7]. On an environmental level, producing these traditional acoustic filters requires extensive use of non—renewable resources while contributing to carbon emissions [7,8,9].

In recent years, a field known as periodic metamaterials has shown great promise in the area of vibration and acoustic attenuation. Metamaterials are specially engineered materials that possess unique properties not commonly found in natural substances. They offer possibilities for reducing vibrations and sound levels in mechanical systems. By harnessing their distinct mechanical and acoustical characteristics, metamaterials have the ability to manipulate and control the transmission of mechanical and sound waves. The key factor behind their effectiveness is their ability to create frequency ranges called bandgaps, within which they either block or attenuate waves passing through them. It is worth noting that these bandgaps differ between vibrational wave attenuation and acoustic wave attenuation within periodic metamaterials.

Initial studies have focused on examining the attenuation of mechanical waves through the presence of bandgaps in periodic metamaterials. These periodic structures include hexagonal honeycomb, square, triangular, kagome lattice, and chiral—based lattices with varying degrees of porosity [10,11,12]. This investigation aims to analyze how changes in the topological parameters affecting porosity can impact both the width and location of these bandgaps. For instance, a triangular periodic metamaterial with different rates of porosity exhibits a single complete bandgap but covers different frequency ranges depending on its level of porosity [10]. Similarly, commercially used hexagonal honeycombs show that, as their level of porosity decreases, fewer bandgaps are formed [11]. These examples demonstrate the influence of topological parameters on bandgap formation and highlight the possibility of tuning bandgaps through adjustments to topology.

Recent research has focused on advancing periodic metamaterials to enhance their ability to attenuate specific frequency ranges through programmable deformations or novel designs. The development of programmable periodic metamaterials has shown promising results as they enable dynamic control over bandgaps and allow for real—time adjustments to different wave frequencies. Essentially, these materials are engineered in a way such that external loads cause changes in their topology, resulting in alterations to the width and locations of the bandgaps. To achieve this functionality, programmable periodic metamaterials utilize factors such as specific or varying Poisson’s ratios and thermally— responsive elastic moduli [11,13,14,15,16]. Another recent improvement involves creating novel designs for periodic metamaterials. Researchers started with an existing base design and systematically applied geometric modification techniques to generate a new class of periodic metamaterial with unique bandgap behavior. The goal is to create these new—class materials that exhibit desired bandgaps within specific frequency ranges. Geometric modifications involve incorporating curvatures, like sinusoidal curves, spline curves, Koch fractals, and hierarchical orders onto the straight ligaments of the base design, which were found to develop new designs of metamaterials with enhanced bandgaps’ characteristics [17,18,19,20,21,22]. Additionally, altering connections between rods and cores in geometric modification methods was proposed for creating optimized periodic metamaterials with enhanced bandgaps [23,24,25,26].

Following the success of periodic metamaterials in applications targeting vibration attenuation, researchers have turned their attention to exploring the potential use of these materials for creating acoustic bandgaps that can attenuate sound waves. Two designs configurations have been proposed: one involves using shell cores and slits as repeating elements in the periodic metamaterial, which act as resonators for acoustic waves and result in different resonating frequencies and bandgaps, while the other approach involves adjusting Co—Continuous complex structures to achieve bandgaps at various frequency ranges. Previous studies focused on this first design configuration to create periodic metamaterials capable of attenuating acoustic waves within the audible frequency range [27,28,29,30,31,32,33]. The variations among these studies primarily stemmed from differences in the shape configuration of shell cores and distributions of slits across the repeating elements of the periodic metamaterial. A parametric analysis was conducted to verify the tunability of acoustic bandgaps with varying parameters controlling the size of the shell—core and slits in these earlier mentioned works. The second proposed design configuration involves the development of a periodic metamaterial that is based on Triply Periodic Minimal Structures (TPMS). These TPMS—based structures have complex architectures with intricate arrangements of pores, resulting in unique interactions with acoustic waves. The performances of different TPMS—based structures, such as simply cubic, body—centered cubic, face—centered cubic, primitive, I—Wrapped Package, and Nevious structures, were analyzed and compared in terms of their bandgap behavior [34,35]. By varying the rate of porosity in each TPMS—based structure, the relationship between topology and bandgaps was investigated.

In order to advance the field of acoustic metamaterials, researchers have proposed further enhancements in bandgap programmability and novel design creation. The primary aim of the aforementioned design configurations was to enhance acoustic bandgaps by developing a new class of acoustic periodic metamaterial. This falls under the category of novel design creation. Another approach is the development of programmable acoustic metamaterials that allow for dynamic tunability of bandgaps through an external tuning mechanism dependent on external loads. By incorporating repeating elements with foldability and compressibility characteristics, it has been possible to achieve programmable periodic acoustic metamaterials, where changes in topology result in variations in both the width and location of the acoustic bandgaps, similar to how topological changes affect mechanical wave attenuation in programmable periodic metamaterials [36,37].

The previous research on the development of acoustic metamaterials has shown promise in creating potential candidate structures to replace conventional acoustic filters. These metamaterials offer advantages over traditional filters in terms of tunability and manufacturability. From a tunability standpoint, previous studies have demonstrated that the acoustic bandgaps in these materials can be adjusted through topological modifications to attenuate specific frequency ranges. This highlights the versatility of acoustic metamaterials, as they can be easily customized to cater to different frequency requirements for noise attenuation. In contrast, conventional acoustic filters typically have a fixed frequency response and cannot be as easily modified or tuned to meet specific needs. From a manufacturing standpoint, acoustic periodic metamaterials offer the advantage of being easily produced using additive manufacturing methods like 3D printing [38], which avoids the toxic fabrication processes involved in conventional filters [39]. It is important to note that additive manufacturing machines are equipped with carbon filters, which help eliminate odors and volatile organic compounds during the printing process. This feature not only ensures a cleaner working environment, but also promotes better health for those involved in the operation.

A fundamental step towards the design of metamaterials for noise attenuation involves obtaining the bandgaps. Despite the extensive work in the literature that examines various geometries [27,28,34,35,37], in terms of their efficiency for bandgap coverage and noise reduction, an important gap that has yet to be addressed is the lack of an extensive parametric analysis and further optimization. This work aims to fill this gap by first selecting some of the pre—existing 1D acoustic periodic metamaterial [28], and then implementing three geometric modifications to this geometry to create several new geometries, namely, Semi—Periodic, Tapered—Diverging, and Tapered—Converging, which are analyzed and further optimized for their bandgap coverage. The tapered geometries allow for a wider range of inlet—to—outlet ratios within a unit cell.

The proposed comprehensive analysis involves considering a set of 505 design parameters used to find the bandgap using FE—Bloch’s theorem. The bandgap results were further normalized, a vital step to facilitate the utilization of a larger set of system parameters. The normalized bandgap results are then used along with a set of 401 various unit cell sizes, making a total of 202,505 study cases for this optimization across all geometries studied in this research. An important filtering process that involves excluding the geometries with high ratios of viscous thermal boundary layer thickness to the smallest diameter within the unit cell was applied. This is essential for ignoring entropy effects and focusing primarily on the acoustic modes for the study and, therefore, the validity of the bandgap theorem. Finally, to verify the effectiveness of the optimal designs in attenuating noise within the audible frequency range, an external sound source was used for actuation at a given location and the acoustic sound pressure levels were found at several sensing locations to confirm the attained bandgaps. As a result, the optimal design identified in this work has shown significant bandgap coverage improvement over the desired audible frequency range compared to previous works [27,28,34,35,37].

2. Materials and Methods

2.1. Design Framework

The initial phase of this study involved the identification of a pre—existing 1D periodic acoustic metamaterial design. Past research primarily focused on examining the influence of different lattice shapes on bandgap sensitivity through some design combinations (i.e., within the range of 9 design combinations), which was enough to prove the bandgaps’ tunability through topological changes [28]. Since this current study’s main objective is to find optimal designs, an extensive parametric analysis for the chosen 1D periodic acoustic metamaterial designs (i.e., including pre—existing designs proposed earlier by Elmadih et al. [28] and the newly developed designs), will be conducted to furtherly and thoroughly investigate their sensitivity to changes in lattice geometry. It should be emphasized that the geometrical variations will specifically occur within the porosity of the acoustic metamaterial designs. In other words, modifications will be made to enhance the air geometrical part since sound waves predominantly propagate through it. For further clarifications, Figure 1 illustrates the solid part of the metamaterial (i.e., the actual shape of the metamaterial). However, Figure 2 and Figure 3 demonstrate the air part of the metamaterial (i.e., the porous part). This concept will be further elucidated in subsequent sections detailing the methodological approach through bandgap determination using FE—Bloch’s theorem methodology sub—section.

The present study introduces an extensive design framework by utilizing an existing 1D periodic acoustic metamaterial design as the base lattice proposed by Elmadih et al. [28]. This base lattice is then modified geometrically to create a new class of 1D periodic acoustic metamaterials, depicted in Figure 4. The figure illustrates the various steps involved in the extensive design framework. In this study, the existing 1D periodic acoustic metamaterial air unit cell was selected as the base structure. The air unit cell for the base structure will be referred to as Basic—Periodic air unit cell in this paper. To explore different design possibilities, three distinct geometrical modifications were applied to the Basic—Periodic air unit cell. The first modification resulted in a new class of air unit cells called Semi—Periodic air unit cells. Similarly, applying the second proposed geometric modification led to a new class named Tapered—Diverging air unit cells, and the third proposed modification resulted in another new class known as Tapered—Converging air unit cells. The proposed modifications in Figure 4 primarily involve changing the uniform cross—sectional holes to non—uniform cross—sectional holes. It is worth mentioning that starting from a pre—existing structure and then applying geometrical modifications on its components to create new structures is a common systematic practice that has been utilized in creating new designs of periodic metamaterials with enhanced mechanical bandgaps [17,18,19,20,40]. Subsequently, the bandgaps of different unit cell configurations (i.e., Basic—Periodic air unit cells, Semi—Periodic air unit cells, Tapered—Diverging air unit cells, and Tapered—Converging air unit cells) were determined using FE—Bloch’s theorem method. The bandgap data obtained from the bandgap determination step were utilized to filter and select optimal designs among various unit cell combinations, including Basic—Periodic air unit cells, Semi—Periodic air unit cells, Tapered—Diverging air unit cells, and Tapered—Converging air unit cells. The filtering process aimed to identify designs that cover a wide range of audible frequencies (i.e., 20 Hz to 20 kHz). Subsequently, the acoustic pressure responses for these four selected designs were computed in order to validate the concept of bandgap determination and choose the most favorable design among them.

2.2. Unit Cell Designs and Geometrical Parameters

The four air unit cells examined in this study, namely, the Basic—Periodic unit cell, Semi—Periodic unit cell, Tapered—Diverging unit cell, and Tapered—Converging unit cell, were characterized by non—dimensional geometrical parameters. The morphology of the Basic—Periodic air unit cell design was determined using the same non—dimensional parameters as previously used [28]. These non—dimensional parameters are $L/C$ and $d/C$, where $L$ is the neck length and $d$ is the diameter of the cylindrical hole. Figure 2 provides a visual representation of these parameters. To make sure that the hole size does not exceed the cube size, the condition $\left(C-2L\right)>d$ should hold (see Figure 2). A total number of 5 $L/C$ values were considered, ranging from 0.1 to 0.3 with an increment of 0.05. A total number of 16 $d/C$ values were considered, ranging from 0.05 to 0.75 with an increment of 0.05. After taking the design rule into account, a total of 55 normalized design combinations of the Basic—Periodic unit cells were considered. Moving forward, we now shift our focus to the newly developed unit cells known as Semi—Periodic unit cells, Tapered—Diverging unit cells, and Tapered—Converging unit cells. These unit cells share a common geometrical characteristic of having non—uniform cross—sectional holes. The non—dimensional geometrical parameters characterizing the morphology of these unit cells were $L/C$, $d_1/C$, and $d_2/d_1$, where $L$ is the neck length and $d_1$ is the diameter of the bigger cylindrical hole in the Semi—Periodic unit cell and the maximum diameter of the non—uniform cross—sectional hole in the Tapered—Diverging and Tapered—Converging unit cells. $d_2$ is the smaller cylindrical hole in the Semi—Periodic unit cell and the minimum diameter of the non—uniform cross—sectional hole in the Tapered—Diverging and Tapered—Converging unit cells. A total number of 5 $L/C$ values were considered, which ranged from 0.1 to 0.3 with an increment of 0.05, while 15 $d_1/C$ values were considered, which ranged from 0.1 to 0.75, and 3 $d_2/d_1$ values were considered, which were 0.5, 0.7, and 0.9. Similar to the design rule implemented for the Basic—Periodic unit cells, a design criterion was applied to these new—class unit cells where only designs that fulfill the requirement of having a sizing rule of $\left(C-2L\right)>d_1$ were considered. After taking into the design rule, a total of 150 normalized design combinations were considered in each new class of unit cell design. A total of 505 normalized design combinations were considered in the parametric analysis. For better visualization of the considered design combination, Figure 5 depicts the normalized design combinations considered for both pre—existing and newly developed configurations as sample points in their respective spaces.

2.3. Bandgaps Determination through Utilization of FE—Bloch’s Theorem Method

The determination of bandgaps using Bloch’s theorem approach with the unit cell as a geometrical domain is a widely used methodology in the field of periodic metamaterials. This is due to Bloch’s theorem utilizing the unit cell as the geometrical domain that physically represents the wave propagation characteristics in periodic structures. In this methodology, the wave equation is solved for a single unit cell of the periodic structure, considering the Bloch boundary conditions, which will be referred to as Periodic Boundary Conditions (P.B.Cs) in this paper. This approach assumes that the periodic structure is infinitely periodic, allowing the use of Bloch’s Theorem to calculate bandgaps. First the acoustic wave propagation equation is defined.

$$\nabla .\left(-{\displaystyle \frac{1}{\rho }}\nabla P\right)-{\displaystyle \frac{1}{c^2\rho }}\ddot{P}=0$$

(1)

The equation governing acoustic wave propagation is a partial differential equation that describes the behavior of sound waves, where $P$ represents the acoustic sound pressure, $c$ is the speed of sound in the medium, and $\rho$ is the density of the medium. Also, $\nabla =\left({\displaystyle \frac{\partial }{\partial x}}+{\displaystyle \frac{d}{dy}}+{\displaystyle \frac{\partial }{\partial z}}\right)$ is the Laplacian operator, which is simply the gradient. In this study, the focus is on sound wave propagation in an air medium due to the assumption of impedance mismatch between solid materials and air, leading to complete reflection of sound waves at their interface (i.e., solid part walls were assumed to be rigid boundaries). It is essential to note that impedance mismatch occurs when both mediums differ significantly in density and speed of sound. Consequently, propagation of sound waves happens dominantly in air medium. The air properties that were considered for this work were $\rho =1.2 \mathrm{k}\mathrm{g}/{\mathrm{m}}^3$ and $c=343 \mathrm{m}/\mathrm{s}$.

Next, it is necessary to solve the eigenvalue problem of Equation (1) in order to determine the bandgaps. As mentioned earlier in the Introduction section, bandgaps refer to frequencies at which sound waves are prevented from propagating. Relating this definition to the eigenvalue problem, bandgaps correspond to frequencies that are not covered by the eigenfrequencies of the air unit cell.

$$\nabla .\left(-{\displaystyle \frac{1}{\rho }}\nabla P\right)+{\displaystyle \frac{{\omega }^2}{c^2\rho }}P=0$$

(2)

Equation (2) represents the eigenvalue problem of the acoustic sound wave propagation (i.e., represented in Equation (1)), where $\omega$ denotes to the eigenfrequencies. For further elaboration, bandgaps will be the frequencies that do not satisfy the eigenvalue equation, meaning that the corresponding eigenfrequencies are not present within the band—structure. The eigenvalue problem is solved numerically using Finite Element Method, this leads to Equation (2) to be rewritten as

$${\omega }^{2}MP+KP=0$$

(3)

Equation (3) represents the finite element equation in relation to Equation (2). In this context, $M$ refers to the mass matrix and $K$ represents the stiffness matrix. To determine the bandgaps through solving Equation (3) with a single unit cell of the periodic structure, the Periodic Boundary Conditions (P.B.Cs) are applied. Bloch’s theorem enables the accurate representation of sound waves in an infinite periodic structure by considering a single unit cell. Figure 3 illustrates the application of Bloch’s theorem, where a unit cell is used as a geometrical domain to represent the wave propagation in the periodic structure. The Periodic Boundary Condition can be defined as

$$P\left(x+C\right)=P\left(x\right)e^{i\mathit{e}.\mathit{k}}$$

(4)

Equation (4) depicts the application of Periodic Boundary Conditions to one of the faces aligned with the direction of tessellation in 1D acoustic metamaterials along the x—axis. Here, $C$ represents the size of the unit cell, $\mathit{e}$ stands for the lattice vector, and $\mathit{k}$ denotes the wave vector. The lattice vector ($\mathit{e}$) signifies structural periodicity by indicating that repeating translations of multiple unit cells using this vector will yield a corresponding periodic structure. The lattice vector for the considered 1D acoustic metamaterial is ($\mathit{e}=Ci+0j+0k$) for the Basic—Periodic, Tapered—Diverging, and Tapered—Converging metamaterials and ($\mathit{e}=2Ci+0j+0k$) for the Semi—Periodic metamaterial. Consequently, it is necessary to determine wave vectors ($\mathit{k}$). These are determined by computing reciprocal vectors based on lattices and defining their relationship within the 1st Brillouin zone [41].

The reciprocal vector ($\mathit{b}$) is expressed as ($\mathit{b}={\displaystyle \frac{2\pi }{C}}i+0j+0k$) for the Basic—Periodic, Tapered—Diverging, and Tapered—Converging metamaterials, and ($\mathit{b}={\displaystyle \frac{\pi }{C}}i+0j+0k$) is for the Semi—Periodic Metamaterial, and it plays a crucial role in determining the first Irreducible Brillouin zone, as shown in Figure 6. Figure 6 depicts the first Irreducible Brillouin zone for the examined one—dimensional periodic metamaterial in the $\mathsf{\Gamma }-X$ direction (i.e., direction of tessellation). The first Irreducible Brillouin zone encompasses the permissible wave vectors through which sound waves can propagate within this periodic structure, specifically within our considered one—dimensional periodic metamaterials. It is worth noting that the first Brillouin zone corresponding to the structures that are one—dimensional periodic is a line in the reciprocal space, as illustrated in Figure 6. This work considered a total of 21 points corresponding to wave vectors ($\mathit{k}$), obtained by sweeping across the first Irreducible Brillouin zone along the $\mathsf{\Gamma }-X$ line.

The eigenvalue problem, described by Equation (3), will be computed for each specific Periodic Boundary Condition within the first Irreducible Brillouin zone. This process is repeated 21 times to account for each design combination and their respective Periodic Boundary Conditions. To solve these eigenvalue problems, the finite element software ABAQUS (ABAQUS 2023) is utilized. The EQUATION function in ABAQUS will incorporate the Periodic Boundary Conditions, while the LANSCOZ SOLVER will compute the eigenfrequencies. In this study, we used acoustic elements (A3D10) as elements comprising of tetrahedron element with 10 nodes. Additionally, a mesh sensitivity analysis was conducted to determine accurate results independent of variations in meshing density. Our findings indicated that when using a maximum mesh size equivalent to 3 mm and a minimum mesh size proportional to 0.05 times the hole diameter (i.e., the hole diameter in the uniform cross—sectional hole of the Basic—Periodic unit cell ($d$) and the smallest diameter ($d_2$) in the non—uniform cross—sectional hole of the Semi—Periodic, Tapered Diverging, and Tapered Converging unit cells) maintained accuracy during simulation.

3. Results

3.1. Benchmarking the Bandgaps through Using FE—Bloch’s Theorem Single Unit Cell—Dependent Methodology

The main emphasis of this study lies in the examination of bandgaps, which serve as indicators for frequency ranges where 1D acoustic metamaterials effectively attenuate sound. In order to achieve accurate and reliable results, it is crucial to benchmark a methodology that determines bandgaps in the considered 1D acoustic metamaterials. Specifically, we employ FE—Bloch’s theorem single unit cell—dependent methodology (i.e., one of the components in the extensive design framework).

This study replicates the bandgap findings for a 1D acoustic metamaterial composed of a Basic—Periodic unit cell with geometrical parameters of $C=30 \mathrm{m}\mathrm{m}, {\displaystyle \frac{L}{C}}=0.2, \mathrm{a}\mathrm{n}\mathrm{d} {\displaystyle \frac{d}{C}}=0.5$, as previously investigated [28]. Figure 7A showcases the replicated bandgap—structure results for this documented unit cell. The bandgaps in this study are represented using a series of plots known as bandgap—structures. These structures display the black regions indicating the presence of bandgaps. Traditionally, band—structure plots were used to depict eigenfrequencies, with frequencies that did not fall within these bands being referred to as bandgaps. However, this research proposes using the bandgap—structure plots instead of the conventional band—structure because attenuation goals for vibrations and acoustics rely on identifying and understanding these essential gaps [17]. Additionally, Figure 7B displays the input signal of the external sound source applied, while Figure 7C displays the acoustic pressure response for a 1D acoustic metamaterial made from seven corresponding repeated Basic—Periodic unit cells within a frequency range of 20 Hz to 12 kHz, while applying rigid boundary conditions on the faces opposite to the tessellation axis (i.e., the $x$ axis). The acoustic pressure response was calculated using modal analysis according to the selected methodology in this study (see Supplementary Materials). The external sound source (i.e., impulse signal sound source with $P=1 \mathrm{P}\mathrm{a}, SPL=94 \mathrm{d}\mathrm{B}$ applied across the calculated frequency range (20 Hz to 12 kHz), and represented in Figure 7B) was applied at one end of the metamaterial and the acoustic sound pressure level was determined at the other end of the metamaterial. By employing FE—Bloch’s theorem single unit— cell—dependent methodology, these re—generated results were obtained with an aim to validate and align with those originally generated by Elmadih et al. [28], confirming their accuracy and consistency as depicted in Figure 7A. Figure 7C demonstrates the presence of bandgap frequency regions by exhibiting significant attenuation within these specific ranges. It is important to acknowledge that by applying rigid boundary conditions or radiation boundary conditions to the faces opposite of the tessellation axis, similar attenuation frequency regions (i.e., bandgaps) can be achieved. This phenomenon has been studied and documented previously [28]. The attenuation regions of these metamaterials depend on their periodicity. Moreover, the bandgaps are influenced by the Periodic Boundary Conditions (PBCs), which are consistent for both a 1D periodic acoustic metamaterial with reflection boundary conditions and one with radiation boundary conditions applied to its opposing tessellated axis faces.

The bandgaps in various 1D acoustic metamaterials, including the Basic—Periodic unit cells, Semi—Periodic unit cells, Tapered Diverging unit cells, and Tapered Converging unit cells, will be determined using the benchmarked FE—Bloch’s theorem methodology. A total of 505 normalized design combinations will be analyzed.

3.2. Bandgaps for All Considered Unit Cells

Based on the previously demonstrated and discussed FE—Bloch’s theorem, the benchmarked methodology dependent on unit cell analysis was employed to determine the bandgaps for different types of unit cells. These included the Basic—Periodic unit cells, Semi—Periodic unit cells, Tapered—Diverging unit cells, and Tapered—Converging unit cells. The benchmarked method was used to evaluate 505 normalized design combinations in order to assess their respective bandgaps as both an initial consideration and core component of this parametric study. It is important to emphasize that this study will examine bandgaps in the normalized format. Computing the normalized bandgaps requires understanding of the normalized eigen—value problem of Equation (2). The normalization of the eigen—value problem is done within its spatial domain. For instance, consider an eigenvalue problem of a one—dimensional acoustic sound wave propagation partial differential equation and normalize its single spatial domain relative to the unit cell size ($C$). Equation (2) can be rewritten as

$$-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{{\partial }^{2}P}{\partial x^2}}+{\displaystyle \frac{4{\pi }^2{f}^2}{\rho c^2}}P=0$$

(5)

Upon normalization, we obtain

$$-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{{\partial }^{2}P}{\partial {x^{*}}^2}}+{\displaystyle \frac{4{\pi }^2}{\rho }}{n}^{2}P=0$$

(6)

where

$$n={\displaystyle \frac{fC}{c}}$$

(7)

Equation (5) represents the eigenvalue problem of the one—dimensional acoustic wave propagation partial differential equation. Equation (6) represents the spatial normalized eigen—value problem of a one—dimensional acoustic sound wave propagation, where $x^{*}={\displaystyle \frac{x}{C}}$ is the normalized spatial domain and $n$ is the normalized frequency. Equation (7) presents the definition of normalized frequency, where $f$ represents the actual frequency, $C$ signifies the size of the unit cell, and $c$ denotes the speed of sound in the medium through which sound wave propagation occurs (i.e., air medium). The normalized frequency is the same for unit cells with the same non—dimensional geometrical parameters but with different unit cell sizes. Further elaboration on the reason behind storing the normalized frequency will be provided in the subsequent section.

Looking at Figure 8, Figure 9, Figure 10 and Figure 11, these figures demonstrate the bandgap—structures of the considered normalized design combinations in each considered unit cell type (i.e., Basic—Periodic unit cells, Semi—Periodic unit cells, Tapered—Diverging unit cells, and Tapered—Converging unit cells), where each figure corresponds to one unit cell type.

Figure 8 illustrates the bandgap—structures of the 55 different normalized design combinations of Basic—Periodic unit cell designs. Each subplot represents normalized design combinations with varying $d/C$ ratios but the same $L/C$ ratio. Therefore, this plot consists of five subplots corresponding to different $L/C$ ratios, ranging from 0.1 to 0.3 in increments of 0.05, as described in this study’s methodology section. The x—axis in each subplot represents the filtered $d/C$ ratio based on the design criterion explained earlier. Meanwhile, the y—axis corresponds to the normalized frequencies. Upon examining the subplots, it becomes apparent that there is a noteworthy increase in bandgap widths as the diameter of the cylindrical hole decreases. This observation aligns with findings presented in previous research [28]. Additionally, it should be noted that the widest bandgap occurs at the first acoustic bandgap.

The bandgap—structures of the 150 considered normalized design combinations of Semi—Periodic unit cell designs is shown in Figure 9. Each subplot in each column represents normalized design combinations with varying $d_1/C$ ratios but the same $L/C$ ratios. Additionally, the morphology of this type of unit cell is influenced by an additional non—dimensional geometrical parameter, specifically the $d_2/d_1$ ratio. Therefore, as analyzed in the figure, each column corresponds to subplots of normalized design combinations with different $d/C$ ratios and $L/C$ ratios but with the same $d_2/d_1$ ratio. Similar to Figure 8, the $x$—axis shows filtered $d_1/C$ ratios based on the design criterion, while the y—axis displays the normalized frequency range. After examining the subplots, it can be observed that the behavior of the Semi—Periodic unit cells’ bandgap is similar to that of the Basic—Periodic unit cells. As the cylindrical holes decrease in size, there is a corresponding increase in bandgap widths. However, there is a narrow bandgap present in the first acoustic bandgap and at a lower normalized frequency range. This narrow bandgap, referred to as the first acoustic bandgap, has limited effectiveness in attenuating the target normalized frequency range for two reasons. One reason is that a small unit cell size is required for this narrow bandgap to be effective, as analyzed from the normalized frequency definition, and this would be advantageous in terms of fabrication. However, this can cause the occurrence of higher frequency bandgaps beyond the audible range, rendering them irrelevant for the desired application. The other reason is that having reliable unit cell sizes relative to the exhibited normalized bandgaps ensures significant behavior within the audible frequency range (i.e., 20 Hz to 20 kHz). In this case, the narrowness of the bandgap prevents attenuation in its corresponding frequency range. Hence, this study will focus on the normalized bandgaps beyond the first acoustic bandgap. This selection will be discussed in detail in the upcoming section, which outlines the optimal choice of a Semi—Periodic unit cell type for constructing a 1D periodic acoustic metamaterial capable of attenuating a wide range of audible frequencies (i.e., 20 Hz to 20 kHz).

The bandgap—structures for the considered 150 normalized design combinations in the Tapered—Diverging unit cell and the Tapered—Converging unit cell are shown in Figure 10 and Figure 11, respectively. These figures follow a similar organization as Figure 9, which depicted the bandgapstructure for the Semi—Periodic unit cell configuration. It is important to note that both Tapered—Diverging and Tapered—Converging designs are controlled by non—dimensional geometrical parameters that also regulate the morphology of the Semi—Periodic unit cell type. Each subplot represents different normalized design combinations with varying $d_1/C$ ratios while maintaining constant $L/C$ ratios, and each column displays subplots with varying $d_1/C$ ratios alongside varying $L/C$ ratios but with a shared $d_2/d_1$ ratio. The $x$—axis reflects filtered $d_1/C$ ratios based on design criteria, whereas the y—axis represents normalized frequency ranges. Upon examination of both Figure 10 and Figure 11, it is apparent that the Tapered—Diverging unit cells and Tapered—Converging unit cells demonstrate a wider bandgap at lower $d_1/C$ and $d_2/d_1$ ratios. This behavior aligns with the bandgap characteristics exhibited by the analyzed Semi—Periodic unit cells. The first bandgap corresponds to the widest bandgap observed, which is consistent with the behavior exhibited by the Basic—Periodic unit cells.

The obtained normalized bandgap data for the four—unit cell types (i.e., Basic—Periodic unit cells, Semi—Periodic unit cells, Tapered—Diverging unit cells, and Tapered—Converging unit cells) will be employed in a proposed filtering method to identify optimal designs for creating 1D acoustic periodic metamaterials that effectively attenuate a wide range of audible frequencies (i.e., 20 Hz to 20 kHz). Details regarding this approach will be further elaborated in the discussion section.

4. Discussion

4.1. Filtering Function of the Optimal Unit Cell Design Combinations in Each Considered Unit Cell Configuration

This study utilized data on bandgaps and key features of the widest bandgap (i.e., size and centroid; see Supplementary Materials) for all normalized design combinations analyzed in the previous parametric analysis. The objective of this work is to filter and identify optimal design combinations for each unit cell configuration. Both the bandgaps’ data and bandgaps’ key features were normalized. By varying the sizes of unit cells in these normalized design combinations, we determined their corresponding frequency ranges of actual bandgaps. Ultimately, the optimal design within each unit cell configuration was selected based on its ability to cover a substantial portion of the audible frequency range (i.e., 20 Hz to 20 kHz) with its bandgaps. The performance evaluation metric for the unit cells’ bandgaps in terms of covering the range of audible frequencies is referred to as bandgap coverage. Bandgap coverage represents the fraction that indicates how much of the audible frequency range (i.e., 20 Hz to 20 kHz) is encompassed by a unit cell’s bandgaps.

The proposed filtering approach in this study determines the optimal unit cell design combination for each configuration through a series of steps, as shown in Figure 12. Firstly, the previously collected normalized bandgaps’ data and key features from the parametric study were used as input. Secondly, a comprehensive sweep of unit cell sizes ($C$) ranging from 10 mm to 50 mm was applied to obtain the actual frequency ranges of the bandgaps for each unit cell design combination. It is important to highlight that the unit cell sizes ($C$) were discretized into 401variants ranging from 10 mm to 50 mm. Considering these 401 unit cells sizes in the 505 normalized design combinations led to 202,505 designs. The actual bandgaps of 202,505 designs were efficiently computed from the stored normalized bandgaps of their corresponding normalized design combinations that share the same non—dimensional geometrical parameters (computations happen within a couple of minutes) instead of going back and utilizing FE—Bloch’s theorem computations, which will require unrealistic computational time. For instance, a Basic—Periodic unit cell with non—dimensional geometrical parameters of ${\displaystyle \frac{L}{C}}=0.2,{\displaystyle \frac{d}{C}}=0.1$ with unit cell size ($C$) of $30 \mathrm{m}\mathrm{m}$ will have the same normalized bandgaps as the Basic—Periodic unit cell with the same non—dimensional geometrical parameters but with a different unit cell size (e.g., $C=50 \mathrm{m}\mathrm{m}).$ Thirdly, a design criterion was established to exclude any unit cells whose cylindrical holes are located too close to the boundary layer thicknesses (i.e., vorticity boundary layer thickness and entropy boundary layer thickness). Excluding these designs was crucial because the presence of closely spaced cylindrical holes near the boundary layer thicknesses leads to a medium flow behavior that encompasses not only the acoustic mode, but also includes the vorticity mode and entropy mode [42]. As a result, the governing equation for sound wave propagation solved in this paper (i.e., Equation (1)) is invalid in this specific scenario. It is essential to note that although having medium flow with additional vorticity mode and entropy mode may be favorable in attenuation, accounting them will lead into solving the coupled system for the specific designs with narrow cylindrical holes, which is a computationally expensive process.

$$l_{vor}={\left({\displaystyle \frac{2\mu }{\omega \rho }}\right)}^{{\displaystyle \frac{1}{2}}}$$

(8)

Equation (8) [42] provides a representation of the vorticity boundary layer thickness. In this equation, $\mu$ represents the kinematic viscosity of air and $\rho$ refers to the density of air. The variable $\omega$ denotes the frequency at which the air is propagating. Upon examining this relationship, it becomes evident that for an air medium with fixed kinematic viscosity $\mu$ and density $\rho$, the vorticity boundary layer is contingent on the propagation frequency. Furthermore, there exists an inversely proportional correlation between these two variables—as the propagation frequency decreases, an increase in vorticity boundary layer thickness is observed.

$$l_{ent}={\displaystyle \frac{l_{vor}}{\sqrt{P_r}}}$$

(9)

The thickness of the entropy boundary layer, introduced by Pierce [42], is represented by Equation (9). This boundary layer depends on the vorticity boundary layer thickness ($l_{vor}$) and the Prandtl number of air ($P_r$). The relationship between the entropy boundary layer thickness and the propagation frequency is indirectly determined through its dependence on the corresponding vorticity boundary layer thickness. It should be noted that this work focuses solely on accounting for the entropy boundary layer thickness during the filtering process, as it exceeds its corresponding vorticity counterpart in size.

The propagation frequency ($\omega$) used to determine the thickness of the entropy boundary layer ($l_{ent}$) in a unit cell with bandgaps pertains to the lower bound of the first acoustic bandgap (i.e., the widest bandgap). The rationale behind choosing this lower limit as the propagation frequency ($\omega$) stems from the understanding that there exists an inverse relationship between propagation frequency and entropy boundary layer thickness. This implies that operating at the lower bound frequency results (i.e., the lower limit frequency in which the metamaterial will be operated) in the maximum thickness for the entropy boundary layer could be attained within metamaterial structures.

$$f_l={\displaystyle \frac{c}{C}}C.O.W.B\left[1-\left({\displaystyle \frac{S.O.W.B}{2C.O.W.B}}\right)\right]$$

(10)

Equation (10) represents the frequency that corresponds to the lower bound of the first acoustic bandgap in Hz. This equation was written in such a way that relates the lower bound of the first acoustic bandgap (i.e., the widest bandgap in all the 505 normalized design combinations) to its corresponding bandgaps’ key features (i.e., the normalized size of the widest bandgap ($S.O.W.B$) and the normalized centroid of the widest bandgap ($C.O.W.B$); see Supplementary Materials). After formulating this equation, the entropy boundary layer ($l_{ent}$) equation was re—written in a way that incorporates Equation (10).

$$l_{ent}=\sqrt{{\displaystyle \frac{a}{f_l}}} ; a={\displaystyle \frac{\mu }{\pi \rho P_r}}$$

(11)

Equation (11) presents the modified form of the entropy boundary layer, incorporating the minimum frequency in Hz for the first acoustic bandgap. The constant variable $a$ represents properties specific to air medium. A filtering criterion is then applied to exclude unit cells with design combinations where either the diameter of a uniform cross—sectional hole in Basic—Periodic unit cell configurations ($d$) or the smallest diameter of a non—uniform cross—sectional hole in Semi—Periodic, Tapered—Diverging, and Tapered—Converging unit cell configurations ($d_2$) is less than 20 times its associated entropy boundary layer thickness ($l_{ent}$).

Fourth, the data corresponding to the filtered bandgaps for unit cells that satisfy the predetermined design criteria were analyzed in order to assess their bandgap coverage. Subsequently, an optimal unit cell design was found for each configuration (i.e., Basic—Periodic, Semi—Periodic, Tapered—Diverging, and Tapered—Converging) based on the evaluation of bandgap coverage.

The optimal combinations for unit cell design in each configuration were determined using the suggested filtering approach. These results are presented visually in Figure 13, showcasing the corresponding bandgap—structures for the optimal unit cell in each unit cell design configuration. Additionally,

Table 1. Bandgaps coverage for the optimal designs.

Metamaterial	Design Combination						Bandgaps Coverage
	$C \left(\mathrm{m}\mathrm{m}\right)$	$L/C$		$d_1/C$		$d_2/{\mathrm{d}}_1$
Semi—Periodic	42.9	0.3		0.1		0.5	0.955
Tapered—Diverging	38.7	0.3		0.1		0.5	0.944
Tapered—Converging	21.1	0.1		0.1		0.9	0.936
Metamaterial	Design Combination						Bandgaps Coverage
	$C \left(\mathrm{m}\mathrm{m}\right)$		$L/C$		$d/C$
Basic—Periodic	21.1		0.1		0.1		0.935

provides detailed information on the optimal unit cell design parameters and the bandgap coverage. The table also organizes the optimal unit cells based on their bandgap coverage from highest to lowest. Upon evaluation of Table 1, it is revealed that the optimal unit cells associated with the new—class unit cell configuration have demonstrated a slightly enhanced bandgap coverage performance compared to the optimal unit cell derived from the morphology of the Basic—Periodic unit cell design configuration. This is evident through their ability to achieve a wider bandgap coverage within the audible frequency range. The efficacy of the optimal newly developed unit cells will be further demonstrated in the actuation scenario subsection. The optimal Semi—Periodic unit cell surpasses other optimal unit cells in achieving the widest bandgap coverage within the audible frequency range, despite what may appear upon initial visual impressions.

The exploration of the new designs presented in the current work and the further optimizations for both these new designs, as well as the ones previously investigated in [28], have led to significant improvements for both the bandgap coverage (95% compared to 60% in [27,34,35,37]), as well as wider frequency range coverage of 20 kHz compared to 16 kHz. This highlights the significance and the necessity of employing an extensive design analysis—based framework that offers the end—user with the optimal design options while ensuring reasonable computational time, which is what this work’s framework provides. Furthermore, it is crucial to emphasize that the substantial data generated by the framework could be furtherly utilized as training input for machine learning models, enabling the development of a machine learning—based design framework for metamaterials’ properties [17,43,44,45].

4.2. Acoustic Pressure Response of 1D Acoustic Metamaterials Made from Repeated Optimal Unit Cell

The acoustic pressure response of a 1D periodic acoustic metamaterial composed of tessellated optimal unit cells, which were determined earlier in the filtered step mentioned in the preceding sub—section, is examined. Computing the acoustic pressure response of 1D metamaterials made from these optimal unit cells serves two purposes. Firstly, it serves as proof—of—concept for determining bandgaps using FE—Bloch’s Theorem (i.e., one of the components in the extensive design framework), and secondly, evaluating the acoustic sound pressure levels of these periodic structures.

One objective of this study is to examine the acoustic sound pressure levels in four different 1D periodic acoustic metamaterials. Each of these metamaterials has a unique unit cell size, resulting in variations in dimensions. To ensure a consistent actuation scenario across all four metamaterials, a building rule was implemented. This involved using a box with dimensions of 500 mm by 45 mm by 45 mm, and filling it with optimal unit cells to create each respective 1D periodic acoustic metamaterial. The number of unit cells required for each metamaterial was determined by dividing the length of the box by the size of the unit cell. This approach allows for the examination of the acoustic pressure response of the 1D periodic acoustic metamaterials and enables comparisons between different metamaterials with varying unit cell sizes. Figure 14A showcases the actuation scenario, where there is a designated actuation location point at one end of the box length and five sensing location points. One sensing location point is positioned at the other end of the box length (i.e., denoted as X), while two are located in the middle and at opposite ends of the tessellation axes (i.e., denoted as Y and Z). Additionally, there are two sensing locations close to one end of the box length, aligned with opposite axes to tessellation (i.e., denoted as ${\mathrm{Y}}^{\prime }$ and ${\mathrm{Z}}^{\prime }$). The external sound source placed at the actuation location was in the form of an impulse signal with $P=1 \mathrm{P}\mathrm{a},SPL=94 \mathrm{d}\mathrm{B}$ applied across the calculated frequency range (i.e., 20 Hz–20 kHz), which is depicted in Figure 14B.

The acoustic pressure responses of the optimal 1D periodic acoustic metamaterials are shown in Figure 15, Figure 16, Figure 17, Figure 18 and Figures S3–S6. These metamaterials consist of tessellated optimal unit cells that follow the introduced building rule and are actuated according to the actuation scenario. To assess the effectiveness of these optimal metamaterials, the acoustic sound pressure levels were compared between five sensing locations within the metamaterials and a block of air. Panels B in Figure 15, Figure 16, Figure 17 and Figure 18 and Panels A to D in Figures S3–S6 illustrate this comparison.

Table 2. The average sound pressure levels attained in the 5 sensing locations by the optimal 1D periodic acoustic metamaterial and the block of air (i.e., no metamaterial added).

Metamaterial	X $(\mathbf{d}\mathbf{B}$)	Y $(\mathbf{d}\mathbf{B}$)	Z $(\mathbf{d}\mathbf{B}$)	${\mathbf{Y}}^{\prime }$ $(\mathbf{d}\mathbf{B}$)	${\mathbf{Z}}^{\prime }$ $(\mathbf{d}\mathbf{B}$)	$\mathbf{Average} (\mathbf{d}\mathbf{B}$)	Reduction to Basic—Periodic (%)
Semi—Periodic	3.46	5.23	5.16	3.04	3.05	3.99	11.33
Tapered— Diverging	3.84	5.07	5.07	3.65	3.69	4.26	5.33
Tapered— Converging	3.77	5.14	5.13	3.68	3.68	4.28	4.89
Basic—Periodic	3.98	5.39	5.40	3.88	3.88	4.50
Block of Air	80.45	70.62	71.45	70.01	70.57	72.62

shows the average sound pressure level in each sensing location (i.e., denoted X, Y, Z, ${\mathbf{Y}}^{\prime }$, ${\mathbf{Z}}^{\prime }$ in the table), which was determined using the centroid of the area under the curve of the acoustic pressure responses (i.e., represented in Panels B–G in Figure 15, Figure 16, Figure 17 and Figure 18 and Panels A to D in Figures S3–S6), as well as the overall average sound pressure level in all the sensing locations of the four optimal metamaterials. Additionally, Table 2 evaluates the efficacy of these metamaterials by illustrating the average sound pressure level achieved at each sensing location (i.e., determined by computing the centroid of the area under the curve of the acoustic pressure responses) and the overall average sound pressure by a block of air. It is important to mention that the sound pressure levels in the frequency regions of bandgaps shown in Figure 15, Figure 16, Figure 17, Figure 18 and Figures S3–S6, demonstrated amplitudes below 0 ($\mathrm{d}\mathrm{B}$). This aligns with the audibility characteristic where a sound pressure level of 0 ($\mathrm{d}\mathrm{B})$ represents the minimum threshold of audibility. Consequently, when calculating average sound pressure levels and average sound pressure reduction values, any negative values for sound pressure level were substituted with a value of 0 ($\mathrm{d}\mathrm{B}$). The reason for replacing any negative sound pressure level with 0 ($\mathrm{d}\mathrm{B}$) was to have a realistic comparison between these metamaterials that is focused on human’s ear. The optimal metamaterials exhibited attenuated sound pressure levels at frequency regions in their acoustic pressure response that corresponded to their bandgap frequency ranges, as demonstrated in Figure 15, Figure 16, Figure 17 and Figure 18 through Panel B and Figures S3–S6 through Panels A to D. This serves as a proof—of—concept for utilizing the FE—Bloch’s theorem unit cell—dependent approach to identify bandgaps. Additionally, the effectiveness of the optimal metamaterials in reducing sound pressure levels at sensing locations compared to a block of air is evident from Figure 15, Figure 16, Figure 17 and Figure 18 through Panel B, Figures S3–S6 through Panels A to D, and Table 2. It is worth noting that Table 2 ranks the four optimal metamaterials based on their ability to attenuate sound, with higher rankings corresponding to greater reductions in sound pressure levels. This ranking aligns with Table 1’s classification of the optimal metamaterials according to bandgap coverage, reinforcing the application of simple— filtering—efficient step in the extensive design framework for determining the optimal designs. According to the analysis of the rankings in Table 2, it is observed that the new class of 1D periodic acoustic metamaterials is more efficient in attenuating sounds within the audible frequency range compared to the existing class of 1D periodic acoustic metamaterials. Additionally, it is found that among all the optimal 1D periodic acoustic metamaterials, the optimal Semi—Periodic acoustic metamaterial shows the highest level of sound attenuation. Upon further examination of Table 2, the analysis reveals that the metamaterial created from tessellating the optimal Semi—Periodic unit cell leads to an overall average acoustic sound pressure level that is 11.33% lower than that achieved by the metamaterial formed from tessellating the optimal Basic—Periodic unit cell.

The choice of material for these optimal 1D periodic acoustic metamaterials should consider the need for an impedance mismatch to effectively perform as simulated. In simpler terms, the selected material should have a much higher impedance than air to ensure that sound propagates primarily through the air medium. Previous studies used materials with significantly higher impedance than air in order to achieve the desired characteristics of impedance mismatch [27,28,37].

5. Conclusions

This work commenced by proposing an extensive design framework for designing 1D periodic acoustic metamaterials for attenuating noises passing within the audible frequency range of 20 Hz to 20 kHz. The focus is on creating and finding optimal designs and evaluating their effectiveness in noise reduction by analyzing the acoustic bandgaps and acoustic pressure response in the audible frequency range (i.e., 20 Hz to 20 kHz). The geometrical configurations are controlled using non—dimensional parameters as outlined earlier in the methodology section. A total of 505 normalized design combinations are considered during the parametric analysis. The noise attenuation for each design combination is evaluated through determining its corresponding bandgaps using the FE—Bloch’s theorem unit cell—dependent approach. In all the design configurations that include the four—unit cell geometries, it is shown that further reduction in the cavity hole diameter results in an increase in the widths of the bandgaps. Furthermore, the widest bandgap for all the periodic designs was found to be the first bandgap; however, the first acoustic bandgap for the Semi—Periodic designs was found to be too narrow to operate in for effective noise attenuation. Two key features (i.e., the normalized size of the widest bandgap and the normalized centroid of the widest bandgap; see Supplementary Materials) are introduced to analyze and evaluate the effectiveness of the noise attenuation in various designs. Analyzing the normalized Size of the Widest Bandgap in relation to non—dimensional geometrical parameters across all designs showed a consistent trend: as the holes’ diameters decrease, the normalized Size of the Widest Bandgap increases. Additionally, analyzing the normalized Centroid of the Widest Bandgap relationship to non—dimensional geometrical parameters led to two main conclusions. The first indicated that designs with lower $L/C$ values saw an decrease in the normalized Centroid of the Widest Bandgap as $d/C$ decreased in the Basic—Periodic designs and $d_1/C$ decreased in the Semi—Periodic, Tapered—Diverging, and Tapered—Converging designs with a fixed $d_2/d_1$ value, while designs with higher $L/C$ values observed a slight decrease or constant centroid with increasing $d/C$ in Basic—Periodic designs and increasing $d_1/C$ in Semi—Periodic, Tapered—Diverging, and Tapered—Converging designs while maintaining a fixed value for $d_2/d_1$. It is expected that the optimal designs must have both a larger size for their widest bandgap as well as a lower centroid value for this bandgap to help reduction for the lower frequency range (i.e., 0–800 Hz), which is more vital for noise reduction. These characteristics are generally found in unit cells with smaller ratios of the hole diameter to the unit cell size ($C$) (${\displaystyle \frac{d}{C}},{\displaystyle \frac{d_1}{C}},{\displaystyle \frac{d_2}{d_1}}$). This leads to hole sizes that may be too close to the visco—thermal boundary layer thickness. Designs that met this criterion were not taken into consideration by introducing a filtering—out rule in the optimal design filtering step. To evaluate the frequency ranges of the bandgaps and calculate their coverage, a comprehensive efficient filtering approach was employed. Various unit cell sizes ranging from 10 mm to 50 mm were considered and led to 202,505 designs, which were systematically examined, considering the established filtering—out rule and the limitations due the visco—thermal layers. The optimal unit cells in each configuration were identified based on their highest bandgap coverage. The results demonstrate that the optimal Semi—Periodic unit cell with $C=42.9 \mathrm{m}\mathrm{m},{\displaystyle \frac{L}{C}}=0.3,{\displaystyle \frac{d_1}{C}}=0.1,{\displaystyle \frac{d_2}{d_1}}=0.5$ exhibits the highest bandgap coverage. To further validate the results from the bandgap analysis using FE—Bloch’s theorem and filtering step, the acoustic pressure responses for all the optimal designs with the suggested actuation scenario were examined. The attenuated frequency regions observed in the acoustic pressure response were in perfect agreement with their respective bandgap frequency ranges from the FE—Bloch’s theorem, further proving the effectiveness of the optimal designs for acoustics attenuation of the desired frequency ranges. The results were further ranked for their effectiveness in sound pressure level reduction; this aligned with rankings determined from the optimal design filtering step, which relied on unit cell—dependent bandgap data.

The present study has provided significant insight into the use of metamaterials for acoustics reduction within the audible frequency range. Through systematic exploration of numerous unit cell configurations, optimal metamaterial designs for noise attenuation have been identified. These designs can be effectively employed in the fabrication of acoustics filters for various applications such as the automotive and aviation industries.