Kriging-assisted multi-objective optimization of locally resonant sonic crystal made of concentric C-shaped cavities for broadened sound attenuation

Abstract

A locally resonant sonic crystal made of acoustically coupled concentric C-shaped cavities is proposed that generates multiple band gaps for usage as ventilated noise barrier. There is lack of comprehensive theoretical model for acoustic response of such concentric resonators, because the inter-scatterer coupling and variations in scatterers’ opening and orientation angles introduce complex nonlinear effects. A semi-empirical theoretical model is proposed and validated for central frequency and bandwidth of the first band gap (BG) of proposed structure. Large-scale numerical analysis evaluates the impact of these angles, and a Kriging-assisted multi-objective genetic algorithm is developed to tune these angles for achieving a target BG with broadened bandwidth. Smaller opening angle and orientation misalignment lowers the central frequency and vice versa, whereas bandwidth varies nonlinearly. Experiments show multiple tunable transmission loss peaks above 20 dB within 1500 Hz, demonstrating a dimension invariant approach for tuning the sound attenuation of this sonic crystal.

Keywords

sonic crystal noise control metamaterial band gaps optimization Kriging model genetic algorithm

Introduction

Traditional methods of environmental noise control involve use of noise barriers and enclosures to partially cover construction site and highways.¹ However, implementing these methods is not always possible due to their visual and spatial restrictions. More specifically, low to mid-frequency sounds (usually between 0 and 1500 Hz) are difficult to attenuate by traditional materials like concrete, steel, wood, etc. In this regard, Sonic Crystals (SC) have drawn high attention as they can significantly control sound wave propagation due to their special stop band properties, simplicity of production and installation, and reduced environmental noise transmission with ventilation.²

SC is a periodic structure with a finite array of highly dense sound reflecting scatterers embedded in air as the host medium.³ It has demonstrated remarkable ability to manipulate sound waves and found practical uses in a variety of applications such as noise barriers, frequency filters, acoustic waveguides, and more.^4–7 SC works on the principles of Bloch’s theorem and the Braggs scattering effect (details can be found in Gupta,³ Chalurkar and Singh,⁸ and Panda and Mohanty⁹). As per these theories, periodicity of SC must be of the same order of magnitude as the wavelength of the target noise. This essentially means that the SC must be at-least several meters in size for attenuating low frequency (long-wavelength) sound waves, typically below 1500 Hz. To overcome this limitation of structural size and to improve the attenuation properties of conventional SC, recently resonance phenomena is being introduced locally at each scatterer.¹⁰ This drastically attenuates sound waves with smaller structural sizes and fewer scatterers, and can boost the sound insulation properties in the low frequency region.^5,11–14

Local resonance phenomenon can be implemented using various shaped resonators, such as C shaped resonator,^15–18 the Quarter, and Helmholtz resonator,^5,19 U profile resonator,²⁰ and other shapes that produce resonance effect. Among these scatterers, the C shaped resonators (or C-resonators) are widely studied because of their higher sound attenuation and ease of fabrication. Recent studies show that, concentric arrangement of sonic scatterers of varying geometric dimensions, enhances broadband attenuation by generating multiple band gaps, while reducing the overall dimension of the structure.^12,21,22 In this paper, sonic crystals whose unit cell is composed of concentrically placed resonators that are acoustically coupled, is referred to as Concentric Coupled Resonator Sonic Crystal (CCRSC).

Each resonator within a unit cell of such CCRSC works on the principle of Helmholtz Resonator (HR). Previous attempts have been made at modeling fundamental frequency of single C-resonator, for which the classical HR equation is tuned to numerous empirical models with open-open end boundary conditions to calculate effective neck length ( $L_{e f f}$ ) with end correction factor (Δ), to improve the accuracy.²³ In this way, several models have been proposed with different $L_{e f f}$ values to predict complex behavior of thin-walled resonator without neck length, such as $(t + Δ w / 2)$ ,²³ $(t + 1 / 2 \sqrt π w),$ ²⁴ $(t + 0.85 w)$ ,²⁵ where $t$ is thickness of the resonator’s wall and $w$ is width of the opening to resonator cavity. There is no uniformity in these models, and they vary based on the resonator configuration. Additionally, the presence of multiple concentric resonators, which are closely spaced, introduces further design complexity and complex acoustic coupling among the scatterers, making the theoretical modeling of CCSRC even more challenging. Therefore, existing investigations on CCRSC have focused on numerical and small scale experimental studies of its sound attenuation response,^12,21,22 without a comprehensive theoretical framework. For example, Elford et al.²² proposed a sonic crystal consisting of six slotted concentric scatterers for multiple low-frequency band gaps. However, this study relied solely on numerical band structure analysis. Similarly, Radosz²¹ studied sonic crystal noise barrier made of multiple concentric resonators, and experimentally demonstrated up to 13 dB transmission loss within 1500 Hz. However, the author was unable to develop a validated theoretical framework to predict the ban gap frequency of the concentric resonators. The lack of comprehensive theoretical model to accurately predict the sound attenuation properties of CCRSC necessitates the need for data driven models.^26–30 Moreover, there is need for optimization techniques for configuring the design of CCRSC for best sound attenuation results.

In such CCRSC, the opening angle of the cavity can effectively govern the neck length and end correction of the scatterer that behaves as Helmholtz like resonator. Additionally, the orientation angles of the scatterers may modify the incident wave field and the inter scatterer coupling. Therefore, these parameters are expected to significantly influence the local resonance. Hence, determining the effects of orientation and opening angle of scatterer would be vital as it can help in tuning the frequency and bandwidth of the band gap of a CCRSC without changing the dimensions of either the unit cell or the overall structure. Although there is research on the effects of scatterer geometry and scatterer size on the band gaps of sonic crystals, the effects of opening angle and orientation angle of the resonant cavity of the scatterers of a CCRSC remain relatively unexplored.

This paper proposes a locally resonant sonic crystal whose unit cell is made of acoustically coupled concentric C-shaped cavities, that is, it proposes a C-shaped CCRSC and develops a semi-empirical theoretical model for its first band gap. It further examines the impact of orientation angle and opening angle of resonant cavity of the scatterers on the band gap. The paper also proposes a Kriging-assisted multi-objective genetic algorithm for optimizing the studied parameters for designing the proposed CCRSC with the first band gap at desired central frequency with broadened bandwidth. The structure comprises of three concentric C-shaped scatterers. Three concentric scatterer design is an engineering choice to provide trade-off between having enough scatterers for multiple resonances, and at the same time avoiding complex design and material requirements. The first band gap is studied because it occurs due to local resonance effect; hence it is heavily influenced by the scatterer’s opening and orientation of resonant cavity (details are presented in section 3).

Theoretical development

Figure 1 shows a typical unit cell of a C shaped resonator (or “C-resonator”) and the unit cell of an CCRSC made of three concentric C-resonators. For demonstration of the theoretical model for this CCRSC, let us choose the inner radii of the outer, middle and inner scatterer as $r_{1} = 48 mm, r_{2} = 38 mm, r_{3} = 23 mm$ respectively (refer to Figure 1(a)). First, the equation for predicting fundamental frequency of each individual C-resonator has been derived, based on which the equation for fundamental frequency of the coupled system in the proposed CCRSC has been developed. A C-resonator works on the principle of Helmholtz Resonator (HR), and the theoretical equation of its fundamental frequency, based on the classical HR theory, can be represented as equation (1).

f = \frac{c}{2 π} \sqrt{\frac{A}{V \times L_{e f f}}}

(1)

where $A$ is the cross-section area of the opening (refer to Figure 1(a)), $c$ is the speed of sound in the host medium, $V$ is the volume of air enclosed within the resonator cavity (refer to Figure 1(a)) and $L_{e f f}$ is the effective length of the opening.

Figure 1.

Schematics used for the theoretical development: (a) 3D and 2D views of a C-resonator, (b) 2D view of a unit cell of Concentric Coupled Resonator Sonic Crystal (CCRSC) made of three concentric C-resonators, (c) calculation of effective surface area inside a C-resonator, and (d) plot of end correction factor as function of opening width of the three concentric C-resonators.

To improve the theoretical model of the C-resonator, a data driven semi-empirical theoretical modeling approach is adopted with HR theory. The 3-dimensional view of the C-resonator whose cylinder’s axial depth is ‘ $D$ ’ is shown in Figure 1(a). The acoustics behavior of this scatterer is approximated using 2D formulation, with the assumptions of plane wave propagation and geometric invariance along the depth $D$ of the scatterer. This allows the expression of the neck area and cavity volume in terms of equivalent parameter per unit depth. Then, the equation (1) can be further simplified by substituting the cavity volume $V = S \times D$ and neck area $A = w \times D$ , where $S = π r^{2}$ is the cross-sectional surface area of the C-shaped cavity, with $r$ as the inner radius of the scatterer, and $w$ is the opening width of the cavity (refer to Figure 1(a)). Additionally, the effective neck length is the thickness of the scatterer wall with some added end-correction. This end correction is represented as a function of $w$ as observed in previous studies.^23–25 Thus, equation (1) can be further solved as follows:

f = \frac{c}{2 π} \sqrt{\frac{w \times D}{S \times D \times L_{e f f}}} = \frac{c}{2 π} \sqrt{\frac{w}{S \times L_{e f f}}}; L_{e f f} = t + Δ^{'} w

(2)

where $t$ is the thickness of the cavity wall. $Δ^{'}$ is the correction factor that is derived empirically. Let us define the opening angle of the scatterer as the angle (θ) subtended at the center of the scatterer by the opening edges of the scatterer as shown in Figure 1. Then the opening width $w = 2 r s i n (θ / 2)$ .

Firstly, each of the C-resonator in the unit cell of CCRSC is considered individually, and their individual end correction factors are determined using empirical data. For this purpose, $L_{e f f}$ of $i^{t h}$ C-resonator is represented as a function of geometric parameters as shown in equation (3).

L_{e f f}^{(i)} = t + {Δ^{'}}_{i} w_{i} = t + {Δ^{'}}_{i} (2 r_{i} s i n (θ_{i} / 2))

(3)

where i = 1, 2, 3 for outer, middle and inner scatterer respectively with $r_{1} = 48 mm$ , $r_{2} = 38 mm$ and $r_{3} = 23 mm$ respectively. (‘ $a$ ’) is the lattice constant, $w_{i} = 2 r_{i} s i n (θ_{i} / 2)$ is the opening width, with $r_{i}$ and $θ_{i}$ being the inner radius and opening angle of $i^{t h}$ C-resonator. ${Δ^{'}}_{i}$ is the proposed empirical end correction factor for the $i^{t h}$ C-resonator. Equation (2) is simplified to represent the semi-empirical theoretical model of the fundamental frequency $f_{i}$ of the $i^{t h}$ C-resonator considered individually as follows.

f_{i} = \frac{c}{2 π} \sqrt{\frac{w_{i}}{S_{i} (t + ({Δ^{'}}_{i}) w_{i})}}

(4)

where $S_{i}$ is the surface area of cavity of $i^{t h}$ C-resonator and is given as follows (refer to Figure 1(c)).

S_{i} = r_{i}^{2} π - (\frac{r_{i}^{2} θ_{i}}{2} - \frac{w_{i} \sqrt{r_{i}^{2} - \frac{w_{i}^{2}}{4}}}{2})

(5)

A series of models were simulated with $a = 150 mm, t = 2 mm$ and $w$ varying from 5 mm–45 mm. This range is chosen because the inner scatterer diameter is 46 mm; therefore, opening width cannot go beyond 46 mm. The central frequency of their first complete band gap were analyzed (refer to methodology in section 3) with sound hard scatterer material and can be denoted as $\hat{f_{i}}$ . Then ${Δ^{'}}_{i}$ was found by fitting the function of equation (4) with numerically determined results using python symbolic and polynomial regression. Equation (6) represents the curve fitting strategy to find the empirical end correction factors, and equations (7)–(9) represent the empirically obtained end correction factors for scatterer with radius $r_{1} = 48 mm$ , $r_{2} = 38 mm$ , and $r_{3} = 23 mm$ respectively.

\min_{Δ_{i}^{'}} \sum_{i = 1}^{n} {(f_{i} - \hat{f_{i}})}^{2}

(6)

Δ_{1}^{'} = 5.279 - w_{1}^{0.125} (6.589 - w_{1})

(7)

Δ_{2}^{'} = - 0.6328 - 2 w_{2} - 0.5025 \ln (w_{2})

(8)

Δ_{3}^{'} = - 0.5559 \ln (w_{3}) - 0.8814

(9)

where, $n$ represents the number of models analyzed numerically for each C-resonator. Figure 1 shows a typical variation of the empirically obtained end correction factors. The regression yielded unique functional forms for each scatterer size, indicating a non-linear size-dependent effect on the end-correction. Each scatterer with different inner diameter creates different acoustic field at opening slit, resulting in different functional forms of $Δ^{'}$ (equations (7)–(9)). Therefore, considering a constant unit cell with different scatterers independently results in a size dependent symbolic regression form. Similar data driven approach is extended to CCRSC. CCRSC involves various design complexities such as enclosed air cavity space and effective length being shared by all concentric scatterers. Moreover, the closeness of the C-resonators leads to acoustic coupling among the individual resonators leading to resonance path ( $L_{e f f})$ elongation and impedance transformation. Due to such complexities, a theoretical model is not available for CCRSC, justifying the need of semi empirical modeling approach. After studying various formulations, the formulation that gave the best fit to the numerical results is presented from hereon. CCRSC is assumed as a parallel combination of the three individual C-resonators with a common equivalent cavity and a common equivalent neck because of the spatial overlap of the three resonators. The fundamental frequency of the combined resonator, that is, CCRSC is given as,

f_{L R M S C} = \frac{c}{2 π} \sqrt{\frac{w_{e q}}{S_{e q} \times L_{e q, e f f}}}

(10)

where the equivalent cavity area $(S_{e q})$ is approximated as follows:

S_{e q} = \sum_{i = 1}^{3} S_{i}

(11)

Further, a parallel impedance analogy is used to calculate the equivalent effective neck length $L_{e q, e f f}$ of the CCRSC as follows:

\frac{1}{L_{e q, e f f}} = \sum_{i = 1}^{3} \frac{1}{L_{e f f}^{(i)}}

(12)

where $L_{e f f}^{(i)}$ is the effective neck length for $i^{t h}$ scatterer as obtained by equation (3) and (7)–(9).

Similarly, equivalent opening width $(w_{e q})$ was modeled using a python symbolic and polynomial regression to best fit the numerical results and is given below:

w_{e q} = {[\frac{w_{1}}{\frac{0.92}{w_{3}} + \frac{1.72}{w_{2}} + \frac{10 y}{w_{1}}}]}^{0.5}

(13)

where $w_{1}, w_{2}, w_{3}$ are the opening slit width of outer, middle, and inner scatterer in meters respectively. This model was validated numerically to predict the fundamental frequency of the proposed CCRSC within the opening range ( $w_{i} =$ 5–45 mm) of $i^{t h}$ scatterer. This model offers a compact formulation and enables more accurate prediction of fundamental frequency by integration of Helmholtz theory with Machine learning approaches.

Due to unavailability of theoretical models for strongly coupled CCRSC, effective geometric parameters are introduced using symbolic regression-based modeling. It is observed that the equivalent cavity area ( $S_{e q}$ ), equivalent effective neck length ( $L_{e q, e f f}$ ), and the equivalent opening width $w_{e q}$ ) deviate from the actual physical geometry as per HR theory. This deviation is because of complex near-field interactions and strong coupling among the concentric scatterers. These empirically derived expressions are used because amongst the various forms that were tested, these come closest to predicting the numerical and the experimental results (refer to section 6). The expression of equivalent cavity area ( $S_{e q}$ ) takes this form, which could mean that during the CCRSC resonance all scatterers are excited simultaneously and the air enclosed in the cavity is under resonance of all three scatterers. Using outer cavity area alone would neglect the contribution of the other concentric scatterers and fail to consider multi resonant behavior in FEM simulations. Hence the total equivalent cavity area of the concentric resonator can be approximated as the sum of the individual areas (equation (11)). Similarly, equivalent effective neck length ( $L_{e q, e f f}$ ) and the equivalent opening width ( $w_{e q}$ ) are introduced to consider combined contribution of the concentric slits. The $L_{e q, e f f}$ formulation is due to the additive nature of the inertial forces of individual resonators (inertial force $F \propto \frac{1}{L_{e f f}}$ ).^31,32 These effective parameters are used solely as physical motivated parameters for python symbolic regression to involve all concentric resonators simultaneously while maintaining consistency with numerical and experimental results.

Using similar approach the bandwidth ( $B_{f}$ ) of the first band gap was obtained as equation (14). Regression analysis showed that the inner scatterer had least effect on the Bandwidth prediction, so $w_{3}$ was ignored to reduce complexity.

B_{f} = 134.76 + \frac{w_{1}}{w_{2}} (9.36 - \frac{0.0734}{w_{2}}) - \frac{1}{w_{1}} (0.377 - \frac{8 \times 10^{- 4}}{w_{1}})

(14)

Methodology of numerical investigations

The CCRSC structure proposed for the studies consisted of three concentric C-shaped scatterers as unit cell as shown in Figure 2(a), whose 2D periodic layout is shown in Figure 2(b). Numerical studies were done using COMSOL Multiphysics version 6.1 to (i) derive the novel semi-empirical theoretical model for this CCRSC (presented in section 2), (ii) study the influence of the parameters (presented in section 4), and (iii) develop the optimization algorithm (presented in section 5). The basic dimensions of the scatterers were chosen to achieve the required band structure within 1000 Hz. The unit cell was meshed using edge mesh with maximum element size of 6 mm, but free triangular mesh was used for finer elements near scatterers’ boundary interface, as shown in Figure 2(c). This meshing was done as per the default mesh conversion criteria (Finer Mesh) of pressure acoustics module in COMSOL Multiphysics. However, maximum size of the element was customized to 6 mm such that it remained significantly less than the standard limit of one-sixth of the minimum incident wavelength at maximum analyzed frequency (1600 Hz), ensuring mesh-independent study.^8,25 Further reduction in the selected mesh size did not significantly change the results but increased computation time, confirming convergence at the chosen meshing. This study analyzed the lower and mid frequencies between 5 and 1500 Hz. Unit cell was considered to calculate the effectiveness of entire CCRSC and its lattice constant. Lattice constant $(a)$ was determined using Bragg’s scattering effect so that CCRSC could have Bragg’s band gap within 1500 Hz. Assuming normal wave incidence and first-order Bragg’s scattering, the relationship is given by equation (15).^33–35

λ = 2 a \Rightarrow a = \frac{c}{2 f_{b}}

(15)

where $a$ is lattice constant, λ is the wavelength of incident sound waves, $c$ is the speed of sound in the host medium, and $f_{b}$ is the central frequency for Bragg’s band gap. The material of the scatterer was Polylactic acid (PLA) with mass density as 1240 $k g m^{- 3}$ and speed of sound as 1860 $m s^{- 1}$ . The host media was air with mass density as 1.21 $k g m^{- 3}$ and speed of sound as 344 $m s^{- 1}$ .³⁶

Figure 2.

Numerical modeling of CCRSC: (a) 2D view of a unit cell showing application of periodic boundary condition, (b) 2D view of periodically arranged CCRSC structure, (c) finite element meshed model of unit cell with free triangular element, and (d) 2D views of unit cell when all scatterers are oriented at the orientation angle $ϕ = - 90 ° a n d + 90 °$ .

Geometric parameters of the studied designs

The radii of outer scatterer ( $r_{1}$ ) was set as 50 mm, middle scatterer ( $r_{2}$ ) as 40 mm, inner scatterer ( $r_{3}$ ) as 25 mm, and thickness for all scatterers $t$ as 2 mm (refer to Figure 2(a)). The lattice constant $(a)$ was set to 150 mm considering Bragg’s effect (equation (15)) for frequencies between 5 and 1500 Hz. The opening angle of each scatterer (θ) were varied across eight angles: 5°, 15°, 30°, 45°, 60°, 120°, 150°, and 178° which resulted in a total of $8^{3} = 512$ unique combinations.

To investigate the effects of orientation of the scatterers, all three scatterers were rotated with respect to the z axis. The orientation angle (ϕ) of a scatterer was defined as the angle made by the line passing mid-way of the resonant cavity opening and direction of incoming sound waves measured anti-clockwise as shown in Figure 2(d). Orientation of each scatterer (ϕ) was varied as: −90°, −60°, −45°, −30°, 0°, 30°, 45°, 60°, and 90°, which resulted in $9^{3} = 729$ unique combinations. All possible combinations of orientation angles and opening angles were studied, which resulted in 373,248 simulation datasets (i.e. $512 \times 729$ datasets). This dataset was generated using COMSOL Multiphysics through a structured automatic parametric sweep in acoustics module. The first two Eigen-frequencies for each of the 373,248 scatterers’ combinations were obtained, based on which the central frequency and the bandwidth of first band gap for each combination were calculated and stored.

Band graph analysis

The band graph was investigated numerically using FEM in COMSOL Multiphysics 6.2. to solve acoustic wave propagation as given in equation (16). Applying Bloch’s Theorem for the acoustic wave propagation in structures with periodic modulation in ρ and/or $B$ , such as in CCRSC, equation (16) converted to equation (17).³⁷

\frac{1}{B} \frac{\partial^{2} p}{\partial t^{2}} + \nabla . (- \frac{1}{ρ} \nabla p) = 0

(16)

\nabla . (\frac{1}{ρ (r)} \nabla p (r, t)) - \frac{1}{B (r)} \frac{\partial^{2} p (r, t)}{\partial t^{2}} = 0

(17)

where $p$ is the acoustic pressure, ρ and $B$ are the density and Bulk modulus of the medium respectively, $r$ is the spatial position vector and $t$ is the time. $ρ (r) = ρ (r + a), B (r) = B (r + a)$ represent periodic change in density and bulk modulus of CCRSC, with lattice constant $(a)$ . A time harmonic pressure wave excitation is considered as equation (18).

p (r, t) = p (r) e^{j (ω t - k . r)}

(18)

where, $p (r)$ is the acoustic pressure, $j$ is the unit imaginary number, $ω = 2 π f$ is the angular wave frequency in $rad / s$ , and $k = ω / c$ is the wave number in $rad / m$ , with $c^{2} = B (r) / ρ (r),$ is the speed of sound in the periodic inhomogeneous medium of CCRSC. Using, this time harmonic pressure wave excitation, equation (17) reduces to the Helmholtz equation as follows:

\nabla . (\frac{1}{ρ (r)} \nabla p (r)) + \frac{ω^{2}}{B (r)} p (r) = 0

(19)

\nabla^{2} p (r) + k^{2} p (r) = 0

(20)

To reduce computational cost, the band structure was calculated using one unit cell of an infinite CCRSC, by applying Floquet–Bloch’s periodic boundary condition, that is, $p (r) = A (r) . e^{j ω t}$ on both opposite sides of the unit cell shown in Figure 2(a). Here, A(r) = A (r + a) $A (r) = A (r + a)$ is the function having periodicity same as the CCRSC. For the construction of band graph, the CCRSC was subjected to eigenfrequency analysis in the pressure acoustic Eigenfrequency domain. This analysis ensured that the pressure was uniformly regulated on both sides of the unit cell (refer to Figure 2(a)). The discrete form of general eigenvalue equation for solving the eigenfrequency is given below.

([K] - ω^{2} [M]) p = 0

(21)

where [K] and [M] are Global stiffness matrix, and mass matrix, ω is angular frequency, and $p$ is mode shape vector which describes the acoustic pressure wave at respective natural frequencies. To get the infinite periodicity of the solution, wave vector ( $k$ ) sweep function was used along the perimeter path of the irreducible Brillouin zone of reciprocal lattice, with sweep directions of ΓX, XM, and MΓ as $(0, \frac{π}{a})$ , $(\frac{π}{a}, \frac{π}{a})$ and $(\frac{π}{a}, 0)$ . Each path was discretized into 50 parts and solved with ARPACK solver.

Results and discussions of numerical investigations

Band gap due to local resonance

Figure 3 compares the band graph of a non-resonant solid cylinder, a resonant hollow C-shaped cylinder and the proposed CCRSC unit cell, where all unit cells have the same outer radius of 50 mm. The first band gap is found missing for solid cylinder, which is due to the absence of local resonance effect. The first band gap appears in other structures that contain resonating elements as scatterers in the unit cell, as shown in Figure 3(b) and (c).

Figure 3.

(a) Unit cell and band graph of a solid cylinder that shows no bandgap in 400–600 Hz due to absence of local resonance. Unit cell and band graph of (b) a single C-resonator and (c) the studied CCRSC, both of which show the first band gap in 400–600 Hz due to local resonance in the scatterers.

The same observation was for all studied designs; hence it can be concluded that the first band gap in CCRSC is due to the local resonance phenomenon at the scatterers. This claim is further confirmed from the sound pressure distributions across an array of these unit cells at 497 Hz that corresponds to the first complete band gap in the proposed CCRSC, as shown in Figure 4. Here also, the local resonance, which is shown by means of the disruption in acoustic pressure inside the resonator, is found only in C-resonators as opposed to the solid cylinder. Moreover, the reduction in sound pressure level is highest when sound waves pass through the array of the proposed CCRSC structure, followed by the array of single C-resonators. Whereas, there is no reduction in sound pressure level through the array of solid cylinders. This highlights that the CCRSC structure of the same dimensions is more effective in low-frequency sound attenuation compared with sonic crystal made of single resonators or solid cylinders.

Figure 4.

Plots of sound pressure distribution at incident frequency of 497 Hz, across an array of unit cells: (a) solid cylinder unit cells show no abrupt pressure changes indicating no acoustic resonance, (b) C-resonator unit cells, and (c) CCRSC unit cells show abrupt pressure changes near the scatterers indicative of acoustic resonance.

The variation in the opening angles and orientation angles of the C-resonators without changing the outer dimensions of the scatterers or the periodicity of the CCRSC, led to a change in the local resonance at the scatterers without significantly affecting the Bragg’s scattering effect. Hence, the variation in opening angles and orientation angles led to changes in the first band gap, which is due to local resonance at the scatterers. So, from hereon only the first band gap is studied in detail.

Validation of proposed theory

Figure 5 shows the numerical versus theoretical plots of central frequency and bandwidth of the first band gap of the studied designs $(for the condition w_{1} = w_{2} = w_{3}, w h e r e 5 m m \leq w_{i} \leq 45 m m)$ . The plots perfectly overlap with the Pearson’s $R^{2} = 0.96$ for central frequency and Pearson’s $R^{2} = 0.92$ for bandwidth. Additionally, the root mean square error (RMSE) values are 6.6 and 2.9 Hz for central frequency and bandwidth respectively, and their corresponding mean absolute error (MAE) values are 5.6 and 2.9 Hz. These values represent the validation of our proposed theory (section 2) against numerical results within opening width range of 5–45 mm of all three scatterers for their all possible combinations. Thus, our proposed theory is numerically validated.

Figure 5.

Comparisons of theoretically predicted and FEM simulation predicted central frequency and bandwidth of the first band gap of the proposed CCRSC.

Effect of opening angle of the resonant cavity

Figure 6 shows the contour plots of the variation in central frequency of the first complete band gap, for varying combinations of opening angles of the three concentric scatterers. Central frequency is strongly influenced by the opening angles, varying from 436 to 1056 Hz for the studied opening angle combinations. In general, the central frequency lowers with a decrease in the opening angles of each of the scatterers and vice versa. To get minimum central frequency of the first band gap the preferred configuration of opening angles is $5 °$ for all scatterers. This could be because decrease in the opening angle θ leads to the decrease in the opening width, which lowers the fundamental frequency of the unit cell, as per the proposed theory in Section 2. Hence, the first band gap (due to local resonance) forms at lower frequencies. Further, the effective path of wave structure interaction increases with increase in the inner circumference of the scatterer (due to reduced opening angle). This leads to attenuation of longer wavelengths (i.e. smaller frequencies), thus causing a lower central frequency of first band gap.

Figure 6.

Contour plots showing the variation in central frequency of the first band gap of CCRSC with variation in the opening angle of the middle and inner scatterers, when opening angle of outer scatterer is fixed.

Figure 7 shows the contour plots of the variation in bandwidth of the first complete band gap, for varying combinations of opening angles of the three scatterers. Bandwidth is strongly influenced by the opening angles, varying from 75 to 241 Hz for the studied opening angle combinations. The bandwidth of the first band gap in general increases with an increase in the opening angle of outer scatterer, except when the C-resonator becomes close to a semi-circle. The bandwidth further increases when at least one of the inner scatterer is close to a semi-circle (opening angle of $178 °$ ). However, the exact combination of opening angles that lead to the highest bandwidth is difficult to predict as it follows a complex non-linear relationship. This complexity likely arises from the internal interaction among the resonators, strength of coupling among them, and nonlinear behavior of the equivalent opening of the CCRSC with change in opening slit width and the size of the resonators. The difference in the requirement of opening angles for minimizing central frequency and maximizing bandwidth, and the complicated nature of bandwidth variation, necessitate the need for an optimization technique to obtain an CCRSC configuration that simultaneously minimizes central frequency with maximizing bandwidth. This is dealt in section 5.

Figure 7.

Contour plots showing the variation in bandwidth of the first band gap of CCRSC with variation in the opening angle of the middle and inner scatterers, when opening angle of outer scatterer is fixed.

Effect of scatterer orientation

The variation in the central frequency and bandwidth of the first complete bandgap with variation in orientation angle of scatterers is given in Figures 8 and 9 respectively. Central frequency is moderately influenced by the orientation angles, varying from 338 to 544 Hz for the studied orientation angle combinations. It is observed that the perfect alignment of all three scatterers, that is, when $ϕ_{1} = ϕ_{2} = ϕ_{3},$ leads to a significant increase in the central frequency and bandwidth. Whereas, increased misalignment of each adjacent pair of scatterers, that is, increase in |φ₁ – φ₂| and $| ϕ_{2} - ϕ_{3} |$ lowers the central frequency of the first complete band gap.

Figure 8.

Contour plots showing the variation in central frequency of the first band gap of CCRSC with variation in the orientation angle of the middle and inner scatterers, when orientation angle of outer scatterer is fixed.

Figure 9.

Contour plots showing the variation in bandwidth of the first band gap of CCRSC with variation in the orientation angle of the middle and inner scatterers, when orientation angle of outer scatterer is fixed.

Figure 10 shows the sound pressure distributions in aligned and misaligned configurations, where the length of adjacent rarefaction and compression that is, $λ / 2$ is traced by black dashed arrows. Misaligned configuration disrupts the direct wave propagation into the unit cell and leads to an increase in the path length traveled by the wave in the resonating structure; thus increasing the θ, and decreasing the fundamental resonance frequency. This is also observed in our previous studies on metamaterials.³⁸ Thus, the first band gap (due to local resonance) forms at lower frequencies. Additionally, the plot exhibits a symmetric behavior for opposite oriented combinations, for example, orientation combination of $- 90 °, 90 °, a n d - 90 °$ has same central frequency as the combination of $90 °, - 90 °, a n d 90 °$ .

Figure 10.

Plots of sound pressure distribution at the central frequency of the first band gap, across an array of unit cells with: (a) aligned configuration and (b) misaligned configuration, which shows elongation of wave path length inside the scatterers in misaligned configuration.

Bandwidth is strongly influenced by the orientation angles, varying from 55 to 119 Hz for the studied opening angle combinations. However, the conditions for increase in bandwidth is opposite to that for decrease in central frequency. In general, the increase in alignment, that is $ϕ_{1} = ϕ_{2} = ϕ_{3}$ , leads to the broadest bandwidth of the first bandgap; but the value of the individual orientation angle does not affect the bandwidth. As observed in section 4.3 and 4.4, minimizing central frequency and maximizing bandwidth have contrasting demands on opening angles and orientation angles, and they follow a complex non-linear relationship. This necessitates the need for an optimization technique to obtain an CCRSC configuration that simultaneously minimizes central frequency with maximizing bandwidth. This is dealt in section 5.

Optimization of the proposed C-shaped CCRSC

The massive 373,248 simulation dataset was used for developing an optimization technique for configuring the opening angles and orientation angles of the three concentric scatterers of the proposed CCRSC structure to obtain a desired first band gap, that is, with desired central frequency and bandwidth. To counter the high computational demand of developing a regression-based model from such extensive dataset, the simulation results were directly used to create surrogate model using Kriging method.^39–42 This Kriging surrogate model was created using Gaussian based regression framework in MATLAB (“fitrgp” function) with default parameters to achieve statistically optimized smooth interpolation. A constant trend function with square exponential kernel was used to establish smooth nonlinear relation between design variables (i.e. opening angle and orientation angle) and the acoustic responses (central frequency and bandwidth). The kernel hyper parameters were estimated through maximum likelihood estimation using quasi-Newton optimization algorithm. The mentioned processes are standard processes in signal processing tools, and their detailed description can be found in Peng et al.,⁴⁰ Cao et al.,⁴¹ and Du and Fu⁴² Kriging is especially effective for complex, nonlinear relationships with high accuracy. It develops an interpolating model of target response based on known sample data points, making it suitable tool for multivariable data analysis. The Kriging method was combined with multi-objective genetic algorithm to discover the optimal solution from the multiple set of best solutions.^27,43,44 To eliminate overfitting of the developed Kriging-assisted model, it was validated against unseen simulation dataset. For this purpose, new simulations were conducted with unseen opening angles of 7°, 40°, and 100° and unseen orientation angles of −70°, −20°, 15°, and 65° resulting in datasets of 1728 (i.e. $3^{3} \times 4^{3}$ ) unique scatterer combinations. This dataset was completely independent of training dataset, whose design variables were selected at random, within the design bounds. The developed surrogate model yielded Pearson $R^{2} = 0.98$ for both central frequency and bandwidth when validated against this unseen dataset. Additionally, the model’s RMSE values were 14 and 5 Hz for central frequency and bandwidth respectively, and their corresponding MAE values were 11 and 4 Hz, when validated against the unseen dataset.

The optimization algorithm was demonstrated for two cases, namely, when the CCRSC needs to be implemented for (i) a low frequency noise source, and (ii) a high frequency noise source. Usually in noise control, it is desired to tune the band gap frequency of a sound attenuating structure to match the spectral peaks of the target noise source. Further, to increase the effectiveness of the sound attenuating structure, the bandwidth at the band gap needs to be broadened. Accordingly, the objectives of the optimization algorithm were set as follows, and the objectives functions are given by equations (22) and (23).

Case 1: Minimize central frequency and maximize bandwidth of the first band gap by configuring the scatterer orientations and opening angles: for low frequency noise source (typically between 100 and 500 Hz).

Case 2: Maximize central frequency and maximize bandwidth of the first band gap by configuring the scatterer orientations and opening angles: for mid to high frequency noise source (typically beyond 1000 Hz).

C a s e 1 : M i n i m i z e f (θ_{1}, θ_{2}, θ_{3}, ϕ_{1}, ϕ_{2}, ϕ_{3}) A N D M a x i m i z e B_{w} (θ_{1}, θ_{2}, θ_{3}, ϕ_{1}, ϕ_{2}, ϕ_{3})

(22)

C a s e 2 : M a x i m i z e f (θ_{1}, θ_{2}, θ_{3}, ϕ_{1}, ϕ_{2}, ϕ_{3}) A N D M a x i m i z e B_{w} (θ_{1}, θ_{2}, θ_{3}, ϕ_{1}, ϕ_{2}, ϕ_{3})

(23)

where, $f$ and $B_{w}$ are the central frequency and bandwidth of the first band gap respectively represented as functions of the opening angles and orientation angles of the three scatterers of the proposed CCRSC.

The optimization algorithm of the proposed Kriging-assisted Multi Objective Genetic Algorithm (K-MOGA) for case 1 is given in Figure 11. K-MOGA was implemented using optimoptions(“gamultiobj”) MATLAB function, with population size as 500, maximum 2400 generations, and the function tolerance of 10⁻⁶. The lower bound ( $L_{i}$ ) and upper bound ( $U_{i}$ ) of the ith scatterer used in K-MOGA implementation is given in equation (24).

5 ° \leq θ_{i} \leq 178 °; - 90 ° \leq ϕ_{i} \leq 90 °; \forall i = 1, 2, 3

(24)

Figure 11.

Algorithm for optimization of scatterers’ orientation angles and opening angles in CCRSC, to minimize central frequency and maximize bandwidth of the first bandgap.

The visualizations of the optimal solutions by K-MOGA implementation in each case is shown in Figures 12 and 13. The optimized values of the response variables and design variables are presented in Table 1. In addition, Table 1 also shows the optimized configurations for individual objectives namely “minimize central frequency,” “maximize central frequency,” and “maximize bandwidth,” and their visualizations are added in Figure 14. The optimized designs were analyzed numerically in COMSOL Multiphysics and their performance compared with the K-MOGA solution.

Figure 12.

Pareto front of the optimization results, together with the optimized unit-cell configuration and its corresponding band graph for the objectives of minimizing the central frequency and maximizing the bandwidth of the first band gap of CCRSC.

Figure 13.

Pareto front of the optimization results, together with the optimized unit-cell configuration and its corresponding band graph for the objectives of maximizing the central frequency and maximizing the bandwidth of the first band gap of CCRSC.

Table 1.

Results for each optimization case (here, Min = Minimize, Max = Maximize).

Objectives	Optimized design variables (in degrees (°))						Output by K-MOGA (Hz)		Output by simulations (Hz)		Mean validation error (%)
Objectives	$θ_{1}$	$θ_{2}$	$θ_{3}$	$ϕ_{1}$	$ϕ_{2}$	$ϕ_{3}$	$f$	$B_{w}$	$f$	$B_{w}$	Mean validation error (%)
Min $f$	5	5	5	10.56	−90	90	321.26	-	327.05	-	1.77
Max $f$	178	178	147.3	−4.4	−3.2	55.7	1048	-	1047	-	0.1
Max $B_{w}$	120	178	171. 7	38.85	−11.4	−27.4	-	270	-	283	4.59
Min $f$ , Max $B_{w}$	7.21	149	107.1	90	−62	21	375	75	389	74	2.47
Max $f$ , Max $B_{w}$	175	178	171	−8.4	51.2	58.5	1010	231	1009	228	0.7

Figure 14.

Optimized unit-cell configuration and its corresponding band graph for the single objectives, namely, minimizing central frequency, maximizing central frequency, and maximizing the bandwidth of the first band gap of CCRSC.

Figure 12 illustrates the optimal opening angle at 7.21°, 149°, 107° and optimal orientation angle at 90, −62, 21° of outer, middle, and inner scatterer respectively, for the combined objective of minimizing central frequency and maximizing bandwidth, which results in 375 Hz as central frequency with 75 Hz bandwidth. Numerical solution of this optimal combination gave central frequency at 389 Hz with 74 Hz bandwidth, with a relative validation error of 3.59% and 1.35% for central frequency and bandwidth respectively. The calculation of this validation error is shown in equation (25), where true value is FEM result and obtained value is optimization result. Similarly, Figure 13 shows the optimal opening angle at 175°, 178°, 171° and optimal orientation at −8.4°, 51.2°, 58.5° of outer, middle, and inner scatterer respectively, for the combined objective of maximizing central frequency and maximizing bandwidth, which gave central frequency and bandwidth of 1010 and 231 Hz respectively. Corresponding numerical results were 1009 and 228 Hz, with validation error of 0.1% and 1.31% in central frequency and bandwidth respectively.

In this way, the optimization results were numerically validated for all cases with mean relative error of 1.39% and 2.42% for central frequency and bandwidth respectively (refer to Table 1), as calculated using equation (26). It is also observed that configurations of “maximizing central frequency and maximizing bandwidth” is very similar to the single objective of just “maximizing central frequency.” It is Case 1 that is tricky due to the tradeoff between minimizing central frequency and maximizing bandwidth.

E r r o r_{i} = \frac{| t r u e v a l u e - o b t a i n e d v a l u e |}{t r u e v a l u e} \times 100

(25)

M e a n e r r o r = \frac{\sum_{i = 1}^{n} E r r o r_{i}}{n}

(26)

Experimental verification

To validate our developed theoretical model and the findings from FEM study, a prototype of the proposed CCRSC was fabricated using commercially available PVC pipes with inner radii of outer, middle, and inner scatterer as 55, 45, and 25 mm respectively with wall thickness of 2 mm. These radii were chosen as they were the only commercially available options that came closest to our studied radii values of 50, 40, and 25 mm respectively. Each pipe was cut to a length of 1000 mm, and a 30 mm longitudinal slot was cut along the length of the pipe using table saw to give each pipe an opening width $w = 30 mm$ . To align and support concentric pipes with flexible orientation and desired lattice constant, a custom platform was built using MDF hardboards with 3D printed holders. These allowed controlled rotation about their vertical axis to change the combination of the orientation and keeps cylindrical scatterers upright. Additionally, 3D printed spacers are used at the top end of the scatterers to ensure concentric placement and prevent physical contact among scatterers. A finite array consisting of three rows and four columns of these scatterers, with the overall dimension of 600 mm × 450 mm × 1000 mm, was arranged for experimental validation and kept at the center of an anechoic chamber at 600 mm above the floor, as shown in Figure 15. This anechoic chamber was acoustically insulated from the outside environment, and had an ambient noise level of 21 dB, and had absorptive foam pyramids on all walls and floor to control reflections (refer to Figure 16). Two omnidirectional free-field pre-polarized piezoelectric GRAS microphones, with model number 40 PH-10 CCP, were kept at a distance of 500 mm from the receiving and transmitting ends of the CCRSC structure (refer to Figure 16(i)).

Figure 15.

Schematic side view of the experimental transmission loss measurement setup inside the anechoic chamber.

Figure 16.

Experimental setup for transmission loss measurement: (a) view from the source side, (b) view from the receiver side, (c) data visualization, (d) experimenter, (e) data acquisition system, (f) 3D printed sample (g) sample with configuration 1, (h) sample with configuration 2, and (i) microphones.

A single diaphragm monopole speaker was kept at a distance of 1 m from the CCRSC and it played harmonic tones at different frequencies, one at a time. The sound pressure levels from the microphone were acquired using a National Instruments make, NI cDAQ-9174 type nine channel acquisition system at the sampling rate of 10,000 samples/second (refer to Figure 16(e)). The height of source, microphones, and sound source were maintained 500 mm above from the CCRSC base as shown in Figure 15. The experiment was conducted for the targeted frequencies from 170 to 1600 Hz and transmission loss values were determined for each frequency as the difference in SPL received at microphone before and microphone after the sample. Two different experimental configurations were analyzed. First with all scatterers oriented at 0° as shown in Figure 16(g), and second with the changed orientation of outer, middle, and inner scatterer at 90°, −90°, 90° respectively as shown in Figure 16(h).

Transmission loss spectrum analysis

For a one-to-one comparison between experimental and numerical results, a 2D model of the same CCRSC structure, as used in the experiments, was modeled in COMSOL Multiphysics with inner radii of 55m, 45, and 25 mm for outer, middle, and inner scatterer respectively as shown in Figure 17. Pressure acoustic frequency module of COMSOL Multiphysics was used to calculate the TL spectra. Perfect matching layer (PML) was applied to minimize the reflection and replicate similarity with experimental study. TL spectra was analyzed using monopole sound source of amplitude 1 Pa (94 dB) with nonlocal integration couplings. The TL was then calculated using the equation (27) using acoustic pressure obtained at the output port ( $p_{o u t p u t}$ ) and input port ( $p_{i n p u t}$ ). The model was meshed using predefined physics controlled free triangular mesh and solved using MUMPS solver.

T L = 20 \log_{10} (\frac{p_{i n p u t}}{p_{o u t p u t}})

(27)

Figure 17.

2D FEM model used for transmission loss analysis for one-to-one comparison with experimentally measured transmission loss. The CCRSC array is placed surrounded by perfectly matched layer (PML) to simulate free field condition equivalent with absorptive lining in anechoic chamber.

Results and discussions

Figure 18 shows the results of experimental verification for configuration 1 and configuration 2 respectively. For configuration 1, the numerical and experimental TL spectra match very closely with each other. Moreover, all the frequency bands containing the TL peaks in the experimental and numerical results correspond to the complete and partial band gaps found in the band graph of the studied CCRSC. The first band gap is obtained between 470 and 595 Hz using numerical simulation and between 450 and 610 Hz (i.e. frequency band between first and third peak) in experimental TL spectra. This gives the central frequency of the first band gap as 532.5 Hz numerically, 530 Hz experimentally, and both of them match very closely with the central frequency predicted from our proposed theoretical model at 531 Hz. Thus, the sound attenuation performance of the proposed CCRSC configuration 1 is experimentally verified.

Figure 18.

Comparisons of experimentally measured transmission loss (TL), FEM predicted TL and theoretically predicted central band gap frequency, of CCRSC configuration 1 ( $0 °$ orientation and 35 mm opening width), as overlaid on the band graph.

For configuration 2 also, the numerical and experimental TL spectra match closely with each other, albeit with slight discrepancies in the first band gap, as shown in Figure 19. Moreover, the frequency bands containing the TL peaks in the experimental and numerical results for most of the cases correspond to the complete and partial band gaps found in the band graph of the studied CCRSC (with slight differences in the first band gap). Thus, the sound attenuation performance of the proposed CCRSC configuration 2 is also experimentally verified. The reason for the slight differences could be slight tilt in the sample that was supposed to be exactly vertical and slight change in orientation from the 90°, −90°, 90° configurations due to unavoidable human errors. For both configurations, experimental results show multiple TL peaks of more than 20 dB, within 1500 Hz; thus highlighting the effectiveness of the proposed CCRSC for low frequency sound attenuation. Experiments further prove that for the same fabricated samples, just a real time change in orientation can drastically change the TL peaks, which demonstrates the capability for tuning the sound attenuation performance in real time of the proposed CCRSC.

Figure 19.

Comparisons of experimentally measured transmission loss (TL), FEM predicted TL and theoretically predicted central band gap frequency, of CCRSC configuration 2 ( $90 °, - 90 °, 90 °$ orientation and 35 mm opening width), as overlaid on the band graph.

Conclusions

This paper proposes and validates the semi-empirical theoretical model of the fundamental frequency of the unit cell of the proposed novel CCRSC structure. It is proved that the first band gap of CCRSC is due to local resonance, and it can be tailored by varying the opening angle and/or orientation angle of the resonant cavity of the CCRSC. This provides a crucial way of real-time tuning of the band gap frequency without changing the dimensions of either the unit cell or the overall structure. Central band gap frequency is found to decrease with decrease in opening angles of each scatterer due to decrease in the fundamental frequency of the unit cell. Whereas the increased misalignment of each adjacent pair of scatterers increases the path length of wave structure interaction, which increases resonance wavelength, lowers resonance frequency, thus the first band gap forms at lower frequencies. Band width of the first band gap follows a complex relationship, but for most cases, it increases with increase in opening angles and orientation alignment. A Kriging-assisted Multi Objective Genetic Algorithm (K-MOGA) is developed and validated for optimizing the scatterers’ opening angles and orientations to obtain desired central frequency of the first band gap with maximized bandwidth. The results are experimentally verified in an anechoic chamber where transmission loss spectra of CCRSC shows multiple tunable peaks of more than 20 dB, within 1500 Hz.

This paper provides a new theoretical model and innovative knowledge on the effects of the geometry of the scatterers’ resonant cavity on sound attenuation by CCRSC. Importantly, it proposes a novel K-MOGA optimization for real-time tuning of the sound attenuation in a Sonic Crystal without changing its dimensions. The presented findings should be interpreted in light of certain limitations. The study was restricted to the optimization of the first band gap and to configurations consisting of three concentric C-shaped scatterers with fixed sizes. Furthermore, the experimental analysis was conducted using a finite 3 × 4 array of CCRSC, and scaling the design for practical large-scale barrier implementation requires further investigation.

Footnotes

Acknowledgements

We would like to thank Professor Rajib Kumar Panigrahi for providing us the access to the Anechoic chamber facility of the Department of Electronics and Communication Engineering, IIT Roorkee. We would also like to thank Mr. Kamveer Maurya, Lab assistant, Microwave lab for his cooperation in conducting experiments done in the anechoic chamber. We also thank Mr. Siddharth Shrivastava and Miss Moh Moh Myint Thu for helping in the experiments.

ORCID iDs

Chetan Ashok Chalurkar

Sneha Singh

Ethical considerations

This article does not contain any studies with human or animal participants.

Consent to participate

There are no human participants in this article and informed consent is not required.

Consent for publication

Authors confirm that consent has been obtained from all individuals who were directly or indirectly involved in this research work.

Author contributions

Chetan A Chalurkar: Methodology, Investigation, Analysis, Writing – Original, Visualization, Experiments. Sneha Singh: Conceptualization, Methodology, Analysis, Writing - Original, Supervision, Project administration.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data availability statement

Data can be made available upon request sent to the corresponding author.

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