On the optimization of sonic crystal acoustic noise barriers

 

David Ramírez-Solana¹

Department of Electric and Information Engineering

Politecnico di Bari, Via E.Orabona 4,70126 Bari, Italy UE

 

Jaime Galiana-Nieves

Universitat Politècnica de València

Camí de Vera, s/n, 46022 Valencia, Valencia, Spain, UE

 

Javier Redondo

Instituto de Investigación para la Gestión Integrada de zonas

Costeras Universitat Politècnica de València, Paranimf 1, 46730, Grao de Gandia, Valencia, Spain, UE

 

Agostino Marcello Mangini

Department of Electric and Information Engineering

Politecnico di Bari, Via E.Orabona 4,70126 Bari, Italy UE

 

Maria Pia Fanti

Department of Electric and Information Engineering

Politecnico di Bari, Via E.Orabona 4,70126 Bari, Italy UE

 

ABSTRACT

 

Acoustic noise barriers based on sonic crystals constitute one of the most promising innovative bets of recent years in the field of environmental noise control. Sonic crystals are defined as met- amaterials formed by periodic arrays of acoustic rigid scatterers embedded in a fluid (usually air). Furthermore, Helmholtz resonators can be added with the purpose of increasing the acoustic isola- tion of the barrier. Helmholtz resonances and multiple scattering are both noise control mechanisms of the sonic crystal noise barrier (SCNB). During the last decades several authors have different techniques to optimize the performance of these devices showing its huge potential. In this work we present new strategies with Particle Swarm Optimization in order to improve the performances of the approaches presented in the related literature.

 

1. INTRODUCTION

 

1.1. Sonic Crystal Noise Barriers

 

Noise is a serious health problem and since several decades ago, noise attenuation methods have been developed to avoid the impact in our life. Regarding to this attenuation methods, when it is not possible to mitigate the noise during the emission phase, the most common mechanism is to use noise barriers during the transmission phase, placing them between the source and the receiver. Classical noise barriers are usually flat continuous walls, with a density not less than 20 Kg/m² and without good discrimination criteria between noise frequency ranges. Also, they don’t let wind, water, or light to pass through them. To improve the noise control performance of the classical noise barriers, the use of metamaterials has brought a huge range of possibilities, in particular, Sonic Crystal Noise Barriers (SCNB) are one of the best and more innovating method with very competitive results com- paring with traditional noise barriers. Sonic Crystals are heterogeneous materials made by periodic arrays of acoustic scatterers embedded in fluids, usually air [1]

 

1.2. Noise control mechanisms on Sonic Crystals Noise Barriers

 

SCNB have been improving their efficacy throw the pass of the years, adding new noise control mechanisms to their main one, multiple scattering. The incorporation of Helmholtz resonators to the scatterers allows to have new band gaps of attenuation frequencies [2].The equation 1 determines which band gaps (BGs) will be attenuated by the multiple scattering (MS) of the metamaterial fol- lowing the Bragg’s law is [3]:

 

 

where c₀ is the sound velocity propagation in the fluid [m/s], 𝑛 is the index of Band gaps, this work is focus on n =1, a is the lattice constant of the Sonic Crystal, the separation between the periodic arrays of scatterers [m].

 

Helmholtz resonators are placed at the centre of the cylindrical scatterers of the SCNB, and the equation 2 shows how the resonance frequency at the 2D model:

 

 

where c₀ is the sound velocity propagation in the fluid [m/s], Λ is the width of the mouth of the resonator [m], L is the length of the neck of the resonator [m], ∆ is the correction factor of the neck length (usually 1.6 or 1.8) and S is the internal surface of the resonator cavity [m² ].

 

1.3. The Particle Swarm Optimization

 

The Particle Swarm Optimization (PSO) is based on animal behavior and belongs to the evolu- tionary algorithm family. The algorithm's original version attempted to imitate animals' social behav- ior by simulating their fly, shifting positions, and discovering the global optimal position. The PSO technique involves placing a number of components, known as particles, in the problem's search space, each of which evaluates the fitness (or objective) function at its current location. Each particle decides its trajectory in the search space by combining some elements of its current and best positions with those of the swarm's closest members. After all particles have been relocated, the next iteration begins.

 

The swarm as a whole, like a flock of birds in search of food, moves close an optimum of the fitness function. Each swarm particle is made up of three D-dimensional vectors: the current position x_i , the previous best position p_i , and the velocity v_i . D is the dimension of the search space. The present position x_i is evaluated as a possible solution to the problem, as a collection of coordinates describing a point in space. The coordinates of this point are recorded in the vector p_i if it proves to be better than the previous one. The resultant best function's value is saved in a variable called previous best pbest_i , which can be compared in later iterations.

 

The algorithm's goal is to update the p_i and pbest_i vectors and find better places. Furthermore, the algorithm updates the velocity vector v_i iteratively and calculates new points by adding v_i co-ordinates to x_i .

 

The following steps can be used to explain a version of the PSO algorithm [4] .

 

Step 1: Create a population array of particles in the search space with random positions and ve-locities on D dimensions.

Step 2: Execute this loop at iteration k:

 

1. Determine the appropriate optimization goal function in D variables for each particle.

2. Compare the particle's pbest_i to its fitness assessment. Set pbest_i equal to the current value and p_i equal to the current location x_i in D-dimensional space if the current value is better than pbest_i .

3. Assign the variable gbest to the particle in the neighborhood that has had the most suc- cess so far.

3. Modify the particle's velocity and position in accordance with the equations:

 

 

 

The factor w , known as inertia weight, was introduced in [5] to soften the impact of v(k) on v(k+1) by reducing the effect of the maximum velocity. While r₁ and r₂ are random values in the range [0,1], c₁ and c₂ are positive constants known as acceleration coefficients that are responsible for weighing both the cognitive and social components correctly. Indeed, particle velocity drives the optimization process through the cognitive component, which is based on particle history, and the social compo- nent, which is based on knowledge about the swarm's optimum solution. To ensure algorithm con- vergence, some parameters must be suitably chosen, depending on the problem to be solved. The maximum velocity of particles is given by the following equation:

 

 

where h is the clamping factor, 0 ≤ℎ≤1. The velocity of each particle is limited by the range [-V_max , + V_max].

 

Usually, the inertia wight w is:

 

 

where itr_max is the algorithm's maximum number of iterations and k is the current iteration. Typi-cally, itr_max =10 and w(k) decrease in the interval [0.9, 0.4], by guaranteeing the equilibrium between exploration and exploitation.

 

The PSO algorithm's convergence is guaranteed by the following relationship between c₁ and c₂ and w :

 

 

Usually, choosing, w=0.7298, c₁ =c₂ =1.49618 gives good convergent behavior [6].

 

2. METHODOLOGY

 

2.1. Finite Elements Method Simulation 2D Model

 

The 2D model of the SCNB to optimize is analyzed after a Finite Elements Method (FEM) simu- lation made with COMSOL Multiphysics software. This model emulates a semi-infinite width bar- rier, because they are usually placed along huge extensions of kilometers in the borders of highways or railways. In the figure 1 it is shown how the incident plane wave (IPW) is travelling from left to right, and Perfectly Matched Layers (PMLs) are located at the vertical contours (left and right) to prevent unwanted reflections (free field condition) [7]. The horizontal contours are periodic condi- tions that replicate the model in the Y axis and the measurement point is always located at 1 meter from the last scatter’s border. Obtaining in this way the Transmission Loss (dB) of the noise Barrier.

 

 

Figure 1 - 2D simulation model of a semi-infinite 3-row SCNB

 

2.2. The Particle Swarm Optimization for Sonic Crystal Noise Barriers

 

The main purpose of this study is to apply PSO to the physical parameters of the model shown in figure 1, with the purpose of maximize the airborne sound insulation for road traffic, DL_SI , (see equation 8 ) obtained with a Sonic Crystal Noise Barrier in the Finite Elements Method Simulations. In doing this it will be obtained the global value that represents the best attenuation for the whole fre- quency range (100-5000 Hz) according to European Standards.

 

Objective Function:

 

The standards EN 1793-6 [8] and EN 1793-3 [9] define the calculation proceeding of the single- number rating of the airborne sound insulation for road traffic (sound insulation index DL_SI ) as follows:

 

 

with m the number of the third octave band spectrum, SI_i is the dB-value of the i-th third-octave band of the sound insulation spectrum and L i is the dB-value of the i-th third-octave band of the traffic noise spectrum, according to EN 1793-3 [9]. The finite elements simulations have been made with 3 frequencies for each third octave band spectrum giving a total of 54 frequencies in the total spectrum of the noise barrier.

 

Input parameters:

 

The four physical parameters to optimize are:

 

• Lattice constant (a) that was shown at equation 1 and it is the separation between rows and columns of the centre of the scatterers the periodic metamaterial.

• External radius of the cylindrical scatter – r_rext

• Internal radius of the cylindrical scatter - r_int

• Width of the mouth ( 𝛬 ) of the resonator embedded in the scatter - 𝛬

 

 

Figure 2 - Physical parameters of SCNB to modify in the PSO

 

In order to make this new metamaterial barrier competitive with the classical ones placed in train environments, a maximum of 1 meter length of the barrier has been defined in the range of variation of the parameters.

 

Optimization Process:

 

Regarding to the calculation/optimization process, it has been made following the flowchart as Figure 3 shows. The complex 4-dimensional search space have been tested with several tests, looking for the best coefficients in the PSO algorithm and changing populations sizes.

 

 

Figure 3 - Flowchart of PSO Process

 

3. RESULTS

 

3.1. Computational performance

 

The Particle Swarm Optimization process have been analysed according to the number of calcu- lations needed to find the fitness objective value. Due to the complex 4-Dimensional search, in some tests the particles have got stuck easily in an extremely big local maximum at 16.797 dBA, but with modifications in the coefficients, population, and iterations, finally the optimal solution has been found.

 

The first tests were made choosing w=0.7298, c 1 =c 2 =1.49618 according to R. Eberhart and Y. Shi proposal [6]. They are “coef. 1” called in figures 4-5 . Although this coefficient provides a bal- anced progression to find a solid convergence, it was not enough for exploration. That is why, after some tests, the accelerations coefficients have been changed to c₁ = 1.53 and c₂ =2.03 giving more importance to the social behavior of the particles rather than the individual, something not usual in the PSO literature but the inertia weight w(k) decreases in the interval [0.9,0.4] as mentioned authors were recommending [4]. These tests were called the “coef. 2” . This modification, have allowed to find a global maximum in the 50 particles case, revealing that not always a bigger population improve the results as both different iteration tests (figure 4 and figure 5) have proved.

 

According to the results in Figure 5 convergence the global best value of 17.012 dBA is found at 99^th iteration after more than 4900 FEM simulations.

Global Best

 

 

Figure 4 - Particle Swarm Optimization with 50 iterations and different population and coefficients

 

 

Figure 5 - Particle Swarm Optimization with 100 iterations and different population and coefficients

 

3.2. Optimal parameters

 

The values that maximize the objective function, obtaining a DL_SI value of 17.01 dBA are in the Table 1 . It is important to remark that the local minimums found were always with the same shape topology than the global best solution but with a different lattice constant (a) which makes this parameter the dominant one and the most important to set in the Sonic Crystal Noise Barrier. The scatter’s parameters (r_ext , r_int , 𝛬 ) find the global maximum setting them to the values close to the upper bounds limits. The external radius is the 44.7% of the lattice constant (a) value, which means a filling factor of 89.4% of the unit cell in the periodic metamaterial. The internal radius is the 95% of the external one and the mouth of the resonator is the 99% of the internal radius value.

 

Table 1: Optimal parameters that maximize the DL SI in the SCNB

 

 

The 2D model representation of the optimal solution (Figure 6) shows the scatterers with the embed- ded Helmholtz resonator. Looking at the shape it is easy to observe how big is the internal resonant air mass and the extremely thin surface of the C-shape scatterers. With a future vision, the small area of rigid material needed will decrease the costs of fabrication of this wide-spectrum SCNB with 3D printing technology.

 

 

Figure 6 - 2D semi-infinite SCNB optimized model simulation

 

3.3. Optimized Sonic Crystal Noise Barrier

 

The single value to optimize in the objective function (equation 8) have been calculated from the sound insulation spectrum (SI_i ), which is also the Transmission Loss of the SCNB that follows the equation:

 

 

Where 𝑃_d is the value of the effective pressure without the barrier and 𝑃_i is the value of the effective pressure with the barrier placed.

 

In figure 5 the result of the optimized SCNB Transmission Loss is got at 1 meter away from last scatter in a Finite Elements simulation following the model used in the cost function and explained at figure 1. The wide-open mouth of the Helmholtz resonator placed in the scatter, improves the lowfrequency insulation of the barrier, especially in the range of 150-550 Hz which is a recurrent problem with difficult solutions for thin noise barriers (less than 1 meter of length).

 

 

Figure 7 - Transmission Loss of SCNB

 

In figure 7 the sound insulation maximum is at 485 Hz which reports a value of 108.6 dB and figure 8a shows a Sound Pressure Level (dB) map of the model representation where it can be clearly checked the attenuation of around 90 dB in the shadow zone of the barrier.

 

The physical topology of the barrier produces the first BG frequency following equation 1 at F Bragg = 548.6 Hz, and in that frequency figure 8b portraits a 40 dB attenuation levels. In this SCNB model, multiple scattering and Helmholtz resonances are the two noise control mechanisms, where usually absorbent materials are often used as a third one to improve higher frequencies attenuation [10]. As it can be observed in figure 7 above 2KHz, the harmonics of BG and Helmholtz resonant frequencies produces huge attenuation peaks with thin frequency ranges than increase the average attenuation of the SCNB. In table 2 European standard classification [11], gives a D2 category to the optimized barrier, rounding DL_SI = 17.01 dBA to 17 dBA.

 

Table 2: Categories of airborne sound insulation

 

 

 

Figure 8 - Sound Pressure Level map of the FEM model at maximum attenuation level (a) and Bragg's frequency (b)

 

4. CONCLUSIONS

 

The main goal of this study was to optimize a SCNB that generally are syntonised to attenuate some selected frequency ranges, to cover the whole traffic noise spectrum. The PSO algorithm have brought an innovative topology for scatterers, without the need of adding absorbent material as wide- band sonic crystal existents in literature were usually using [2]. By using a PSO combined with FEM simulations in a periodic model, this study has exposed how metamaterials like Sonic crystals can have a good performance as a noise barrier competing with the classical barriers attenuation levels and evaluated with the European Standards. In particular, according to European standards EN 17936 [8] and EN 1793-3 [9] the single-number rating of the airborne sound insulation for road traffic is DL_SI = 17.01 dBA . This optimized SCNB has a length less than 1 meter (0.765 m) which is always a space exigence to place it in the borders of highways but even that, it produces a very good low- frequency insulation level thanks to the Helmholtz resonances. The sound insulation single value (DL_SI) of 17.01dBA is rounded to 17 dBA which is a D2 category barrier according to table 2 and European standards classification.

 

6. REFERENCES

 

1. J. V. Sánchez-Pérez, C. Rubio, R. Martinez-Sala, R. Sánchez-Grandía and V. Gomez, “Acoustic Barriers Based on periodic arrays of scatterers,” Applied Physics Letters, vol. 27, no. 81, pp. 5240-5242, 2002.
2. V. Romero-Garcia, S. Castiñeira-Ibañez, J.V. Sanchez-Perez and L.M. Garcia-Raffi, “Design of wideband attenuation devices based on Sonic Crystals made of multi- phenomena scatterers,” in Société Française d’Acoustique. Acoustics 2012 , Nantes, France, 2012.
3. C. Kittel, Introduction to Solid State Physics, 8a Ed., 2004.
4. R. Poli, J. Kennedy, T. Blackwell, “Particle Swarm Optimization, an Overview,” Swarm Intelligence, vol. 1, no. 1, pp. 33-57, 2007.
5. Y. Shi, R. Eberhart, “A Modified Particle Swarm Optimizer,” in World Congress on Computational Intelligence and IEEE International Conference on Evolutionary Computation , 1998.
6. R. Eberhart, Y. Shi, “Comparing inertia weights and constriction factors in particle swarm optimization,” in Proceedings of the IEEE Congress on Evolutionary Computation , San Diego, USA, 2000.
7. J. P. Berenguer, “A perfectly matched layer for the absorption of electromagnetic waves,” Journal of Computational Physics, vol. 114, no. 2, pp. 185-200, 1994.
8. European Committee for Standardization. EN 1793-6:2018 , Road traffic noise reducing devices - Test method for determining the acoustic performance - Part 6: Intrinsic characteristics - In situ values of airborne sound insulation under direct sound field conditions, 2018.
9. European Committee for Standardization. EN 1793-3:1997, Road traffic noise reducing devices – test method for determining the acoustic performance – Part 3: Normalized traffic noise spectrum, 1997.
10. Castiñeira-Ibáñez, S., Rubio,C., Romero-García, V., Sánchez-Pérez, J.V., García-Raffi, L.M., “Design, Manufacture and Characterization of an Acoustic Barrier Made of Multi- Phenomena Cylindrical Scatterers Arranged in a Fractal-Based Geometry,” Archives of Acoustics, vol. 37, no. 4, 2012.
11. European Committee for Standardization. EN 1793-6:2011, Road traffic noise reducing devices - Test method for determining the acoustic performance - Part 6: Intrinsic characteristics - In situ values of airborne sound insulation under direct sound field conditions, 2018.