Sound & Vibration

DOI: 10.32604/sv.2021.09729

ARTICLE

Improving the Morphological Parameters of Aluminum Foam for Maximum Sound Absorption Coefficient using Genetic Algorithm

Mohammad Javad Jafari1, Mohsen Niknam Sharak2, Ali Khavanin3, Touraj Ebadzadeh4, Mahmood Fazlali5 and Rohollah Fallah Madvari6,*

1Department of Occupational Health Engineering, School of Public Health and Safety, Shahid Beheshti University of Medical Sciences, Tehran, Iran
2Mechanical engineering, Department of Mechanical Engineering, University of Birjand, Birjand, Iran
3Department of Occupational Health Engineering, Faculty of Medical Sciences, Tarbiat Modares University, Tehran, Iran
4Department of Ceramics, Materials and Energy Research Center, Alborz, Iran
5Department of Computer Science, Shahid Beheshti University, Tehran, Iran
6Student Research Committee, Department of Occupational Health Engineering, School of Public Health and Safety, Shahid Beheshti University of Medical Sciences, Tehran, Iran
*Corresponding Author: Rohollah Fallah Madvari. Email: fallah134@gmail.com
Received: 16 January 2020; Accepted: 13 April 2020

1  Introduction

Noise control is generally carried out using passive and active methods [1]. In the passive method, the absorbing materials are used to control the noise [1]. The porous absorbent materials can be classified as foam, fibrous or granular absorbents [2]. In this regard, the metal foam has been considered as a sound absorbent since 1950 [3]. Due to its interesting features, it has attracted a lot of attention in various fields of the industry [4]. Low density, high mechanical strength, high heat resistance, and anti-corrosion are among these attractive characteristics [4–7]. The use of metal foams are rapidly expanding in engineering applications [3,4,8,9]. Generally, there are two kinds of open-cell and closed-cell metal foams [10]. Lu et al. [11], Han et al. [7]. and Hakamada et al. [12]. stated that open-cell metal foam has a better sound absorption capacity compared with closed cells; mainly because the sound wave is easily transmitted to the material in open cell foam [7]. Given that the air movement which is controlled by the morphological parameters of the cell and the cell structure, also affects the sound absorption capacity. The significant property of metal foams is their sound absorption [7].

Many parameters may affect the sound absorption coefficient (SAC) in metal foams, including porosity, pore size, pore opening, thickness, static flow resistance, and etc [13]. Some of These parameters including morphological parameters can be considered while fabricating metal foams. In other words, it is possible to fabricate metal foams with desired morphological parameters. So far, the trial and error methods are applied in laboratory to select the best metal foam for the highest sound absorption from the available metal foams. If the optimized morphological parameters could be determined in advance it would be possible to fabricate the desired metal foam for the best sound absorption using the sintering technology.

Therefore, to have a metal foam with best sound absorption, it is necessary to determine The optimized morphological parameters of the foam. For this purpose, theoretical models which describe the relationship of the sound absorption coefficient with morphological parameters are required. [14,15]. So far, such models which describe the relationship between cell structure and the sound absorption coefficient in porous materials have been proposed [7,11,16–18]. Lu model of sound absorption describes the relationship between 3 morphological parameters of porosity (Ω), pore size (D) and pore opening size (d) with sound absorption coefficient (SAC). Thus this model seems to be the best candidate for this purpose. It is expected to improve the morphological parameters of a metal foam for the best absorption coefficient using the optimization algorithms.

In the previous study (Article titled “Optimization of the Morphological Parameters of a Metal Foam for the Highest Sound Absorption Coefficient Using Local Search Algorithm” in journal of Archives of Acoustics, under publishing Vol.45, No3(2020)), the morphological parameters of metal foam were optimized using local search algorithm.

Different optimization algorithms including General Local Search (GLS) and genetic algorithm (GA) [19], Ant Colony Optimization [20] and Particle Swarm Optimization [21] have been proposed so far. GLS algorithm explores neighborhood by iteratively expanding searching domain in the neighborhood of the current solution and moving from current solution to an enhancing neighborhood [22]. This process is repeated until the current solution cannot be improved anymore. This process continues until the optimum locale is reached. The GLS approach may stuck and trapped in a local optimum disregarding other optimums in the neighboring [23]. So there need to be avoided from trapping in these local optimums [22] . Dumitrescu et al. [24] believe that the disadvantages of local search algorithms are typically 1. they cannot prove optimality, 2. they cannot provably reduce the search space, and 3. they often have problems with highly constrained problems where feasible areas of the solution space are disconnected.

For more complex problems, meta-heuristics such as genetic algorithm(GA) [19], Ant Colony Optimization [20] and Particle Swarm Optimization [21] can be considered. The meta heuristic algorithm becomes very popular because of its stability, flexibility and its ability in avoiding from trapping in local optimums [25]. The meta-heuristic GA is inspired by Darwinian evolution theory [26,27]. Today, it is widely used in solving optimization problems and learning processes [28].

The GA is a method independent from the problem domain and quickly seeks the search space for the optimized point with a qualitative function [29]. GA has a distinguished advantage compared to other stochastic methods. In this method it is extremely easy to parallelize the algorithm. It is due to the fact that the calculations of each iteration are independent of one another [30,31]. Due to different design parameters and constraints, the optimization of sound absorbents has always been a challenge in acoustic engineering [32].

Usually, the pore size, pore opening, porosity, and thickness are not known for fabricating the foam for sound absorption. Therefore, available foams with approximate dimensions are investigated through trial and error method to find the best foam absorbent. However, if it is possible to predetermine the morphology of metal foam to absorb a specific sound with best SAC, it can be claimed that a great step has been taken to fabricate porous foams intelligently. The study aimed to optimize the morphological parameters affecting the sound absorption coefficient (α) in metal foams. For this purpose, first the Genelarl Local Search approach was applied to optimize the morphological parameters of a metal foam (Article titled “Optimization of the Morphological Parameters of a Metal Foam for the Highest Sound Absorption Coefficient Using Local Search Algorithm” in journal of Archives of Acoustics, under publishing Vol.45, No3(2020)). This part of the study, discusses the applicability of the GA in improving the optimal set of metal foam parameters, including porosity, pore size and pore opening at any thickness and frequency for fabrication of an optimized foam.

2  Structure

The study was carried out according to the following flowchart (Fig. 1).

Figure 1: Study process

2.1 Lu Model

The relationship between sound absorption coefficient (α) and morphological parameters, including pore size, pore opening, and porosity are well described [11]. In present study, the equations presented in Lu model [11] were numerically solved to optimize the morphological parameters for the highest value of SAC. These equations are shown in Tab. 1.

Table 1: Equations of the Lu model

2.2 Optimization by GA

The porosity, pore size, and pore opening parameters were optimized for a frequency range of 250 Hz to 8000 Hz and a thickness of 5 mm to 40 mm. Avoiding from trapping in any local search and affirming the global optimization was essential. A typical GA flowchart is shown in Fig. 2.

Figure 2: A typical GA flowchart [33,34]

The summary of the steps taken by the GA used in this study were [35–37]:

Step 1. Read input data

Step 2. Create primary population randomly

Step 3. Consider initial parents

Step 4. Determine mutation (considering Pm)

Step 5. Calculate all parameters

Step 6. Select the best answer

Step 7. End; if SAC=1 or the number of iteration is <150

Step 8. Else; go back to step 4

Roulette Wheel and tournament methods were used to select the members for making generation and mutation. The population included 50; the generation rate 80% (Pc = 0.8); the percentage of mutation 30% (Pm = 0.3); the mutation rate 0.02, the stopping criteria was SAC=1 or the number of iterations=150 [38]. Furthermore, the uniform crossover method was used for making generation. In order to obtain each number in the last tables, the problem solving process was used 5 times using the roulette wheel method and 5 times with the tournament method and the best result was noted. The results showed that the tournament method is better for this particular problem. The number of specified iterations (150) was used to prevent the program from hanging if there is no improvement in calculated SAC (Fitness Function).

The input data required for GA to run are summarized in Tab. 2. These parameters were selected by repeating the algorithm many times and in accordance with the best results.

Table 2: Parameters setting for GA

The searching domain for optimization of each parameter was considered according to Tab. 3. All three morphological parameters were optimized at frequencies of 0.5, 1, 2, 3, 3.5, 4, 6 and 8 kHz.

Table 3: The searching domain of each parameter

The verification and validation of the proposed codes were carried out as in Jafari et al. (Article titled “Optimization of the Morphological Parameters of a Metal Foam for the Highest Sound Absorption Coefficient Using Local Search Algorithm” in journal of Archives of Acoustics, under publishing Vol. 45, No3(2020))

3  Results

In Fig. 3, for a thickness of 5 mm to 40 mm at the frequency of 4 kHz, the convergence chart of the GA is drawn. Considering the convergence chart, it is clear that the value of the answer does not change from one iteration to the next one. Therefore, the number of iterations chosen is appropriate for achieving the desired answer. The highest value of the sound absorption coefficient is 1. Therefore, the answers must converge to the number 1.

Figure 3: Convergence authentication of GA for different thicknesses at 4000 Hz

Fig. 4 depicts the optimized morphological parameters at different frequencies for the highest sound absorption coefficient (α) in an Aluminum foam barrier with 5mm thickness. Fig. 4 shows that at high frequencies (3 kHz to 8 kHz), when porosity (Ω) is between 60 to 95 percent (0.6–0.95 in figure), the pore size is 0.33 mm to 2.5 mm, the pore opening size is 0.09 mm to 0.2 mm, and the sound absorption coefficient(α) is 0.9 and 1. Furthermore, at low frequencies from 250 Hz to 2 kHz, the α is not increased; since according to previous studies, the main limitation is low thickness. Therefore, the parameters have been improved only at high frequencies. According to the results of Fig. 4 at optimized parameters, the pore size decreases as the frequency increases. Meanwhile, the porosity and pore opening do not follow a special correlation.

Figure 4: The optimized morphological parameters at different frequencies for the highest sound absorption coefficient (α) in an Al foam barrier with 5 mm Thickness

The optimized morphological parameters at different frequencies for the highest sound absorption coefficient (α) in an Aluminum foam barrier with 10mm thickness are shown in Fig. 5. The sound absorption coefficient (α) is close to 1 at the frequency range of 2 kHz–8 kHz when the porosity is between 70 and 95% (0.7–0.95 in figure), pore size is 0.1 mm to 2.3 mm, and pore opening is 0.07 mm to 0.3 mm. The results show that at optimized values, the pore size decreases by increasing frequency, and the porosity and pore opening size do not follow any special correlation.

Figure 5: The optimized morphological parameters at different frequencies for the highest sound absorption coefficient (α) in an Al foam barrier with 10 mm Thickness

The optimized morphological parameters at different frequencies for the highest sound absorption coefficient (α) in an Aluminum foam barrier with 20 mm thickness are shown in Fig. 6. This figure shows that the sound absorption coefficient (α) is close to 1 at the frequency range of 2 kHz–8 kHz when the porosity is between 61% to 95 % (0.61–0.95 in figure), the pore size is 0.1 mm to 4.4 mm, and the pore opening is 0.08 mm to 0.5 mm. Fig. 6 shows that the pore size decreases by increasing frequency, while the porosity and pore opening do not follow any special correlation.

Figure 6: The optimized morphological parameters at different frequencies for the highest sound absorption coefficient (α) in an Al foam barrier with 20 mm Thickness

The results revealed that at 30 mm thickness, when the porosity is between 0.6 to 0.95, The pore size is 0.1 mm to 2.08 mm, and the pore opening is 0.09 mm to 0.43 mm, the sound absorption coefficient (α) is almost equal to 1 at the frequency range of 1 kHz to 8 kHz (Fig. 7). For most frequencies, the pore opening is 0.1 mm. Porosity, pore size, and pore opening do not follow a certain correlation, and their values vary at each studied frequency.

Figure 7: The optimized morphological parameters at different frequencies for the highest sound absorption coefficient (α) in an Al foam barrier with 30 mm Thickness

Fig. 8 shows the optimized morphological parameters at different frequencies for the highest sound absorption coefficient (α) in an Aluminum foam barrier with 35 mm thickness. This figure reveals that in frequencies of 0.5 kHz–8 kHz, when the porosity is between 0.5 to 0.85, the pore size is from 0.1 mm to 4.4 mm, and the pore opening is between 0.09 mm and 0.6 mm, the sound absorption coefficient (α) is close to one. Porosity, pore size, and pore opening do not follow any special correlation, and their values vary at each studied frequency.

Figure 8: The optimized morphological parameters at different frequencies for the highest sound absorption coefficient (α) in an Al foam barrier with 35 mm Thickness

The results show that when the thickness is 40mm, the sound absorption is close to 1 at frequencies of 0.5-8kHz with a porosity of 0.51 to 0.95, a pore size of 0.1 mm to 4.5 mm, and a pore opening of 0.1 mm and 0.6 mm (Fig. 9). It is worth mentioning that in the thickness