Optimization of Screw Mufflers Equipped with Two Inlets and One Outlet Using Neural Network Model, Finite Element Method, and Genetic Algorithm

Noise abatement by using efficient mufflers is compulsory, as venting noise, a type of huge noise in the industry, has a serious impact on human hearing. In order to reduce the noise abatement cost, the idea of using a muffler to suppress two kinds of venting noise sources arises. In this study, a muffler internally inserted with a screwed plate was proposed with two inlets and an outlet for exhaust venting. An analysis of the finite element method (FEM) was performed to estimate the muffler’s acoustical performance using the COMSOL program. A simplified objective function established by an artificial neural network (ANN) was trained to shorten the optimization procedure and linked to a genetic algorithm (GA). During the muffler analysis, both Rx (the outline dimension of the muffler) and D (the diameter of a straight and perforated tube) were chosen as design parameters. In addition, two target frequencies (1500Hz and 2000Hz) were specified during the optimization process. Consequently, the result reveals that the optimization of a screw muffler having two inlets and one outlet was efficiently assessed.


Introduction
Sound absorbing wool and acoustical tube are commonly used as essential materials in noise abatement. Concerning the research of acoustical wool, Delany and Bazley [1] analyzed the sound absorption coe cient for acoustical wool in 1969. Johnson et al. [2], in 1987, continued to investigate the sound absorption coe cient using acoustical ow resistance, porosity, curvature, and viscous characteristics length. In addition, Champoux and Allard [3], in 1991, explored the sound absorbing property using a new parameter of thermal characteristics length. Lafarge et al. [4] developed a Johnson-Champoux-Allard model to evaluate the sound absorbing coe cient in 1995.
Regarding the research of curved acoustical duct, Cummings [5] assessed the acoustical performance of curved tube equipped with various sections (rectangular and circular) in 1974. Rosta nski [6] developed a model to analyze the sound propagation within a curved duct in 1974. In 1978, Fuller and Bies [7,8] tried to improve the acoustical performance by adjusting the duct shape and section area. In 1999, Kim and Ih [9] used a four-pole matrix to estimate the sound transmission loss of a curved expansion chamber. As mentioned above, both the curve-shaped tube and acoustical wool have a great in uence in the noise elimination.

Research Gate
Mu ers have been customarily used in dealing with venting noise. Studies on mu er optimization were focused on the use of plane wave theory which is valid below the cuto frequency, a low frequency depending on the muffler's geometry [10][11][12]. However, the muffler mechanism mentioned above is simple, and the acoustical performance in a high-order sound wave is ignored. A muffler with a complicated mechanism is required for better acoustical performance. Considering the highfrequency effect for a muffler with a complicated acoustical mechanism, an FEM simulation was thus adopted for acoustical simulation [13][14][15][16]. While dealing with the optimization issue, Bhunia and Sahoo assessed the reliability optimization in an interval environment by genetic algorithm [17]. Mahato et al. optimized system reliability for series system with fuzzy component reliabilities using a genetic algorithm [18]. Sahoo et al. used an efficient GA-PSO approach to solve mixed-integer nonlinear programming problem in reliability optimization [19]. As seen in previous studies [16,[20][21][22], to enable the optimization process in muffler design, an artificial neural network (ANN) was established to serve as a simplified objective function and linked to the GA during the optimization. Despite the scientific progress in establishing efficient mufflers, it is always the ultimate goal to invent a more efficient muffler at a low cost.

Motivation and Novelty
In the real world, there are many venting noise sources in chemical industries. For the economic purpose, an idea of using one muffler to simultaneously reduce two venting noise sources arises. erefore, a muffler composed of the screwed shell (an efficient acoustical element) and hybridized with two inlets and one outlet is proposed to develop an efficient muffler to reduce the multiple noises. Here, a muffler with two inlets was connected to two noise sources to direct the venting sound waves into the muffler, before the exhaust was vented through a single outlet. e muffler proposed in this study was optimized by applying the artificial neural network (ANN) and GA method. In addition, two target frequencies (1500 Hz and 2000 Hz), which are very sensitive to human hearing, were also specified during the muffler optimization.

Theoretical FEM Model (on COMSOL)
As indicated in Figure 1, a muffler internally inserted by a screwed shell and hybridized with two inlets and one inlet was introduced. e acoustical boundary condition of a solid boundary analyzed in the acoustical model of COMSOL is where q is a dipole sound source, c is the sound speed, and ρ is the air density and are set at zero, 343 (m/s), and 1.293 (kg/m 3 ), respectively. e acoustical boundary condition of the perforated tube (a solid boundary) analyzed in the COMSOL model is Using the Johnson-Champoux-Allard model, the analysis of the acoustical behavior of porous acoustical wool yields where α ∞ is the shearing viscosity, η is the curvature level, and φ is the porosity of the material, and σ 0 is the flowing impedance expressed as follows: In addition, the bulk factor (K eff ) yields e vicious character length (Λ) and thermal character length (Λ e governing equation of the sound wave propagating inside the muffler is where e acoustical transmission loss (TL) is

Model Check
Before the shape was optimized, the FEM analysis by COMSOL was checked for correctness based on experimental data and other theoretical data. For a muffler hybridized with a straight and perforated tube, the simulated TL verified by experimental data [23] was proved to be acceptable, as indicated in Figure 2.
In addition, as exemplified in Figure 3, the simulated TL of a muffler having two inlets and one outlet was similar to the theoretical data by the green solution method [24]. erefore, the FEM model was correct. e acoustical simulation of the muffler using the FEM is performed in the following section.

Artificial Neural Network Model
e mathematical function of ANN was implicit when using the hidden layers inside the ANN structure, which was inconvenience during calculation. erefore, an explicit form using a polynomial neural network was compulsory. As seen in Ivakhnenko's research [25], the relationship between the neurons' layers was shortened, thus allowing an easy weight adjustment and assessment of the factors of the polynomial functions when the polynomial neural network was adopted with a regression process. e polynomial neural network illustrated in Figure 4 includes an input layer, a hidden layer, Σ(summation), and an output layer (product).
For h's unit number of the hidden layer, the neural network's output is  Figure 1: e mechanism of a screw muffler equipped with two inlets and one outlet. Developing equation (12) yields where x i , x j , x k are the input data, y k is the output value, and B 0 , B i , B ij , and B ijk are the node function factors. e ANN model was trained by importing the training data bank of the muffler's parameters and theoretical TL calculated by the COMSOL. e polynomial together with the PSE standard was also calculated. e PSE, a mean square's deviation, yields where σp 2 , CPM, and QQ represent the error variation, the product of the penalty function, and the number of the network factors, respectively. NN, y i , and y i are the number of training data, data required by ANN, and data predicted by the model, also respectively. e related TL was predicted by substituting an arbitrary value of the muffler's geometrical data (design data) into the trained ANN model. e muffler was then optimally shaped using the ANN model (simplified OBJ function) and the genetic algorithm.

Genetic Algorithm
Holland [26] developed a genetic algorithm (GA) according to the concept of Darwinian natural selection. Later, Jone [27] extended the GA theory in practical application. anks to the excellent ability in searching for the optimal solution, the application of GA was prosperously developed. In previous studies [28,29], GA was adopted in solving engineering's space-constrained problem. Seven control parameters were chosen for GA in the optimization process for the purpose of the study. e GA's control parameters are gene population (pop), a length of chromosome (bit), a selection strategy (elitism), a mutation ratio (pm), a crossover ratio (pc), and a maximum iteration (iter max ).
Each pair of candidate parents was chosen by using the coding and decoding process in conjunction with the simplified OBJ function. e precision of the parameter search (MM) is where P max and P min are the parameters' maximum and minimum ranges, respectively. N p is calculated as 2 m , where m is the number of muffler parameters. e related GA optimization process is illustrated in Figure 5. As seen in Figure 5, the optimization procedure terminated when the generation number reached iter max .

Sensitivity Analysis
As indicated in Figure 6, R, an acoustical flowing resistance, has been selected as a design parameter in the sensitivity analysis. e simulation result shown in Figure 7 reveals that the TL increased with R. Similarly, as depicted in Figure 8, considering that the muffler was lined with acoustical wool of 500 kg/m 3 .s internally, σ (the perforation ratio of a straight and perforated tube in muffler) was chosen as a  design parameter. e simulation result is plotted in Figure 9. As observed in Figure 9, the uctuation of TL below the frequencies of 1000 Hz was not obvious as the σ value varied. Moreover, the Rx (the outline diameter of the mu er) was also selected as a design parameter and is depicted in Figure 10. e impact of TL with regard to Rx was captured and is illustrated in Figure 11. As shown in Figure 11, the uctuation of TL at the frequency of 300 Hz above was obvious as the value of Rx varied. Subsequently, as depicted in Figure 12, the D (the diameter of the straight and perforated tube) was selected as a design parameter. e e ect of TL related to D was investigated and is shown in Figure 13. As seen in Figure 13, the uctuation of TL was roughly clear when the value of D varied.

Case Study
A mu er with a screwed shell inside was presented in order to enhance the acoustical e ciency of the mu er used to depress the venting noise. In addition, a concept of two inlets was also adopted in the mu er design for the cost-saving purpose.
As mentioned in section 6, the e ect of TL with regard to four kinds of geometric factors, including R, σ, D (the diameter of a straight and perforated tube within the mu er), and Rx, was assessed. e results in Figures 7,9,11,and 13 indicate that the TL was proportional to R. e tendency of TL with respect to σ was trivial and not clear. e uctuation of TL with respect to Rx and D was large. erefore, the

Mathematical Problems in Engineering
geometrical design data of Rx and D shown in Figure 14 were then chosen as a design parameter in the muffler optimization. e parameters' range and schedule levels are illustrated in Table 1. e simulated TL in relation to sixteen training data sets is depicted in Table 2. With Rx and D as the input data and the TL (simulated by COMSOL) as the output data, the ANN model was established via a training process and testing process. e ANN model, a simplified objective function, regarding the specified frequencies of 1500 Hz and 2000 Hz was obtained and is listed as follows.    e GA control parameters adopted in the study are provided in Table 3. e comparison of design data sets (before and after optimization at targeted frequencies of 1500 Hz and 2000 Hz) is shown in Tables 4-5. e accuracy of TL predicted by the ANN model was also verified by the exact solution run on the COMSOL. As indicated in Tables 6-7 1  20  125  2  20  130  3  20  135  4  20  140  5  25  125  6  25  130  7  25  135  8  25  140  9  30  125  10  30  130  11  30  135  12  30  140  13  35  125  14  35  130  15  35  135  16  35  140   8 Mathematical Problems in Engineering By substituting the original data and the optimal design data into the COMSOL's calculation, the theoretical TL profiles (before and after optimization) are illustrated in Figures 15-16. As illustrated in Figure 15, the TLs at the targeted frequency of 1500 Hz before and after executing the optimization were 16.0 dB and 39.0 dB, respectively. Moreover, as depicted in Figure 16, the TLs at the specified frequency of 2000 Hz before and after optimization were 9.7 dB and 21.9 dB, respectively.

Target Frequency-1500 Hz
10.2. Discussion. As seen in section 8, the effect of the TL with regard to the geometric data of R, Rx, and D was significant. e tendencies of the TL with respect to Rx and D were oblique. However, the fluctuation of TL with respect to Rx and D was obvious. erefore, both the Rx and the D were then chosen as the design parameters during the optimization procedure in order to find appropriate design data.
e numerical assessment of the muffler using ANN along with the GA method was performed. e simulation results were obtained and are shown in Tables 6-7        and 12.2 dB, respectively, when using the optimization process. Moreover, the accuracy between the ANN model and the FEM shown in Tables 6-7 was between 12.8% and 17.9%.

Conclusion
A screw muffler internally inserted with a screwed shell and hybridized with two inlets and one outlet was introduced to advance the acoustical efficiency of the muffler. In addition, the result of sensitivity analysis indicates that R, Rx, and D had a significant influence on the muffler's acoustical efficiency. Here, the TL is relational to the R value, but the tendencies of TL with respect to Rx and D were not clear. In order to purchase a best design data for the muffler, the optimization using Rx and D as the design parameters was necessary. e trained neural network (ANN) model was adopted and served as a simplified objective function for the muffler optimization. e muffler optimization applying the ANN model in combination with the GA method was then performed. Simulation results in this study reveal that the TLs at the target frequencies of 1500 Hz and 2000 Hz were improved by 23 dB and 12.2 dB. e ratio of crossover Pm: e ratio of mutation P max : e parameter's maximum range P min : e parameter's minimum range pop: e population number R:

Abbreviations
Acoustical impedance of acoustical wool (kg/ m 3 .s) Rx: Outline diameter of the screw muffler (m) x i , x j , x k : e ANN's input data y k : e ANN's output value y i : e required data in the ANN y i : e predicted data for the ANN TL: Transmission loss (dB) σ: Perforation rate of a perforated tube (%) σp 2 : e ANN's error variation.

Data Availability
We already include all the data in the manuscript.

Conflicts of Interest
e authors declare that they have no conflicts of interest.