Numerical Techniques (others): Paper ICA2016-45Numerical Techniques (others): Paper ICA2016-45 ... Transmission loss optimization using genetic algorithm in Helmholtz resonator under

Buenos Aires – 5 to 9 September, 2016 Acoustics for the 21

st Century…

PROCEEDINGS of the 22nd International Congress on Acoustics

Numerical Techniques (others): Paper ICA2016-45

Transmission loss optimization using genetic algorithm in Helmholtz resonator under space

constraints

Gabriela Cristina Cândido da Silva(a), Maria Alzira de Araújo Nunes(b)

(a) University of Brasilia - Gama College, Brazil, [email protected]

(b) University of Brasilia - Gama College, Brazil, [email protected]

Abstract

The development of high-performance mufflers associated to a compact volume it is of great

importance in industrial field in order to obtain duct noise reduction in an economically and

efficiently way. The Helmholtz resonator (HR), a classical reactive muffler, is considered in this

work. The main purpose of this paper is to optimize numerically the sound transmission loss

(TL) of the HR, since TL is an intrinsic characteristic of the muffler and does not depend on the

source or termination impedances. The optimization methodology consists in maximizing the TL

obtained for a pure tone frequency under dimensional constraints when the HR is considered

mounted in a duct system. In order to search for the optimal dimensions of the HR cavity, an

evolutionary search algorithm has been used. The sound pressure data is obtained from a finite

element (FE) model (Ansys®) of the system HR/duct which communicates with the genetic

algorithm (GA) implemented in Matlab® software. Many numeric simulations were performed

varying GA parameters: number of generations and population size. The results show a strong

match dependency between these parameters as well the importance of the bounds constraints

range. Optimized HR has a gain of approximately 35 db attenuation in the frequency of interest

when compared with the baseline.

Keywords: Helmholtz resonator, Transmission Loss, Optimization, Finite elements, Genetic algorithm.

22

nd International Congress on Acoustics, ICA 2016

Buenos Aires – 5 to 9 September, 2016

Acoustics for the 21st

Century…

2

Transmission loss optimization using genetic algorithm in Helmholtz resonator under space

constraints

1 Introduction

High noise levels are an inherent problem to industrial equipment like compressors, blowers and

fans which are connected with ducts or exhaust systems. These latter are important devices

due to transmit or to guide the noise through the external environmental. In general the noise

generated by these is low frequency tonal and it may cause a considerable annoyance to the

human who can actually cause physical harm leading to both psychological and physiological

symptoms [1]. The most usual low frequency control in order to reduce the noise transmitted by

exhaust systems is the use of reactive mufflers (like side branch elements) or active noise

system [2].

Reactive silencers provide a feasible alternative when compared to the others conventional

dissipative systems which use acoustic materials. The simplest reactive silencer is known as

Helmholtz Resonators (HR) and it is largest indicated to duct noise mitigation. When the HR is

mounted on a duct it reduces the radiated acoustical power primarily through the use of the

cross-sectional discontinuities that reflect the sound back toward the source [3].

Although this acoustic filter is considered a classical device and its operating principle is widely

known from many times ago, it continues be used in recently researches [4; 5; 6] due to its high

efficiency, easy application and low cost when compared with dissipative filters.

According to [3] and [7] the acoustic performance of the HR is directly related with its geometric

dimensions: the cavity volume, neck’s radius and length. So the physical space available to

install this device should be an important parameter. The conditions of operation and

maintenance in practical engineering works lead to a constrained problem and the necessity of

to optimize the HR acoustical performance within a limited operation space [1].

In this context, in the HR design should be considerate, beyond the geometrical characteristics,

the maximization acoustic performance in the desired frequency in function of a limited space

for installation. So an optimization technique becomes crucial in order to enhance a proficiently

acoustic performance of such device. Many researchers have been used the genetic algorithm,

a heuristic optimization method, obtaining good results when applied to mufflers [8; 9; 10; 11;

12]. In addition, the TL parameter is indicated for use in muffler shape optimization once it

considers only the acoustic filter influence [2].

In this sense, it is proposed in this paper to use GA optimization in order to achieve the

maximum acoustic performance of a HR under space constraints at a desired noise control

frequency. In this way a manual adjust can be eliminated from the design step of a HR saving

meanly time. Here the considered constraint values were the radius and length of the cylindrical

cavity of the HR simulating real dimensions for application in industry for example. The sound

pressure data is obtained from a finite element (FE) model, using commercial software

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(Ansys®), of the system HR/duct which communicates with the GA implemented in Matlab® code

in order to maximize the TL. In order to evaluate this optimization technique applied to this

specific device, many simulations were performed varying some GA parameters like the

generations and population numbers and the results analysed.

2 Methodology

The first step of the methodology is to model the system duct/HR using the finite element

method. This model must attend the configuration shown in Figure 1 in order to use the

decomposition method for TL estimation [13]. A three-dimensional finite element model was

obtained using the software Ansys®. The advantage of use the acoustic numerical model of the

system duct/HR when compared with this same analytical model is that the first automatically

determines and incorporate the end-correction factors during it solution step, providing feasible

results [2].

Figure 1: Setup of decomposition theory.

It was assumed a uniform circular duct with rigid wall (no fluid-structure interaction) and a planar

sound source located in the left end. At the duct right end all the acoustic energy is absorbed by

the anechoic termination (unitary absorption coefficient), such that the reflected wave back to

the resonator section is negligible. The HR has a cylindrical cavity and neck.

The system duct/HR modelled and its respective dimensions are shown in Figure 2. The

dimensions and position of the resonator along the duct were chosen in order to represent a

practical application. The dimensions: Length and Radius are the variables which will be

optimized in order to achieve the maximum TL.

Figure 2: System HR/duct model.

The acoustic element used in the numerical model is Fluid30 (tetrahedral element) available in

Ansys® library for three-dimensional analysis. These elements are pressure formulated

elements which model the pressure variation in the element associated with an acoustic wave.

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The element has eight corner nodes with four degrees of freedom per node: translations in the

nodal x, y and z directions and pressure.

In the model pre-processor, the parameter Keyopt(2) is used to specify the presence or

absence of structure in contact with the acoustic fluid. In this work, it was used Keyopt(2)=1 that

indicates pure acoustic elements with no fluid structure interaction. Therefore, the specified

elements have only a single degree-of-freedom at each node which is pressure.

The reference pressure and the isotropic material properties (density, speed of sound and

boundary admittance) must be defined by user. The reference pressure is used to calculate the

element sound pressure level (dB). For the referred model it was defined the following

constants: reference pressure is 20 x 10-6 N/m², air density is 1.21 m/s² and speed of sound is

344 m/s. The anechoic termination must be characterized imposing the boundary admittance

equal to the unity (represents full sound absorption) on the elements located at the right end of

the duct. The surface load label impedance with a value of the unity should be used to include

damping (sound absorption lining).

The elements were created with mean size of 30 mm. This size was defined due to

recommendations from classical literature and researches in the numerical acoustic area which

establish a minimum of six elements per wavelength in order to achieve reliable results [2].

The sound source is modelled applying a pressure field on all nodes present in the left duct-

end. In each node was specified a pressure of 1 Pa. In the Figure 3 it is possible to observe the

pressure field (sound source) in green symbols and the boundary condition representing the

anechoic termination. The red lines are the active impedance applied. The mesh of the duct and

resonator is in blue (acoustic elements).

In order to estimate the sound transmission loss a harmonic analysis must be conducted in the

frequency of interest. The output model is the sound pressure (real and imaginary parts in

Pascal) in the nodes of interest. In this case the sound pressure is obtained in three nodes

(according to Figure 1): 2 nodes upstream the HR and 1 node downstream the HR as shown in

the Figure 4. These data were exported to Matlab® software where the TL is estimated.

Figure 3: Finite Element Model of the system duct/HR.

Figure 4: Selected nodes for pressure acoustic data.

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In order to verify the feasibility of the TL maximization through the HR shape optimization a

sensibility analysis was performed. This analysis was prior executed to verify the dependency of

the HR acoustic performance, the TL, with the resonator cavity dimension. Considering a cavity

volume of 0.001 m3, the TL of two cylindrical HR with different length/radius rate was

numerically estimated like proceeding described above. Both HR was in the same duct position

(longitudinal axis).

Using the classical formula [14], considering end-correction factors and the dimensions

described in Table 1 the analytical resonance frequency of the HR is 219 Hz for both cases

once by analytical formulation this parameter is only dependent of the cavity volume and neck

dimensions. For this sensitive analysis the dimensions illustrated in Figure 2 was used too. For

each case (1 and 2), the TL was numerically estimated from a harmonic analysis at 219 Hz

(FEM by Ansys®) and using the decomposition method [13] implemented in Matlab®. The results

of this analysis will be discussed in the next topic which demonstrates that the shape

optimization is a practical and effective manner to obtain a HR with maximum TL.

Table 1: HR Cavity Dimensions for sensitive analysis.

Cylindrical volume cavity = 0.0010 m3

Case Radius [m] Length [m]

1 0.0478 0.150

2 0.0665 0.075

The optimization methodology applied in this paper is illustrated in the flowchart of the Figure 5.

The GA starts with the specification of the following parameters by the user: the population size,

number of generations, crossover and mutations parameters and the boundary constrains. As

mentioned the parameters: population size (called as pops) and number of generation (called

as genno), were varied in order to investigate their influence in the optimization result. It was

attributed four values for each one of both parameters: 50; 100; 150 and 200. The combination

between them resulted in 24=16 cases. For each case the mutation and crossover parameters

were keeping constant. Other parameter which was varied is the boundary constraint matrix.

Two matrixes with different constraints (radius and length of the cavity) limits were investigated.

For each matrix the 16 cases (combination of genno and pops) was simulated.

After specified all input parameters the GA algorithm generates a random population (according

to the bounds specified) and it is initialized the optimization process. For evaluation of the

fitness equation is necessary that the Matlab® function communicates with the Ansys® software,

where the pressure data necessary to compute the TL (objective function) is obtained. The

number of generation is verified after fitness evaluation. If this number is reached the

optimization process is stopped. If no, the crossover and mutation parameters are applied and a

new generation is created. The process starts again until the stop condition to be reached. The

convergence of the results was analysed after complete optimization process.

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Figure 5: Optimization methodology flowchart.

3 Results

The results of the sensibility analysis described in the topic 2 are showed in Figure 6. The TL

curve of each HR described in the Table 1 is shown in this graph. The red curve (dot)

corresponds to the case 1 and the blue curve (continuous line) corresponds to the case 2. It is

important to note that both have the same cavity volume and they are positioned in the same

local in the duct, the only difference between them is the length and radius dimensions of the

cavity.

It is possible to observe in Figure 6 that the maximum TL is obtained in the resonance

frequency of the HR (219 Hz) as estimated by the analytical equation. Therefore the values of

this maximum attenuation are different between them (case 1: 50 dB and case 2: 37 dB)

because the HR has different length/radius rate. We can conclude that the optimization process

is a good tool to estimate the maximum TL in a defined range of HR cavity shape. So, a manual

process aiming HR shape adjustment can be unfeasibly.

The GA optimization was conducted at two different conditions: 1) the constraint bounds were

defined between 30 mm to 200 mm for both, radius and length of the HR cavity; 2) the

constraint bounds were defined between 100 mm to 150 mm for the length of the cavity and

from 30 mm to 60 mm for the radius. These limits were established considering a practical

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installation in an industrial environment. For each situation, the GA parameters defined as

genno (number of generation) and pops (population size) were combined totalling 16

simulations. All the results were obtained in the frequency of interest of 219 Hz (from analytical

formulation) as explained before.

Figure 6: Sensibility analysis results.

The Table 2 shows the results obtained in the condition 1, where the two design variables

(length and radius of the HR cavity) are submitted to the same upper and lower bounds. For

space saving only the main results (5 from the 16 simulations) are shown in this table. Just the

results closest to the optimized value were exhibit in this table. The line filled with grey colour is

the optimized dimensions. The others combinations of the genno and pops parameters

generated a TL below 40 dB.

Table 2 – Results obtained with the condition 1.

CONDITION 1

Length and Radius bounds = [ 30 mm a 200 mm]

GA parameters Objective function Optimized variables

genno pops TL [dB] Length [mm] Radius[mm]

50 50 47.04 100.16 57.09 100 150 44.43 114.95 53.09 200 150 71.29 142.17 44.89 150 150 53.43 35.02 93.06 200 150 53.47 34.72 92.53

The Table 3 shows the results obtained with the condition 2. Note that in this condition the

length and the radius of the cavity have different limits for each one and the constraint interval

have a narrow range when compared with the specified in the condition 1. In Table 2 is possible

to note that the maximum TL was encountered for 12 of the 16 combinations of the genno and

pops parameters. All they are filled with grey colour.

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Table 3 – Results obtained with the situation 2.

CONDITION 2

Length bounds = [ 100 mm a 150 mm]

Radius bounds = [ 30 mm a 60 mm]

GA Variable parameters Objective function Optimized parameters

genno pops TL Length Radius

50 50 57.79 115.44 50.97

50 100 71.29 142.11 44.93

50 150 71.29 141.88 45.45

50 200 71.29 142.36 45.16

100 50 57.79 115.03 50.92

100 100 71.29 142.18 45.05

100 150 71.29 142.18 45.05

100 200 71.29 142.42 45.08

150 50 57.79 141.75 45.13

150 100 71.29 141.84 45.45

150 150 71.29 141.84 45.45

150 200 71.29 142.48 45.06

200 50 57.79 114.53 50.99

200 100 71.29 142.09 45.45

200 150 71.29 141.75 45.13

200 200 71.29 142.01 45.00

As indicated in Table 2 many configurations of GA parameters converge for optimal design

getting a TL of 71.29 dB at 219 Hz. In Table 1 just one combination converged for this optimized

value. For both conditions the results converge for the same value for the length and radius

dimensions, approximately 142 mm and 45 mm respectively.

Analysing Table 1 it is possible to note that it is necessary to specify a large number of

generations (genno) in order to obtain the optimized parameters. In this case, the maximum

considered here were 200. So in condition 2 the search space was defined smaller than

condition 1 for both variables, then the optimized value was encountered using the number of

generation equal 50, but it was possible combining this one with a population number of 100.

The Figure 7 can help to analysis the results described in the Table 2. This figure illustrates the

fitness evaluation at condition 2 for the number of generation and population size equal to 100

for both. The curves show the maximum (red line) and average (blue line) functional value of the

population for each generation. As can be observed, the optimization process converges to the

optimal value at low values of generation (approximately 40). It means that for condition 2 the

number of generation could be reduced decreasing the computational time but it will be able to

maintain reliable results.

Comparing the results obtained with the GA optimization (Table 1 and 2) and the results shown

in the sensibility analysis simulation (Figure 6) we can conclude that GA optimization for this

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application problem is a great design tool for acoustic muffler. In Table 1, the case 1 has the

dimensions closer to the optimized dimensions shown in Table 1 and 2, but a difference of more

than 20 dB is achieved in the maximum attenuation adjusting the geometrical dimensions using

GA optimization. This proof the importance of the methodology proposed.

Figure 9: GA simulation convergence for condition 2 (genno=100 and pops=100).

4 Conclusions

This paper proposes the application of GA optimization in the design of acoustic mufflers,

specifically Helmholtz resonator (HR). The methodology consists in maximizing the TL in a pure

tone frequency by space constraints when the HR is applied in a duct system. All the

methodology is numerically where the sound pressure data is obtained from a finite element

model (Ansys®) of the system HR/duct which communicates with the genetic algorithm (GA)

implemented in Matlab® code. Simulations were performed varying the number of generations

and population size of the GA.

It was evaluated two conditions with different bounds constraints range. The condition 2

considerate in this paper where the search space was defined smaller than condition 1 for both

variables (length and radius of the HR cavity), encountered the maximum TL of 71 dB using the

number of generation equal 50 combined with a population number of 100. The convergence

curve of condition 2 showed that the optimization process converges to the optimal value at low

values of generation (approximately 40). It means that the number of generation could be

reduced decreasing the computational time but it will be able to maintain reliable results.

Comparing the results obtained from GA optimization and the results obtained from analytical

formulation (sensibility analysis in this paper) we can conclude that GA optimization for this

application problem is a great design tool for acoustic muffler because a difference of more than

35 dB is achieved in the maximum attenuation.

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Acknowledgments

The authors would like to thanks the University of Brasilia and FAPDF (Support Research of the

Federal District Foundation) for the financial support.

References

[1] Chang Y. C.; Chiu, M. C. Shape optimization of one-chamber perforated plug/non-plug mufflers by simulated annealing method. International Journal for Numerical Methods in Engineering, Vol.74, 2008,1592-1620.

[2] Singh, S. Tonal noise attenuation in ducts by optimizing adaptive Helmholtz resonators. Dissertation of Master Degree of Engineering Science, School of Mechanical Engineering, The University of Adelaide, Australia, 2006.

[3] Beranek, L. L. Noise and Vibration Control Engineering: Principles and Applications, Wiley, New York (USA), 2st Edition, 2005.

[4] Jyun-Hong L.; Chung-Chun K.; Fu-Li H.; Chii-Chang C. Acoustic filter based on Helmholtz resonator array. Appl. Phys. Lett, Vol. 101, 2012.

[5] Yang, D.; Zhu, M.; Wang, X. Nonlinear effects of Helmholtz resonators with perforated ceramics at the neck. Proceedings of Meetings on Acoustics-POMA, Vol. 19, Session 4pNSb: Noise Control, pp. 040139, Montreal, Canada, 2013.

[6] Shi, X.; Ming Mak, C. Helmholtz resonator with a spiral neck. Applied Acoustics, Vol.99, pp 68-71, 2015.

[7] Kinsler, L. E.; Frey, A. R.; Coppens, A. B. Fundamentals of Acoustics, Wiley, New York (USA), 4th Edition, 1999.

[8] Yeh, L. J.; Chang, Y.C; Chiu, M. C. Numerical studies on constrained venting system with reactive mufflers by GA optimization. International Journal for Numerical Methods in Engineering, Vol.65, pp.1165-1185, 2001.

[9] Barbieri, R.; Barbieri, N. Finite element acoustic simulation based shape optimization of a muffler. Applied Acoustics, Vol. 67, pp. 346–357, 2006.

[10] Chiu, M.C. Shape optimization of multi-chamber mufflers with plug-inlet tube on a venting process by genetic algorithms. Applied Acoustics, Vol.71 (6), pp.495-505, 2010.

[11] Lima, K. F.; Lenzi, A.; Barbieri, R. The study of reactive silencers by shape and parametric optimization techniques. Applied Acoustics, Vol.72 (4), pp.142-150, 2011.

[12] Airaksinen, T.; Heikkola, E. Multiobjective muffler shape optimization with hybrid acoustics modeling. The Journal of the Acoustical Society of America, Vol.130 (3), pp.1359-1369, 2011.

[13] Seybert, A. F. Two-sensor Methods for the Measurement of Sound Intensity and Acoustic Properties in Ducts. J. Acoust. Soc. Am, Vol. 83, pp. 2233- 2239, 1988.

[14] Helmholtz, H. V. Theorie der Luftschwingungen in Röhren mit offenen Enden. Journal für die reine und angewandte Mathematik, Vol 57, Heft 1, S. 1-72, 1860.

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Numerical Techniques (others): Paper ICA2016-45Numerical Techniques (others): Paper ICA2016-45 ... Transmission loss optimization using genetic algorithm in Helmholtz resonator under