CC BY
40Electronic Journal «Technical Acoustics» http://webcenter.ru/~eeaa/ejta/
2003, 18
Long-Jyi Yeh*, Ying-Chun Chang and Min-Chie Chiu
Department of Mechanical Engineering, Tatung University, Taipei, Taiwan 104, R.O.C e-mail: [email protected]
Shape optimization on constrained linearly expanded tubes by using genetic algorithm
Received 09.10.2003, published 19.11.2003
One of the most important practical considerations in muffler design is the constrain problems in a confined place. In addition, to release the pressure drop in a muffler system, a new silencer of linearly expanded tube is proposed and investigated in this paper. The genetic algorithm (GA), a stochastic algorithm, is used as an optimizer by mimicking the genetic drift and Darwinian strife for survival.
To approach this study effectively, the linearly inclined tube is divided into several segments of straight tube with different diameters. Four-pole transfer matrix is then in use, accordingly. Not only the theoretical derivation in sound transmission loss (STL) but also the GA searching technique is discussed. Additionally, a numerical case on the expanded tube is introduced. To achieve the best optimization in terms of STL of a muffler, the GA parameters are on trial in various values.
Results show that when the divided elements of the tube are more than sixteen segments, the modeled segmental tube is similar to the linearly expanded tube. In addition, the STL in muffler becomes to be stable.
Keywords: shape optimization; linearly expanded tube; muffler; transfer matrix method; space constraints; genetic algorithm
1. INTRODUCTION
As the report by America Petroleum Institute (API) [1] shown that reactive silencers, accessory of a gas venting system, are particularly effective in eliminating the noise wave whose spectrum characteristics is either at low frequency or limited bandwidth. Whilst most of the electric machines almost belong to this kind of low frequency type [2], the muffler system is thus adopted in the gas venting system in which the space volume is constrained in usual.
In practical engineering design, the shape optimization to maximize the muffler’s performance is essential when the space volume of mufflers in a venting system is constrained inside a building. In the previous study [3], the graphical analysis of optimal shape design to improve the performance of STL on a constrained single expansion muffler was discussed. However, the abrupt shape of a muffler often increases the pressure drop [4] and results in the fact that the performance of machine such as engine will more or less be influenced. Therefore, the interest in developing a high performance of muffler with lower pressure drop under space constraints is thus arising in the field.
‘Corresponding author. E-mail: [email protected],mailing address: Department of Mechanical Engineering, Tatung University, 40, Chungshan N. Rd., 3rdSec., Taipei, Taiwan 104, R.O.C.
Recently, genetic algorithms [5] have been applied successfully in many fields of optimal problems. In addition, GA optimizers are global searching methods modeled on the concepts of natural selection and evolution. With no need of any other information regarding the objective function, GA is able to locate the global optimum of objection function in finitely many generations. Therefore, GA is applied and coupled together with the transfer matrix method [6, 7, 8] to optimize the value of STL by adjusting the slope, length and the divided elements in the linearly expanded tube.
2. THEORETICAL BACKGROUND
To simulate the STL for the linearly expanded tube, the tube is divided as several discrete segments with different diameters. Five kinds of division are graphically depicted in Figures 1-5. To brief the introduction in mathematical description, an assumption of a 3-D linearly expanded tube with four divided elements is thus demonstrated and shown in Figure 6. The related flow condition and location represented by nine chosen nodes are specified in Figure 7. The theoretical derivation is illustrated as follows:
2.1 Straight duct
As derived by Prasad and Crocker [6, 7], the four-pole matrix between inlet and outlet of an element with mean flow can be written as equations (1) - (4). The detailed expressions of the following coefficients of matrix are shown in the Appendix.
A
Pocoui
= e
- jMikLA /(1-M/)
' F. '
p c u3
y r o o 3 y
P 5
KPoCoU5 ,
P 7 PoCoU7
e
- jM2klA /(1-M2Z)
- jM 3kLA /(1-M 32)
=e
- jM AkLA /(1-M 42)
b11 b12 ] <N V
b21 b22 _ KPocoU2 j
" c11 c12~ ■"t
c21 c22 KPocoU4 J
" dll d12 i P6 '
d 21 d 22 p c u6 o o 6 J
' ell e12" ' P8 ^ .
e21 e22 p c u8 i o o 8 J
(1)
(2)
(3)
(4)
2.2 Expansion/Contraction duct
As derived by Munjal [8], the continuity of pressure and volume velocity between point 2 and point 3 with mean flow is expressed in equation (5).
P2 = P3; U2 = U3. (5)
With the replacement of Uby u, the transfer matrix for equation (5) is thus illustrated as
(6)
"l 0 "
<N = 0 S 2 P3
p c u 2 \* o o 2 J Sl _ pocoU3 ,
e
Similarly, the transfer matrix for other elements can be expressed as
(7)
' P6 '
p c u6
o o 6 j
1 0
0 Sl
S 3
P7
pocou7
(8)
/ ^ A "1 0 " / \
P8 = 0 S5 Ch P
yPoCoU8 y \J S 4 _ p C U9 y' o o 9 J
(9)
2.3 Combination of system matrix
Using the matrix substitution on Eqs. (1) - (9), one has
< b11 b12 "1 0 " 1 1 1c 2 1 "1 0 " 1 1 2 1
p C U1 \^r o o 1 J _b21 b22 _ _0 S2/S1 _ _C21 C22 _ _0 Ss/S4 _ 2 2 21 J
P9
pocou9
Hl T12 T21 T22
P9
pocou9
(10)
The sound transmission loss (STL) of a muffler is defined as [8]
STL (Du La ) = 20 log
|T 11 + T12 + T 21 + T 22|
+10log
S'
VS5,
(11)
where La = L / N.
2
3. GENETIC ALGORITHM
GA accomplishes the task of optimization by starting with a random “population” of values for the parameters of an optimization problem. Thereafter, a new “generation” with improved value of objection function is then produced. In order to achieve the evolution in new generation, the binary system, a representation of real numbers and integers is in use. In addition, by manipulating the strings, the operators of reproduction, crossover, mutation and elitism are thus under work sequentially. A brief description in GA operators and its components is made as followings:
A. Populations and Chromosomes: The initial population is randomly built. The parameter set is encoded to form a string that represents the chromosome. By evaluation of the object function, each chromosome is assigned a fitness.
B. Parents: By using the probabilistic computation weighted by the relative fitness, pairs of chromosome are selected as parents. Each individual in the population is assigned space on the roulette wheel, which is proportional to the individual relative fitness. Individuals with the largest portion on the wheel have the greatest probability to be selected as parent generation
for the next generation. A typical selection scheme of a weighted roulette wheel is depicted in Figure 8.
C. Offspring: One pair of offspring is generated from the selected parent by utilizing a crossover. Crossover occurs with a probability ofpc. The random selection of a crossover and the combination of the two parent’s genetic data are then ensued. The scheme of single-point crossover is chosen for the GA’s optimization. Recombination and parent selection are the principle method that is used for the evolution in GA. A typical scheme of single-point crossover is depicted in Figure 9.
D. Mutation: Mutation operator is used to provide the necessary diversity to the population to search in different area. Genetically, mutation occurs with a probability of pm, of which the new and unexpected point will be brought into the GA optimizer’s search domain. It’s an essential operator to introduce diversity in the population thus preventing the GA from getting saturated with solutions in local optimum. A typical scheme of mutation is depicted in Figure 10.
E. Elitism: Elitism, in which the best candidate encountered is reintroduced in each generation, can prevent the best gene from the disappearance and improve the accuracy of optimization during reproduction.
F. New Generation: Reproduction includes selection, crossover, mutation and elitism. Reduplication continues until a new generation is constructed and the original generation is substituted. Highly fit characteristics produce more copies of themselves in subsequent generation resulting in a movement of the population towards an optimal direction. The process can then be terminated when number of generations exceeds a pre-selected value.
The optimization procedure for mufflers is depicted in Figure 11.
4. CASE STUDY
A noise control of fan noise with pure tone effect at 500 Hz is introduced as the numerical case. The confined dimension for the muffler is 0.3 meter in width, 0.3 meter in height and
1.0 meter in length. An attempt of sound elimination by using the linearly expanded tube at the pure tone of 500Hz is thus proposed. GA optimization of this muffler’s shape is to proceed. By equation (11), six design parameters are chosen in GA optimization. The allowable D1 and L are specified in 0.0762 m - 0.5 m and 0.5 m - 3.0 m in individual.
The space constraints for the muffler are shown in Figs. 1-5, and the design volume flow rate is confined to 0.8 m3/s.
5. RESULTS AND DISCUSSION
5.1 Results
To obtain the accurate design by GA optimization, the numbers of population (popuSize), maximum generation (genno) and bit length (bit_n) are set as 60, 500, 40 individually. According to Johnson and Yahya [9], both the typical ratio crossover (pc) and mutation ratio (pm) used in following GA optimization are chosen as 0.8 and 0.05 individually. In addition, six kinds of segment division are made as N=2, 4, 8, 16, 32 and 64 individually.
To achieve a better approach in GA, both of dividing number (N) and GA parameters (pc,
pm and eltno) are on trial, respectively. Other parameters such as bitn, popuSize, genno are fixed at the same value. The optimization system is programmed by MATLAB and run in IBM PC - Pentium IV.
The results with respect to the chosen cases performed individually are summarized in Table 1. The optimal STL for the six kinds of segment divisions are plotted as Figure 12.
5.2 Discussion
As illustrated in Table 1, the optimal value of STL occurred in the case of pc=0.8, pm=0.05 and elt_no=1 in GA optimization. In addition, when the number of dividing segments is small (less than 8), the discrete chamber effects on a muffler are obvious and result in a larger STL at the concerned frequency of 500 Hz. On the country, the STL will be decreased to 7 dB when the segment number of N is greater than 16 in which the profile of inlet and outlet tube is most equivalent to the linearly inclined tube.
As shown in Figure 11, the peak values of STL at 500 Hz will be lower with increasing dividing segments. When the segment number reaches to 64, the profile of the modeled muffler is much more similar to the prototype muffler with linearly inclined tube at inlet/outlet. In the meantime, the STL remains 6-7 dB.
6. CONCLUSIONS
It has been shown that GA can be used in the shape optimization of the constrained mufflers. Due to the inexistence of starting design, the GA becomes easier to be used. The case study reveals that the crossover, mutation and elitism are essential in GA optimization. In addition, the proposed method of dividing linearly expanded tube into several segments makes the simulation easier. Studies indicate that the values of STL will be decreased when the segment number in duct is increased. In other words, to release the pressure loss in a system, the ability of noise reduction of a linearly expanded tube is then reduced. However, the GA method surely provides a quick and easier way in the optimization of a linearly expanded muffler, which is benefit for the pressure’s conservation in a system.
APPENDIX
b11 = cos
c11 = cos
kL,
1 - m 2
d11 = cos
kL,
1 - M 22
e11 = cos
kL,
1 - M 22
b12 = j sin
c12 = j sin
kL,
1 - M 22
d12 = j sin
e12 = j sin
kL,
1 - M 22
kL,
1 - M22
b21 = j sin
c21 = j sin
kL,
1 - M 22
d21 = j sin
' kL, ' 1 - M 22
e21 = j sin
1 - m 22
b22 = cos
1 - m 2
c22 = cos
kL,
1 - M 22
d22 = cos
e22 = cos
kL,
1 - M 22
kL,
1 - M 2
NOMENCLATURE
bit n bit length
Co sound speed, m-s-1
D diameter, m
elt no selection of elite (1 for yes and 0 for no)
gen no maximum no. of generation
j V-I
k wave number
L length, m
M, mean flow Mach number at i
pc crossover ratio
F, pressure; acoustic pressure at i,. Pa
pm mutation ratio
popuSize no. of population
St section area at i , m2
STL sound transmission loss, dB
u acoustic particle velocity at i, m-s-1
U 3 1 acoustic volume velocity at i , m s
Po -3 air density, kg-m'
REFERENCES
[1] Tracor Inc. Guidelines on Noise. American Petroleum Institute, 1973.
[2] P. L. Timar. Noise and Vibration of Electrical Machines. Elsevier Science, New York, 1989.
[3] L. J. Yeh, M. C. Chiu, G. J. Lay. Computer aided design on single expansion muffler under space constraints. Proceedings of the 19th National Conference on Mechanical Engineering (The Chinese Society of Mechanical Engineers), C7: pp.625-633, Yun Lin, R.O.C., 2002.
[4] Mark E. Schaffer. A Practical Guide to Noise and Vibration Control for HVAC Systems. ASHRAE, 1991.
[5] D. E. Goldberg. Genetic Algorithm in Search, Optimization and Machine Learning. Addison-Wesley, Massachusets, 1988.
[6] M. G. Prasad. A note on acoustic plane waves in a uniform pipe with mean flow. Journal of Sound and Vibration. 1984, 95(2), pp. 284-290.
[7] M. G. Prasad, M. J. Crocker. Studies of acoustical performance of a multi-cylinder engine exhaust muffler system. Journal of Sound and Vibration. 1983, 90(4), pp. 491-508.
[8] M. L. Munjal. Acoustics of Ducts and Mufflers with Application to Exhaust and Ventilation System Design. John Wiley & Sons, New York, 1987.
[9] J. M. Johnson, R. S. Yahya. Genetic algorithm optimization and its application to antenna design. IEEE. 1994, pp. 326-329.
Do
Figure 1. Linearly expanded tube approached by two elements (Do=0.5 m; Lo=3.0 m)
Lo
Do
Figure 2. Linearly expanded tube approached by four elements (Do=0.5 m; Lo=3.0 m)
Lo
Do
Figure 3. Linearly expanded tube approached by eight elements (Do=0.5 m; Lo=3.0 m)
Lo
L
Figure 4. Linearly expanded tube approached by sixteen elements (Do=0.5 m; Lo=3.0 m)
LO
L
Figure 5. Linearly expanded tube approached by thirty-two elements (Do=0.5 m; Lo=3.0 m)
Figure 6. 3-D muffler, N=4
Figure 7. Flow condition for a muffler, N=4
Figure 8. Weighted roulette wheel with ratios to individual relative fitness
C | -■ U S S O V e-1 "■ Ul'uSSQVer
Parent 1
^ Point
Point
Parent c
1 0 1 1 n 0 1 0 1 1 0 1 0 1 1 0 1 Cl 1 1 1 Cl 1 1
1 2 3 4 5 6 7 8 9 10^4^ Yd 1 2 3 4 5 6 7 8—?" 10 11 IS
1 0 1 1 0 Ü 1 1 1 u 1 1 Ü 1 1 0 1 Cl 1 Cl i 1 CI 1
1 2 3 4 5 6 7 8 9 10 11 12
□ ffspring 1
1 2 3 4 5 6 7 8 9 10 11 12
□ ffspring 2
Figure 9. Scheme of single-point crossover
1 U 1 1 Ü 0 1 Cl 1 1 n 1
1 2 3 4 5 6 /\ 7 8 9 10 11 18
IT Mli ta"tio n
* F’ o lut
1 Cl 1 1 Ü M 1 1 i Cl 1 i
1 2 3 4 5 6 7 8 9 10 11 18
Figure 10. Scheme of mutation
Figure 11. Optimization procedure for mufflers
Figure 12. Profiles of STL on muffler at different dividing elements Table 1. Results for the variations of control parameters and segment numbers
Segment Number GA parameters Optimal Results Calculation time in PC
NA pc pm elt no D:(m) L(m) STL t (min)
2 0.8 0.05 1 0.0762 1.3715 8.6 0.53
2 0.8 0.05 0 0.0762 1.3748 8.6 0.57
2 0.8 01 0.0762 0.6562 8.5 0.58
2 0 0.05 1 0.0768 2.0493 8.5 0.49
4 0.8 0.05 1 0.0762 1.3715 11.0 0.78
4 0.8 0.05 0 0.0762 2.7437 11.0 0.80
4 0.8 01 0.0762 1.3385 10.9 0.79
4 0 0.05 1 0.0770 1.3701 11.0 0.77
8 0.8 0.05 1 0.0762 2.7439 12.8 1.21
8 0.8 0.05 0 0.0762 2.7484 12.8 1.24
8 0.8 01 0.0788 2.7257 12.6 1.24
8 0 0.05 1 0.0771 2.7646 12.7 1.19
16 0.8 0.05 1 0.0762 0.5000 7.0 2.10
16 0.8 0.05 0 0.0795 0.5021 6.8 2.10
16 0.8 01 0.0812 0.5016 6.8 2.23
16 0 0.05 1 0.0767 0.5060 6.9 2.05
32 0.8 0.05 1 0.0762 0.5000 7.1 3.79
32 0.8 0.05 0 0.0766 0.5014 7.0 3.78
32 0.8 01 0.0795 0.5097 6.7 3.77
32 0 0.05 1 0.0767 0.7897 6.8 3.72
64 0.8 0.05 1 0.0762 0.5000 7.1 7.65
64 0.8 0.05 0 0.0787 0.5001 7.0 7.62
64 0.8 0 1 0.0766 1.7821 6.3 7.63
64 0 0.05 1 0.0766 0.8020 6.8 7.65
