CC BY
34Electronic Journal «Technical Acoustics» http://webcenter.ru/~eeaa/ejta/
2004, 4
Ying-Chun Chang1, Long-Jyi Yeh, Min-Chie Chiu
Department of Mechanical Engineering, Tatung University, Taipei, Taiwan 104, R.O.C. e-mail: ycchang@ttu.edu.tw
Optimization on constrained single-layer absorbers by simulated annealing
Received27.02.2004, published 07.04.2004
As the thickness of sound absorber is often confined by the necessity of worker’s maintenance and access in the practical engineering work, the consideration of maximal sound absorption on the sound absorber with limited thickness becomes important and essential. In this paper, by using the simulated annealing (SA), a stochastic relaxation technique based on the analogy of the physical process of annealing metal, a single-layer absorber, including (1) one layer of perforated plate; (2) one layer of acoustic fiber; (3) one layer of air space; and
(4) a rigid-backing wall, is optimized under the thickness constraints. A numerical case in dealing with pure tone noise of 350 Hz was exemplified. Before optimization, a sample was tested and compared with the experimental data for accuracy check of mathematical model. Results proved that SA optimization provides a quick and efficient approach on the design of single-layer sound absorbers under fixed thickness.
Keywords: sound absorption, transfer matrix method, shape optimization, simulated annealing
1. INTRODUCTION
Many researches of sound absorber were well developed; however, the discussion of optimization on sound absorption of the absorber under the thickness constraints was rarely accomplished even though the thickness of absorber confined by the demand of maintenance and access occurred intermittently. And so, the interest to optimize sound absorption of the absorber by adjusting the shape parameters, including (1) the perforated ratio, the depth, the face density and the holes of the perforated front plate; (2) the depth of sound absorbing material and air space, at the limited thickness of sound absorber is arising in the field of acoustics.
Yeh et al. [1] developed the shape optimization of the constrained single-layer sound absorbers by the graphic sensitivity analysis. But it is still troublesome and tedious. To solve efficiently the optimal design of sound absorbers, simulated annealing (SA) by Kirkpatrick et al. [2], based on the work of Metropolis et al. [3], was applied. The matrix transfer form for sound absorption deduced in the previous work [4] was applied for the derivation of normal sound absorption coefficient. In addition, the half-experienced formula of specific normal impedance by Delany and Bazley [5] as well as by Bolt and Ingard [6, 7] were both included in the model derivation simultaneously.
This paper provides a quick and efficient method to optimize the shape of single-layer absorber by using the matrix transfer method together with the SA searching technique.
Corresponding author: Ying-Chun Chang, e-mail: ycchang@ttu.edu.tw, mailing address: Department of Mechanical Engineering, Tatung University, 40 Chungshan N. Rd., 3rdSec., Taipei, Taiwan 104, R.O.C.
2. THEORETICAL BACKGROUND
For a plane wave propagating through a partitioned and uniform section with stationary medium, the relationships of acoustic pressure and particle velocity between point 1 and point 2 can be expressed in matrix form (see Appendix) as follow:
cos(kL) j sin(kL)/Za
fiotr sin(kL)Yp2 '
otr
cos(kL)
u2
2
(1)
For an acoustic wave propagating within the arbitrary medium symbolized by mm, the general matrix form between point 1 and point 2 is then expressed as
' Pl "
u, 1
cos(kmmL )
V mm / j Sin(kmmL)/Z m
jZmm Sin(kmmL) COs(kmmL )
p2
u2
2
(2)
The absorber’s acoustic impedance on the perforated front plate is obtained from the bottom wall of the infinity of impedance [8]. The composition of single-layer perforated absorber is illustrated in Fig. 1. As derived in Eq. (2), the relation of acoustic pressure p and acoustic particle velocity u between point 0 and point 1 is expressed as the transfer matrix and shown as below.
(3)
where p1 is the acoustic pressure at the surface of the air layer, ul is the acoustic particle velocity at the surface of the air layer, pa is the acoustic pressure at the absorber’s bottom, and uo is the acoustic particle velocity at the back plate (wall).
For a structure of “partitioned rigid wall + L thickness of air + Df thickness of the acoustic
fiber”, the relation of acoustic pressure p and acoustic particle velocity u between point 1 and point 2 is expressed as the transfer matrix:
Pl cos(œL / co ) jPoc sin(œL / co) Po
u, _j sin(œL / co)/ Poco cos(œL / co ) _ uo
" P 2 " " cos(kfiberDf ) jZfiber sin(kfiberDf ) " Pl
_ u2 _ j sin(kfiberDf )/ Zfiber cos(kfiberDf ) _ _ u1
(4)
By adopting the formula of specific normal impedance and wave number, which is derived by Delany and Bazley [5] and valid in fibrous porous materials at the flow resistivity range R of 103~5x104 (MKS rayls/m), Eq. (4) can thus be rearranged as
Z2 (R fiber + jX fiber )
fiber
sinh(k2 ) cos(k, ) - j sin(k, ) cosh(k2 ) cos(k, ) cosh(k2 ) - j sinh(k2 ) sin(k, )
where kl = (œ / ^ )[1 + 0.0978(pof / R) -0 700 ]; k2 = (œ / co )[-0.189(pof / R ) ™];
-0.595 •
R
fiber
poco [1 + 0.0571(pof / R) -0 754)]; X fiber = poco [-0.087(pof / R)-0732].
(5a)
(5b)
Next, by analyzing structure of “partitioned rigid wall + L thickness of air + Df thickness of acoustic fiber + q thickness of the perforated front plate”, the normal impedance Z3 at the surface of perforated front plate is expressed in the matrix form:
P3
u3
P 2 u 2
(6)
cos(kpq) jZp sin(kpq)
j sin( kpq )/ Zp cos( kpq)
By developing Eq. (6) and adopting the formula of specific normal impedance and wave number of perforated plate derived by Bolt and Ingard [6, 7], the normal impedance Z 3 at the surface of the perforated front plate is simplified into the complex form
Z2 + ap tan(kpq)
7p +172 tan(kpq)
(7a)
where 7 p = j
32nfMh
16M 1 + h
2/4
\
mNn d
(nk2 d4 )
; Mh = Po
4
3
For normal incidence, the sound absorption coefficient [9] is
a(Df, L, f, R, q, d, m, p %) = 1 -
73 -PoC
3 7 + PoC
(7b)
(8)
2
3. MODEL CHECK
Before performing the SA optimal simulation, the accuracy check of a fundamental mathematical model on single-layer sound absorber was made by an experiment, as follows.
3.1. Experimental Setup
An experiment for measuring the normal sound absorption coefficient is designed and depicted in Fig. 2. As indicated in Fig. 2, the acoustic impedance tube has been used by emitting the normal incident plane wave and by detecting the standing-wave with a movable microphone. Therefore, on the plane wave bases, the measurement is valid only when the frequency performed is under the cutoff frequency, fc.
According to Rayleigh [10], the cutoff frequency for circular pipe is derived as 0.586 co / d where co is the sound speed and d is the diameter of acoustic impedance tube respectively.
3.2. Sound Absorption Coefficient
A sample of a single-layer sound absorber was simulated and tested by experiment. The accuracy comparisons between theoretical and experiment for the model were made and illustrated in Fig. 3, accordingly. As shown in Fig. 3, the result conveys that they are in good agreement. Therefore, the proposed fundamental mathematical model is acceptable. Thereafter, the developed model of single-layer sound absorber linked with SA optimization method on the constrained sound absorption system is applied in the following sections.
4. SIMULATED ANNEALING
Simulated annealing (SA) algorithm imitates the cooling process of material in a heat bath. In the physical system, annealing is the process of heating a solid to its melting phase, followed by a cooling operation until the material has crystallized. The slow cooling (annealing) allows the lowest energy state (i.e. minimum) to be reached; however, the fast cooling rate (quenching) will result in higher energy condition with the large internal energy stored inside the imperfect lattice. The basic concept behind SA was first introduced by Metropolis et al. [3]. and developed by Kirkpatrick et al. [2]. The purpose of using SA algorithm is to avoid the local optimal solution during the optimization.
The algorithm starts by generating a random initial solution and by initializing both the initial temperature (To) and the frozen temperature (Tfiml). The scheme of SA is a variation of
the hill-climbing algorithm. All down hill movements for improvements are accepted for the decrement of the system energy. Simultaneously, SA also allows moves resulting in worse-quality solutions (uphill moves) than the current solution in order to escape from local optimum. At higher temperature, the uphill move is already in a good change. However, the change of going up hill is decreased when the temperature gets lower.
To simulate the evolution of the SA algorithm, a new random solution is chosen from the neighborhood of the current solution. If the change in energy (or objection function) is negative, the new solution is accepted to be the new current solution. Otherwise, the transition property (pb(T)) of accepting the increase is computed by evaluating the Boltzmann’s factor (pb(T)=exp(AF/CT)) where AF , C and T are the difference of object function, Boltzmann constant and current temperature respectively. If the probability is greater than a random number in the interval of [0,1], the new solution is then accepted. If not, it is rejected. The algorithm iterates by perturbing the current solution and measuring the change in objection function.
Each successful replacement of the new current solution leads the decrement of new current temperature by multiplying current temperature by the cooling rate (kk), which is normally within 0.95 to 0.98. The process is repeated until the frozen temperature (Tfinal) is
reached or until there are no changes in the objection function.
The flow diagram regarding SA optimization is described and shown in Fig. 4. As indicated in Fig. 4, to maximize the sound’s absorbing coefficient, the objection function in SA is represented by the negative value (-a) in SA optimization.
5. CASE STUDY
The noise control of a machine fan room was exemplified. Because of the periodic disturbance of the in-flow acting on the fan, the phenomenon of siren effect occurred. A pure tone of 350 Hz was identified by the formula off=Nbn where Nb and n are the number and rotational speed of blades, respectively.
For the necessity of maintenance and access, the thickness of absorber was limited to be
3
0.2 m. A sound absorbing material of rockwool with density 80 kg/m was chosen. According to Wang’s experiment [11], the flow resistance of above rockwool is about 22000 rayls/m.
For the purpose of lightness on sound absorber, the thickness (q) and face density (m) of the perforated front plate are pre-selected as q=0.0006 m; m=2.0 kg/m . A series of constrained conditions in the design are specified as 5.0% < p% < 50.0%;
0.003 m<d<0.015 m; 0.01 m< Df <0.184 m;f=350 Hz; Do =0.3 m; R=22000 rayls/m.
6. RESULTS AND DISCUSSION
In this case, both the initial temperature (To) and lower frozen temperature (Tfinal) were set
as 120 and 0.00001 respectively. According to Cave et al. [12], the cooling rate (kk) was chosen as 0.95 firstly. To achieve a better approach in SA, four kinds of initial temperature were tested. In addition, the cooling rate was varied from 0.95 to 0.98 gradually.
The comparison of results for the seven sets of SA’s control parameters is illustrated in Table 1. It shows that in the 1st case, the initial temperature (To) and the cooling rate (kk) were
taken as 120 and 0.95 respectively. It has an optimal value of 0.886 in sound absorption coefficient compared to other cases.
The profiles of sound absorption curves with respect to spectrum in various initial temperature (To) are plotted together in Fig. 5. In addition, the profiles of sound absorption
curves with respect to spectrum in various cooling rate (kk) are plotted together in Figure 6. As indicated in Figures 5 and 6, each of the sound absorption at the desired frequency of 350 Hz was very close even though the deviations of profiles existed between them.
As indicated in Table 1, the iteration will increase at the higher value of cooling rate (i.e. the slower annealing) at the same initial temperature (To). Otherwise, the iteration will decrease at the lower cooling rate (kk). However, the deviation of iteration time with respect to different initial temperature (To) at the same cooling rate (kk) is not obvious.
7. CONCLUSIONS
It has been shown that SA can be used in the optimization of sound absorber by adjusting the shape’s parameters, including (1) the perforated ratio, the depth, the face density and the holes of the perforated front plate; (2) the depth of sound absorbing material and air space, at the fixed thickness of sound absorber.
SA becomes easier to be used comparing to traditional gradient methods in which both the complicated calculation in mathematic derivation and the selection of good starting point are required during optimal process [1]. The case study indicates that the SA operators of initial temperature (To), cooling rate (kk) and the frozen temperature (Tfinal) play essentially. The
iteration in SA highly depends on the value of cooling rate. In addition, by the exemplification of pure tone’s elimination, the expected sound absorption at the desired frequency of 350 Hz is well-optimized as 0.886. Consequently, the SA optimization can provide a quick and efficient approach on the design of single-layer absorbers under thickness constraints.
REFERENCES
[1] L. J. Yeh, M. C. Chiu, G. J. Lai. Computer aided design on single-layer perforated absorbers under space constraints. Proceedings of the 19-th National Conference on the Chinese Society of Mechanical Engineers. 2002, pp. 635-643.
[2] S. Kirkpatrick, C. D. Jr. Gelatt, M. P. Vecchi. Optimization by simulated annealing.
Science, 1983, vol. 220, N°4598, pp. 671-680.
[3] A. Metropolis, W. Rosenbluth, M. N. Rosenbluth, H. Teller, E. Teller. Equation of static calculations by fast computing machines. The Journal of Chemical Physics, 1953, vol. 21, N°6, pp. 1087-1092.
[4] M. C. Chiu. Compact acoustic board for low frequencies: experimental study and theoretical analysis. The 180-th National Conference on C.S.M.E. 2001, C3, pp. 719-724.
[5] M. E. Delany and E. N. Bazley. Acoustical properties of fibrous absorbent materials. Applied Acoustics, 1969, 13, pp. 105-116.
[6] K. U. Ingard and R. H. Bolt. Absorption characteristics of acoustic material with
perforated facings. Journal of the Acoustical Society of America, 1951, 23, pp. 533-540.
[7] R. H. Bolt. On the design of perforated facings for acoustic materials. Journal of the Acoustical Society of America, 1947, 19, pp. 917-921.
[8] D. A. Bies and C. H. Hansen. Engineering Noise Control. Unwin Hyman, 1988.
[9] Jinkyo Lee, W. Swenson, Jr. George. Compact sound absorbers for low frequencies. Noise Control Engineering Journal, 1992, 38(3), pp. 109-117.
[10] Rayleigh, Lord. The Theory of Sound, Macmillan. London, 1896.
[11] C. N. Wang and J. H. Torng. Experimental study of the absorption characteristics of some porous fibrous materials. Applied Acoustics, 2001, 62, pp. 447-459.
[12] Alex Cave, Saeid Nahavandi, Abbas Kouzani. Simulation optimization for process scheduling through simulated annealing. Proceedings of the 2002 Winter Simulation Conference, pp. 1909-1913.
APPENDIX
For three dimensional wave with moving medium, three kinds of governing equations are:
I. Mass continuity equation:
poV-U + Dp = 0. o Dt
II. Momentum equation:
DU „ n
Po-+ Vp = 0 .
y° Dt
III. Energy equation (isentropic):
(A1)
(A2)
r(po + p) = c2 or P = c2
Po +P Po 0 P 0
(A3)
By partial derivation and substitution in Eqs. (A1), (A2) and (A3), the wave governing equation yields
( D2
- c 2V2
2 o V
Dt
P = 0,
(A4)
where D = — + Vn -V is the total derivative. Vn is the mean velocity Dt dt o o y
For Cartesian coordinate system (for rectangular ducts)
n d2 d2 d2
V2 = — + — + - .
dx2 dy2 dz2
By using separation of variables method, it yields
(A5)
P( X, y, Z,t) = XX C0s COsfh (1mne " + C 2,m,ne
m=0 n=0 b h
+jkz,m,nZ I e I
(A6)
^ \ 1 ^ ^ mnx . nny
uz (X, y, Z, t) =--------X X cos----------------sin ——
PoCo m=0 n=0 b h
k+
k — Mk
o z,m,n
C1,m,ne
—Jkz,m,nZ
+ 1,m,n
k—
+
' Z,m,n C e + Jkz,m,nZ
r+ ^2,m,n
>o ■ * z,m,n
ko + MkZ
-,jwt
with the compatibility condition k2 + k2 + k2 =(k + Mk )2
x,m y,n z,m,n \ o z,m,n /
Kmn =(ko + Mk,^n )2 —
or
(A7)
(A8a)
(A8b)
2
+ Mk„ +
k] -(1 -M2)
“ / \ 2 2
mn + nn
b h
L v ^ \ _
1/2
(A8c)
k±
1 - M£
Consequently, only the plane wave at fundamental mode of (m=0, n=0) would propagate if the frequency is small enough so that
cr , V/2
2h
(A9)
where h is the larger value of the transverse dimensions of rectangular duct.
For one dimensional plane wave propagating perpendicularly through to a partitioned and uniform section with moving medium, the acoustic pressure is reduced to
p(z,t) = (C1e-]k°z/(1+M) + C2e+]k°z/(1-M))ejwt.
The corresponding acoustic particle velocity is
Q
poCo PoCo y
Considering the boundary conditions of point 1 (z=0) and point 2 (z=L), it yields
u(z, t) :
,jwt
' P1 ' "1 1" f C1 ^
p c u, o o 1^ 1 -1 C2 2
(A10)
(A11)
(A12)
/ P2 '
p c U2
o o 2
e -Jk+L eJk _L
e -jk+l — e+Jk - L
ko ko
k
C2
1+M 1 — M Combination of Eqs. (A12) and (A13) carries that
(A13)
P1
. MkL
- J----2
, 1-M2
V /
/
cos
kL
J sin
1 - M 2
f kL A
1 - M 2
JPoCo sin
kL
1 - M 2
/poco
cos
1 — M
For a plane wave in a quiescent air, Eq. (A14) is reduced to
u2
2
f Pi" cos(kL) JZa!> sin(kL) f P2"
u1 v 1 y J sin (kL)/Za!> cos(kL) u2 2
(A14)
(A15)
Therefore, for a acoustic wave propagating normally into a arbitrary quiescent medium symbolized by “mm”, the general matrix form between point 1 and point 2 is then expressed as
13
u
cos(k L) jZ sin(k L)lfp2 ^
\ mm / J mm V mm / r 2
j sin(k L)/Z cos(k L)
u2
2
(A16)
u
NOMENCLATURE
This paper is constructed on the basis of the following notations:
c0 sound speed (m-s"1)
d diameter of perforated hole on the front plate (m)
D0 thickness of absorber (m)
Df thickness of acoustic fiber (m)
fc cut-off frequency (Hz)
j imaginary part (=V-1).
k wave number of air
k 11 fiber complex propagation constant of acoustic fiber
kk cooling rate
kmm wave constant of arbitrary medium
kp complex propagation constant of perforated front plate
kl real part of complex kfiber
k2 image part of complex kfiber
L air depth of sound absorber (m)
m surface density (kg-m"2)
N hole’s number on the perforated front plate per m2
Pb (T) transition probability
p% perforated ratio of front plate (%)
Pr acoustic pressure at i (Pa)
q thickness of perforated front plate (m)
R acoustic flow resistance of acoustic fiber (MKS rayls-m"1)
Rfiber real part of complex Zfiber
T current temperature (oC)
T+1 next temperature (oC)
To initial temperature (oC)
T 1 final final temperature (=5-10"5 oC)
u. 1 acoustic particle velocity at i (kg-s"1)
vo mean velocity of medium (m-s"1)
a angular frequency (rad s"1)
Xfiber image part of complex Zfibrr
Z . ai r characteristic impedance of air ( = poco )
Z mm characteristic impedance of arbitrary medium
Zi specific normal impedance at i
Z fiber characteristic impedance of acoustic fiber
ZP characteristic impedance of perforated front plate
a sound absorption coefficient of absorber
Po air density (kg-m"3)
Fig. 1. Composition of a single-layer sound absorber ( Do = 0.2 m)
Fig. 2. Experimental setup for measurement of normal sound absorption coefficient using
impedance tube
Fig. 3. Accuracy check for theoretical mode of single-layer sound absorber (Z=0.05 m, Df =0.046 m, p%=19.4%, d=0.003 m, R=8000 rayls/m, m=2 kg/m2, ^=0.001 m)
Fig. 4. A flow diagram of SA optimization
Sound Absorption Coefficient versus Frequency by Simulated Annealing 1 -----------1-----------1----------1----------1----------1-----------1----------1---------
0 _________I_________I________I____t____I________I_________I________I_________
0 100 200 300 400 500 B00 700 800
Frequency [Hz]
Fig. 5. Sound absorption coefficients with respect to various initial temperature ( To )
Sound Absorption Coefficient versus Frequency by Simulated Annealing 1 -----------1-----------1----------1----------1-----------1----------1----------1--------
0 ■] __________I__________I__________I_____It___I__________I__________I__________I__________
' 0 100 200 300 400 500 B00 700 800
Frequency [Hz]
Fig. 6. Sound absorption coefficients with respect to various cooling rate (kk)
Table 1. Comparison of results for the variations of control parameters
SA parameters Results Iterations in simulation
"¡5 k to kk P (%) d (m) Df (m) a
Case 1 0.00005 120 0.95 37.0 0.0033 0.1716 0.886 319
Case 2 0.00005 120 0.96 35.5 0.0035 0.1737 0.884 401
Case 3 0.00005 120 0.97 30.9 0.0047 0.1735 0.877 537
Case 4 0.00005 120 0.98 35.3 0.0037 0.1712 0.883 808
Case 5 0.00005 115 0.95 20.1 0.0063 0.1732 0.868 318
Case 6 0.00005 125 0.95 47.4 0.0130 0.1728 0.871 320
Case 7 0.00005 130 0.95 43.7 0.0037 0.1757 0.885 321
