A BOUNDARY-ELEMENT-BASED OPTIMIZATION TECHNIQUE FOR DESIGN OF ENCLOSURE ACOUSTICAL TREATMENTS

A boundary-element-based

optimization

technique for design

of enclosure

acoustical

treatments

T. C. Yang

Center for Measurement Standards, Industrial Technology Research Institute, 321 Sec. 2, Kuang Fu Road, Hsinchu 30042, Taiwan, Republic of China

C. H. Tseng

Department of Mechanical Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 30050, Taiwan, Republic of China

S. F. Ling

School of Mechanical and Production Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 2263, Singapore

(Received 7 September 1994; revised 31 January 1995; accepted 16 February 1995)

An effective design tool was developed in this study for solving the acoustical treatments in enclosures at low frequencies. The boundary element method was used for accurately predicting sound fields of an acoustical system. The sequential quadratic programming was selected as the continuous design variable optimizer as a result of its robustness and rapid convergence. In coping with the noncontinuous design variables, the optimizer was enhanced by a modified branch and bound procedure. A small two-dimensional cavity and an irregularly shaped car cabin were applied to demonstrate the utilities of the proposed design tool. Computer simulations indicated that at a low frequency, where enclosures can properly be termed as tightly coupled, purely reactive acoustical

treatments

are

generally

preferred

at resonances

or near-resonances

of the

two-dimensional

cavity

as

previous work [R. J. Bernhard and S. Takeo, J. Acoust. Soc. Am. 83, 2224-2232 (1988)] could be

considered adequate for the proposed tool to be utilized for preliminary design studies of acoustical treatments in enclosures. Furthermore, in the case of multiple patching, the optimal locations of acoustical treatments would always contain the region of dominantly acoustical influence. However, simultaneous capability of optimizing locations and impedances of acoustical material were not considered in the previous work. ¸ 1995 Acoustical Society of America.

PACS numbers: 43.50.Gf, 43.55.Ka, 43.55Dt

INTRODUCTION

The design of effective noise control treatments for en- closures where typical dimensions are comparable with acoustic wavelengths, and/or sound sources and enclosures are geometrically complex, is relatively difficult. Such acoustical cavity systems are complicated by the near-field source effects, standing wave characteristics, and low- frequency behavior of typical sound treatments. The passen- ger compartments of a car and a propeller-driven aircraft, and enclosures of close-fitting appliances and business ma- chine enclosures are such important applications where ei- ther the noise source is of low frequency or the enclosure is quite small. However, effective design procedures for such

applications

are rarely

due to the complexity

of the prob-

lems. Therefore, a reliable modeling procedure capable of accurately predicting the acoustical behaviors in enclosures as well as providing the design optimization information is fundamentally necessary.

Absorption of the low-frequency components of un- wanted noise in cavities is difficult and expensive by con- ventional methods of using porous materials because of the thickness required. Through the extensive investigations of

Maa

1'2 and Lee and Swenson,

3 however,

noise

control

for

low frequencies by passive techniques has become practical.

Thus the possibility of employing passive techniques to ef- fectively control low-frequency sound radiated from some- where inside the cavity appears worthy of investigation.

Previous studies on the modeling techniques were useful for enclosures where either high-frequency statistical behav- ior can be assumed or the geometrical configuration is rela-

tively

simple.

4-7 In this

study,

simulation

results

obtained

by

dratic

programming

(SQP)

algorithm

8 is integrated

with the

ous

work,

e.g.,

Bernhard

and

Takeo,

7 only

the acoustical

im-

dd

,% ,©%

FIG. 1. Domain of acoustic system utilized for interior Helmholtz wave equation.

tical impedance and location of the absorbing material are simultaneously determined in this study for making the po- tential technique useful. Constraints, i.e., geometrical and/or functional, are also included as required due to practical con- siderations. From practical and economic concerns, only some boundaries of the cavity can be reserved for acoustical treatments. The acoustical material is not positioned any- where but is rather patched at specified regions during the optimization process. In coping with this difficulty, noncon- tinuous design variables capable of fully describing the pa- rameter of specified regions must therefore be introduced in addition to continuous ones. As a consequence of manipulat- ing the problems associated with continuous and noncontinu- ous design variables, the SQP scheme is enhanced by a

modified

branch

and

bound

method

(BBM)

9-11

to effectively

An optimization problem is formed in this study which optimizes allocations of acoustical materials on the boundary in cavities by minimizing the objective function which is subjected to suitable constraints. Total acoustic potential en- ergy of the control volume, similar to those in noise control problems, is selected here as the objective function. The op- timization is performed in the frequency domain by assum- ing that the noise source is of harmonic excitation. Next, computer simulations of acoustical treatments in enclosures are performed and discussed for different arrangements. In those simulations, a two-dimensional cavity is used to verify the BEM-based optimization design tool. An irregularly shaped car cabin is then modeled for further studies.

I. BOUNDARY ELEMENT FORMULATION IN

ACOUSTICS

Indirect

BEM as derived

by Chen

and

Schweikert

12

is a

derivation of optimal passive noise controllers can be formu- lated by using the indirect BEM technique. The numerical implementation of the indirect B EM technique in this study is customized from the general boundary element formula- tion. In summary, the process can be divided into the follow- ing steps.

Step 1: The nonhomogeneous Helmholtz equation,

V2p

+ k2p

= ½s(Xs),

is utilized

in forming

the

problem,

where

Step 2: The general form of indirect BEM integral equa- tions for pressure, velocity, and impedance (locally reacting) boundary conditions, respectively, are written as

p(•)=

fBrr(b)p*(b,•)dB+

fz>½s(Xs)p*(Xs,•)dD,

(1)

U(•)=CbCr(•)+

fbCr(b)u*(b,•)dB

+ zOs(Xs)U*(Xs,e)dD,

z(e)=p(e)/u(e), (3)

where o'(b) represents the fictitious source density function at the boundary point b, C b is an integration constant as a

result

of the

integral

singularity,

13

and

• is a dummy

variable.

Green's

function,

p*(b,•)= 1/[b-•l e-je b-e and

the

fun-

damental

velocity

solution

u* can

be related

TM

to p* by Eu-

Step 3: A noncompatible, rectangular, linear element is selected to discretize the boundary of the acoustic system. Consequently, results can be obtained with reasonable com- puter capacity and time as a result of the need of optimiza- tion and model with a large number of elements. The volume

integrals of ½s in Eqs. (1), (2), and (3) are then replaced by

the summation term in light of the point noise sources. Hence, Eqs. (1), (2), and (3) are rewritten as

N e

P(bi)

=

E fB

ø'(bj)p*(bj'bi)dBj

q-

E •tsl(Xsl)p*(Xsl,bi),

u(bi):Cbø'(bi)q-jE

1

fB

ø'(bJ)u*(bj'bi)dBj

q-

E •tsl(Xsl)lt*(Xsl,bi),

0=

rr(bj)p*(bj

,bi)dBjn

t-

• ½sl(Xsl)P*(Xsl,bi)

-z(bi)(CbO'(bi)-•-••BO'(bj)Ig*(bj,bi)dB

j

)

-3-

• IPsl(Xsl)l•lV•(Xsl

,bi) ,

(6)

where N e is the number of elements, Ns is the number of

noise

sources,

Bj is the boundary

contained

by the jth ele-

ment, and b i is the i th node.

Step 4: A system of N e equations for the N e unknown o-'s is obtained by writing Eqs. (4), (5), and/or (6) for each element according to the given boundary conditions. Thus a compact matrix form of Eq. (7) is solved for the fictitious source strength o- of each element:

Atr= a-bus, (7)

where matrices A and B are computed from Eqs. (4), (5), and/or (6), and vector a contains the values of boundary conditions. For instance, if element i has a pressure boundary condition, then

Aij-

fBjP*(bj,bi)dBj, (8)

Bil=P*(Xsl ,bi), (9)

o•i=p(bi). (10)

Step 5: Finally, Eqs. (4) and (5) are used to find the acoustic pressure and particle velocity of the interior points, in which the boundary point b i is replaced by the domain point xi.

II. MATHEMATICAL MODEL FOR OPTIMIZATION PROBLEM

Generally, three elements are included in a well-defined mathematical statement of the constrained design optimiza- tion, i.e., design variables, objective function, and design constraints. The primary objective of this study is to form an optimization problem capable of optimizing placements of acoustical materials on the boundary of a cavity. Each ele- ment of optimization is described in the following.

A. Design variables

Assuming that the acoustical materials is only locally reacting, the normal impedance has the form R(f )-jX(f ), where R and X are the resistance and reactance parts, respec-

tively,

and

are dependent

on frequency

f.]5 From

a practical

characteristic of the noise source is known and the sound field is only controlled by the acoustical materials, the acous- tic pressure at field points of the cavity can be represented by

pi=Pi(n,x,y,z,R,X,X), i- 1,Nfp,

(11)

where

Nfp is the total number

of field points.

B. Objective function

Various objective functions have been used for the opti- mization of noise control, i.e., the total acoustic potential

energy

by Jo and

Elliott,

•6 weighted

sum

of the magnitudes

of the squared

pressures

by Mollo and

Bernhard,

17

average

of acoustic

pressures

ratio

by Tanaka

et al. 18

total

acoustic

power

by Cunefare

and Koopmann,

19 and space-average

sound

energy

density

by Yang

et al.

2ø

All of these

functions

are deemed effective as a performance index, even though a slight difference occurs among them. The total acoustic po- tential energy of the control volume is, however, selected as the objective function in this study. Two control strategies,

local

and

global,

2• can

be applied

where

necessary.

The ob-

'

(I)--

4pc2 Ip(n,x,y,z

g,x,x)l

2 dV,

(12)

where c and V are the speed of sound and the control vol- ume, respectively. The (I) can only be measured by ideal distributed sensors, which may not be practical for applica- tion purposes. In practice, a reasonable number of acoustical pickups are uniformly positioned in the cavity to accumulate the response for formulating the discrete form of the objec- tive function, as illustrated by Eq. (13)'

V Nfp

V-

2 • IPi(n,x,Y,z,R,X,K)I

2.

(13)

4pc i=i

This form is implemented in the numerical simulations. The control volume V can notably be taken out of the summation only if the acoustical pickups are uniformly distributed throughout the cavity. Otherwise, the contribution of each pickups must be weighted.

C. Design constraints

All engineering systems are designed to perform within a given set of constraints which include limitations on re- sources, material failure, response of the system, and mem- ber sizes. Three types of constraints are introduced in this study, i.e., design bounds, equality constraints, and inequality constraints. Possible candidates for each type are defined in the following. 1. Design bounds O•< n•< Ne , (14) x•<xi•<Xu, i=l,n, (15) y•<yi•<yu, i-l,n, (16) z•<zi•<zu, i= l,n, (17)

Rl•<Ri•<R u, i=l,n, (18)

Xi•<Xi•<Xu, i= 1,n, (19)

kj•{0,1},

j=l,Ne,

(20)

where the subcripts 1 and u represent the lower and upper bound of the design variables, respectively.

2. Equality constraints

Xi+X2+'" +XiVe=n.

(21)

This constraint implies that the sum of the patched codes must be equal to the number of pieces of acoustical materials utilized.

3. Inequality constraints

If the user wishes to maintain the specified sound pres-

sure level (SPL) at particular locations, e.g., conforming to

mandatory regulations, the following equation can be consid- ered:

SPL at (xj,yj,;zj)-Lp•<O, j= 1,Nsp

1,

(22)

where

Lp and Nsp

1 are the specified

SPL and number

of se-

lected locations, respectively.

D. Summary of optimization model

In summary, the design optimization model of this study finds the number of pieces (n), location (x,y,z), and imped- ance (R,X) of acoustical materials so as to minimize the total acoustic potential energy, which satisfies the required constraints. Generally, the aforementioned constrained opti- mization problem can be described mathematically while minimizing the objective function

f ( x) = f (x i ,x2 ... X2v), subject to the constraints

(23)

h i(x) = 0, i = 1 ,Neql,

(24)

gj(x) •< 0, j = 1 ,Niq•,

(25)

and the design bounds

x•<x•<Xu, (26)

where

Neq

1 and

Niq

1 are the number

of equality

and

inequality

constraints, respectively. It is a general mathematical model for the nonlinear single-objective constrained optimization problem. The optimum design x can be obtained by employ- ing efficient minimization algorithms capable of solving the related subproblem for determining the search direction, as well as determining the step size by minimizing the descent function. 8

E. Optimization algorithm

Multiple minima basically exist in the feasible domain of the preceding optimization problem along with a number of numerical nonlinear programming (NLP) methods which are capable of solving it. However, the sequential quadratic

programming (SQP) is selected in this study because of its

robustness

and

rapid

convergence.

22-24

The SQP

algorithm

is

a generalized gradient-descent optimization method and sub- sequently converges to a local rather than global optimum. This optimization method solves the quadratic programming subproblem to produce the direction of design improvement, and a step size along the search direction. The subproblem is obtained by using a quadratic approximation of the Lagrang- ian and linearizing the constraints as follows:

min •7f(xk)

r Axk+0.5 Ax•I-I

r Ax •,

subject to

and

hi(xk)

+ •7hi(xk)

r Axk=0,

g/(x•)

+ Vg/(x•)

r Ax•<0,

(27)

i= 1 ,Neql,

(28)

j= 1 ,Niql,

(29)

x•< x • + Axk•<

Xu,

(30)

where H is a positive definite approximation of the Hessian matrix, which is composed of the second partial derivatives of the Lagrangian function with respect to each of the design

variables,

and

x • is the

current

iterate.

The

Lagrang•an

func-

constraints and is defined as L(x,l•)=f(x)+•/•ihi(x )

+•/•jgj(x), where

/z is the Lagrange

multiplier.

If Ax

• is

selected as the solution of the subproblem, a line search can

be utilized

in finding

the new point

x • + •, i.e.,

x•+ •=xk + •7• Axe,

(31)

where

the step

size •7•

is selected

as 0.5

J with J as the small-

est positive integer which satisfies a descent function having

a lower value at the new point.

8 The descent

function

is

basically formed by the objective function and maximum

constraint

violation

8 due

to its simplicity

and

success

in solv-

ing a large

number

of engineering

design

problems.

25'26

This

iteration process is continued until the convergence test is passed; otherwise, the design variables are updated and the new iteration is executed.

To make use of well-established continuous optimiza- tion algorithms, most discrete optimization techniques are based on the assumption of transforming the noncontinuous solution space into multiple continuous solution sub-spaces. The optimization problems in each of these continuous sub- spaces are solved sequentially by imposing constraints on discreteness of the design variables. The optimal discrete so- lution is selected from among the continuous solutions ob- tained. The conventional discrete optimization techniques re- quire too many executions of the continuous optimization

scheme

and,

thus,

is very

time consuming.

9

Here,

an enhanced

branch

and

bound

method

(BBM)

•ø'•

is employed to cope with the obstacles arising from noncon- tinuous design variables. In order to locate a discrete optimal solution using continuous optimization schemes, the BBM repeatedly delete portions of the original design space that do not contain allowable values of the noncontinuous design variables. This procedure is referred to as "branching." As illustrated in Fig. 2, the original design space is first divided into three sub-spaces with the allowable noncontinuous de- sign values nearest the continuous optimal solution as the upper and lower bound of the left and right sub-spaces, re-

Continuous

optimizer

SQP

• Sc

Solve continuous Continuous

conversion problem optimizer

[ SQP

•[ Evaluate values of criteria

-•for selection

of

branch

object

I

Choose

branch design variables Branch procedure:

Separate original design space into three parts

Pa[rtl

I [ Pa[rt2

] [ Paint3

Adding a ] [ Without [ } Ad•ng a

new upper I

.bound

[ lallowable

I value

] !new

bot

lower[

nd

optimizer SQP

Ch,,e;i•e..c•

C

_ost

Infeasible

_>

_Bound

•[

_{• Delete}

_I

node{

]

I Choo

branch

I

No

,:.•:.(• ...

FIG. 2. Conceptual flow chart of the noncontinuous optimizer BBM.

spectively. The branching procedure must be repeated in each of the sub-spaces until a feasible optimal design is lo- cated. Each design sub-space is depicted as a "node" and a diagram of the branching is referred to as the "tree" of BBM. Theoretically, a discrete solution can be found if an exhaustive search of the tree is made.

If a feasible noncontinuous solution is obtained in the process of branching, the corresponding objective function value can be taken as a bound. Any other design sub-spaces that possess a continuous minimum cost larger than this bound need not be further branched since it would only gen- erate higher cost values. This strategy is referred to as "bounding" and can be used to select a branching route in- telligently, thereby avoiding complete and impartial search- ing through the tree.

Furthermore, all the procedures of the constrained opti- mization model defined in this study are incorporated into an architectural framework for the proposed design tool, as il- lustrated in Fig. 3. The communication between the BEM acoustical analysis and the BBM enhanced SQP optimization during the solution procedure is also shown in this architec- tural framework.

Input f'fle

INPUT

Read input data

BBM Non-continuous design variable treatment Optimization

Input

f'de

Boundary element analysis

Output f'de

FIG. 3. Architectural framework of the proposed design tool, where cuserxx routines are user supplied functions. The cusermf and cusermg compute values of the objective and constraint functions, respectively. The cusercf and cusercg calculate gradients of the objective and constraint functions, respectively. However, the cuserou can provide additional data for verifica-

tion.

III. COMPUTER SIMULATIONS

A. Verification of the BEM-based optimization algorithm

Numerical characteristics of the developed BEM-based optimization technique are investigated by conducting a computer simulation for design of acoustical treatments in a two-dimensional cavity (Fig. 4). The cavity with two open- ings has a dimension of 0.56X0.32X0.08 m, i.e., slightly

different

from the one utilized

by Bernhard

and Takeo.

7 A

point

noise

source

of volume

velocity

1 m3/s

is positioned

at

Two configurations of acoustical treatments, C1 and C2 (Fig. 4), are first simulated to make a qualitative comparison of the results with the concluding remarks in the previous

work,

7 i.e., (1) it would

appear

that

at a low frequency,

where

C1

Noise source

Openings

FIG. 4. Schematic diagram of a two-dimensional cavity utilized in Sec.

IIIA

• 600

•

•

1000

•

-500

FIG. 7. Optimal acoustical •eatments versus •equency at C2 configuration. FIG. 5. Optimal acoustical •eatments versus •equency at C1 configurationß

enclosures can properly be termed as tightly coupled, purely reactive noise control treatments are preferred; (2) there is also a particular optimal reactive impedance depending on the frequency, geometry, and design objective function. For illustrative purposes, the impedance (R,X) of acoustical ma- terials is selected as the only design variable (i.e., acoustical material being fixed at specified location). The design bounds are set to 0•<R•<1000 Pa s/m and -2000<•X•<2000 Pa s/m. No physical constraints are involved in this case. Consequently, acoustical impedance at fixed configuration (i.e., n = 1) is optimized with the noise source of resonant or off-resonant excitation. Typical results are summarized in Figs. 5-8. The following general statements can be drawn from these numerical simulations.

(1) The optimal impedances of acoustical material vary with the exciting frequencies of the noise source and also change with locations of acoustical treatments. At resonances (i.e., 1540, 1610, and 1840 Hz in the frequency of interest) or near resonances, the optimal noise control treatments in this small cavity are more or less purely reactive (see Figs. 5 and 7) as a result of attenuation of the noise level by altering the mode shape. These remarks are quite similar to those

concluded

by Bernhard

and

Takeo.

7

(2) In light of no obvious mode shape, however, the mechanism of controlling the noise level at off-resonances of

the cavity can be different from that described at resonances. As also shown in Figs. 5 and 7, resistances (i.e., absorption) of the acoustical material are required in the off-resonant excitations for globally dissipating the acoustic energy in the control volume for better noise reduction.

(3) The objective function at resonances or near- resonances of the cavity decreases dramatically while the optimal impedances are obtained (Figs. 6 and 8). This can be considered as a consequence of a large reduction of SPL in the control volume. However, the objective function is insen- sitive to the variation of impedances of acoustical materials at off-resonances. Thus, the global minimum of this situation is difficult to be located. The local minimum can definitely be accepted from an engineering perspective if the improve- ment of design is obvious.

In summary, the major remarks of this simulation corre-

late sufficiently

with those

of Bernhard

and

Takeo.

7 Further

studies of acoustical treatments in enclosures are thus sup- ported by utilizing the proposed design tool of this work. Due to the powerfulness of the BBM enhanced SQP algo- rithm, the location and impedance of acoustical materials as design variables are simultaneously solved for the aforemen- tioned case. Two more configurations, C3 and C4, are added to the current case for further investigations (Fig. 9). Addi- tionally, design bounds of impedance are restricted to

-o• Optimal

•

Optimal

ß 3000 ß 3000 • 2000 c5 2000 ß

1000

.•

•-x

o

-

--

o

,

FIG. 6. Optimal objective function versus frequency at C1 configuration. FIG. 8. Optimal objective function versus frequency at C2 configuration.

C1 Openings C2 C3 Noise source C4

FIG. 9. Schematic diagram of a two-dimensional cavity utilized in Sec.

IiI A.

0•R•1000 and 100•X•2000 Pa s/m as a result of avoid-

ing the possibility of zero impedance, which is approximated

for a waterborne

sound

by the free surface

of the ocean?

Here, only noise source of the resonance of the cavity is focused on, i.e., at frequencies of 1540, 1610, and 1840 Hz, in light of having a large acoustic potential energy. Three cases, i.e., patching the acoustical material at a single, two,

and three configuration; • are simulated, as indicated in Table

I. As previously mentioned, optimal impedances for these cases are generally purely reactive. Also, optimal locations and impedances of acoustical treatments alter with the fre- quency of excitation. If the characteristic of the noise source is a narrow band with the specific resonance of •..cavity, this information can be useful for better acoustical material patching in the preliminary design stage. Performance of us- ing three pieces of acoustical material is deemed more gen- erally effective than that of a single and double one, i.e., a 16% to 97% lower objective function. Table II indicates that the typical history of iterations while location and impedance of acoustical material are simultaneously considered as de- sign variables.

B. Acoustical treatments in an irregularly shaped cavity

A car cabin

of its dimensions

shown

in Fig. 1028

is se-

lected as a practical situation in which an irregularly shaped boundary occurs. The engine firing frequency dominates the internal noise level in a number of cars, particularly at higher

TABLE I. Optimal acoustical treatments in the resonant excitation of the two-dimensional cavity with n = 1, 2, and 3, respectively.

Objective function Impedance X 10 -3, joule Frequency Location Pa s/m n Hz Ci (R,X) Original Optimal 1540 C2 (0, - 119) 3821 208 1610 C 1 (0, -100) 1388 116 1840 C1 (0, - 180) 32441 6324 1540 C2/C 4 (0, -429) 3821 171 1610 C•/C 3 (0, -294) 1388 91 1840 C2/C4 (0, -183) 32441 217 1540 C1/C2/C 4 (0, -942) 3821 137 1610 C1/C3/C 4 (0, - 100) 1388 67 1840 C2/C3/C 4 (0, -220) 32441 185 ,

engine

speeds.

29 Therefore,

control

of engine-induced

noise

inside the car is of vital interest in this study. A frequency range of 20 to 200 Hz is focused upon as a result of the likelihood of a low-frequency, engine noise boom phenom- enon which is increased by the current trend toward lighter car bodies and more powerful engines. The control volume

utilized

to compute

the squared

acoustic

pressure

(p2) of the

objective function is located in the vicinity of the driver's and passengers' heads in the cavity. For convenience, the seats in the cabin are removed, and rigid walls are assumed on the boundary unless otherwise stated. A noise source of

volume

velocity

1 m3/s

is positioned

at (0.5, 0.14, 0.34) to

simply simulate a single transmission path of the engine- induced noise. Five candidate regions are selected for patch- ing acoustical matehals, i.e., roof, floor, left door, right door, and back panels. Design bounds of impedance are 0•<R •< 1000 and 100•<X•<2000 Pa s/m. SPL at passenger heads, i.e., locations of (2.3, 0.6, 1.2), (2.3, 1.4, 1.2), (3.2, 0.6, 1.2), and (3.2, 1.4, 1.2), must be smaller than 80 dB. This SPL value can be practically assigned if the character- istics of the noise source are precisely defined.

First, acoustical influences in the frequency range from 20 to 200 Hz at five regions are verified at specified imped- ance, as shown in Fig. 11. Acoustical influence is defined here as the average insertion loss (IL) in the frequency range of interest. However, the IL is known to be the difference of SPL in the control volume with and without the acoustical treatments. The thickness of the shading near the panel or the number of the dB, indicates the relative influence on the sound pressure reduction of the control volume qualitatively or quantitatively. This reduction is a result of patching acous- tical matehal with fixed impedances to that particular panel across the frequency band of interest. This figure reveals that the major contributing panel to noise reduction is situated in the floor region. Of course, these results depend strongly upon the frequency range, control volume, characteristics of noise source, and specified impedance of acoustical material. However, computations such as these can provide useful in- formation regarding contributing panels at frequencies char- actedzing important excitation of engine firing. From Fig. 11, acoustical influence of the floor panel (region) is obvi- ously more dominant than others, i.e., a 6 to 7 dB higher IL.

In order to more fully comprehend the characteristics of acoustical treatments in the region of floor, a simulation of results shown in Figs. 12 and 13 is conducted. Again, the purely reactive impedance is preferred at the optimal acous- tical treatments in this low frequency range. Reasons for this trend are similar to those mentioned in Sec. III A. However, a lower bound tendency of the optimal reactance can be re- lated to the geometrical configuration of the cavity, position of noise source, and complexity of sound fields in the cabin. Furthermore, effectiveness of acoustical treatments is quite high at the resonant excitation of the car cabin due to a lower objective function (0.4%-6% of original one) and a higher insertion loss (13 dB on average).

Finally, acoustical treatments in the resonant excitation of the car cabin with location and impedance as design vari- ables simultaneously are illustrated in Table III for n = 1, 2

TABLE II. The typical history of iterations while using the location of the acoustical material as a design

variable.

Acoustical treatments for a cavity of dimensions 0.56X0.32X0.08 m The input data file in pnc. inp

NUMBER OF DESIGN VARIABLES --6

NUMBER OF OBJECTIVE FUNCTIONS -1

NUMBER OF EQUALITY CONSTRAINTS -1 NUMBER OF INEQUALITY CONSTRAINTS =0

MAXIMUM NUMBER OF ITERATIONS -100

INDEX OF PRINTING CODE =0

GRADIENT CALCULATION INDICATOR -1 NO. OF CONSECUTIVE ITER. FOR ACT =5 TOL. IN CONSTR. VIOL. AT OPT. = 1.0000e-04

CONVERGENCE PARAMETER VALUE = 1.0000e-04

DEL FOR F. D. GRAD. CALCULATION = 1.0000e-01 ACCEPTABLE CHANGE IN COST FUNC. = 1.0000e-09

ß :::' Type, Starting design and its limits ...

No. Type Design Lower lim Upper lim

1 Zero-one 0.0000e +00 0.0000e +00

2 Zero-one 1.0000e +00 0.0000e +00

3 Zero-one 0.0000e +00 0.0000e +00

4 Zero-one 1.0000e +00 0.0000e +00

5 Continue 0.0(J00e +00 0.0000e +00

6 Continue i.0000e +02 1.0000e +02

First cycle: Initial design:

X[1] 0.0000e +00 X[2] 1.0000e +00 X[3] 0.0000e +00 X[4] 1.0000e +00

X[5] 0.0000e+00 X[6] 1.0000e+02

*** Final design:: ***

*X[1] 0.0000e+00 *X[2] 5.0000e-01 *X[3] 1.0000e+00

*X[4] 5.0000e-01 *X[5] 0.0000e+00 *X[6] 1.9686e+02

*** Cost function at optimum = 1.8992628 e + 02 ***

No. of calls for cost function evaluation (usermf) =700

No. of calls for cost function gradient evaluation (usermg) =0

No. of calls for constraint function evaluation (usercf) =700

No. of calls for constraint function gradients evaluation (usercg) =0

No. of total gradient evaluations = 100

The following cycles are the BRANCH-AND-BOUND cycles:

Initial design:

X[1] 0.0000e +00 X[2] 1.0000e +00 X[3] 5.0000e-01 X[4] 0.0000e +00 X[5] 1.9686e+02

*** Final design:: ***

*X[1] 4.4432e-02 *X[2] 1.0000e+00 *X[3] 9.5557e-01

*X[4] 0.0000e+00 *X[5] 1.9318e+02

*** Cost function at optimum=2.3261524e +02 ***

No. of calls for cost function evaluation (usermf)=71 No. of calls for cost function gradient evaluation (usermg)=0 No. of calls for constraint function evaluation (usercf)=69 No. of calls for constraint function gradients evaluation (usercg)=0 No. of total gradient evaluations=6

1.0000e+00 1.0000e +00 1.0000e+00 1.0000e +00 1.0000e+03 2.0000e+03

TABLE II. (Continued.)

(The middle steps are deleted) Initial design:

X[1] 0.0000e+00 X[2] 0.0000e+00 X[3] 0.0000e+00 X[4] 1.8345e+02

*** Final design:: ***

*X[1] 0.0000e+00 *X[2] 0.0000e+00 *X[3] 0.0000e+00 *X[4] 1.8345e+02 *** Cost function at optimum=2.1746219e +02 ***

No. of calls for cost function evaluation (usermf)=30 No. of calls for cost function gradient evaluation (usermg)=0 No. of calls for constraint function evaluation (usercf)=29 No. of calls for constraint function gradients evaluation (usercf)=0 No. of calls gradient evaluations=2

Final results:

Number of iterations= 1

Maximum constraint violation = 0.00000e + 00

Convergence parameter= 1.48308e-01 The value of supercrition= 2.17462e + 02 The design variables:

X[1] 0.0000e+00 x[2] 1.0000e+00 X[3] 0.0000e+00

X[4] 1.0000e+00 X[5] 0.0000e+00 X[6] 1.8345e+02

Total number of calls of USERMF=2308

Total number of calls of USERMG=0 Total number of calls of USERCF=2304

Total number of calls of USERCG=0

No. of total eval. of grad=386

and 3, respectively. Some remarks can be drawn from these

typical simulations.

(1) The optimal location of acoustical patching for the n = 1 case is all of positioning in the floor region. Also, one of the optimal treatments for all of n =2 and 3 cases contains the floor panel. These results can directly correlate with the analysis of acoustical influence in the aforementioned sec-

tion. ß

(2) In light of dominance of acousti6al treatment in the

floor region, performance of the n =2 and 3 cases does not

have much of an improvement as compared with the n-1

case, except for the resonant excitations of 39 and 66 Hz. (3) However, purely-reactive impedance preferred is not

true for some particular situations. For instance, resistance is

1.75: 2.28m 0.64m 1.3111 0.34m (o,o,o] 5m

FIG. 10. The mesh and dimensions of a car cabin utilized in Sec. III B.

310 d. Acoust. Soc. Am., Vol. 98, No. 1, July 1995

also required

to optimally

attenuate

the noise

at the excita-

tions of 75 and 103 Hz for the n =2 case. This can be as a result of either the objective function being insensitive to the variation of impedance in the neighborhood of global mini- mum or other mechanisms which are not clearly known.

4.2

dB

Roof

•

ß -•

FIG. 11. Acoustical influence diagram: (a) side view, (b) top view.

0 • o o

-tOO

X-X-X--

FIG. 12. Optimal impedance in the region of floor as a function of fre- quency for the car cabin.

(4) Performance for the n = 3 case is not necessarily bet-

ter than that for the n-- 1 and 2 cases as a result of complexi- ties of primary sound fields in the cavity, dominance of acoustical treatment in the floor region, and local minimum searching of the algorithm. This knowledge implies that the number of piece of acoustical materials can be optimized if it is selected as a design variable.

IV. CONCLUSIONS

Combining the numerical BE acoustical analysis with the SQP optimization solver could be an effective tool for designing acoustical treatments in cavities at low frequency. The SQP optimization solver was enhanced in this study by the modified BBM procedure to simultaneously treat the practical applications of continuous and noncontinuous de- sign variables. This would be especially useful while the conventional combinatorial design has become impractical. A small two-dimensional cavity was verified as well as the utility of the proposed design tool demonstrated by an irregu- larly shaped car cabin. Computer simulations indicated that at a low frequency, where enclosures can properly be termed as tightly coupled, purely reactive acoustical treatments are generally preferred at resonances or near-resonances of the two-dimensional cavity as well as at the frequency of interest

1200 1000

800

400

i

•

original

FIG. 13. Objective function in the region of floor as a function of frequency

for the car cabin.

TABLE III. Optimal acoustical treatments in the resonant excitation of the car cabin with n = 1, 2 and 3, where C, F, L, and R represent the regions of ceiling, floor, left, and right door panel, respectively.

Objective function

Impedance X 10

-3, joule Insertion

Frequency Pa s/m loss

n Hz Location (R,X) Original Optimal dB

39 F (0, - 159) 1092 5 21 66 F (0, -300) 339 19 12 1 75 F (0, - 100) 445 3 23 84 F (0, -100) 332 7 14 103 F (0, - 100) 616 22 15 39 C/F (0, - 100) 1092 0.4 32 66 C/F (0, - 100) 339 3 20 2 75 C/F (240, - 100) 445 2 24 84 C/F (0, - 100) 332 5 18 103 F/R (272, - 100) 616 7 19 39 C/F/L (0, - 100) t 092 0.1 38 66 F/L/R (0, - 100) 339 0.7 27 3 75 C/F/L (0, -100) 445 9 18 84 C/F/L (0, - 100) 332 5 18 103 C/F/L (0, - 100) 616 19 15

in the car cabin. This general agreement between the present

study

and the previous

work of Bernhard

and Takeo

7 was

considered adequate for the proposed tool to be utilized for the preliminary design study of acoustical treatments in en- closures. Furthermore, the optimal locations of acoustical treatments would always contain the region of dominant acoustical influence in the situation of multiple patching. However, capability of optimizing locations and impedances of acoustical material simultaneously were not considered in the previous work.

ACKNOWLEDGMENTS

The authors acknowledge the National Science Council, Taiwan, R.O.C. for partial support of this work under grant

NSC 82-0401-E009-085. The Center for Measurement Stan-

dards of Industrial Technology Research Institute, Taiwan, R.O.C. is also appreciated for financially supporting T. C. Yang.

1D. Y. Maa, "Theory and Design of Microperoforated Panel Sound-

Absorbing Constructions," Sci. Sinica 18, 55-71 (1975).

D. Y. Maa, "Microperoforated Panel Wideband Absorbers," Noise Control

Eng. J. 29 (3), 77-84 (1987).

3j. Lee and G. W. Swenson, Jr., "Compact Sound Absorbers for Low

Frequencies," Noise Control Eng. J. 38 (3), 109-117 (1992).

4R. S. Jackson, "The Performance of Acoustic Hoods at Low Frequen-

cies," Acusfica 12, 139-152 (1962).

M. C. Junger, "Sound Transmission Through an Elastic Enclosure Acous-

tically Closely Coupled to a Noise Source," ASME Paper No. 70-WA/ DE-t2 (1970).

L. W. Tweed and D. R. Tree, "Three Methods for Predicting the Insertion

Loss of Close-Fitting Acoustical Enclosures," Noise Control Eng. 10 (2), 74-79 (1978).

7R. J. Bernhard and S. Takeo, "A Finite Element Procedure for Design of Cavity Acoustical Treatments," J. Acoust. Soc. Am. 83, 2224-2230 (1988).

8j. S. Arora, Introduction to Optimum Design (McGraw-Hill, New York,

1989).

9C. H. Tseng, L. W. Wang, and S. E Ling, "Numerical Study of Branch-

and-Bound Method in Structural Optimization," Technical Report AODL-

90-02, National Chiao Tung University, Taiwan, ROC (1990).

løC. H. Tseng, L. W. Wang, and S. E Ling, "Enhancing the Branch-and-

Bound Structural Optimization," Proc. Int. Conf. Comput. Methods Eng.,

Singapore, 1180-1185 (1992).

• C. H. Tseng, L. W. Wang, and S. F. Ling, "Enhancing the Branch-and-

Bound Method for Structural Optimization," to appear in the ASCE J. Struct. Eng. (1995).

12L. H. Chen and D. G. Schweikert, "Sound Radiation from an Arbitrary Body," J. Acoust. Soc. Am. 35, 1626-1632 (1963).

13 R. D. Ciskowski and C. A. Brebbia, Boundary Element Methods in Acous- tics (Elsevier Applied Science, London, 1991).

14L. E. Kinsler, A. R. Frey, A. B. Coppens, and J. V. Sanders, Fundamentals of Acoustics. (Wiley, New York, 1982).

•5 A. Craggs, "A Finite Element Method for Modeling Dissipative Mufflers with a Locally Reactive Lining," J. Sound Vib. 54 (2), 285-296 (1977). •6C. H. Jo and S. J. Elliott, "Active Control of Low Frequency Sound

Transmission Between Rooms," J. Acoust. Soc. Am. 92, 1461-1472 (1992).

•7 C. G. Mollo and R. J. Bernhard, "Generalized Method of Predicting Op-

timal Performance of Active Noise Controllers," AIAA J. 27 (11), 1473- 1478 (1989).

•8M. Tanaka, Y. Yamada, and M. Shirotori, "Boundary Element Method

Applied to Simulation of Active Noise Control in Ducts," JSME Int. J. 35 (3), 387-392 (1992).

19 K. A. Cunefare and G. H. Koopmann, "A Boundary Element Approach to

Optimization of Active Noise Control Sources on Three-Dimensional Structures," ASME J. Vib. Acoust. 113, 387-394 (1991).

2øT. C. Yang, C. H. Tseng, and C. H. Huang, "An Interface Coupler for

Finite Element Analysis and Optimization," Proc. Sixteenth National Conf. Theoretical and Applied Mechanics, Keelung, Taiwan, ROC, 887- 894 (1992).

2•T. C. Yang, C. H. Tseng and S. F. Ling, "Constrained Optimization of

Active Noise Control Systems in Enclosures," J. Acoust. Soc. Am. 95, 3390-3399 (1994).

22C. H. Tseng and J. S. Arora, "On Implementation of Computational Al-

gorithms for Optimal Design, Part I: Preliminary Investigation," Int. J. Num. Methods Eng. 26, 1365-1384 (1988).

23 C. H. Tseng and J. S. Arora, "On Implementation of Computational Al-

gorithms for Optimal Design, part II: Extensive Numerical Investigation," Int. J. Num. Methods Eng. 26, 1385-1402 (1988).

24C. U. Tseng and W. C. Liao, "Integrated Software for Multifunctional

Optimization," Tech. Report AODL-90-01, Dept. of Mechanical Eng., Na- tional Chiao Tung Univ., Taiwan, ROC (1990).

25 B. N. Pshenichny, "Algorithms for the General Problem of Mathematical Programming," Kibemetica, No. 5 (1978).

26 A.D. Belegundu and J. S. Arora, "A Recursive Quadratic Programming

Algorithm with Active Set Strategy for Optimal Design," Int. J. Num. Methods Eng. 20, 803-816 (1984).

27 M. C. Junger and D. Feit, Sound, Structures, and Their Interaction (MIT,

Cambridge, 1986).

28M. R. Bai, "Study of Acoustic Resonance in Enclosures Using Eigenan- alysis Based on Boundary Element Methods," J. Acoust. Soc. Am. 91,

2529-2538 (1992).

29p. A. Nelson and S. J. Elliott, Active Noise Control of Sound (Academic,

London, 1992).