# Structural-acoustic optimization of a rectangular plate: A tabu search approach

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ments. A numerical model of FEA was used to construct the transfer

function matrices, which related the primary (an acoustic source)

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for a wide range of optimization categories without considering

Contents lists available at SciVerse ScienceDirect

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Finite Elements in An

Finite Elements in Analysis and Design 50 (2012) 142146problems, is its slow rate of convergence when a very accurateMranjbar@pardisiau.ac.ir (M. Ranjbar).strict assumptions [10]. Also the meta-heuristic methods can avoidlocal traps.

Chiou [10] and, also, Mashinchi et al. [9] reported that thecommon drawback of TS method for global continues optimization

0168-874X/$ - see front matter & 2011 Elsevier B.V. All rights reserved.

doi:10.1016/j.nel.2011.09.005

n Corresponding author. Tel.: 989354503598; fax: 982176281010.E-mail addresses: mostafa_ranjbar1380@yahoo.com,available nite element analysis (FEA) code, ANSYS, for active soundcontrol around a uid-loaded plate by multiple piezoelectric ele-

be addressed [9]. However, meta-heuristic approaches do notguarantee nding global optimum solutions, but they can be usedpromising area of the solution space to nd the optimum design.It has been observed that tabu search is more likely to nd theglobal minimum than many other methods.

Khan et al. [3] presented a method, which used a least-squaresapproach and tabu search technique together with the commercially

applied to multimodal test functions [7,8].Tabu search as a meta-heuristic method does not need to s

strict requirements of differentiability. It is advantageous as ireal world applications, we usually end up with optimizationnon-differentiable system as an example signal setting problemGlovers basic approach. In order to cover a wide domain ofpossible solutions, this algorithm rst performs the diversica-tion: it locates the most promising areas by tting size of theneighborhood structure to the objective function and its deni-tion domain. When the most promising areas are located, thealgorithm continues the search by intensication within one

with respect to passive design parameters, such as material con-stants, geometric parameters, cellular core geometry and boundaryconditions.

Hajela [6] reviewed heuristic methods including tabu (or: taboo)search, design classier systems, and a hybrid method that combinedan expert system and numerical optimization. Also, tabu search was1. Introduction

Tabu search (TS) was originallsuccessfully applied to a variety oproblems [1]. However, very few won the global minimization of functivariables. An algorithm called enhaby Chelouah and Siarry [2] was promulti-minima functions. It resultednatorial tabu search, which aimed tloped by Glover andinatorial optimizationeal with its applicationpending on continuousontinuous tabu searchfor the optimization ofn adaptation of combi-w, as close as possible,

and the secondary (piezoelectric elements) sources to the acousticpressures in the eld of interest with consideration of environmen-tal effects. With the transfer function matrices, the least-squaresalgorithm and tabu search technique were applied for subsequentactive sound control.

Denli and Sun [4,5] reviewed recent advances in the area ofcomposite sandwich modeling, sensitivity analyses, optimizationtechniques and applications, with the focus on structural acousticproblems. The optimization of sandwich structures was studiedStructural-acoustic optimization of a reA tabu search approach

Mostafa Ranjbar a,n, Steffen Marburg b, Hans-Jurgena Department of Mechanical Engineering, Pardis Branch, Islamic Azad University, Pardib LRT4, Institut fur Mechanik, Universitat der Bundeswehr Munchen, 85577 Neubiberg,c Institut fur Festkorpermechanik Technische Universitat Dresden, 01062 Dresden, Germ

a r t i c l e i n f o

Article history:

Received 19 November 2010

Received in revised form

6 September 2011

Accepted 7 September 2011Available online 28 September 2011

Keywords:

Structural acoustic optimization

Sound power level

a b s t r a c t

In this paper, the applic

introduced. A shape modi

of the nite element mode

of structure borne sound f

with other previously publ

for using it in structural a

journal homepage: wwwangular plate:

ardtke c

an

many

n of tabu search method for optimization in structural acoustics is

tion concept is used to change the vibrational and acoustical properties

bjective of optimization process is to minimize the root mean square level

quadratic plate made of steel. The comparison of results from this study

ed works shows that tabu search method can give a suitable performance

stic optimization applications.

& 2011 Elsevier B.V. All rights reserved.

lsevier.com/locate/finel

alysis and Design

solution is required. Indeed, TS can nd a promising local minimumarea but getting to the bottom of the valley is a time consumingprocess. To overcome this issue, meta-heuristic approaches aremostly hybridized with Local Optimizer Methods (LOM).

A general concept for design modication of shell meshesin structural-acoustic optimization was presented by Marburg[11,12]. Also, topology optimization of nonlinear structures wasinvestigated by Jung and Gea [13]. Furthermore, in the paper byEl-Sabbagh et al. [14], a topology optimization approach is used tomaximize the fundamental natural frequency of Mindlin plateswhile enforcing periodicity.

After careful review of previous works in the eld of structuralacoustic optimization, it is experienced that there is a lack of studieson the application of TS optimization method in structural acousticswhen a geometry modication approach is being used. Anotherscope of this study is to perform a short comparison between tabusearch method and other optimization methods with respect to astructural acoustics application. The results, which are presented inthis work stem from Ranjbar [15]. This work contains more detailsand comparison between additional methods, which are not dis-

interaction would set up a very similar optimization problem sincethe uid does not add any addition resonance peaks and, a light uidassumed, adds weak damping only. Usage of ERP has been commonin structural acoustic optimization, see for example Bos [21].

The LS constitutes a spectrum, i.e. it is a function of circularfrequency o. To obtain some single global measure of thevibrational behavior of a structure in a given frequency range ofinterest, the root mean square level of structure borne sound overthat frequency band, known hereafter as RMSL, is calculated as

RMSLRomaxomin LS

2odoomaxomin

sdB 1

In Eq. (1), omax and omin are the lower and upper bounds ofthe circular frequency range under consideration, respectively.The RMSL is the root mean square level of the area beneath thelevel of structure borne sound spectrum divided by the width ofthe frequency band and can be computed numerically. In thispaper, the RMSL is considered as the objective function to beminimized.

r pla

M. Ranjbar et al. / Finite Elements in Analysis and Design 50 (2012) 142146 143cussed in detail in this paper.

2. Optimization procedure

A reliable measure for the noise emitted from the structure ormachine part is the level of radiated sound power or sound powerlevel, which is a function of circular frequency [16]. In this paper,only structure borne sound is considered. The level of structureborne sound, which is from now on referred to as LS for convenience,can be interpreted as a measure of the vibrational sensitivity of astructure when subjected to some excitation.

In the review paper [17], it was concluded that, in particular forexterior acoustic problems, solution of the uid part of the structuralacoustic problem appears as the bottle neck for optimization since itis very time consuming. This remark holds for uid structureinteraction problems as well as for one way coupled sequentialevaluations, i.e. the structure excites the uid but the uid does notact back on the structure. Several methods are known to circumventsolution of the acoustic boundary value problem, see for examplethe comparison in [20]. Based on these results and owing to itssimplicity and efciency, the radiated sound power is approximatedby a quantity, which is known as equivalent radiated sound power(ERP), i.e. the radiation efciency is set to 1.0. Clearly, this is not arealistic assumption but, qualitatively, a full uid structure

Fig. 1. Modeling of the rectangular plate. (A) 3-Dimensinal model of the rectangula

plate along with 400 nite elements and nine design keypoints (showed with symThe structure to be optimized in this paper is a square platemade of steel. At least, the initial design is a plate. After modica-tion, the plate changes into a shallow shell. There is no couplingbetween the plate and the uid, which might be air herein. Thedamping is assumed to be independent of frequency with a constantdamping coefcient of 0.3%. The concept of a geometry basedmodeling technique is used; cf. Refs. [11,22]. The commercial niteelement code Ansys is used to build the FEmodel and to perform theFE analysis. Furthermore, the plate is modeled as bafed and a half-space acoustical radiation problem is considered. However, asmentioned above, the equivalent radiated power is just an approx-imation, which does not consider the uid in detail.

The example for which the optimization is demonstrated ischosen arbitrarily but coincides with the example in Ref. [18],i.e. it is based on a square plate. Fig. 1 depicts the modeling of theplate. Its geometry is dened by 9 design (key) points, which areconsidered over the surface of a square with 1 m edge length; seeFig. 1(A). These points are connected by lines, which are actuallycreated by Spline interpolation. Finally, the plate is composed of 16,later on, non-planar areas. Each area is meshed by 55 quadrilat-erals, eight node Serendipity shell elements, i.e. the plates meshconsists of 400 nite elements, see Fig. 1(B). In ANSYS, the shellelements are called shell93 [19]. The plate is simply supported. It is1 mm thick. Excitation is applied by local harmonic pressure loads.There are three uniform harmonic pressure excitations on the surface

te with nine design key-points. (B) Excitation (hatched) areas on the surface of thebols).

of the plate, which are shown by hatched areas in Fig. 1(B). All ofthem act at the same amplitude and phase. An excitation frequencyrange of 0100 Hz is considered. The excitation pressures act at thelocations where presumably all relevant mode shapes of the struc-ture in the frequency range of interest are excited.

Herein, the optimization problem is dened as follows:

FW RMSLW-min 2

while the design variables Wk remain within a prescribed intervalof lower and upper modication values as

10mmrWkr10mm 3

Eq. (3) denes the design space just as a nine-dimensionalcube since all parameters are allowed to take values within thesame xed interval and all parameters are independent of eachother. There are no additional equality and inequality constraints.

As mentioned above, the design variables, Wkk 1,2,. . .,9, arethe vertical positions of specic points, in other words, the normalgeometry modications at these movable points. If the shape ofthe surface is varied by means of a spline function, then thepositions of the spline key-points are the design variables.

The optimization method basically works as follows: build aninitial approximation based on a given sample set, use an optimiza-tion method to nd the minimum of the objective function to get anew design, evaluate the new design with the full analysis code, usethe new results to improve the accuracy of the optimum designsand continue until termination criterion is met. The main termina-tion criterion used in this study is the maximum number ofobjective function evaluations, which is set to 500.

The parameter setting used in TS program is similar to whatwas considered in Ref. [2] by Chelouah and Siarry. The dimensionof tabu list is considered as ve. The other required parameters in

the computer program are being set with respect to dimension ofobjective function, automatically.

3. Optimization results

The optimization should not be started from the at plate,i.e. all design variables are equal to zero. Therefore, ten randomlyselected designs are compared and the design with the lowest RMSLvalue is used as the initial design. The initial design improves theobjective function by 6.6 dB. Also, other indicators of a better designshow improvements, i.e. the maximum sound power level isdecreased by more than 11 dB and the fundamental frequencyincreases from 4.9 to 21.8 Hz. Both designs are compared with theoptimum design in Table 1. The geometry of the initial design isshown in Fig. 2. In both the initial design and the optimum design,the maximum geometry modications are greater than 10 mm. In atechnical sense, this is in contradiction to the aim of our optimiza-tion. However, the design variables remain within the limits of710 mm. These larger modications result from the spline approx-imation between the supports. Herein, they should be accepted toavoid further and more difcult constraint conditions.

Fig. 3 shows the reduction of sound power level for tabu searchmethod. The fundamental frequency of the modied model isincreased to 31.2 Hz. The maximum level of structure borne sound

M. Ranjbar et al. / Finite Elements in Analysis and Design 50 (2012) 142146144Table 1Optimization results for tabu search method after maximum 500 function

evaluations.

Property Original at

plate

Initial

design

Optimized

design

RMSL (dB) 49.3 42.7 28.6

Minimum geometry

modication (mm)

0 7.58 10.54

Maximum geometry

modication (mm)

0 10.95 11.16

Fundamental frequency (Hz) 4.9 21.8 31.2

Maximum LS (dB) 80.6 69.1 55.0Fig. 2. Initial design for the rectangular plate. (A) 2-Dimensinal view of moFig. 3. LS spectra of the original and the optimized rectangular plate by TS.del (values in mm). (B) 3-Dimensional view of model (values in mm).

sina

M. Ranjbar et al. / Finite Elements in Analysis and Design 50 (2012) 142146 145Fig. 4. Geometry distribution of the modied rectangular plate by TS. (A) 2-Dimenof the optimized structure is decreased to 55 dB at the newfundamental frequency. Interestingly, the number of natural fre-quencies within the frequency range of interest is reduced to three.

The contour plot of the plates optimized geometry in 2-and 3-dimensional views are shown in Fig. 4, respectively. Theoptimized geometry distribution can also be interpreted as astiffening rib across the diagonal of the plate, which efcientlysuppresses vibrations.

Fig. 5 shows the iteration history for tabu search methods after500 objective function evaluations. Similar to the comparison ofapproximative methods in Ref. [18], the iteration has been stoppedafter 500 evaluations of the opjective function. It is shown that theTS algorithm has been able to reduce the value of objective functionup to 14 dB. However, it is...

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