Finite Elements in Analysis and Design 50 (2012) 142–146

Contents lists available at SciVerse ScienceDirect

Finite Elements in Analysis and Design

0168-87

doi:10.1

n Corr

E-m

Mranjba

journal homepage: www.elsevier.com/locate/finel

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

Mostafa Ranjbar a,n, Steffen Marburg b, Hans-Jurgen Hardtke c

a Department of Mechanical Engineering, Pardis Branch, Islamic Azad University, Pardis, Iranb LRT4, Institut fur Mechanik, Universitat der Bundeswehr Munchen, 85577 Neubiberg, Germanyc Institut fur Festkorpermechanik Technische Universitat Dresden, 01062 Dresden, Germany

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

Tabu search

4X/$ - see front matter & 2011 Elsevier B.V.

016/j.finel.2011.09.005

esponding author. Tel.: þ989354503598; fax

ail addresses: mostafa_ranjbar1380@yahoo.co

r@pardisiau.ac.ir (M. Ranjbar).

a b s t r a c t

In this paper, the application of tabu search method for optimization in structural acoustics is

introduced. A shape modification concept is used to change the vibrational and acoustical properties

of the finite element model. Objective of optimization process is to minimize the root mean square level

of structure borne sound for a quadratic plate made of steel. The comparison of results from this study

with other previously published works shows that tabu search method can give a suitable performance

for using it in structural acoustic optimization applications.

& 2011 Elsevier B.V. All rights reserved.

1. Introduction

Tabu search (TS) was originally developed by Glover andsuccessfully applied to a variety of combinatorial optimizationproblems [1]. However, very few works deal with its applicationon the global minimization of functions depending on continuousvariables. An algorithm called enhanced continuous tabu searchby Chelouah and Siarry [2] was proposed for the optimization ofmulti-minima functions. It resulted from an adaptation of combi-natorial tabu search, which aimed to follow, as close as possible,Glover’s basic approach. In order to cover a wide domain ofpossible solutions, this algorithm first performs the diversifica-tion: it locates the most promising areas by fitting size of theneighborhood structure to the objective function and its defini-tion domain. When the most promising areas are located, thealgorithm continues the search by intensification within onepromising area of the solution space to find the optimum design.It has been observed that tabu search is more likely to find 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 commerciallyavailable finite element analysis (FEA) code, ANSYS, for active soundcontrol around a fluid-loaded plate by multiple piezoelectric ele-ments. A numerical model of FEA was used to construct the transfer

All rights reserved.

: þ982176281010.

m,

function matrices, which related the primary (an acoustic source)and the secondary (piezoelectric elements) sources to the acousticpressures in the field 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 studiedwith 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 classifier systems, and a hybrid method that combinedan expert system and numerical optimization. Also, tabu search wasapplied to multimodal test functions [7,8].

Tabu search as a meta-heuristic method does not need to satisfystrict requirements of differentiability. It is advantageous as in thereal world applications, we usually end up with optimization of anon-differentiable system as an example signal setting problem canbe addressed [9]. However, meta-heuristic approaches do notguarantee finding global optimum solutions, but they can be usedfor a wide range of optimization categories without consideringstrict 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 optimizationproblems, is its slow rate of convergence when a very accurate

M. Ranjbar et al. / Finite Elements in Analysis and Design 50 (2012) 142–146 143

solution is required. Indeed, TS can find 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 modification 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 field of structuralacoustic optimization, it is experienced that there is a lack of studieson the application of TS optimization method in structural acousticswhen a geometry modification 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-cussed 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 fluid part of the structuralacoustic problem appears as the bottle neck for optimization since itis very time consuming. This remark holds for fluid structureinteraction problems as well as for one way coupled sequentialevaluations, i.e. the structure excites the fluid but the fluid 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 efficiency, the radiated sound power is approximatedby a quantity, which is known as equivalent radiated sound power(ERP), i.e. the radiation efficiency is set to 1.0. Clearly, this is not arealistic assumption but, qualitatively, a full fluid structure

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

plate along with 400 finite elements and nine design keypoints (showed with ‘‘�’’ sym

interaction would set up a very similar optimization problem sincethe fluid does not add any addition resonance peaks and, a light fluidassumed, 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

RMSL¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRomax

ominLS2ðoÞdo

omax�omin

sdB ð1Þ

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.

The structure to be optimized in this paper is a square platemade of steel. At least, the initial design is a plate. After modifica-tion, the plate changes into a shallow shell. There is no couplingbetween the plate and the fluid, which might be air herein. Thedamping is assumed to be independent of frequency with a constantdamping coefficient of 0.3%. The concept of a geometry basedmodeling technique is used; cf. Refs. [11,22]. The commercial finiteelement code Ansys is used to build the FE model and to perform theFE analysis. Furthermore, the plate is modeled as baffled 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 fluid 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 defined 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 5�5 quadrilat-erals, eight node Serendipity shell elements, i.e. the plate’s meshconsists of 400 finite 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 the

bols).

M. Ranjbar et al. / Finite Elements in Analysis and Design 50 (2012) 142–146144

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 0–100 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 defined as follows:

FðWÞ ¼ RMSLðWÞ-min ð2Þ

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

�10mmrWkr10mm ð3Þ

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

As mentioned above, the design variables, Wkðk¼ 1,2,. . .,9Þ, arethe vertical positions of specific points, in other words, the normalgeometry modifications 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 find 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 five. The other required parameters in

Table 1Optimization results for tabu search method after maximum 500 function

evaluations.

Property Original flat

plate

Initial

design

Optimized

design

RMSL (dB) 49.3 42.7 28.6

Minimum geometry

modification (mm)

0 �7.58 �10.54

Maximum geometry

modification (mm)

0 10.95 11.16

Fundamental frequency (Hz) 4.9 21.8 31.2

Maximum LS (dB) 80.6 69.1 55.0

Fig. 2. Initial design for the rectangular plate. (A) 2-Dimensinal view of mo

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

3. Optimization results

The optimization should not be started from the flat 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 modifications 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 modifications result from the spline approx-imation between the supports. Herein, they should be accepted toavoid further and more difficult constraint conditions.

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

del (values in mm). (B) 3-Dimensional view of model (values in mm).

Fig. 3. LS spectra of the original and the optimized rectangular plate by TS.

Fig. 4. Geometry distribution of the modified rectangular plate by TS. (A) 2-Dimensinal view of model (values in mm). (B) 3-Dimensional view of model (values in mm).

Fig. 5. Minimization of RMSL for a rectangular plate made of steel by tabu search

method.

M. Ranjbar et al. / Finite Elements in Analysis and Design 50 (2012) 142–146 145

of 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 plate’s 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 efficientlysuppresses 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 visible that the most improvement(around 9.5 dB) is achieved after just 150 function evaluations andthen 350 function evaluations are spent to gain the more 4.5 dB.

4. Conclusions

It has been shown that the optimization procedure used forthis study is able to reduce significantly the value of RMSL for theplate. The performance of TS is comparable with the methods,which were considered in Ref. [18]. However, this method shouldbe more investigated by considering of more initial design setsand different termination criteria and internal parameter settings.

Considering the results published in Refs. [12,14] and theiteration history, which is presented in Fig. 5 in this study, whena same initial set of design variables is being used, then, it can beexpected that tabu search method performs relatively faster thansome previously investigated methods such as sequential quad-ratic programming, limited memory Broyden–Fletcher–Goldfarb–Shanno for bound constrained optimization, simulated annealingand Newton method to reduce the value of objective function,cf. [18]. However, it should be also indicated that tabu search hasbe slower than other methods discussed therein. These have beenthe mid-range multi-points method and method of movingasymptotes. Clearly, the setting and the implementation of thesemethods have a great influence on the final optimization results.

The nature of TS method is to explore the entire design space(diversification) and then to focus on most promising areas to findthe best designs (intensification). This overwhelming process makesthe TS method in general a slow optimization method. However, it isshown that the most improvement in the value of objective functionmay occur in the first stages of objective function calculations.In this paper, an interesting reduction of 67% in the value ofobjective function is occurred just after 30% of total number offunction evaluations.

Although tabu search is generally a slow optimization methodbut this method, can cover a wide solution space and decrease thepossibility of trapping in local minima. Therefore, usage of thisadvantage in the development of effective hybrid optimizationalgorithm is useful. In this regard, Mashinchi et al. [9] proposed amethod based on tabu search and Nelder-Mead search strategy inapplication to global continuous optimization problems. The resultsshowed better performance especially for the objective functionswith maximum four design variables.

The main part of computation time in optimization processhas been consumed by finite element analysis part of optimiza-tion procedure for the calculation of surface nodal velocitiesof the FE model. In this regard, it is possible to carry out thecomputations in parallel when using the optimization algorithm.Furthermore, development and implementation of fast analysismethods for the calculation of objective function remains anessential issue.

Acknowledgments

The SGI Altix 3700-Merkur of the Zentrum fur Informations-dienste und Hochleistungsrechnen (ZIH) of the Technische Uni-versitat Dresden was used to carry out the computational work.

M. Ranjbar et al. / Finite Elements in Analysis and Design 50 (2012) 142–146146

References

[1] F. Glover, Tabu search: part I, ORSA J. Comput. 110 (1989) 190–206.[2] R. Chelouah, P. Siarry, Tabu search applied to global optimization, Eur. J. Oper.

Res. 123 (2) (2000) 256–270.[3] M.S. Khan, C. Cai, K.C. Hung, Vijay K. Varadan, Active control of sound around

a fluid-loaded plate using multiple piezoelectric elements, J. Smart Mater.Struct. 13 (2002) 314–346.

[4] H. Denli, J.Q. Sun, Structural-acoustic optimization of composite sandwichstructures: a review, Shock Vib. Dig. 39 (2007) 189–200.

[5] H. Denli, J.Q. Sun, Structural-acoustic optimization of sandwich cylindricalshells for minimum interior sound transmission, J. Sound Vib. 316 (2008)32–49.

[6] P. Hajela, Nongradient methods in multidisciplinary design optimization—

status and potential, J. Aircr. 36 (1) (1999) 255–265.[7] R. Battiti, G. Tecchiolli, The continuous reactive tabu search: blending

combinatorial optimization and stochastic search for global optimization,Ann. Oper. Res. 63 (1996) 153–188.

[8] D. Cvijovic, J. Klinowski, Taboo search: an approach to the multiple minimaproblem, Science 667 (1995) 664–666.

[9] M.H. Mashinchi, M.A. Orgun, W. Pedrycz, Hybrid optimization with improvedtabu search, Appl. Soft Comput. 11 (2) (2011) 1993–2006.

[10] S.-W. Chiou, Optimization for signal setting problems using non-smoothtechniques, Inf. Sci. 179–17 (2009) 2985–2996.

[11] St. Marburg, A general concept for design modification of shell meshes instructural-acoustic optimization, Part I: Formulation Concept, Finite Elem.Anal. Des. 38 (8) (2002) 725–735.

[12] St. Marburg, H.-J. Hardtke, A general concept for design modification of shellmeshes in structural-acoustic optimization, Part II: Appl. Vehicle Floor Panel,Finite Elem. Anal. Des. 38 (8) (2002) 737–754.

[13] D. Jung, H.C. Gea, Topology optimization of nonlinear structures, Finite Elem.Anal. Des. 40 (11) (2004) 1417–1427.

[14] A. El-Sabbagh, W. Akl, A. Baz, Topology optimization of periodic Mindlinplates, Finite Elem. Anal. Des. 44 (2008) 439–449.

[15] M. Ranjbar, A Comparative Study on Optimization in Structural Acoustics,Ph.D. Thesis, Department of Mechanical Engineering, Technische UniversitatDresden, Germany, 2010.

[16] G.H. Koopmann, J.B. Fahnline, Designing Quiet Structures: A Sound PowerMinimization Approach, Academic Press, San Diego, London, 1997.

[17] St. Marburg, Developments in structural-acoustic optimization for passivenoise control, Arch. Comput. Methods Eng. State Art Rev. 9 (2002) 291–370.

[18] M. Ranjbar, H.-J. Hardtke, D. Fritze, St. Marburg, Finding the best designwithin limited time: a comparative case study on methods for optimizationin structural acoustics, J. Comput. Acoust. 18 (2010) 149–164.

[19] ANSYSs Academic Research, Release 11.0, Help System, ANSYS, Inc.[20] D. Fritze, S. Marburg, H.-J. Hardtke, Estimation of radiated sound power: a

case study on common approximation methods, Acta Acustica UnitedAcustica 95 (2009) 833–842.

[21] J. Bos, Numerical optimization of the thickness distribution of three-dimen-sional structures with respect to their structural acoustic properties, Struct.Multidiscip. Optim. 32 (2006) 12–30.

[22] St. Marburg, Efficient optimization of a noise transfer function by modifica-tion of a shell structure geometry, Part I: Theory Struct. Multidiscip. Optim.24 (2002) 51–59.