Materials and Design 52 (2013) 207–213

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Materials and Design

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A study on the application of material selection optimization approachfor structural-acoustic optimization q

0261-3069/$ - see front matter � 2013 The Authors. Published by Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.matdes.2013.05.031

q This is an open-access article distributed under the terms of the CreativeCommons Attribution-NonCommercial-No Derivative Works License, which per-mits non-commercial use, distribution, and reproduction in any medium, providedthe original author and source are credited.⇑ Corresponding author. Tel./fax: +86 21 34207165.

E-mail address: cluyun@sjtu.edu.cn (L. Chen).

Luyun Chen ⇑, Yufang ZhangState Key Laboratory of Ocean Engineering, Shanghai Jiaotong University, Shanghai 200240, China

a r t i c l e i n f o

Article history:Received 20 March 2013Accepted 12 May 2013Available online 24 May 2013

Keywords:Material selection optimization (MSO)Laminated compositeStacking sequenceStructural-acoustic radiation optimization

a b s t r a c t

Issues of application of the material selection optimization approach for structural-acoustic optimizationis investigated herein. By introducing the stacking sequence hypothesis of metal material, the mechanicalproperties parameters and plies’ numbers of the metal material or composite material are defined asdesign variables; the mathematical formulation about material selection optimization approach is estab-lished. Finally, a hexahedral box structure is taken as an example, and the material selection optimizationis conducted. By introducing genetic algorithm (GA), the optimization problem is solved. The numericalexample shows the effectiveness of the proposed stacking sequence hypothesis of metal material.

� 2013 The Authors. Published by Elsevier Ltd. All rights reserved.

1. Introduction

Noise and vibration control becomes more and more importantin engineering design and manufacturing. The optimization analy-sis of acoustic radiation properties is considered with the objectiveof minimizing the total acoustic power radiated from the vibratingstructure surface into a surrounding acoustic medium. On theother hand, many new materials are used in dynamic environment,vibration control and noise reduction which have great technicalsignificance. The material properties and material selection meth-od are very important in product design and manufacturing. Com-posite materials begin to play important roles in vibration andnoise reduction in recent years. Meanwhile, a lot of researcheshave been conducted on composite material structure for acousticradiation problem. The discrete material optimization formulationhas been applied to achieve the design optimization of fiber orien-tation angles, plies number and material selection of for compositelaminated plates [1]. The optimization study on cylindrical sand-wich shell to minimize the transmitted sound which is into theinterior induced by the exterior acoustic excitation was analyzed[2]. By introducing solid isotropic material with penalization(SIMP) model, the topology optimization of laminated compositestructures for the minimization of the acoustic power radiationhas been studied [3]. The acoustic radiation power is defined as

the objective function, in which the influence of fiber orientationsangle and plies number were compared [4]. To reduce the acousticpressure in the acoustic domain, the thickness distribution optimi-zation problem of a multilayered structure was analyzed, in which,continuous approximation of thickness distribution is assumed [5].To minimize the acoustic radiation which transmits into the inte-rior induced by the exterior acoustic excitations, the thickness ofskins and core were defined as the design variables, and the sand-wich structure optimization has been studied [6].

The structural-acoustic optimization approach includes sizingoptimization, shape optimization, topology optimization, and soon. With the development of technology, material selection meth-od becomes a new optimization tool in manufacturing process andlife cycle. In structure design, the right selection of available mate-rial is critical for the success and competitiveness of the manufac-turing organization. Main superiority of reliable materials selectionis to take advantage of best mechanical behavior of each materialcandidate of a structure under given load and boundary conditions.The material selection optimization (MSO) is the method formaterial conversion in optimization process, which includesconversion between composite material and metal materials; con-version between different kinds of composite material; conversionbetween different kinds of metal material, and so on. Numerousstudies have been published on this topic during the last decade.This MSO problem is solved by using the so-called discrete mate-rial optimization approach, in which the structure constituentsare chosen among a given set of different candidate materials [7].Different glass/carbon ratios and stacking sequences of a staticloading problem were investigated in a hybrid composite lami-nated structure [8]. An integrated approach was used to optimize

208 L. Chen, Y. Zhang / Materials and Design 52 (2013) 207–213

the multi-objectives problem by material selection, concerningmaterial characteristics and sustainable strategies [9]. With adesign index, the discrete problem in material selection can bedefined as a continuous optimization problem, and the gradient-based optimization algorithm is used in numerical analysis [10].A novel multi-material selection method for lightweight design isdiscussed, which incorporates recyclability for an automotive bodyassembly [11]. The material section optimization issue wasanalyzed in the car’s structure design; minimization weight ofthe entire bottom structure defining as objective function, con-straints including stiffness, strength and buckling parameter [12].The environmental impact was considered as a constraint, theMSO approach were applied a pressure vessel design, in which,four pressure vessel steels and three aluminum alloys can be se-lected [13]. In the automotive design, it is widely accepted thatusing material selection theory can reduced weight withoutincreasing cost [14]. The acoustic pressure on a prescribed refer-ence plane/domain in the acoustic field that is generated by thevibrating structure is minimized by two-phase damping materialdistribution optimization over the structure domain [15]. Materialsselection optimization formulations for hybrid steel-compositewith 0–1 topological design variables were presented, and themethod of continuation with variables and mapping functiontransformation was discussed in optimization formulations [16].A mapping transformation function approach is presented toconvert the mixed variable formulation to the one with onlycontinuous variables, and the vibration problem of a hybridsteel-composite floating raft is discussed for example [17]. Themulti-objective optimization on the basis of ratio analysis(MOORA) method was applied to solve common material selectionproblems, in which, three mathematical approaches are applied[18].

The objective of present study is the minimization of structural-acoustic radiated power from a hexahedral box structure underbroadband excitation by using material selection optimizationapproach. Based on the structural optimization theory and metalstacking sequence hypothesis, the acoustic radiation power is ta-ken as the optimization objective function, the material mechani-cal properties (density, Young’s modulus, shear modulus, lossfactor, etc.) and plies number of hybrid metal-composite structureare taken as design variables, the genetic algorithm (GA) is em-ployed to solve the relationship between acoustic radiation perfor-mances and design variables, the materials selection optimizationfunction is analyzed in present paper.

This paper is organized as follows: a brief literature review ofstructural-acoustic radiation and material selection optimizationis given in Section 1; the basic structural-acoustic radiation formu-lation is reviewed in Section 2; the basic concept of composite lam-inated structure is performed in Section 3; the metal materialstacking sequence hypothesis is introduced in Section 4; the mate-rial selection optimization mathematical formulation for acousticoptimization is established in Section 5; the numerical result andthe comparison of material selection optimization are presentedin Section 6; finally, the conclusion is drawn in Section 7.

2. Structural-acoustic radiation formulations

In this section, structural-acoustic radiation formulations arediscussed. Finite element method (FEM) is used to obtain the struc-tural frequency response analysis and boundary element method(BEM) is applied to deal with the exterior acoustic radiation prob-lem. Taking the structural response (harmonic normal velocity) asboundary condition, the boundary element method is used to cal-culate the acoustic radiation behavior (acoustic pressure andacoustic radiation power).

2.1. Structural vibration formulations

The structural vibration analysis is the premise and basis ofstructure optimization analysis. Considering a continuum struc-ture with an external harmonic loading F(x, t), it is assumed thatthe material damping can be regarded as a proportional dampingand the fluid–structural coupling is weak coupling that can beneglected, especially for air and wide open spaces. In the structuredomain XS, the differential equation governs the behavior of thisstructural dynamic system can be expressed as:

½M�f€Ug þ ½C�f€Ug þ ½K�fUg ¼ Fðx; tÞ; x 2 XS; t > 0: ð1Þ

where XS indicates the structural domain, {U} indicates the nodaldisplacement vector matrix, [M] is the structural mass matrix, [K]is the structural stiffness matrix, and [C] is the viscous damping ma-trix. Considering the excitation force F(x, t) is a harmonic timedependence load, it can be expressed as:

Fðx; tÞ ¼ f ðxÞeixt ; ð2Þ

where f(x) is the magnitude of the harmonic load, x the circularfrequency, and it is considered as a constant. Using the complexvariable method, the nodal displacement vector can be expressedas: Uðx; tÞ ¼ uðxÞeixt , where, [u(x)] is a column matrix of the nodaldisplacement vectors, and i ¼

ffiffiffiffiffiffiffi�1p

.Substituting the displacement vector equation and harmonic

loading equation into Eq. (1), and there is obtaining the spatialstate operator equation:

f�x2½M� þ ix½C� þ ½K�g½uðxÞ� ¼ f ðxÞ: ð3Þ

In addition, the frequency response equation can be written asshorthand ½AðxÞ�½uðxÞ� ¼ f ðxÞ, where ½AðxÞ� ¼ �x2½M� þ ix½C�þ½K�, and the nodal displacement vector matrix ½uðxÞ� ¼½AðxÞ��1f ðxÞ.

Defining the nodal velocity vector v(x), it can be expressed as:vðxÞ ¼ ixuðxÞ. At the interface between the structure and thefluid, the nodal particle normal velocity vector can be written as:

vnðxÞ ¼ ixNA�1ðxÞf ðxÞ: ð4Þ

where [N] is the nodal normal vector matrix; it associates to struc-tural surface shape. The nodal particle normal velocity is used asboundary condition in acoustic boundary element method analysis.

2.2. Acoustic power formulations

In solving the acoustic radiation problem, the boundary ele-ment method (BEM) has many advantages. It is unnecessary togenerate a complicated three-dimensional acoustic model. Onlythe low-middle frequency domains exterior acoustic radiationproblem of the continuum structure was analyzed present paper.The standard acoustic wave equation is reduced to the Helmholtzequation in the harmonic response problem. For an arbitrary shapestructure, the governing differential equation in steady-state linearacoustics is the classical Helmholtz equation as follows:

r2pþ k2p ¼ 0: ð5Þ

where p is the acoustic pressure of the acoustic field point, k(=x/c)denotes the wave number, x and c are the angular frequency andspeed of sound, respectively. r2 is the Laplace operator. The acous-tic wave has assumed harmonic time variations throughout withe�ixt dependence suppressed for simplicity.

At the interface of the structure-fluid X, the acoustic pressuremust satisfy the Neumann boundary condition: @p

@n ¼ �ixqvn,where vn is the nodal normal velocity of structure, q is the densityof fluid medium and n is the outer-normal units vector of thestructure surface. Moreover, the acoustic pressure p satisfies

ic ttic

2

planeMid-

o(90 )o(0 )1

1(0 )o

2

2(90 )o

3(0 )o

4(90 )o

5(0 )o

o(0 )3o(90 )4

o(0 )5

n

n

Fig. 1. Composite laminated with symmetrically stacking.

L. Chen, Y. Zhang / Materials and Design 52 (2013) 207–213 209

Sommerfeld condition at infinity: limr!1½rð@p@r � ikpÞ� ¼ 0: The

acoustic pressure at any position within the acoustic domain canbe gained from the Helmholtz integral equation. The solution hastwo steps: firstly evaluating the pressure variable on the acousticboundary by using the structural surface normal velocity, and thencalculating the pressure variable within the acoustic domain byusing the boundary pressure information.

2.3. Acoustic power formulations

It assumes that the air as the acoustic medium in this paper, andfeedback coupling between the acoustic medium and the structurecan be neglected. The acoustic radiation field quantities are rele-vant to acoustic radiation pressure and acoustic radiation power.The acoustic power reveals itself as the most adequate means forquantifying the radiation on the structure’s surface. It does notchange with spatial field position and is only related to the charac-teristics of structure vibration. Therefore, acoustic radiation poweris more suitable than acoustic pressure for structural-acousticanalysis and evaluation. In the acoustic field domain, the acousticradiation power describes the energy flow of an assumed integralsurface, and it can be defined as:

Y¼ 1

2

ZX

Re pð~rÞ � v�nð~rÞ� �

dX: ð6Þ

where Re () indicates the real part of a complex value, pð~rÞ is theacoustic pressure at the acoustic field point; v�nð~rÞ is the field pointnormal complex conjugate operator velocity of fluid-particle; X isthe integral surface of acoustic domain. Omitting the acoustic trans-mission loss and the acoustic absorption of the boundary andsource point, the acoustic radiation power of exterior acoustic fieldis equal to the source point surface acoustic radiation power. Thestructure surface radiation acoustic power can be written as:

Y¼ 1

2ReZ

Xpf v

�n dX: ð7Þ

where, X is the structure-fluid interface, pf is acoustic pressure onthe structural surface, v�n is the nodal normal complex conjugatevelocity on the structural surface. For the vibration structure, thereexists pf ¼ qcvn, and the phase angle between acoustic pressure pf

and the normal velocity v�n is 0.The acoustic power radiated from the structure surface is a

function of radiation frequency, which varies over the band. Then,the objective function is minimizing the radiated power over thisband in the acoustic radiation power formulation. This frequencyaveraged acoustic radiation power over this band can be obtainedby integrating G over the frequency band:

Ya

¼ 1x2 �x1

Z x2

x1

PðxiÞdx ffi 1x2 �x1

Xm

i¼1

PðxiÞDxi ð8Þ

where x2 and x1 are the upper and lower bounds of the frequencyband respectively. PðxiÞ is the acoustic radiation power in fre-quency xi.

3. Basic theory of composite laminated

In the following, a rectangular Cartesian coordinate system x, yand z is used to describe the loadings and global strains of a2n-layer laminated plate. A layerwise material coordinate system,denoted by 1, 2 and 3, is employed to analyze lamina failure. The 1-axis and 2-axis are aligned parallel and perpendicular to the fiberorientation direction in the x-y plane, respectively, and the 3-axisis aligned parallel to the global z-axis.

The infinitesimal deformation of thin orthotropic laminate isanalyzed using the classical lamination theory. In the composite

laminated design problem, such as laminates for various strengthand stiffness requirements, is how to select a required number ofplies for each orientation directions and determine an optimalstacking sequence. Generally, the thickness of each lamina is thesame and not varied during the stacking sequence optimization.The material stack sequence of composite laminated includes sym-metrical ply form, anti-symmetrical ply form and random ply form.For the most applications, the stack sequence is symmetrical plyform, and the stacking sequence should be balanced (same numberof +ti.c/2 plies than �ti.c/2 plies, with ti.c/2 has different from 0�,±45� or 90�) and symmetric about the mid-plane in order to avoidshear extension coupling.

There is a laminate having lamina oriented at +ti.c/2 to the lam-inate coordinate axes on one side of the mid-plane and corre-sponding equal-thickness lamina oriented at �ti.c/2 on the otherside. This paper presents stacks of plies symmetrically arrangedabout the mid-plane. Defining the symmetric composite materiallaminated with thickness variable ti.c, and it can be written as fol-lowing equations:

ti�c ¼ 2 � n � tc0 ð9Þ

where tc0 is the thickness of the lamina for the composite laminatedmaterial, n is the number of one side of the mid-plane. Fiber orien-tations angle had also to be chosen within a finite set of angles dueto manufacturing constraints. Since the stacking sequence is sym-metric, there are n distinct fiber orientation angle, and it definedas [h1/h2/. . ./hn�1/hn], where hn is the fiber orientation angle of theoutermost lamina and h1 is the innermost lamina below the mid-plane, as shown in Fig. 1. The composite laminated consists of a cer-tain number of plies lamina with a given thickness. In this study,there are n distinct fiber orientation angles, which defining as [0�/90�/0�/90�/0�. . .].

4. Hybrid metal-composite model

The metal material was extensively used in the machine frame-work design. However, the metal material has many shortcomings,such as high density, poor designable performance, low dampingfactor. Instead, composite material is extensively used in engineer-ing structures due to its excellent mechanical properties, such ashigh damping loss factor, low density, high special strength, highspecial stiffness, flexible capability in physical performance design.In order to take advantage of the two kinds materials, the hybridmetal-composite structure has become extensively used in themechanical design. In fact, it is essential of the material selectionoptimization design.

210 L. Chen, Y. Zhang / Materials and Design 52 (2013) 207–213

4.1. Hybrid metal-composite structure model

Hybrid structures consist of many components made of differ-ent materials. Material types in hybrid structures are assumed toconsist of metal material plies and composite material plies. Gen-erally speaking, the material selection optimization in hybridstructural design has three kinds of models about the differentchoices of design variables, i.e. (1) taking the elastic modulus ofmaterial as discrete design variable in the optimization model;(2) introducing cellular material theory, taking the thickness ofmetal structure or composite laminate structure as continuous de-sign variables in optimization model; and (3) introducing cellularmaterial theory, taking the topology distribution and thickness ofstructure as hybrid variable in optimization model. In presentstudy, the material elastic modulus and thickness (total numberof plies) of structure are taken as discrete design variables; thepower flow of structure is taken as the objective function.

The elastic modulus (including shear modulus) is the mainmaterial mechanical properties. In addition, Poisson’s ratio, lossfactor and density of the material are important for the materialmechanical properties.

In the process of material selection optimization, the metalmaterial and composite material can be defined as candidate mate-rials. In this paper, each of design variables are defined as a mate-rial cellular, the material will be arbitrary between metal andcomposite laminated. The elastic modulus of the metal is writtenas ES and the elastic modulus of the composite laminated is writtenas Ec. According to the cellular material theory, the structure mate-rial design variables can be expressed as Ei e {ES, EC}, if Ei = ES, thestructure will be metal; if Ei = EC, the structure will be compositelaminated.

4.2. Stacking sequence hypothesis of metal material

Material mechanical performance of metal, such as steel andaluminum, is treated as isotropic material. The stack sequence con-cept is not suitable for the metal material, and thickness is the onlyparameter for metal material structure. In order to use the materialselection optimization approach in which the material elastic mod-ulus is defined as design variable, the hypothesis of metal materialis shown as follows:

(1) The metal structure is defined as laminated with multi lam-ina metal layer. Shown in Fig. 1, there are 2n-plies laminasmetal with stack sequence form. Then, the metal plate thick-ness variable ti,m of the metal laminated is written as:

ti�m ¼ 2 � n � tm�0 ð10Þ

where tm.0 is the thickness of metal material lamina, n is the

half plies number of the symmetrical plies. In this paper, thethickness of metal material lamina is equal to thickness ofcomposite material lamina, namely: tc�0 ¼ tm�0:

(2) The composite material fiber orientation angle theory isapplied to the metal material structure, the stackingsequence angle of material lamina meet with [0�/90�/0�/90�/0�. . .] in the metal laminated plate.

With the stacking sequence hypothesis of metal material, thematerial selection optimization problem is formulated. The hybridstructural design problems are formulated as a concurrent thick-ness and material topology optimization problem with all materi-als considered for selection within one model. The optimizationmodel for material selection is established by defining the designvariables of materials topological distribution and the thickness,the design objective and constraints that capture the relationshipbetween them.

5. Material selection optimization formulations

Since the structural acoustic radiation power equation is de-fined, the structural-acoustic optimization problem can be con-verted into searching a group of design variables in design scopewhich minimize the structural-acoustic radiation power. Definingthe acoustic radiation power instead of acoustic pressure as objec-tive function has many advantages in structural-acoustic optimiza-tion. It turns the vector parameter analysis into scalar quantityparameter analysis; it does not need to optimize the pressure ofa defined field point. In this paper, the excitation frequency variesover a band which may include resonant frequencies of thestructure.

In the hybrid metal-composite structure material selectionoptimization, it needs to consider the influence of overall struc-tural properties of various indexes and constraints. Discontinuityin the properties of materials for the given materials’ choice setmakes material selection designs an inherent discrete optimizationproblem. The optimal design of hybrid structures is more compli-cated than conventional structure designs. The objective is to min-imize the acoustic radiated power over the frequency band, whichcan be obtained by integrating P over the frequency band. Then,the structural-acoustic optimization mathematical formulationbased on material selection optimization approach can be statedas:

Find E ¼ ½E1; E2; . . . ; Ek�T; i ¼ 1;2; . . . k; ð11Þ

N ¼ ½N1;N2; . . . ; Nk�T; i ¼ 1;2; . . . k;

MinY

a

¼ 1x2 �x1

Xm

i¼1

PðxiÞDxi;

s:t:Xk

i¼1

ðtim � qm � sim þ ticqc � sicÞ 6W0;

rmax � ½r� 6 0;

dmax � ½d� 6 0;

tlm 6 tim 6 tu

m;

tlc 6 tic 6 tu

c ;

Ei 2 fES; ECg;�xf ðbÞ þx0 6 0: ð12Þ

where a multi-objective function is derived from the list of criteriafor the best composite structure. ½E1; E2; . . . ; Ek�T is k dimensionalelastic modulus design variable vector. Ei 2 fES; ECg is the allowableelastic modulus for the materials set, in this case, steel and compos-ite material, respectively. ½N1;N2; . . . ;Nk�T is k dimensional designvariables of plies, which define the thickness parameters tim andtic, respectively.

Qa is the acoustic radiation power at the band fre-

quency, which is defined by Eq. (8). tim, qm and sim are the metalstructure plate thickness, material density and plate area, respec-tively. tic, qc and sic are the composite laminated structure platethickness, material density and plate area, respectively. W0 is thestructure mass in initial design. rmax is the maximum Von Misesdynamic stress, and [r] is corresponding to the allowable designstress. dmax is the maximum dynamic displacement, and [d] is cor-responding to the allowable linear displacement. tu

i and tli are corre-

sponding to the lower thickness bounds and upper thicknessbounds of the ith design variables ti, respectively. Parameter xf ðbÞis the fundamental frequency of the optimal design. Parameter x0

is the allowable minimum fundamental frequency (the fundamen-tal frequency in initial design). The above model is a discreteoptimization problem, the conventional optimization approachescannot be applied in solution search.

L. Chen, Y. Zhang / Materials and Design 52 (2013) 207–213 211

6. Numerical examples

To demonstrate the structural-acoustic optimization process, ahexahedral box structure which under external harmonic excita-tion in the air is presented. Both finite element method and bound-ary element method are used, and the genetic algorithm isemployed in the integrating optimization platform.

6.1. Model description

There is a hexahedral box, shown as Fig. 2. The design parame-ter of the hexahedral box structure is shown as follows: the heightis 0.8 m, the length of x direction is 0.6 m, and the length of y direc-tion is 0.5 m. In the FEM model of the hexahedral box structure, thezero point of overall rectangular Cartesian coordinate system is thecenter of the hexahedral box bottom.

The precisely discretized hexahedral box structure with finiteelement is shown in Fig. 2; there are 3296 quadrilateral elementsand 3341 nodes totally in the finite element model.

In the hexahedral box structure, there are five plate structuresand each plate is defined as a design variable, which is denotedwith D-ban, X-xiao, X-da, Y-xiao and Y-da (shown in Fig. 2) respec-tively. Moreover, in the initial design, thickness of design variableD-ban is 10 mm, the thickness of X-xiao, X-da, Y-xiao and Y-da are12 mm.

In material selection optimization of the hexahedral boxstructure, the structure material is steel in initial design, and thereare two kinds of materials for selection: steel material and carbonfiber reinforced plastics (CFRP) material. Moreover, the paste of thecomposite material and steel material structure is ideal connectionwithout any structure defects.

According to the metal material layers hypothesis, stackingsequence orientation is symmetrical, the thickness of compositelamina is 0.5 mm, and the thickness of metal lamina is 0.5 mmtoo. The mechanical properties of the steel material and carbon fi-ber reinforced plastics material are shown in Table 1.

6.2. Calculation parameter

The finite element method is used to analyze the structural dy-namic response of the hexahedral box structure. In this paper, thenumerical analysis process has two steps. The first step is materialselection optimization, then the material distribution and thick-ness is defined; the second step is fiber orientation angle optimiza-tion. The variation of radiated sound power in frequency rangesfrom 0 Hz to 400 Hz, and the frequency step size is 2 Hz.

Fig. 2. The FEM model of the hexahedral box.

In the hexahedral box structure dynamic response analysis, thecorner of hexahedral box is fixed supported. A time-harmonic con-centrated loading p(t) = P sin (2p ft) with prescribed amplitudePh�100,�100,�200iN and force frequency f = 121 Hz is appliedat the corner of the hexahedral box, shown in Fig. 2. The cornerof the hexahedral box structure is simply supported.

The acoustic properties of the fluid (i.e., air) are: densityq = 1.225 kg/m3; the speed of sound is c = 343 m/s; the acousticpower reference value is W0 = 1 � 10�12 Watt, the acoustic pres-sure reference value is P0 = 2 � 10�5 Pa.

6.3. Acoustic optimization of the hexahedral box

In addition, the total mass constraint satisfies W � 188:2 6 0.The fundamental frequency meets the following equation:xf � 126.7 P 0. According to Eq. (11), the structural-acoustic opti-mization of the hexahedral box structure is expressed as:

Find E ¼ ½E1; E2; E3; E4; E5�T; ð13ÞN ¼ ½N1;N2;N3;N4;N5�T;

MinY

a

¼ 1x2 �x1

Xm

i¼1

PðxiÞDxi;

s:t:Xk

i¼1

ðtim � qm � sim þ tic � qc � sicÞ 6 188:2;

rmax 6 80 MPa;

dmax 6 0:002;

0:008 6 tim 6 0:02;0:01 6 tic 6 0:04;

xf � 126:7 P 0;

Ei 2 fES; ECg:

where i ¼ 1;2; . . . ;5, The other parameters of Eq. (12) are the samemeaning as those in Eq. (11).

In the present study, in order to search for the globallyoptimum, genetic algorithm (GA) is proposed. GA simultaneouslyconsiders multiple candidate solutions to the problem of minimiz-ing the objective function and iterating by moving this populationof solutions toward a global optimum. GA has received increasingattention in a wide range of engineering applications because ofthe versatile capabilities for both continuous and discrete optimi-zation problems. Research has demonstrated that GA can be robustin global solutions and multimodal design spaces, even whennumerous design variables and constraints are employed.

According to Eq. (12), material selection optimization is carriedout on the iSIGHT optimization platform which is combined withfinite element numerical analysis code and boundary elementanalysis code.

In material selection optimization, if the elastic modulus Ei

changes, then the other mechanical properties (shear modulus,Poisson’s ratio, loss factor, density, etc.) will change too. Withmaterial selection optimization, the optimal structure will beobtained finally. After the optimization, the vibration and acousticradiation of hexahedral box structure are reduced. Table 2 showsthe various design variables optimization results.

In Table 2, the symbol M represents the steel material variables,and the symbol C represents the carbon fiber reinforced plasticsmaterial variables. In the material selection optimization, thestructure material includes steel and composite material. The com-posite laminated structure lists the thickness and plies numberparameter, and steel structure lists thickness parameter only.

According to Fig. 3, it is noted that the peak amplitude value ofacoustic power has been reduced by 15 dB compared with the ini-tial design, the optimization effect is obvious.

Table 1The mechanical properties of steel and carbon fiber reinforced plastics.

Elasticity modulus(GPa)

Elasticity modulus(GPa)

Shear modulus(GPa)

Shear modulus(GPa)

Shear modulus(GPa)

Poisson’sratio

Lossfactor

Density (kg/m3)

Steel E = 210 E = 210 G = 81 G = 81 G = 81 0.3 g = 0.001 7800Carbon fiber reinforced

plasticsE11 = 30.0 E22 = 26.0 G12 = 8.0 G23 = 4.5 G13 = 8.0 0.25 g = 0.01 1900

Table 2Results of the initial design and optimal design.

Designvariables

Initialmaterial

Initialthickness(m)

Optimummaterial

Optimumthickness(m)

Numberof ply

D-ban M 0.010 M 0.012X-xiao M 0.012 M 0.015X-da M 0.012 C 0.030 60Y-xiao M 0.012 M 0.015Y-da M 0.012 C 0.033 66

Fundamentalfrequency

126.7 Hz 166.2 Hz

Structuremass

188.2 kg 183.9 kg

Note: symbol M represents the steel material variables, and symbol C represents thecarbon fiber reinforced plastics material variables.

0 100 200 300 40040

50

60

70

80

90

100

110

120

130

140

Aco

ustic

Pow

er (

dB)

Frequency /Hz

Initial designMSO optimal

Fig. 3. Comparison of acoustic power levels with material selection optimization.

0 100 200 300 40020

40

60

80

100

120

140

Aco

ustic

Pre

ssur

e /d

B

Frequency /Hz

Initial design MSO optimal

Fig. 4. Comparison of acoustic pressure in field with material selectionoptimization.

0 50 100 150 200110

115

120

125

130

135

Aco

ustic

Pow

er /d

BIteration Number

Optimal History

Fig. 5. Iteration history of the acoustic power optimization.

Table 3Optimal orientation angle of X-da and Y-da.

Optimal orientation angle

X-da [0�/�45�/90�/45�/0�/45�/90�/�45�/0�/45�/90�/�45�/0�/45�/90�/45�/0�/�45�/90�/45�/0�/45�/90�/�45�/0�/�45�/0�/�45�/0�/0�]

Y-da [0�/�45�/90�/45�/0�/45�/90�/�45�/0�/45�/90�/�45�/0�/45�/90�/45�/0�/�45�/90�/45�/0�/45�/90�/�45�/0�/�45�/90�/0�/�45�/0�/�45�/0�/0�]

0 100 200 300 40040

50

60

70

80

90

100

110

120

130

140

Aco

ustic

pow

er /d

B

Frequency /Hz

Initial design MSO optimal FOA optimal

Fig. 6. Comparison of acoustic power with material selection optimization and fiberorientation angle optimization.

212 L. Chen, Y. Zhang / Materials and Design 52 (2013) 207–213

Fig. 4 shows the acoustic pressure magnitude of the field point(0,0,5). Compared with the initial design, the peak amplitude valueof acoustic pressure has been reduced by about 17 dB, the optimi-zation effect is obvious.

L. Chen, Y. Zhang / Materials and Design 52 (2013) 207–213 213

The iteration history for the composite integrating optimizationwas shown in Fig. 5, and the result achieved convergence afterabout 200 times optimization iterations.

6.4. The orientation angle optimization

In order to obtain better optimization effect, the fiber orienta-tion angle optimization of carbon fiber reinforced plastics materialis done. The fiber orientation angles were chosen as design vari-ables. The fiber orientation angle is limited to a small set variableof angles here.

In the fiber orientation angle optimization, only covers thestructure designs variable X-da and Y-da. For the structure designvariable X-da, which is satisfied to the constrain �90� < hm 6 90�,m = 1,2, . . .,30, the angle variable set can be written as a discreteform:(�45�/0�/45�/90�). For the structure design variable Y-da,-which is satisfied to the constrain �90� < hn 6 90�, n = 1,2, . . .,33,the angle variable set can be written as a discrete form: (�45�/0�/45�/90�). For the stacking sequence orientation angle optimiza-tion function, the other constrains condition is the same as Eq. (12).

After fiber orientation angle optimization, the angle of fiber ori-entation of carbon fiber reinforced plastics material for structure X-da and structure Y-da are shown in Table 3.

Fig. 6 shows the structural-acoustic radiation power magnitudeof the initial design, material selection optimization (MSO optimal)and fiber orientation angle optimization (FOA optimal). Comparedwith the material selection optimization, the peak amplitude valueof acoustic power has been reduced by 3.3 dB by fiber orientationangle optimization, and the angle optimization has a certain effect.

Compared with Figs. 3–5, it shows that the acoustic power hasbeen reduced significantly after material selection and fiber orien-tation angle optimization, the optimization has achieved the pur-pose of reducing vibration and noise.

7. Conclusions

In this paper, acoustic radiation power is proposed as a perfor-mance index. With the stacking sequence hypothesis of the metalmaterial, the structural-acoustic optimization problem of hybridmetal-composite structure is discussed by material selection opti-mization approach. In the numerical example, the peak values ofthe optimum design parameter are reduced significantly at the fre-quencies of interest. The best mechanical behavior of metal andcomposite laminated is full playing them role by material selectionoptimization. In addition, if we implement the fiber orientation an-gle optimization in material selection optimization, there will be abetter result.

Acknowledgement

This work was partially supported by the Youth Fund of StateKey Laboratory of Ocean Engineering (Grant No: GKZD010059-22). This support is gratefully acknowledged by the authors.

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