Research ArticleThree-Dimensional Dynamic Topology Optimization withFrequency Constraints Using Composite Exponential Functionand ICM Method

Hongling Ye1 Ning Chen1 Yunkang Sui1 and Jun Tie2

1College of Mechanical Engineering and Applied Electronics Beijing University of Technology Beijing 100124 China2Department of Mathematics Tianjin University of Finance and Economics Tianjin 300222 China

Correspondence should be addressed to Jun Tie tielaoshisinacomcn

Received 4 September 2014 Accepted 5 January 2015

Academic Editor Chenfeng Li

Copyright copy 2015 Hongling Ye et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The dynamic topology optimization of three-dimensional continuum structures subject to frequency constraints is investigatedusing Independent Continuous Mapping (ICM) design variable fields The composite exponential function (CEF) is selected to bea filter function which recognizes the design variables and to implement the changing process of design variables from ldquodiscreterdquoto ldquocontinuousrdquo and back to ldquodiscreterdquo Explicit formulations of frequency constraints are given based on filter functions first-order Taylor series expansion And an improved optimal model is formulated using CEF and the explicit frequency constraintsDual sequential quadratic programming (DSQP) algorithm is used to solve the optimal model The program is developed on theplatform of MSC Patran amp Nastran Finally numerical examples are given to demonstrate the validity and applicability of theproposed method

1 Introduction

Structural topology optimization is to find optimal materiallayout within a given design space for a given set of loadsand boundary conditions such that the resulting layout meetsa prescribed set of performance targets The essence oftopology optimization lies in searching for the optimum pathof transferring loads therefore the computational results oftopology optimization are usually more attractive and morechallenging than the results of cross-sectional and shapeoptimization Although topology optimization is only inconceptual design phase in engineering the design resultssignificantly impact the performance of the final structure

In the last decades since the landmark paper of Bendsoslasheand Kikuchi [1] numerical methods for topology optimiza-tion of continuum structures have been developed quickly inapplication [2ndash4] Homogenizationmethods and SIMP (solidisotropic material with penalization) method are especiallypopular in continuum topology optimization though theirdesign variables are ldquofictionsrdquo In homogenization method [56] the design variables are microscale void and the optimal

problem is defined by seeking the optimal porosity of porousmedium using the optimality criteria But sometimes thedesigns result in infinitesimal pores in the materials and thestructure is not easy to manufacture SIMP method [7ndash9] isemployed to describe the relationship between the relativedensity and the material elastic modules of grey elementsby a nonlinear function with continuous variables Thesegrey elements have intermediate densities by introducinga penalty coefficient ESO method is another engineeringapproach which is investigated and used widely in recentyears The optimal topology is generated by deleting thegroup of elements with low strain energy from entire domainsystematically [10ndash12] Later a new development in ESOis the recent introduction approach named bidirectionalevolutionary structural optimization (BESO) where elementsare allowed to be added as well as removed [13 14] Thevalidity and new-look of ESO type methods are stated byHuang and Xie [15ndash17] and a critical evaluation as well ascomparison of SIMP and ESO method is discussed in detailby Rozvany [18] Other promising newmethods include level

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 491084 10 pageshttpdxdoiorg1011552015491084

2 Mathematical Problems in Engineering

set method [19ndash23] and phase field method [24ndash26] and theyare used widely in multiple material topology optimization

Compared with static topology optimization the studyon dynamic topology optimization is more complicated andthere are few investigations Frequency topology optimiza-tion is of great importance in dynamic topology optimizationand engineering fields Topology optimization with respectto frequencies of structural vibration was first consideredby Dıaaz and Kikuchi [27] who studied the topology opti-mization of eigenvalues by using the homogenizationmethodwhere reinforcement of a structure was optimized to max-imize eigenvalues Subsequently many researches focus onexpanding topology optimization in dynamic problems Maet al [28 29] studiedmulti-eigenvalue optimization problemsand frequency response optimization problems for vibratingplate structure An optimization algorithm is derived tomax-imize a set of eigenvalues Kosaka and Swan [30] investigatedthe optimization of the first five eigenfrequencies of platestructure and obtained the elastic properties by using a pureReuss formulation Krog and Olhoff [31] and Jensen andPedersen [32] utilized a variable bound formulation that facil-itates proper treatment of multiple frequencies Pedersen [33]dealt with maximum fundamental frequency design of platesand included a technique to avoid spurious localized modesJensen and Pedersen [32] presented a method to maximizethe separation of two adjacent frequencies in structures withtwo material components Zhu and Zhang [34] emphasizedon layout design which was related to optimization ofboundary conditions and it was studied to maximize naturalfrequency of structures In 2007 Du and Olhoff [35] intro-duced SIMPmethod for maximization of first eigenvalue andfrequency gaps Recently Niu et al [36] proposed a two-scale optimizationmethod and found the optimal figurationsof macrostructure-microstructure of cellular material withmaximum structural fundamental frequency Huang et al[17] investigated the maximization of fundamental frequencyof beam plane and three-dimensional block by applyinga new bidirectional evolutionary structural optimization(BESO)method and dealt with localizedmodes bymodifyingthe traditional penalization function of SIMP method Xiaet al [37 38] presented a level set based shape and topologyoptimization method for maximizing the simple or repeatedfirst eigenvalues of structure vibration Further developmenton frequency topology optimization is seen in [3 38ndash41]

However in many researches concerning the model ofdynamic topology optimization for continuum structurefrequency has been used as objective instead of constraintsIn this paper we extend our previous researches [42 43] pri-marily on static topology optimization issues of continuumstructures to dynamic topology optimization field Usingstructural weight as objective function we build topologyoptimization model for continuum structure with frequencyconstraints This method is beneficial to maintain the consis-tency of objective and constraint in cross-sectional optimiza-tion shape optimization and topology optimization

Among the methods of mathematic optimization modelsolving mathematical programming (MP) method is pop-ular Sequential quadratic programming (SQP) [44 45] inthe MP method is widely used SQP algorithm is a local

quadratic approximation A Lagrange function is neededto build for the global nonlinear optimization problemHowever a large amount of calculation is required if thenumber of optimization variables increases in the problembecause of the repeated building of the Hessian of theLagrange function In this paper dual sequential quadraticprogramming (DSQP) algorithm is employed to solve thetopology optimization model constrained by frequency Inthe process of solving optimal model the dual theory isused to convert the constrained optimization model to onewith reduced number of design variables and the solvingefficiency is greatly improved

This paper is organized as follows In Section 2 Inde-pendent Continuous Mapping (ICM) method is introducedIn Section 3 an improved frequency topology optimizationmodel based on ICM method is built Optimal algorithmto solve the mathematical optimization problem is givenin Section 4 Two numerical simulations are presented inSection 5 In Section 6 conclusions are given

2 Independent ContinuousMapping (ICM) Method

Independent Continuous Mapping (ICM) method is pro-posed by Sui [42] for skeleton and continuum structuraltopology optimization It generalizes topological variablesabstractly independent of the design variables such as sec-tional sizes geometrical shape density or Young moduleof material Using filter functions it maps the topologicalvariables essentially with discrete values of zero and oneto continuous independent topological variables with valuesbelonging to [0 1]

Based on the idea of ICM the smooth model for struc-tural topology optimization is established and it can be solvedby the traditional algorithms in mathematical programmingAfterwards these continuous variables are discretized to 1 or0 according to the criterionwhich the corresponding elementis conserved or removed The essential idea of the processis selecting the map from ldquodiscreterdquo to ldquocontinuousrdquo and theinverse map from ldquocontinuousrdquo to ldquodiscreterdquo

For structural cross-section and shape optimization nat-ural frequency of structure is often taken as constraint Wedenote 119891

119894as the frequency of 119894th order and 119891

119894 119891119894are the

low and up bound of frequency respectively They satisfy thefollowing inequality

(i) 1198911ge 1198911

(ii) 119891119894

le 119891119894and 119891

119894+1ge 119891119894+1

in nonfrequency bandconstraints

Here 1198911and 119891

1are natural and low bound of the first order

frequency respectivelyFor elastic structure the usual relation between frequency

f and eigenvalue 120582 is 120582 = (2120587119891)2 Therefore the frequency

constraints can apparently be transformed into eigenvalueconstraints using the formula Here we uniformly use 119892(120582) le

120582 to generalize (i) and (ii) based on 120582 = (2120587119891)2

Mathematical Problems in Engineering 3

Thus the model of continuum topology optimizationwith frequency constraints can be formulated as follows

find t isin 119864119873

make 119882 =

119873

sum119894=1

119908119894997888rarr min

st 119892 (120582119895) le 120582119895 (119895 = 1 119869)

0 le 119905119894le 1 (119894 = 1 119873)

(1)

where t 119882 and 119908119894denote the topological design variable

vector the total weight of structure and the 119894th elementweight 119894 and 119895 are the 119894th element and the 119895th order frequencyrespectively and 119869 and 119873 represent the total number ofconstraints and elements

3 Improved Model Based on ICM Method

31 Properties of Filter Function In order to develop themodel ICM method we firstly investigate the essential partof ICMmdashthe filter function Its definition and choosingdetermine the establishment of optimization model and itssolving and further it will make great impact on the finalperformance of topology optimization In order to map thetopological variables from ldquodiscreterdquo to ldquocontinuousrdquo Sui[42] studied the filter function 119891(119905

119894) and summarized the

following properties

(1) Continuity filter function 119891(119905119894) is continuous about

topology variable 119905119894in the range of [0 1]

(2) Differentiability if 119905119894isin [0 1] 119891(119905

119894) is continuously

differential(3) Monotonicity 119891(119905

119894) is strictly monotonous

(4) Convexity 119891(119905119894) is strictly convex in [0 1]

(5) Approximation the approximation is given as follows

119891 (119905119894)

997888rarr 1 (119905119894gt 120576)

= 0 (119905119894997888rarr 0)

(2)

here 120576 is an arbitrary small positiveHowever the approximation property is not easy to

control it is a limit process To achieve the approximationwe introduce the following formula

119891 (119905119894) isin [0 1] 119891 (0) = 0 119891 (1) = 1 (3)

Consider int10119891(119905119894)119889119909 lt 120576 here 120576 is a given arbitrary small

positive and 0 lt 120576 lt 12

32 Improved Filter Functions Several types of filter functionare suggested in ICMmethod among which Power Function(PF) is used frequently in application [46] and is as follows

119891 (119905119894) = 119905120572

119894 120572 ge 1 (4)

Here 119905119894denotes 119894th design variable 120572 is a positive constant

We introduce a new filter function-composite exponen-tial function (CEF) to take the place of the old one and it is asfollows

119891 (119905119894) =

119890119905119894120574 minus 1

1198901120574 minus 1 120574 gt 0 (5)

120574 is a given positive constant and it is determined by numer-ical experiments with different problems In Section 5 wecompare the performance of the two types of filter function

33 Improved Model of Frequency Topology OptimizationDenote 119891

119908(119905119894) 119891119896(119905119894) and 119891

119898(119905119894) as filter functions for fre-

quency topology optimization and they are given as follows

119908119894= 119891119908(119905119894) 1199080

119894 k

119894= 119891119896(119905119894) k0119894 m

119894= 119891119898(119905119894)m0119894

(6)

Here1199080119894 k0119894 andm0

119894are the element weight element stiffness

matrix and element mass matrix of original structure beforethe process of topology optimization respectively119908

119894 k119894 and

m119894are the ones in the process of topology optimization

respectivelyBased on the above analysis we build the model of

topology optimization with frequency constraints as follows

find t isin 119864119873

make 119882 =

119873

sum119894=1

119891119908(119905119894) 1199080

119894997888rarr min

st 119892 (120582119895(119891119896(119905119894) 119891119898(119905119894))) le 120582

119895 (119895 = 1 119869)

0 le 119905119894le 1 (119894 = 1 119873)

(7)

In what follows we face the problem of approximating thefunction of frequency constraints explicitly

In the finite element analysis the dynamic behavior ofa continuum structure can be represented by the followinggeneral eigenvalue problem

(K minus 120582119895M) u119895= 0 (8)

whereK is the global stiffnessmatrix andM is the globalmassmatrix 120582

119895is the 119895th eigenvalue and u

119895is the eigenvector cor-

responding to 120582119895 The eigenvalue 120582

119895and the corresponding

eigenvector u119895are related to each other by Rayleigh quotient

120582119895=

u119879119894Ku119894

u119879119894Mu119894

(9)

Since eigenvalue 120582119895is implicitly related with topology

variable t we use first-order Taylor series expansion foreigenvalue to express their relationship explicitly

Take the reciprocal of stiffness filter function as designvariables as follows

119909119894=

1

119891119896(119905119894) (10)

4 Mathematical Problems in Engineering

We have

119905119894= 119891minus1

119896(119909119894) (11)

Therefore the stiffness matrix filter function mass matrixfilter function and weight filter function are given as follows

119891119896(119905119894) =

1

119909119894

119891119898(119905119894) = 119891119898[119891minus1

119896(

1

119909119894

)]

119891119908(119905119894) = 119891119908[119891minus1

119896(

1

119909119894

)]

(12)

The global stiffness matrix K and the mass matrix M can becalculated by

K =

119873

sum119894=1

k119894=

119873

sum119894=1

119891119896(119905119894) k119894=

119873

sum119894=1

1

119909119894

k0119894

M =

119873

sum119894=1

m119894=

119873

sum119894=1

119891119898(119905119894)m0119894=

119873

sum119894=1

119891119898[119891minus1

119896(

1

119909119894

)]m0119894

(13)

In view of (9) we have the derivative of 120582119895to design variable

as follows120597120582119895

120597119909119894

= u119879119895

120597K120597119909119894

u119895minus 120582119895u119879119895

120597M120597119909119894

u119895 (14)

Substituting (13) to (14) we have

120597120582119895

120597119909119894

= minusu119879119895

2k119894

2119909119894

u119895+ 120573 (119909

119894) 120582119895u119879119895

2m119894

2119909119894

u119895

= minus2

119909119894

(119880119894119895minus 120573 (119909

119894) 119881119894119895)

(15)

where

120573 (119909119894) =

1198911015840

119898[119891minus1

119896(1119909119894)] 119891119896(1119909119894)

119891119898[119891minus1119896

(1119909119894)] 1198911015840119896(1119909119894)=

1198911015840

119898(119905119894) 119891119896(119905119894)

119891119898(119905119894) 1198911015840119896(119905119894) (16)

In (15) 119880119894119895

= (12)u119879119895k119894u119895and 119881

119894119895= (12)120582

119895u119879119894m119894u119895

represent the strain energy and the kinetic energy of 119894thelement corresponding to the jth eigenmode respectively Atthis moment the derivatives of eigenvalue with respect toall design variables can be obtained by subtracting the strainenergy and kinetic energy for element mode from the resultsof modal analyses

Using the first-order Taylor series expansion the approx-imate expression of eigenvalue 120582

119895(t) can be obtained

120582119895 (t) = 120582

119895 (x) = 120582119895(x(])) +

119873

sum119894=1

120597120582119895

120597119909119894

(119909119894minus 119909(])119894

)

= 120582119895(x(])) +

119873

sum119894=1

2 (119880(])119894119895

minus 120573 (119909(])119894

)119881(])119894119895

)

minus

119873

sum119894=1

2

119909(])119894

(119880(])119894119895

minus 120573 (119909(])119894

)119881(])119894119895

) 119909119894

(17)

where superscript ] is the number at the ]th iteration

If we define 119863 as

119863 =

1 for 120582119895le 120582119895

minus1 for 120582119895ge 120582119895

119895=

120582119895 (120582

119895ge 120582119895)

120582119895 (120582

119895le 120582119895)

(18)

and further define

119860119894119895= 119880(])119894119895

minus 120573 (119909(])119894

)119881(])119894119895

119888119894119895= minus

2

119909(])119894

119860119894119895 119888

119894119895= minus119863119888

119894119895

119889119895= minus120582119895(x(])) minus

119873

sum119894=1

2119860119894119895 119889

119895= 119863 times (

119895+ 119889119895)

(19)

then frequency constraints can be simplified by the followinginequality

119873

sum119894=1

119888119894119895119909119894le 119889119895 (20)

Thus ends the process of explicit approximation of thefrequency constraints

4 Solution of the Improved TopologyOptimization Model

However model (7) is a programming with nonlinear objec-tive and linear constraints following the explicit process offrequency constraints We use second-order Taylor seriesexpansion to approximate the objective function and ignorethe constant item The model is transformed into the follow-ing quadratic programming model

find t isin 119864119873

make 119882 =

119873

sum119894=1

(1198861198941199092

119894+ 119887119894119909119894) 997888rarr min

st119873

sum119894=1

119888119894119895119909119894le 119889119895 (119895 = 1 119869)

1 le 119909119894le 119909119894 (119894 = 1 119873)

(21)

As objective function varies with different filter functionsinvestigation of the different cases following different typesof filter functions is necessary Here we focus on PF and CEF

When PF is applied as the filter function it is given asfollows

119891119908(119905119894) = 119905120574119908

119894 119891

119896(119905119894) = 119905120574119896

119894 119891

119898(119905119894) = 119905120574119898

119894 (22)

where 120574119908 120574119896 and 120574

119898 are all arbitrarily integers greater thanzero In view of (22) we have 119905

119894= 1119909

1120574119896

119894 119905120574119908119894

= 1119909120574119908120574119896

119894=

1119909120572

119894 and 120572 = 120574

119908120574119896

Mathematical Problems in Engineering 5

Therefore the objective function (7) can be rewritten as

119882 =

119873

sum119894=1

119905120574119908

1198941199080

119894=

119873

sum119894=1

1199080

119894

119909120572119894

asymp

119873

sum119894=1

(1198861198941199092+ 119887119894119909) (23)

where 119886119894= 120572(120572+1)119908

0

1198942(119909119894)120572+2 and 119887

119894= minus120572(120572+2)119908

0

119894(119909119894)120572+1

are undetermined parametersWhen CEF is applied as the filter function it is given as

follows

119891119908(119905119894) =

119890119905119894120574119908 minus 1

1198901120574119908 minus 1 119891

119896(119905119894) =

119890119905119894120574119896 minus 1

1198901120574119896 minus 1

119891119898(119905119894) =

119890119905119894120574119898 minus 1

1198901120574119898 minus 1

(24)

We have 119909119894= 1119891

119896(119905119894) = (119890

1120574119896 minus 1)(119890119905119894120574119896 minus 1) and therefore

119891119908(119905119894) =

((1198901120574119896 minus 1) 119909

119894+ 1)120574119896120574119908

minus 1

1198901120574119908 minus 1

119891119898(119905119894) =

((1198901120574119896 minus 1) 119909

119894+ 1)120574119896120574119898

minus 1

1198901120574119898 minus 1

(25)

Similarly the objective function in (7) can be expressed as

119873

sum119894=1

((1198901120574119896 minus 1) 119909

119894+ 1)120574119896120574119908

minus 1

1198901120574119908 minus 11199080

119894asymp

119873

sum119894=1

(1198861198941199092

119894+ 119887119894119909119894) (26)

where

119886119894=

1

2

120574119896

120574119908

1199080

119894

(119909(V)119894

)4

1198901120574119896 minus 1

1198901120574119908 minus 1(

1198901120574119896 minus 1

119909(V)119894

+ 1)

120574119896120574119908minus2

sdot [(120574119896

120574119908

+ 1) (1198901120574119896 minus 1) + 2119909

(V)119894

]

119887119894= minus

120574119896

120574119908

1199080

119894

(119909(V)119894

)3

1198901120574119896 minus 1

1198901120574119908 minus 1(

1198901120574119896 minus 1

119909(V)119894

+ 1)

120574119896120574119908minus2

sdot [(120574119896

120574119908

+ 2) (1198901120574119896 minus 1) + 3119909

(V)119894

]

(27)

They are two undetermined parametersThe dual model of (21) is

find zisin E119869

make minus Φ (z) 997888rarr min

st z119895ge 0 (119895 = 1 119869)

(28)

where z is the design variables in dual model

Φ (z) = min1le119905119894le119909

(119871 (x z))

119871 (x z) =

119873

sum119894=1

(1198861198941199092

119894+ 119887119894119909119894) +

119869

sum119895=1

(119911119895(

119873

sum119894=1

119888119894119895119909119894minus 119889119895))

(29)

Here 119871(x z) is a Lagrange augmented function in (28)

Start

Construct the finite element model and input the optimal parameters

Finite element analysis with MSCNastran

Formulate the mathematical optimal model (7)

Solve (21) with DSQP and update variables

End

No

Yes

|W(+1) minus W()|

|W()|le 120576

Figure 1 Flow chart of the iterative solution procedure

Then DSQP is employed to solve (28) and the precisionof convergence is controlled to be 0001 With the aboveanalysis and solving of (28) the optimal topology structurewith continuous design variables is obtained

5 Numerical Simulation

Based on the above analysis we outline the following compu-tational procedure with a flow chart (Figure 1) it is used formaximizing theweight of structure constrained by frequency

Following the above flow chart we develop a programbased on the platform of MSC Patran amp Nastran to solve theimproved topology optimal model In order to demonstratethe effectiveness of the proposed method two structuresare presented as inputs The first one is a cantilever withfundamental frequency constraint the second one is a 3Dbeam with two concentrated masses by multiple frequencyconstraints For the computation the initial values of topol-ogy variables are all given as unit (119905 = 1) the lowest bounds oftopology variables and the convergence precision values are001 and 0001 respectively

51 Example 1 Example 1 is a cantilever with size 100 times 50times 50mm3 It is fixed at one end as shown in Figure 2 and aconcentrated mass Mc = 50 g is attached at the center of theother end Youngrsquos modulus 119864 = 100GPa Poissonrsquos ratio 120583 =

03 andmass density 120588= 1000 kgm3The structure is dividedinto 30 times 16 times 16 eight-node hexahedron elements The5010Hz fundamental frequency of the cantilever is calculatedby finite element analysis andwe set the frequency constraintfor the design problem as 119891

1ge 3500Hz

The solving topology configuration of the cantilever withdifferent filter functions is given in Figure 3 Figures 3(a) and3(b) show the whole optimal structure and (c)-(d) show the

6 Mathematical Problems in Engineering

50mm

50mm

100mm

Mc

Figure 2 A cantilever structure

(a) (b)

(c) (d)

Figure 3 Topology configuration with different filter functions (a) Topology configuration with PF (b) topology configuration with CEF(c)-(d) the half optimal structure of (a)-(b)

half structure corresponding to (a)-(b) The performance oftopology optimization with different filter functions is givenin Table 1 The iterative curve of computation with differentfilter functions is described in Figure 4

52 Example 2 The beam (40 times 10 times 10 cm3) with two endsfixed is presented in Figure 5 Two concentrated masses with1198721198881

= 1198721198882

= 5 kg are located on the upper surface of 13and 23 span Youngrsquos modulus E = 100GPa Poissonrsquos ratio120583 = 03 and mass density 120588 = 1000 kgm3 The structure

is divided into 30 times 8 times 8 eight-node hexahedron elementsThe first computed fundamental frequency of the beam is13435Hz and the second one is 14951 Hz The set constraintfrequencies are 119891

1ge 1000Hz 119891

2ge 1100Hz

The solving topology configuration of the beam withdifferent filter functions is given in Figure 6 The perfor-mances of topology optimization with different filter func-tions are given in Table 2 To describe the dynamics ofoptimal structure the first three modal shapes of optimalstructure with two different filter functions are computedand displayed in Figure 7 The frequency changing with time

Mathematical Problems in Engineering 7

260

240

220

200

180

160

140

120

100

80

60

Mas

s (g)

PFCEF

0 10 20 30 40

Iteration

(a)

34

36

38

40

42

44

46

48

50

52

Freq

uenc

y (H

z)

times103

0 10 20 30 40

Iteration

PFCEF

(b)

Figure 4 Iteration process with different filter function

L

H

Mc1 Mc2

Figure 5 The beam with two ends fixed

Table 1 Performance of topology optimization with differentalgorithms and filter functions

Method DSQP

PFIteration 411198911(Hz) 35020270996

Mass (g) 73033882813

CEFIteration 341198911(Hz) 35019787598

Mass (g) 70509476563

in the optimization process is presented in Figure 8 withdifferent filter functions

6 Conclusion

In this paper an improved frequency topology optimizationmodel of three-dimensional continuum structure is devel-oped based on ICM method CEF a new filter function isselected to recognize the design variables as well as to imple-ment much better the changing process of design variablesfrom ldquodiscreterdquo to ldquocontinuousrdquo and back to ldquodiscreterdquo Explicit

Table 2 Performance of topology optimization with differentalgorithms and filter functions

Method DSQP

PF

Iteration 331198911(Hz) 1000392

1198912(Hz) 1100647

Mass (g) 1760004

CEF

Iteration 301198911(Hz) 1000551

1198912(Hz) 1100791

Mass (g) 1682534

formulations of frequency constraints are given based onfilter functions and first-order Taylor series expansion andby extracting structural strain and structural kinetic energyfrom the results of structural modal analysis An improvedoptimal model is formulated using CEF and the explicitfrequency constraints The program based on DSQP forsolving the optimal model is developed on the platform ofMSC Patran amp Nastran

Finally two examples of three-dimensional continuumstructure show that clear and stable configurations can beobtained using ICM method We find that configurationscomputed with DSQP combined PF and DSQP combinedCEF are similar between one and the other in the case of fun-damental frequency constraint But we can find that DSQPcombined CEF has the best performance for the optimizationexample from the point of view of optimal objective anditerative numbers We also compute the first three ordermodal shapes of optimal structure with two different filterfunctions in multiple frequency constraints The vibrationmodes from two filter functions are the same and the iteration

8 Mathematical Problems in Engineering

(a) (b)

Figure 6 Topology configuration with different filter functions (a) Topology configuration with PF (b) topology configuration with CEF

Mod

e

First

PF

(a)

Modal shapesSecond

(b)

Third

(c)

CEF

Mod

e

(d) (e) (f)

Figure 7 The first three modal shapes of optimal structure with two filter functions

0 5 10 15 20 25 30 35

Eige

nfre

quen

cy (H

z)

Iterationminus5

f1f2f3

18171616161515141413131211111110

095

times103

(a)

0 5 10 15 20 25 30Iteration

f1f2f3

Eige

nfre

quen

cy (H

z)

18171616161515141413131211111110

095

times103

(b)

Figure 8 Iteration process of the first three frequencies with different filter functions (a) PF (b) CEF

Mathematical Problems in Engineering 9

curves of the first three order frequencies show that structuralfrequencies satisfy the frequency constraints Therefore allthe computing results indicate that the proposed method isalso feasible for multiple frequency constraints

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of the paper

Acknowledgment

The work of this project was supported by National ScienceFoundation of China (ContractGrant no 11072009 andContractGrant no 11172013)

References

[1] M P Bendsoslashe and N Kikuchi ldquoGenerating optimal topologiesin structural design using a homogenization methodrdquo Com-puterMethods in AppliedMechanics and Engineering vol 71 no2 pp 197ndash224 1988

[2] H A Eschenauer and N Olhoff ldquoTopology optimization ofcontinuum structures a reviewrdquo Applied Mechanics Reviewsvol 54 no 4 pp 331ndash390 2001

[3] J D Deaton and R V Grandhi ldquoA survey of structuraland multidisciplinary continuum topology optimization post2000rdquo Structural and Multidisciplinary Optimization vol 49no 1 pp 1ndash38 2014

[4] G I N Rozvany ldquoAims scope methods history and unifiedterminology of computer-aided topology optimization in struc-tural mechanicsrdquo Structural and Multidisciplinary Optimiza-tion vol 21 no 2 pp 90ndash108 2001

[5] M P Bendsoslashe andO Sigmund Topology OptimizationTheoryMethods and Applications Springer Berlin Germany 2ndedition 2003

[6] B Hassani and E Hinton ldquoA review of homogenization andtopology optimization Imdashhomogenization theory for mediawith periodic structurerdquo Computers amp Structures vol 69 no 6pp 707ndash717 1998

[7] G I N Rozvany and M Zhou ldquoApplications of the COCalgorithm in layout optimizationrdquo in Engineering Optimizationin Design Processes Proceedings of the International ConferenceKarlsruhe Nuclear Research Center Germany September 3-41990 H Eschenauer C Matteck and N Olhoff Eds vol 63of Lecture Notes in Engineering pp 59ndash70 Springer BerlinGermany 1991

[8] R J Yang and C H Chuang ldquoOptimal topology design usinglinear programmingrdquo Computers and Structures vol 52 no 2pp 265ndash275 1994

[9] M P Bendsoslashe andO Sigmund ldquoMaterial interpolation schemesin topology optimizationrdquoArchive of AppliedMechanics vol 69no 9-10 pp 635ndash654 1999

[10] YM Xie andG P Steven ldquoA simple evolutionary procedure forstructural optimizationrdquo Computers and Structures vol 49 no5 pp 885ndash896 1993

[11] YMXie andG P StevenEvolutionary StructuralOptimizationSpringer Berlin Germany 1997

[12] G I N Rozvany O M Querin Z Gaspar and V PomezanskildquoWeight-increasing effect of topology simplificationrdquo Structural

and Multidisciplinary Optimization vol 25 no 5-6 pp 459ndash465 2003

[13] O M Querin G P Steven and Y M Xie ldquoEvolutionarystructural optimisation (ESO) using a bidirectional algorithmrdquoEngineering Computations vol 15 no 8 pp 1031ndash1048 1998

[14] J H ZhuWH Zhang andK P Qiu ldquoBi-directional evolution-ary topology optimization using element replaceable methodrdquoComputational Mechanics vol 40 no 1 pp 97ndash109 2007

[15] X Huang and Y M Xie ldquoA new look at ESO and BESOoptimization methodsrdquo Structural and Multidisciplinary Opti-mization vol 35 no 1 pp 89ndash92 2008

[16] X Huang and YM Xie ldquoA further review of ESO typemethodsfor topology optimizationrdquo Structural and MultidisciplinaryOptimization vol 41 no 5 pp 671ndash683 2010

[17] X Huang Z H Zuo and Y M Xie ldquoEvolutionary topologicaloptimization of vibrating continuum structures for naturalfrequenciesrdquoComputers and Structures vol 88 no 5-6 pp 357ndash364 2010

[18] G I N Rozvany ldquoA critical review of established methods ofstructural topology optimizationrdquo Structural and Multidisci-plinary Optimization vol 37 no 3 pp 217ndash237 2009

[19] S Osher and R P Fedkiw ldquoLevel set methods an overview andsome recent resultsrdquo Journal of Computational Physics vol 169no 2 pp 463ndash502 2001

[20] G Allaire F Jouve and A M Toader ldquoStructural optimizationusing sensitivity analysis and a level-set methodrdquo Journal ofComputational Physics vol 194 no 1 pp 363ndash393 2004

[21] J Luo Z Luo S Chen L Tong and M Y Wang ldquoA newlevel set method for systematic design of hinge-free compliantmechanismsrdquo Computer Methods in Applied Mechanics andEngineering vol 198 no 2 pp 318ndash331 2008

[22] M Y Wang S Chen X Wang and Y Mei ldquoDesign ofmultimaterial compliant mechanisms using level-set methodsrdquoTransactions of the ASME Journal of Mechanical Design vol 127no 5 pp 941ndash956 2005

[23] M Y Wang X Wang and D Guo ldquoA level set methodfor structural topology optimizationrdquo Computer Methods inApplied Mechanics and Engineering vol 192 no 1-2 pp 227ndash246 2003

[24] B Bourdin and A Chambolle ldquoDesign-dependent loads intopology optimizationrdquo ESAIM Control Optimisation andCalculus of Variations vol 9 pp 19ndash48 2003

[25] S Zhou and M Y Wang ldquoMultimaterial structural topologyoptimization with a generalized Cahn-Hilliard model of mul-tiphase transitionrdquo Structural and Multidisciplinary Optimiza-tion vol 33 no 2 pp 89ndash111 2007

[26] A Takezawa S Nishiwaki and M Kitamura ldquoShape andtopology optimization based on the phase field method andsensitivity analysisrdquo Journal of Computational Physics vol 229no 7 pp 2697ndash2718 2010

[27] A R Dıaaz and N Kikuchi ldquoSolutions to shape and topologyeigenvalue optimization problems using a homogenizationmethodrdquo International Journal for Numerical Methods in Engi-neering vol 35 no 7 pp 1487ndash1502 1992

[28] Z D Ma H C Cheng and N Kikuchi ldquoStructural design forobtaining desired eigenfrequencies by using the topology andshape optimizationmethodrdquoComputing Systems in Engineeringvol 5 no 1 pp 77ndash89 1994

[29] Z DMa N Kikuchi andH-C Cheng ldquoTopological design forvibrating structuresrdquo Computer Methods in Applied Mechanicsand Engineering vol 121 no 1ndash4 pp 259ndash280 1995

10 Mathematical Problems in Engineering

[30] I Kosaka and C C Swan ldquoA summetry reduction methodfor continuum structural topology optimizationrdquo Computers ampStructures vol 70 no 1 pp 47ndash61 1999

[31] L A Krog and N Olhoff ldquoOptimum topology and reinforce-ment design of disk and plate structures with multiple stiffnessand eigenfrequency objectivesrdquoComputers amp Structures vol 72no 4 pp 535ndash563 1999

[32] J S Jensen and N L Pedersen ldquoOn maximal eigenfrequencyseparation in two-material structures the 1D and 2D scalarcasesrdquo Journal of Sound andVibration vol 289 no 4-5 pp 967ndash986 2006

[33] N L Pedersen ldquoMaximization of eigenvalues using topologyoptimizationrdquo Structural and Multidisciplinary Optimizationvol 20 no 1 pp 2ndash11 2000

[34] J H Zhu and W H Zhang ldquoMaximization of structuralnatural frequency with optimal support layoutrdquo Structural andMultidisciplinaryOptimization vol 31 no 6 pp 462ndash469 2006

[35] J Du and N Olhoff ldquoTopological design of freely vibratingcontinuum structures for maximum values of simple andmultiple eigenfrequencies and frequency gapsrdquo Structural andMultidisciplinary Optimization vol 34 no 2 pp 91ndash110 2007

[36] B Niu J Yan and G Cheng ldquoOptimum structure with homo-geneous optimum cellular material for maximum fundamentalfrequencyrdquo Structural and Multidisciplinary Optimization vol39 no 2 pp 115ndash132 2009

[37] Q Xia T Shi and M Y Wang ldquoA level set based shape andtopology optimization method for maximizing the simple orrepeated first eigenvalue of structure vibrationrdquo Structural andMultidisciplinary Optimization vol 43 no 4 pp 473ndash485 2011

[38] Q Xia T Shi M Y Wang and S Liu ldquoA level set basedmethod for the optimization of cast partrdquo Structural andMultidisciplinary Optimization vol 41 no 5 pp 735ndash747 2010

[39] A K Nandy andC S Jog ldquoOptimization of vibrating structuresto reduce radiated noiserdquo Structural and MultidisciplinaryOptimization vol 45 no 5 pp 717ndash728 2012

[40] J Zheng S Long and G Li ldquoTopology optimization forfree vibrating continuum structures based on the element freeGalerkin methodrdquo Structural and Multidisciplinary Optimiza-tion vol 45 no 1 pp 119ndash127 2012

[41] T D Tsai and C C Cheng ldquoStructural design for desired eigen-frequencies and mode shapes using topology optimizationrdquoStructural and Multidisciplinary Optimization vol 47 no 5 pp673ndash686 2013

[42] K Sui Y Modeling Transformation and Optimization NewDevelopments of Structural Synthesis Method Dalian Universityof Technology Press Dalian China 1996

[43] YK Sui J X LiuH L Ye and J ZDu ldquoApplication ofmodifiedsigmoid function in structural topology optimizationrdquo Journalof BeijingUniversity of Technology vol 34 supplement 1 pp 1ndash62008

[44] P E Gill W Murray and M A Saunders ldquoSNOPT anSQP algorithm for large-scale constrained optimizationrdquo SIAMReview vol 47 no 1 pp 99ndash131 2005

[45] J Nocedal and S J Wright Numerical Optimization SpringerBerlin Germany 1999

[46] Y K Sui H L Ye J X Liu S Chen and H P Yu ldquoAstructural topological optimization method based on exploringconceptual rootrdquo Engineering Mechanics vol 25 no 2 pp 7ndash192008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

set method [19ndash23] and phase field method [24ndash26] and theyare used widely in multiple material topology optimization

Compared with static topology optimization the studyon dynamic topology optimization is more complicated andthere are few investigations Frequency topology optimiza-tion is of great importance in dynamic topology optimizationand engineering fields Topology optimization with respectto frequencies of structural vibration was first consideredby Dıaaz and Kikuchi [27] who studied the topology opti-mization of eigenvalues by using the homogenizationmethodwhere reinforcement of a structure was optimized to max-imize eigenvalues Subsequently many researches focus onexpanding topology optimization in dynamic problems Maet al [28 29] studiedmulti-eigenvalue optimization problemsand frequency response optimization problems for vibratingplate structure An optimization algorithm is derived tomax-imize a set of eigenvalues Kosaka and Swan [30] investigatedthe optimization of the first five eigenfrequencies of platestructure and obtained the elastic properties by using a pureReuss formulation Krog and Olhoff [31] and Jensen andPedersen [32] utilized a variable bound formulation that facil-itates proper treatment of multiple frequencies Pedersen [33]dealt with maximum fundamental frequency design of platesand included a technique to avoid spurious localized modesJensen and Pedersen [32] presented a method to maximizethe separation of two adjacent frequencies in structures withtwo material components Zhu and Zhang [34] emphasizedon layout design which was related to optimization ofboundary conditions and it was studied to maximize naturalfrequency of structures In 2007 Du and Olhoff [35] intro-duced SIMPmethod for maximization of first eigenvalue andfrequency gaps Recently Niu et al [36] proposed a two-scale optimizationmethod and found the optimal figurationsof macrostructure-microstructure of cellular material withmaximum structural fundamental frequency Huang et al[17] investigated the maximization of fundamental frequencyof beam plane and three-dimensional block by applyinga new bidirectional evolutionary structural optimization(BESO)method and dealt with localizedmodes bymodifyingthe traditional penalization function of SIMP method Xiaet al [37 38] presented a level set based shape and topologyoptimization method for maximizing the simple or repeatedfirst eigenvalues of structure vibration Further developmenton frequency topology optimization is seen in [3 38ndash41]

However in many researches concerning the model ofdynamic topology optimization for continuum structurefrequency has been used as objective instead of constraintsIn this paper we extend our previous researches [42 43] pri-marily on static topology optimization issues of continuumstructures to dynamic topology optimization field Usingstructural weight as objective function we build topologyoptimization model for continuum structure with frequencyconstraints This method is beneficial to maintain the consis-tency of objective and constraint in cross-sectional optimiza-tion shape optimization and topology optimization

Among the methods of mathematic optimization modelsolving mathematical programming (MP) method is pop-ular Sequential quadratic programming (SQP) [44 45] inthe MP method is widely used SQP algorithm is a local

quadratic approximation A Lagrange function is neededto build for the global nonlinear optimization problemHowever a large amount of calculation is required if thenumber of optimization variables increases in the problembecause of the repeated building of the Hessian of theLagrange function In this paper dual sequential quadraticprogramming (DSQP) algorithm is employed to solve thetopology optimization model constrained by frequency Inthe process of solving optimal model the dual theory isused to convert the constrained optimization model to onewith reduced number of design variables and the solvingefficiency is greatly improved

This paper is organized as follows In Section 2 Inde-pendent Continuous Mapping (ICM) method is introducedIn Section 3 an improved frequency topology optimizationmodel based on ICM method is built Optimal algorithmto solve the mathematical optimization problem is givenin Section 4 Two numerical simulations are presented inSection 5 In Section 6 conclusions are given

2 Independent ContinuousMapping (ICM) Method

Independent Continuous Mapping (ICM) method is pro-posed by Sui [42] for skeleton and continuum structuraltopology optimization It generalizes topological variablesabstractly independent of the design variables such as sec-tional sizes geometrical shape density or Young moduleof material Using filter functions it maps the topologicalvariables essentially with discrete values of zero and oneto continuous independent topological variables with valuesbelonging to [0 1]

Based on the idea of ICM the smooth model for struc-tural topology optimization is established and it can be solvedby the traditional algorithms in mathematical programmingAfterwards these continuous variables are discretized to 1 or0 according to the criterionwhich the corresponding elementis conserved or removed The essential idea of the processis selecting the map from ldquodiscreterdquo to ldquocontinuousrdquo and theinverse map from ldquocontinuousrdquo to ldquodiscreterdquo

For structural cross-section and shape optimization nat-ural frequency of structure is often taken as constraint Wedenote 119891

119894as the frequency of 119894th order and 119891

119894 119891119894are the

low and up bound of frequency respectively They satisfy thefollowing inequality

(i) 1198911ge 1198911

(ii) 119891119894

le 119891119894and 119891

119894+1ge 119891119894+1

in nonfrequency bandconstraints

Here 1198911and 119891

1are natural and low bound of the first order

frequency respectivelyFor elastic structure the usual relation between frequency

f and eigenvalue 120582 is 120582 = (2120587119891)2 Therefore the frequency

constraints can apparently be transformed into eigenvalueconstraints using the formula Here we uniformly use 119892(120582) le

120582 to generalize (i) and (ii) based on 120582 = (2120587119891)2

Mathematical Problems in Engineering 3

Thus the model of continuum topology optimizationwith frequency constraints can be formulated as follows

find t isin 119864119873

make 119882 =

119873

sum119894=1

119908119894997888rarr min

st 119892 (120582119895) le 120582119895 (119895 = 1 119869)

0 le 119905119894le 1 (119894 = 1 119873)

(1)

where t 119882 and 119908119894denote the topological design variable

vector the total weight of structure and the 119894th elementweight 119894 and 119895 are the 119894th element and the 119895th order frequencyrespectively and 119869 and 119873 represent the total number ofconstraints and elements

3 Improved Model Based on ICM Method

31 Properties of Filter Function In order to develop themodel ICM method we firstly investigate the essential partof ICMmdashthe filter function Its definition and choosingdetermine the establishment of optimization model and itssolving and further it will make great impact on the finalperformance of topology optimization In order to map thetopological variables from ldquodiscreterdquo to ldquocontinuousrdquo Sui[42] studied the filter function 119891(119905

119894) and summarized the

following properties

(1) Continuity filter function 119891(119905119894) is continuous about

topology variable 119905119894in the range of [0 1]

(2) Differentiability if 119905119894isin [0 1] 119891(119905

119894) is continuously

differential(3) Monotonicity 119891(119905

119894) is strictly monotonous

(4) Convexity 119891(119905119894) is strictly convex in [0 1]

(5) Approximation the approximation is given as follows

119891 (119905119894)

997888rarr 1 (119905119894gt 120576)

= 0 (119905119894997888rarr 0)

(2)

here 120576 is an arbitrary small positiveHowever the approximation property is not easy to

control it is a limit process To achieve the approximationwe introduce the following formula

119891 (119905119894) isin [0 1] 119891 (0) = 0 119891 (1) = 1 (3)

Consider int10119891(119905119894)119889119909 lt 120576 here 120576 is a given arbitrary small

positive and 0 lt 120576 lt 12

32 Improved Filter Functions Several types of filter functionare suggested in ICMmethod among which Power Function(PF) is used frequently in application [46] and is as follows

119891 (119905119894) = 119905120572

119894 120572 ge 1 (4)

Here 119905119894denotes 119894th design variable 120572 is a positive constant

We introduce a new filter function-composite exponen-tial function (CEF) to take the place of the old one and it is asfollows

119891 (119905119894) =

119890119905119894120574 minus 1

1198901120574 minus 1 120574 gt 0 (5)

120574 is a given positive constant and it is determined by numer-ical experiments with different problems In Section 5 wecompare the performance of the two types of filter function

33 Improved Model of Frequency Topology OptimizationDenote 119891

119908(119905119894) 119891119896(119905119894) and 119891

119898(119905119894) as filter functions for fre-

quency topology optimization and they are given as follows

119908119894= 119891119908(119905119894) 1199080

119894 k

119894= 119891119896(119905119894) k0119894 m

119894= 119891119898(119905119894)m0119894

(6)

Here1199080119894 k0119894 andm0

119894are the element weight element stiffness

matrix and element mass matrix of original structure beforethe process of topology optimization respectively119908

119894 k119894 and

m119894are the ones in the process of topology optimization

respectivelyBased on the above analysis we build the model of

topology optimization with frequency constraints as follows

find t isin 119864119873

make 119882 =

119873

sum119894=1

119891119908(119905119894) 1199080

119894997888rarr min

st 119892 (120582119895(119891119896(119905119894) 119891119898(119905119894))) le 120582

119895 (119895 = 1 119869)

0 le 119905119894le 1 (119894 = 1 119873)

(7)

In what follows we face the problem of approximating thefunction of frequency constraints explicitly

In the finite element analysis the dynamic behavior ofa continuum structure can be represented by the followinggeneral eigenvalue problem

(K minus 120582119895M) u119895= 0 (8)

whereK is the global stiffnessmatrix andM is the globalmassmatrix 120582

119895is the 119895th eigenvalue and u

119895is the eigenvector cor-

responding to 120582119895 The eigenvalue 120582

119895and the corresponding

eigenvector u119895are related to each other by Rayleigh quotient

120582119895=

u119879119894Ku119894

u119879119894Mu119894

(9)

Since eigenvalue 120582119895is implicitly related with topology

variable t we use first-order Taylor series expansion foreigenvalue to express their relationship explicitly

Take the reciprocal of stiffness filter function as designvariables as follows

119909119894=

1

119891119896(119905119894) (10)

4 Mathematical Problems in Engineering

We have

119905119894= 119891minus1

119896(119909119894) (11)

Therefore the stiffness matrix filter function mass matrixfilter function and weight filter function are given as follows

119891119896(119905119894) =

1

119909119894

119891119898(119905119894) = 119891119898[119891minus1

119896(

1

119909119894

)]

119891119908(119905119894) = 119891119908[119891minus1

119896(

1

119909119894

)]

(12)

The global stiffness matrix K and the mass matrix M can becalculated by

K =

119873

sum119894=1

k119894=

119873

sum119894=1

119891119896(119905119894) k119894=

119873

sum119894=1

1

119909119894

k0119894

M =

119873

sum119894=1

m119894=

119873

sum119894=1

119891119898(119905119894)m0119894=

119873

sum119894=1

119891119898[119891minus1

119896(

1

119909119894

)]m0119894

(13)

In view of (9) we have the derivative of 120582119895to design variable

as follows120597120582119895

120597119909119894

= u119879119895

120597K120597119909119894

u119895minus 120582119895u119879119895

120597M120597119909119894

u119895 (14)

Substituting (13) to (14) we have

120597120582119895

120597119909119894

= minusu119879119895

2k119894

2119909119894

u119895+ 120573 (119909

119894) 120582119895u119879119895

2m119894

2119909119894

u119895

= minus2

119909119894

(119880119894119895minus 120573 (119909

119894) 119881119894119895)

(15)

where

120573 (119909119894) =

1198911015840

119898[119891minus1

119896(1119909119894)] 119891119896(1119909119894)

119891119898[119891minus1119896

(1119909119894)] 1198911015840119896(1119909119894)=

1198911015840

119898(119905119894) 119891119896(119905119894)

119891119898(119905119894) 1198911015840119896(119905119894) (16)

In (15) 119880119894119895

= (12)u119879119895k119894u119895and 119881

119894119895= (12)120582

119895u119879119894m119894u119895

represent the strain energy and the kinetic energy of 119894thelement corresponding to the jth eigenmode respectively Atthis moment the derivatives of eigenvalue with respect toall design variables can be obtained by subtracting the strainenergy and kinetic energy for element mode from the resultsof modal analyses

Using the first-order Taylor series expansion the approx-imate expression of eigenvalue 120582

119895(t) can be obtained

120582119895 (t) = 120582

119895 (x) = 120582119895(x(])) +

119873

sum119894=1

120597120582119895

120597119909119894

(119909119894minus 119909(])119894

)

= 120582119895(x(])) +

119873

sum119894=1

2 (119880(])119894119895

minus 120573 (119909(])119894

)119881(])119894119895

)

minus

119873

sum119894=1

2

119909(])119894

(119880(])119894119895

minus 120573 (119909(])119894

)119881(])119894119895

) 119909119894

(17)

where superscript ] is the number at the ]th iteration

If we define 119863 as

119863 =

1 for 120582119895le 120582119895

minus1 for 120582119895ge 120582119895

119895=

120582119895 (120582

119895ge 120582119895)

120582119895 (120582

119895le 120582119895)

(18)

and further define

119860119894119895= 119880(])119894119895

minus 120573 (119909(])119894

)119881(])119894119895

119888119894119895= minus

2

119909(])119894

119860119894119895 119888

119894119895= minus119863119888

119894119895

119889119895= minus120582119895(x(])) minus

119873

sum119894=1

2119860119894119895 119889

119895= 119863 times (

119895+ 119889119895)

(19)

then frequency constraints can be simplified by the followinginequality

119873

sum119894=1

119888119894119895119909119894le 119889119895 (20)

Thus ends the process of explicit approximation of thefrequency constraints

4 Solution of the Improved TopologyOptimization Model

However model (7) is a programming with nonlinear objec-tive and linear constraints following the explicit process offrequency constraints We use second-order Taylor seriesexpansion to approximate the objective function and ignorethe constant item The model is transformed into the follow-ing quadratic programming model

find t isin 119864119873

make 119882 =

119873

sum119894=1

(1198861198941199092

119894+ 119887119894119909119894) 997888rarr min

st119873

sum119894=1

119888119894119895119909119894le 119889119895 (119895 = 1 119869)

1 le 119909119894le 119909119894 (119894 = 1 119873)

(21)

As objective function varies with different filter functionsinvestigation of the different cases following different typesof filter functions is necessary Here we focus on PF and CEF

When PF is applied as the filter function it is given asfollows

119891119908(119905119894) = 119905120574119908

119894 119891

119896(119905119894) = 119905120574119896

119894 119891

119898(119905119894) = 119905120574119898

119894 (22)

where 120574119908 120574119896 and 120574

119898 are all arbitrarily integers greater thanzero In view of (22) we have 119905

119894= 1119909

1120574119896

119894 119905120574119908119894

= 1119909120574119908120574119896

119894=

1119909120572

119894 and 120572 = 120574

119908120574119896

Mathematical Problems in Engineering 5

Therefore the objective function (7) can be rewritten as

119882 =

119873

sum119894=1

119905120574119908

1198941199080

119894=

119873

sum119894=1

1199080

119894

119909120572119894

asymp

119873

sum119894=1

(1198861198941199092+ 119887119894119909) (23)

where 119886119894= 120572(120572+1)119908

0

1198942(119909119894)120572+2 and 119887

119894= minus120572(120572+2)119908

0

119894(119909119894)120572+1

are undetermined parametersWhen CEF is applied as the filter function it is given as

follows

119891119908(119905119894) =

119890119905119894120574119908 minus 1

1198901120574119908 minus 1 119891

119896(119905119894) =

119890119905119894120574119896 minus 1

1198901120574119896 minus 1

119891119898(119905119894) =

119890119905119894120574119898 minus 1

1198901120574119898 minus 1

(24)

We have 119909119894= 1119891

119896(119905119894) = (119890

1120574119896 minus 1)(119890119905119894120574119896 minus 1) and therefore

119891119908(119905119894) =

((1198901120574119896 minus 1) 119909

119894+ 1)120574119896120574119908

minus 1

1198901120574119908 minus 1

119891119898(119905119894) =

((1198901120574119896 minus 1) 119909

119894+ 1)120574119896120574119898

minus 1

1198901120574119898 minus 1

(25)

Similarly the objective function in (7) can be expressed as

119873

sum119894=1

((1198901120574119896 minus 1) 119909

119894+ 1)120574119896120574119908

minus 1

1198901120574119908 minus 11199080

119894asymp

119873

sum119894=1

(1198861198941199092

119894+ 119887119894119909119894) (26)

where

119886119894=

1

2

120574119896

120574119908

1199080

119894

(119909(V)119894

)4

1198901120574119896 minus 1

1198901120574119908 minus 1(

1198901120574119896 minus 1

119909(V)119894

+ 1)

120574119896120574119908minus2

sdot [(120574119896

120574119908

+ 1) (1198901120574119896 minus 1) + 2119909

(V)119894

]

119887119894= minus

120574119896

120574119908

1199080

119894

(119909(V)119894

)3

1198901120574119896 minus 1

1198901120574119908 minus 1(

1198901120574119896 minus 1

119909(V)119894

+ 1)

120574119896120574119908minus2

sdot [(120574119896

120574119908

+ 2) (1198901120574119896 minus 1) + 3119909

(V)119894

]

(27)

They are two undetermined parametersThe dual model of (21) is

find zisin E119869

make minus Φ (z) 997888rarr min

st z119895ge 0 (119895 = 1 119869)

(28)

where z is the design variables in dual model

Φ (z) = min1le119905119894le119909

(119871 (x z))

119871 (x z) =

119873

sum119894=1

(1198861198941199092

119894+ 119887119894119909119894) +

119869

sum119895=1

(119911119895(

119873

sum119894=1

119888119894119895119909119894minus 119889119895))

(29)

Here 119871(x z) is a Lagrange augmented function in (28)

Start

Construct the finite element model and input the optimal parameters

Finite element analysis with MSCNastran

Formulate the mathematical optimal model (7)

Solve (21) with DSQP and update variables

End

No

Yes

|W(+1) minus W()|

|W()|le 120576

Figure 1 Flow chart of the iterative solution procedure

Then DSQP is employed to solve (28) and the precisionof convergence is controlled to be 0001 With the aboveanalysis and solving of (28) the optimal topology structurewith continuous design variables is obtained

5 Numerical Simulation

Based on the above analysis we outline the following compu-tational procedure with a flow chart (Figure 1) it is used formaximizing theweight of structure constrained by frequency

Following the above flow chart we develop a programbased on the platform of MSC Patran amp Nastran to solve theimproved topology optimal model In order to demonstratethe effectiveness of the proposed method two structuresare presented as inputs The first one is a cantilever withfundamental frequency constraint the second one is a 3Dbeam with two concentrated masses by multiple frequencyconstraints For the computation the initial values of topol-ogy variables are all given as unit (119905 = 1) the lowest bounds oftopology variables and the convergence precision values are001 and 0001 respectively

51 Example 1 Example 1 is a cantilever with size 100 times 50times 50mm3 It is fixed at one end as shown in Figure 2 and aconcentrated mass Mc = 50 g is attached at the center of theother end Youngrsquos modulus 119864 = 100GPa Poissonrsquos ratio 120583 =

03 andmass density 120588= 1000 kgm3The structure is dividedinto 30 times 16 times 16 eight-node hexahedron elements The5010Hz fundamental frequency of the cantilever is calculatedby finite element analysis andwe set the frequency constraintfor the design problem as 119891

1ge 3500Hz

The solving topology configuration of the cantilever withdifferent filter functions is given in Figure 3 Figures 3(a) and3(b) show the whole optimal structure and (c)-(d) show the

6 Mathematical Problems in Engineering

50mm

50mm

100mm

Mc

Figure 2 A cantilever structure

(a) (b)

(c) (d)

Figure 3 Topology configuration with different filter functions (a) Topology configuration with PF (b) topology configuration with CEF(c)-(d) the half optimal structure of (a)-(b)

half structure corresponding to (a)-(b) The performance oftopology optimization with different filter functions is givenin Table 1 The iterative curve of computation with differentfilter functions is described in Figure 4

52 Example 2 The beam (40 times 10 times 10 cm3) with two endsfixed is presented in Figure 5 Two concentrated masses with1198721198881

= 1198721198882

= 5 kg are located on the upper surface of 13and 23 span Youngrsquos modulus E = 100GPa Poissonrsquos ratio120583 = 03 and mass density 120588 = 1000 kgm3 The structure

is divided into 30 times 8 times 8 eight-node hexahedron elementsThe first computed fundamental frequency of the beam is13435Hz and the second one is 14951 Hz The set constraintfrequencies are 119891

1ge 1000Hz 119891

2ge 1100Hz

The solving topology configuration of the beam withdifferent filter functions is given in Figure 6 The perfor-mances of topology optimization with different filter func-tions are given in Table 2 To describe the dynamics ofoptimal structure the first three modal shapes of optimalstructure with two different filter functions are computedand displayed in Figure 7 The frequency changing with time

Mathematical Problems in Engineering 7

260

240

220

200

180

160

140

120

100

80

60

Mas

s (g)

PFCEF

0 10 20 30 40

Iteration

(a)

34

36

38

40

42

44

46

48

50

52

Freq

uenc

y (H

z)

times103

0 10 20 30 40

Iteration

PFCEF

(b)

Figure 4 Iteration process with different filter function

L

H

Mc1 Mc2

Figure 5 The beam with two ends fixed

Table 1 Performance of topology optimization with differentalgorithms and filter functions

Method DSQP

PFIteration 411198911(Hz) 35020270996

Mass (g) 73033882813

CEFIteration 341198911(Hz) 35019787598

Mass (g) 70509476563

in the optimization process is presented in Figure 8 withdifferent filter functions

6 Conclusion

In this paper an improved frequency topology optimizationmodel of three-dimensional continuum structure is devel-oped based on ICM method CEF a new filter function isselected to recognize the design variables as well as to imple-ment much better the changing process of design variablesfrom ldquodiscreterdquo to ldquocontinuousrdquo and back to ldquodiscreterdquo Explicit

Table 2 Performance of topology optimization with differentalgorithms and filter functions

Method DSQP

PF

Iteration 331198911(Hz) 1000392

1198912(Hz) 1100647

Mass (g) 1760004

CEF

Iteration 301198911(Hz) 1000551

1198912(Hz) 1100791

Mass (g) 1682534

formulations of frequency constraints are given based onfilter functions and first-order Taylor series expansion andby extracting structural strain and structural kinetic energyfrom the results of structural modal analysis An improvedoptimal model is formulated using CEF and the explicitfrequency constraints The program based on DSQP forsolving the optimal model is developed on the platform ofMSC Patran amp Nastran

Finally two examples of three-dimensional continuumstructure show that clear and stable configurations can beobtained using ICM method We find that configurationscomputed with DSQP combined PF and DSQP combinedCEF are similar between one and the other in the case of fun-damental frequency constraint But we can find that DSQPcombined CEF has the best performance for the optimizationexample from the point of view of optimal objective anditerative numbers We also compute the first three ordermodal shapes of optimal structure with two different filterfunctions in multiple frequency constraints The vibrationmodes from two filter functions are the same and the iteration

8 Mathematical Problems in Engineering

(a) (b)

Figure 6 Topology configuration with different filter functions (a) Topology configuration with PF (b) topology configuration with CEF

Mod

e

First

PF

(a)

Modal shapesSecond

(b)

Third

(c)

CEF

Mod

e

(d) (e) (f)

Figure 7 The first three modal shapes of optimal structure with two filter functions

0 5 10 15 20 25 30 35

Eige

nfre

quen

cy (H

z)

Iterationminus5

f1f2f3

18171616161515141413131211111110

095

times103

(a)

0 5 10 15 20 25 30Iteration

f1f2f3

Eige

nfre

quen

cy (H

z)

18171616161515141413131211111110

095

times103

(b)

Figure 8 Iteration process of the first three frequencies with different filter functions (a) PF (b) CEF

Mathematical Problems in Engineering 9

curves of the first three order frequencies show that structuralfrequencies satisfy the frequency constraints Therefore allthe computing results indicate that the proposed method isalso feasible for multiple frequency constraints

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of the paper

Acknowledgment

The work of this project was supported by National ScienceFoundation of China (ContractGrant no 11072009 andContractGrant no 11172013)

References

[1] M P Bendsoslashe and N Kikuchi ldquoGenerating optimal topologiesin structural design using a homogenization methodrdquo Com-puterMethods in AppliedMechanics and Engineering vol 71 no2 pp 197ndash224 1988

[2] H A Eschenauer and N Olhoff ldquoTopology optimization ofcontinuum structures a reviewrdquo Applied Mechanics Reviewsvol 54 no 4 pp 331ndash390 2001

[3] J D Deaton and R V Grandhi ldquoA survey of structuraland multidisciplinary continuum topology optimization post2000rdquo Structural and Multidisciplinary Optimization vol 49no 1 pp 1ndash38 2014

[4] G I N Rozvany ldquoAims scope methods history and unifiedterminology of computer-aided topology optimization in struc-tural mechanicsrdquo Structural and Multidisciplinary Optimiza-tion vol 21 no 2 pp 90ndash108 2001

[5] M P Bendsoslashe andO Sigmund Topology OptimizationTheoryMethods and Applications Springer Berlin Germany 2ndedition 2003

[6] B Hassani and E Hinton ldquoA review of homogenization andtopology optimization Imdashhomogenization theory for mediawith periodic structurerdquo Computers amp Structures vol 69 no 6pp 707ndash717 1998

[7] G I N Rozvany and M Zhou ldquoApplications of the COCalgorithm in layout optimizationrdquo in Engineering Optimizationin Design Processes Proceedings of the International ConferenceKarlsruhe Nuclear Research Center Germany September 3-41990 H Eschenauer C Matteck and N Olhoff Eds vol 63of Lecture Notes in Engineering pp 59ndash70 Springer BerlinGermany 1991

[8] R J Yang and C H Chuang ldquoOptimal topology design usinglinear programmingrdquo Computers and Structures vol 52 no 2pp 265ndash275 1994

[9] M P Bendsoslashe andO Sigmund ldquoMaterial interpolation schemesin topology optimizationrdquoArchive of AppliedMechanics vol 69no 9-10 pp 635ndash654 1999

[10] YM Xie andG P Steven ldquoA simple evolutionary procedure forstructural optimizationrdquo Computers and Structures vol 49 no5 pp 885ndash896 1993

[11] YMXie andG P StevenEvolutionary StructuralOptimizationSpringer Berlin Germany 1997

[12] G I N Rozvany O M Querin Z Gaspar and V PomezanskildquoWeight-increasing effect of topology simplificationrdquo Structural

and Multidisciplinary Optimization vol 25 no 5-6 pp 459ndash465 2003

[13] O M Querin G P Steven and Y M Xie ldquoEvolutionarystructural optimisation (ESO) using a bidirectional algorithmrdquoEngineering Computations vol 15 no 8 pp 1031ndash1048 1998

[14] J H ZhuWH Zhang andK P Qiu ldquoBi-directional evolution-ary topology optimization using element replaceable methodrdquoComputational Mechanics vol 40 no 1 pp 97ndash109 2007

[15] X Huang and Y M Xie ldquoA new look at ESO and BESOoptimization methodsrdquo Structural and Multidisciplinary Opti-mization vol 35 no 1 pp 89ndash92 2008

[16] X Huang and YM Xie ldquoA further review of ESO typemethodsfor topology optimizationrdquo Structural and MultidisciplinaryOptimization vol 41 no 5 pp 671ndash683 2010

[17] X Huang Z H Zuo and Y M Xie ldquoEvolutionary topologicaloptimization of vibrating continuum structures for naturalfrequenciesrdquoComputers and Structures vol 88 no 5-6 pp 357ndash364 2010

[18] G I N Rozvany ldquoA critical review of established methods ofstructural topology optimizationrdquo Structural and Multidisci-plinary Optimization vol 37 no 3 pp 217ndash237 2009

[19] S Osher and R P Fedkiw ldquoLevel set methods an overview andsome recent resultsrdquo Journal of Computational Physics vol 169no 2 pp 463ndash502 2001

[20] G Allaire F Jouve and A M Toader ldquoStructural optimizationusing sensitivity analysis and a level-set methodrdquo Journal ofComputational Physics vol 194 no 1 pp 363ndash393 2004

[21] J Luo Z Luo S Chen L Tong and M Y Wang ldquoA newlevel set method for systematic design of hinge-free compliantmechanismsrdquo Computer Methods in Applied Mechanics andEngineering vol 198 no 2 pp 318ndash331 2008

[22] M Y Wang S Chen X Wang and Y Mei ldquoDesign ofmultimaterial compliant mechanisms using level-set methodsrdquoTransactions of the ASME Journal of Mechanical Design vol 127no 5 pp 941ndash956 2005

[23] M Y Wang X Wang and D Guo ldquoA level set methodfor structural topology optimizationrdquo Computer Methods inApplied Mechanics and Engineering vol 192 no 1-2 pp 227ndash246 2003

[24] B Bourdin and A Chambolle ldquoDesign-dependent loads intopology optimizationrdquo ESAIM Control Optimisation andCalculus of Variations vol 9 pp 19ndash48 2003

[25] S Zhou and M Y Wang ldquoMultimaterial structural topologyoptimization with a generalized Cahn-Hilliard model of mul-tiphase transitionrdquo Structural and Multidisciplinary Optimiza-tion vol 33 no 2 pp 89ndash111 2007

[26] A Takezawa S Nishiwaki and M Kitamura ldquoShape andtopology optimization based on the phase field method andsensitivity analysisrdquo Journal of Computational Physics vol 229no 7 pp 2697ndash2718 2010

[27] A R Dıaaz and N Kikuchi ldquoSolutions to shape and topologyeigenvalue optimization problems using a homogenizationmethodrdquo International Journal for Numerical Methods in Engi-neering vol 35 no 7 pp 1487ndash1502 1992

[28] Z D Ma H C Cheng and N Kikuchi ldquoStructural design forobtaining desired eigenfrequencies by using the topology andshape optimizationmethodrdquoComputing Systems in Engineeringvol 5 no 1 pp 77ndash89 1994

[29] Z DMa N Kikuchi andH-C Cheng ldquoTopological design forvibrating structuresrdquo Computer Methods in Applied Mechanicsand Engineering vol 121 no 1ndash4 pp 259ndash280 1995

10 Mathematical Problems in Engineering

[30] I Kosaka and C C Swan ldquoA summetry reduction methodfor continuum structural topology optimizationrdquo Computers ampStructures vol 70 no 1 pp 47ndash61 1999

[31] L A Krog and N Olhoff ldquoOptimum topology and reinforce-ment design of disk and plate structures with multiple stiffnessand eigenfrequency objectivesrdquoComputers amp Structures vol 72no 4 pp 535ndash563 1999

[32] J S Jensen and N L Pedersen ldquoOn maximal eigenfrequencyseparation in two-material structures the 1D and 2D scalarcasesrdquo Journal of Sound andVibration vol 289 no 4-5 pp 967ndash986 2006

[33] N L Pedersen ldquoMaximization of eigenvalues using topologyoptimizationrdquo Structural and Multidisciplinary Optimizationvol 20 no 1 pp 2ndash11 2000

[34] J H Zhu and W H Zhang ldquoMaximization of structuralnatural frequency with optimal support layoutrdquo Structural andMultidisciplinaryOptimization vol 31 no 6 pp 462ndash469 2006

[35] J Du and N Olhoff ldquoTopological design of freely vibratingcontinuum structures for maximum values of simple andmultiple eigenfrequencies and frequency gapsrdquo Structural andMultidisciplinary Optimization vol 34 no 2 pp 91ndash110 2007

[36] B Niu J Yan and G Cheng ldquoOptimum structure with homo-geneous optimum cellular material for maximum fundamentalfrequencyrdquo Structural and Multidisciplinary Optimization vol39 no 2 pp 115ndash132 2009

[37] Q Xia T Shi and M Y Wang ldquoA level set based shape andtopology optimization method for maximizing the simple orrepeated first eigenvalue of structure vibrationrdquo Structural andMultidisciplinary Optimization vol 43 no 4 pp 473ndash485 2011

[38] Q Xia T Shi M Y Wang and S Liu ldquoA level set basedmethod for the optimization of cast partrdquo Structural andMultidisciplinary Optimization vol 41 no 5 pp 735ndash747 2010

[39] A K Nandy andC S Jog ldquoOptimization of vibrating structuresto reduce radiated noiserdquo Structural and MultidisciplinaryOptimization vol 45 no 5 pp 717ndash728 2012

[40] J Zheng S Long and G Li ldquoTopology optimization forfree vibrating continuum structures based on the element freeGalerkin methodrdquo Structural and Multidisciplinary Optimiza-tion vol 45 no 1 pp 119ndash127 2012

[41] T D Tsai and C C Cheng ldquoStructural design for desired eigen-frequencies and mode shapes using topology optimizationrdquoStructural and Multidisciplinary Optimization vol 47 no 5 pp673ndash686 2013

[42] K Sui Y Modeling Transformation and Optimization NewDevelopments of Structural Synthesis Method Dalian Universityof Technology Press Dalian China 1996

[43] YK Sui J X LiuH L Ye and J ZDu ldquoApplication ofmodifiedsigmoid function in structural topology optimizationrdquo Journalof BeijingUniversity of Technology vol 34 supplement 1 pp 1ndash62008

[44] P E Gill W Murray and M A Saunders ldquoSNOPT anSQP algorithm for large-scale constrained optimizationrdquo SIAMReview vol 47 no 1 pp 99ndash131 2005

[45] J Nocedal and S J Wright Numerical Optimization SpringerBerlin Germany 1999

[46] Y K Sui H L Ye J X Liu S Chen and H P Yu ldquoAstructural topological optimization method based on exploringconceptual rootrdquo Engineering Mechanics vol 25 no 2 pp 7ndash192008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Thus the model of continuum topology optimizationwith frequency constraints can be formulated as follows

find t isin 119864119873

make 119882 =

119873

sum119894=1

119908119894997888rarr min

st 119892 (120582119895) le 120582119895 (119895 = 1 119869)

0 le 119905119894le 1 (119894 = 1 119873)

(1)

where t 119882 and 119908119894denote the topological design variable

vector the total weight of structure and the 119894th elementweight 119894 and 119895 are the 119894th element and the 119895th order frequencyrespectively and 119869 and 119873 represent the total number ofconstraints and elements

3 Improved Model Based on ICM Method

31 Properties of Filter Function In order to develop themodel ICM method we firstly investigate the essential partof ICMmdashthe filter function Its definition and choosingdetermine the establishment of optimization model and itssolving and further it will make great impact on the finalperformance of topology optimization In order to map thetopological variables from ldquodiscreterdquo to ldquocontinuousrdquo Sui[42] studied the filter function 119891(119905

119894) and summarized the

following properties

(1) Continuity filter function 119891(119905119894) is continuous about

topology variable 119905119894in the range of [0 1]

(2) Differentiability if 119905119894isin [0 1] 119891(119905

119894) is continuously

differential(3) Monotonicity 119891(119905

119894) is strictly monotonous

(4) Convexity 119891(119905119894) is strictly convex in [0 1]

(5) Approximation the approximation is given as follows

119891 (119905119894)

997888rarr 1 (119905119894gt 120576)

= 0 (119905119894997888rarr 0)

(2)

here 120576 is an arbitrary small positiveHowever the approximation property is not easy to

control it is a limit process To achieve the approximationwe introduce the following formula

119891 (119905119894) isin [0 1] 119891 (0) = 0 119891 (1) = 1 (3)

Consider int10119891(119905119894)119889119909 lt 120576 here 120576 is a given arbitrary small

positive and 0 lt 120576 lt 12

32 Improved Filter Functions Several types of filter functionare suggested in ICMmethod among which Power Function(PF) is used frequently in application [46] and is as follows

119891 (119905119894) = 119905120572

119894 120572 ge 1 (4)

Here 119905119894denotes 119894th design variable 120572 is a positive constant

We introduce a new filter function-composite exponen-tial function (CEF) to take the place of the old one and it is asfollows

119891 (119905119894) =

119890119905119894120574 minus 1

1198901120574 minus 1 120574 gt 0 (5)

120574 is a given positive constant and it is determined by numer-ical experiments with different problems In Section 5 wecompare the performance of the two types of filter function

33 Improved Model of Frequency Topology OptimizationDenote 119891

119908(119905119894) 119891119896(119905119894) and 119891

119898(119905119894) as filter functions for fre-

quency topology optimization and they are given as follows

119908119894= 119891119908(119905119894) 1199080

119894 k

119894= 119891119896(119905119894) k0119894 m

119894= 119891119898(119905119894)m0119894

(6)

Here1199080119894 k0119894 andm0

119894are the element weight element stiffness

matrix and element mass matrix of original structure beforethe process of topology optimization respectively119908

119894 k119894 and

m119894are the ones in the process of topology optimization

respectivelyBased on the above analysis we build the model of

topology optimization with frequency constraints as follows

find t isin 119864119873

make 119882 =

119873

sum119894=1

119891119908(119905119894) 1199080

119894997888rarr min

st 119892 (120582119895(119891119896(119905119894) 119891119898(119905119894))) le 120582

119895 (119895 = 1 119869)

0 le 119905119894le 1 (119894 = 1 119873)

(7)

In what follows we face the problem of approximating thefunction of frequency constraints explicitly

In the finite element analysis the dynamic behavior ofa continuum structure can be represented by the followinggeneral eigenvalue problem

(K minus 120582119895M) u119895= 0 (8)

whereK is the global stiffnessmatrix andM is the globalmassmatrix 120582

119895is the 119895th eigenvalue and u

119895is the eigenvector cor-

responding to 120582119895 The eigenvalue 120582

119895and the corresponding

eigenvector u119895are related to each other by Rayleigh quotient

120582119895=

u119879119894Ku119894

u119879119894Mu119894

(9)

Since eigenvalue 120582119895is implicitly related with topology

variable t we use first-order Taylor series expansion foreigenvalue to express their relationship explicitly

Take the reciprocal of stiffness filter function as designvariables as follows

119909119894=

1

119891119896(119905119894) (10)

4 Mathematical Problems in Engineering

We have

119905119894= 119891minus1

119896(119909119894) (11)

Therefore the stiffness matrix filter function mass matrixfilter function and weight filter function are given as follows

119891119896(119905119894) =

1

119909119894

119891119898(119905119894) = 119891119898[119891minus1

119896(

1

119909119894

)]

119891119908(119905119894) = 119891119908[119891minus1

119896(

1

119909119894

)]

(12)

The global stiffness matrix K and the mass matrix M can becalculated by

K =

119873

sum119894=1

k119894=

119873

sum119894=1

119891119896(119905119894) k119894=

119873

sum119894=1

1

119909119894

k0119894

M =

119873

sum119894=1

m119894=

119873

sum119894=1

119891119898(119905119894)m0119894=

119873

sum119894=1

119891119898[119891minus1

119896(

1

119909119894

)]m0119894

(13)

In view of (9) we have the derivative of 120582119895to design variable

as follows120597120582119895

120597119909119894

= u119879119895

120597K120597119909119894

u119895minus 120582119895u119879119895

120597M120597119909119894

u119895 (14)

Substituting (13) to (14) we have

120597120582119895

120597119909119894

= minusu119879119895

2k119894

2119909119894

u119895+ 120573 (119909

119894) 120582119895u119879119895

2m119894

2119909119894

u119895

= minus2

119909119894

(119880119894119895minus 120573 (119909

119894) 119881119894119895)

(15)

where

120573 (119909119894) =

1198911015840

119898[119891minus1

119896(1119909119894)] 119891119896(1119909119894)

119891119898[119891minus1119896

(1119909119894)] 1198911015840119896(1119909119894)=

1198911015840

119898(119905119894) 119891119896(119905119894)

119891119898(119905119894) 1198911015840119896(119905119894) (16)

In (15) 119880119894119895

= (12)u119879119895k119894u119895and 119881

119894119895= (12)120582

119895u119879119894m119894u119895

represent the strain energy and the kinetic energy of 119894thelement corresponding to the jth eigenmode respectively Atthis moment the derivatives of eigenvalue with respect toall design variables can be obtained by subtracting the strainenergy and kinetic energy for element mode from the resultsof modal analyses

Using the first-order Taylor series expansion the approx-imate expression of eigenvalue 120582

119895(t) can be obtained

120582119895 (t) = 120582

119895 (x) = 120582119895(x(])) +

119873

sum119894=1

120597120582119895

120597119909119894

(119909119894minus 119909(])119894

)

= 120582119895(x(])) +

119873

sum119894=1

2 (119880(])119894119895

minus 120573 (119909(])119894

)119881(])119894119895

)

minus

119873

sum119894=1

2

119909(])119894

(119880(])119894119895

minus 120573 (119909(])119894

)119881(])119894119895

) 119909119894

(17)

where superscript ] is the number at the ]th iteration

If we define 119863 as

119863 =

1 for 120582119895le 120582119895

minus1 for 120582119895ge 120582119895

119895=

120582119895 (120582

119895ge 120582119895)

120582119895 (120582

119895le 120582119895)

(18)

and further define

119860119894119895= 119880(])119894119895

minus 120573 (119909(])119894

)119881(])119894119895

119888119894119895= minus

2

119909(])119894

119860119894119895 119888

119894119895= minus119863119888

119894119895

119889119895= minus120582119895(x(])) minus

119873

sum119894=1

2119860119894119895 119889

119895= 119863 times (

119895+ 119889119895)

(19)

then frequency constraints can be simplified by the followinginequality

119873

sum119894=1

119888119894119895119909119894le 119889119895 (20)

Thus ends the process of explicit approximation of thefrequency constraints

4 Solution of the Improved TopologyOptimization Model

However model (7) is a programming with nonlinear objec-tive and linear constraints following the explicit process offrequency constraints We use second-order Taylor seriesexpansion to approximate the objective function and ignorethe constant item The model is transformed into the follow-ing quadratic programming model

find t isin 119864119873

make 119882 =

119873

sum119894=1

(1198861198941199092

119894+ 119887119894119909119894) 997888rarr min

st119873

sum119894=1

119888119894119895119909119894le 119889119895 (119895 = 1 119869)

1 le 119909119894le 119909119894 (119894 = 1 119873)

(21)

As objective function varies with different filter functionsinvestigation of the different cases following different typesof filter functions is necessary Here we focus on PF and CEF

When PF is applied as the filter function it is given asfollows

119891119908(119905119894) = 119905120574119908

119894 119891

119896(119905119894) = 119905120574119896

119894 119891

119898(119905119894) = 119905120574119898

119894 (22)

where 120574119908 120574119896 and 120574

119898 are all arbitrarily integers greater thanzero In view of (22) we have 119905

119894= 1119909

1120574119896

119894 119905120574119908119894

= 1119909120574119908120574119896

119894=

1119909120572

119894 and 120572 = 120574

119908120574119896

Mathematical Problems in Engineering 5

Therefore the objective function (7) can be rewritten as

119882 =

119873

sum119894=1

119905120574119908

1198941199080

119894=

119873

sum119894=1

1199080

119894

119909120572119894

asymp

119873

sum119894=1

(1198861198941199092+ 119887119894119909) (23)

where 119886119894= 120572(120572+1)119908

0

1198942(119909119894)120572+2 and 119887

119894= minus120572(120572+2)119908

0

119894(119909119894)120572+1

are undetermined parametersWhen CEF is applied as the filter function it is given as

follows

119891119908(119905119894) =

119890119905119894120574119908 minus 1

1198901120574119908 minus 1 119891

119896(119905119894) =

119890119905119894120574119896 minus 1

1198901120574119896 minus 1

119891119898(119905119894) =

119890119905119894120574119898 minus 1

1198901120574119898 minus 1

(24)

We have 119909119894= 1119891

119896(119905119894) = (119890

1120574119896 minus 1)(119890119905119894120574119896 minus 1) and therefore

119891119908(119905119894) =

((1198901120574119896 minus 1) 119909

119894+ 1)120574119896120574119908

minus 1

1198901120574119908 minus 1

119891119898(119905119894) =

((1198901120574119896 minus 1) 119909

119894+ 1)120574119896120574119898

minus 1

1198901120574119898 minus 1

(25)

Similarly the objective function in (7) can be expressed as

119873

sum119894=1

((1198901120574119896 minus 1) 119909

119894+ 1)120574119896120574119908

minus 1

1198901120574119908 minus 11199080

119894asymp

119873

sum119894=1

(1198861198941199092

119894+ 119887119894119909119894) (26)

where

119886119894=

1

2

120574119896

120574119908

1199080

119894

(119909(V)119894

)4

1198901120574119896 minus 1

1198901120574119908 minus 1(

1198901120574119896 minus 1

119909(V)119894

+ 1)

120574119896120574119908minus2

sdot [(120574119896

120574119908

+ 1) (1198901120574119896 minus 1) + 2119909

(V)119894

]

119887119894= minus

120574119896

120574119908

1199080

119894

(119909(V)119894

)3

1198901120574119896 minus 1

1198901120574119908 minus 1(

1198901120574119896 minus 1

119909(V)119894

+ 1)

120574119896120574119908minus2

sdot [(120574119896

120574119908

+ 2) (1198901120574119896 minus 1) + 3119909

(V)119894

]

(27)

They are two undetermined parametersThe dual model of (21) is

find zisin E119869

make minus Φ (z) 997888rarr min

st z119895ge 0 (119895 = 1 119869)

(28)

where z is the design variables in dual model

Φ (z) = min1le119905119894le119909

(119871 (x z))

119871 (x z) =

119873

sum119894=1

(1198861198941199092

119894+ 119887119894119909119894) +

119869

sum119895=1

(119911119895(

119873

sum119894=1

119888119894119895119909119894minus 119889119895))

(29)

Here 119871(x z) is a Lagrange augmented function in (28)

Start

Construct the finite element model and input the optimal parameters

Finite element analysis with MSCNastran

Formulate the mathematical optimal model (7)

Solve (21) with DSQP and update variables

End

No

Yes

|W(+1) minus W()|

|W()|le 120576

Figure 1 Flow chart of the iterative solution procedure

Then DSQP is employed to solve (28) and the precisionof convergence is controlled to be 0001 With the aboveanalysis and solving of (28) the optimal topology structurewith continuous design variables is obtained

5 Numerical Simulation

Based on the above analysis we outline the following compu-tational procedure with a flow chart (Figure 1) it is used formaximizing theweight of structure constrained by frequency

Following the above flow chart we develop a programbased on the platform of MSC Patran amp Nastran to solve theimproved topology optimal model In order to demonstratethe effectiveness of the proposed method two structuresare presented as inputs The first one is a cantilever withfundamental frequency constraint the second one is a 3Dbeam with two concentrated masses by multiple frequencyconstraints For the computation the initial values of topol-ogy variables are all given as unit (119905 = 1) the lowest bounds oftopology variables and the convergence precision values are001 and 0001 respectively

51 Example 1 Example 1 is a cantilever with size 100 times 50times 50mm3 It is fixed at one end as shown in Figure 2 and aconcentrated mass Mc = 50 g is attached at the center of theother end Youngrsquos modulus 119864 = 100GPa Poissonrsquos ratio 120583 =

03 andmass density 120588= 1000 kgm3The structure is dividedinto 30 times 16 times 16 eight-node hexahedron elements The5010Hz fundamental frequency of the cantilever is calculatedby finite element analysis andwe set the frequency constraintfor the design problem as 119891

1ge 3500Hz

The solving topology configuration of the cantilever withdifferent filter functions is given in Figure 3 Figures 3(a) and3(b) show the whole optimal structure and (c)-(d) show the

6 Mathematical Problems in Engineering

50mm

50mm

100mm

Mc

Figure 2 A cantilever structure

(a) (b)

(c) (d)

Figure 3 Topology configuration with different filter functions (a) Topology configuration with PF (b) topology configuration with CEF(c)-(d) the half optimal structure of (a)-(b)

half structure corresponding to (a)-(b) The performance oftopology optimization with different filter functions is givenin Table 1 The iterative curve of computation with differentfilter functions is described in Figure 4

52 Example 2 The beam (40 times 10 times 10 cm3) with two endsfixed is presented in Figure 5 Two concentrated masses with1198721198881

= 1198721198882

= 5 kg are located on the upper surface of 13and 23 span Youngrsquos modulus E = 100GPa Poissonrsquos ratio120583 = 03 and mass density 120588 = 1000 kgm3 The structure

is divided into 30 times 8 times 8 eight-node hexahedron elementsThe first computed fundamental frequency of the beam is13435Hz and the second one is 14951 Hz The set constraintfrequencies are 119891

1ge 1000Hz 119891

2ge 1100Hz

The solving topology configuration of the beam withdifferent filter functions is given in Figure 6 The perfor-mances of topology optimization with different filter func-tions are given in Table 2 To describe the dynamics ofoptimal structure the first three modal shapes of optimalstructure with two different filter functions are computedand displayed in Figure 7 The frequency changing with time

Mathematical Problems in Engineering 7

260

240

220

200

180

160

140

120

100

80

60

Mas

s (g)

PFCEF

0 10 20 30 40

Iteration

(a)

34

36

38

40

42

44

46

48

50

52

Freq

uenc

y (H

z)

times103

0 10 20 30 40

Iteration

PFCEF

(b)

Figure 4 Iteration process with different filter function

L

H

Mc1 Mc2

Figure 5 The beam with two ends fixed

Table 1 Performance of topology optimization with differentalgorithms and filter functions

Method DSQP

PFIteration 411198911(Hz) 35020270996

Mass (g) 73033882813

CEFIteration 341198911(Hz) 35019787598

Mass (g) 70509476563

in the optimization process is presented in Figure 8 withdifferent filter functions

6 Conclusion

In this paper an improved frequency topology optimizationmodel of three-dimensional continuum structure is devel-oped based on ICM method CEF a new filter function isselected to recognize the design variables as well as to imple-ment much better the changing process of design variablesfrom ldquodiscreterdquo to ldquocontinuousrdquo and back to ldquodiscreterdquo Explicit

Table 2 Performance of topology optimization with differentalgorithms and filter functions

Method DSQP

PF

Iteration 331198911(Hz) 1000392

1198912(Hz) 1100647

Mass (g) 1760004

CEF

Iteration 301198911(Hz) 1000551

1198912(Hz) 1100791

Mass (g) 1682534

formulations of frequency constraints are given based onfilter functions and first-order Taylor series expansion andby extracting structural strain and structural kinetic energyfrom the results of structural modal analysis An improvedoptimal model is formulated using CEF and the explicitfrequency constraints The program based on DSQP forsolving the optimal model is developed on the platform ofMSC Patran amp Nastran

Finally two examples of three-dimensional continuumstructure show that clear and stable configurations can beobtained using ICM method We find that configurationscomputed with DSQP combined PF and DSQP combinedCEF are similar between one and the other in the case of fun-damental frequency constraint But we can find that DSQPcombined CEF has the best performance for the optimizationexample from the point of view of optimal objective anditerative numbers We also compute the first three ordermodal shapes of optimal structure with two different filterfunctions in multiple frequency constraints The vibrationmodes from two filter functions are the same and the iteration

8 Mathematical Problems in Engineering

(a) (b)

Figure 6 Topology configuration with different filter functions (a) Topology configuration with PF (b) topology configuration with CEF

Mod

e

First

PF

(a)

Modal shapesSecond

(b)

Third

(c)

CEF

Mod

e

(d) (e) (f)

Figure 7 The first three modal shapes of optimal structure with two filter functions

0 5 10 15 20 25 30 35

Eige

nfre

quen

cy (H

z)

Iterationminus5

f1f2f3

18171616161515141413131211111110

095

times103

(a)

0 5 10 15 20 25 30Iteration

f1f2f3

Eige

nfre

quen

cy (H

z)

18171616161515141413131211111110

095

times103

(b)

Figure 8 Iteration process of the first three frequencies with different filter functions (a) PF (b) CEF

Mathematical Problems in Engineering 9

curves of the first three order frequencies show that structuralfrequencies satisfy the frequency constraints Therefore allthe computing results indicate that the proposed method isalso feasible for multiple frequency constraints

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of the paper

Acknowledgment

The work of this project was supported by National ScienceFoundation of China (ContractGrant no 11072009 andContractGrant no 11172013)

References

[1] M P Bendsoslashe and N Kikuchi ldquoGenerating optimal topologiesin structural design using a homogenization methodrdquo Com-puterMethods in AppliedMechanics and Engineering vol 71 no2 pp 197ndash224 1988

[2] H A Eschenauer and N Olhoff ldquoTopology optimization ofcontinuum structures a reviewrdquo Applied Mechanics Reviewsvol 54 no 4 pp 331ndash390 2001

[3] J D Deaton and R V Grandhi ldquoA survey of structuraland multidisciplinary continuum topology optimization post2000rdquo Structural and Multidisciplinary Optimization vol 49no 1 pp 1ndash38 2014

[4] G I N Rozvany ldquoAims scope methods history and unifiedterminology of computer-aided topology optimization in struc-tural mechanicsrdquo Structural and Multidisciplinary Optimiza-tion vol 21 no 2 pp 90ndash108 2001

[5] M P Bendsoslashe andO Sigmund Topology OptimizationTheoryMethods and Applications Springer Berlin Germany 2ndedition 2003

[6] B Hassani and E Hinton ldquoA review of homogenization andtopology optimization Imdashhomogenization theory for mediawith periodic structurerdquo Computers amp Structures vol 69 no 6pp 707ndash717 1998

[7] G I N Rozvany and M Zhou ldquoApplications of the COCalgorithm in layout optimizationrdquo in Engineering Optimizationin Design Processes Proceedings of the International ConferenceKarlsruhe Nuclear Research Center Germany September 3-41990 H Eschenauer C Matteck and N Olhoff Eds vol 63of Lecture Notes in Engineering pp 59ndash70 Springer BerlinGermany 1991

[8] R J Yang and C H Chuang ldquoOptimal topology design usinglinear programmingrdquo Computers and Structures vol 52 no 2pp 265ndash275 1994

[9] M P Bendsoslashe andO Sigmund ldquoMaterial interpolation schemesin topology optimizationrdquoArchive of AppliedMechanics vol 69no 9-10 pp 635ndash654 1999

[10] YM Xie andG P Steven ldquoA simple evolutionary procedure forstructural optimizationrdquo Computers and Structures vol 49 no5 pp 885ndash896 1993

[11] YMXie andG P StevenEvolutionary StructuralOptimizationSpringer Berlin Germany 1997

[12] G I N Rozvany O M Querin Z Gaspar and V PomezanskildquoWeight-increasing effect of topology simplificationrdquo Structural

and Multidisciplinary Optimization vol 25 no 5-6 pp 459ndash465 2003

[13] O M Querin G P Steven and Y M Xie ldquoEvolutionarystructural optimisation (ESO) using a bidirectional algorithmrdquoEngineering Computations vol 15 no 8 pp 1031ndash1048 1998

[14] J H ZhuWH Zhang andK P Qiu ldquoBi-directional evolution-ary topology optimization using element replaceable methodrdquoComputational Mechanics vol 40 no 1 pp 97ndash109 2007

[15] X Huang and Y M Xie ldquoA new look at ESO and BESOoptimization methodsrdquo Structural and Multidisciplinary Opti-mization vol 35 no 1 pp 89ndash92 2008

[16] X Huang and YM Xie ldquoA further review of ESO typemethodsfor topology optimizationrdquo Structural and MultidisciplinaryOptimization vol 41 no 5 pp 671ndash683 2010

[17] X Huang Z H Zuo and Y M Xie ldquoEvolutionary topologicaloptimization of vibrating continuum structures for naturalfrequenciesrdquoComputers and Structures vol 88 no 5-6 pp 357ndash364 2010

[18] G I N Rozvany ldquoA critical review of established methods ofstructural topology optimizationrdquo Structural and Multidisci-plinary Optimization vol 37 no 3 pp 217ndash237 2009

[19] S Osher and R P Fedkiw ldquoLevel set methods an overview andsome recent resultsrdquo Journal of Computational Physics vol 169no 2 pp 463ndash502 2001

[20] G Allaire F Jouve and A M Toader ldquoStructural optimizationusing sensitivity analysis and a level-set methodrdquo Journal ofComputational Physics vol 194 no 1 pp 363ndash393 2004

[21] J Luo Z Luo S Chen L Tong and M Y Wang ldquoA newlevel set method for systematic design of hinge-free compliantmechanismsrdquo Computer Methods in Applied Mechanics andEngineering vol 198 no 2 pp 318ndash331 2008

[22] M Y Wang S Chen X Wang and Y Mei ldquoDesign ofmultimaterial compliant mechanisms using level-set methodsrdquoTransactions of the ASME Journal of Mechanical Design vol 127no 5 pp 941ndash956 2005

[23] M Y Wang X Wang and D Guo ldquoA level set methodfor structural topology optimizationrdquo Computer Methods inApplied Mechanics and Engineering vol 192 no 1-2 pp 227ndash246 2003

[24] B Bourdin and A Chambolle ldquoDesign-dependent loads intopology optimizationrdquo ESAIM Control Optimisation andCalculus of Variations vol 9 pp 19ndash48 2003

[25] S Zhou and M Y Wang ldquoMultimaterial structural topologyoptimization with a generalized Cahn-Hilliard model of mul-tiphase transitionrdquo Structural and Multidisciplinary Optimiza-tion vol 33 no 2 pp 89ndash111 2007

[26] A Takezawa S Nishiwaki and M Kitamura ldquoShape andtopology optimization based on the phase field method andsensitivity analysisrdquo Journal of Computational Physics vol 229no 7 pp 2697ndash2718 2010

[27] A R Dıaaz and N Kikuchi ldquoSolutions to shape and topologyeigenvalue optimization problems using a homogenizationmethodrdquo International Journal for Numerical Methods in Engi-neering vol 35 no 7 pp 1487ndash1502 1992

[28] Z D Ma H C Cheng and N Kikuchi ldquoStructural design forobtaining desired eigenfrequencies by using the topology andshape optimizationmethodrdquoComputing Systems in Engineeringvol 5 no 1 pp 77ndash89 1994

[29] Z DMa N Kikuchi andH-C Cheng ldquoTopological design forvibrating structuresrdquo Computer Methods in Applied Mechanicsand Engineering vol 121 no 1ndash4 pp 259ndash280 1995

10 Mathematical Problems in Engineering

[30] I Kosaka and C C Swan ldquoA summetry reduction methodfor continuum structural topology optimizationrdquo Computers ampStructures vol 70 no 1 pp 47ndash61 1999

[31] L A Krog and N Olhoff ldquoOptimum topology and reinforce-ment design of disk and plate structures with multiple stiffnessand eigenfrequency objectivesrdquoComputers amp Structures vol 72no 4 pp 535ndash563 1999

[32] J S Jensen and N L Pedersen ldquoOn maximal eigenfrequencyseparation in two-material structures the 1D and 2D scalarcasesrdquo Journal of Sound andVibration vol 289 no 4-5 pp 967ndash986 2006

[33] N L Pedersen ldquoMaximization of eigenvalues using topologyoptimizationrdquo Structural and Multidisciplinary Optimizationvol 20 no 1 pp 2ndash11 2000

[34] J H Zhu and W H Zhang ldquoMaximization of structuralnatural frequency with optimal support layoutrdquo Structural andMultidisciplinaryOptimization vol 31 no 6 pp 462ndash469 2006

[35] J Du and N Olhoff ldquoTopological design of freely vibratingcontinuum structures for maximum values of simple andmultiple eigenfrequencies and frequency gapsrdquo Structural andMultidisciplinary Optimization vol 34 no 2 pp 91ndash110 2007

[36] B Niu J Yan and G Cheng ldquoOptimum structure with homo-geneous optimum cellular material for maximum fundamentalfrequencyrdquo Structural and Multidisciplinary Optimization vol39 no 2 pp 115ndash132 2009

[37] Q Xia T Shi and M Y Wang ldquoA level set based shape andtopology optimization method for maximizing the simple orrepeated first eigenvalue of structure vibrationrdquo Structural andMultidisciplinary Optimization vol 43 no 4 pp 473ndash485 2011

[38] Q Xia T Shi M Y Wang and S Liu ldquoA level set basedmethod for the optimization of cast partrdquo Structural andMultidisciplinary Optimization vol 41 no 5 pp 735ndash747 2010

[39] A K Nandy andC S Jog ldquoOptimization of vibrating structuresto reduce radiated noiserdquo Structural and MultidisciplinaryOptimization vol 45 no 5 pp 717ndash728 2012

[40] J Zheng S Long and G Li ldquoTopology optimization forfree vibrating continuum structures based on the element freeGalerkin methodrdquo Structural and Multidisciplinary Optimiza-tion vol 45 no 1 pp 119ndash127 2012

[41] T D Tsai and C C Cheng ldquoStructural design for desired eigen-frequencies and mode shapes using topology optimizationrdquoStructural and Multidisciplinary Optimization vol 47 no 5 pp673ndash686 2013

[42] K Sui Y Modeling Transformation and Optimization NewDevelopments of Structural Synthesis Method Dalian Universityof Technology Press Dalian China 1996

[43] YK Sui J X LiuH L Ye and J ZDu ldquoApplication ofmodifiedsigmoid function in structural topology optimizationrdquo Journalof BeijingUniversity of Technology vol 34 supplement 1 pp 1ndash62008

[44] P E Gill W Murray and M A Saunders ldquoSNOPT anSQP algorithm for large-scale constrained optimizationrdquo SIAMReview vol 47 no 1 pp 99ndash131 2005

[45] J Nocedal and S J Wright Numerical Optimization SpringerBerlin Germany 1999

[46] Y K Sui H L Ye J X Liu S Chen and H P Yu ldquoAstructural topological optimization method based on exploringconceptual rootrdquo Engineering Mechanics vol 25 no 2 pp 7ndash192008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

We have

119905119894= 119891minus1

119896(119909119894) (11)

Therefore the stiffness matrix filter function mass matrixfilter function and weight filter function are given as follows

119891119896(119905119894) =

1

119909119894

119891119898(119905119894) = 119891119898[119891minus1

119896(

1

119909119894

)]

119891119908(119905119894) = 119891119908[119891minus1

119896(

1

119909119894

)]

(12)

The global stiffness matrix K and the mass matrix M can becalculated by

K =

119873

sum119894=1

k119894=

119873

sum119894=1

119891119896(119905119894) k119894=

119873

sum119894=1

1

119909119894

k0119894

M =

119873

sum119894=1

m119894=

119873

sum119894=1

119891119898(119905119894)m0119894=

119873

sum119894=1

119891119898[119891minus1

119896(

1

119909119894

)]m0119894

(13)

In view of (9) we have the derivative of 120582119895to design variable

as follows120597120582119895

120597119909119894

= u119879119895

120597K120597119909119894

u119895minus 120582119895u119879119895

120597M120597119909119894

u119895 (14)

Substituting (13) to (14) we have

120597120582119895

120597119909119894

= minusu119879119895

2k119894

2119909119894

u119895+ 120573 (119909

119894) 120582119895u119879119895

2m119894

2119909119894

u119895

= minus2

119909119894

(119880119894119895minus 120573 (119909

119894) 119881119894119895)

(15)

where

120573 (119909119894) =

1198911015840

119898[119891minus1

119896(1119909119894)] 119891119896(1119909119894)

119891119898[119891minus1119896

(1119909119894)] 1198911015840119896(1119909119894)=

1198911015840

119898(119905119894) 119891119896(119905119894)

119891119898(119905119894) 1198911015840119896(119905119894) (16)

In (15) 119880119894119895

= (12)u119879119895k119894u119895and 119881

119894119895= (12)120582

119895u119879119894m119894u119895

represent the strain energy and the kinetic energy of 119894thelement corresponding to the jth eigenmode respectively Atthis moment the derivatives of eigenvalue with respect toall design variables can be obtained by subtracting the strainenergy and kinetic energy for element mode from the resultsof modal analyses

Using the first-order Taylor series expansion the approx-imate expression of eigenvalue 120582

119895(t) can be obtained

120582119895 (t) = 120582

119895 (x) = 120582119895(x(])) +

119873

sum119894=1

120597120582119895

120597119909119894

(119909119894minus 119909(])119894

)

= 120582119895(x(])) +

119873

sum119894=1

2 (119880(])119894119895

minus 120573 (119909(])119894

)119881(])119894119895

)

minus

119873

sum119894=1

2

119909(])119894

(119880(])119894119895

minus 120573 (119909(])119894

)119881(])119894119895

) 119909119894

(17)

where superscript ] is the number at the ]th iteration

If we define 119863 as

119863 =

1 for 120582119895le 120582119895

minus1 for 120582119895ge 120582119895

119895=

120582119895 (120582

119895ge 120582119895)

120582119895 (120582

119895le 120582119895)

(18)

and further define

119860119894119895= 119880(])119894119895

minus 120573 (119909(])119894

)119881(])119894119895

119888119894119895= minus

2

119909(])119894

119860119894119895 119888

119894119895= minus119863119888

119894119895

119889119895= minus120582119895(x(])) minus

119873

sum119894=1

2119860119894119895 119889

119895= 119863 times (

119895+ 119889119895)

(19)

then frequency constraints can be simplified by the followinginequality

119873

sum119894=1

119888119894119895119909119894le 119889119895 (20)

Thus ends the process of explicit approximation of thefrequency constraints

4 Solution of the Improved TopologyOptimization Model

However model (7) is a programming with nonlinear objec-tive and linear constraints following the explicit process offrequency constraints We use second-order Taylor seriesexpansion to approximate the objective function and ignorethe constant item The model is transformed into the follow-ing quadratic programming model

find t isin 119864119873

make 119882 =

119873

sum119894=1

(1198861198941199092

119894+ 119887119894119909119894) 997888rarr min

st119873

sum119894=1

119888119894119895119909119894le 119889119895 (119895 = 1 119869)

1 le 119909119894le 119909119894 (119894 = 1 119873)

(21)

As objective function varies with different filter functionsinvestigation of the different cases following different typesof filter functions is necessary Here we focus on PF and CEF

When PF is applied as the filter function it is given asfollows

119891119908(119905119894) = 119905120574119908

119894 119891

119896(119905119894) = 119905120574119896

119894 119891

119898(119905119894) = 119905120574119898

119894 (22)

where 120574119908 120574119896 and 120574

119898 are all arbitrarily integers greater thanzero In view of (22) we have 119905

119894= 1119909

1120574119896

119894 119905120574119908119894

= 1119909120574119908120574119896

119894=

1119909120572

119894 and 120572 = 120574

119908120574119896

Mathematical Problems in Engineering 5

Therefore the objective function (7) can be rewritten as

119882 =

119873

sum119894=1

119905120574119908

1198941199080

119894=

119873

sum119894=1

1199080

119894

119909120572119894

asymp

119873

sum119894=1

(1198861198941199092+ 119887119894119909) (23)

where 119886119894= 120572(120572+1)119908

0

1198942(119909119894)120572+2 and 119887

119894= minus120572(120572+2)119908

0

119894(119909119894)120572+1

are undetermined parametersWhen CEF is applied as the filter function it is given as

follows

119891119908(119905119894) =

119890119905119894120574119908 minus 1

1198901120574119908 minus 1 119891

119896(119905119894) =

119890119905119894120574119896 minus 1

1198901120574119896 minus 1

119891119898(119905119894) =

119890119905119894120574119898 minus 1

1198901120574119898 minus 1

(24)

We have 119909119894= 1119891

119896(119905119894) = (119890

1120574119896 minus 1)(119890119905119894120574119896 minus 1) and therefore

119891119908(119905119894) =

((1198901120574119896 minus 1) 119909

119894+ 1)120574119896120574119908

minus 1

1198901120574119908 minus 1

119891119898(119905119894) =

((1198901120574119896 minus 1) 119909

119894+ 1)120574119896120574119898

minus 1

1198901120574119898 minus 1

(25)

Similarly the objective function in (7) can be expressed as

119873

sum119894=1

((1198901120574119896 minus 1) 119909

119894+ 1)120574119896120574119908

minus 1

1198901120574119908 minus 11199080

119894asymp

119873

sum119894=1

(1198861198941199092

119894+ 119887119894119909119894) (26)

where

119886119894=

1

2

120574119896

120574119908

1199080

119894

(119909(V)119894

)4

1198901120574119896 minus 1

1198901120574119908 minus 1(

1198901120574119896 minus 1

119909(V)119894

+ 1)

120574119896120574119908minus2

sdot [(120574119896

120574119908

+ 1) (1198901120574119896 minus 1) + 2119909

(V)119894

]

119887119894= minus

120574119896

120574119908

1199080

119894

(119909(V)119894

)3

1198901120574119896 minus 1

1198901120574119908 minus 1(

1198901120574119896 minus 1

119909(V)119894

+ 1)

120574119896120574119908minus2

sdot [(120574119896

120574119908

+ 2) (1198901120574119896 minus 1) + 3119909

(V)119894

]

(27)

They are two undetermined parametersThe dual model of (21) is

find zisin E119869

make minus Φ (z) 997888rarr min

st z119895ge 0 (119895 = 1 119869)

(28)

where z is the design variables in dual model

Φ (z) = min1le119905119894le119909

(119871 (x z))

119871 (x z) =

119873

sum119894=1

(1198861198941199092

119894+ 119887119894119909119894) +

119869

sum119895=1

(119911119895(

119873

sum119894=1

119888119894119895119909119894minus 119889119895))

(29)

Here 119871(x z) is a Lagrange augmented function in (28)

Start

Construct the finite element model and input the optimal parameters

Finite element analysis with MSCNastran

Formulate the mathematical optimal model (7)

Solve (21) with DSQP and update variables

End

No

Yes

|W(+1) minus W()|

|W()|le 120576

Figure 1 Flow chart of the iterative solution procedure

Then DSQP is employed to solve (28) and the precisionof convergence is controlled to be 0001 With the aboveanalysis and solving of (28) the optimal topology structurewith continuous design variables is obtained

5 Numerical Simulation

Based on the above analysis we outline the following compu-tational procedure with a flow chart (Figure 1) it is used formaximizing theweight of structure constrained by frequency

Following the above flow chart we develop a programbased on the platform of MSC Patran amp Nastran to solve theimproved topology optimal model In order to demonstratethe effectiveness of the proposed method two structuresare presented as inputs The first one is a cantilever withfundamental frequency constraint the second one is a 3Dbeam with two concentrated masses by multiple frequencyconstraints For the computation the initial values of topol-ogy variables are all given as unit (119905 = 1) the lowest bounds oftopology variables and the convergence precision values are001 and 0001 respectively

51 Example 1 Example 1 is a cantilever with size 100 times 50times 50mm3 It is fixed at one end as shown in Figure 2 and aconcentrated mass Mc = 50 g is attached at the center of theother end Youngrsquos modulus 119864 = 100GPa Poissonrsquos ratio 120583 =

03 andmass density 120588= 1000 kgm3The structure is dividedinto 30 times 16 times 16 eight-node hexahedron elements The5010Hz fundamental frequency of the cantilever is calculatedby finite element analysis andwe set the frequency constraintfor the design problem as 119891

1ge 3500Hz

The solving topology configuration of the cantilever withdifferent filter functions is given in Figure 3 Figures 3(a) and3(b) show the whole optimal structure and (c)-(d) show the

6 Mathematical Problems in Engineering

50mm

50mm

100mm

Mc

Figure 2 A cantilever structure

(a) (b)

(c) (d)

Figure 3 Topology configuration with different filter functions (a) Topology configuration with PF (b) topology configuration with CEF(c)-(d) the half optimal structure of (a)-(b)

half structure corresponding to (a)-(b) The performance oftopology optimization with different filter functions is givenin Table 1 The iterative curve of computation with differentfilter functions is described in Figure 4

52 Example 2 The beam (40 times 10 times 10 cm3) with two endsfixed is presented in Figure 5 Two concentrated masses with1198721198881

= 1198721198882

= 5 kg are located on the upper surface of 13and 23 span Youngrsquos modulus E = 100GPa Poissonrsquos ratio120583 = 03 and mass density 120588 = 1000 kgm3 The structure

is divided into 30 times 8 times 8 eight-node hexahedron elementsThe first computed fundamental frequency of the beam is13435Hz and the second one is 14951 Hz The set constraintfrequencies are 119891

1ge 1000Hz 119891

2ge 1100Hz

The solving topology configuration of the beam withdifferent filter functions is given in Figure 6 The perfor-mances of topology optimization with different filter func-tions are given in Table 2 To describe the dynamics ofoptimal structure the first three modal shapes of optimalstructure with two different filter functions are computedand displayed in Figure 7 The frequency changing with time

Mathematical Problems in Engineering 7

260

240

220

200

180

160

140

120

100

80

60

Mas

s (g)

PFCEF

0 10 20 30 40

Iteration

(a)

34

36

38

40

42

44

46

48

50

52

Freq

uenc

y (H

z)

times103

0 10 20 30 40

Iteration

PFCEF

(b)

Figure 4 Iteration process with different filter function

L

H

Mc1 Mc2

Figure 5 The beam with two ends fixed

Table 1 Performance of topology optimization with differentalgorithms and filter functions

Method DSQP

PFIteration 411198911(Hz) 35020270996

Mass (g) 73033882813

CEFIteration 341198911(Hz) 35019787598

Mass (g) 70509476563

in the optimization process is presented in Figure 8 withdifferent filter functions

6 Conclusion

In this paper an improved frequency topology optimizationmodel of three-dimensional continuum structure is devel-oped based on ICM method CEF a new filter function isselected to recognize the design variables as well as to imple-ment much better the changing process of design variablesfrom ldquodiscreterdquo to ldquocontinuousrdquo and back to ldquodiscreterdquo Explicit

Table 2 Performance of topology optimization with differentalgorithms and filter functions

Method DSQP

PF

Iteration 331198911(Hz) 1000392

1198912(Hz) 1100647

Mass (g) 1760004

CEF

Iteration 301198911(Hz) 1000551

1198912(Hz) 1100791

Mass (g) 1682534

formulations of frequency constraints are given based onfilter functions and first-order Taylor series expansion andby extracting structural strain and structural kinetic energyfrom the results of structural modal analysis An improvedoptimal model is formulated using CEF and the explicitfrequency constraints The program based on DSQP forsolving the optimal model is developed on the platform ofMSC Patran amp Nastran

Finally two examples of three-dimensional continuumstructure show that clear and stable configurations can beobtained using ICM method We find that configurationscomputed with DSQP combined PF and DSQP combinedCEF are similar between one and the other in the case of fun-damental frequency constraint But we can find that DSQPcombined CEF has the best performance for the optimizationexample from the point of view of optimal objective anditerative numbers We also compute the first three ordermodal shapes of optimal structure with two different filterfunctions in multiple frequency constraints The vibrationmodes from two filter functions are the same and the iteration

8 Mathematical Problems in Engineering

(a) (b)

Figure 6 Topology configuration with different filter functions (a) Topology configuration with PF (b) topology configuration with CEF

Mod

e

First

PF

(a)

Modal shapesSecond

(b)

Third

(c)

CEF

Mod

e

(d) (e) (f)

Figure 7 The first three modal shapes of optimal structure with two filter functions

0 5 10 15 20 25 30 35

Eige

nfre

quen

cy (H

z)

Iterationminus5

f1f2f3

18171616161515141413131211111110

095

times103

(a)

0 5 10 15 20 25 30Iteration

f1f2f3

Eige

nfre

quen

cy (H

z)

18171616161515141413131211111110

095

times103

(b)

Figure 8 Iteration process of the first three frequencies with different filter functions (a) PF (b) CEF

Mathematical Problems in Engineering 9

curves of the first three order frequencies show that structuralfrequencies satisfy the frequency constraints Therefore allthe computing results indicate that the proposed method isalso feasible for multiple frequency constraints

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of the paper

Acknowledgment

The work of this project was supported by National ScienceFoundation of China (ContractGrant no 11072009 andContractGrant no 11172013)

References

[1] M P Bendsoslashe and N Kikuchi ldquoGenerating optimal topologiesin structural design using a homogenization methodrdquo Com-puterMethods in AppliedMechanics and Engineering vol 71 no2 pp 197ndash224 1988

[2] H A Eschenauer and N Olhoff ldquoTopology optimization ofcontinuum structures a reviewrdquo Applied Mechanics Reviewsvol 54 no 4 pp 331ndash390 2001

[3] J D Deaton and R V Grandhi ldquoA survey of structuraland multidisciplinary continuum topology optimization post2000rdquo Structural and Multidisciplinary Optimization vol 49no 1 pp 1ndash38 2014

[4] G I N Rozvany ldquoAims scope methods history and unifiedterminology of computer-aided topology optimization in struc-tural mechanicsrdquo Structural and Multidisciplinary Optimiza-tion vol 21 no 2 pp 90ndash108 2001

[5] M P Bendsoslashe andO Sigmund Topology OptimizationTheoryMethods and Applications Springer Berlin Germany 2ndedition 2003

[6] B Hassani and E Hinton ldquoA review of homogenization andtopology optimization Imdashhomogenization theory for mediawith periodic structurerdquo Computers amp Structures vol 69 no 6pp 707ndash717 1998

[7] G I N Rozvany and M Zhou ldquoApplications of the COCalgorithm in layout optimizationrdquo in Engineering Optimizationin Design Processes Proceedings of the International ConferenceKarlsruhe Nuclear Research Center Germany September 3-41990 H Eschenauer C Matteck and N Olhoff Eds vol 63of Lecture Notes in Engineering pp 59ndash70 Springer BerlinGermany 1991

[8] R J Yang and C H Chuang ldquoOptimal topology design usinglinear programmingrdquo Computers and Structures vol 52 no 2pp 265ndash275 1994

[9] M P Bendsoslashe andO Sigmund ldquoMaterial interpolation schemesin topology optimizationrdquoArchive of AppliedMechanics vol 69no 9-10 pp 635ndash654 1999

[10] YM Xie andG P Steven ldquoA simple evolutionary procedure forstructural optimizationrdquo Computers and Structures vol 49 no5 pp 885ndash896 1993

[11] YMXie andG P StevenEvolutionary StructuralOptimizationSpringer Berlin Germany 1997

[12] G I N Rozvany O M Querin Z Gaspar and V PomezanskildquoWeight-increasing effect of topology simplificationrdquo Structural

and Multidisciplinary Optimization vol 25 no 5-6 pp 459ndash465 2003

[13] O M Querin G P Steven and Y M Xie ldquoEvolutionarystructural optimisation (ESO) using a bidirectional algorithmrdquoEngineering Computations vol 15 no 8 pp 1031ndash1048 1998

[14] J H ZhuWH Zhang andK P Qiu ldquoBi-directional evolution-ary topology optimization using element replaceable methodrdquoComputational Mechanics vol 40 no 1 pp 97ndash109 2007

[15] X Huang and Y M Xie ldquoA new look at ESO and BESOoptimization methodsrdquo Structural and Multidisciplinary Opti-mization vol 35 no 1 pp 89ndash92 2008

[16] X Huang and YM Xie ldquoA further review of ESO typemethodsfor topology optimizationrdquo Structural and MultidisciplinaryOptimization vol 41 no 5 pp 671ndash683 2010

[17] X Huang Z H Zuo and Y M Xie ldquoEvolutionary topologicaloptimization of vibrating continuum structures for naturalfrequenciesrdquoComputers and Structures vol 88 no 5-6 pp 357ndash364 2010

[18] G I N Rozvany ldquoA critical review of established methods ofstructural topology optimizationrdquo Structural and Multidisci-plinary Optimization vol 37 no 3 pp 217ndash237 2009

[19] S Osher and R P Fedkiw ldquoLevel set methods an overview andsome recent resultsrdquo Journal of Computational Physics vol 169no 2 pp 463ndash502 2001

[20] G Allaire F Jouve and A M Toader ldquoStructural optimizationusing sensitivity analysis and a level-set methodrdquo Journal ofComputational Physics vol 194 no 1 pp 363ndash393 2004

[21] J Luo Z Luo S Chen L Tong and M Y Wang ldquoA newlevel set method for systematic design of hinge-free compliantmechanismsrdquo Computer Methods in Applied Mechanics andEngineering vol 198 no 2 pp 318ndash331 2008

[22] M Y Wang S Chen X Wang and Y Mei ldquoDesign ofmultimaterial compliant mechanisms using level-set methodsrdquoTransactions of the ASME Journal of Mechanical Design vol 127no 5 pp 941ndash956 2005

[23] M Y Wang X Wang and D Guo ldquoA level set methodfor structural topology optimizationrdquo Computer Methods inApplied Mechanics and Engineering vol 192 no 1-2 pp 227ndash246 2003

[24] B Bourdin and A Chambolle ldquoDesign-dependent loads intopology optimizationrdquo ESAIM Control Optimisation andCalculus of Variations vol 9 pp 19ndash48 2003

[25] S Zhou and M Y Wang ldquoMultimaterial structural topologyoptimization with a generalized Cahn-Hilliard model of mul-tiphase transitionrdquo Structural and Multidisciplinary Optimiza-tion vol 33 no 2 pp 89ndash111 2007

[26] A Takezawa S Nishiwaki and M Kitamura ldquoShape andtopology optimization based on the phase field method andsensitivity analysisrdquo Journal of Computational Physics vol 229no 7 pp 2697ndash2718 2010

[27] A R Dıaaz and N Kikuchi ldquoSolutions to shape and topologyeigenvalue optimization problems using a homogenizationmethodrdquo International Journal for Numerical Methods in Engi-neering vol 35 no 7 pp 1487ndash1502 1992

[28] Z D Ma H C Cheng and N Kikuchi ldquoStructural design forobtaining desired eigenfrequencies by using the topology andshape optimizationmethodrdquoComputing Systems in Engineeringvol 5 no 1 pp 77ndash89 1994

[29] Z DMa N Kikuchi andH-C Cheng ldquoTopological design forvibrating structuresrdquo Computer Methods in Applied Mechanicsand Engineering vol 121 no 1ndash4 pp 259ndash280 1995

10 Mathematical Problems in Engineering

[30] I Kosaka and C C Swan ldquoA summetry reduction methodfor continuum structural topology optimizationrdquo Computers ampStructures vol 70 no 1 pp 47ndash61 1999

[31] L A Krog and N Olhoff ldquoOptimum topology and reinforce-ment design of disk and plate structures with multiple stiffnessand eigenfrequency objectivesrdquoComputers amp Structures vol 72no 4 pp 535ndash563 1999

[32] J S Jensen and N L Pedersen ldquoOn maximal eigenfrequencyseparation in two-material structures the 1D and 2D scalarcasesrdquo Journal of Sound andVibration vol 289 no 4-5 pp 967ndash986 2006

[33] N L Pedersen ldquoMaximization of eigenvalues using topologyoptimizationrdquo Structural and Multidisciplinary Optimizationvol 20 no 1 pp 2ndash11 2000

[34] J H Zhu and W H Zhang ldquoMaximization of structuralnatural frequency with optimal support layoutrdquo Structural andMultidisciplinaryOptimization vol 31 no 6 pp 462ndash469 2006

[35] J Du and N Olhoff ldquoTopological design of freely vibratingcontinuum structures for maximum values of simple andmultiple eigenfrequencies and frequency gapsrdquo Structural andMultidisciplinary Optimization vol 34 no 2 pp 91ndash110 2007

[36] B Niu J Yan and G Cheng ldquoOptimum structure with homo-geneous optimum cellular material for maximum fundamentalfrequencyrdquo Structural and Multidisciplinary Optimization vol39 no 2 pp 115ndash132 2009

[37] Q Xia T Shi and M Y Wang ldquoA level set based shape andtopology optimization method for maximizing the simple orrepeated first eigenvalue of structure vibrationrdquo Structural andMultidisciplinary Optimization vol 43 no 4 pp 473ndash485 2011

[38] Q Xia T Shi M Y Wang and S Liu ldquoA level set basedmethod for the optimization of cast partrdquo Structural andMultidisciplinary Optimization vol 41 no 5 pp 735ndash747 2010

[39] A K Nandy andC S Jog ldquoOptimization of vibrating structuresto reduce radiated noiserdquo Structural and MultidisciplinaryOptimization vol 45 no 5 pp 717ndash728 2012

[40] J Zheng S Long and G Li ldquoTopology optimization forfree vibrating continuum structures based on the element freeGalerkin methodrdquo Structural and Multidisciplinary Optimiza-tion vol 45 no 1 pp 119ndash127 2012

[41] T D Tsai and C C Cheng ldquoStructural design for desired eigen-frequencies and mode shapes using topology optimizationrdquoStructural and Multidisciplinary Optimization vol 47 no 5 pp673ndash686 2013

[42] K Sui Y Modeling Transformation and Optimization NewDevelopments of Structural Synthesis Method Dalian Universityof Technology Press Dalian China 1996

[43] YK Sui J X LiuH L Ye and J ZDu ldquoApplication ofmodifiedsigmoid function in structural topology optimizationrdquo Journalof BeijingUniversity of Technology vol 34 supplement 1 pp 1ndash62008

[44] P E Gill W Murray and M A Saunders ldquoSNOPT anSQP algorithm for large-scale constrained optimizationrdquo SIAMReview vol 47 no 1 pp 99ndash131 2005

[45] J Nocedal and S J Wright Numerical Optimization SpringerBerlin Germany 1999

[46] Y K Sui H L Ye J X Liu S Chen and H P Yu ldquoAstructural topological optimization method based on exploringconceptual rootrdquo Engineering Mechanics vol 25 no 2 pp 7ndash192008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Therefore the objective function (7) can be rewritten as

119882 =

119873

sum119894=1

119905120574119908

1198941199080

119894=

119873

sum119894=1

1199080

119894

119909120572119894

asymp

119873

sum119894=1

(1198861198941199092+ 119887119894119909) (23)

where 119886119894= 120572(120572+1)119908

0

1198942(119909119894)120572+2 and 119887

119894= minus120572(120572+2)119908

0

119894(119909119894)120572+1

are undetermined parametersWhen CEF is applied as the filter function it is given as

follows

119891119908(119905119894) =

119890119905119894120574119908 minus 1

1198901120574119908 minus 1 119891

119896(119905119894) =

119890119905119894120574119896 minus 1

1198901120574119896 minus 1

119891119898(119905119894) =

119890119905119894120574119898 minus 1

1198901120574119898 minus 1

(24)

We have 119909119894= 1119891

119896(119905119894) = (119890

1120574119896 minus 1)(119890119905119894120574119896 minus 1) and therefore

119891119908(119905119894) =

((1198901120574119896 minus 1) 119909

119894+ 1)120574119896120574119908

minus 1

1198901120574119908 minus 1

119891119898(119905119894) =

((1198901120574119896 minus 1) 119909

119894+ 1)120574119896120574119898

minus 1

1198901120574119898 minus 1

(25)

Similarly the objective function in (7) can be expressed as

119873

sum119894=1

((1198901120574119896 minus 1) 119909

119894+ 1)120574119896120574119908

minus 1

1198901120574119908 minus 11199080

119894asymp

119873

sum119894=1

(1198861198941199092

119894+ 119887119894119909119894) (26)

where

119886119894=

1

2

120574119896

120574119908

1199080

119894

(119909(V)119894

)4

1198901120574119896 minus 1

1198901120574119908 minus 1(

1198901120574119896 minus 1

119909(V)119894

+ 1)

120574119896120574119908minus2

sdot [(120574119896

120574119908

+ 1) (1198901120574119896 minus 1) + 2119909

(V)119894

]

119887119894= minus

120574119896

120574119908

1199080

119894

(119909(V)119894

)3

1198901120574119896 minus 1

1198901120574119908 minus 1(

1198901120574119896 minus 1

119909(V)119894

+ 1)

120574119896120574119908minus2

sdot [(120574119896

120574119908

+ 2) (1198901120574119896 minus 1) + 3119909

(V)119894

]

(27)

They are two undetermined parametersThe dual model of (21) is

find zisin E119869

make minus Φ (z) 997888rarr min

st z119895ge 0 (119895 = 1 119869)

(28)

where z is the design variables in dual model

Φ (z) = min1le119905119894le119909

(119871 (x z))

119871 (x z) =

119873

sum119894=1

(1198861198941199092

119894+ 119887119894119909119894) +

119869

sum119895=1

(119911119895(

119873

sum119894=1

119888119894119895119909119894minus 119889119895))

(29)

Here 119871(x z) is a Lagrange augmented function in (28)

Start

Construct the finite element model and input the optimal parameters

Finite element analysis with MSCNastran

Formulate the mathematical optimal model (7)

Solve (21) with DSQP and update variables

End

No

Yes

|W(+1) minus W()|

|W()|le 120576

Figure 1 Flow chart of the iterative solution procedure

Then DSQP is employed to solve (28) and the precisionof convergence is controlled to be 0001 With the aboveanalysis and solving of (28) the optimal topology structurewith continuous design variables is obtained

5 Numerical Simulation

Based on the above analysis we outline the following compu-tational procedure with a flow chart (Figure 1) it is used formaximizing theweight of structure constrained by frequency

Following the above flow chart we develop a programbased on the platform of MSC Patran amp Nastran to solve theimproved topology optimal model In order to demonstratethe effectiveness of the proposed method two structuresare presented as inputs The first one is a cantilever withfundamental frequency constraint the second one is a 3Dbeam with two concentrated masses by multiple frequencyconstraints For the computation the initial values of topol-ogy variables are all given as unit (119905 = 1) the lowest bounds oftopology variables and the convergence precision values are001 and 0001 respectively

51 Example 1 Example 1 is a cantilever with size 100 times 50times 50mm3 It is fixed at one end as shown in Figure 2 and aconcentrated mass Mc = 50 g is attached at the center of theother end Youngrsquos modulus 119864 = 100GPa Poissonrsquos ratio 120583 =

03 andmass density 120588= 1000 kgm3The structure is dividedinto 30 times 16 times 16 eight-node hexahedron elements The5010Hz fundamental frequency of the cantilever is calculatedby finite element analysis andwe set the frequency constraintfor the design problem as 119891

1ge 3500Hz

The solving topology configuration of the cantilever withdifferent filter functions is given in Figure 3 Figures 3(a) and3(b) show the whole optimal structure and (c)-(d) show the

6 Mathematical Problems in Engineering

50mm

50mm

100mm

Mc

Figure 2 A cantilever structure

(a) (b)

(c) (d)

Figure 3 Topology configuration with different filter functions (a) Topology configuration with PF (b) topology configuration with CEF(c)-(d) the half optimal structure of (a)-(b)

half structure corresponding to (a)-(b) The performance oftopology optimization with different filter functions is givenin Table 1 The iterative curve of computation with differentfilter functions is described in Figure 4

52 Example 2 The beam (40 times 10 times 10 cm3) with two endsfixed is presented in Figure 5 Two concentrated masses with1198721198881

= 1198721198882

= 5 kg are located on the upper surface of 13and 23 span Youngrsquos modulus E = 100GPa Poissonrsquos ratio120583 = 03 and mass density 120588 = 1000 kgm3 The structure

is divided into 30 times 8 times 8 eight-node hexahedron elementsThe first computed fundamental frequency of the beam is13435Hz and the second one is 14951 Hz The set constraintfrequencies are 119891

1ge 1000Hz 119891

2ge 1100Hz

The solving topology configuration of the beam withdifferent filter functions is given in Figure 6 The perfor-mances of topology optimization with different filter func-tions are given in Table 2 To describe the dynamics ofoptimal structure the first three modal shapes of optimalstructure with two different filter functions are computedand displayed in Figure 7 The frequency changing with time

Mathematical Problems in Engineering 7

260

240

220

200

180

160

140

120

100

80

60

Mas

s (g)

PFCEF

0 10 20 30 40

Iteration

(a)

34

36

38

40

42

44

46

48

50

52

Freq

uenc

y (H

z)

times103

0 10 20 30 40

Iteration

PFCEF

(b)

Figure 4 Iteration process with different filter function

L

H

Mc1 Mc2

Figure 5 The beam with two ends fixed

Table 1 Performance of topology optimization with differentalgorithms and filter functions

Method DSQP

PFIteration 411198911(Hz) 35020270996

Mass (g) 73033882813

CEFIteration 341198911(Hz) 35019787598

Mass (g) 70509476563

in the optimization process is presented in Figure 8 withdifferent filter functions

6 Conclusion

In this paper an improved frequency topology optimizationmodel of three-dimensional continuum structure is devel-oped based on ICM method CEF a new filter function isselected to recognize the design variables as well as to imple-ment much better the changing process of design variablesfrom ldquodiscreterdquo to ldquocontinuousrdquo and back to ldquodiscreterdquo Explicit

Table 2 Performance of topology optimization with differentalgorithms and filter functions

Method DSQP

PF

Iteration 331198911(Hz) 1000392

1198912(Hz) 1100647

Mass (g) 1760004

CEF

Iteration 301198911(Hz) 1000551

1198912(Hz) 1100791

Mass (g) 1682534

formulations of frequency constraints are given based onfilter functions and first-order Taylor series expansion andby extracting structural strain and structural kinetic energyfrom the results of structural modal analysis An improvedoptimal model is formulated using CEF and the explicitfrequency constraints The program based on DSQP forsolving the optimal model is developed on the platform ofMSC Patran amp Nastran

Finally two examples of three-dimensional continuumstructure show that clear and stable configurations can beobtained using ICM method We find that configurationscomputed with DSQP combined PF and DSQP combinedCEF are similar between one and the other in the case of fun-damental frequency constraint But we can find that DSQPcombined CEF has the best performance for the optimizationexample from the point of view of optimal objective anditerative numbers We also compute the first three ordermodal shapes of optimal structure with two different filterfunctions in multiple frequency constraints The vibrationmodes from two filter functions are the same and the iteration

8 Mathematical Problems in Engineering

(a) (b)

Figure 6 Topology configuration with different filter functions (a) Topology configuration with PF (b) topology configuration with CEF

Mod

e

First

PF

(a)

Modal shapesSecond

(b)

Third

(c)

CEF

Mod

e

(d) (e) (f)

Figure 7 The first three modal shapes of optimal structure with two filter functions

0 5 10 15 20 25 30 35

Eige

nfre

quen

cy (H

z)

Iterationminus5

f1f2f3

18171616161515141413131211111110

095

times103

(a)

0 5 10 15 20 25 30Iteration

f1f2f3

Eige

nfre

quen

cy (H

z)

18171616161515141413131211111110

095

times103

(b)

Figure 8 Iteration process of the first three frequencies with different filter functions (a) PF (b) CEF

Mathematical Problems in Engineering 9

curves of the first three order frequencies show that structuralfrequencies satisfy the frequency constraints Therefore allthe computing results indicate that the proposed method isalso feasible for multiple frequency constraints

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of the paper

Acknowledgment

The work of this project was supported by National ScienceFoundation of China (ContractGrant no 11072009 andContractGrant no 11172013)

References

[1] M P Bendsoslashe and N Kikuchi ldquoGenerating optimal topologiesin structural design using a homogenization methodrdquo Com-puterMethods in AppliedMechanics and Engineering vol 71 no2 pp 197ndash224 1988

[2] H A Eschenauer and N Olhoff ldquoTopology optimization ofcontinuum structures a reviewrdquo Applied Mechanics Reviewsvol 54 no 4 pp 331ndash390 2001

[3] J D Deaton and R V Grandhi ldquoA survey of structuraland multidisciplinary continuum topology optimization post2000rdquo Structural and Multidisciplinary Optimization vol 49no 1 pp 1ndash38 2014

[4] G I N Rozvany ldquoAims scope methods history and unifiedterminology of computer-aided topology optimization in struc-tural mechanicsrdquo Structural and Multidisciplinary Optimiza-tion vol 21 no 2 pp 90ndash108 2001

[5] M P Bendsoslashe andO Sigmund Topology OptimizationTheoryMethods and Applications Springer Berlin Germany 2ndedition 2003

[6] B Hassani and E Hinton ldquoA review of homogenization andtopology optimization Imdashhomogenization theory for mediawith periodic structurerdquo Computers amp Structures vol 69 no 6pp 707ndash717 1998

[7] G I N Rozvany and M Zhou ldquoApplications of the COCalgorithm in layout optimizationrdquo in Engineering Optimizationin Design Processes Proceedings of the International ConferenceKarlsruhe Nuclear Research Center Germany September 3-41990 H Eschenauer C Matteck and N Olhoff Eds vol 63of Lecture Notes in Engineering pp 59ndash70 Springer BerlinGermany 1991

[8] R J Yang and C H Chuang ldquoOptimal topology design usinglinear programmingrdquo Computers and Structures vol 52 no 2pp 265ndash275 1994

[9] M P Bendsoslashe andO Sigmund ldquoMaterial interpolation schemesin topology optimizationrdquoArchive of AppliedMechanics vol 69no 9-10 pp 635ndash654 1999

[10] YM Xie andG P Steven ldquoA simple evolutionary procedure forstructural optimizationrdquo Computers and Structures vol 49 no5 pp 885ndash896 1993

[11] YMXie andG P StevenEvolutionary StructuralOptimizationSpringer Berlin Germany 1997

[12] G I N Rozvany O M Querin Z Gaspar and V PomezanskildquoWeight-increasing effect of topology simplificationrdquo Structural

and Multidisciplinary Optimization vol 25 no 5-6 pp 459ndash465 2003

[13] O M Querin G P Steven and Y M Xie ldquoEvolutionarystructural optimisation (ESO) using a bidirectional algorithmrdquoEngineering Computations vol 15 no 8 pp 1031ndash1048 1998

[14] J H ZhuWH Zhang andK P Qiu ldquoBi-directional evolution-ary topology optimization using element replaceable methodrdquoComputational Mechanics vol 40 no 1 pp 97ndash109 2007

[15] X Huang and Y M Xie ldquoA new look at ESO and BESOoptimization methodsrdquo Structural and Multidisciplinary Opti-mization vol 35 no 1 pp 89ndash92 2008

[16] X Huang and YM Xie ldquoA further review of ESO typemethodsfor topology optimizationrdquo Structural and MultidisciplinaryOptimization vol 41 no 5 pp 671ndash683 2010

[17] X Huang Z H Zuo and Y M Xie ldquoEvolutionary topologicaloptimization of vibrating continuum structures for naturalfrequenciesrdquoComputers and Structures vol 88 no 5-6 pp 357ndash364 2010

[18] G I N Rozvany ldquoA critical review of established methods ofstructural topology optimizationrdquo Structural and Multidisci-plinary Optimization vol 37 no 3 pp 217ndash237 2009

[19] S Osher and R P Fedkiw ldquoLevel set methods an overview andsome recent resultsrdquo Journal of Computational Physics vol 169no 2 pp 463ndash502 2001

[20] G Allaire F Jouve and A M Toader ldquoStructural optimizationusing sensitivity analysis and a level-set methodrdquo Journal ofComputational Physics vol 194 no 1 pp 363ndash393 2004

[21] J Luo Z Luo S Chen L Tong and M Y Wang ldquoA newlevel set method for systematic design of hinge-free compliantmechanismsrdquo Computer Methods in Applied Mechanics andEngineering vol 198 no 2 pp 318ndash331 2008

[22] M Y Wang S Chen X Wang and Y Mei ldquoDesign ofmultimaterial compliant mechanisms using level-set methodsrdquoTransactions of the ASME Journal of Mechanical Design vol 127no 5 pp 941ndash956 2005

[23] M Y Wang X Wang and D Guo ldquoA level set methodfor structural topology optimizationrdquo Computer Methods inApplied Mechanics and Engineering vol 192 no 1-2 pp 227ndash246 2003

[24] B Bourdin and A Chambolle ldquoDesign-dependent loads intopology optimizationrdquo ESAIM Control Optimisation andCalculus of Variations vol 9 pp 19ndash48 2003

[25] S Zhou and M Y Wang ldquoMultimaterial structural topologyoptimization with a generalized Cahn-Hilliard model of mul-tiphase transitionrdquo Structural and Multidisciplinary Optimiza-tion vol 33 no 2 pp 89ndash111 2007

[26] A Takezawa S Nishiwaki and M Kitamura ldquoShape andtopology optimization based on the phase field method andsensitivity analysisrdquo Journal of Computational Physics vol 229no 7 pp 2697ndash2718 2010

[27] A R Dıaaz and N Kikuchi ldquoSolutions to shape and topologyeigenvalue optimization problems using a homogenizationmethodrdquo International Journal for Numerical Methods in Engi-neering vol 35 no 7 pp 1487ndash1502 1992

[28] Z D Ma H C Cheng and N Kikuchi ldquoStructural design forobtaining desired eigenfrequencies by using the topology andshape optimizationmethodrdquoComputing Systems in Engineeringvol 5 no 1 pp 77ndash89 1994

[29] Z DMa N Kikuchi andH-C Cheng ldquoTopological design forvibrating structuresrdquo Computer Methods in Applied Mechanicsand Engineering vol 121 no 1ndash4 pp 259ndash280 1995

10 Mathematical Problems in Engineering

[30] I Kosaka and C C Swan ldquoA summetry reduction methodfor continuum structural topology optimizationrdquo Computers ampStructures vol 70 no 1 pp 47ndash61 1999

[31] L A Krog and N Olhoff ldquoOptimum topology and reinforce-ment design of disk and plate structures with multiple stiffnessand eigenfrequency objectivesrdquoComputers amp Structures vol 72no 4 pp 535ndash563 1999

[32] J S Jensen and N L Pedersen ldquoOn maximal eigenfrequencyseparation in two-material structures the 1D and 2D scalarcasesrdquo Journal of Sound andVibration vol 289 no 4-5 pp 967ndash986 2006

[33] N L Pedersen ldquoMaximization of eigenvalues using topologyoptimizationrdquo Structural and Multidisciplinary Optimizationvol 20 no 1 pp 2ndash11 2000

[34] J H Zhu and W H Zhang ldquoMaximization of structuralnatural frequency with optimal support layoutrdquo Structural andMultidisciplinaryOptimization vol 31 no 6 pp 462ndash469 2006

[35] J Du and N Olhoff ldquoTopological design of freely vibratingcontinuum structures for maximum values of simple andmultiple eigenfrequencies and frequency gapsrdquo Structural andMultidisciplinary Optimization vol 34 no 2 pp 91ndash110 2007

[36] B Niu J Yan and G Cheng ldquoOptimum structure with homo-geneous optimum cellular material for maximum fundamentalfrequencyrdquo Structural and Multidisciplinary Optimization vol39 no 2 pp 115ndash132 2009

[37] Q Xia T Shi and M Y Wang ldquoA level set based shape andtopology optimization method for maximizing the simple orrepeated first eigenvalue of structure vibrationrdquo Structural andMultidisciplinary Optimization vol 43 no 4 pp 473ndash485 2011

[38] Q Xia T Shi M Y Wang and S Liu ldquoA level set basedmethod for the optimization of cast partrdquo Structural andMultidisciplinary Optimization vol 41 no 5 pp 735ndash747 2010

[39] A K Nandy andC S Jog ldquoOptimization of vibrating structuresto reduce radiated noiserdquo Structural and MultidisciplinaryOptimization vol 45 no 5 pp 717ndash728 2012

[40] J Zheng S Long and G Li ldquoTopology optimization forfree vibrating continuum structures based on the element freeGalerkin methodrdquo Structural and Multidisciplinary Optimiza-tion vol 45 no 1 pp 119ndash127 2012

[41] T D Tsai and C C Cheng ldquoStructural design for desired eigen-frequencies and mode shapes using topology optimizationrdquoStructural and Multidisciplinary Optimization vol 47 no 5 pp673ndash686 2013

[42] K Sui Y Modeling Transformation and Optimization NewDevelopments of Structural Synthesis Method Dalian Universityof Technology Press Dalian China 1996

[43] YK Sui J X LiuH L Ye and J ZDu ldquoApplication ofmodifiedsigmoid function in structural topology optimizationrdquo Journalof BeijingUniversity of Technology vol 34 supplement 1 pp 1ndash62008

[44] P E Gill W Murray and M A Saunders ldquoSNOPT anSQP algorithm for large-scale constrained optimizationrdquo SIAMReview vol 47 no 1 pp 99ndash131 2005

[45] J Nocedal and S J Wright Numerical Optimization SpringerBerlin Germany 1999

[46] Y K Sui H L Ye J X Liu S Chen and H P Yu ldquoAstructural topological optimization method based on exploringconceptual rootrdquo Engineering Mechanics vol 25 no 2 pp 7ndash192008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

50mm

50mm

100mm

Mc

Figure 2 A cantilever structure

(a) (b)

(c) (d)

Figure 3 Topology configuration with different filter functions (a) Topology configuration with PF (b) topology configuration with CEF(c)-(d) the half optimal structure of (a)-(b)

half structure corresponding to (a)-(b) The performance oftopology optimization with different filter functions is givenin Table 1 The iterative curve of computation with differentfilter functions is described in Figure 4

52 Example 2 The beam (40 times 10 times 10 cm3) with two endsfixed is presented in Figure 5 Two concentrated masses with1198721198881

= 1198721198882

= 5 kg are located on the upper surface of 13and 23 span Youngrsquos modulus E = 100GPa Poissonrsquos ratio120583 = 03 and mass density 120588 = 1000 kgm3 The structure

is divided into 30 times 8 times 8 eight-node hexahedron elementsThe first computed fundamental frequency of the beam is13435Hz and the second one is 14951 Hz The set constraintfrequencies are 119891

1ge 1000Hz 119891

2ge 1100Hz

The solving topology configuration of the beam withdifferent filter functions is given in Figure 6 The perfor-mances of topology optimization with different filter func-tions are given in Table 2 To describe the dynamics ofoptimal structure the first three modal shapes of optimalstructure with two different filter functions are computedand displayed in Figure 7 The frequency changing with time

Mathematical Problems in Engineering 7

260

240

220

200

180

160

140

120

100

80

60

Mas

s (g)

PFCEF

0 10 20 30 40

Iteration

(a)

34

36

38

40

42

44

46

48

50

52

Freq

uenc

y (H

z)

times103

0 10 20 30 40

Iteration

PFCEF

(b)

Figure 4 Iteration process with different filter function

L

H

Mc1 Mc2

Figure 5 The beam with two ends fixed

Table 1 Performance of topology optimization with differentalgorithms and filter functions

Method DSQP

PFIteration 411198911(Hz) 35020270996

Mass (g) 73033882813

CEFIteration 341198911(Hz) 35019787598

Mass (g) 70509476563

in the optimization process is presented in Figure 8 withdifferent filter functions

6 Conclusion

In this paper an improved frequency topology optimizationmodel of three-dimensional continuum structure is devel-oped based on ICM method CEF a new filter function isselected to recognize the design variables as well as to imple-ment much better the changing process of design variablesfrom ldquodiscreterdquo to ldquocontinuousrdquo and back to ldquodiscreterdquo Explicit

Table 2 Performance of topology optimization with differentalgorithms and filter functions

Method DSQP

PF

Iteration 331198911(Hz) 1000392

1198912(Hz) 1100647

Mass (g) 1760004

CEF

Iteration 301198911(Hz) 1000551

1198912(Hz) 1100791

Mass (g) 1682534

formulations of frequency constraints are given based onfilter functions and first-order Taylor series expansion andby extracting structural strain and structural kinetic energyfrom the results of structural modal analysis An improvedoptimal model is formulated using CEF and the explicitfrequency constraints The program based on DSQP forsolving the optimal model is developed on the platform ofMSC Patran amp Nastran

Finally two examples of three-dimensional continuumstructure show that clear and stable configurations can beobtained using ICM method We find that configurationscomputed with DSQP combined PF and DSQP combinedCEF are similar between one and the other in the case of fun-damental frequency constraint But we can find that DSQPcombined CEF has the best performance for the optimizationexample from the point of view of optimal objective anditerative numbers We also compute the first three ordermodal shapes of optimal structure with two different filterfunctions in multiple frequency constraints The vibrationmodes from two filter functions are the same and the iteration

8 Mathematical Problems in Engineering

(a) (b)

Figure 6 Topology configuration with different filter functions (a) Topology configuration with PF (b) topology configuration with CEF

Mod

e

First

PF

(a)

Modal shapesSecond

(b)

Third

(c)

CEF

Mod

e

(d) (e) (f)

Figure 7 The first three modal shapes of optimal structure with two filter functions

0 5 10 15 20 25 30 35

Eige

nfre

quen

cy (H

z)

Iterationminus5

f1f2f3

18171616161515141413131211111110

095

times103

(a)

0 5 10 15 20 25 30Iteration

f1f2f3

Eige

nfre

quen

cy (H

z)

18171616161515141413131211111110

095

times103

(b)

Figure 8 Iteration process of the first three frequencies with different filter functions (a) PF (b) CEF

Mathematical Problems in Engineering 9

curves of the first three order frequencies show that structuralfrequencies satisfy the frequency constraints Therefore allthe computing results indicate that the proposed method isalso feasible for multiple frequency constraints

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of the paper

Acknowledgment

The work of this project was supported by National ScienceFoundation of China (ContractGrant no 11072009 andContractGrant no 11172013)

References

[1] M P Bendsoslashe and N Kikuchi ldquoGenerating optimal topologiesin structural design using a homogenization methodrdquo Com-puterMethods in AppliedMechanics and Engineering vol 71 no2 pp 197ndash224 1988

[2] H A Eschenauer and N Olhoff ldquoTopology optimization ofcontinuum structures a reviewrdquo Applied Mechanics Reviewsvol 54 no 4 pp 331ndash390 2001

[3] J D Deaton and R V Grandhi ldquoA survey of structuraland multidisciplinary continuum topology optimization post2000rdquo Structural and Multidisciplinary Optimization vol 49no 1 pp 1ndash38 2014

[4] G I N Rozvany ldquoAims scope methods history and unifiedterminology of computer-aided topology optimization in struc-tural mechanicsrdquo Structural and Multidisciplinary Optimiza-tion vol 21 no 2 pp 90ndash108 2001

[5] M P Bendsoslashe andO Sigmund Topology OptimizationTheoryMethods and Applications Springer Berlin Germany 2ndedition 2003

[6] B Hassani and E Hinton ldquoA review of homogenization andtopology optimization Imdashhomogenization theory for mediawith periodic structurerdquo Computers amp Structures vol 69 no 6pp 707ndash717 1998

[7] G I N Rozvany and M Zhou ldquoApplications of the COCalgorithm in layout optimizationrdquo in Engineering Optimizationin Design Processes Proceedings of the International ConferenceKarlsruhe Nuclear Research Center Germany September 3-41990 H Eschenauer C Matteck and N Olhoff Eds vol 63of Lecture Notes in Engineering pp 59ndash70 Springer BerlinGermany 1991

[8] R J Yang and C H Chuang ldquoOptimal topology design usinglinear programmingrdquo Computers and Structures vol 52 no 2pp 265ndash275 1994

[9] M P Bendsoslashe andO Sigmund ldquoMaterial interpolation schemesin topology optimizationrdquoArchive of AppliedMechanics vol 69no 9-10 pp 635ndash654 1999

[10] YM Xie andG P Steven ldquoA simple evolutionary procedure forstructural optimizationrdquo Computers and Structures vol 49 no5 pp 885ndash896 1993

[11] YMXie andG P StevenEvolutionary StructuralOptimizationSpringer Berlin Germany 1997

[12] G I N Rozvany O M Querin Z Gaspar and V PomezanskildquoWeight-increasing effect of topology simplificationrdquo Structural

and Multidisciplinary Optimization vol 25 no 5-6 pp 459ndash465 2003

[13] O M Querin G P Steven and Y M Xie ldquoEvolutionarystructural optimisation (ESO) using a bidirectional algorithmrdquoEngineering Computations vol 15 no 8 pp 1031ndash1048 1998

[14] J H ZhuWH Zhang andK P Qiu ldquoBi-directional evolution-ary topology optimization using element replaceable methodrdquoComputational Mechanics vol 40 no 1 pp 97ndash109 2007

[15] X Huang and Y M Xie ldquoA new look at ESO and BESOoptimization methodsrdquo Structural and Multidisciplinary Opti-mization vol 35 no 1 pp 89ndash92 2008

[16] X Huang and YM Xie ldquoA further review of ESO typemethodsfor topology optimizationrdquo Structural and MultidisciplinaryOptimization vol 41 no 5 pp 671ndash683 2010

[17] X Huang Z H Zuo and Y M Xie ldquoEvolutionary topologicaloptimization of vibrating continuum structures for naturalfrequenciesrdquoComputers and Structures vol 88 no 5-6 pp 357ndash364 2010

[18] G I N Rozvany ldquoA critical review of established methods ofstructural topology optimizationrdquo Structural and Multidisci-plinary Optimization vol 37 no 3 pp 217ndash237 2009

[19] S Osher and R P Fedkiw ldquoLevel set methods an overview andsome recent resultsrdquo Journal of Computational Physics vol 169no 2 pp 463ndash502 2001

[20] G Allaire F Jouve and A M Toader ldquoStructural optimizationusing sensitivity analysis and a level-set methodrdquo Journal ofComputational Physics vol 194 no 1 pp 363ndash393 2004

[21] J Luo Z Luo S Chen L Tong and M Y Wang ldquoA newlevel set method for systematic design of hinge-free compliantmechanismsrdquo Computer Methods in Applied Mechanics andEngineering vol 198 no 2 pp 318ndash331 2008

[22] M Y Wang S Chen X Wang and Y Mei ldquoDesign ofmultimaterial compliant mechanisms using level-set methodsrdquoTransactions of the ASME Journal of Mechanical Design vol 127no 5 pp 941ndash956 2005

[23] M Y Wang X Wang and D Guo ldquoA level set methodfor structural topology optimizationrdquo Computer Methods inApplied Mechanics and Engineering vol 192 no 1-2 pp 227ndash246 2003

[24] B Bourdin and A Chambolle ldquoDesign-dependent loads intopology optimizationrdquo ESAIM Control Optimisation andCalculus of Variations vol 9 pp 19ndash48 2003

[25] S Zhou and M Y Wang ldquoMultimaterial structural topologyoptimization with a generalized Cahn-Hilliard model of mul-tiphase transitionrdquo Structural and Multidisciplinary Optimiza-tion vol 33 no 2 pp 89ndash111 2007

[26] A Takezawa S Nishiwaki and M Kitamura ldquoShape andtopology optimization based on the phase field method andsensitivity analysisrdquo Journal of Computational Physics vol 229no 7 pp 2697ndash2718 2010

[27] A R Dıaaz and N Kikuchi ldquoSolutions to shape and topologyeigenvalue optimization problems using a homogenizationmethodrdquo International Journal for Numerical Methods in Engi-neering vol 35 no 7 pp 1487ndash1502 1992

[28] Z D Ma H C Cheng and N Kikuchi ldquoStructural design forobtaining desired eigenfrequencies by using the topology andshape optimizationmethodrdquoComputing Systems in Engineeringvol 5 no 1 pp 77ndash89 1994

[29] Z DMa N Kikuchi andH-C Cheng ldquoTopological design forvibrating structuresrdquo Computer Methods in Applied Mechanicsand Engineering vol 121 no 1ndash4 pp 259ndash280 1995

10 Mathematical Problems in Engineering

[30] I Kosaka and C C Swan ldquoA summetry reduction methodfor continuum structural topology optimizationrdquo Computers ampStructures vol 70 no 1 pp 47ndash61 1999

[31] L A Krog and N Olhoff ldquoOptimum topology and reinforce-ment design of disk and plate structures with multiple stiffnessand eigenfrequency objectivesrdquoComputers amp Structures vol 72no 4 pp 535ndash563 1999

[32] J S Jensen and N L Pedersen ldquoOn maximal eigenfrequencyseparation in two-material structures the 1D and 2D scalarcasesrdquo Journal of Sound andVibration vol 289 no 4-5 pp 967ndash986 2006

[33] N L Pedersen ldquoMaximization of eigenvalues using topologyoptimizationrdquo Structural and Multidisciplinary Optimizationvol 20 no 1 pp 2ndash11 2000

[34] J H Zhu and W H Zhang ldquoMaximization of structuralnatural frequency with optimal support layoutrdquo Structural andMultidisciplinaryOptimization vol 31 no 6 pp 462ndash469 2006

[35] J Du and N Olhoff ldquoTopological design of freely vibratingcontinuum structures for maximum values of simple andmultiple eigenfrequencies and frequency gapsrdquo Structural andMultidisciplinary Optimization vol 34 no 2 pp 91ndash110 2007

[36] B Niu J Yan and G Cheng ldquoOptimum structure with homo-geneous optimum cellular material for maximum fundamentalfrequencyrdquo Structural and Multidisciplinary Optimization vol39 no 2 pp 115ndash132 2009

[37] Q Xia T Shi and M Y Wang ldquoA level set based shape andtopology optimization method for maximizing the simple orrepeated first eigenvalue of structure vibrationrdquo Structural andMultidisciplinary Optimization vol 43 no 4 pp 473ndash485 2011

[38] Q Xia T Shi M Y Wang and S Liu ldquoA level set basedmethod for the optimization of cast partrdquo Structural andMultidisciplinary Optimization vol 41 no 5 pp 735ndash747 2010

[39] A K Nandy andC S Jog ldquoOptimization of vibrating structuresto reduce radiated noiserdquo Structural and MultidisciplinaryOptimization vol 45 no 5 pp 717ndash728 2012

[40] J Zheng S Long and G Li ldquoTopology optimization forfree vibrating continuum structures based on the element freeGalerkin methodrdquo Structural and Multidisciplinary Optimiza-tion vol 45 no 1 pp 119ndash127 2012

[41] T D Tsai and C C Cheng ldquoStructural design for desired eigen-frequencies and mode shapes using topology optimizationrdquoStructural and Multidisciplinary Optimization vol 47 no 5 pp673ndash686 2013

[42] K Sui Y Modeling Transformation and Optimization NewDevelopments of Structural Synthesis Method Dalian Universityof Technology Press Dalian China 1996

[43] YK Sui J X LiuH L Ye and J ZDu ldquoApplication ofmodifiedsigmoid function in structural topology optimizationrdquo Journalof BeijingUniversity of Technology vol 34 supplement 1 pp 1ndash62008

[44] P E Gill W Murray and M A Saunders ldquoSNOPT anSQP algorithm for large-scale constrained optimizationrdquo SIAMReview vol 47 no 1 pp 99ndash131 2005

[45] J Nocedal and S J Wright Numerical Optimization SpringerBerlin Germany 1999

[46] Y K Sui H L Ye J X Liu S Chen and H P Yu ldquoAstructural topological optimization method based on exploringconceptual rootrdquo Engineering Mechanics vol 25 no 2 pp 7ndash192008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

260

240

220

200

180

160

140

120

100

80

60

Mas

s (g)

PFCEF

0 10 20 30 40

Iteration

(a)

34

36

38

40

42

44

46

48

50

52

Freq

uenc

y (H

z)

times103

0 10 20 30 40

Iteration

PFCEF

(b)

Figure 4 Iteration process with different filter function

L

H

Mc1 Mc2

Figure 5 The beam with two ends fixed

Table 1 Performance of topology optimization with differentalgorithms and filter functions

Method DSQP

PFIteration 411198911(Hz) 35020270996

Mass (g) 73033882813

CEFIteration 341198911(Hz) 35019787598

Mass (g) 70509476563

in the optimization process is presented in Figure 8 withdifferent filter functions

6 Conclusion

In this paper an improved frequency topology optimizationmodel of three-dimensional continuum structure is devel-oped based on ICM method CEF a new filter function isselected to recognize the design variables as well as to imple-ment much better the changing process of design variablesfrom ldquodiscreterdquo to ldquocontinuousrdquo and back to ldquodiscreterdquo Explicit

Table 2 Performance of topology optimization with differentalgorithms and filter functions

Method DSQP

PF

Iteration 331198911(Hz) 1000392

1198912(Hz) 1100647

Mass (g) 1760004

CEF

Iteration 301198911(Hz) 1000551

1198912(Hz) 1100791

Mass (g) 1682534

formulations of frequency constraints are given based onfilter functions and first-order Taylor series expansion andby extracting structural strain and structural kinetic energyfrom the results of structural modal analysis An improvedoptimal model is formulated using CEF and the explicitfrequency constraints The program based on DSQP forsolving the optimal model is developed on the platform ofMSC Patran amp Nastran

Finally two examples of three-dimensional continuumstructure show that clear and stable configurations can beobtained using ICM method We find that configurationscomputed with DSQP combined PF and DSQP combinedCEF are similar between one and the other in the case of fun-damental frequency constraint But we can find that DSQPcombined CEF has the best performance for the optimizationexample from the point of view of optimal objective anditerative numbers We also compute the first three ordermodal shapes of optimal structure with two different filterfunctions in multiple frequency constraints The vibrationmodes from two filter functions are the same and the iteration

8 Mathematical Problems in Engineering

(a) (b)

Figure 6 Topology configuration with different filter functions (a) Topology configuration with PF (b) topology configuration with CEF

Mod

e

First

PF

(a)

Modal shapesSecond

(b)

Third

(c)

CEF

Mod

e

(d) (e) (f)

Figure 7 The first three modal shapes of optimal structure with two filter functions

0 5 10 15 20 25 30 35

Eige

nfre

quen

cy (H

z)

Iterationminus5

f1f2f3

18171616161515141413131211111110

095

times103

(a)

0 5 10 15 20 25 30Iteration

f1f2f3

Eige

nfre

quen

cy (H

z)

18171616161515141413131211111110

095

times103

(b)

Figure 8 Iteration process of the first three frequencies with different filter functions (a) PF (b) CEF

Mathematical Problems in Engineering 9

curves of the first three order frequencies show that structuralfrequencies satisfy the frequency constraints Therefore allthe computing results indicate that the proposed method isalso feasible for multiple frequency constraints

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of the paper

Acknowledgment

The work of this project was supported by National ScienceFoundation of China (ContractGrant no 11072009 andContractGrant no 11172013)

References

[1] M P Bendsoslashe and N Kikuchi ldquoGenerating optimal topologiesin structural design using a homogenization methodrdquo Com-puterMethods in AppliedMechanics and Engineering vol 71 no2 pp 197ndash224 1988

[2] H A Eschenauer and N Olhoff ldquoTopology optimization ofcontinuum structures a reviewrdquo Applied Mechanics Reviewsvol 54 no 4 pp 331ndash390 2001

[3] J D Deaton and R V Grandhi ldquoA survey of structuraland multidisciplinary continuum topology optimization post2000rdquo Structural and Multidisciplinary Optimization vol 49no 1 pp 1ndash38 2014

[4] G I N Rozvany ldquoAims scope methods history and unifiedterminology of computer-aided topology optimization in struc-tural mechanicsrdquo Structural and Multidisciplinary Optimiza-tion vol 21 no 2 pp 90ndash108 2001

[5] M P Bendsoslashe andO Sigmund Topology OptimizationTheoryMethods and Applications Springer Berlin Germany 2ndedition 2003

[6] B Hassani and E Hinton ldquoA review of homogenization andtopology optimization Imdashhomogenization theory for mediawith periodic structurerdquo Computers amp Structures vol 69 no 6pp 707ndash717 1998

[7] G I N Rozvany and M Zhou ldquoApplications of the COCalgorithm in layout optimizationrdquo in Engineering Optimizationin Design Processes Proceedings of the International ConferenceKarlsruhe Nuclear Research Center Germany September 3-41990 H Eschenauer C Matteck and N Olhoff Eds vol 63of Lecture Notes in Engineering pp 59ndash70 Springer BerlinGermany 1991

[8] R J Yang and C H Chuang ldquoOptimal topology design usinglinear programmingrdquo Computers and Structures vol 52 no 2pp 265ndash275 1994

[9] M P Bendsoslashe andO Sigmund ldquoMaterial interpolation schemesin topology optimizationrdquoArchive of AppliedMechanics vol 69no 9-10 pp 635ndash654 1999

[10] YM Xie andG P Steven ldquoA simple evolutionary procedure forstructural optimizationrdquo Computers and Structures vol 49 no5 pp 885ndash896 1993

[11] YMXie andG P StevenEvolutionary StructuralOptimizationSpringer Berlin Germany 1997

[12] G I N Rozvany O M Querin Z Gaspar and V PomezanskildquoWeight-increasing effect of topology simplificationrdquo Structural

and Multidisciplinary Optimization vol 25 no 5-6 pp 459ndash465 2003

[13] O M Querin G P Steven and Y M Xie ldquoEvolutionarystructural optimisation (ESO) using a bidirectional algorithmrdquoEngineering Computations vol 15 no 8 pp 1031ndash1048 1998

[14] J H ZhuWH Zhang andK P Qiu ldquoBi-directional evolution-ary topology optimization using element replaceable methodrdquoComputational Mechanics vol 40 no 1 pp 97ndash109 2007

[15] X Huang and Y M Xie ldquoA new look at ESO and BESOoptimization methodsrdquo Structural and Multidisciplinary Opti-mization vol 35 no 1 pp 89ndash92 2008

[16] X Huang and YM Xie ldquoA further review of ESO typemethodsfor topology optimizationrdquo Structural and MultidisciplinaryOptimization vol 41 no 5 pp 671ndash683 2010

[17] X Huang Z H Zuo and Y M Xie ldquoEvolutionary topologicaloptimization of vibrating continuum structures for naturalfrequenciesrdquoComputers and Structures vol 88 no 5-6 pp 357ndash364 2010

[18] G I N Rozvany ldquoA critical review of established methods ofstructural topology optimizationrdquo Structural and Multidisci-plinary Optimization vol 37 no 3 pp 217ndash237 2009

[19] S Osher and R P Fedkiw ldquoLevel set methods an overview andsome recent resultsrdquo Journal of Computational Physics vol 169no 2 pp 463ndash502 2001

[20] G Allaire F Jouve and A M Toader ldquoStructural optimizationusing sensitivity analysis and a level-set methodrdquo Journal ofComputational Physics vol 194 no 1 pp 363ndash393 2004

[21] J Luo Z Luo S Chen L Tong and M Y Wang ldquoA newlevel set method for systematic design of hinge-free compliantmechanismsrdquo Computer Methods in Applied Mechanics andEngineering vol 198 no 2 pp 318ndash331 2008

[22] M Y Wang S Chen X Wang and Y Mei ldquoDesign ofmultimaterial compliant mechanisms using level-set methodsrdquoTransactions of the ASME Journal of Mechanical Design vol 127no 5 pp 941ndash956 2005

[23] M Y Wang X Wang and D Guo ldquoA level set methodfor structural topology optimizationrdquo Computer Methods inApplied Mechanics and Engineering vol 192 no 1-2 pp 227ndash246 2003

[24] B Bourdin and A Chambolle ldquoDesign-dependent loads intopology optimizationrdquo ESAIM Control Optimisation andCalculus of Variations vol 9 pp 19ndash48 2003

[25] S Zhou and M Y Wang ldquoMultimaterial structural topologyoptimization with a generalized Cahn-Hilliard model of mul-tiphase transitionrdquo Structural and Multidisciplinary Optimiza-tion vol 33 no 2 pp 89ndash111 2007

[26] A Takezawa S Nishiwaki and M Kitamura ldquoShape andtopology optimization based on the phase field method andsensitivity analysisrdquo Journal of Computational Physics vol 229no 7 pp 2697ndash2718 2010

[27] A R Dıaaz and N Kikuchi ldquoSolutions to shape and topologyeigenvalue optimization problems using a homogenizationmethodrdquo International Journal for Numerical Methods in Engi-neering vol 35 no 7 pp 1487ndash1502 1992

[28] Z D Ma H C Cheng and N Kikuchi ldquoStructural design forobtaining desired eigenfrequencies by using the topology andshape optimizationmethodrdquoComputing Systems in Engineeringvol 5 no 1 pp 77ndash89 1994

[29] Z DMa N Kikuchi andH-C Cheng ldquoTopological design forvibrating structuresrdquo Computer Methods in Applied Mechanicsand Engineering vol 121 no 1ndash4 pp 259ndash280 1995

10 Mathematical Problems in Engineering

[30] I Kosaka and C C Swan ldquoA summetry reduction methodfor continuum structural topology optimizationrdquo Computers ampStructures vol 70 no 1 pp 47ndash61 1999

[31] L A Krog and N Olhoff ldquoOptimum topology and reinforce-ment design of disk and plate structures with multiple stiffnessand eigenfrequency objectivesrdquoComputers amp Structures vol 72no 4 pp 535ndash563 1999

[32] J S Jensen and N L Pedersen ldquoOn maximal eigenfrequencyseparation in two-material structures the 1D and 2D scalarcasesrdquo Journal of Sound andVibration vol 289 no 4-5 pp 967ndash986 2006

[33] N L Pedersen ldquoMaximization of eigenvalues using topologyoptimizationrdquo Structural and Multidisciplinary Optimizationvol 20 no 1 pp 2ndash11 2000

[34] J H Zhu and W H Zhang ldquoMaximization of structuralnatural frequency with optimal support layoutrdquo Structural andMultidisciplinaryOptimization vol 31 no 6 pp 462ndash469 2006

[35] J Du and N Olhoff ldquoTopological design of freely vibratingcontinuum structures for maximum values of simple andmultiple eigenfrequencies and frequency gapsrdquo Structural andMultidisciplinary Optimization vol 34 no 2 pp 91ndash110 2007

[36] B Niu J Yan and G Cheng ldquoOptimum structure with homo-geneous optimum cellular material for maximum fundamentalfrequencyrdquo Structural and Multidisciplinary Optimization vol39 no 2 pp 115ndash132 2009

[37] Q Xia T Shi and M Y Wang ldquoA level set based shape andtopology optimization method for maximizing the simple orrepeated first eigenvalue of structure vibrationrdquo Structural andMultidisciplinary Optimization vol 43 no 4 pp 473ndash485 2011

[38] Q Xia T Shi M Y Wang and S Liu ldquoA level set basedmethod for the optimization of cast partrdquo Structural andMultidisciplinary Optimization vol 41 no 5 pp 735ndash747 2010

[39] A K Nandy andC S Jog ldquoOptimization of vibrating structuresto reduce radiated noiserdquo Structural and MultidisciplinaryOptimization vol 45 no 5 pp 717ndash728 2012

[40] J Zheng S Long and G Li ldquoTopology optimization forfree vibrating continuum structures based on the element freeGalerkin methodrdquo Structural and Multidisciplinary Optimiza-tion vol 45 no 1 pp 119ndash127 2012

[41] T D Tsai and C C Cheng ldquoStructural design for desired eigen-frequencies and mode shapes using topology optimizationrdquoStructural and Multidisciplinary Optimization vol 47 no 5 pp673ndash686 2013

[42] K Sui Y Modeling Transformation and Optimization NewDevelopments of Structural Synthesis Method Dalian Universityof Technology Press Dalian China 1996

[43] YK Sui J X LiuH L Ye and J ZDu ldquoApplication ofmodifiedsigmoid function in structural topology optimizationrdquo Journalof BeijingUniversity of Technology vol 34 supplement 1 pp 1ndash62008

[44] P E Gill W Murray and M A Saunders ldquoSNOPT anSQP algorithm for large-scale constrained optimizationrdquo SIAMReview vol 47 no 1 pp 99ndash131 2005

[45] J Nocedal and S J Wright Numerical Optimization SpringerBerlin Germany 1999

[46] Y K Sui H L Ye J X Liu S Chen and H P Yu ldquoAstructural topological optimization method based on exploringconceptual rootrdquo Engineering Mechanics vol 25 no 2 pp 7ndash192008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

(a) (b)

Figure 6 Topology configuration with different filter functions (a) Topology configuration with PF (b) topology configuration with CEF

Mod

e

First

PF

(a)

Modal shapesSecond

(b)

Third

(c)

CEF

Mod

e

(d) (e) (f)

Figure 7 The first three modal shapes of optimal structure with two filter functions

0 5 10 15 20 25 30 35

Eige

nfre

quen

cy (H

z)

Iterationminus5

f1f2f3

18171616161515141413131211111110

095

times103

(a)

0 5 10 15 20 25 30Iteration

f1f2f3

Eige

nfre

quen

cy (H

z)

18171616161515141413131211111110

095

times103

(b)

Figure 8 Iteration process of the first three frequencies with different filter functions (a) PF (b) CEF

Mathematical Problems in Engineering 9

curves of the first three order frequencies show that structuralfrequencies satisfy the frequency constraints Therefore allthe computing results indicate that the proposed method isalso feasible for multiple frequency constraints

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of the paper

Acknowledgment

The work of this project was supported by National ScienceFoundation of China (ContractGrant no 11072009 andContractGrant no 11172013)

References

[1] M P Bendsoslashe and N Kikuchi ldquoGenerating optimal topologiesin structural design using a homogenization methodrdquo Com-puterMethods in AppliedMechanics and Engineering vol 71 no2 pp 197ndash224 1988

[2] H A Eschenauer and N Olhoff ldquoTopology optimization ofcontinuum structures a reviewrdquo Applied Mechanics Reviewsvol 54 no 4 pp 331ndash390 2001

[3] J D Deaton and R V Grandhi ldquoA survey of structuraland multidisciplinary continuum topology optimization post2000rdquo Structural and Multidisciplinary Optimization vol 49no 1 pp 1ndash38 2014

[4] G I N Rozvany ldquoAims scope methods history and unifiedterminology of computer-aided topology optimization in struc-tural mechanicsrdquo Structural and Multidisciplinary Optimiza-tion vol 21 no 2 pp 90ndash108 2001

[5] M P Bendsoslashe andO Sigmund Topology OptimizationTheoryMethods and Applications Springer Berlin Germany 2ndedition 2003

[6] B Hassani and E Hinton ldquoA review of homogenization andtopology optimization Imdashhomogenization theory for mediawith periodic structurerdquo Computers amp Structures vol 69 no 6pp 707ndash717 1998

[7] G I N Rozvany and M Zhou ldquoApplications of the COCalgorithm in layout optimizationrdquo in Engineering Optimizationin Design Processes Proceedings of the International ConferenceKarlsruhe Nuclear Research Center Germany September 3-41990 H Eschenauer C Matteck and N Olhoff Eds vol 63of Lecture Notes in Engineering pp 59ndash70 Springer BerlinGermany 1991

[8] R J Yang and C H Chuang ldquoOptimal topology design usinglinear programmingrdquo Computers and Structures vol 52 no 2pp 265ndash275 1994

[9] M P Bendsoslashe andO Sigmund ldquoMaterial interpolation schemesin topology optimizationrdquoArchive of AppliedMechanics vol 69no 9-10 pp 635ndash654 1999

[10] YM Xie andG P Steven ldquoA simple evolutionary procedure forstructural optimizationrdquo Computers and Structures vol 49 no5 pp 885ndash896 1993

[11] YMXie andG P StevenEvolutionary StructuralOptimizationSpringer Berlin Germany 1997

[12] G I N Rozvany O M Querin Z Gaspar and V PomezanskildquoWeight-increasing effect of topology simplificationrdquo Structural

and Multidisciplinary Optimization vol 25 no 5-6 pp 459ndash465 2003

[13] O M Querin G P Steven and Y M Xie ldquoEvolutionarystructural optimisation (ESO) using a bidirectional algorithmrdquoEngineering Computations vol 15 no 8 pp 1031ndash1048 1998

[14] J H ZhuWH Zhang andK P Qiu ldquoBi-directional evolution-ary topology optimization using element replaceable methodrdquoComputational Mechanics vol 40 no 1 pp 97ndash109 2007

[15] X Huang and Y M Xie ldquoA new look at ESO and BESOoptimization methodsrdquo Structural and Multidisciplinary Opti-mization vol 35 no 1 pp 89ndash92 2008

[16] X Huang and YM Xie ldquoA further review of ESO typemethodsfor topology optimizationrdquo Structural and MultidisciplinaryOptimization vol 41 no 5 pp 671ndash683 2010

[17] X Huang Z H Zuo and Y M Xie ldquoEvolutionary topologicaloptimization of vibrating continuum structures for naturalfrequenciesrdquoComputers and Structures vol 88 no 5-6 pp 357ndash364 2010

[18] G I N Rozvany ldquoA critical review of established methods ofstructural topology optimizationrdquo Structural and Multidisci-plinary Optimization vol 37 no 3 pp 217ndash237 2009

[19] S Osher and R P Fedkiw ldquoLevel set methods an overview andsome recent resultsrdquo Journal of Computational Physics vol 169no 2 pp 463ndash502 2001

[20] G Allaire F Jouve and A M Toader ldquoStructural optimizationusing sensitivity analysis and a level-set methodrdquo Journal ofComputational Physics vol 194 no 1 pp 363ndash393 2004

[21] J Luo Z Luo S Chen L Tong and M Y Wang ldquoA newlevel set method for systematic design of hinge-free compliantmechanismsrdquo Computer Methods in Applied Mechanics andEngineering vol 198 no 2 pp 318ndash331 2008

[22] M Y Wang S Chen X Wang and Y Mei ldquoDesign ofmultimaterial compliant mechanisms using level-set methodsrdquoTransactions of the ASME Journal of Mechanical Design vol 127no 5 pp 941ndash956 2005

[23] M Y Wang X Wang and D Guo ldquoA level set methodfor structural topology optimizationrdquo Computer Methods inApplied Mechanics and Engineering vol 192 no 1-2 pp 227ndash246 2003

[24] B Bourdin and A Chambolle ldquoDesign-dependent loads intopology optimizationrdquo ESAIM Control Optimisation andCalculus of Variations vol 9 pp 19ndash48 2003

[25] S Zhou and M Y Wang ldquoMultimaterial structural topologyoptimization with a generalized Cahn-Hilliard model of mul-tiphase transitionrdquo Structural and Multidisciplinary Optimiza-tion vol 33 no 2 pp 89ndash111 2007

[26] A Takezawa S Nishiwaki and M Kitamura ldquoShape andtopology optimization based on the phase field method andsensitivity analysisrdquo Journal of Computational Physics vol 229no 7 pp 2697ndash2718 2010

[27] A R Dıaaz and N Kikuchi ldquoSolutions to shape and topologyeigenvalue optimization problems using a homogenizationmethodrdquo International Journal for Numerical Methods in Engi-neering vol 35 no 7 pp 1487ndash1502 1992

[28] Z D Ma H C Cheng and N Kikuchi ldquoStructural design forobtaining desired eigenfrequencies by using the topology andshape optimizationmethodrdquoComputing Systems in Engineeringvol 5 no 1 pp 77ndash89 1994

[29] Z DMa N Kikuchi andH-C Cheng ldquoTopological design forvibrating structuresrdquo Computer Methods in Applied Mechanicsand Engineering vol 121 no 1ndash4 pp 259ndash280 1995

10 Mathematical Problems in Engineering

[30] I Kosaka and C C Swan ldquoA summetry reduction methodfor continuum structural topology optimizationrdquo Computers ampStructures vol 70 no 1 pp 47ndash61 1999

[31] L A Krog and N Olhoff ldquoOptimum topology and reinforce-ment design of disk and plate structures with multiple stiffnessand eigenfrequency objectivesrdquoComputers amp Structures vol 72no 4 pp 535ndash563 1999

[32] J S Jensen and N L Pedersen ldquoOn maximal eigenfrequencyseparation in two-material structures the 1D and 2D scalarcasesrdquo Journal of Sound andVibration vol 289 no 4-5 pp 967ndash986 2006

[33] N L Pedersen ldquoMaximization of eigenvalues using topologyoptimizationrdquo Structural and Multidisciplinary Optimizationvol 20 no 1 pp 2ndash11 2000

[34] J H Zhu and W H Zhang ldquoMaximization of structuralnatural frequency with optimal support layoutrdquo Structural andMultidisciplinaryOptimization vol 31 no 6 pp 462ndash469 2006

[35] J Du and N Olhoff ldquoTopological design of freely vibratingcontinuum structures for maximum values of simple andmultiple eigenfrequencies and frequency gapsrdquo Structural andMultidisciplinary Optimization vol 34 no 2 pp 91ndash110 2007

[36] B Niu J Yan and G Cheng ldquoOptimum structure with homo-geneous optimum cellular material for maximum fundamentalfrequencyrdquo Structural and Multidisciplinary Optimization vol39 no 2 pp 115ndash132 2009

[37] Q Xia T Shi and M Y Wang ldquoA level set based shape andtopology optimization method for maximizing the simple orrepeated first eigenvalue of structure vibrationrdquo Structural andMultidisciplinary Optimization vol 43 no 4 pp 473ndash485 2011

[38] Q Xia T Shi M Y Wang and S Liu ldquoA level set basedmethod for the optimization of cast partrdquo Structural andMultidisciplinary Optimization vol 41 no 5 pp 735ndash747 2010

[39] A K Nandy andC S Jog ldquoOptimization of vibrating structuresto reduce radiated noiserdquo Structural and MultidisciplinaryOptimization vol 45 no 5 pp 717ndash728 2012

[40] J Zheng S Long and G Li ldquoTopology optimization forfree vibrating continuum structures based on the element freeGalerkin methodrdquo Structural and Multidisciplinary Optimiza-tion vol 45 no 1 pp 119ndash127 2012

[41] T D Tsai and C C Cheng ldquoStructural design for desired eigen-frequencies and mode shapes using topology optimizationrdquoStructural and Multidisciplinary Optimization vol 47 no 5 pp673ndash686 2013

[42] K Sui Y Modeling Transformation and Optimization NewDevelopments of Structural Synthesis Method Dalian Universityof Technology Press Dalian China 1996

[43] YK Sui J X LiuH L Ye and J ZDu ldquoApplication ofmodifiedsigmoid function in structural topology optimizationrdquo Journalof BeijingUniversity of Technology vol 34 supplement 1 pp 1ndash62008

[44] P E Gill W Murray and M A Saunders ldquoSNOPT anSQP algorithm for large-scale constrained optimizationrdquo SIAMReview vol 47 no 1 pp 99ndash131 2005

[45] J Nocedal and S J Wright Numerical Optimization SpringerBerlin Germany 1999

[46] Y K Sui H L Ye J X Liu S Chen and H P Yu ldquoAstructural topological optimization method based on exploringconceptual rootrdquo Engineering Mechanics vol 25 no 2 pp 7ndash192008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

curves of the first three order frequencies show that structuralfrequencies satisfy the frequency constraints Therefore allthe computing results indicate that the proposed method isalso feasible for multiple frequency constraints

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of the paper

Acknowledgment

The work of this project was supported by National ScienceFoundation of China (ContractGrant no 11072009 andContractGrant no 11172013)

References

[1] M P Bendsoslashe and N Kikuchi ldquoGenerating optimal topologiesin structural design using a homogenization methodrdquo Com-puterMethods in AppliedMechanics and Engineering vol 71 no2 pp 197ndash224 1988

[2] H A Eschenauer and N Olhoff ldquoTopology optimization ofcontinuum structures a reviewrdquo Applied Mechanics Reviewsvol 54 no 4 pp 331ndash390 2001

[3] J D Deaton and R V Grandhi ldquoA survey of structuraland multidisciplinary continuum topology optimization post2000rdquo Structural and Multidisciplinary Optimization vol 49no 1 pp 1ndash38 2014

[4] G I N Rozvany ldquoAims scope methods history and unifiedterminology of computer-aided topology optimization in struc-tural mechanicsrdquo Structural and Multidisciplinary Optimiza-tion vol 21 no 2 pp 90ndash108 2001

[5] M P Bendsoslashe andO Sigmund Topology OptimizationTheoryMethods and Applications Springer Berlin Germany 2ndedition 2003

[6] B Hassani and E Hinton ldquoA review of homogenization andtopology optimization Imdashhomogenization theory for mediawith periodic structurerdquo Computers amp Structures vol 69 no 6pp 707ndash717 1998

[7] G I N Rozvany and M Zhou ldquoApplications of the COCalgorithm in layout optimizationrdquo in Engineering Optimizationin Design Processes Proceedings of the International ConferenceKarlsruhe Nuclear Research Center Germany September 3-41990 H Eschenauer C Matteck and N Olhoff Eds vol 63of Lecture Notes in Engineering pp 59ndash70 Springer BerlinGermany 1991

[8] R J Yang and C H Chuang ldquoOptimal topology design usinglinear programmingrdquo Computers and Structures vol 52 no 2pp 265ndash275 1994

[9] M P Bendsoslashe andO Sigmund ldquoMaterial interpolation schemesin topology optimizationrdquoArchive of AppliedMechanics vol 69no 9-10 pp 635ndash654 1999

[10] YM Xie andG P Steven ldquoA simple evolutionary procedure forstructural optimizationrdquo Computers and Structures vol 49 no5 pp 885ndash896 1993

[11] YMXie andG P StevenEvolutionary StructuralOptimizationSpringer Berlin Germany 1997

[12] G I N Rozvany O M Querin Z Gaspar and V PomezanskildquoWeight-increasing effect of topology simplificationrdquo Structural

and Multidisciplinary Optimization vol 25 no 5-6 pp 459ndash465 2003

[13] O M Querin G P Steven and Y M Xie ldquoEvolutionarystructural optimisation (ESO) using a bidirectional algorithmrdquoEngineering Computations vol 15 no 8 pp 1031ndash1048 1998

[14] J H ZhuWH Zhang andK P Qiu ldquoBi-directional evolution-ary topology optimization using element replaceable methodrdquoComputational Mechanics vol 40 no 1 pp 97ndash109 2007

[15] X Huang and Y M Xie ldquoA new look at ESO and BESOoptimization methodsrdquo Structural and Multidisciplinary Opti-mization vol 35 no 1 pp 89ndash92 2008

[16] X Huang and YM Xie ldquoA further review of ESO typemethodsfor topology optimizationrdquo Structural and MultidisciplinaryOptimization vol 41 no 5 pp 671ndash683 2010

[17] X Huang Z H Zuo and Y M Xie ldquoEvolutionary topologicaloptimization of vibrating continuum structures for naturalfrequenciesrdquoComputers and Structures vol 88 no 5-6 pp 357ndash364 2010

[18] G I N Rozvany ldquoA critical review of established methods ofstructural topology optimizationrdquo Structural and Multidisci-plinary Optimization vol 37 no 3 pp 217ndash237 2009

[19] S Osher and R P Fedkiw ldquoLevel set methods an overview andsome recent resultsrdquo Journal of Computational Physics vol 169no 2 pp 463ndash502 2001

[20] G Allaire F Jouve and A M Toader ldquoStructural optimizationusing sensitivity analysis and a level-set methodrdquo Journal ofComputational Physics vol 194 no 1 pp 363ndash393 2004

[21] J Luo Z Luo S Chen L Tong and M Y Wang ldquoA newlevel set method for systematic design of hinge-free compliantmechanismsrdquo Computer Methods in Applied Mechanics andEngineering vol 198 no 2 pp 318ndash331 2008

[22] M Y Wang S Chen X Wang and Y Mei ldquoDesign ofmultimaterial compliant mechanisms using level-set methodsrdquoTransactions of the ASME Journal of Mechanical Design vol 127no 5 pp 941ndash956 2005

[23] M Y Wang X Wang and D Guo ldquoA level set methodfor structural topology optimizationrdquo Computer Methods inApplied Mechanics and Engineering vol 192 no 1-2 pp 227ndash246 2003

[24] B Bourdin and A Chambolle ldquoDesign-dependent loads intopology optimizationrdquo ESAIM Control Optimisation andCalculus of Variations vol 9 pp 19ndash48 2003

[25] S Zhou and M Y Wang ldquoMultimaterial structural topologyoptimization with a generalized Cahn-Hilliard model of mul-tiphase transitionrdquo Structural and Multidisciplinary Optimiza-tion vol 33 no 2 pp 89ndash111 2007

[26] A Takezawa S Nishiwaki and M Kitamura ldquoShape andtopology optimization based on the phase field method andsensitivity analysisrdquo Journal of Computational Physics vol 229no 7 pp 2697ndash2718 2010

[27] A R Dıaaz and N Kikuchi ldquoSolutions to shape and topologyeigenvalue optimization problems using a homogenizationmethodrdquo International Journal for Numerical Methods in Engi-neering vol 35 no 7 pp 1487ndash1502 1992

[28] Z D Ma H C Cheng and N Kikuchi ldquoStructural design forobtaining desired eigenfrequencies by using the topology andshape optimizationmethodrdquoComputing Systems in Engineeringvol 5 no 1 pp 77ndash89 1994

[29] Z DMa N Kikuchi andH-C Cheng ldquoTopological design forvibrating structuresrdquo Computer Methods in Applied Mechanicsand Engineering vol 121 no 1ndash4 pp 259ndash280 1995

10 Mathematical Problems in Engineering

[30] I Kosaka and C C Swan ldquoA summetry reduction methodfor continuum structural topology optimizationrdquo Computers ampStructures vol 70 no 1 pp 47ndash61 1999

[31] L A Krog and N Olhoff ldquoOptimum topology and reinforce-ment design of disk and plate structures with multiple stiffnessand eigenfrequency objectivesrdquoComputers amp Structures vol 72no 4 pp 535ndash563 1999

[32] J S Jensen and N L Pedersen ldquoOn maximal eigenfrequencyseparation in two-material structures the 1D and 2D scalarcasesrdquo Journal of Sound andVibration vol 289 no 4-5 pp 967ndash986 2006

[33] N L Pedersen ldquoMaximization of eigenvalues using topologyoptimizationrdquo Structural and Multidisciplinary Optimizationvol 20 no 1 pp 2ndash11 2000

[34] J H Zhu and W H Zhang ldquoMaximization of structuralnatural frequency with optimal support layoutrdquo Structural andMultidisciplinaryOptimization vol 31 no 6 pp 462ndash469 2006

[35] J Du and N Olhoff ldquoTopological design of freely vibratingcontinuum structures for maximum values of simple andmultiple eigenfrequencies and frequency gapsrdquo Structural andMultidisciplinary Optimization vol 34 no 2 pp 91ndash110 2007

[36] B Niu J Yan and G Cheng ldquoOptimum structure with homo-geneous optimum cellular material for maximum fundamentalfrequencyrdquo Structural and Multidisciplinary Optimization vol39 no 2 pp 115ndash132 2009

[37] Q Xia T Shi and M Y Wang ldquoA level set based shape andtopology optimization method for maximizing the simple orrepeated first eigenvalue of structure vibrationrdquo Structural andMultidisciplinary Optimization vol 43 no 4 pp 473ndash485 2011

[38] Q Xia T Shi M Y Wang and S Liu ldquoA level set basedmethod for the optimization of cast partrdquo Structural andMultidisciplinary Optimization vol 41 no 5 pp 735ndash747 2010

[39] A K Nandy andC S Jog ldquoOptimization of vibrating structuresto reduce radiated noiserdquo Structural and MultidisciplinaryOptimization vol 45 no 5 pp 717ndash728 2012

[40] J Zheng S Long and G Li ldquoTopology optimization forfree vibrating continuum structures based on the element freeGalerkin methodrdquo Structural and Multidisciplinary Optimiza-tion vol 45 no 1 pp 119ndash127 2012

[41] T D Tsai and C C Cheng ldquoStructural design for desired eigen-frequencies and mode shapes using topology optimizationrdquoStructural and Multidisciplinary Optimization vol 47 no 5 pp673ndash686 2013

[42] K Sui Y Modeling Transformation and Optimization NewDevelopments of Structural Synthesis Method Dalian Universityof Technology Press Dalian China 1996

[43] YK Sui J X LiuH L Ye and J ZDu ldquoApplication ofmodifiedsigmoid function in structural topology optimizationrdquo Journalof BeijingUniversity of Technology vol 34 supplement 1 pp 1ndash62008

[44] P E Gill W Murray and M A Saunders ldquoSNOPT anSQP algorithm for large-scale constrained optimizationrdquo SIAMReview vol 47 no 1 pp 99ndash131 2005

[45] J Nocedal and S J Wright Numerical Optimization SpringerBerlin Germany 1999

[46] Y K Sui H L Ye J X Liu S Chen and H P Yu ldquoAstructural topological optimization method based on exploringconceptual rootrdquo Engineering Mechanics vol 25 no 2 pp 7ndash192008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

[30] I Kosaka and C C Swan ldquoA summetry reduction methodfor continuum structural topology optimizationrdquo Computers ampStructures vol 70 no 1 pp 47ndash61 1999

[31] L A Krog and N Olhoff ldquoOptimum topology and reinforce-ment design of disk and plate structures with multiple stiffnessand eigenfrequency objectivesrdquoComputers amp Structures vol 72no 4 pp 535ndash563 1999

[32] J S Jensen and N L Pedersen ldquoOn maximal eigenfrequencyseparation in two-material structures the 1D and 2D scalarcasesrdquo Journal of Sound andVibration vol 289 no 4-5 pp 967ndash986 2006

[33] N L Pedersen ldquoMaximization of eigenvalues using topologyoptimizationrdquo Structural and Multidisciplinary Optimizationvol 20 no 1 pp 2ndash11 2000

[34] J H Zhu and W H Zhang ldquoMaximization of structuralnatural frequency with optimal support layoutrdquo Structural andMultidisciplinaryOptimization vol 31 no 6 pp 462ndash469 2006

[35] J Du and N Olhoff ldquoTopological design of freely vibratingcontinuum structures for maximum values of simple andmultiple eigenfrequencies and frequency gapsrdquo Structural andMultidisciplinary Optimization vol 34 no 2 pp 91ndash110 2007

[36] B Niu J Yan and G Cheng ldquoOptimum structure with homo-geneous optimum cellular material for maximum fundamentalfrequencyrdquo Structural and Multidisciplinary Optimization vol39 no 2 pp 115ndash132 2009

[37] Q Xia T Shi and M Y Wang ldquoA level set based shape andtopology optimization method for maximizing the simple orrepeated first eigenvalue of structure vibrationrdquo Structural andMultidisciplinary Optimization vol 43 no 4 pp 473ndash485 2011

[38] Q Xia T Shi M Y Wang and S Liu ldquoA level set basedmethod for the optimization of cast partrdquo Structural andMultidisciplinary Optimization vol 41 no 5 pp 735ndash747 2010

[39] A K Nandy andC S Jog ldquoOptimization of vibrating structuresto reduce radiated noiserdquo Structural and MultidisciplinaryOptimization vol 45 no 5 pp 717ndash728 2012

[40] J Zheng S Long and G Li ldquoTopology optimization forfree vibrating continuum structures based on the element freeGalerkin methodrdquo Structural and Multidisciplinary Optimiza-tion vol 45 no 1 pp 119ndash127 2012

[41] T D Tsai and C C Cheng ldquoStructural design for desired eigen-frequencies and mode shapes using topology optimizationrdquoStructural and Multidisciplinary Optimization vol 47 no 5 pp673ndash686 2013

[42] K Sui Y Modeling Transformation and Optimization NewDevelopments of Structural Synthesis Method Dalian Universityof Technology Press Dalian China 1996

[43] YK Sui J X LiuH L Ye and J ZDu ldquoApplication ofmodifiedsigmoid function in structural topology optimizationrdquo Journalof BeijingUniversity of Technology vol 34 supplement 1 pp 1ndash62008

[44] P E Gill W Murray and M A Saunders ldquoSNOPT anSQP algorithm for large-scale constrained optimizationrdquo SIAMReview vol 47 no 1 pp 99ndash131 2005

[45] J Nocedal and S J Wright Numerical Optimization SpringerBerlin Germany 1999

[46] Y K Sui H L Ye J X Liu S Chen and H P Yu ldquoAstructural topological optimization method based on exploringconceptual rootrdquo Engineering Mechanics vol 25 no 2 pp 7ndash192008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of