Computers and Structures 89 (2011) 445–454

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Computers and Structures

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

Evolutionary level set method for structural topology optimization

Haipeng Jia a,⇑, H.G. Beom a, Yuxin Wang a, Song Lin a, Bo Liu b

a Department of Mechanical Engineering, Inha University, 253 Yonghyun-dong, Incheon 402-751, South Koreab Department of Mechanical Engineering, Hebei University of Technology, 8 Dingzigu, Tianjin 300130, PR China

a r t i c l e i n f o a b s t r a c t

Article history:Received 11 August 2009Accepted 3 November 2010Available online 13 January 2011

Keywords:Evolutionary structure optimizationStructure topology optimizationIntelligent computationLevel set method

0045-7949/$ - see front matter Crown Copyright � 2doi:10.1016/j.compstruc.2010.11.003

⇑ Corresponding author. Tel.: +82 32 860 7310; faxE-mail addresses: jiahaipeng@gmail.com, jhp@h

m@inha.ac.kr (H.G. Beom), wyxphd@dlut.edu.cn (Y. W(S. Lin), liubo@hebut.edu.cn (B. Liu).

This paper proposes an evolutionary accelerated computational level set algorithm for structure topologyoptimization. It integrates the merits of evolutionary structure optimization (ESO) and level set method(LSM). Traditional LSM algorithm is largely dependent on the initial guess topology. The proposedmethod combines the merits of ESO techniques with those of LSM algorithm, while allowing new holesto be automatically generated in low strain energy within the nodal neighboring region during optimiza-tion. The validity and robustness of the new algorithm are supported by some widely used benchmarkexamples in topology optimization. Numerical computations show that optimization convergence isaccelerated effectively.

Crown Copyright � 2010 Published by Elsevier Ltd. All rights reserved.

1. Introduction

In recent years, structure optimization has become one of themost important topics of engineering applications. Structure opti-mization has been an interesting area of research in engineeringdesign field for its ability to shorten the design cycle and to en-hance the product quality. Significant research activity has oc-curred in the area of structural optimization during the lastdecade. Especially for topology optimization of structure, manynew theoretical, algorithmic, and computational contributionshave resulted from researchers and engineers. Topology optimiza-tion is a powerful tool for global and multi-scale design of macro-structures, microstructures, and the cell topology of prescribedcomposite materials. The bio-inspired evolutionary algorithmshave emerged as powerful mechanism for finding optimum solu-tions of complex optimization problems in engineering duringthe last two decades. Evolutionary computation is the study ofnumerical computation systems which uses ideas and gets inspira-tion from natural evolution and adaptation [1,2].

Optimization of structures can be classified into three catego-ries: size, shape, and topology optimization. Topology optimizationis usually referred to as layout optimization or general shape opti-mization [3]. Topology optimization has been identified as one ofthe most challenging tasks in structural design. Various techniquesand approaches have been established during the last two decades.It lets engineers get the optimal topology of structure or new con-

010 Published by Elsevier Ltd. All r

: +82 32 868 1716.ebut.edu.cn (H. Jia), hgbeo-

ang), lsshg2000@naver.com

figurations during product design phase, as they are implementingthe optimal design of the size and shape of structure. In the lasttwo decades, topology optimization has been becoming increas-ingly popular in industrial applications [4–6].

In order to improve efficiency in global optimization search fortopology of engineering problems, many heuristic algorithms [7,8]have been developed, such as evolutionary algorithm [9,10], genet-ic algorithms [11,12], ant algorithm and simulated annealing algo-rithm. In recent years, many biologically inspired methods come tobe used in topology optimization of structure. Evolutionary algo-rithms are a popular and robust strategy for solving structure opti-mization problems. Especially the evolutionary structureoptimization (ESO) has been applied widely in solving structuraltopology optimization problems [13–15]. These methods have spe-cial characteristics, such as parallel computing and global optimumsearching ability. With improved template variety and recognitionreliability to depict topology, Lin and Lin [7] gave two-stage artifi-cial neural networks for topology and shape optimization of struc-ture. Salami and Hendtlass [8] proposed a fast evolutionaryalgorithm that does not evaluate all new individuals, in which fit-ness and associated reliability value are assigned to each new indi-vidual that is evaluated using the true fitness function only if thereliability value is below a threshold. Xie and Steven [10] proposedthe ESO and bidirectional ESO (BESO) approach [14] for topologyoptimization and applied to the optimization of structures success-fully. For its simplicity in implementation and convenience in cop-ing with the local buckling, displacement constraint and localstress constraint, ESO algorithm has wide applications in thedynamic modification, topology optimization thermal-structurecoupling problems with different criteria, for further please seeRefs. [15–17]. Based on the principle of evolutionary optimization,

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446 H. Jia et al. / Computers and Structures 89 (2011) 445–454

Cervera and Trevelyan [18,19] developed Non-uniform rational B-spines topology optimization algorithm. Victoria et al. [20] pro-posed the isolines topology design algorithm, and in essence it isa variant of evolutionary structure topology optimization ap-proach. Based on the thinking of a perfect state of harmony in mu-sical processing, Lee and Geem [21] developed the implementationof harmony searching algorithm for structural optimization.Eschenauer et al. [22] put forward the bubble method. The mainidea is firstly to introduce a new small circle and then implementa shape optimization by a conventional fixed topology shape opti-mization to get the size, shape and position of the hole. Osher andSethian [23] proposed the concept of level set, which soon has beenproven to be phenomenally successful as a numerical device. Sinceits appearance, level set method has wide applications rangingfrom capturing multiphase fluid dynamical flows [24], to specialeffects in Hollywood to visualization, image processing [25,26],topology optimization of structure [27,28], computer vision andmany more. Wang and his coauthors [29,30] proposed level setmethod for structural topology optimization and many other vari-ants of this algorithm to improve its solving ability. Wei and Wang[30] proposed piecewise constant level set method to nucleateholes during optimization and some benchmark problems showthe validity of the algorithm. Level set method attracts many inter-ests of engineers and researchers in structural topology optimiza-tion field for the smoothness of the boundary and having nointermediate density in the final topology [27–33].

The studies all aimed at developing a robust and efficient algo-rithm for searching global optimum solution for engineering appli-cations. Usually optimization procedure composed of structuralanalysis, in which finite element analysis (FEA) is used and costlarge percent of total CPU time for each iteration. For each optimi-zation iteration needs at least one finite element analysis, the com-putational cost for engineering optimization is often very high ifthe first order derivative is needed. The programming optimizationneeds at least the first order derivative and the second order deriv-ative is needed in some optimization algorithm. Because the finiteelement analysis for engineering model takes lots of time in findingthe required data for calculating the parameters of objective func-tion in optimization problem and that of constraint function. Var-ious mathematical programming methods have been used to solveengineering optimization problems. But these methods need calcu-lation of the first or second order differentiation that will increasethe difficulty in searching optimum solution. The number of itera-tion is considered as an important efficiency indicator of optimiza-tion algorithm. The less the number of iteration, the less time needin optimization and structure analysis. With the proposed algo-rithm, the number of iterations is reduced considerably for bench-mark problems in structural topology optimization. On the otherhand, the mathematical programming methods are easy to fall intolocal optimum for non-convexity of topology optimization prob-lems. The traditional level set method algorithms for topology opti-mization use a Hamilton–Jacobi equation to connect the evolutionof the scalar function with the boundary of the topology contours.For this reason, it can hardly create new holes during evolvingotherwise other measures have been taken.

This paper proposes an evolutionary accelerated LSM algorithm.The newly modified method integrates the ESO bio-inspired hole-inserting technique with LSM method and overcomes the short-comings of traditional LSM approach for structure topology optimi-zation. Traditional LSM algorithm is largely dependent on theinitial guess topology. Proposed algorithm in this need no initiali-zation and can get good result. Using this algorithm, new holes canbe inserted at low strain energy regions during the optimization todetermine the optimal topology. From the point of view of groundstructure, the proposed method of topology optimization enlargesthe searching space of level set method. On the other hand, if the

initialization topology is given and the hole is evenly distributedthe proposed algorithm can get the same topology in less iteration.At the same time, the computational efficiency is enhanced consid-erably for less iteration needed for some benchmark problems.

2. Mathematical formulation of evolutionary acceleratedoptimization algorithm

2.1. Mathematical formulation of structural topology optimization

The problem of topology optimization of structure to maximizestiffness can be specified as

minimize : JðuÞ ¼Z

XFðuÞdX ð1Þ

subject to : aðu; vÞ ¼ LðvÞ; ð2ÞujCd¼ u0 8v 2 U; ð3Þ

V ¼Z

XdX 6 Vmax; ð4Þ

where the solid design domain is represented by X with its bound-ary C. The linearly elastic equilibrium equation is written in itsweak variation form, u is the nodal displacement field functionand v is the adjoint displacement field function in the space U ofkinematically admissible displacement fields. Field function u0 pre-scribes displacement field on partial boundary Cd. The design do-main of the structure is denoted by X. J(u) is the objectivefunction, and F(u) is the specific physical or geometric type on de-sign domain and Vmax prescribes the maximum volume constraintof structure. In structural optimization F(u) is usually the compli-ance of structure and the optimization is to find the minimum ofit, let the structure be the stiffest under all kinds of constraint.

The essence of structural topology optimization is to find thebest material distribution under given amount of material understatically equilibrium governing equation. In terms of the energybilinear form a(u, v), the load linear form L(v) and eij(u) described by

aðu;vÞ ¼Z

XEijkleijðuÞeklðvÞdX; ð5Þ

LðvÞ ¼Z

Xpv dXþ

ZC

tv ds; ð6Þ

eijðuÞ ¼12

@ui

@xjþ @uj

@xi

� �: ð7Þ

The purpose of the topology optimization is to optimize the objec-tive function by rationally distributing the prescribed amount ofmaterial in design domain. Level set method is one of the mostimportant methods for its advantages over other similar ones forstructural topology optimization. But when this kind of algorithmis applied to structure topology optimization there are some draw-backs, such as low computational efficiency for more iterations.

2.2. Traditional level set method for topology optimization

The level set models may also be referred to as implicit movingboundary models. In the level set method, the boundary of struc-ture is described by zero level set and can easily represent compli-cated surface shapes that can form holes, split to form multipleboundaries, or merge with other boundaries to form a single sur-face. Based on the concept of propagation of the level set surface,the design changes are carried out to solve the problem of struc-tural topology optimization. Zero level sets are decided by theobjective function, such as energy of deformation, stress, eigen-value, and the optimal structure can be got through the movement,amalgamation of the external boundary of the structure. Comparedwith the homogenization method and solid isotropic material

Fig. 1. Diagram of level set function and its implicit boundary at some time.

H. Jia et al. / Computers and Structures 89 (2011) 445–454 447

penalty (SIMP) method [31], the LSM has some excellent aspects:no check-board phenomena, no mesh-dependency problems, andgood numerical stability. For an easy embedding of the level setmodels, as shown in Fig. 1 we define a larger, fixed reference do-main X such that it fully contains the current structure solid designdomain, the structural boundary can be represented implicitly by alevel set model S as an embedding through a higher dimensionalfunction

S ¼ fx : /ðxÞ ¼ kg: ð8Þ

Furthermore, we define an inside–outside function for U such that

/ðxÞ ¼> 0 8x 2 solid;

¼ 0 8x 2 boundary;

< 0 8x 2 void:

8><>: ð9Þ

Herein, we use the convention that the zero level sets /ðx; tÞ ¼ 0give the information on structural boundaries at virtual time t asin Fig. 1. The LSM describes the topology of structure implicitly,and the course of the topology optimization of continuum struc-tures is achieved by solving the Hamilton–Jacobi equation given as

@/@tþr/ � V ¼ 0: ð10Þ

To make the value of level set function decrease, the normal veloc-ity V is chosen to let level set function change, t is virtual time. Theequation can be solved using finite differential method and the dif-ference points are consistent with nodes used in finite elementanalysis. Time step selection should be satisfied with Courant–Friedrichs–Levy (CFL) condition which makes difference calculationstability. The up-wind solutions produce the motion of level setmodels over the design domain. In numerical implementation onlythe zero level set is considered. With the level set models the math-ematical formulation of structural topology optimization problemcan be specified as

minimize : Jðu;/Þ ¼Z

XFðuÞHð/ÞdX ð11Þ

subject to : aðu;v ;/Þ ¼ Lðv;/Þ; ð12ÞujCd¼ u0 8v 2 U; ð13Þ

V ¼Z

XHð/ÞdX 6 Vmax: ð14Þ

In terms of the energy bilinear form aðu;v;/Þ, the load linear formLðv;/Þ, and the volume Vð/Þ of the structure can be formulated as

aðu;v ;/Þ ¼Z

XEijkleijðuÞeklðvÞHð/ÞdX; ð15Þ

Lðv;/Þ ¼Z

XpvHð/ÞdXþ

ZC

tvdð/Þjr/jds; ð16Þ

Vð/Þ ¼Z

XHð/ÞdX: ð17Þ

where d(x) is Dirichlet function and H(x) Heaviside function [23].Generally speaking, the displacement and adjoint displacementfields are obtained through finite element analysis. To avoid the sin-gularity in numerical calculation in finite element analysis of thewhole structure with different topology, the Heaviside function isintroduced. In this paper the Heaviside function is approximated as

HðxÞ ¼a for x < �D;3ð1�aÞ

4xD� x3

3D3

� �þ 1þa

2 for � D < x < D;

1 for x > D;

8>><>>:

ð18Þ

in this paper, to get clear boundary the value of D is 1 and a is aninfinite small positive number for void region. For more details onthe numerical implementation of level set methods, readers can re-fer to related papers [23,29].

For its permanent characteristics, the level set method for struc-tural topology optimization depends on the initialization of guesstopology. For similar benchmark problems in structure topologyoptimization the computer iteration in level set method is morethan that of mathematical programming, such as feasible directionmethod (FDM) and sequential quadratic programming (SQP). Sothe efficiency improvement is inevitable for engineering applica-tion and much achievement has been got during the last years. Thispaper proposed an evolutionary enhanced level set method.

2.3. Evolutionary accelerated topology optimization algorithm

Only the moving and merging of holes can be implemented dur-ing the LSM topology optimization, no new holes can be generatedthrough the optimization iteration. The disadvantage of level setbased topology optimization is apparent for the emergence ofnew topology especially applying to some engineering problems.To conquer the difficulty, one method is to initialize the guess de-sign with enough holes in order to include as many topologies aspossible. To get a good result, we should comply with the followingtwo fundamental principles to initialize the guess configurationbefore carrying out the topology optimization. Then we shouldhave the candidate topology prepared in advance on the start ofthe optimization. On the one hand, the number of holes must beenough to include all the possible topology. On the other hand,the layout of the holes should be rationally positioned for theevolving of the boundary in order to obtain the best topology ofstructure.

Cantilever beam is a benchmark problem in topology optimiza-tion. As shown in Fig. 2, it has the length of L = 64 mm and heightH = 40 mm, thickness of the plate t = 1 mm and is subjected to aconcentrated load P = 80 N at the middle point of the free end.The objective function of the problem is strain energy of the struc-ture with material volume constraint. The Young’s modulus andPoisson’s ratio of the material used in the example are 200 GPaand 0.3, respectively. To eliminate the numerical singularityparameters in Eq. (18), a = 10�9, D = 1.0 are used in the numericalapproximation of d(x) and H(x). The volume ratio is limited to 25%of the design domain. The cantilever beam is meshed by 64 � 40quadrilateral isoperimetric plane stress elements.

Fig. 2. Geometry parameters and boundary conditions of cantilever plate.

Fig. 3. Computational flow of the structural topology optimization.

Fig. 4. Computational diagram for strain energy of node neighboring region.

448 H. Jia et al. / Computers and Structures 89 (2011) 445–454

Numerical computation shows that good topology can be ob-tained if the initialized configuration includes sufficient numberof holes [29]. It consists with the result in paper [33]. And asbenchmark problems, the final topology is almost the same nomatter optimization criteria (OC) approach is used or mathemati-cal programming is applied [34]. For a more detailed descriptionon the theory and numerical implementation of the level set basedstructure topology optimization, readers can refer to the paper[29]. Numerical experiments show that the optimal topology de-pends on the initialization considerably. In fact, the optimal topol-ogy is only a subset of the candidate topology set of initialization.The more the topologies are included, the higher the possibility of agood design to be obtained for optimization algorithm.

To illustrate the invalidity of the level set based topology opti-mization algorithm, let us design the topology of a cantileveredplate, a classical benchmark problem for topology optimization,from an initial guess topology with no hole. The design result indi-cates that the optimal topology is a two-bar-truss-like structure,which is apparently different from the real optimum topology.From this example, we can safely come to the conclusion thatthe optimal topology highly depends on the initial guess design,and that LSM can only find the best topology in the given topologysets in advance.

To circumvent the obstacle of independence of initialization,new criteria should be introduced to generate new holes at theright position during the optimization iteration then the candidatetopology should be extended. The procedure for material removal/addition plays a crucial role in finding the optimal design. This isthe emphasis of this paper. The proposed method is based on thenode neighboring strain energy as illustrated in Fig. 3, and the per-formance index bn can be calculated through

bn ¼Z

XNEijkleijðuÞeklðvÞdX; ð19Þ

in which XN indicates the node neighboring region as shown inFig. 4, and bn is the performance index of the nth node relative tothe whole structure. This value indicates the effect of nth nodeneighboring region on strain energy of whole structure when it isremoved from structure. In numerical computation, the subscriptn indicates the loop for each node in the design domain.

Based on the principle of evolutionary structure optimization,the proposed algorithm finds a small percentage of the loweststrain energy of all nodes within solid material region and gener-ates a hole during optimization iteration. In the implicit embed-ding description of the topology of structure through level setfunction. The node value of node can be got through solving the

Hamilton–Jacobi equation. Points not on the node can be calcu-lated through interpolation functions as

/ðx; tÞ ¼X

j

/jðtÞNjðxÞ; ð20Þ

herein /jðtÞ is the nodal value of the level set function and Nj(x) isthe standard interpolation functions, j indicates the node numberfor each element, j = 1, 2, 3, 4 in the proposed algorithm. The shapefunction of quadrilateral isoparametric element is adopted in thisalgorithm [35,36].

The nodal values that are updated during the optimization pro-cedure then result in the variation of the structural topology. Thenthe new evolving boundary can be inserted through setting a neg-ative constant at nodes with low value of strain energy within itsneighboring region. The newly generated structure boundary ap-peared as shown in Fig. 5. The next newly expanding velocitycan be calculated based on the new boundary.

The initial value can be changed a little to get better result fordifferent problems and it has little effect on the final topology.

H. Jia et al. / Computers and Structures 89 (2011) 445–454 449

The numerical implementation of the proposed algorithm is asfollows:

Step 1: Initialize the guess topology of the structure with signeddistance function in terms of the external boundary.

Fig. 5. Node strain energy averaged from four neighboring gauss points.

Fig. 6. Updated topology boundary after inserting a new hole during evolvingoptimization.

Step 2: Solve the equilibrium equation of the structure throughfinite element analysis (FEA).Step 3: Compute the performance index of the candidatenode and the value is the strain energy of a node-neighboringregion bn.Step 4: Insert hole in the material region according to the valueof bn, generally the remove rate is 2–3% of all nodes.Step 5: Evolve the topology of the structure. Solve the level setequation to update the embedding function and get newtopology.Step 6: Check the optimization convergence. If the volume con-straint is met, then the optimization iteration is terminated orrepeated from Step 2 to Step 6 until convergence.

For clarity, Fig. 6 gives the numerical implement diagram for theevolutionary level set algorithm proposed in this paper.

In Step 3 and Step 4, the proposed method can control the posi-tion of the inserted hole adaptively. Apart from that the number ofholes and the iteration number can be carried out individually incomputational implementation. For different fields of topologyoptimization problems, corresponding parameters should be ad-justed accordingly.

3. Numerical examples

To illustrate the reliability and the validity of the nodal evolu-tionary accelerated level set method for topology optimization,the classical cantilever beam in Fig. 2 is optimized and gets a goodresult. At the same time, to show the efficiency of the evolutionarylevel set algorithm, guess topology having no hole problems iscomputed to show the characteristics. In order to illustrate theability and efficiency of the proposed method, topology evolvinghistory of Michell structure and Messerschmitt–Bölkow–Blohm(MBB) beam is given.

3.1. Cantilever beam

To solve the problem with initialization configuration having noholes as in Fig. 7a, one cannot get the optimal topology using tra-ditional level set method algorithm easily. According to the en-hanced theory of evolutionary level set method for structureoptimization, this paper gives the automated hole-inserting ap-proach. Proposed evolutionary level set method solved the cantile-ver beam with 72 iterations.

The evolution procedure of structural topology is shown fromFig. 7b to f. The final topology of the cantilever shows the validityof the proposed method. From topology Fig. 7b we can see the gen-erated hole that appeared. Then it is evolving, enlarged andemerged with the new boundary to form a bigger hole. Othertwo new holes appeared in Fig. 7d and evolved until they formthe final topology in Fig. 7f. It is clear that the topology is similarto the widely accepted ones in the literature.

Fig. 8 gives the variation history of structural strain energy dur-ing optimization. Due to the volume constraint, the compliancefunction is increasing with the decreasing usage of material instructure until satisfying the volume constraint, which is 25% ofdesign domain. After that compliance is minimized while holingthe constraint. Due to drastic topology changes, such as boundarymerging or breaking there are small bumps in compliance at theiteration number 21 and 34. Fig. 9 shows the iteration history ofmaterial usage within the design domain during topology evolving.From the history curve, we can see that the curve nearly leveledout at constant value at the iteration number 55. This demon-strates that the optimization in the iteration from 55 until 72 is a

Fig. 7. The proposed algorithm: topology evolving process with initialization having no holes.

0 10 20 30 40 50 60 70 800.0004

0.0005

0.0006

0.0007

0.0008

0.0009

Str

ain

Ene

rgy

(N.m

)

Iteration number

Fig. 8. History of strain energy of the structure and iteration number.

0 10 20 30 40 50 60 70 80-200

0

200

400

600

800

1000

1200

1400

1600

Con

stra

int f

unct

ion

(mm

3 )

Iteration number

Fig. 9. History value of constraint function and iteration number, the valueindicated the gross material usage during the optimization.

450 H. Jia et al. / Computers and Structures 89 (2011) 445–454

procedure of redistribution of the used available material withinthe design domain.

If the given guess topology has some distributed holes in ad-vance, the evolutionary level set method can find the best topology

Fig. 10. Initialized topology with uniformly distributed holes.

0 5 10 15 20 25 30 35 400.0004

0.0005

0.0006

0.0007

0.0008

0.0009

Stra

in E

nerg

y (N

.m)

Iteration number

Fig. 12. History of strain energy of the structure and iteration number.

H. Jia et al. / Computers and Structures 89 (2011) 445–454 451

with less iterations. To illustrate the high computational efficiencyof the proposed algorithm, the same benchmark problem is solved,but the initialization has initialized holes as in Fig. 10. Using theproposed algorithm, holes can be newly generated when possible

(b) Iteration number 13

(a) Iteration number 6

(c) Iteration number 20

Fig. 11. The proposed algorithm: topology evolv

during the optimization iteration at low strain energy region andcan be evolved with the initialized holes to reach the final topol-ogy. Fig. 11 gives the key intermediate evolving topology duringoptimization. Optimization history shows that the iterationnumber decreased from 72 to 39. Fig. 12 gives the structural strain

(e) Iteration number 33

(d) Iteration number 27

(f) Iteration number 39

ing process with initialization having holes.

0 5 10 15 20 25 30 35 40-200

0

200

400

600

800

1000

1200

1400

1600

Con

stra

int f

unct

ion

(mm

3 )

Iteration number

Fig. 13. History value of constraint function and iteration number, the valueindicated the gross material usage during optimization.

H

L

P

Fig. 14. Michell structure: geometry model and boundary conditions.

452 H. Jia et al. / Computers and Structures 89 (2011) 445–454

energy variation history during optimization. Fig. 13 shows theiteration history of material usage within the design domain dur-ing topology evolving.

For the two cases of different initial designs, we have got thesimilar resultant topology as in Figs. 7f and 11f. The topology issame as that in papers for the benchmark problem. From the pointof view optimization iteration, the iteration numbers are all less

Fig. 15. Topology evolving his

than those through traditional level set algorithm and the valuesare 72 and 39 for two initializations, respectively, using the pro-posed evolutionary level set method in this paper. From the figureof iteration history we know that the compliance of structure is8.69 � 10�4 and 8.71 � 10�4, respectively. Twofold advantages ofevolutionary level set method can be got through the numericalexamples, one is it can solve any problem without initializationin advance and another is its high efficiency compared with otheralgorithm for benchmark problems. The lower iteration number

tory of Michell structure.

H. Jia et al. / Computers and Structures 89 (2011) 445–454 453

illustrates the capability and high efficiency of proposed evolution-ary accelerated level set method.

3.2. Michell structure

Another widely occuring problem is solved to further show thecomputational ability of the evolutionary level set method. Asshown in Fig. 14 is the geometry model of Michell structure, inwhich the length of L = 80 mm and height H = 40 mm, thicknessof the plate t = 1 mm. The vertical load is located at the middlepoint of lower border and the value of P = 80 N. The Young’s mod-ulus of the material is 200 GPa and the Possion’s ratio is 0.3. Foroptimization of the topology, the volume constraint is 25% of theoriginal design domain. In level set mathematical formulation theparameters are selected as D = 1, a = 10�9. The design domain is di-vided into 80 � 40 quadrilateral isoparametric plane stresselement.

The topology optimization is carried out using the proposedalgorithm and the topology evolving history is shown as inFig. 15. The history shows that the iteration number is 62. The finalreport is the same as that obtained through optimal criteria andmathematical programming.

3.3. MBB beam

MBB beam is the standard problem for topology optimization.Simply supported MBB beam whose vertical displacement but

Fig. 16. MBB beam: geometry model and boundary conditions, for symmetry halfmodel optimized.

Fig. 17. Topology evolving

not the moment is fixed at the end points is subject to a down-ward-directed point load at the mid point of the beam. The MBBbeam has the function of carrying the floor in the fuselage of an air-bus passenger carrier. Fig. 16 gives the geometry model and load-ing conditions of MBB beam. The dimension of the design domainis as follows, the length of L = 240 mm and height H = 40 mm,thickness of the plate t = 1 mm. The downward external load is lo-cated at the middle point of the upper border, and the forceP = 120 N. Considering the symmetry of the structure and bound-ary conditions, we compute the topology using half of it as in rightshadow part to save computational time. The Young’s modulus ofthe material is 200 GPa and the Possion’s ratio is 0.3, and the vol-ume constraint is 35%, parameter D = 1, a = 10�9, the design do-main is divided into 60 � 20 quadrilateral isoperimetric planestress element.

The topology optimization is carried out using the proposedalgorithm and the topology evolving history shown as in Fig. 17.The final topology is the same as that obtained through optimalcriteria method and modified feasible direction method.

Compared with evolutionary structure optimization (ESO) algo-rithm, the proposed algorithm is highly efficient. Generally speak-ing, the proposed algorithm can get final topology within 80iterations, which cannot be implemented by ESO method. On theother hand, compared with traditional level set method (LSM) for-mulation, the proposed algorithm can compute to get the finaltopology without initialized topology. As we all know, the initial-ized topology is given without any theoretical foundation. It is verydifficult to give initial topology except benchmark problems fortopology optimization. The evolutionary enhanced algorithm hasno need for initialization even for complicated design domain forits optimum searching ability implicitly.

4. Conclusions

This paper proposed an evolutionary LSM algorithm for topol-ogy optimization, which integrated with hole-inserting strategiesused in ESO algorithm. The algorithm integrated the merits oftwo methods and eliminated the weaknesses of conventional levelset method. Smooth boundary of the final topology can be got

history of MBB beam.

454 H. Jia et al. / Computers and Structures 89 (2011) 445–454

directly and the topology optimization method needs no post-processing for the manufacturability. The evolutionary level setfor topology optimization needs no explicit description of the var-iation of the structural topology, i.e. all the merits of LSM methodsare kept and topology searching ability is enhanced considerably.In conclusion, the nodal ESO integrated level set methods for topol-ogy optimization has the following characteristics: Enlarged opti-mum searching scope of the ground structure and solved thetopology optimization without holes in initialization of guess con-figuration. For it cannot get a satisfied topology within proper iter-ation for traditional LSM method. With the proposed acceleratedalgorithm proposed in this paper to solve benchmark problems,the computational efficiency can be improved considerably. Fur-thermore it can get the optimal topology in less iteration for initial-ization with enough initialized holes. Additionally, the proposedalgorithm can be used to solve other engineering problems easilyif given different optimization criteria, such as local stress con-straint, eigenvalue optimization and design of compliantmechanism.

Acknowledgments

This work was supported by Inha University Research Grant,Natural Science Foundation of Hebei (Grant No. 2008000087) andNatural Science Foundation of China (Grant No. 60973079). Theirfinancial supports are greatly appreciated.

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