a new topology optimization scheme for nonlinear structures

Journal of Mechanical Science and Technology 28 (7) (2014) 2779~2786

www.springerlink.com/content/1738-494x DOI 10.1007/s12206-014-0319-8

A new topology optimization scheme for nonlinear structures†

Young-Sup Eom1 and Seog-Young Han2,* 1Department of Mechanical Engineering, Graduate School, Hanyang University, 17 Haengdang-Dong, Seongdong-Gu, Seoul, 133-791, Korea

2School of Mechanical Engineering, Hanyang University, 17 Haengdang-Dong, Seongdong-Gu, Seoul, 133-791, Korea

(Manuscript Received October 18, 2012; Revised June 25, 2013; Accepted January 15, 2014)

----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract A new topology optimization algorithm based on artificial bee colony algorithm (ABCA) was developed and applied to geometrically

nonlinear structures. A finite element method and the Newton-Raphson technique were adopted for the nonlinear topology optimization. The distribution of material is expressed by the density of each element and a filter scheme was implemented to prevent a checkerboard pattern in the optimized layouts. In the application of ABCA for long structures or structures with small volume constraints, optimized topologies may be obtained differently for the same problem at each trial. The calculation speed is also very slow since topology optimi-zation based on the roulette-wheel method requires many finite element analyses. To improve the calculation speed and stability of ABCA, a rank-based method was used. By optimizing several examples, it was verified that the developed topology scheme based on ABCA is very effective and applicable in geometrically nonlinear topology optimization problems.

Keywords: Artificial bee colony algorithm; Nonlinear finite element analysis; Rank-based method; Sensitivity number; Topology optimization ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction

To date, topology optimization has been applied for various linear structural problems [1-9]. However, when a very large load is applied or the structural deformation is very large, geometric nonlinearity may occur. In this case, the nonlinear-ity must be considered in the analysis and design in order to obtain a more stable and robust structural topology. Stiffness designs of geometrically nonlinear structures using topology optimization based on the SIMP (solid isotropic material with penalization) method have been performed [10]. It was found that minimization of complementary elastic work rather than compliance as an objective function, should be used for geo-metrically nonlinear topology optimization. In addition, it was verified that there is no difference between the obtained to-pologies of four- and nine-node rectangular elements when a mesh-independent filtering scheme is employed, while the computation speed of the four-node element is much faster than that of the nine-node element. Huang and Xie [11, 12] also carried out nonlinear topology optimization with load and displacement constraints using a bidirectional evolutionary structural optimization (BESO) method.

It is known that the SIMP and BESO methods widely used in topology optimization may provide different topologies under the same constraints due to the parameters used in each

method. For example, even in the case of linear topology op-timization in the SIMP and BESO methods, the obtained op-timized topologies may be different due to parameters of the filtering scheme or the element removal ratio [1, 3]. It is ex-pected that these phenomena may be severe in nonlinear to-pology optimization. Therefore, a new nonlinear topology optimization scheme is needed to obtain a stable and robust optimal topology regardless of the filtering parameter or re-moval ratio.

Nature-inspired computation methods have been applied for optimum designs using natural or physical phenomena with an evolutionary optimization process. As opposed to local search methods, these techniques are close to global search methods. In general, stochastic and probabilistic methods are imple-mented. The artificial bee colony algorithm (ABCA) sug-gested by Karaboga [13] is one of the nature-inspired compu-tation methods, which imitates bee colony behavior of gather-ing nectar. Recently, it has been applied to determine a vari-able of concrete dam [14], and in linear topology optimization for static [15] and dynamic stiffness problems [16].

Since topology optimization methods are based on an itera-tive optimization process, the computing time is very impor-tant. It is known that ABCA is a very fast and accurate algo-rithm among evolutionary algorithm (EA), particle swarm optimization (PSO) and differential evolution (DE) ap-proaches for multi-dimensional numeric problems [17], and the genetic algorithm (GA), particle swarm algorithm (PSO) and particle swarm-inspired evolutionary algorithm (PS-EA)

*Corresponding author. Tel.: +82 2 2220 0456, Fax.: +82 2 2220 2299 E-mail address: [email protected]

† Recommended by Associate Editor Jongsoo Lee © KSME & Springer 2014

2780 Y.-S. Eom and S.-Y. Han / Journal of Mechanical Science and Technology 28 (7) (2014) 2779~2786

in finding a global minimum for multivariable functions [17, 18]. It was also verified that a topology optimization method based on ABCA is faster than SIMP for static stiffness prob-lems [15] and BESO for dynamic stiffness problems [16]. Furthermore, a topology optimization method based on ABCA has not been applied to topology optimization for geometrically nonlinear structures. For these reasons, ABCA was adopted as a topology optimization algorithm in this study.

In this study, a new topology optimization algorithm based on ABCA is proposed for nonlinear structural problems. The algorithm can be applied to both linear and nonlinear struc-tures. The distribution of material is expressed by the density of each element in the application of ABCA. In order to pre-vent a checkerboard pattern in optimized layouts, a filter scheme was implemented. In order to improve the conver-gence rate and stability of the optimization process, a rank-based method was employed. The optimized topologies were compared with those of the SIMP [19] and soft-killed BESO methods [12] to evaluate the effectiveness and applicability of the proposed algorithm.

2. Topology optimization

2.1 Linear topology optimization

Topology optimization is used to obtain material distribu-tion to minimize an objective function. In order to find the material distribution, the design domain must be divided into finite elements. Commonly, topology optimization for a struc-ture is performed to obtain the stiffest structure under a pre-scribed volume. Hence, the compliance, C , is employed as an objective function and the topology optimization can be formulated as follows.

Minimize : .Subjected to : equilibrium and . s s

CV V£

(1)

Here, sV is the target volume and sV is the volume for

the present topology. The compliance is defined as follows:

11 12 2

T TC F U U K U-= = (2)

where F is the load vector, U is the displacement vector, and K is the stiffness matrix.

If a method using the density distribution such as the SIMP method is applied to topology optimization, the optimized topology naturally includes a grey region. Thus, it is very dif-ficult to apply the optimal topology to a real structure. In order to produce the material distribution without a grey region, a discrete function, c , is introduced [12].

1 if is a solid element

( )0 if is an empty element .

cì

= íî

xx

x (3)

If the discrete function is applied to the stiffness matrix, the following is obtained.

0( )K Kc= x (4)

where 0K is the stiffness matrix in the design domain.

2.2 Geometrically nonlinear topology optimization

2.2.1 Formulation for geometrically nonlinear topology op-timization

In geometrically nonlinear topology optimization under a load constraint, the complementary work, CW , shown in Fig. 1 as an objective function, provides a better result than the compliance [10]. Nonlinear topology optimization used com-plementary work as an objective function can be formulated as follows:

Minimize : ( ) .Subjected to : equilibrium and .

C

s s

f x WV V

=

£ (5)

where complementary work can be expressed as the follows.

( )( )1 11

1lim .2

nC T T

i i i ini

W F F U U- -®¥=

= - +å (6)

2.2.2 Sensitivity number for load constraint

The sensitivity number, ea , indicates the effectiveness of the i-th element on the objective function. The sensitivity number for a load constraint, ea , in geometrically nonlinear topology optimization can be expressed as the following equa-tion [12].

An adjoint equation is introduced by adding a series of

( ) 11

1

( ) 1lim2

nT T i i

e i inie e e

f x U UF Fx x x

a --®¥

=

é ùæ ö¶ ¶ ¶= = - ´ +ê úç ÷

¶ ¶ ¶ê úè øë ûå (7)

vectors of Lagrangian multipliers into the objective function as follows:

Fig. 1. Load-deflection curves in nonlinear finite element analysis.

Y.-S. Eom and S.-Y. Han / Journal of Mechanical Science and Technology 28 (7) (2014) 2779~2786 2781

( ) ( )( )

( )

1 11

1

1lim2

nT T

i i i ini

i i i

f x F F U U

R Rl

- -®¥=

-

é= - +ë

+ + ùû

å (8)

where iR and 1iR - are residual forces at the i and 1i - , respectively, defined as follows.

int int

1 1 1 0.i i i i i iR R F F F F- - -+ = - + - = (9)

The sensitivity number for the modified objective function

of Eq. (8) can be expressed as follows.

( )

( )

11

1

11 1

1

1lim2

.

nT T i i

e i ini e e

i ii i i ii

i e i e e

U UF Fx x

R RR U R UU x U x x

a

l

--®¥

=

-- -

-

é æ ö¶ ¶= - +ê ç ÷

¶ ¶ê è øëùæ ¶ + ö¶ ¶ ¶ ¶

+ + + úç ÷¶ ¶ ¶ ¶ ¶ úè øû

å (10)

If the increment of the load is small enough, the relationship between the load and displacement can be assumed to be lin-ear, where /i iR U¶ ¶ and 1 1/i iR U- -¶ ¶ can be expressed us-ing the tangential stiffness matrix, iK , at the i-th increment as follows.

1

1

.ti ii

i i

R R KU U

-

-

¶ ¶= = -

¶ ¶ (11)

The sensitivity number for the objective function can be

written using the following equation.

( )

( )

11

1

1

1lim2

.

nT T t i i

e i i i ini e e

i iTi

e

U UF F Kx x

R Rx

a l

l

--®¥

=

-

é æ ö¶ ¶= - - ´ +ê ç ÷

¶ ¶ê è øë¶ + ù

+ ú¶ û

å (12)

In order to remove the ( )1/ /i e i eU x U x-¶ ¶ + ¶ ¶ term, the

Lagrangian multiplier is calculated as follows.

( )1 1t T T t

i i i i i i iK F F K U Ul - -= - = - (13)

1 .i i iU Ul -= - (14)

Using Eqs. (13) and (14), the sensitivity number can be ex-

pressed as follows.

( )

( )

1

1

int int1

11

1lim2

1lim .2

ni iT

e ini e

nT T i ii in

i e e

R Rx

F FU Ux x

a l -

®¥=

--®¥

=

é ¶ + ù= ê ú¶ë û

é ùæ ö¶ ¶= - - ´ +ê úç ÷

¶ ¶ê úè øë û

å

å (15)

3. A new topology algorithm

3.1 Artificial bee colony algorithm (ABCA)

The fitness, fiti, plays the role of the objective function in ABCA. As the fitness value increases, the candidate solution improves. The fitness of the i-th element is defined using the sensitivity number of each element as follows [15, 16].

1 0

11 ( ) 0 : sensitivity number of the -th element .

iii

i i

i

fitabs

i

aa

a aa

ì ³ï += íï + <î

(16)

All employed bees, that leave the hive to search for promis-

ing flower patches, are initialized randomly over the full de-sign domain as shown in the following equations [14].

( ) if

otherwise

: random number in the range [ 1,1] .

ij ij ij kj ij kjij

ij

ij

x x x x xx

fn

f

+ - ¹ìï= íïî

-

(17)

In the above equation, we can ignore the index j , because

we can express the two-dimensional index number as an inte-ger in MATLAB programming. Therefore, the variables iv ,

ix and kx indicate a candidate solution and two arbitrary initialized solutions, respectively. The subscripts i and k indicate the index of each discretized element randomly chosen in the full design domain. The temporary candidate solution, iv , is obtained using the random search method shown in Eq. (17). Then, the positions of the employed bees are determined by comparing the fitness values corresponding to iv , ix and kx .

An onlooker bee chooses a food source depending on the probability, ip , associated with the i-th element. ip is de-termined by transforming the i-th element’s fitness into a probability, ranging between 0 and 1. The procedure for searching for a candidate solution, iv , is performed by choos-ing ix and kx only in the case of satisfying the condition, with a random number [0,1] < ip for each element in the full design domain. That is, an improved candidate solution is searched by raising the probability of the employed bee’s movement to a food source with a larger fitness by increasing the number of random trials (roulette-wheel method [20]). A candidate solution, iv , from the old solutions, ix and kx , can be generated as follows [17, 18].

1

( ) : random number in the range [ 1,1] .

ii n

jj

i i i i k

i

fitpfit

x x xn ff

=

=

= + -

-

å (18)

After each candidate solution is produced and evaluated by


the artificial bee model, its performance is compared to that of its old solution. If the new solution has equal or better quality than the old solution, the old one is replaced by the new one. Otherwise, the old solution is retained.

The topology optimization procedure using the suggested ABCA proceeds as follows [15].

(1) Establish the design domain and parameters of topology optimization using ABCA.

(2) Calculate the fitness values for the initial design domain using finite element analysis.

(3) Perform the employed bee phase. Determine the posi-tions of the ix and kx elements and the randomly chosen temporary candidate solution ( )iv . The positions are deter-mined by applying the following equations.

[ [0,1] (number of all food source)][ [0,1] (number of all food source)]

[ [0,1]( )]

i

k

i i i k

x INT randx INT rand

x INT rand x xn

= ´

= ´

= + - (19)

interger value, random number .INT rand= = (4) Perform the onlooker bee phase. Search for the new em-

ployed bee position determined by an onlooker bee. Calculate the probability of each element and select the temporary can-didate solution.

1

ii n

jj

fitpfit

=

=å

(20a)

[ [0,1]( )] .i i i kx INT rand x xn = + - (20b) (5) Perform the scout bee phase. Search occupied elements

(employed bee colony) and abandoned elements (scout bee colony). Compare the fitness of employed bee colony with that of the scout bee colony.

(6) Obtain a candidate solution satisfying all constraints. (7) Calculate the total strain energy for the updated design

domain using finite element analysis. In this step, a developed filter scheme [21] is implemented to prevent a checkerboard pattern which can occur when the finite element method is used for topology optimization.

1min

1

( ) , ( ) ( 1,2,... ) .

( )

Mn

ij jj

i ij ijM

ijj

w rw r r r j M

w r

aa =

=

= = - =å

å (21)

Here, M is the total number of nodes in the circular sub-

domain, ( )ijw r is the linear weight factor, nja is the nodal

sensitivity number of the j-th node, minr is the length scale parameter, and ijr is the distance between the center of the element i and the j-th node.

Check whether the updated topology converged to the op-timized topology using the following convergence criterion [21].

'

'

'

1 11

11

( ).

NC C

k i k N ii

NC

k ii

W Werror

Wt

- + - - +=

- +=

-= £å

å (22)

Here, CW is the complementary work, t is an allowable

convergence error, k is the present iteration number, and N is the integral number which results in a stable complementary work in at least ten successive iterations.

If the updated topology is not converged, calculate fitness values for the updated design domain with the density of en-tire elements updated by the density updating scheme using finite element analysis. Return to step 3 and repeat the above steps until an optimized topology is obtained. A schematic diagram of the suggested ABCA is described in Fig. 2.

3.2 A new topology algorithm for geometrically nonlinear

problems

A new topology algorithm for both linear and nonlinear problems is proposed based on ABCA in the previous section. The main characteristics of ABCA were modified because of the following reasons. When topology optimization based on ABCA is carried out, the effects of the topology and density at the previous iteration must be considered.

Since onlooker bees search for nectars probabilistically us-ing the roulette-wheel method [20] in step 4 of the ABCA, they may be located at bad positions. This results in providing

Fig. 2. A flowchart of the bee colony algorithm [16].


a different topology at each trial. In the case of a short or sim-ple structure, the bees at bad positions easily move to good positions using the roulette-wheel method. As a result, an optimal topology can be obtained. However, in the case of linear or nonlinear topology optimization for a long or com-plex structure, since the effects of the topology and density at the previous iteration are very severe, the optimized topology is very differently obtained at each trial. In order to overcome this problem, the rank-based method [20] is implemented instead of the roulette-wheel method. The rank-based method ranks a temporary candidate solution by the probability ip calculated by each element’s fitness value. It is assumed that the result of the rank-based method would be the same as that obtained when the roulette-wheel method is infinitely per-formed.

Fig. 3 shows the results of topology optimizations for a long cantilever beam using various techniques. When the rou-lette-wheel method was used, a different optimal topology was obtained at each trial. It is far from the optimized topol-ogy obtained from the SIMP and BESO methods. When the rank-based method was employed, the optimized topology coincided with that of the SIMP and BESO methods. If the

rank-based method is implemented with ABCA, the scout bee phase for escaping the local minimum in step 5 can be omitted since the probability is not used to search for a new solution in the proposed algorithm.

The density reflection ratio, Cr , is introduced to make the fitness values stable. If the density reflection ratio is not prop-erly chosen, the fitness values are severely varied, and as a result, the topology may not converge. The density update rule is modified to Eq. (23) using the density reflection ratio, Cr , and the densities at the previous and present iterations.

1 1 (1 ) .k k kC Cr rr r r+ -= ´ + - ´ (23)

Here, kr is the density at the present iteration, and 1kr -

is the density at the previous iteration. It is found that Cr is appropriate in the range of 0.85~0.9 for a fine mesh size for both linear and geometrically nonlinear problems from nu-merical experiments like those shown in Tables 1 and 2.

4. Numerical examples

4.1 Example 1: a clamped long beam under a given load

A clamped beam having dimensions of 0.2 m×1.6 m× 0.01 m was subjected to 30N at the center of the bottom surface, as shown in Fig. 4. When a filter scheme was applied to topology optimization, there was no difference of the final results be-tween four- and nine-node rectangular elements, while the computing time decreased by a factor of 16 when the four-node rectangular element was used [10]. Thus, the design domain was divided by the four-node rectangular element in this study. The material was assumed to have a Young’s modulus of 30 MPa and a Poisson’s ratio of 0.3. The allow-able convergence error, t , was set at 0.001. The objective was to obtain the stiffest structure under a volume constraint of 20% of the original volume.

When the proposed method is applied for nonlinear topol-ogy optimization, the effects of the density reflection ratio, Cρ , the filtering parameter, minr , and the size of the finite element must be considered. In order to examine these effects, the values of Cρ used were 0.75, 0.8, 0.85, 0.9, and 0.95, the val-ues of minr used were 1.5, 2, and 2.5, and the element sizes evaluated were 120×15, 240×30, and 360×45. The linear and nonlinear topology optimization results of each combination are shown in Tables 1 and 2, where “-” indicates that the op-timal topology for the case was not obtained.

By comparing the results of the proposed method with those [19] of the SIMP method shown in Fig. 5, it can be seen that

(a) Design domain

(b) The SIMP method

(c) The BESO method

(d) The proposed method with the roulette-wheel method

(e) The proposed method with rank-based method

Fig. 3. Design domain and optimal layouts for various topology opti-mization methods.

Fig. 4. A clamped long beam with a given load.


the optimal topologies for linear and geometrically nonlinear cases are very similar to those of SIMP. The complementary works were calculated to be 0.23~0.24J for the linear case and 0.22~0.24J for the geometrically nonlinear case. It was also found that the appropriate density reflection ratio, Cρ, was 0.95 for the mesh size 240×30 and 0.85~0.9 for the mesh size 360×45 for the geometrically nonlinear case.

Consequently, the appropriate density reflection ratio, Cρ, is 0.85~0.9 for the mesh size of 360×45 for geometrically

nonlinear cases. In other words, the density reflection ratio, Cρ, must be in the range of 0.85~0.9 and the mesh size must be fine enough in order to obtain an optimized topology regard-less of the filtering parameter, minr .

4.2 Example 2: a simply supported beam under a given load

A simply supported beam with dimensions of 0.8 m× 0.2 m×0.001 m was subjected to 200N at the center of the top

Table 1. Results of example 1 for the linear case.

Element size rmin Cρ = 0.75 Cρ = 0.80 Cρ = 0.85 Cρ = 0.90 Cρ = 0.95

1.5 - - - - -

2.0 - - - - - 120 × 15 2.5 - - - - -

1.5 -

2.0 -

240 × 30

2.5 -

1.5

2.0 -

360 × 45

2.5 -

Table 2. Results of example 1 for the geometrically nonlinear case.

Element size rmin Cρ = 0.75 Cρ = 0.80 Cρ = 0.85 Cρ = 0.90 Cρ = 0.95

1.5 - - - -

2.0

- - -

120 × 15

2.5 - - -

1.5 -

2.0

240 × 30

2.5

1.5

2.0

360 × 45

2.5 -


surface as shown in Fig. 6. The design domain was divided into a mesh size of 160×40 by the four-node rectangular ele-ment, considering the results of Sec. 4.1. The material is as-sumed to have a Young’s modulus of 1 GPa and a Poisson’s ratio of 0.3. The allowable convergence error,t , was set at 0.001. The objective was to obtain the stiffest structure under a volume constraint of 20% of the original volume. The den-sity reflection ratio, Cρ , and the filtering parameter, minr were set at 0.85 and 2, respectively. The results are shown in Fig. 7. The allowable convergence error,t , was set at 0.001. The objective was to obtain the stiffest structure under a volume constraint of 20% of the original volume.

From comparisons of the results of the proposed method with those [19] of the SIMP method shown in Fig. 7 for the linear and geometrically nonlinear cases, it is seen.

5. Conclusions

In this study, a new topology optimization algorithm base on ABCA was proposed. Topology optimization using the proposed algorithm was performed for linear, and geometri-cally nonlinear problems. From the results, the following con-clusions can be made.

(1) The proposed method provides a clear optimized topol-ogy consisting of only solid elements, whereas those from the SIMP consist of grey and solid elements.

(2) It was verified that the proposed algorithm can be effec-tively applied for topology optimization based on a compari-son of the optimized topologies of the SIMP method.

(3) It was found that the density reflection ratio, Cρ, must be in the range of 0.85~0.9 and the mesh size must be fine enough in order to obtain an optimal topology regardless of filtering parameter ( minr ).

References

[1] M. P. Bendsoe and N. Kikuchi, Generating optimal topolo-gies in structural design using a homogenization method, Comput. Meth. Appl. Mech. Eng., 71 (28) (1988) 197-224.

[2] H. P. Mlejek and R. Schirmacher, An engineer's approach to optimal material distribution & shape finding, Comput. Meth. Appl. Mech. Eng., 106 (1993) 1-6.

[3] Y. M. Xie and G. P. Steven, A simple evolutionary proce-dure for structural optimization, Comput. Struct., (1993) 885-896.

[4] O. M. Querin, G. P. Steven and Y. M. Xie, Evolutionary structural optimization (ESO) using a bidirectional algorithm, Eng. Comput., 15 (1998) 1031-1048.

[5] Q. Q. Liang and G. P. Steven, A performance-based optimi-zation method for topology design of continuum structures with mean compliance constraints, Comput. Meth. Appl. Mech. Eng., 191 (2002) 1471-1489.

[6] Q. Q. Liang and G. P. Steven, Performance-based optimiza-tion of structures: Theory and applications, Spon press, Tay-lor and Francis Group, London, New York (2005).

[7] J. Sethian and A. Wiegmann, Structural boundary design via level set and immersed interface methods, J. Comput. Phys., 163 (2000) 489-528.

[8] T. Belytschko, S. P. Xiao and C. Parimi, Topology optimiza-tion with implicit functions and regularization, Int. J. Numer. Meth. Engng., 57 (2003) 1177-1196.

[9] K. X. Zou, C. M. Chan, G. Li and Q. Wang, Multiobjective Optimization for Performance-Based Design of Reinforced Concrete Frames, J. Struct. Eng., 133 (2007) 1462-1474.

[10] T. Buhl and C. B. W. Pedersen and O. Sigmund, Stiffness design of geometrically nonlinear structures using topology optimization, Struct. Multidisc. Optim., 19 (2000) 93-104.

[11] X. Huang and Y. M. Xie, Topology optimization of nonlin-ear structures under displacement loading, Eng. Struct., 30 (2008) 2057-2068.

[12] X. Huang and Y. M. Xie, Evolutionary topology optimiza-tion of Continuum Structures, John Wiley & Sons. Ltd.,

(a) Linear case

(b) Geometrically nonlinear case

Fig. 5. Optimal topology using the SIMP method [19].

Fig. 6. Design domain of example 2.

(a) Linear case

(b) Geometrically nonlinear case

Fig. 7. Optimized topologies using the SIMP method [19] and the proposed method.


(2010) 121-150. [13] D. Karaboga, An idea based on honey bee swarm for nu-

merical optimization, Engineering Faculty, Computer Engi-neering Department, Erciyes University, Technical report-TR06, (2005).

[14] F. Kang, J. Li and Q. Xu, Structural inverse analysis by hybrid simplex artificial bee colony algorithms, Comput. Struct., 87 (2009) 861-870.

[15] J. Y. Park and S. Y. Han, Swarm intelligence topology optimization based on artificial bee colony algorithm, Int. J. Precis. Eng, Man., 14 (1) 2013) 115-121.

[16] J. Y. Park and S. Y. Han, Application of artificial bee col-ony algorithm to topology optimization for dynamic stiffness problems, Comput. Math. with Appl. (2013) article in press.

[17] D. Karaboga and B. Basturk, On the performance of artifi-cial bee colony (ABC) algorithm, Appl. Soft Comput., 8 (2008) 687-697.

[18] D. Karaboga and B. Basturk, A powerful and efficient algo-rithm for numerical function optimization: artificial bee col-ony (ABC) algorithm, J. of Glob. Optim., 39 (2010) 459-471.

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

[20] Z. Gu^^rdal, R. T. Haftka and P. Hajela, Design and opti-mization of laminated composite materials, John Wiley & Sons, Inc. (1999) 191-201.

[21] X. Huang and Y. M. Xie, Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method, Finite Elem. Anal. Des., 43 (2007) 1039-1049.

Seog-Young Han received his M.S. and Doctor Degree in Mechanical Engineer-ing from Oregon State University, Cor-vallis, USA, in 1984 and 1989, respec-tively. Now he is a professor in School of Mechanical Engineering of Hanyang University, Korea. His reseach area is topology and shape optimization, na-

ture-inspired optimization algorithm and structural strength analysis.

a new topology optimization scheme for nonlinear structures

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