topology optimization of structures using modified binary

Struct Multidisc OptimDOI 10.1007/s00158-010-0523-9

RESEARCH PAPER

Topology optimization of structures using modified binarydifferential evolution

Chun-Yin Wu · Ko-Ying Tseng

Received: 11 February 2009 / Revised: 27 February 2010 / Accepted: 29 March 2010c© Springer-Verlag 2010

Abstract Differential evolution (DE) is an efficient pop-ulation based algorithm used to solve real-valued opti-mization problems. It has the advantage of incorporatingrelatively simple and efficient mutation and crossover oper-ators. However, the DE operator is based on floating-pointrepresentation only, and is difficult to use when solv-ing combinatorial optimization problems. In this paper, amodified binary differential evolution (MBDE) based on abinary bit-string framework with a simple and new binarymutation mechanism is proposed. Two test functions areapplied to verify the MBDE framework with the new binarymutation mechanism, and four structural topology optimiza-tion problems are used to study the performance of theproposed MBDE algorithm. The experimental studies showthat the proposed MBDE algorithm is not only suitable forstructural topology optimization, but also has high viabilityin terms of solving numerical optimization problems.

Keywords Modified binary differential evolution ·Numerical optimization · Structural topology optimization

1 Introduction

The increased use of topology optimization in industrialdesign is notable, as topology optimization can facilitate theautomation of preliminary design steps and propose inno-vative designs to satisfy diverse multidisciplinary designrequirements. Topology optimization methods allow for theintroduction or removal of boundaries in designs through

C.-Y. Wu (B) · K.-Y. TsengDepartment of Mechanical Engineering, Tatung University,Taipei, 104, Taiwan, Republic of Chinae-mail: [email protected]

additions or deletions of material within the design domain.This results in improved performance and may lead toentirely new design options during product development.

Several methods for solving topology optimizationproblems have been explored in recent decades. The homog-enization method, a popular method for topology optimiza-tion, was first introduced by Bendsøe and Kikuchi (1988).The structural form in this method is represented by amicrostructure that redistributes the void and the material.The disadvantage of the homogenization design method isthat the evaluation of microstructure and its orientationsis often cumbersome (Sigmund 2001). Another importantapproach originally introduced by Bendsøe (1989), referredto as solid isotropic microstructure with penalization(SIMP) or the power law approach (Rozvany et al. 1992),has been recognized as a computationally efficient tool forstructural topology optimization problems (Rozvany 2001).The material property of each element in the SIMP methodis constant and the density of each element is defined asa design variable. Many commercial software packagesuse the SIMP approach as a structural topology optimiza-tion tool due to its high efficiency. As a result, SIMP hasreceived considerable attention in recent years. However,SIMP cannot directly solve 1/0 problems (Bendsøe andSigmund 2003), and tends to converge to a local optimumwith a blurry boundary (Wang and Tai 2005). It requires asuitable filtering method to resolve the final solution, whichin turn can increase the application feasibility for struc-tural problems in real world applications (Wang and Tai2005). Evolutionary structural optimization (ESO) was pro-posed by Xie and Steven (1993) for optimizing structure.The topology of structure in ESO was optimized through astep by step removal of the material with the lowest sensi-tivity value of stress or strain energy (Xie and Steven 1993;Hinton and Sienz 1995; Querin et al. 1998). Although it is

C.-Y. Wu, K.-Y. Tseng

Table 1 Mutation mechanisms of differential evolution

Mechanism of Mathematical equation

DE mutation

Best/1/exp vi,G+1 = xbest,G + F(xr1,G − xr2,G

)

Rand/1/exp vi,G+1 = xr1,G + F(xr2,G − xr3,G

)

Rand-to-Best/ vi,G+1 = xi,G + F(xr1,G − xr2,G

)

1/exp

Best/2/exp vi,G+1 = xbest,G + F(xr1,G + xr2,G − xr3,G − xr4,G

)

Rand/2/exp vi,G+1 = xr1,G + F(xr2,G + xr3,G − xr4,G − xr5,G

)

a fully heuristic method, there is no proof of optimality andit may easily result in a non-optimal design (Rozvany 2001;Zhou and Rozvany 2001).

In addition to the methods mentioned above, popula-tion based approaches have also been invented to solvetopology optimization problems. It is well known thatpopulation based approaches are stochastic global optimiza-tion methods (Wang and Tai 2005; Storn and Price 1997;Luh and Chueh 2004). These approaches usually employthe binary code 1/0 to represent the solid/void materialdistribution, which represents a direct approach to dealwith structural topology optimization problems. Compar-ing the performance between population based and gradientbased approaches (i.e. SIMP, ESO) when solving struc-tural topology optimization problems, population basedapproaches are generally less efficient than methods thatemploy gradients (Wang and Tai 2005; Campelo et al. 2006;Balamurugan et al. 2008). However, population basedapproaches not only allow for global search and provide dis-tinct solutions (void and material elements only), but also donot require a good initial design to tackle structural topol-ogy optimization problems (Wang and Tai 2005; Campeloet al. 2006; Balamurugan et al. 2008). The final solutions

obtained by gradient based approaches for the same prob-lem usually differ based on different starting solutions. Assuch, a global optimization method is generally desirable,because a good initial design for many real world problemsis usually difficult to obtain (Wang and Tai 2005).

For the reasons listed above, many types of populationbased approaches have been applied to discover the opti-mal design of topological structures. Genetic algorithms(GAs) have been used to solve topology optimization prob-lems for more than two decades (Goldberg and Samtani1986; Chapman et al. 1994; Chapman and Jakiela 1996;Jakiela et al. 2000; Rajeev and Krishnamoorthy 1992; Wuand Chow 1995; Jenkins 1991; Woon et al. 2001; Kane andSchoenauer 1996; Wang and Tai 2005; Balamurugan et al.2008). In genetic algorithms, three operating mechanisms—reproduction, crossover and mutation—are used to changethe binary 1/0 chromosome codes. GAs can determine theoptimal structure without violating constraints in the designdomain by processing large numbers of 1/0 binary stringcombinations. Luh and Chueh (2004) applied artificialimmune systems (AIS), a heuristic algorithm inspired bytheoretical immunology and observed immune functions, tofind multi-modal optimal solutions of topological structures(Luh and Chueh 2004). The Ant algorithm (Dorigo et al.1996) was inspired by the behavior of real ant colonies.Researchers (Kaveh et al. 2008; Wu et al. 2009; Luh andLin 2009) have also employed this approach in topologyoptimization problems.

Differential evolution (DE) (Storn and Price 1995, 1997)is a simple and efficient population based global optimiza-tion method. It has been used successfully in many fieldsof engineering optimization (Cheng et al. 2008; Qian et al.2008; Gao and Liu 2008). However, it cannot be directlyapplied to binary-valued optimization problems due to thefact that this algorithm is based on continuous-valued searchspaces only. To alleviate this issue, various researches

Fig. 1 DE crossovermechanism

Topology optimization of structures using modified binary differential evolution

Fig. 2 Flowchart of differential evolution

proposed new strategies to transfer continuous-valued DEinto binary-valued DE. Pampará et al. (2006) proposed abinary DE that uses trigonometric functions to transformthe design variables from continuous-space to binary-space.Although this method is easily applied using the originalDE to solve topology optimization problems, the processcannot be reversed by converting the transformed binarydata back into real value data. This may lead to problemswhen the modification of binary data is called for in termsof transforming binary code to real values. Another binarystrategy for differential evolution as proposed by Xingshiand Lin (2007) uses logical operators to replace the origi-nal operations of the mutation mechanism in DE, and thusthe individual can be represented by a binary bit-stringdirectly. However, this method may restrict the diversity ofthe search process because the probabilities of generating

Fig. 3 Example of individuals and value decoding

codes “1” and “0” when using parts of logical operators arenot equivalent.

This paper presents a modified binary differential evolu-tion (MBDE) for solving topology optimization problems.The individual in this study is represented by a binary bit-string, while a novel binary mutation mechanism of MBDEis developed to solve optimization problems. Several topol-ogy optimization examples, including minimum weightdesign problems and minimum compliance design prob-lems, are presented to demonstrate the high viability andperformance of the proposed algorithm. In addition to topol-ogy optimization problems, certain mathematical functionsare also applied to demonstrate the feasibility of MBDE interms of solving parametric optimization problems.

2 Differential evolution

Differential evolution (DE) (Storn and Price 1995, 1997)was developed as a population based algorithm for tack-ling real-value numerical optimization problems. The initialpopulation of DE is generated randomly between the lowerand upper bounds for each design variable. DE uses areal-value vector for the design variable with three controlparameters: crossover rate (CR), scaling factor (F), and pop-ulation size (NP). There are three main DE steps, includingmutation, crossover, and selection. The mutation operatorof DE is executed by adding a weighted difference vec-tor between two individuals to a third individual. Aftercrossover and selection, the mutated individuals produceoffspring. The whole search process is iterated until aconvergence state is reached.

Fig. 4 Mutation procedure in Xingshi and Lin (2007)


Fig. 5 The new binary mutationmechanism of MBDE

2.1 Mutation operator

The mutation operation in DE serves to generate a newmutated vector according to the following scheme equation:

vi,G+1 = xbest,G + F(xr1,G − xr2,G

),

i = 1, 2, 3 . . . . . . NP (1)

The index G is used as an index of the current generation.In (1), r1 and r2 are random numbers generated between1 and NP, and two random target vectors, xr1,G and xr2,G,

are chosen from the population of the current generationG. Further, xbest,G and vi,G+1 are the best target vectorand mutated vector of the current generation. Scaling fac-tor F is a real value between 0 and 1 that will enlarge orreduce the differential variation between the two randomvectors. Equation (1) is a type of DE mutation mecha-nism, and various other mutation mechanisms are listed inTable 1. Further, the individual vectors listed in Table 1,xr1,G, xr2,G, xr3,G, xr4,G, and xr5,G, are randomly selectedfrom the population; these individuals all differ from oneanother.

Fig. 6 Binary crossovermechanism in MBDE


Table 2 Parameter settings ofMBDE for experimentalapproach

Parameter Value

Population size 10

Bit-string length 30

F1 0.5

F2 0.005

CR 0.8

2.2 Crossover operator

In the DE crossover operator, the trial vector ui,G+1 is gen-erated by a combination of the mutated vector vi,G+1 and thetarget vector xi,G. The DE crossover operator is illustratedin Fig. 1 and (2), where CR represents the crossover proba-bility and j is the variable index. If the random number R issmaller than the CR value, the variable of the mutation vec-tor vi,G+1 is chosen as the variable of the trial vector ui,G+1.Otherwise, the variable of the target vector xi,G is selectedas the variable of the trial vector ui,G+1. The mutation andcrossover operators are used to diversify the search space interms of the optimization problems:

ui,G+1 = ui,G+1, if R ≤ CR or j �= i

ui,G+1 = xi,G, if R ≤ CR or j �= i (2)

2.3 Selection operator

After the execution of the mutation and crossover oper-ations, all trial vectors are considered as candidates forselection operations. The trial vector ui,G+1 is then com-pared with the target vector xi,G to select individuals for thenext generation. The selection operator is as follows:

xi,G+1 = ui,G+1, if f(ui,G+1

)> f

(xi,G

)

xi,G+1 = xi,G, if f(ui,G+1

) ≤ f(xi,G

),

i = 1, 2, 3 . . . . . . NP (3)

The trial vector ui,G+1 is chosen as the new target vectorxi,G+1 for the next generation when the objective functionvalue of the trial vector is better than the value of the target

Table 3 Numerical results for test functions

Test D Cal MBDE MBDE FADE FADE

function objective deviation objective deviation

f 1(x) 2 2,000 0.0 0.0 1.131e−7 4.767e−13

f 1(x) 50 5,000,000 2.344e+1 3.26e+1 2.584e+2 9.168e+1

f 2(x) 3 1,500 1.221e−7 0.0 0.156 0.012

f 2(x) 50 2,500,000 1.221e−7 1.287e−7 5.9e-2 1.231e−6

Fig. 7 The translation of binary code to topology structure

vector. Otherwise, the original target vector xi,G becomesthe new target vector xi,G+1. The stop criteria used here callsfor terminating the DE after a pre-specified maximum num-ber of generations, or when no improvement is observedover a pre-specified number of consecutive generations—whichever is encountered first. A DE flowchart is shown inFig. 2.

3 Modified binary differential evolution

Modified binary differential evolution (MBDE) as proposedin this paper is based on both bit-string representationand the novel binary mutation mechanism. The proposedapproach uses a binary bit-string to represent the individualswithin a DE population. The initial population of MBDE israndomly generated by uniform random numbers coded “0”

(a) (b)

Fig. 8 a Continuous structure, b discontinuous structure


Fig. 9 Filling in or removingelements in the structuralmodification

(a) (b) (c) (d)

or “1”. The objective function value of all individuals of thepopulation is calculated and a novel binary mutation mech-anism is used to generate mutated individuals. The binarycrossover mechanism is then applied to build trial solutions.If the objective function value of the trial solution is betterthan original solution, the trial solution will be selected forthe next generation. Otherwise, the new solution for the nextiteration is replaced by the original solution.

3.1 Binary mutation operator

Logical operations were first introduced by Xingshi and Lin(2007) to replace the original operations of the mutationmechanism in DE. The idea of using logical operations isto replace the subtraction, multiplication and addition oper-ations in the mutation procedure by XOR, AND and ORoperators. By using these logical operators, this approachcan directly evolve the individuals using binary strings.The binary mutation equation proposed by Xingshi and Lin(2007) is shown in (4):

vi,G+1 = xr1,G � F ⊗ (xr2,G ⊗ xr3,G

)

i = 1, 2, 3 . . . . . . NP (4)

The subtraction of two randomly selected vectors wasrepresented by the logical operation XOR of the two binarystrings, denoted as “⊕”. The multiplication of factor F wasreplaced by the AND operator and is represented by “⊗”.The OR operator of the two binary strings replaced the addi-tion operation of the two vectors, and is represented by“�”. The traditional vector mutation mechanism of DE in

Fig. 10 Cantilever structure ofcase 1

continuous space was transformed into discrete space usingbinary string logical operations. However, as this methodhas a higher probability of producing code “1” in the evo-lutionary process when using the OR operator, it restrictsthe search diversity of the optimum solution. The followingdescriptions explain the effect of using the OR operator inthe binary mutation mechanism.

From the truth table of the OR logical operator, the prob-ability of a result of truth, coded “1”, is three times higherthan that of false, coded “0”. Code “1” can be easily accu-mulated within the binary string vi,G+1 of the trial solutionafter the OR operator, �, is used in (4). Thus, the num-ber of bits with code “1” may increase as the evolutionaryprocess continues. A simple example can demonstrate this.Consider an integer test function f (x) = x , where the rangeof variable x is set from 0 to 15, and four binary stringsare used to represent the values of x for four individualsas shown in Fig. 3. The individuals a, b and c shown inFig. 4 are selected as xr2,G, xr3,G and xr1,G, respectively.The bits with code “1” of the (xr2,G ⊕ xr3,G) string representthe bits with different binary codes between xr2,G and xr3,G

after the XOR operation of xr2,G and xr3,G. Factor F is setto binary string “1100”. The random string F emulates thedifference vector F in the continuous space. The final trialsolution vi,G+1 is obtained by the OR operation on xr1,G andF. The number of bits with code “1” is increased by one. Itis not easy to change from code “1” to code “0” becausethere is no true mutation operation used in the logical muta-tion operation proposed by Xingshi and Lin (2007). Whenthe code of the third position of all individuals is “1” inFig. 3, use of the logical mutation of (4) leads to a prob-lem. If the optimum solution needs code “0” in this position,

Table 4 Parameter settings ofMBDE for topologyoptimization of structure

Parameter Value

Population size 10

Bit-string length Element

number

F1 0.5

F2 0.005

CR 0.8

Maximum 3,000

generation

Consecutive 150

generation


(a) (b)

Fig. 11 Optimum results of a Wang and Tai (2005), weight = 0.2,b MBDE, weight = 0.19

the code of this position will never be changed to code “0”due to the execution of the OR operator in the final stageof the binary mutation strategy. From the truth table, if thecode of this position for all individuals is “1”, the result ofthese bits after executing the OR operator must be “1”. Evenwhen using a crossover operator to rebuild the binary stringof the individuals, this situation remains because the code“1” exists in the same position among all individuals.

The novel binary mutation mechanism developed in thisstudy is also based on bit-string frameworks and logicaloperations. The binary mutation is truly applied to twogroups of bits determined using an XOR logical operation.The main idea behind the proposed approach is to find thecommon and difference feature patterns of two individu-als and then apply a mutation operation to mutate bits inthe different feature patterns using diverse mutation rates.The bits coded as “0” in the string after the XOR opera-tion represent the common bits in two selected strings. Bitscoded as “1” represent different bits with differing codes in

two selected strings. The new binary mutation equation isillustrated in (5) and (6):

vi,G+1 = F1(xi,G ⊕ xr1,G

) + F2! (xi,G ⊕ xr1,G),

i = 1, 2, 3 . . . . . . NP (5)

vi,G+1 = F1(xbest,G ⊕ x1,G

) + F2! (xbest,G ⊕ xbest,G),

i = 1, 2, 3 . . . . . . NP (6)

Equation (5) is developed to represent the rand-to-best/1/exp mutation mechanism and (6) to express thebest/1/exp mutation mechanism of continuous DE, as shownin Table 1. In (5), the difference bits and the common bitsbetween the xi,G and xr1,G solutions are determined usingthe XOR operation. The difference bits of xi,G are mutatedfrom “0” to “1” or “1” to “0” using mutation rate F1. Sim-ilarly, the common bits of xi,G are mutated with mutationrate F2. Generally, the mutation rate F1 for the differencebits is higher than the mutation rate F2 for the common bits,because there is a higher probability that the common bitsmay become the feature pattern of a final optimum solu-tion. In the final stage, we combine these binary strings,including the common and difference patterns, to repre-sent mutated individuals. The flowchart of the new binarymutation mechanism is shown in Fig. 5. Though it uses log-ical operations with binary strings, the proposed algorithmattempts to follow only the process of DE in searching forthe optimum topology, without any enhancement by otheroptimization algorithms.

3.2 Binary crossover operator

The binary crossover operator is used to build a trial solu-tion ui,G+1 from the mutated solution vi,G and the original

Fig. 12 Convergence history ofMBDE, case 1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

1 19 37 55 73 91 109 127 145 163 181 199 217 235 253 271 289 307 325 343 361 379 397 415 433 451Number of iterations

Wei

ght


solution xi,G. The flowchart of the binary crossover mech-anism is shown in Fig. 6, where j represents the designvariable number of individuals. The concept of the binarycrossover mechanism of MBDE is the same as the crossovermechanism of the original DE, although there is a differencein terms of the component data type. In MBDE, the bit-string data is chosen from the mutated solution vi,G if arandom number is larger than the crossover rate. If thecrossover rate is zero, all bit-strings of the original solu-tion are selected to build the trial solution ui,G+1. Afterbinary mutation and crossover operations, the better solu-tion between the original solution and the trial solutionis selected for the next generation after comparing theobjective function values.

4 Experimental verification

Two numerical test functions are employed in this study toverify the framework of the program and examine the searchcapability of MBDE in real-valued optimization problems.The test functions, Rastrigin’s function f 1 in (7) andAckley’s path function f 2 in (8) (Liu and Lampinen 2005),are used to illustrate the feasibility of MBDE for minimumoptimization problems. The equations of the test functionsare:

f 1 (x) =D∑

j=1

(x j − 10 cos

(2π × x j

)) + 10D

x j ∈ [−5.12, 5.12] , j = 1 : D (7)

f 2 (x) = −20.0 × exp

⎛

⎜⎝−0.2 ×

√√√√√

1

D

D∑

j=1

x2j

⎞

⎟⎠

− exp

⎛

⎝ 1

D×

D∑

j=1

cos(2π × x j

)⎞

⎠

+ 20 + exp (1)

x j ∈ [−32.768, 32.768] , j = 1 : D (8)

Fig. 13 Cantilever structure of case 2

(a) (b)

Fig. 14 Optimum results of a Wang and Tai (2005), weight = 0.325,b Balamurugan et al. (2008), weight = 0.34

where D is the dimension of the test functions and theglobal optimum of these two test functions is 0. The param-eter settings of MBDE are given in Table 2. Mutation rateF1 and crossover rate CR are determined by referencingsuggestions from the literature (Storn and Price 1997) andexecuting many numerical experiments. Parametric settingsof this study are similar to the settings found in the liter-ature. The value of F1 is 0.5 and CR is 0.8. The searchperformance for the optimum solution will decrease if thevalue of F1 is too large or too small. The convergence rateis affected by crossover rate. A large value for the crossoverrate CR can increase convergence speed, but it may also trapinto the local optimum, while a small CR can slow the con-vergence speed and requires a greater number of objectivefunction calculations. A good choice for mutation rate F2 isone percent of the F1 value, as we found through multiplesimulation runs.

The test population size of MBDE is 10 individuals,where each x j of a single individual used 30 bits to rep-resent a real value. The mutation rates F1 and F2 for thenew binary mutation mechanism are 0.5 and 0.005, respec-tively. The crossover rate of MBDE for examination of thetest function is 0.8. In order to test the performance ofMBDE in a real-valued optimization, the results of a fuzzyadaptive differential evolution (FADE) developed by Liuand Lampinen (2005) are introduced to compare the MBDEsolutions across the same number of objective calculations.The results of the comparison between MBDE and FADEare given in Table 3, where Objective represents the objec-tive function value, and Deviation is the standard deviationbased on the results within the best objective function val-ues of 100 runs. Further, D stands for the dimension of thetest function, while Cal is the number of objective functioncalculations. The optimized solutions found using MBDEare very close to the global optimum of the test functionsf 1 and f 2 in the case of lower dimensionality. The MBDEsolution with a dimensionality of 50 in test function f 2 isnot only close to the global optimum, but also better than

Fig. 15 Optimum results ofcase 2 as derived by MBDE,weight = 0.275



0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

1 14 27 40 53 66 79 92 105 118 131 144 157 170 183 196 209 222 235 248 261 274 287 300 313Number of iterations

Wei

ght

the results of FADE (Liu and Lampinen 2005). Thoughthe final MBDE results for test function f 1 with a dimen-sionality of 50 are not identical to the global optimum, theoptimized results are still better than the results given in Liuand Lampinen (2005). These comparison results demon-strate that MBDE, a modified binary differential evolutionbased on a bit-string framework, can be used as a tool fornumerical optimization.

5 Topology optimization using modified binarydifferential evolution

Topology optimization (Yang et al. 2002; Gersborg-Hansenet al. 2006; Du et al. 2008; Canfield and Frecker 2000;Maute and Frangopol 2003) is useful for concept designduring development of an innovative product. When usingthe finite element method to analyze structure, each elementlacking constraints can be eliminated or retained withinthe design domain. This freedom of choice may enablethe creation of entirely new designs outside of the design-ers’ previous experience and professional knowledge. Thedifficulty of applying topology optimization in many real-world applications is often due to the long computationtimes needed during the analysis of complex designs. In

Fig. 17 Design domain for case 3

topology optimization for a practical and feasible structure,the binary code 1/0 is used to find the optimum structure. Inthis study, MBDE is proposed to solve structural topologyoptimization problems, where the bit-string of individualsrepresents the element distribution for the structure topol-ogy. The finite element method is used to analyze thestructure for the data required in calculating the objectivefunction value. In the current study, the computer programintegrating MBDE and finite element analysis for topologyoptimization was developed using the C++ programminglanguage.

The design domain of a two-dimensional continuousstructure is subdivided into elements for the finite elementmodel. A binary string is used to store the shape informationfor each design variable. The code “1” in the string refers tothe corresponding solid element of the structure in the finiteelement model, while the code “0” in the string representsthe corresponding void element of the structure. The stringcan be mapped into a two-dimensional design domain asshown in Fig. 7.

Discontinuous structure may occur when the topologicrepresentation of the structure is changed due to mutationor crossover operation during the search process. It is nec-essary to develop an algorithm to transform discontinuousstructures into continuous structures, as only continuousstructures can be analyzed using the finite element method.

(a) (b)

Fig. 18 Optimum results of a GA (Chapman and Jakiela 1996), com-pliance = 4.469, b Homogenization method (Chapman and Jakiela1996), compliance = 3.987


Fig. 19 Optimum result ofMBDE, compliance = 3.701

The definition of the criterion for the continuity conditionis that any element of the structure must share at least onecommon edge with any other element in the finite elementmodel, as shown in Fig. 8.

To check the continuity of the structure before finite ele-ment analysis, a special algorithm is developed and statedas follows:

1. First, elements with support or loading should beselected as the starting element, because these elementsmust exist in the structure to maintain the coherence ofthe physical model. Code “2” is assigned to the startingelement for the continuity check.

2. The structural configuration is decoded from the binarystring and only elements coded as “1” will be checkedfor continuity. The checking procedure begins at thestarting element. If there is an element with the code“1” sharing a common edge with an element with thecode “2”, the coded “1” is changed into a coded “2”.This procedure is repeated until no remaining elementschange from a code “1” to a code “2”.

3. If the elements of structure coded as “1” are all changedinto elements coded as “2”, this represents a continu-ous structure. If any element coded as “1” still exists inthe structure at the end of the continuity check, this rep-resents a discontinuous structure, as this element doesnot share a common edge with any elements coded as“2”. Finally, all elements coded as “2” in the continuousstructure are changed and coded as “1” for the restora-tion of the binary string representation of the structuretopology.

A topology structure containing discontinuous elementswill undergo a structural modification procedure. Duringthis procedure, removing discontinuous elements from thestructure or filling in the void element of a neighbor of adiscontinuous element is randomly performed to improvethe continuity of the structure, as shown in Fig. 9. Figure9b and c show the element filling procedure, while Fig. 9ddepicts the removal procedure. The continuous structure canbe analyzed using the finite element method to calculate thecompliance or weight of the structure. If the structure is stilldiscontinuous after several discontinuity modification trials,the objective function value of the discontinuous structure incases 1 and 2 is set to a minimal value for the maximizationproblem. In cases 3 and 4, the compliance of the discontinu-ous structure is set to a maximal value for the minimizationproblem.

In order to normalize each term of the objective func-tion, data pertaining to stress, displacement, and volume ofthe full structure without voids is used as the basis for theobjective function. The objective function is defined by (9):

Fobj = Vfull

V× Dfull

D× 1

Penalty(9)

Penalty =n∑

i=1

(Di − Dallow) (10)

where V f ull is the number of elements of the full structure,V is the number of elements of the current structure, D f ull

represents the maximum displacement of the full structure,and D shows the maximum displacement of the currentstructure. The volume and maximum displacement of thefull structure are used to normalize the values for the vol-ume and maximum displacement of the current structure.The definition of the penalty factor of the displacement con-straint is listed in (10) for the subsequent case 1 and case 2.The penalty factor depends on the nodal displacement of


3

4

5

6

7

8

9

10

11

12

1 28 55 82 109 136 163 190 217 244 271 298 325 352 379 406 433 460 487 514 541 568 595 622 649 676Number of iterations

Com

plia

nce


(a) (b)

Fig. 21 Optimum results of Wang and Tai (2005) a without symmetry,compliance = 69.39, b with symmetry, compliance = 65.26

the structures that violate the maximum displacement limit:it represents the summation of the differences between thenodal displacement and the maximum displacement limitfor the structural nodes. If there is no displacement violationfor any of the nodes, the value of Penalty is set to 1.0. Theobjective of the optimization is to search for low volumedesigns that do not violate displacement constraints.

5.1 Case 1

This case (Wang and Tai 2005) involves finding a suitabletopology structure for a minimum weight that satisfies themaximum displacement constraints. The geometric dimen-sions of the cantilever structure are a 1-in. width and a 2-in.height. The design domain is divided into 10 × 20 quadri-lateral elements for finite element analysis. It is subjected toa load F whose value is 1 at the center on the right side. Thenodes with all degrees of freedom fixed are located at thetop and bottom left, as shown in Fig. 10. It is assumed thatEt = 1, with E referring to the Young’s modulus and treferring to the thickness of the plate. The maximum dis-placement is 20. The parameter setting is listed in Table 4and the optimum results derived by Wang and Tai (2005)are shown in Fig. 11a. The results obtained by MBDEare shown in Fig. 11b, and the convergence histogram isillustrated in Fig. 12.

Figure 11a and b show that the weight derived by MBDEis smaller than that by GA. Further, Wang and Tai (2005)required about 20,000 function evaluations to obtain thesolution, while MBDE needed only 4,520 (10 individualswith 452 generations). This shows that MBDE can obtain abetter solution with fewer function evaluations.

(a) (b)

Fig. 22 Optimum results of using MBDE a without symmetry, com-pliance = 65.787, b with symmetry, compliance = 64.44

5.2 Case 2

Case 2 involves GAs (Wang and Tai 2005; Balamuruganet al. 2008). The width and height are 2 and 1. The designdomain is divided into 20 × 10 quadrilateral elements andthe maximum displacement is 220. The downward load-ing F is applied at the center on the right side, while theremaining conditions and parameter settings are identicalto those in case 1. The model of the cantilever structureis shown in Fig. 13, and GA results (Wang and Tai 2005;Balamurugan et al. 2008) are shown in Fig. 14a and b. Theoptimized result derived by MBDE is shown in Fig. 15, andthe convergence histogram is shown in Fig. 16.

Figures 14 and 15 show that the weight of the topologyfound by MBDE is less than that found by Wang and Tai(2005) or Balamurugan et al. (2008). MBDE needs far fewerobjective function calculations than the GA approaches:3,230 as compared to 20,000 (Wang and Tai 2005) and99,800 (Balamurugan et al. 2008). The results of cases 1and 2 demonstrate that MBDE can offer better performancein terms of solving minimum weight design problemsthan GA.

5.3 Case 3

Case 3 is based on an example proposed by Chapman andJakiela (1996) for minimizing a cantilevered plate’s com-pliance with material limitations. In this case, the materialproperties and boundary conditions are illustrated in Chuet al. (1996). The nodes at the top and bottom left are setas fixed nodal displacements, as shown in Fig. 17. The geo-metric dimensions of the cantilever structure are 0.16 m inwidth, 0.1 m in height and 0.001 m in thickness. The designdomain is divided into 32 × 20 quadrilateral elements forfinite element analysis. The vertical load applied at the mid-dle of the right side is 3,000N. The Young’s modulus E is207 GPa and the Poisson’s ratio ν is 0.3. The populationsize of MBDE in this case is 30 because the element numberin this case is triple those of cases 1 and 2. The remainingsettings are the same as in Table 4.

The objective in this case is to find the minimum com-pliance of the topology structure subjected to a maximum

(a) (b)

Fig. 23 Topological results using SIMP a without filtering, b withfiltering (Sigmund 2001)



50

100

150

200

250

300

1 99 197 295 393 491 589 687 785 883 981 1079 1177 1275 1373 1471 1569 1667 1765 1863 1961

Number of iterations

Com

plia

nce

Without symmetry

With symmetry

volume constraint of 25%. The (11), (12), and (13) selectedfrom Chapman and Jakiela (1996) are definitions of theobjective function. If the structure’s volume (V ) is greaterthan the maximum volume constraint (Vmax), the structureis assigned the value of the objective function using (12)or (13). If the structure volume is less than the maximumvolume constraint, the structure is assigned the value of theobjective function using (11). However, if the number ofgenerations executed is <175, (12) is used as the objectivefunction; otherwise, (13) is used. Equations (11), (12), (13)and (14) are listed below:

ObjectiveFunction = 1.0

ln (compliance)(11)

ObjectiveFunction

={

1.0 − generation

175× 0.6 × V − Vmax

Vmax

}

× 1.0

ln (compliance)(12)

ObjectiveFunction ={

1.0 − 0.6 × V − Vmax

Vmax

}

× 1.0

ln (compliance)(13)

Compliance = δmax · FAPPLIED (14)

The definition of compliance listed in (14) is equal to theinner product of the maximum displacement (δmax) at thepoint of the load application with the applied load FAPPLIED.To compare the results of this study with the optimizedresults in Chapman and Jakiela (1996), the objective func-tion definition used in MBDE is the same as that usedin Chapman and Jakiela (1996). The solutions from GAand homogenization method (Chapman and Jakiela 1996)are re-calculated using the commercial software ANSYS toobtain the compliance values under the same constraints andmaterial properties. The results of the references are shownin Fig. 18, while the MBDE results are shown in Fig. 19.

Fig. 25 Convergence history ofSIMP, case 4

0

50

100

150

200

250

300

350

400

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31Number of iterations

Com

plia

nce


Fig. 26 Comparison of bestresult found by SIMP withdifferent meshes; a 40 × 20mesh, compliance = 67.024;b 50 × 25 mesh, compliance =66.483; c 60 × 30 mesh,compliance = 65.307

(a) (b) (c)

Figure 20 demonstrates the convergence history of thebest solution found using MBDE for case 3. The totalnumber of objective function calculations is 20,580 (686generations with 30 individuals) and the compliance of thefinal topology is 3.701. They are similar in volume, but thecompliance of the MBDE solution is 20.7% less than theGA-based solution and 7.7% less than the solution providedby the homogenization method. The number of finite ele-ment analyses performed by MBDE in this case is <21,000,whereas for GA the figure is 24,750 (Chapman and Jakiela1996) due to the fact that only continuous structures canbe analyzed using the finite element approach. In compari-son with the homogenization method, the efficiency of thehomogenization method is better than both GA and MBDE(the number of FE analyses is 50 for the homogenizationmethod). Nevertheless, the compliance found by MBDE issmaller than those determined by both the homogenizationmethod and GA (Chapman and Jakiela 1996).

5.4 Case 4

The example in case 4 is based on Wang and Tai (2005)and sets out to minimize a 2 × 1 cantilevered plate in com-pliance with material limitations. The design domain ofcase 4 is the same as that of case 2 (Fig. 13). The fixednodes are located at the top and bottom left. The thick-ness is 1 and the force is 1 located at center on the rightside. The design domain is divided into 24 × 12 quadri-lateral elements for finite element analysis. The Young’smodulus is 1, and Poisson’s ratio is 0.3. The objective isto find the minimum compliance of the topology structuresubjected to a maximum volume constraint of 50%. Fur-ther, the topological results are compared with the results ofthe SIMP method in Sigmund (2001) to check the correct-ness of the MBDE solution. The SIMP parameter settingassumes that the penalization power is 3.0 and the filter sizeis 1.2. The parameter setting of MBDE is the same as in

Table 5 Minimum compliance using SIMP with filtering (Sigmund2001) across different mesh sizes

Mesh 24 × 12 40 × 20 50 × 25 60 × 30

Compliance 74.12 67.024 66.483 65.307

cases 1 and 2. The results from the literature and MBDE areshown in Figs. 21 and 22, while the SIMP results are shownin Fig. 23. The convergence history of the final topologystructure using MBDE is shown in Fig. 24.

Comparing results without symmetry in Figs. 21a and22a, the compliance obtained by MBDE is better: just19,800 function evaluations are required by MBDE (10individuals with 1,980 generations), which is less than the28,105 required by Wang and Tai (2005). The results forsymmetry in Figs. 21b and 22b show that the MBDE com-pliance is better than that of Wang and Tai (2005), whileMBDE requires only 15,730 function evaluations, com-pared to 20,000 in Wang and Tai (2005). The results of cases3 and 4 demonstrate that MBDE can obtain a better solutionwith fewer objective function evaluations than GA (Wangand Tai 2005).

The results derived by the SIMP method (Sigmund 2001)are shown in Fig. 23. Figures 22b and 24b show that the twofinal topologies are quite similar. The convergence historiesdepicted in Figs. 24 and 25 show that SIMP has much bettercomputational efficiency in terms of solving for topologystructure optimization. However, SIMP results must still befiltered to eliminate the checkerboard phenomenon shownin Fig. 23a. The SIMP design after filtering may sufferfrom blurry boundaries due to the smoothing filtering acrossthe edges (Wang and Wang 2005), resulting in boundaryidentification problems.

Figure 26 displays the optimized solutions using theSIMP approach with different meshes. The minimum com-pliance values corresponding to the solutions in Fig. 26 areshown in Table 5. It can be seen that the optimum solutionsare very similar in shape, and that more distinct boundariescan be obtained using a finer mesh. Although the two finaltopologies in Figs. 22b and 26c are similar, the solutionprovided by MBDE is a 1/0 based design with voids andmaterial only, analogous with the other population basedmethods. Even through a finer mesh and proper filtering,the SIMP results still show fewer gray elements around theboundaries of the topology.

6 Conclusions

In this study, an investigation into structural topology opti-mization using a modified binary differential evolution with


a newly proposed binary mutation operator is performed.The proposed binary mutation operator is simple andsuitable in terms of finding solutions for numerical opti-mization and topology optimization problems. The opti-mized MBDE results are close or superior to the FADEresults (Liu and Lampinen 2005) in numerical optimiza-tion problems. MBDE also offers better performance thanGA approaches (Chapman and Jakiela 1996; Wang andTai 2005; Balamurugan et al. 2008) in solving structuraltopology optimization problems for either minimum weightor minimum compliance design problems. A performancecomparison among the present MBDE, the homogeniza-tion method, and the popular SIMP method is also carriedout. Results show that MBDE can find better solutions thanSIMP and the homogenization method with more compu-tational cost due to the global search nature. The solutionsderived by MBDE, like other population based approaches,are crisp optimum topologies consisting of voids and mate-rial only. The new MBDE applied in topology optimiza-tion is acceptable and the final results are feasible withoutthe necessity for recourse to other enhancement strategiesduring the evolution process.

References

Balamurugan R, Ramakrishnan CV, Singh N (2008) Performanceevaluation of a two stage adaptive genetic algorithm (TSAGA)in structural topology optimization. Applied Soft Computing8:1607–1624

Bendsøe MP (1989) Optimal shape design as a material distributionproblem. Struct Optim 1:193–202

Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in struc-tural design using a homogenization method. Comput MethodsAppl Mech Eng 71:197–224

Bendsøe MP, Sigmund O (2003) Topology optimization: theory, meth-ods and applications. Springer, Berlin

Campelo F, Guimarães FG, Igarashi H, Watanabe K, Ramirez JA(2006) An immune-based algorithm for topology optimization.In: Proceedings of IEEE world congress on computational intelli-gence, Vancouver, BC, Canada, pp 10973–10980

Canfield S, Frecker M (2000) Topology optimization of compliantmechanical amplifiers for piezoelectric actuators. Struct MultidiscOptim 20:269–279

Chapman CD, Jakiela MJ (1996) Genetic algorithm-based structuraltopology design with compliance and topology simplificationconsiderations. ASME J Mech Des 118:89–98

Chapman CD, Saitou K, Jakiela MJ (1994) Genetic algorithms as anapproach to con figuration and topology design. ASME J MechDes 116:1005–1012

Cheng CC, Tsai TD, Lin DW, Chiu CC (2008) Design and analysis ofshock-absorbing structure for flat panel display. IEEE Trans AdvPackaging 31(1):135–142

Chu DN, Xie YM, Hira A, Steven GP (1996) Evolutionary structuraloptimization for problems with stiffness constraints. Finite ElemAnal Des 21:239–251

Dorigo M, Maniezzo V, Coloni A (1996) The ant system: optimizationby a colony of cooperating agents. IEEE Trans Syst Man CybernPart B 26(1):1–13

Du Y, Fang Z, Wu Z, Tian Q (2008) Thermomechanical compliantactuator design using meshless topology optimization. In: Pro-ceedings of the 7th international conference on system simulationand scientific computing, pp 1018–1025

Gao Y, Liu J (2008) A modified differential evolution algorithm andits application in the training of BP neural network. In: Advancedintelligent mechatronics (AIM 2008), pp 1373–1377

Gersborg-Hansen A, Bendsøe MP, Sigmund O (2006) Topology opti-mization of heat conduction problems using the finite volumemethod. Struct Multidisc Optim 31:251–259

Goldberg DE, Samtani MP (1986) Engineering optimization viagenetic algorithm. In: Electronic computation—Proceedings ofthe ninth conference on electronic computation. American Soci-ety of Civil Engineers, Alabama at Birmingham, pp 471–482

Hinton E, Sienz J (1995) Fully stressed topological design of structuresusing an evolutionary procedure. Eng Comput 12:229–244

Jakiela MJ, Chapman CD, Duda J, Adewuya A, Saitou K (2000)Continuum structural topology design with genetic algorithms.Comput Methods Appl Mech Eng 186:339–356

Jenkins W (1991) Structural optimization with the genetic algorithm.Struct Eng 69:418–422

Kane C, Schoenauer M (1996) Topological optimum using geneticalgorithm. Contr Cybern 25:1059–1088

Kaveh A, Hassani B, Shojaee S, Tavakkoli SM (2008) Structuraltopology optimization using ant colony methodology. Eng Struct30:2559–2565

Liu J, Lampinen J (2005) A fuzzy adaptive differential evolutionalgorithm. Soft Comput 9:448–462

Luh GC, Chueh CH (2004) Multi-modal topological optimization ofstructure using immune algorithm. Comput Methods Appl MechEng 193:4035–4055

Luh GC, Lin CY (2009) Structural topology optimization usingant colony optimization algorithm. Applied Soft Computing9(4):1343–1353

Maute K, Frangopol DM (2003) Reliability-based design of MEMSmechanisms by topology optimization. Comput Struct 81:813–824

Pampará G, Engelbrecht AP, Franken N (2006) Binary differentialevolution. In: Proceedings of the IEEE congress on evolutionarycomputation, pp 1873–1879

Qian B, Wang L, Huang DX, Wang X (2008) Scheduling multi-objective job shops using a memetic algorithm based ondifferential evolution. Int J Adv Manuf Technol 35:1014–1027

Querin OM, Steven GP, Xie YM (1998) Evolutionary structural opti-mization (ESO) using a bidirectional algorithm. Eng Comput15:1031–1048

Rajeev S, Krishnamoorthy CS (1992) Discrete optimization of struc-tures using genetic algorithms. Struct Eng 118:1233–1250

Rozvany GIN (2001) Aims, scope, methods, history and unified ter-minology of computer-aided topology optimization in structuralmechanics. Struct Multidisc Optim 21(2):90–108

Rozvany GIN, Zhou M, Birker T (1992) Generalized shape optimiza-tion without homogenization. Struct Optim 4:250–254

Sigmund O (2001) A 99 line topology optimization code written inMATLAB. Struct Multidisc Optim 21(2):120–127

Storn R, Price K (1995) Differential evolution—a simple and efficientadaptive scheme for global optimization over continuous space.Technical Report TR-95-012

Storn R, Price K (1997) Differential evolution—a simple and efficientheuristic for global optimization over continuous spaces. J GlobOptim 11:341–359

Wang SY, Tai K (2005) Structural topology design optimizationusing genetic algorithm with a bit-array representation. ComputMethods Appl Mech Eng 194:3749–3770

Wang MY, Wang SY (2005) Bilateral filtering for structural topologyoptimization. Int J Numer Methods Eng 63(13):1911–1938


Woon SY, Querin OM, Steven GP (2001) Structural application of ashape optimization method based on a genetic algorithm. StructMultidisc Optim 22:57–64

Wu SJ, Chow PT (1995) Steady-state genetic algorithms for discreteoptimization of trusses. Comput Struct 56:979–991

Wu CY, Zhang CB, Wang CJ (2009) Topology optimization of struc-tures using ant colony optimization. In: Proceedings of the firstACM/SIGEVO summit on genetic and evolutionary computation,Shanghai, China, pp 601–607

Xie YM, Steven GP (1993) A simple evolutionary procedure forstructural optimization. Comput Struct 49:885–896

Xingshi H, Lin H (2007) A novel binary differential evolution algo-rithm based on artificial immune system. In: Proceedings ofthe IEEE congress on evolutionary computation, pp 2267–2272

Yang KW, Liu AG, Cheng CC, Lo YL (2002) Topology andshape optimizations of substrates for chirp fiber Bragggrating spectrum tuning. J Lightwave Technol 20(7):1182–1187

Zhou M, Rozvany GIN (2001) On the validity of ESO type meth-ods in topology optimization. Struct Multidisc Optim 21:80–83

topology optimization of structures using modified binary

Documents