An adaptive elitist differential evolution for optimization of truss structures with discrete design variables

Computers and Structures 165 (2016) 59–75

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Computers and Structures

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

An adaptive elitist differential evolution for optimization of trussstructures with discrete design variables

http://dx.doi.org/10.1016/j.compstruc.2015.11.0140045-7949/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: Division of Computational Mathematics and Engi-neering, Institute for Computational Science, Ton Duc Thang University, Vietnam.Tel.: +84 933 666 226.

E-mail addresses: [email protected], [email protected](V. Ho-Huu), [email protected], [email protected](T. Nguyen-Thoi), [email protected] (T. Vo-Duy), [email protected] (T. Nguyen-Trang).

V. Ho-Huu, T. Nguyen-Thoi ⇑, T. Vo-Duy, T. Nguyen-TrangDivision of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, VietnamFaculty of Civil Engineering, Ton Duc Thang University, Vietnam

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

Article history:Received 12 October 2015Accepted 29 November 2015

Keywords:Differential evolution (DE)Adaptive elitist differential evolution (aeDE)Optimization of truss structuresOptimization with discrete design variables

This paper proposes an adaptive elitist differential evolution (aeDE) for optimization of truss structureswith discrete design variables. The aeDE algorithm is a newly improved version of the differential evolu-tion (DE) algorithm with three modifications. Firstly, in the mutation phase, an adaptive technique basedon the deviation of objective function between the best individual and the whole population in the pre-vious generation is proposed to select a suitable mutation operator. This technique helps preserve thebalance between global and local searching abilities in the DE. Secondly, in the selection phase, an elitistselection technique which helps choose the best individuals for the next generation is utilized to increasethe convergence rate. Finally, a rounding technique is integrated into the aeDE for solving optimizationproblems with discrete design variables. The efficiency and reliability of the proposed method aredemonstrated through six optimization problems of truss structures with discrete design variables.Numerical results reveal that in most of the test cases, the aeDE is more efficient than the DE and someother methods in the literature in terms of the quality of solution and convergence rate.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Structural optimization is one of the interesting subjects formany researchers. In this research direction, besides proposingthe effective optimization algorithms which help reduce the costand improve the performance of structures, the researchers havealso tried to find the effective solutions for handling highly nonlin-ear constraints or discrete design variables. During the past dec-ades, many optimization techniques have been developed andsuccessfully applied to a wide range of structural optimizationproblems. Some of them can be mentioned here such as sequentiallinear programming (SLP) [1–3], sequential quadratic program-ming (SQP) [4], optimality criterion (OC) [5,6], and forced method[7,8]. Most of them are applied to the continuous design variables.However, in many practical engineering problems, the designvariables such as the cross-section areas or thickness are usuallydiscrete values which can be selected from a set of available

discrete values. For the optimization problems with discrete designvariables, the round-off techniques based on continuous solutionswere initially proposed. Nonetheless, the rounded-off solutionsmay be far from the optimum solution, or they may even be infea-sible, when the number of variables increases [9].

To overcome the computational shortcomings of conventionaloptimization methods, many direct searching methods based onthe model of biological evolution and social interaction such asgenetic algorithm (GA), differential evolution (DE), and particleswarm optimization (PSO) have been proposed. They were quiteeffective and robust for solving truss optimization problems withdiscrete design variables. These can be illustrated from the publica-tions by Rajeev and Krishnamoorthy [10], Wu and Chow [11] usingGA and a modified GA, Lee et al. [9] using harmony search (HS), Liet al. [12] using heuristic particle swarm optimization (HPSO),Sadollah et al. [13] using mine blast algorithm (MBA), Sonmez[14] using artificial bee colony algorithm (ABC), Dede [15] using teaching–learning-based-optimization (TBLO), and Kaveh and Mah-davi [16] using colliding bodies optimization (CBO).

Although these direct search methods have occasionally over-come several restrictions of traditional optimization methods,structural design engineers are still concerned much in findingmore simple, effective, and robust methods for structural opti-mization problems with discrete design variables. Among directsearch optimization algorithms, the differential evolution (DE)

60 V. Ho-Huu et al. / Computers and Structures 165 (2016) 59–75

firstly introduced by Storn and Price 1997 [17] is one of the mostpopular algorithms. The DE is a population-based method anddemonstrated excellent performance in solving many differentengineering problems with continuous design variables [18–22].Nevertheless, similar to many methods in the evolution algorithmfamily, the DE is still costly in computational source to seek theglobal optimal solution. Especially, in many real-world problemswhere the evaluation of a candidate solution is a computationallyexpensive operation and consequently finding the global optimumor a good suboptimal solution with the original differential evolu-tion algorithm is too time-consuming, or even impossible withinthe time available [23]. Hence, the improvement of the DE withthe aim of accelerating the convergence rate, enhancing the qualityof solutions and integrating a technique for handling discretevariables is an attractive subject in the optimization field.

In the present study, the DE is employed for the first time tosolve truss optimization with discrete design variables. In orderto enhance the DE, three modifications are implemented to givea so-called the adaptive elitist differential evolution (aeDE). Firstly,an adaptive technique based on the deviation of objective functionbetween the best individual and the whole population in the pre-vious generation is proposed to choose a suitable mutation opera-tor. This technique helps preserve the balance between global andlocal searching abilities in the DE. Secondly, an elitist selectiontechnique which helps choose the best individuals for the nextgeneration is utilized to speed up the convergence. Thirdly, a tech-nique for handling discrete variables is integrated into the aeDE inwhich individuals in the population are rounded to the nearestvalue in set of discrete values. The detail of these improvementswill be presented in following sections of the paper. The robustnessand performance of the proposed aeDE are verified through sixnumerical examples. Numerical results indicate that in most ofthe cases, the aeDE is more efficient than the DE and many othermethods in the literature in terms of both the quality of solutionand convergence rate.

The remainder of the paper is organized as follows. Section 2presents mathematical model of structural optimization withdiscrete design variables. Section 3 provides a brief of basic DEalgorithm. The detail of the proposed aeDE is described in Section 4.Some numerical examples are considered in Section 5. Finally,some conclusions are drawn in Section 6.

2. Structural optimization with discrete design variables

A structural optimization problem with discrete design vari-ables is known as a nonlinear programming problem with multinonlinear constraints related to structural behavior. For the sizingoptimization of truss structures, the cross-section areas of the trussbars are often discrete design variables. All of them are selectedfrom a list of discrete cross-sections based on production standard.The optimization problem aims to minimize the weight of thestructure and to satisfy constraints about structural behavior andlimitations of design variables. The truss optimization problemwith discrete design variables can be expressed as follows:

Minimize weightðAÞ ¼Xe

i¼1

qiliAi; i ¼ 1;2; . . . ; e

subject to dmin 6 di 6 dmax; i ¼ 1;2; . . . ; nrmin 6 ri 6 rmax; i ¼ 1;2; . . . ; e

rbi 6 ri 6 0; i ¼ 1;2; . . . ; nc

A 2 S ¼ fA1;A2; . . . ;Adg

ð1Þ

where A is the design variable vectors containing the bar cross-sectional areas Ai; weightðAÞ is the weight of the whole truss struc-ture; qi and li are the material density and the length of the ith

member, respectively; e is the total number of bars in the truss; nis the number of nodes; nc is the number of elements subjectingto compression; di and ri are the nodal displacement and the stress,respectively; rb

i is the allowable buckling stress in the ith bar whenit is in compression and S is set of discrete value of areas.

To handle constraints for the problem (1) in the proposed algo-rithm, we use the penalty function method defined as follows [24]:

f penaltyðAÞ ¼ ð1þ e1 � vÞe2 �weightðAÞ; v ¼Xq

i¼1

maxf0; giðAÞg

ð2Þwhere v denotes the sum of the design violated constraints; q is thenumber of constraints in the problem; giðAÞ is the ith constraint ofthe optimization problem; e1 and e2 are constants presented theexploration and the exploitation rate of the search space. In thispaper, the value of the e1 is set 1, and e2 starts from 20 and thenlinearly increases to 40.

3. The differential evolution (DE) algorithm

The differential evolution (DE) algorithm firstly proposed byStorn and Price (1997) [17] was proven to be one of the mostpromising global search methods and widely used to solveoptimization problems in many fields such as communication[25], pattern recognition [26], and mechanical engineering[19–22]. The DE includes four main phases as follows.

3.1. Initialization

Initially, an initial population, includes NP individuals, isgenerated by means of randomly sampling from the search space.Each individual is a vector containing D design variablesxi ¼ ðx1; x2; . . . ; xDÞ and is created by

xi;j ¼ xlj þ rand½0;1� � ðxuj � xljÞ i ¼ 1;2; . . . ;NP; j ¼ 1;2; . . . ;D

ð3Þwhere xlj and xuj are the lower and upper bound of xj, respectively;rand½0;1� is a uniformly distributed random number in [0,1]; NPis the population size; and D is the number of design variables.

3.2. Mutation

Secondly, each individual called the target vector xi in thepopulation is used to generate a mutant vector vi via mutationoperators. Some popular mutation operators are usually used inthe DE as follows

� rand=1: vi ¼ xr1 þ F � ðxr2 � xr3 Þ ð4Þ� rand=2: vi ¼ xr1 þ F � ðxr2 � xr3 Þ þ F � ðxr4 � xr5 Þ ð5Þ� best=1: vi ¼ xbest þ F � ðxr1 � xr2 Þ ð6Þ� best=2: vi ¼ xbest þ F � ðxr1 � xr2 Þ þ F � ðxr3 � xr4 Þ ð7Þ� current� to� best=1: vi ¼ xi þ F � ðxbest � xiÞ þ F � ðxr1 � xr2 Þwhere integers r1; r2; r3; r4; r5 are randomly selected fromf1;2; . . . ;NPg such that r1 – r2 – r3 – r4 – r5 – i; the scale factor Fis randomly chosen within [0,1]; and xbest is the best individual inthe current population.

After this phase, the jth components v ij of mutant vector vi arereflected back to allowable region if the boundary constraints areviolated. This procedure is conducted as follows:

v ij ¼2xlj � v ij if v ij < xlj2xuj � v ij if v ij > xujv ij otherwise

8><>: ð8Þ

Fig. 1. A 10-bar planar truss structure.

0.4 0.6 0.8 1.0 [0.4,1.0]

5500

5600

5700

5800

5900

Mea

n w

eigh

t (lb

)

F

0.4 0.6 0.8 1.0 [0.4,1.0]1000

2000

3000

4000

5000

Num

ber o

f ana

lyse

s

F

Fig. 2. Influence of the mutant factor F on the mean optimal solution.

Table 2Influence of the crossover control parameter CR on the optimal solution.

Crossover control parameterCR

0.7 0.8 0.9 [0.7,1.0]

Best weight (lb) 5490.738 5490.738 5490.738 5490.738

V. Ho-Huu et al. / Computers and Structures 165 (2016) 59–75 61

3.3. Crossover

Thirdly, some elements of the target vector xi are replaced bysome elements of the mutant vector vi to create a trial vector usingbinomial crossover operation

uij ¼v ij if rand½0;1� 6 CR or j ¼ jrandxij otherwise

�ð9Þ

where i 2 f1;2; . . . ;NPg; j 2 f1;2; . . . ;Dg; jrand is an integer selectedfrom 1 to D; CR is the crossover control parameter.

Worst weight (lb) 5569.51 5538.342 5604.433 5540.407Mean weight (lb) 5525.123 5512.096 5529.1 5506.824Standard deviation (lb) 24.06045 21.40296 31.48818 20.34532

Average of number ofanalyses

2517 2599 2666 2561

3.4. Selection

Finally, based on the value of objective function, the trial vectorui is compared to the target vector xi. The better one having lowerobjective function value will survive to the next generation

xi ¼ui if f ðuiÞ 6 f ðxiÞxi otherwise

�ð10Þ

4. The adaptive elitist differential evolution algorithm (aeDE)

In the DE, the parameters such as mutant factor F and crossovercontrol parameter CR and trial vector generation strategies havesignificant influence on its performance [27]. Commonly, toacquire the most satisfactory optimization performance for a par-ticular problem, they are often defined using a trial-and-errorsearch. Obviously, it may lead to a huge amount of the computa-tional cost. In addition, a rigorous selection mechanism in theselection phase may make the DE computationally expensive.Therefore, to overcome these limitations, the paper introducestwomodifications, the first one for the mutation phase and the sec-ond one for the selection phase in order to enhance the searchcapability as well as the convergence speed of the DE algorithm.Their details are presented in the next subsections.

Table 1Influence of the mutant factor F on the optimal solution.

Mutant factor F 0.4 0.6

Best weight (lb) 5573.543 5490.738Worst weight (lb) 6787.291 5567.509Mean weight (lb) 5897.428 5522.658Standard deviation (lb) 317.3342 25.714

Average of number of analyses 1238 2579

4.1. Modification of the mutation phase

As pointed out by Das et al. [28], the balance between globalexploration and local exploitation significantly influence on thesuccess of the population-based search methods. In the DE, themutation scheme plays a pivotal role in its searching ability andconvergence rate. There are at least five mutation operators havebeen proposed for the DE with different purposes. For example,for the mutation operator ‘‘rand/1”, the DE is good at global search,but bad at local search and hence slowly converges to the globaloptimal solution [29]. In contrast, for the mutation operator ‘‘current-to-best/1”, the DE is good at local search, but bad at globalsearch and easily trapped at the local optimal solutions [30]. Thus,to equalize the global and local search capabilities of the DE,Mohamed & Sabry [31] introduced a new mutation operator whichbased on the weighted difference vector between the best and theworst individuals at a particular generation. This operator is thencombinedwith the ‘‘rand/1” to generate trial vectors with the equalprobability of 0.5. Compared with other versions of the DE, the

0.8 1.0 [0.4,1.0]

5490.738 5491.717 5490.7385545.762 5574.054 5549.5475512.038 5524.887 5512.46818.127 23.250 18.758

3582 4108 2530

0.7 0.8 0.9 [0.7,1.0]

5510

5520

5530

Mea

n w

eigh

t (lb

)

CR

0.7 0.8 0.9 [0.7,1.0]2500

2550

2600

2650

2700

Num

ber o

f ana

lyse

s

CR

Fig. 3. Influence of the crossover control parameter CR on the mean optimalsolution.

0.1 0.01 0.001 0.0001 0.000015500

5550

5600

5650

Mea

n w

eigh

t (lb

)

Threshold

0.1 0.01 0.001 0.0001 0.000011000

1500

2000

2500

3000

Num

ber o

f ana

lyse

s

Threshold

Fig. 4. Influence of the threshold on the mean optimal solution.


modified method gives good solutions; however, it is only limitedon benchmark test functions. Wang et al. [27] investigated the DEwith composite trial vector generation strategies and controlparameters for unconstraint optimization problems of benchmarktest functions. In particular, for each target vector, three trial vec-tors are created by randomly combining between three mutationoperators ‘‘rand/1”, ‘‘rand/2” and ‘‘current-to-rand/1” with threecontrol parameter settings [F = 1.0, CR = 0.1], [F = 1.0, CR = 0.9],and [F = 0.8, CR = 0.2]. Although the improved method outperformsother variants of the DE, its computational cost is relatively highdue to increasing of the population in each generation. Followingthe trend, in this work, a new adaptive mutation scheme for themutation phase of the DE is proposed. In this scheme, we use twomutation operators, the first one is ‘‘rand/1” which aims to ensurediversity of the population and prohibits the population from get-ting stuck in a local optimum, and the second one is ‘‘current-to-best/1” which aims to accelerate convergence speed of the popula-tion by mean of guiding the population toward the best individual.These two mutation operators are adaptively chosen based on theabsolute value of deviation of objective function between the bestindividual and the entire population in the previous generation(denoted as delta). More specifically, the value of delta is defined by

delta ¼ jfmean=f best � 1j ð11Þwhere f best is the objective function value of the best individual andf mean is the mean objective function value of the whole population.The new mutation scheme is described as follows

if ðdelta > thresholdÞ ð12Þvi ¼ xr1 þ F � ðxr2 � xr3 Þ %rand=1 ð13Þ

elsevi ¼ xi þ F � ðxbest � xiÞ þ F � ðxr2 � xr3 Þ

%current � to� best=1 ð14Þend

Table 3Influence of the threshold on the optimal solution.

Threshold 10�1 10�2

Best weight (lb) 5490.738 5491.717Worst weight (lb) 5878.434 5818.897Mean weight (lb) 5630.111 5555.033Standard deviation (lb) 111.544 70.070

Average of number of analyses 1175 1879

where F is the mutant factor which is randomly created in the inter-val [0.4,1]; threshold is a criterion value which is chosen based onthe stopping criterion of the algorithm. This definition will be dis-cussed in more detail in Section 4.3.

In the proposed adaptive mechanism, for each individual, onlyone of two mutation operators is utilized for producing the currenttrial vector. If the value of delta is bigger than the threshold, themutation operator ‘‘rand/1” is used. Otherwise, the mutation oper-ator ‘‘current-to-best/1” is employed. It should be noted that thevalue of delta can self-adjust during the searching process for vari-ous optimization problems, which leads to the change in the imple-mentation of the mutation operators. For example, for unimodalproblems, the value of delta usually reduces gradually during thesearching process. As a result, in the first generations, the value ofdelta is bigger than threshold. This leads to the mutation operator‘‘rand/1” is applied. However, after many generations, the diversityof the population is gradually stable, and then once the value ofdelta is smaller than threshold, the mutation operator ‘‘current-to-best/1” will be utilized. More generally, for multimodal problems,the value of delta is jumpy and becomes unpredictable during thesearching process. Consequently, the implementation of the muta-tion operators is also unpredictable. In addition, the variety in glo-bal and local search abilities is also enriched by randomlygenerating the mutant factor F within the range [0.4, 1.0] which isrecommended as a good choice from many previous studies [27].Summarily, with the proposed adaptive mutation mechanism, theglobal and local search capabilities as well as the convergence rateof the DE may be enhanced significantly.

4.2. Modification of the selection phase

The selection mechanism in Section 3.4 shows that each trialvector ui created after crossover phase will be compared with thetarget vector xi to choose a better individual for the next genera-tion. This may lead to a disadvantage that some good information

10�3 10�4 10�5

5490.738 5491.717 5490.7385590.561 5593.065 5761.6935521.059 5524.099 5542.20230.319 28.556 63.533

2442 2454 2657

Table 4Influence of the population size on the optimal solution.

The population size NP 15 20 30 30 35

Best weight (lb) 5490.738 5490.738 5490.738 5490.738 5490.738Worst weight (lb) 5959.898 5549.204 5536.965 5540.407 5538.093Mean weight (lb) 5582.669 5502.623 5505.469 5506.541 5497.593Standard deviation (lb) 119.4543 20.780 18.17706 20.59047 14.94384

Average of number of analyses 1820 2550 3406 4014 5100

15 20 25 30 355450

5500

5550

5600

Mea

n w

eigh

t (lb

)

NP

15 20 25 30 350

2000

4000

6000

Num

ber

of a

naly

ses

NP

Fig. 5. Influence of the population size on the mean optimal solution.

0 2000 4000 6000

5500

6500

7500

8500

Wei

ght (

lb)

Number of analyses

DE: Mean valueDE: Best valueaeDE: Mean valueaeDE: Best value

Fig. 6. Comparison of convergence of the DE and aeDE for the 10-bar planar trussstructure.


of unselected individuals can be neglected. Although an individualis not good compared to its target individual in the pair, it can bestill better than other individuals in the entire population. Thus,to save good information for the next generation, we use the elitistselection technique introduced by Padhye et al. [32] for the selec-tion progress instead of the basic selection in the DE. This newmechanism is performed as follows: firstly, the children populationC consisting of trial vectors is combined with parent population Pof target vectors to create a combined population Q. Then, NP bestindividuals are chosen from the Q to construct the population forthe next generation. In this way, the best individual of the whole

Table 5Comparison of optimized designs for the 10-bar planar truss structure.

Design variable (area in.2) Rajeev et al. [10] Li et al. [1

GA HPSO

A1 33.5 30A2 1.62 1.62A3 22 22.9A4 15.5 13.5A5 1.62 1.62A6 1.62 1.62A7 14.2 7.97A8 19.9 26.5A9 19.9 22A10 2.62 1.8Weight (lb) 5613.84 5531.98Number of analyses – –

Worst weight (lb) – –Mean weight (lb) – –Standard deviation (lb) – 3.8402

Average of number of analyses – –

population are always stored for the next generation. This helpsthe algorithm obtain better convergence rate. The elitist operatoris depicted as in Algorithm 1.

Algorithm 1: Elitist selection operator

1: Input: Children population C and parent population P2: Assign Q = C [ P3: Select NP best individuals from Q and assign to P4: Output: P

2] Sadollah et al. [13] This study

MBA DE aeDE

30 33.5 33.51.62 1.62 1.6222.9 22.9 22.916.9 14.2 14.21.62 1.62 1.621.62 1.62 1.627.97 7.97 7.9722.9 22.9 22.922.9 22 221.62 1.62 1.625507.75 5490.738 5490.7383600 6440 2380

5536.965 5546.685 5549.2045527.296 5501.547 5502.62311.38 19.521 20.780

– 5111 2550


4.3. The proposed aeDE algorithm

By integrating two above-mentioned improvements into theDE, a so-called adaptive elitist differential evolution (aeDE) algo-rithm is proposed. This method is summarily shown as in Algo-rithm 2 below:

Algorithm 2: The proposed algorithm (aeDE)

1: Generate the initial population2: Evaluate the fitness for each individual in the population3: while delta > tolerance or MaxIter is not reached do4: for i = 1 to NP do5: F = rand[0.4,1]6: CR = rand[0.7,1]7: jrand = randint(1,D)8: for j = 1 to D do9: if rand[0,1] < CR or j == jrand then10: if delta > threshold then11: Select randomly r1 – r2 – r3 – i;

8i 2 f1; . . . ;NPg12: ui;j ¼ xr1;j þ F � ðxr2;j � xr3;jÞ13: else14: Select randomly r1 – r2 – best – i;

8i 2 f1; . . . ;NPg15: ui;j ¼ xi;j þ F � ðxbest;j � xi;jÞ þ F � ðxr2 ;j � xr3 ;jÞ16: end if17: else18: ui;j ¼ xi;j19: end if20: end for21: Evaluate the trial vector ui

22: end for23: Do selection phase based on Algorithm 124: Define f best; f mean

25: delta ¼ jfmean=f best � 1j26: end while

where tolerance is the allowed error; MaxIter is the maximum num-ber of iterations; and randint(1,D) is the function which returns auniformly distributed random integer number between 1 and D.

According to the Algorithm 2, the aeDE will finish the searchingprogress either when the absolute value of deviation of the objec-tive function of the best individual and the whole population(delta) is less than or equal to the previously assigned value of

(a)

0 2 4 6 8 10 12

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Dis

plac

men

t (in

.)

Number of displacement

Allowable limitCase 1Case 2

Fig. 7. Constraint boundaries evaluated a

the tolerance or when the maximum number of iterations (MaxIter)is achieved. It can be seen that the threshold is chosen based on thetolerance and obviously, it must be bigger than the tolerance. More-over, it is important to know that choosing the value of thresholdwill directly impact on the global and local search capabilities ofthe aeDE. If the value of threshold is too big compared to the valueof tolerance, the aeDE will have a priority in global searching;otherwise, if it is too small, the aeDE will have a priority in localsearching. Therefore, it can be set to an adequate value based onthe characteristic of a particular problem. For example, for highlynonlinear and complex problems, the value of threshold shouldbe small (10�4 or 10�5 for threshold and 10�6 for tolerance). Onthe contrary, for small and simple problems, the value of thresholdshould be large (10�2 or 10�3 for threshold and 10�6 for tolerance).

4.4. Handling discrete variables for the aeDE

A simple method with a rounding function which permits tochange the continuous value of a result to the nearest discretevalue is utilized. The method is described as follows [16]

xdiscretei ¼ fixðxcontinuous

i Þ ð15Þwhere fix(x) is a function which rounds each element of x to thenearest permissible discrete value. In the same way, in the aeDE(Algorithm 2), before evaluating the fitness function, each individ-ual in the population will be rounded to the nearest discrete valuein the set of available discrete values by using the Eq. (15).

5. Numerical examples

In this section, the proposed aeDE method in Algorithm 2 isused to solve six truss optimization problems with discrete designvariables. The first three examples consist of the planar trusseshaving number of bars of 10, 52 and 200, respectively and the nextthree examples include space trusses having the number of bars of25, 72, and 160, respectively. The influence of the mutant factor F,crossover control parameter CR, population size NP and the thresh-old of the aeDE algorithm on the optimal solution is investigatedfor all examples; however, to avoid wordiness these investigationsis presented only for the first example of the 10 bar-truss problem.Based on obtained results of the investigations, some adequateparameters are recommended. In details, for the 10, 25, 52 and72 bar truss problems, the NP and threshold are set 20, and 10�3,respectively and for the 200 and 160 bar truss problems, the NPand threshold are respectively set 25, and 10�4. Besides, the values

(b)

0 2 4 6 8 10

-25

-15

-5

5

15

25

Str

ess

(ksi

)

Element number

Allowable limitCase 1Case 2

t the optimized design by the aeDE.



of F and CR are set within the range [0.4,1.0] and [0.7,1.0], respec-tively, for all examples. The aeDE algorithm will be stopped eitherwhen the value of delta is greater than the value of tolerance orwhen the maximum number of iterations (MaxIter) is reached.Here, the values of tolerance and MaxIter are respectively set 10-6

and 2000 for all examples. The behavior of trusses is analyzed byfinite element method using two-node linear elements. All theproblems are implemented in Matlab and are run with 20 indepen-dent times. The gained results are verified in comparison to those

Table 6The available cross-section areas of the ASIC code.

No. in.2 mm2 No. in.2 mm2 N

1 0.111 71.613 17 1.563 1008.385 32 0.141 90.968 18 1.620 1045.159 33 0.196 126.451 19 1.80 1161.288 34 0.25 161.29 20 1.990 1283.868 35 0.307 198.064 21 2.130 1374.191 36 0.391 252.258 22 2.380 1535.481 37 0.442 285.161 23 2.620 1690.319 38 0.563 363.225 24 2.630 1696.771 49 0.602 388.386 25 2.880 1858.061 4

10 0.766 494.193 26 2.930 1890.319 411 0.785 506.451 27 3.090 1993.544 412 0.994 641.289 28 3.130 729.031 413 1.000 645.16 29 3.380 2180.641 414 1.228 792.256 30 3.470 2238.705 415 1.266 816.773 31 3.550 2290.318 416 1.457 939.998 32 3.630 2341.931 4

obtained by other researches in the literature. The Matlab code ofall the problems can be obtained by sending an email to the firstauthor.

5.1. A 10-bar planar truss structure

In the first example, a simple 10-bar truss, as shown in Fig. 1 isconsidered. This work was previously investigated by manyresearchers such as Rajeev and Krishnamoorthy [10] using GeneticAlgorithm (GA), Ringertz [33] using Branch and Bound (BB), Li et al.[12] using Heuristic Paritcal Swarm Optimization (HPSO), andSadollah et al. [13] using Mine Blast Algorithm (MBA). The materialdensity is 0.1 lb/in.3 and the modulus of elasticity is 104ksi. Themembers are subjected to stress limitations of ±25 ksi. All nodesare subjected to displacement limitations of ±2.0 in. in both xand y directions. The vertical load P is 105 lbs. There are 10 designvariables corresponding to 10 bars and selected from the setS = {1.62,1.80,1.99,2.13,2.38,2.62, 2.63,2.88,2.93, 3.09,3.13,3.38,3.47,3.55,3.63,3.84,3.87,3.88,4.18,4.22,4.49,4.59,4.80,4.97,5.12,5.74,7.22,7.97,11.50,13.50,13.90,14.20,15.50,16.00,16.90,18.80,19.90,22.00,22.90,26.50,30.00,33.50} (in.2).

5.1.1. Influence of the parameters F, CR, NP and threshold on theoptimal solution

To get satisfactory parameters of CR, F, NP and threshold of theaeDE for this problem, the influence of these factors on the optimalsolutions is evaluated.

Table 1 and Fig. 2 show the affection of the mutant factor F,Table 2 and Fig. 3 describe influence of the crossover controlparameter CR. It can be found that if the value of F and CR are ran-domly generated in the interval [0.4,1.0] and [0.7,1.0] respectively,a capability of equalizing between the solution quality and compu-tational cost of the aeDE are better than those obtained by othervalues.

The impact of threshold on the optimum results is presented inTable 3 and Fig. 4. It can be seen that for the large value of threshold(e.g. 10�1, 10�2), the algorithm has fast convergences rate with lessnumber of structural analyses, but the results are not stable. Incontrast, with a smaller value of threshold (e.g. 10�4, 10�5), thealgorithm convergences more slowly, but the results are morestable. From Table 3 and Fig. 4, it can be recognized that the thresh-old = 10-3 is a good choice for this problem.

The influence of the population size NP on the results of theproblem is provided in Table 4 and Fig. 5. It can be realizedthat with the cases of NP > 15, the deviation of the mean weightsis not significant; however, the computational cost increases

o. in.2 mm2 No. in.2 mm2

3 3.840 2477.414 49 11.500 7419.3404 3.870 2496.769 50 13.500 8709.6605 3.880 2503.221 51 13.900 8967.7246 4.180 2696.769 52 14.200 9161.2727 4.220 2722.575 53 15.500 9999.9808 4.490 2896.768 54 16.000 10322.5609 4.590 2961.284 55 16.900 10903.2040 4.800 3096.768 56 18.800 12129.0081 4.970 3206.445 57 19.900 12838.6842 5.120 3303.219 58 22.000 14193.5203 5.740 3703.218 59 22.900 14774.1644 7.220 4658.055 60 24.500 15806.4205 7.970 5141.925 61 26.500 17096.7406 8.530 5503.215 62 28.000 18064.4807 9.300 5999.988 63 30.000 19354.8008 10.850 6999.986 64 33.500 21612.860


Design variable (area mm2) Wu and Chow [11] Li et al. [12] Sadollah et al. [13] Kaveh et al. [16] Sadollah et al. [34] This study

HS HPSO MBA CBO WCA IMBA DE aeDE

A1 4658.055 4658.055 4658.055 4658.055 4658.055 4658.055 4658.055 4658.055A2 1161.288 1161.288 1161.288 1161.288 1161.288 1161.288 1161.288 1161.288A3 645.16 363.225 494.193 388.386 494.193 494.193 494.193 494.193A4 3303.219 3303.219 3303.219 3303.219 3303.219 3303.219 3303.219 3303.219A5 1045.159 940.000 940.000 939.998 940.000 940.000 939.998 939.998A6 494.193 494.193 494.193 506.451 494.193 494.193 494.193 494.193A7 2477.414 2238.705 2283.705 2238.705 2283.705 2283.705 2238.705 2238.705A8 1045.159 1008.385 1008.385 1008.385 1008.385 1008.385 1008.385 1008.385A9 285.161 388.386 494.193 506.451 494.193 494.193 494.193 494.193A10 1696.771 1283.868 1283.868 1283.868 1283.868 1283.868 1283.868 1283.868A11 1045.159 1161.288 1161.288 1161.288 1161.288 1161.288 1161.288 1161.288A12 641.289 792.256 494.193 506.451 494.193 494.193 494.193 494.193Weight (lb) 1970.142 1905.49 1902.605 1899.35 1902.605 1902.605 1902.605 1902.605Number of analyses 60000 100000 5450 3840 7100 4750 13240 3720

Worst weight (lb) – – 1912.646 2262.8 1912.646 1904.83 1918.777 1925.714Mean weight (lb) – – 1906.076 1963.12 1909.856 1903.076 1906.626 1906.735Standard deviation (lb) – – 4.09 106.01 7.09 1.13 4.991 6.679

Average of number of analyses – – – – – – 11314 3402

0 2000 4000 6000 8000 100001000

3000

5000

7000

9000

Wei

ght (

kg)

Number of analyses



0 10 20 30 40 50

-180

-80

0

80

180

Element number

Stre

ss (M

Pa)

Allowable limit

Fig. 10. Stress constraint boundaries of the example 5.2 evaluated at the optimizeddesign by the aeDE.



significantly when the value of NP increases. For this problem, itcan be found that NP = 20 is a proper selection which has an abilityto effectively balance the computational cost and the quality of thesolution.


Element number Members in the group Toǧan andDaloǧlu [35]

Talebpouret al. [36]

Azad and Hasançebi [37] This study

IGA HACOHS-T ARCGA MABC ESASS DE aeDE

1 1, 2, 3, 4 0.347 0.1 0.1 0.1 0.1 0.1 0.12 5, 8, 11, 14, 17 1.081 1.081 1.081 1.333 0.954 0.954 0.9543 19, 20, 21, 22, 23, 24 0.1 0.347 0.1 0.1 0.1 0.347 0.3474 18, 25, 56, 63, 94, 101, 132, 139, 170, 177 0.1 0.1 0.1 0.1 0.1 0.1 0.15 26, 29, 32, 35, 38 2.142 2.142 2.142 2.697 2.142 2.142 2.1426 6, 7, 9, 10, 12, 13, 15, 16, 27, 28, 30, 31, 33,

34, 36, 370.347 0.347 0.347 0.347 0.347 0.539 0.347

7 39, 40, 41, 42 0.1 0.1 0.1 0.1 0.1 0.1 0.18 43, 46, 49, 52, 55 3.565 3.131 3.131 3.131 3.131 3.565 3.1319 57, 58, 59, 60, 61, 62 0.347 0.1 0.1 0.1 0.1 0.347 0.34710 64, 67, 70, 73, 76 4.805 4.805 4.805 4.805 4.805 4.805 4.80511 44, 45, 47, 48, 50, 51, 53, 54, 65, 66, 68, 69,

71, 72, 74, 750.44 0.44 0.347 0.44 0.347 0.539 0.539

12 77, 78, 79, 80 0.44 0.1 0.1 0.539 0.1 0.1 0.34713 81, 84, 87, 90, 93 5.952 5.952 5.952 5.952 5.952 5.952 5.95214 95,96, 97, 98, 99, 100 0.347 0.1 0.1 0.1 0.1 0.347 0.115 102, 105, 108, 111, 114 6.572 6.572 6.572 6.572 6.572 6.572 6.57216 82, 83, 85, 86, 88, 89, 91, 92, 103, 104, 106,

107, 109, 110, 112, 1130.954 0.539 0.539 1.081 0.44 0.954 0.954

17 115, 116, 117, 118 0.347 1.174 1.081 0.347 0.539 0.347 0.4418 119, 122, 125, 128, 131 8.525 8.525 7.192 8.525 7.192 8.525 8.52519 133, 134, 135, 136, 137, 138 0.1 0.1 0.539 0.1 0.44 0.1 0.120 140, 143, 146, 149, 152 9.3 9.3 8.525 9.3 8.525 9.3 9.321 120, 121, 123, 124, 126, 127, 129, 130, 141,

142, 144, 145, 147, 148, 150, 1510.954 1.333 1.333 0.954 0.954 0.954 0.954

22 153, 154, 155, 156 1.764 0.539 1.081 1.764 1.174 1.333 1.08123 157, 160, 163, 166, 169 13.3 13.33 10.85 13.33 10.85 13.33 13.3324 171, 172, 173, 174, 175, 176 0.347 1.174 0.1 0.44 0.44 0.347 0.53925 178, 181, 184, 187, 190 13.3 13.33 13.33 13.33 10.85 13.33 14.2926 158, 159, 161, 162, 164, 165, 167, 168, 179,

180, 182, 183, 185, 186, 188, 1892.142 2.697 1.488 2.142 1.764 2.142 2.142

27 191, 192, 193, 194 4.805 3.565 5.952 3.813 8.525 3.813 3.81328 195, 197, 198, 200 9.3 8.525 13.33 8.525 13.33 8.525 8.52529 196, 199 17.17 17.17 14.29 19.18 13.33 17.17 17.17Weight (lb) 28544.014 28030.20 28347.594 28366.365 28075.488 27901.583 27858.500Number of analyses 51360 – 25000 40000 11156 41475 12325

Worst weight (lb) – – – – – 29652.891 29415.000Mean weight (lb) – – – – – 28470.114 28425.871Standard deviation

(lb)– – – – – 457.467 481.590

Average number ofstructural analyses

– – – – – 45740 11644

10000 20000 30000 40000 50000

0.3

0.6

0.9

1.2

1.5

1.8x 10

5

Wei

ght (

lb)

Number of analyses



0 50 100 150 200

-10

-5

0

5

10

Stre

ss (k

si)

Element number

Allowable limitLoad case 1Load case 2Load case 3

Fig. 13. Stress constraint boundaries of the example 5.3 evaluated at the optimizeddesign by the aeDE.


Fig. 14. A 25-bar space truss structure.

Table 9Load condition for the 25-bar space truss structure.

Nodes Loads

Px (kips) Py (kips) Pz (kips)

1 1 �10 �102 0 �10 �103 0.5 0 06 0.6 0 0

0 500 1000 1500 2000

500

550

600

650

700

Wei

ght (

kg)

Number of analyses


Fig. 15. Comparison of convergence of the DE and aeDE for the 25-bar space trussstructure.


5.1.2. Comparison with other methods in the literatureA comparison of the obtained results by the present work and

some other methods in the literature is given in Table 5. The resultsshow that the optimumweights of both the DE and aeDE are betterthan those of other methods (5490.738 lb for the DE and aeDE,5613.84 lb for GA, 5531.98 lb for HPSO and 5507.75 lb for MBA).The aeDE requires less number of structural analyses than the DEand MBA (2380 structural analyses for the aeDE, 6440 analysesfor the DE and 3600 analyses for MBA). However, the MBA is more

Table 10Comparison of optimized designs for the 25-bar space truss structure.

Design variable (area in.2) Wu and Chow [11] Lee et al. [9] Li et al. [12] Kaveh and Ghazaan [38] Sadollah et al. [13] This study

SGA HS HPSO ECBO MBA DE aeDE

A1 0.1 0.1 0.1 0.1 0.1 0.1 0.1A2 0.5 0.3 0.3 0.3 0.3 0.3 0.3A3 3.4 3.4 3.4 3.4 3.4 3.4 3.4A4 0.1 0.1 0.1 0.1 0.1 0.1 0.1A5 1.5 2.1 2.1 2.1 2.1 2.1 2.1A6 0.9 1.0 1.0 1.0 1.0 1.0 1.0A7 0.6 0.5 0.5 0.5 0.5 0.5 0.5A8 3.4 3.4 3.4 3.4 3.4 3.4 3.4Weight (lb) 486.29 484.85 484.85 484.85 484.85 484.854 484.854Number of analyses 40000 13523 3750 7050 2150 3500 1440

Worst weight (lb) – – – – 485.048 485.380 486.100Mean weight (lb) – – – 485.89 484.885 484.910 485.014Standard deviation (lb) – – – – 0.072 0.131 0.273

Average of number of analyses – – – – – 3736 1678

(a) (b)

0 5 10 15 20 25 30

-0.35

-0.15

0

0.15

0.35

Dis

plac

men

t (in

.)

Number of displacement

Allowable limit

0 5 10 15 20 25

-40

-20

0

20

40

Stre

ss (k

si)

Element number

Allowable limit

Fig. 16. Constraint boundaries evaluated at the optimized design by the aeDE. (a) Displacement constraints. (b) Stress constraints.


Table 11Load cases for the 72-bar space truss structure.

Nodes Load case 1 Load case 2

Px (kips) Py (kips) Pz (kips) Px (kips) Py (kips) Pz (kips)

17 5.0 5.0 �5.0 0.0 0.0 �5.018 0.0 0.0 0.0 0.0 0.0 �5.019 0.0 0.0 0.0 0.0 0.0 �5.020 0.0 0.0 0.0 0.0 0.0 �5.0


stable than the aeDE and DE with the smallest standard deviation(11.38 lb for MBA, 19.521 lb for the DE and 20.780 lb for the aeDE).As shown in Fig. 6, the convergence of the aeDE is considerably fas-ter than that of the DE. After about 2500 structural analyses, theaeDE reaches the optimal result, while to obtain the same optimalresult, the DE needs about 5000 structural analyses. Fig. 7 demon-strates that constraint boundaries evaluated at the optimumdesign by the aeDE are satisfied the requirements of the problem.


The second example considers the optimization problem for a52-bar planar truss structure as shown in Fig. 8 This problemwas previously solved by Lee et al. [9], Li et al. [12], Sadollahet al. [13,34], Kaveh and Mahdavi [16], etc. The modulus of elastic-ity and the mass density are E = 207 GPa and q = 7860 kg/m3,respectively. All the truss members are subject to stress limitationsof 180 MPa in both tension and compression. The vertical loadswere set equal to Px = 100 kN and Py = 200 kN. The members of thisstructure are divided into 12 groups: (1) A1–A4, (2) A5–A10, (3) A11–A13, (4) A14–A17, (5) A18–A23, (6) A24–A26, (7) A27–A30, (8) A31–A36,(9) A37–A39, (10) A40–A43, (11) A44–A49, and (12) A50–A52 corre-sponding to 12 design variables. The design variables are selectedfrom set of discrete value listed in Table 6.

The comparison of optimal results for this example is providedin Table 7. It can be found that the optimal weight gained by theaeDE agrees well with those of the DE and other methods. How-ever, the aeDE requires the least number of structural analyseswith only 3720, while the number of structural analyses obtainedby using the DE, HS, HPSO, MBA, CBO, WCA and IMBA are 13240,60000, 100000, 5450, 3840, 7100 and 4750, respectively. Fig. 9compares the convergence of the DE and aeDE. It again indicatesthat the aeDE converges much faster than the DE. Constraintboundaries of the problem evaluated at the optimum design areshown in Fig. 10. It exhibited that all constraints are satisfied.


The third problem is a 200-bar truss as shown in Fig. 11. Thematerial density and the elasticity of modulus are 30000 ksi and0.283 lb/in.3, respectively. The 200 bars of the truss are categorizedinto 29 groups corresponding to 29 design variables. Stress limita-tions of ±10 ksi is adopted for the truss members. The truss is sup-ported to three loading conditions: (1) 1 kip acting in the positive xdirection at nodes 1, 6, 15, 20, 29, 34, 43, 48, 57, 62, and 71; (2) 10kips acting in the negative y direction at nodes 1, 2, 3, 4, 5, 6, 8, 10,12, 14, 15, 16, 17, 18, 19, 20, 22, 24, 26, 28, 29, 30, 31, 32, 33, 34, 36,38, 40, 42, 43, 44, 45, 46, 47, 48, 50, 52, 54, 56, 57, 58, 59, 60, 61, 62,64, 66, 68, 70, 71, 72, 73, 74, and 75; and (3) cases 1 and 2 are com-bined together. The available set of area is S = {0.100, 0.347, 0.440,0.539, 0.954, 1.081, 1.174, 1.333, 1.488, 1.764, 2.142, 2.697, 2.800,3.131, 3.565, 3.813, 4.805, 5.952, 6.572, 7.192, 8.525, 9.300, 10.850,13.330, 14.290, 17.170, 19.180, 23.680, 28.080, 33.700 in.2}. This


Design variable (areamm2)

Wu et al.[11]

Kaveh et al.[39]

Li et al.[12]

Sadollah et al.[13]

Kaveh et al.[16]

Kaveh and Ghazaan[38]

Sadollah et al. [34] This study

SGA DHPSACO HPSO MBA CBO ECBO WCA IMBA DE aeDE

A1 0.196 1.800 4.970 0.196 1.620 1.990 1.990 1.990 1.990 1.990A2 0.602 0.442 1.228 0.563 0.563 0.563 0.442 0.442 0.563 0.563A3 0.307 0.141 0.111 0.442 0.111 0.111 0.111 0.111 0.111 0.111A4 0.766 0.111 0.111 0.602 0.111 0.111 0.111 0.111 0.111 0.111A5 0.391 1.228 2.880 0.442 1.457 1.228 1.228 1.228 1.228 1.228A6 0.391 0.563 1.457 0.442 0.442 0.442 0.563 0.563 0.442 0.442A7 0.141 0.111 0.141 0.111 0.111 0.111 0.111 0.111 0.111 0.111A8 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111A9 1.800 0.563 1.563 1.266 0.602 0.563 0.563 0.563 0.563 0.563A10 0.602 0.563 1.228 0.563 0.563 0.563 0.563 0.563 0.563 0.563A11 0.141 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111A12 0.307 0.250 0.196 0.111 0.111 0.111 0.111 0.111 0.111 0.111A13 1.563 0.196 0.391 1.800 0.196 0.196 0.196 0.196 0.196 0.196A14 0.766 0.563 1.457 0.602 0.602 0.563 0.563 0.563 0.563 0.563A15 0.141 0.442 0.766 0.111 0.391 0.391 0.391 0.391 0.391 0.391A16 0.111 0.563 1.563 0.111 0.563 0.563 0.563 0.563 0.563 0.563Weight (lb) 427.203 393.380 933.09 390.73 391.07 389.33 389.334 389.334 389.334 389.334Number of analyses 60000 5330 50000 11600 4500 17010 4600 6250 11920 4160

Worst weight (lb) – – – 399.49 495.97 – 393.778 389.457 394.170 393.325Mean weight (lb) – – – 395.432 403.71 391.59 389.941 389.823 390.531 390.913Standard deviation (lb) – – – 3.04 24.8 – 1.43 0.84 1.400 1.161

Average of number ofanalyses

– – – – – – 12973 4101

0 4000 8000 120000

3000

6000

9000

Wei

ght (

kg)

Number of analyses



(a)

0 10 20 30 40 50 60

-0.25

-0.15

-0.05

0.05

0.15

0.25

Dis

plac

men

t (in

.)

Number of displacements

Allowable limitLoad case 1Load case 2

Fig. 19. Constraint boundaries evaluated at the optimized design by


example was previously solved using an improved genetic algo-rithm (IGA) by Toǧan and Daloǧlu [35], a hybridized ant colony–harmony search-genetic algorithm called HACOHS-T by Talebpouret al. [36], and an elitist self-adaptive step-size search (ESASS) byAzad and Hasançebi [37].

Table 8 shows a comparison between the optimal designobtained by the present work and those obtained by some othermeta-heuristic algorithms in the literature. It can be observed thatthe aeDE outperforms most of the considered methods in both thequality of solution and number of structural analyses except for theESASS by Azad and Hasançebi [37]. The aeDE takes the best resultwith 27858.5 lb after 12325 analyses, while the IGA is28544.014 lb with 51360 analyses, the DE is 27901.583 lb with41475 analyses, the ARCGA is 28347.594 lb with 25000 analyses,the MABC is 28366.365 lb with 40000 analyses and the ESASS is28075.488 lb with 11156 analyses. Fig. 12 compares the conver-gence of the DE and aeDE. It indicates that the aeDE convergesmuch faster than the DE. Fig. 13 displays the stress constraintboundary of the problem evaluated at the optimum design. It exhi-bits that all constraints are satisfied.

(b)

0 10 20 30 40 50 60 70

-25

-15

-5

5

15

25

Stre

ss (k

si)

Element number

Allowable limitLoad case 1Load case 2

the aeDE. (a) Displacement constraints. (b) Stress constraints.



5.4. A 25-bar space truss structure

The fourth example considers the optimization problem for a25-bar space truss structure as shown in Fig. 14. This problemwas previously studied by Wu and Chow [11], Lee et al. [9], Liet al. [12], Sadollah et al. [13], Kaveh and Ghazaan [38], etc. Themodulus of elasticity is 104 ksi and the material density is 0.1 lb/in.3. All nodes are subjected to displacement limitations of ±0.35in. in three directions x, y, z. The stress limitations of the membersare ±40000 psi. The structure includes 25 members which aredivided into 8 groups as follows: (1) A1, (2) A2–A5, (3) A6–A9, (4)A10–A11, (5) A12–A13, (6) A14–A17, (7) A18–A21 and (8) A22–A25. Thedesign variables are selected from the set S = {0.1, 0.2, 0.3, 0.4,0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9,2.0, 2.1, 2.2, 2.3, 2.4, 2.6, 2.8, 3.0, 3.2, 3.4} (in.2). The loads areshown in Table 9.

Table 13Load cases for the 160-bar space truss structure.

Load case Node Px Py Pz

1 52 �868 0 �49137 �996 0 �54625 �1091 0 �54628 �1091 0 �546

2 52 �493 1245 �36337 �996 0 �54625 �1091 0 �54628 �1091 0 �546

3 52 �917 0 �49137 �951 0 �54625 �1015 0 �54628 �1015 0 �546

4 52 �917 0 �54637 �572 1259 �42825 �1015 0 �54628 �1015 0 �546

Table 10 compares the optimal results of the aeDE and othermethods. It can be seen that all methods give the same value ofthe minimum weight with 484.85 lb except SGA with 486.29 lb.The number of structural analyses of the aeDE is the smallestamong all considered methods. It takes the optimal weight after1440 analyses, while the SGA, HS, HSPO, MBA, ECBO and DE take40000, 13523, 3750, 2150, 7050 and 3500, respectively. The con-vergence of the aeDE and DE and constraint boundaries evaluatedat the optimum design by the aeDE are shown in Fig. 15 and Fig. 16,respectively. It can be seen that the aeDE converges faster than theDE and no constraints are violated.


The five example executes the optimization problem for a 72-bar space truss structure as shown in Fig. 17. This problem waspreviously examined by Wu and Chow [11], Lee et al. [9], Li et al.[12], Sadollah et al. [13,34], Kaveh and Mahdavi [16], Kaveh andTalatahari [39], Kaveh and Ghazaan [38], etc. The material densityis 0.1 lb/in.3 and the modulus of elasticity is 104 ksi. The stress lim-itations of the members are ±25,000 psi. All nodal displacementsmust be smaller than ±0.25 in. There are 72 truss elements whichare divided into 16 groups: (1) A1–A4, (2) A5–A12, (3) A13–A16, (4)A17–A18, (5) A19–A22, (6) A23–A30, (7) A31–A34, (8) A35–A36, (9) A37–A40, (10) A41–A48, (11) A49–A52, (12) A53–A54, (13) A55–A58, (14)A59–A66 (15), A67–A70, and (16) A71–A72. The design variables areselected from Table 6. This structure was designed for two separateload conditions given in Table 11.

The results gained by this work in comparison with previousstudies is presented in Table 12. It can be seen that the aeDEacquires the optimal weight which agrees well with those acquiredby different methods and has the least number of structural anal-yses. Fig. 18 demonstrates that the aeDE reaches the optimumsolutions much faster than the DE. Fig. 19 shows that all con-straints of the problem evaluated at the optimum results by theaeDE are satisfied.


The last example is a 160-bar truss shown in Fig. 20. This prob-lem has been solved using selective dynamic rounding (SDR) byGroenwold and Stander [40], the regional genetic algorithm(RGA) by Groenwold et al. [41], and rank-based ant system (RBAS)by Capriles et al. [42]. The material density is 0.00785 kg/cm3 andthe modulus of elasticity is 2.047 � 106 kgf/cm2. The 160 membersof the truss are linked to 38 independent design variables. The

Load case Node Px Py Pz

5 52 �917 0 �49137 �951 0 �54625 �1015 0 �54628 �636 1259 �428

6 52 �917 0 �49137 �572 1303 �42825 �1015 0 �54628 �1015 0 �546

7 52 �917 0 �49137 �951 0 �54625 �1015 0 �54628 �636 1303 �428

8 52 �498 1460 �36337 �951 0 �54625 �1015 0 �54628 �1015 0 �546


Design variable (area cm2) Groenwold and Stander [40] Groenwold et al. [41] Capriles et al. [42] This study

SDR RGA RBAS DE aeDE

A1 19.03 19.03 19.03 19.03 19.03A2 5.27 5.27 5.27 5.27 5.27A3 19.03 19.03 19.03 19.03 19.03A4 5.27 5.27 5.27 5.27 5.27A5 19.03 19.03 19.03 19.03 19.03A6 5.75 5.75 5.75 5.75 5.75A7 17.03 15.39 15.39 17.03 15.39A8 6.25 5.75 5.75 5.75 5.75A9 13.79 13.79 13.79 13.79 13.79A10 6.25 5.75 5.75 5.75 5.75A11 5.75 5.75 5.75 6.84 5.75A12 12.21 13.79 12.21 12.21 12.21A13 6.84 6.25 6.25 7.44 6.25A14 5.75 5.75 5.75 5.75 5.75A15 2.66 2.66 3.47 6.84 3.88A16 7.44 7.44 7.44 8.66 7.44A17 1.84 1.84 1.84 2.26 1.84A18 8.66 8.66 9.40 12.21 8.66A19 2.66 2.66 2.66 3.88 2.66A20 3.07 3.07 3.47 3.88 3.07A21 2.66 2.66 3.07 3.88 2.66A22 8.06 8.06 8.06 8.66 8.06A23 5.27 5.27 5.75 6.25 5.75A24 7.44 6.25 6.25 7.44 6.25A25 6.25 5.75 5.75 9.4 5.75A26 1.84 1.84 2.26 4.79 2.26A27 4.79 4.79 4.79 6.25 4.79A28 2.66 2.66 3.07 4.79 2.66A29 3.47 3.47 3.47 4.79 3.47A30 1.84 1.84 1.84 1.84 1.84A31 2.26 2.26 3.88 2.66 2.26A32 3.88 3.88 3.88 3.88 3.88A33 1.84 1.84 1.84 2.26 1.84A34 1.84 1.84 2.26 2.66 1.84A35 3.88 3.88 3.88 4.79 3.88A36 1.84 1.84 2.66 2.26 1.84A37 1.84 1.84 3.47 3.88 1.84A38 3.88 3.88 3.88 4.79 3.88Weight (kg) 1359.781 1337.442 1348.905 1448.306 1336.634Number of analyses – – 90000 50025 23925

Worst weight (kg) – – 1401.6323 1743.596 1410.611Mean weight (kg) – – 1367.5275 1617.346 1355.875Standard deviation (kg) – – – 81.930 18.805

Average of number of analyses – – – 50025 21265

0 10000 20000 30000 40000 50000

2000

4000

6000

8000

Wei

ght (

kg)

Number of analyses




members’ linkage and the nodal coordinates of this truss is given inAppendix A. The structure is designed to subject to the eight inde-pendent load cases given in Table 13. Buckling stress constraintsfor compression members are considered. For a member undercompression, the buckling stress is calculated as: rb = 1300 –(kl/r)2/24 if kl/r 6 120, and rb = 107/(kl/r)2 if kl/r > 120, where l isthe length of the member, r is the radius of gyration, and k is theeffective length factor. For this problem k is supposed to be 1.0.The cross-section areas and the corresponding radii of gyrationfor the 42 prescribed discrete sections are S = {1.84, 2.26, 2.66,3.07, 3.47, 3.88, 4.79, 5.27, 5.75, 6.25, 6.84, 7.44, 8.06, 8.66, 9.40,10.47, 11.38, 12.21, 13.79, 15.39, 17.03, 19.03, 21.12, 23.20,25.12, 27.50, 29.88, 32.76, 33.90, 34.77, 39.16, 43.00, 45.65,46.94, 51.00, 52.10, 61.82, 61.90, 68.30, 76.38, 90.60, 94.13 cm2},and r = {0.47, 0.57, 0.67, 0.77, 0.87, 0.97, 0.97, 1.06, 1.16, 1.26,1.15, 1.26, 1.36, 1.46, 1.35, 1.36, 1.45, 1.55, 1.75, 1.95, 1.74, 1.94,2.16, 2.36, 2.57, 2.35, 2.56, 2.14, 2.33, 2.97, 2.54, 2.93, 2.94, 2.94,2.92, 3.54, 3.96, 3.52, 3.51, 3.93, 3.92, 3.92 cm}.

The best solution vector for the thirty-eight design variablesacquired by this work and some methods in the literature are

(a) (b)

0 50 100 150

0

0.2

0.4

0.6

0.8

1

Stre

ss ra

tio

Element number

Load case 1Load case 2Load case 3Load case 4

0 50 60 150

0

0.2

0.4

0.6

0.8

1

Stre

ss ra

tio

Element number

Load case 5Load case 6Load case 7Load case 8

Fig. 22. Constraint boundaries evaluated at the optimized design by the aeDE. (a) Load case 1–4. (b) Load case 5–8.


presented in Table 14. The results show that the aeDE is the mosteffective algorithm in all considered methods. The optimizationweight of the aeDE is 1336.634 kg. It is much lighter than thosegained by the DE with 1448.306 kg, SDR with 1359.781 kg, RGAwith 1337.442 kg and RBAS with 1348.905 kg. Moreover, the aeDEis much efficient than the DE in term of computational cost. TheaeDE only needs 23925 analyses to reach the best solution, whilethe DE requires 50025 analyses to reach the best solution. Fig. 21shows comparison of the convergence of the DE and aeDE.Fig. 22 describes the existing buckling stress ratios of the problemevaluated at the optimum design by the aeDE. It shows that noconstraints are violated.

6. Conclusion

In this work, three modifications of the DE are proposed to givea so-called the adaptive elitist differential evolution (aeDE) forsolving truss optimization problems with discrete design variables.

Table 15Coordinate data for the 160-bar space truss structure.

Node no. X-Coord. Y-Coord. Z-Coord.

1 �105 �105 02 105 �105 20003 105 105 04 �105 105 05 �93.929 �93.929 1756 93.929 �93.929 1757 93.929 93.929 1758 �93.929 93.929 1759 �82.859 �82.859 350

10 82.859 �82.859 35011 82.859 82.859 35012 �82.859 82.859 35013 �71.156 �71.156 53514 71.156 �71.156 53515 71.156 71.156 53516 �71.156 71.156 53517 �60.085 �60.085 71018 60.085 �60.085 71019 60.085 60.085 71020 �60.085 60.085 71021 �49.805 �49.805 872.522 49.805 �49.8O50 872.523 49.805 49.805 872.524 �49.805 49.805 872.525 �214 0 1027.526 �40 �40 1027.5

Firstly, an adaptive technique based on the absolute deviation ofobjective function between the best individual and the entire pop-ulation in the previous generation is proposed to select a suitablemutation operator. This technique helps preserve the balancebetween global and local searching abilities. Secondly, an elitistselection technique is used in the selection phase to enhance theconvergence rate of the algorithm. Thirdly, a rounding techniqueis integrated into the aeDE to deal with optimization problemswith discrete design variables.

The aeDE algorithm is applied to solve six optimization prob-lems of truss structures with discrete design variables. The numer-ical results demonstrate that in most of the cases the aeDE caneffectively attain the optimum solutions with less iterations thanthe DE and some other methods in the literature. Particularly,the number of structural analysis obtained by the aeDE is approx-imately a third of that obtained by the DE. Especially, for the largeproblems of 200 and 160 bar trusses, the optimal solutionsobtained by the aeDE are better than those found by the DE and

Node no. X-Coord. Y-Coord. Z-Coord.

27 40 �40 1027.528 214 0 1027.529 40 40 1027.530 �40 40 1027.531 �40 �40 1105.532 40 �40 1105.533 40 40 1105.534 �40 40 1105.535 �40 �40 1256.536 40 �40 1256.537 �207 0 1256.538 40 40 1256.539 �40 40 1256.540 �40 �40 1346.541 40 �40 1346.542 40 40 1346.543 �40 40 1346.544 �26.592 �26.592 1436.545 26.592 �26.592 1436.546 26.592 26.592 1436.547 �26.592 26.592 1436.548 �12.737 �12.737 1526.549 12.737 �12.737 1526.550 12.737 12.737 1526.551 �12.737 12.737 1526.552 0 0 1615

Table 16Element data for the 160-bar space truss structure.

Elem. no. Node Ai Elem. no. Node Ai Elem. no. Node Ai Elem. no. Node Ai

1 2 1 2 1 2 1 2

1 1 5 1 41 13 18 8 81 25 31 17 121 36 40 292 2 6 1 42 14 17 8 82 28 32 17 122 38 41 293 3 7 1 43 14 19 8 83 28 33 17 123 39 42 294 4 8 1 44 15 18 8 84 25 34 17 124 35 43 295 1 6 2 45 15 20 8 85 26 31 18 125 40 41 306 2 5 2 46 16 19 8 86 27 32 18 126 41 42 307 2 7 2 47 16 17 8 87 29 33 18 127 42 43 308 3 6 2 48 13 20 8 88 30 34 18 128 43 40 309 3 8 2 49 17 21 9 89 26 32 19 129 35 36 31

10 4 7 2 50 18 22 9 90 27 31 19 130 36 38 3111 4 5 2 51 19 23 9 91 29 34 19 131 38 39 3112 1 8 2 52 20 24 9 92 30 33 19 132 39 35 3113 5 9 3 53 17 22 10 93 27 33 20 133 40 44 3214 6 10 3 54 18 21 10 94 29 32 20 134 41 45 3215 7 11 3 57 19 24 10 95 30 31 20 135 42 46 3216 8 12 3 58 20 23 10 96 26 34 20 136 43 47 3217 5 10 4 55 18 23 11 97 26 29 21 137 40 45 3318 6 9 4 56 19 22 11 98 27 30 21 138 41 46 3319 6 11 4 59 20 21 11 99 31 35 22 139 42 47 3320 7 10 4 60 17 24 11 100 32 36 22 140 43 44 3321 7 12 4 61 21 26 12 101 33 38 22 141 44 45 3422 8 11 4 62 22 27 12 102 34 39 22 142 45 46 3423 8 9 4 63 23 29 12 103 33 39 23 143 46 47 3424 5 12 4 64 24 30 12 104 32 35 23 144 44 47 3425 9 13 5 65 21 27 13 105 31 36 23 145 44 48 3526 10 14 5 66 22 26 13 106 34 38 23 146 45 49 3527 11 15 5 67 23 30 13 107 32 38 24 147 46 50 3528 12 16 5 68 24 29 13 108 33 36 24 148 47 51 3 529 9 14 6 69 22 29 14 109 34 35 24 149 45 48 3630 10 13 6 70 23 27 14 110 31 39 24 150 46 49 3631 10 15 6 71 24 26 14 111 37 35 25 151 47 50 3632 11 14 6 72 21 30 14 112 37 39 25 152 44 51 3633 11 16 6 73 26 27 15 113 37 40 26 153 48 49 3734 12 15 6 74 27 29 15 114 37 43 26 154 49 50 3735 12 13 6 75 29 30 15 115 35 40 27 155 50 51 3736 9 16 6 76 30 26 15 116 36 41 27 156 48 51 3737 13 17 7 77 25 26 16 117 38 42 27 157 48 52 3838 14 18 7 78 27 28 16 118 39 43 27 158 49 52 3839 15 19 7 79 25 30 16 119 35 38 28 159 50 52 3840 16 20 7 80 29 28 16 120 36 39 28 160 51 52 38


some methods in the literature. These illustrate that with properimprovements, the aeDE can offer a robust, effective and reliableoptimization method for solving optimizations of truss structureswith discrete design variables.

In addition, the aeDE is quite similar to the standardDE. Thus, it issimple tounderstandand implement.Hence, it canbe easy to extendfor various engineering optimization problems such as optimizationof frame, composite plates/shell structures, and stiffened plates/shell structures. Furthermore, with fast convergence rate, the aeDEcan be also applied for the reliability-based design optimizationproblems where the computational cost is always a concern.

Acknowledgements

This research is funded by Vietnam National Foundation forScience and Technology Development (NAFOSTED) under grantnumber 107.99-2014.11.

Appendix A

See Tables 15 and 16.

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An adaptive elitist differential evolution for optimization of truss structures with discrete design variables

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