An improved constrained differential evolution using discrete variables (D-ICDE) for layout optimization of truss structures

Expert Systems with Applications 42 (2015) 7057–7069

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An improved constrained differential evolution using discrete variables (D-ICDE) for layout optimization of truss structures V. Ho-Huu, T. Nguyen-Thoi ⇑, M.H. Nguyen-Thoi, L. Le-Anh Division of Computational Mathematics and Engineering (CME), Institute for Computational Science (INCOS), Ton Duc Thang University, Hochiminh City, Viet Nam Faculty of Civil Engineering, Ton Duc Thang University, Hochiminh City, Viet Nam

a r t i c l e

i n f o

Article history: Available online 8 May 2015 Keywords: (l + k) Improved differential evolution (IDE) Archiving-based adaptive tradeoff model (ArATM) Improved (l + k) constrainted differential evolution (ICDE) Discrete-ICDE (D-ICDE) Discrete variables Truss layout optimization

a b s t r a c t Recently, an improved (l + k) constrainted differential evolution (ICDE) has been proposed and proven to be robust and effective for solving constrainted optimization problems. However, so far, the ICDE has been developed mainly for continuous design variables, and hence it becomes inappropriate for solving layout truss optimization problems which contain both discrete and continuous variables. This paper hence fills this gap by proposing a novel discrete variables handling technique and integrating it into original ICDE to give a so-called Discrete-ICDE (D-ICDE) for solving layout truss optimization problems. Objective functions of the optimization problems are minimum weights of the whole truss structures and constraints are stress, displacement and buckling limitations. Numerical examples of five classical truss problems are carried out and compared to other state-of-the-art optimization methods to illustrate the reliability and effectiveness of the proposed method. The D-ICDE’s performance shows that it not only successfully handles discrete variables but also significantly improves the convergence of layout truss optimization problem. The D-ICDE is promising to extend for determining the optimal solution of other structural optimization problems which contain both discrete and continuous variables. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Truss layout optimization is one of the most important and challenging areas in the structural optimization field. By considering the size and shape variables simultaneously, the layout optimization problem can give more accurate design due to the coupling influences between two variables and also achieve more material savings than purely size optimization (Gholizadeh, 2013). The problem is considered to be more challenging owing to the different natures of the variables. The main difficulty lies in treating the discrete variables together with continuous variables, whereas traditional optimization methods normally treat design variables as continuous ones. So far, many meta-heuristic algorithms belonging to the evolutionary algorithm family have yielded practical and improved solutions to many structural optimization problems dealing with discrete variables. The most popular methods include genetic algorithm (GA) (Dede, Bekirog˘lu, & Ayvaz, 2011; Rajeev & Krishnamoorthy, 1992), ant colony ⇑ Corresponding author at: Institute for Computational Science (INCOS), Ton Duc Thang University, Hochiminh City, Viet Nam. E-mail addresses: [email protected] (V. Ho-Huu), nguyenthoitrung@tdt. edu.vn, [email protected] (T. Nguyen-Thoi), [email protected] (M.H. Nguyen-Thoi), [email protected] (L. Le-Anh). http://dx.doi.org/10.1016/j.eswa.2015.04.072 0957-4174/Ó 2015 Elsevier Ltd. All rights reserved.

optimization (ACO) (Camp & Bichon, 2004), harmony search (HS) (Lee, Geem, Lee, & Bae, 2005), evolutionary strategy (ES) (Chen & Chen, 2008), particle swarm optimization (PSO) (Kaveh & Talatahari, 2009), firefly algorithm (FA) (Gandomi, Yang, & Alavi, 2011), etc. The efficiency of the above methods for solving the structural optimization problems has also been investigated by many researchers. For example, Wu and Chow (1995) utilized the GA with discrete size and continuous configuration variables. Hasançebi and Erbatur (2001) proposed an improved GA by combining the GA with annealing perturbation and adaptive design space reduction strategies. Fourie and Groenwold (2002) improved the PSO by introducing new elite operators. Recently, Kaveh and Khayatazad (2013) proposed a new method termed the ray optimization method in dealing with the similar problem, while Gholizadeh (2013) introduced a hybrid algorithm integrating a cellular automata and the PSO. Among the members of the evolutionary algorithm family, the differential evolution (DE) proposed by Storn and Price in 1995 (Storn & Price, 1997) is a robust and reliable technique. The DE is outstanding with three main advantages including: (1) the true global minimum is always found in the search space regardless of initial points; (2) the convergence is fast; and (3) there is less tunable parameters compared to the GA (Karaboga & Cetinkaya,

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2005). As many other evolutionary algorithms, the DE is a population-based method, which models the stochastic evolution processes of the nature including mutation, cross-over and selection process which enable the population to iteratively evolve to the best solutions. The mutation and cross-over mechanism create the necessary diversity of the population, while the selection facilitates the exploitation for better candidates in the search region. During the last decades, the DE has continuously been improved and demonstrated much potential in addressing complex structural constrained optimization problems (COPs), motivating many research works in this field. For instance, Zhaoliang, Hesheng, and Pengfei (2009) applied the DE for designing optimal truss structures with continuous and discrete variables. Wu and Tseng (2010) applied a multi-population differential evolution with a penalty-based, self-adaptive strategy to solve the COP of the truss structures. Recently, Wang and Cai (2010) introduced a new variant of DE termed an (l + k) constraint differential evolution methods (CDE) for solving general COPs. They also proposed a mean to enhance the search capability of the DE via the orthogonal crossover operator (Wang, Cai, & Zhang, 2012). Hernandez, Leguizamon, and Mezura-Montes (2013) developed a hybrid version of the DE based on two novel mutation operators. In general, researchers mainly focused on two central improvement strategies; increasing the diversification mechanism and efficiently handling the constraints violations. Following this trend, Jia, Wang, Cai, and Jin (2013) proposed an improved (l + k)-constrained differential evolution (ICDE). This version of the DE combines an improved (l + k)-differential evolution (IDE) with an archiving-based adaptive tradeoff model (ArATM). The IDE search engine is to enhance the population diversity by generating three new offsprings from the current population. Three different mutation strategies are employed in this process. The ArATM then defines three different selection mechanisms to deal with the constraint violations, in which infeasible individuals in the promising area can be selected for the next generation. The combination of IDE and ArATM in the ICDE foster both the diversity and the convergence of the population, improving efficiently the performance of conventional DE. However, since the ICDE method are currently designed for optimization problem with continuous variables. In applications related to discrete or integer variables, the ICDE becomes inappropriate owing to two main obstacles: (i) the obtained results may be far from the permissible value and (ii) the continuous search space contains large amount of inadmissible values, leading to waste in computational cost. This paper hence proposed a novel variables handling technique to help the ICDE overcome these two disadvantages in solving optimization problems with both discrete and continuous variables. The new method is called Discrete-ICDE (D-ICDE). In the proposed technique, the set of discrete variables is transformed into a set of continuous integer variables accordingly. The formulation of the technique is applied to the initial and mutation phases in a manner that new individuals are ensured to be admissible while the diversification of the population is maintained. The D-ICDE is then applied for the truss layout optimization problem under constraints of stress, displacement and buckling limitations. Five numerical examples are performed and compared to state-of-the-art methodologies to illustrate the reliability and effectiveness of the D-ICDE. The paper is organized as follows. General concepts related to the truss layout optimization problems and the DE are introduced in Section 2. The ICDE is briefly described in Section 3, and the ICDE using discrete variables handling technique (D-ICDE) is presented in Section 4. Section 5 performs numerical examples, and some conclusions are withdrawn in Section 6.

2. Basic concepts of truss layout optimization problem and differential evolution (DE) algorithm This section will shortly introduce the mathematical model of the general truss layout optimization problem and the principal of the differential evolution (DE) algorithm. 2.1. Truss layout optimization problem The truss layout optimization problem can be described in mathematical formulations as below:

min

f ðxÞ ¼

j P iþ1

( s:t

qi xi li

DðxÞ 6 ½D; rðxÞ 6 ½r; kðxÞ 6 ½k x ¼ fxli 6 xi 6 xui g;

ð1Þ

i ¼ 1; 2 . . . ; j; . . . ; n

where f ðxÞ is the objective function measuring the weight of the structure; x is a D-dimensional vector of n design variables, containing the size and shape variables of the truss elements; qi is the material density of the ith member; li is the length of the ith member; DðxÞ is the displacement, determined within the allowable displacement ½D; rðxÞ is the element stress, determined within the allowable stress ½r; kðxÞ is the buckling stress, determined within the allowable buckling stress ½k; xi is the ith design variable, determined between the lower bound xli and upper bound xui , i ¼ 1; 2; . . . ; j are the indices representing the area design variables while i ¼ j þ 1; . . . ; n are the indices representing the nodal coordinates. 2.2. Differential evolution (DE) algorithm Differential evolution (DE) proposed by Storn and Price (1997) was proven to be one of the most promising global search methods and widely used to solve continuous optimization problems for many kinds of structures. This paper hence employs the DE in association with some novel techniques to solve the problem of layout truss optimization. An original scheme of the DE consists of four main phases as 2.2.1. Phase 1: initialization Create an initial population Pt of NP individuals by randomly sampling from the search space 2.2.2. Phase 2: mutation Generate a new mutant vector v i from each current individual xi based on mutation operations. Four popular mutation operations used in the DE algorithm as follows

- Rand=1 : v i ¼ xr1 þ F  ðxr2  xr3 Þ

ð2Þ

- Rand=2 : v i ¼ xr1 þ F  ðxr2  xr3 Þ þ F  ðxr4  xr5 Þ

ð3Þ

- Current  to  rand=1 : v i ¼ xi þ F  ðxr1  xi Þ þ F  ðxr2  xr3 Þ

ð4Þ

- Current  to  best=1 : v i ¼ xi þ F  ðxbest  xi Þ þ F  ðxr1  xr2 Þ

ð5Þ

where integers r1 ; r 2 ; r 3 ; r4 ; r 5 are randomly selected from f1; 2; . . . ; NPg such that r 1 –r 2 –r3 –r 4 –r 5 ; the weighting factor F is randomly chosen between 0 and 1; and xbest is the current best individual in the population; After mutation, the components v ij of mutant vectors vi are modified if the boundary constraints are violated. The modified procedure is conducted as follows:

V. Ho-Huu et al. / Expert Systems with Applications 42 (2015) 7057–7069

v ij

8 l > < 2xj  v ij ¼ 2xuj  v ij > :

v ij

xlj

v ij < xlj if v ij > xuj if

ð6Þ

otherwise

xuj

where and are respectively the lower bound and upper bound of jth variables in the layout truss optimization problem. 2.2.3. Phase 3: crossover Create a trial vector ui by replacing some elements of the mutant vector v i via the crossover operation. The most common crossover operation is binomial crossover, stated by

uij ¼



v ij

if rand 6 CR or j ¼ jrand

xij

otherwise

ð7Þ

where i 2 f1; 2; . . . ; NPg; j 2 f1; 2; . . . ; ng; rand is a uniformly distributed random number between 0 and 1; jrand is an integer selected from 1 to n; CR is the crossover control parameter. 2.2.4. Phase 4: selection Compare the trial vector ui with the target vector xi . One with lower objective function value will survive in the next generation:

xi ¼



ui

if f ðui Þ 6 f ðxi Þ

xi

otherwise

ð8Þ

3. Improved constraint differential evolution (ICDE) method In general, the ICDE is similar to the DE which utilizes the mutation and crossover operations to diversify the evolving population. But better still, the ICDE employs the improved (l + k)-differential evolution (IDE) to significantly improve the diversity of the conventional DE and an ArATM to efficiently select potential individuals for the next generation. Further information and details are available at the reference Jia et al. (2013), but the most important implementing terms are presented here to make the paper self-explaining. 3.1. Improved (l + k)-differential evolution (IDE) In the IDE, from l individuals in the parent population Pt, a new offspring population Qt, with k individuals is generated by simultaneously adopting the ‘‘rand/1’’, the ‘‘rand/2’’ and the ‘‘current-to-r and/best/1’’ mutation strategies. The last mutation strategy is the new point in the IDE to speed up the evolvement of the last offspring to the optimal value. The formulation for ‘‘current-to-rand/b est/1’’ mutation strategy can be found in the reference Jia et al. (2013) for interested readers. The IDE scheme can be summarized by two main phases as follows. 3.1.1. Phase 1: initialization Set Q t ¼ ; (which is an empty set) Create an initial population Pt of NP individuals, via randomly sampling from the search space. 3.1.2. Phase 2: mutation + crossover For each individual xi ði ¼ 1; . . . ; lÞ in Pt, simultaneously generate three offspring based on the following mutation and crossover combinations  Generate the first offspring y1 by using the ‘‘rand/1’’ strategy Eq. (2) and binomial crossover.  Generate the second offspring y2 by using the ‘‘rand/2’’ strategy Eq. (3) and binomial crossover.

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 Generate the third offspring y3 by using a new strategy named ‘‘current-to-rand/best/1’’ and improved breeder genetic algorithm mutation. Update the new offspring Q t ¼ y1 [ y2 [ y3 If the current generation number is greater than a threshold generation number, the last mutation strategy ‘‘current-to-rand/b est/1’’ is implemented by switching from the ‘‘current-to-rand/1’’ Eq.(4) to the ‘‘current-to-best/1’’ Eq. (5) during the evolution procedure. The first equation chooses the information randomly from the search space to enhance the global search ability, while the second equation exploits the information of the best individuals in the current population to speed up the convergence to the global optimum. Hence, the combined equation ‘‘current-to-rand/best/1’’ can efficiently maintain the balance between the diversity of the population and the convergence of the evolution. 3.2. Archiving-based adaptive tradeoff model (ArATM) There are three situations existing in the combined population Ht = Pt + Qt including the infeasible, the semi-feasible, and the feasible situations. Individuals in the infeasible region may carry valuable information that lead to the optimal solution. The ArATM thus provides different schemes to handle the constraint violations in each situation properly in order to select potential candidates (both feasible and infeasible) to the next population. 3.2.1. Infeasible situation In this case, when all individuals in Ht violate the constraints, the original COP is transformed into a bi-objective optimization problem. The objective function f(x) and the degree of constraint violation G(x) are considered as two different objective functions with equal importance. The aim of the ArATM in this situation is to quickly direct the population to enter into the feasible region while maintaining the diversity of the population. The pseudo-code of this procedure is listed in algorithm 1. Algorithm 1: ArATM scheme for the infeasible situation Input parameter: the combined population Ht If (A–;) then Select randsize individuals randomly from A and insert them into Ht , (randsize is a random integer between 0 and jAj) . endif Set A ¼ 0 and Ptþ1 ¼ 0 While jPtþ1 j < l do Identify non-dominated individuals on Ht based on Pareto dominance (Kalyanmoy & Deb, 2001); Sort the non-dominated individuals in ascending order based on their degree of constraint violations; Select the first half of the sorting list and put them into the next parent population Ptþ1 ; Remove the selected individuals from Ht . endwhile If jPtþ1 j < l then Delete the last (jPtþ1 j  l) individuals in Ptþ1 , and move them into Ht . endif Store all remaining individuals in Ht into A.

3.2.2. Semi-feasible situation In this case, the combined population Ht contains individuals in both regions. Since some certain infeasible individuals may contain

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Table 1 Parameters of all examples in the D-ICDE. Number of generation

Parent population

Offspring population

F

CR

Threshold number in Section 3.2

g = 500

l = 15

k = 45

0.8

0.9

g = 200

(1)

(2)

1 10

2

y 11 x

8

9

5

(6)

120 in

12 0 in

15

13

4

(4)

3 14

12 7

(5)

(3)

(7)

120 in

6

crucial information to find the optimal solution, it is not reasonable to get rid of all infeasible individuals in the semi-feasible situation. An adaptive normalization scheme is used to select not only some feasible individuals with small objective function values but also some infeasible individuals with both small degree of constraint violation and small objective function values. Details of normalization scheme can be found in the reference Jia et al. (2013) for interested readers.

(8) 10 kips

120 in

Fig. 1. The 15-bar planar truss structure.

3.2.3. Feasible situation In this case, when all individuals in Ht has entered into the feasible region, the problem becomes the unconstraint optimization problem where objective function value is the only important factor. The procedure now is just simply selecting l individuals with the smallest objective function f ðxi Þ value to create the next parent generation Ptþ1 . 4. The proposed ICDE for discrete variable (D-ICDE) The above-mentioned ICDE was originally designed to handle problems with continuous search space. It hence becomes

Table 2 Data for design of the 15-bar planar truss structure. P15

Objective function

min

Stress constraint

ðrt Þi 6 25ðksiÞ; jðrc Þji 6 25ðksiÞ; i ¼ 1; 2; . . . ; 15 Geometry variables x2 ¼ x6 ; x3 ¼ x7 ; y2 ; y3 ; y4 ; y6 ; y7 ; y8

Area variables Ai ; i ¼ 1; 2; . . . ; 15

wðxÞ ¼

i¼1

qi xi l i

Permissible area variables and side constraints for geometry variables Ai 2 S ¼ f0:111; 0:141; 0:174; 0:220; 0:270; 0:287; 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; 19:180gðin2 Þ100 6 x2 6 140; 220 6 x3 6 260; 100 6 y2 6 140; 100 6 y3 6 140; 50 6 y4 6 90; 20 6 y6 6 20; 20 6 y7 6 20; 20 6 y8 6 60 ðinÞ Young modulus E = 104 (ksi) Material density q = 0.1 (lb/in3)

Table 3 Optimal discrete areas and geometry of the 15-bar planar truss structure. Design variables

Tang et al. (2005)

Rahami et al. (2008)

Gholizadeh (2013)

Present research

PSO

CPSO

SCPSO

R-ICDE

D-ICDE

1.174 0.539 0.347 0.954 0.954 0.141 0.141 0.111 1.174 0.141 0.44 0.44 0.141 0.141 0.347 102.2873 240.505 112.584 108.0428 57.7952 6.4299 1.8006 57.7987 77.6153

0.954 0.539 0.27 0.954 0.539 0.174 0.111 0.111 0.287 0.347 0.347 0.22 0.22 0.174 0.27 137.2216 259.9093 123.5006 110.002 59.9356 5.1799 4.2193 57.8829 72.5143

1.081 0.539 0.270 0.954 0.954 0.220 0.111 0.111 0.287 0.220 0.440 0.440 0.174 0.174 0.347 117.4983 242.9729 112.3731 101.2684 54.6397 12.3953 14.3909 54.6396 80.5688

1.081 0.539 0.141 0.954 0.539 0.287 0.111 0.111 0.141 0.347 0.440 0.270 0.270 0.287 0.174 100.0309 238.7010 132.8471 125.3669 60.3072 10.6651 12.2457 59.9931 74.6818

A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 x2 x3 y2 y3 y4 y6 y7 y8 Weight (lb)

1.081 0.539 0.287 0.954 0.954 0.220 0.111 0.111 0.287 0.220 0.440 0.440 0.111 0.220 0.347 133.612 234.752 100.449 104.738 73.762 10.067 1.339 50.402 79.820

1.081 0.539 0.287 0.954 0.539 0.141 0.111 0.111 0.539 0.440 0.539 0.270 0.220 0.141 0.287 101.5775 227.9112 134.7986 128.2206 54.8630 16.4484 13.3007 54.8572 76.6854

0.954 1.081 0.27 1.081 0.539 0.287 0.141 0.111 0.347 0.44 0.27 0.111 0.347 0.44 0.22 106.0521 239.0245 130.3556 114.273 51.9866 1.8135 9.1827 46.9087 82.2344

Constraint tolerance

0.00089

0.0

0.0

0.0

0.0

0.0

0.0

Structural analyses

8000

8000

4500

4500

4500

7980

7980

V. Ho-Huu et al. / Expert Systems with Applications 42 (2015) 7057–7069

7061

Fig. 2. Geometry and optimal topology of the 15-bar planar truss structure.

900 Best value Average value

Weight (lb)

700

500

300

100 0

0

50

100 Generation

150

200

Fig. 3. Convergence of the 15-bar planar truss structure problem.

inappropriate for solving the layout truss optimization problem which contains both continuous variable (coordinates of truss’s nodes) and discrete variable (a set of pre-determined cross section areas). Therefore, it is necessary to propose a new technique which could be embedded into the conventional ICDE to handle both discrete and continuous variables for solving the layout truss optimization. So far, some researches for transforming optimization algorithms to fit with integer or discrete variables have been conducted. These algorithms could be categorized into two groups of approaches: (i) indirect approaches that generate integer values after optimization process by using posterior handling techniques and (ii) direct approaches that directly generate integer values right in steps of the optimization process. In the indirect approaches, variables are handled as in original form and they are converted to discrete values after optimization

process. These approaches are simple and easy to carry out. However, the main deficiency of the indirect approaches is the low-convergence caused by continuous search space throughout optimization process. In the direct approaches, algorithms work with integer-valued variables without any posterior conversion. So far, only some limited work in this direction could be found in specialized literature. Chen and Chen (2009) proposed three different strategies alternatively to deal with various optimization problems for integer and discrete variables for evolution strategy algorithm. Datta and Figueira (2011, 2013), utilized a binary-coded technique to handle discrete variables for PSO and DE optimization algorithms. In spite of gaining relatively good results, the above techniques are still costly or difficult to execute. The main weakness of these binary-coded techniques is the high computational cost caused by the complexity of algorithm and unnecessary transformation between binary variable and integer variable in the optimization process. In order to overcome these issues, this paper proposed a new technique to handle discrete variables more accurately by integrating a discrete variable handling technique to the existing ICDE. The new method is hence called Discrete-ICDE (D-ICDE) for convenience. Note that the proposed technique will focus on modifying the initial phase and mutation phase, because the affection of the solution due to the type of design variables only happens in these two phases. Other phases of the D-ICDE are similar to those in original ICDE. The formulation of the adjustment is as follows.

4.1. Initialization phase Firstly, all discrete variables are stored to the set E in ascending order and indexed correspondingly with their order in the set E. The indexes are then stored to a set D. The representative set D starts from 1 to number of permissive discrete areas. By doing so, the original set of discrete variables E is transformed into a new set of continuous integer variables D which represents the corresponding discrete variables. Then, the initial population P0

Fig. 4. The 18-bar planar truss structure.

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Table 4 Data for design of the 18-bar planar truss structure. P18

Objective function

min

Stress constraints

ðrt Þi 6 25ðksiÞ; jðrc Þji 6 25ðksiÞ; i ¼ 1; 2; . . . ; 18

Buckling constraints

jðrc Þji 6 KEAi =li ; i ¼ 1; 2; . . . ; 18 Geometry variables x3 ; y3 ; x5 ; y5 ; x7 ; y7 ; x9 ; y9

wðxÞ ¼

i¼1

qi xi li

2

Size variables A1 ¼ A4 ¼ A8 ¼ A12 ¼ A16 ; A2 ¼ A6 ¼ A10 ¼ A14 ¼ A18 A3 ¼ A7 ¼ A11 ¼ A15 ; A5 ¼ A9 ¼ A13 ¼ A17 : Permissible size variables and side constraints for geometry variables Ai 2 S ¼ f2:00; 2:25; 2:50; . . . ; 21:25; 21:50; 21:75g ðin2 Þ; 775 6 x3 6 1225; 525 6 x5 6 975; 275 6 x7 6 725; 25 6 x9 6 475; 225 6 y3 ; y5 ; y7 ; y9 6 245 ðinÞ Young modulus E = 104 (ksi) Buckling coefficient K = 4 Material density q = 0.1 (lb/in3)

Table 5 Optimal discrete areas and geometry of the 18-bar planar truss structure. Design variables

Hasançebi and Erbatur (2002)

Kaveh and Kalatjari (2004)

Rahami et al. (2008)

Present research R-ICDE

D-ICDE

A1 A2 A3 A5 x3 y3 x5 y5 x7 y7 x9 y9 Weight (lb) Constraint tolerance Structural analyses

12.5 18.25 5.50 3.75 933 188 658 148 422 100 205 32 4574.28 0.0

12.25 18 5.25 4.25 913 186.8 650 150.5 418.8 97.40 204.8 26.70 4547.9 0.0

12.75 18.50 4.75 3.25 917.4475 193.7899 654.3243 159.9436 424.4821 108.5779 208.4691 37.6349 4530.68 0.0

12.25 18 5.5 4.5 909.52 184.02 646.71 147.73 416.45 96.46 204.03 25.32 4591.42 0.0

13 17.5 6.5 3 914.06 183.46 640.53 133.74 406.12 92.63 196.69 37.06 4554.29 0.0

–

–

8000

8025

8025 Fig. 6. Convergence of the 18-bar planar truss structure problem.

4.2. Mutation phase

Fig. 5. Geometry and optimal topology of the 18-bar planar truss structure.

is generated by randomly selecting l individuals from the representative continuous integer set via the formula

In the mutation phase, for the conventional ICDE, the main purpose is to generate mutant vectors which are different from target vectors. This enhances the diversity of the population, and ICDE could promptly find the optimum solution. For the layout truss optimization problem, in addition to the requirement of difference, the variables have to be integer value between 1 and jDj. In this phase, all mutation strategies are modified to create the integer valued mutant vectors by: + rand/1

v i ¼ xr xij ¼ xlj þ round½randð0; 1Þðxuj  xlj Þ;

i ¼ 1; . . . ; NP; j ¼ 1; . . . ; jDj ð9Þ

1

þ roundðF  ðxr2  xr3 ÞÞ

+ rand/2

v i ¼ xr

1

þ roundðF  ðxr2  xr3 Þ þ F  ðxr4  xr5 ÞÞ

xuj

in which, jDj is the length of vector D, ¼ 1 and ¼ jDj are respectively the lower bound and upper bound value of the design variable xj . From Eq. (9), it is seen that an integer value between 0 and jDj  1 (difference between lower and upper limits) is added to the lower bound xlj (which have the value of 1). As a result, all initial individuals are integer while the diversity of the initial population is ensured.

ð11Þ

+ current-to-rand/1

vi ¼ xi þ roundðF  ðxr xlj

ð10Þ

1

 xi Þ þ F  ðxr2  xr3 ÞÞ

ð12Þ

+ current-to-best/1

vi ¼ xi þ roundðF  ðxbest  xi Þ þ F  ðxr

1

 xr2 ÞÞ

ð13Þ

As seen from the Eqs. (10)–(13), because ri ði ¼ 1; 2; . . . ; 5Þ are integer and chosen such that satisfying r 1 –r2 –r3 –r 4 –r 5 , the values of ðxr2  xr3 Þ, ðxr4  xr5 Þ, ðxr1  xi Þ, ðxbest  xi Þ or ðxr1  xr2 Þ are integer. Because the scale factor F in this paper is chosen to be greater than or equal to 0.8, the rounded additional amount to target vectors is

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Fig. 8. Geometry and optimal topology of the 47-bar planar truss structure.

Fig. 7. The 47-bar planar truss structure.

larger than or equal to 1. As a result, the mutant vectors are highly different from the target vectors and still have integer values between 1 and jDj. After the process, the resulting mutant vectors contain all integer components, each of which represents the corresponding discrete variables.

These integer valued mutant vectors are also used in crossover phases as presented in Section 2.2. Because the nature of crossover process is the information exchange between target vectors and mutant vector in order to create trial vectors, and hence it’s unnecessary to use the discrete handling technique in this phase. For the selection phase, the integer valued trial vectors are then transformed back to corresponding discrete variables in the set E. As seen from the process, the discrete variable handling technique only affects the treating variables methodology, the evolution of population and constraints handling are almost untouched. Hence, the population diversity brought by the improved (l + k)-differential evolution (IDE) and effective constraints handling from the archiving-based adaptive tradeoff model (ArATM) are maintained. As a result, the discrete variables handling technique not only preserves the advantage of the

Table 6 Data for design of the 47-bar planar truss structure. Objective function Stress constraints Buckling constraints Size variables A3 ¼ A1 ; A4 ¼ A2 ; A5 ¼ A6 ; A7 ; A8 ¼ A9 ; A10 ; A12 ¼ A11 ; A14 ¼ A13 ; A15 ¼ A16 ; A18 ¼ A17 ; A20 ¼ A19 ; A22 ¼ A21 ; A24 ¼ A23 ; A26 ¼ A25 ; A27 ; A28 ; A30 ¼ A29 ; A31 ¼ A32 ; A33 ; A35 ¼ A34 ; A36 ¼ A37 ; A38 ; A40 ¼ A39 ; A41 ¼ A42 ; A43 ; A45 ¼ A44 ; A46 ¼ A47

P min wðxÞ ¼ 47 i¼1 qi xi li ðrt Þi 6 20ðksiÞ; jðrc Þji 6 15ðksiÞ; i ¼ 1; 2; . . . ; 47 2

jðrc Þji 6 KEAi =li ; i ¼ 1; 2; . . . ; 47 Geometry variables x2 ¼ x1 ; x4 ¼ x3 ; y4 ¼ y3 ; x6 ¼ x5 ; y6 ¼ y5 ; x8 ¼ x7 ; y8 ¼ y7 ; x10 ¼ x9 ; y10 ¼ y9 ; x12 ¼ x11 ; y12 ¼ y11 ; x14 ¼ x13 ; y14 ¼ y13 ; x20 ¼ x19 ; y20 ¼ y19 ; x21 ¼ x18 ; y21 ¼ y18

Discrete set of size variables and side constraints of geometry variables Ai 2 S ¼ f0:1; 0:2; 0:3; . . . ; 4:8; 4:9; 5:0g; xi ; yi 2 R Loads Node: 17, 22 Young modulus E = 3  104 (ksi) Buckling coefficient K = 3.96 Material density q = 0.3 (lb/in3)

Fx = 6 (ksi) Fy = 14 (ksi)

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4500

Weight (lb)

searching algorithm of IDE, but also specifies accurately the discrete area for the optimal layout. More importantly, the discrete variables handling technique significantly improves the ability of the ICDE in solving layout truss optimization problem by considerably narrowing down the search space. As presented in the technique procedure, instead of searching the optimal solution on infinite real valued space, the optimization problem is now solved in a finite search space of continuous integer variables. As a result, the convergence speed and exploitation mechanisms of the D-ICDE are significantly improved for layout truss optimization problem.

Best value Average value

3500

2500

1500

5. Numerical examples

0

100

200 Generation

300

This section studies five examples, in which the discrete variables are handled by both the round-off treatment and the discrete variable handling technique. Based on the structure’s characteristic, the examples are classified into 2 groups: the first group consists of 3 different planar truss systems with fifteen, eighteen, and forty-seven truss elements, respectively; the second group

400

Fig. 9. Convergence of the 47-bar planar truss structure problem. Table 7 Optimal discrete area and geometry of the 47-bar planar truss structure. Design variables

A3 A4 A5 A7 A8 A10 A12 A14 A15 A18 A20 A22 A24 A26 A27 A28 A30 A31 A33 A35 A36 A38 A40 A41 A43 A45 A46 x2 x4 y4 x6 y6 x8 y8 x10 y10 x12 y12 x14 y14 x20 y20 x21 y21 Weight (lb) Constraint tolerance Structural analyses

Salajegheh and Vanderplaats (1993)

Hasançebi and Erbatur (2001)

Hasançebi and Erbatur (2002)

Gholizadeh (2013) PSO

CPSO

SCPSO

Present research R-ICDE

D-ICDE

2.61 2.56 0.69 0.47 0.80 1.13 1.71 0.77 1.09 1.34 0.36 0.97 1.00 1.03 0.88 0.55 2.59 0.84 0.25 2.86 0.92 0.67 3.06 1.04 0.10 3.13 1.12 107.76 89.15 137.98 66.75 254.47 57.38 342.16 49.85 417.17 44.66 475.35 41.09 513.15 17.90 597.92 93.54 623.94 1900 0.0

2.5 2.2 0.7 0.1 1.3 1.3 1.8 0.5 0.8 1.2 0.4 1.2 0.9 1.0 3.6 0.1 2.4 1.1 0.1 2.7 0.8 0.1 2.8 1.3 0.2 3.0 1.2 114 97 125 76 261 69 316 56 414 50 463 54 524 1.0 587 99 631 1925.79 0.0

2.5 2.5 0.8 0.1 0.7 1.3 1.8 0.7 0.9 1.2 0.4 1.3 0.9 0.9 0.7 0.1 2.5 1.0 0.1 2.9 0.8 0.1 3.0 1.2 0.1 3.2 1.1 104 87 128 70 259 62 326 53 412 47 486 45 504 2.0 584 89 637 1871.7 0.0

2.80 2.70 0.80 1.10 0.80 1.30 1.80 0.90 1.20 1.40 0.30 1.40 1.10 1.20 1.60 1.00 2.80 0.80 0.10 3.00 0.90 0.10 3.30 0.90 0.10 3.30 1.20 98.8628 78.6595 146.7331 66.5231 239.0901 55.6936 327.7882 48.8641 398.6775 43.1400 464.7831 37.8993 511.0450 18.2341 594.0710 90.9369 621.3943 1975.839 0.0

2.60 2.50 0.70 0.30 1.20 1.10 1.60 0.80 1.10 1.30 0.30 0.80 1.00 1.00 0.90 0.10 2.70 0.90 0.10 3.00 1.00 0.20 3.30 0.90 0.10 3.30 1.10 99.3630 83.4439 126.3845 69.5148 218.2013 58.0004 322.2272 51.4015 401.5626 46.8605 458.3021 46.8885 527.8575 16.2354 610.8496 98.3239 624.9580 1908.831 0.0

2.5 2.5 0.8 0.1 0.7 1.4 1.7 0.8 0.9 1.3 0.3 0.9 1.0 1.1 5.0 0.1 2.5 1.0 0.1 2.8 0.9 0.1 3.0 1.0 0.1 3.2 1.2 101.3393 85.9111 135.9645 74.7969 237.7447 64.3115 321.3416 53.3345 414.3025 46.0277 489.9216 41.8353 522.4161 1.0005 598.3905 97.8696 624.0552 1864.10 0.0

2.6 2.0 0.7 0.5 1.3 1.0 2.0 0.7 0.7 1.6 0.4 1.1 1.4 1.3 1.0 1.1 2.8 1.0 0.3 3.3 0.5 0.2 3.1 0.5 0.2 3.1 0.4 114.6321 94.0579 109.1345 64.3973 263.2525 52.6935 319.3121 46.7773 399.6352 34.7722 473.7424 41.8471 511.9268 17.6853 588.2343 81.2657 616.4535 1759.88 0.0

2.7 3.0 0.5 1.1 0.7 1.5 2.1 0.9 0.8 1.8 0.4 1.0 1.3 1.6 1.0 0.4 3.0 0.9 0.2 3.3 0.4 0.1 3.3 0.3 0.1 3.3 0.5 106.2986 82.4936 136.9634 62.7192 244.4495 47.5630 332.7201 42.7377 401.7876 32.8229 468.0985 27.0026 500.4160 11.9079 581.5046 82.6543 611.0089 1744.80 0.0

–

100,000

–

25,000

25,000

25,000

17,745

17,745

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Fig. 10. The 25-bar space truss structure.

Table 8 Data for design of the 25-bar space truss structure. P min wðxÞ ¼ 25 i¼1 qi xi li Dðx;y;zÞi 6 0:3ðinÞ; i ¼ 1; 2; . . . ; 6 ðrt Þi 6 40ðksiÞ; jðrc Þji 6 40ðksiÞ; i ¼ 1; 2; . . . ; 25 Geometry variables x4 ¼ x5 ¼ x3 ¼ x6 ; x8 ¼ x9 ¼ x7 ¼ x10 ; y3 ¼ y4 ¼ y5 ¼ y6 ; y7 ¼ y8 ¼ y9 ¼ y10 ; z3 ¼ z4 ¼ z5 ¼ z6

Objective function Displacement constraints Stress constraints Size variables A1 ; A2 ¼ A3 ¼ A4 ¼ A5 ; A6 ¼ A7 ¼ A8 ¼ A9 ; A10 ¼ A11 ; A12 ¼ A13 ; A14 ¼ A15 ¼ A16 ¼ A17 ; A18 ¼ A19 ¼ A20 ¼ A21 ; A22 ¼ A23 ¼ A24 ¼ A25 Discrete size variables and side constraints for geometry variables Ai 2 S ¼ ½0:1; 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; 2:5; 2:6; 2:8; 3:0; 3:2; 3:4ðin2 Þ 20 6 x4 6 60; 40 6 x8 6 80; 40 6 y4 6 80; 100 6 y8 6 140; 90 6 z4 6 130 ðinÞ Loads Node 1 2 3 6 Young modulus E = 104 (ksi) Material density q = 0.1 (lb/in3)

Fx (kips)

Fy (kips)

Fz (kips)

1.0 0.0 0.5 0.6

10 10 0.0 0.0

10 10 0.0 0.0

includes 2 different space truss systems twenty-five and thirty-nine truss elements, respectively. The parameters in the D-ICDE used in all examples are given in Table 1. The performance of the D-ICDE is verified by comparing its results with those by the round-off ICDE (R-ICDE) and those by previous authors. 5.1. Planar truss structure 5.1.1. A 15-bar planar truss structure The optimization problem considers a 15-bar planar truss subjected to a traversal load F = 10 (kip) as shown in Fig. 1. The objective function and input data of the truss is given in Table 2. In this example, there are 15 discrete design variables for the cross-section areas and 8 continuous design variables for the nodal coordinates. All bars are subjected to tension stress limitation of ±25 (ksi). The comparison of the optimal designs with those of other references is provided in Table 3. It’s worth noting that in this case, both R-ICDE and D-ICDE are governed to converge within 8000 iterations. This limitation of iterations is to make a fair comparison

Fig. 11. Geometry and optimal topology of the 25-bar space truss structure.

with other authors whose works are done within 8000 iterations. It is found that the optimal weight obtained from the D-ICDE is relatively better than those from other methods. More specifically, it can be seen that for the same analysis steps, the optimal weight

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Table 9 Optimal discrete areas and geometry of the 25-bar space truss structure. Design variables

Wu and Chow (1995)

Kaveh and Kalatjari (2004)

Tang et al. (2005)

Rahami et al. (2008)

A1 A2 A3 A4 A5 A6 A7 A8 x4 y4 z4 x8 y8 Weight (lb) Constraint tolerance Structural analyses

0.1 0.2 1.1 0.2 0.3 0.1 0.2 0.9 41.07 53.47 124.6 50.80 131.48 136.20 0.0 –

0.1 0.1 1.1 0.1 0.1 0.1 0.1 1.0 36.23 58.56 115.59 46.46 127.95 124.0 0.0 –

0.1 0.1 1.1 0.1 0.1 0.2 0.2 0.7 35.47 60.37 129.07 45.06 137.04 124.943 0.0 6000

0.1 0.1 1.1 0.1 0.1 0.1 0.2 0.8 33.0487 53.5663 129.9092 43.7826 136.8381 120.115 0.0 10,000

R-ICDE

D-ICDE

0.2 0.2 0.9 0.2 0.2 0.2 0.2 1.0 36.380 57.080 126.62 48.200 139.90 145.275 0.0 6000

0.1 0.1 0.9 0.1 0.1 0.1 0.1 1.0 36.83 58.53 122.67 49.21 136.74 118.76 0.0 6000

13

Best value Average value

350

Authors

15

14

Weight (lb)

300

10

11

250

12

200

7 8

9

150 100

4

5 0

20

40

60 80 Generation

100

6

120

1

Fig. 12. Convergence of the 25-bar space truss structure.

2

3 of the D-ICDE is 74.6818 (lb), while the result obtained from the R-ICDE is 80.5688, those by Tang, Tong, and Gu (2005) is 79.82 (lb) and those by Rahami, Kaveh, and Gholipour (2008) is 76.685 (lb). This proves that discrete variable handling technique significantly improves the ICDE in terms of solving layout truss optimization problem. The geometry of the truss before and after optimization is shown in Fig. 2. The final shape of the structure exposed significant changes versus the original version. This implies that if a certain truss structure is shaped appropriately, the loading capacity of the structure could be enhanced significantly. The convergence of the D-ICDE for the 15-bar planar truss problem is shown in Fig. 3. It is seen that the D-ICDE has fast convergence. In detail, the results start to converge after 50 iterations and completely converge from the 150th iteration. 5.1.2. A 18-bar planar truss structure We now consider the optimization problem for an 18-bar planar truss structure subjected to external load as shown in Fig. 4. Similar to the 15-bar planar truss structure, the objective function is to minimize the truss weight. There are 4 discrete area variables and 8 nodal coordinate variables in the system. In addition to the tension stress constraints, the system is also

Fig. 13. The 39-bar space truss structure.

Table 10 Nodal coordinates of bottom and top nodes of the 39-bar space truss structures. Bottom nodes

Top nodes

Number

x (m)

y (m)

z (m)

Number

x (m)

y (m)

z (m)

1 2

0

1 0.5

0 0

13 14

0 pffiffi  0:42 3

0.28 0.14

4 4

0.5

0

15

0:42 pffiffi 3

0.14

4

3

pffiffi  23 pffiffi 3 2

restricted within buckling constraints. Input data is given in Table 4. The comparison of the optimal designs with those of the other references is presented in Table 5. The optimal weight of the D-ICDE after 250 iterations is 4554.29 (lb) which agrees well with those obtained by Hasançebi and Erbatur (2002), Kaveh and Kalatjari (2004) and Rahami et al. (2008). The differences among the objective functions are inconsiderable. The geometry of the

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truss before and after optimization is shown in Fig. 5. The final shape of the structure also exposes significant changes versus the original version. It can be seen that the optimal shape has larger overall section at the beginning whose degree of freedom is constrained and becomes smaller at the end of truss. This is consistent with structural principals of loading resistance. For this case, the truss structure could be considered as beam clamped at the beginning end. As a result, the bending moment has the biggest value at clamped section and become smaller on the way to free end of the beam. This moment diagram has exactly the same shape with truss shape after optimization. This proves that truss structures restructure itself in the layout optimization process to exploit loading capacity in the most efficient way. In addition, the changes of the cross-sectional areas reflect reasonable principle, in which bars with greater size are subjected to greater loads. The convergence of the D-ICDE applying on the 18-bar planar truss structure is shown in Fig. 6. Similar with the 15-bar case, the D-ICDE has fast convergence compared with the average data.

5.1.3. A 47-bar planar truss structure The optimization problem for a 47-bar planar truss structure subjected to vertical and horizontal loads as shown in Fig. 7 is considered. Members of this structure are categorized into 27 groups. As a result, there are 27 discrete area variables and 17 nodal coordinate variables in the system. Constraints are stress limitation and buckling limitation. Input data for the system is given in Table 6. Fig. 8 demonstrates the optimal design of the 47-bar planar truss structure solved by the D-ICDE. The result shows considerable changes of the topology compared with the original design. This design show that since the loading distribution is various among the truss elements, the size of each bar element should be chosen accordingly to save the material usage while maintaining the functions of the structure. In Fig. 9, the method also shows its fast convergence for the 47-bar 2D truss problems. The comparison of the optimal designs with those of other references is provided in Table 7. It shows that both D-ICDE and R-ICDE converge with the fastest rate, however the optimal result of the D-ICDE is still better than that of the R-ICDE. More specifically, the optimal weight of the D-ICDE obtained after 394 iterations and 17745 structural analyses is 1744.8 (lb) whereas one by the R-ICDE is 1759.88 (lb). These results are much better than that by Hasançebi and Erbatur (2001) where optimal weight is 1925.79 (lb) corresponding to 100,000 structural analyses and that by Gholizadeh (2013) where optimal weight is 1864.10 (lb) corresponding to 25,000 structural analyses.

5.2. Space truss 5.2.1. A 25-bar space truss structure We now consider the optimization problem for a space truss structure with 25 bar elements subjected to loads as shown in Fig. 10. Members of this structure are categorized into 8 groups, resulting in 8 discrete size variables and 5 geometry variables. The displacements of all nodes are limited within ±0.3 (in) in the x, y, z directions. Constraints also include tension stress limitation of ±40 (ksi). Input data is given in Table 8. Fig. 11 demonstrates the optimal design of the 25-bar space truss structure solved by the D-ICDE. The result shows considerable changes of the topology compared with original design. The comparison of the optimal designs with those of the other references is provided in Table 9. It shows that the result obtained by the D-ICDE for mixed discrete–continuous 25-bar space truss is the best compared to the references, while the R-ICDE arrived at the worst result. Fig. 12 demonstrates that the convergence of the D-ICDE on the 25-bar space truss agrees well with the average result. This proves the robustness and effectiveness of using D-ICDE for layout space truss optimization problem with both discrete and continuous variables.

10

7

4

Fig. 14. Geometry and optimal topology of the 39-bar space truss structure.

Table 11 Data for design of the 39-bar space truss structures. P min wðxÞ ¼ ij qi xi li Dyð13Þ 6 4 ðmmÞ ðrt Þi 6 240 ðMPaÞ; jðrc Þji 6 240 ðMPaÞ; i ¼ 1; 2; . . . ; 39 Geometry variables z4 ; y4 ; y7 ; z7 ; y10 ; z10

Objective function Displacement constraints Stress constraints Size variables A1 ½ð1; 4Þ; ð2; 5Þ; ð3; 6Þ; A2 ½ð4; 7Þ; ð5; 8Þ; ð6; 9Þ A (the remaining) A3 ½ð7; 10Þ; ð8; 11Þ; ð9; 12Þ; A4 ½ð10; 13Þ; ð11; 14Þ; ð12; 15Þ; 5 Discrete set of size variables and side constraints for geometry variables Ai 2 S ¼ ½0:1; 0:2; . . . ; 13 ðcm2 Þ 0:28 6 y4 6 1; 0 6 z4 6 20:28 6 y7 6 11 6 z7 6 30:28 6 y10 6 1; 2 6 z10 6 4 ðmÞ Loads Node Fy (kN) Young modulus E = 210 (MPa) Material density q = 7800 (kg/m3)

13 10

14 10

15 10

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individuals converge to the best individual in the population and the convergence criteria is satisfied.

Table 12 Optimal discrete areas and geometry of the 39-bar space truss structure. Design variables

Wang, Zhang, and Jiang (2002)

Shojaee, Arjomand, and Khatibinia (2013)

Present research R-ICDE

D-ICDE

A1 A2 A3 A4 A5 y4 z4 y7 z7 y10 z10 Weight (kg) Constraint tolerance Structural analyses

11.01 8.63 6.69 4.11 4.37 0.805 1.186 0.654 2.204 0.466 3.092 203.18 0.0

10.12 9.91 8.56 3.92 3.44 0.6683 1.9 0.4732 2.8734 0.3002 3.4415 176.834 0.0

11.8 11.5 11.6 2.6 1.7 0.9549 0.8589 0.9258 2.0154 0.7160 3.1011 142.29 0.0

13 12.9 9 2.7 1.6 0.9232 0.5380 0.7958 2.1637 0.5105 3.4134 140.35 0.0

–

–

1140

1140

350 Best value Average value

Weight (lb)

300

250

200

150

100

0

10

20

30 40 Generation

50

60

6. Conclusion The paper further extends the existing ICDE for both discrete and continuous design variables by integrating a novel discrete variable handling technique into the ICDE to give a so-called D-ICCDE. The D-ICDE shows great enhancement in both diversification and exploitation, two key mechanisms of a meta-heuristic optimization algorithms. The present technique helps narrow down the search space of the discrete variables to contain only admissible values. This thus enhances the search engine in the D-ICDE, and provides accurate values for the optimal results. The D-ICDE is then applied for the truss layout optimization problem under the constraint of stress, displacement and buckling limitations. Numerical results demonstrate that the D-ICDE can effectively attain the best optimum solutions with less iteration than other methods in the literature. This proves that with a proper adjustment, the D-ICDE can offer a robust, effective and reliable optimization method for solving complex COPs, which are used popularly in engineering. On the other font of engineering applications, the solving of layout truss optimization problem provides the designer a general point of view about truss structural design. In this optimization design, both cross sectional areas and structural shape are adjusted simultaneously to reach the minimum structural weight while loading capacity is still warranted. In other words, the layout truss optimization give a rise for a novel methodology of designing which brings stable truss structures with lower cost. For future development, the D-ICDE is promising to extend for determining the optimal solution of other structural optimization problems which contain both discrete and continuous variables such as shell roof, transmission tower, composite plates. In addition, the current research can be extended to establish and solve the reliability-based design optimization problems which give the optimal design of structures with a certain reliability requirement. Acknowledgments

Fig. 15. Convergence of the 39-bar space truss structure problem.

5.2.2. A 39-bar space truss structure Last, we consider an optimization problem for a tower truss structure with 39 bar elements as shown in Fig. 13. The truss is subjected to vertical loads at three top nodes. The coordination node data is given in Table 10. Members of this structure are categorized into 3 groups, resulting in 3 discrete size variables and 6 geometry variables. Displacement of node 13 is limited within ±4 (mm) in y directions. Constraints also include tension stress limitation of ±40 (ksi) for all elements. Input data is given in Table 11 below. Fig. 14 demonstrates the optimal design of the 39-bar space truss structure solved by the D-ICDE. The result again shows considerable changes of the topology compared with original design. The comparison of the optimal designs with those of other references is provided in Table 12. It shows that the result obtained by the D-ICDE for mixed discrete–continuous 39-bar space truss is much better than those from other methods. In this problem, it is noted that the performance of the D-ICDE and the R-ICDE are nearly the same. However, results by D-ICDE are still pretty better than those by R-ICDE. Fig. 15 shows that the convergence of the D-ICDE on the 39-bar 3D truss agrees well with the average result at the end of optimization process. This shows that all

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