Multimodal size, shape, and topology optimisation of truss structures using the Firefly algorithm

Advances in Engineering Software 56 (2013) 23–37

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Multimodal size, shape, and topology optimisation of truss structures using the Firefly algorithm Leandro Fleck Fadel Miguel a,⇑,1, Rafael Holdorf Lopez a,1, Letícia Fleck Fadel Miguel b,2 a b

Department of Civil Engineering, Federal University of Santa Catarina, Florianópolis, Brazil Department of Mechanical Engineering, Federal University of Rio Grande do Sul, Porto Alegre, Brazil

a r t i c l e

i n f o

Article history: Received 24 August 2012 Received in revised form 19 September 2012 Accepted 12 November 2012 Available online 17 December 2012 Keywords: Truss size Shape and topology optimisation Firefly algorithm Multimodal optimisation Mixed continuous-discrete variable problems Structural optimisation Single-stage procedure

a b s t r a c t This paper presents an efficient single-stage Firefly-based algorithm (FA) to simultaneously optimise the size, shape and topology of truss structures. The optimisation problem uses the minimisation of structural weight as its objective function and imposes displacement, stress and kinematic stability constraints. Unstable and singular topologies are disregarded as possible solutions by checking the positive definiteness of the stiffness matrix. Because cross-sectional areas are usually defined by discrete values in practice due to manufacturing limitations, the optimisation algorithm must assess a mixed-variable optimisation problem that includes both discrete and continuous variables at the same time. The effectiveness of the FA at solving this type of optimisation problem is demonstrated with benchmark problems, the results for which are better than those reported in the literature and obtained with lower computational costs, emphasising the capabilities of the proposed methodology. In addition, the procedure is capable of providing multiple optima and near-optimal solutions in each run, providing a set of possible designs at the end of the optimisation process. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Methods for the size optimisation of truss structures, in which the member areas are taken as design variables, are fully established in the literature [1–3]. However, it is well-known that better results can be achieved when size, shape, and topology optimisation are performed simultaneously [4]. In this case, the problem generally begins with a ground structure to determine the best element topology, and truss geometry can also be altered by taking nodal coordinates as design variables [5]. Many structural design variables are chosen based on discrete values due to manufacturing constraints. Thus, the size, shape and topology optimisation of truss structures becomes a mixed variable optimisation problem, in that it deals simultaneously with discrete and continuous design variables [6]. Such problems are usually nonconvex by nature [7–9], and therefore must be solved by optimisation methods capable of handling this type of problem.

⇑ Corresponding author. E-mail addresses: [email protected] (L.F.F. Miguel), [email protected] (R.H. Lopez), [email protected] (L.F.F. Miguel). 1 Department of Civil Engineering, Federal University of Santa Catarina, Rua João Pio Duarte da Silva, CEP 88040-970, Florianópolis, SC, Brazil. 2 Department of Mechanical Engineering, Federal University of Rio Grande do Sul, Av. Sarmento Leite 425, 2° andar, CEP 90050-170, Porto Alegre, RS, Brazil. 0965-9978/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.advengsoft.2012.11.006

Metaheuristic algorithms are well suited to solving such optimisation problems. Known advantages of these algorithms include the following: (i) they do not require gradient information and can be applied to problems in which the gradient is difficult to obtain or simply does not exist; (ii) they do not become stuck in local minima if correctly tuned; (iii) they can be applied to non-smooth or discontinuous functions; (iv) they furnish a set of optimal solutions instead of a single solution, giving the designer a set of options from which to choose; and (v) they can be easily employed to solve mixed variable optimisation problems. Genetic algorithms (GA) have been applied within this context for truss optimisation by several research studies. The initial emphasis on GA algorithms in the literature can be explained by the fact that they were one of the first developed heuristic algorithms, and have been successfully applied in different engineering fields, such as scheduling problems [10,11], pipe network optimisation [12], and laminated composite structures [13–15], to name just a few. Size optimisation has been performed by Goldberg and Samtani [16] and Rajeev and Krishnamoorthy [17], while size and shape optimisation has been employed by Wu and Chow [18], Galante [19], Soh and Yang [20] and Kelesoglu [21]. Fixed shapes with size and topology optimisation have been studied by Hajela et al. [22] and Sakamoto and Oda [23], while Grierson and Pak [24], Rajan [25], Hajela and Lee [26], Shrestha and Ghaboussi [27], Deb and Gulati [28] and Tang et al. [29] have studied size, shape, and topology optimisation. However, the use of GA has pre-

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sented some drawbacks linked mainly to the long computational time required when dealing with large computational models. Hence, alternate approaches have been developed to reduce the computational demand. Several researchers have developed and improved robust search techniques in the last decade that simulate the paradigm of a biological, chemical, or social system and are known as metaheuristic algorithms. This alternative can be efficient at dealing with the challenges that traditional heuristic optimisation algorithms (e.g., GA or SA) have faced for years, being particularly suitable to achieving fast and accurate solutions in the field of structural optimisation [30]. For example, the truss optimisation problem has been solved using alternate heuristic methods such as the simulated annealing (SA) [31], ant colony optimisation (ACO) [32], the harmony search (HS) [33] and the cuckoo search algorithm (CS) [34]. For a comprehensive review of the optimisation of truss structures using metaheuristics, the reader is referred to Saka [35], Saka [36], Lamberti and Pappalettere [37] and the references therein. The particular emphasis in more recent research [38–41] has been on so-called multimodal optimisation, which implies that multiple locally optimal or near-optimal solutions can be identified in each run of an algorithm. The ability to furnish the designer with a set of options is very attractive because it is impossible to account for construction aspects or aesthetics in the fitness functions. The alternatives presented by multimodal optimisation allow engineers to evaluate those characteristics based on external priorities. In this context, the Firefly algorithm (FA) developed recently by Yang [42] has proved to be more accurate and efficient than wellestablished heuristic algorithms such as the GA and Particle Swarm Optimisation (PSO). FA is capable of effectively and simultaneously finding the global as well as local optima and is particularly suitable for parallel implementation. Several researchers have focused their attention on solving optimisation problems using FA in a growing number of papers [43–50]. However, its implementation in the field of structural optimisation is still fairly recent and requires a substantial amount of further study [51,52]. This paper employs the FA in the simultaneous optimisation of size, shape, and topology in truss structures. Unstable and singular topologies are disregarded as possible solutions by checking the positive definiteness of the stiffness matrix. The procedure is capable of providing multiple optimal and near-optimal solutions in each run, presenting a set of options at the end of the design stage. The effectiveness of the FA is demonstrated through a selection of benchmark problems. Section 2 presents the formulation of the optimisation problem and a description of the FA is given in Section 3. A series of numerical examples is presented in Section 4, while Section 5 presents the main conclusions of the study.

2. Problem formulation The ground structure approach is followed in the proposed methodology. This scheme, initially proposed by Dorn et al. [53], starts with a universal truss containing all (or almost all) possible member connections n among all nodes q in the structure. The topology optimisation procedure is then employed to discard the unnecessary members. In other words, the algorithm chooses, from all possible members n, the m members that remain in the structure for which m 6 n. Simultaneously, the algorithm performs the size optimisation of the truss by changing the cross-sectional area (A 2 Rm ) of the remaining structural members and the shape optimisation by modifying the nodal coordinates of the nodes q0 con0 sidered as design variables (n 2 Rq ). This optimisation procedure seeks the minimum structural weight of the truss subjected to stress, displacement and kinematic stability constraints. For conve-

nience of notation, the design variables A and n are grouped into the vector x = [A1, . . ., Am, n1, . . ., nq’]. Thus, the optimisation problem can be posed as:

Find

x

Minimise

WðxÞ ¼

m X

qj ‘j ðnÞAj

j¼1

Subject to G1  truss is kinematically stable G2  dk ðxÞ  dmax 6 0; k

k ¼ 1; . . . ; q

ð1Þ

G3  tension : jrj ðxÞj  rtj 6 0 or compression : jrj ðxÞj  rcj 6 0; G4 

Amin j

G5 

nmin i

6 Aj 6

Amax ; j

6 ni 6

nmax ; i

j ¼ 1; . . . ; m

j ¼ 1; . . . ; m i ¼ 1; . . . ; q0

in which W is the structural weight, m is the number of members in the design, q is the specific weight of the bar material, ‘ is the length of each bar (which is a function of the nodal coordinates) and G is the set of constraints. dk and dmax are the displacement and maxik mum allowable displacement at node k, respectively, rj is the stress of the jth bar, rtj and rcj are respectively the maximum allowable stress in tension and compression of the jth bar, Amin and Amax are j j the lower and upper bounds of the cross-sectional area of the jth bar, respectively, and nmin and nmax are the lower and upper bounds i i of the allowable movement of the ith design variable nodal coordinate, respectively. The displacement constraints are formulated by considering that the deflection in a specified coordinate direction of a node must be lower than an allowable displacement chosen by the designer. Likewise, the stress in the element must be lower than the allowable stress in tension and compression of the material. Furthermore, the supports and nodes carrying loads cannot be disregarded in the final solution, which is, in fact, based on the classical concept of basic and non-basic nodes. Finally, kinematic stability is achieved by checking the positive definiteness of the stiffness matrix. If buckling constraints are included in the formulation, the allowable stress in compression is the most critical case between the maximum stress in compression of the material and the critical stress given by the Euler’s equation (see Eq. (6) in Section 4.4.3). The following scheme is applied to address the constraints in this optimisation problem. First, the feasibility of the solution is checked (constraint G1 in Eq. (1)). In other words, if the truss is not kinematically stable, it is discarded and a new solution is attempted. Thus, the member forces and node displacements are calculated only if this first check is verified. Next, if one or more of the displacement and/or stress constraints are violated (constraints G2 and G3 in Eq. (1)), a penalty Pt is added to the objective function of the current design. In this case, the penalty magnitude is proportional to the violation, and takes the form:

Pt ðxÞ ¼ h

"  þ q X dk ðxÞ  dmax k¼1

dmax k

k

þ

m X jrj ðxÞj  rij j¼1

rij

!þ # ;

ð2Þ

in which ðÞþ ¼ ½ðÞþjðÞj ; jðÞj stands for the absolute value, h is a posi2 tive parameter, i is equal to t if the member is in tension and i is equal to c if the member is in compression. Finally, constraints G4 and G5 are addressed by a coding approach. These bounds are imposed by not sampling infeasible designs in the computer code. In general, two classical approaches have been employed in the literature to address the simultaneous optimisation of size, topology and geometry of trusses. The first alternative attempts to perform a single-stage procedure, in which all variables including topology, shape and size are determined together. The second alternative is a two-stage procedure. The main goal is to search for a set of stable topologies, after which the best size and shape are determined.

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However, because the problems are not linearly separable, it is impossible to always provide a global optimal design. Thus, a principal goal of this paper is to achieve better performance by incorporating the advantages of FA with a single-stage procedure. A specific member is eliminated from the ground structure following the criteria proposed by Deb and Gulati [28]. The cross-sectional area of a member is compared to a user-defined small critical cross-sectional area h. If the member area is smaller than h, this element is eliminated from the ground structure. This method defines how different topologies can be obtained in a continuous optimisation procedure. Note that the value of h and the lower (Amin) and upper (Amax) bounds of the cross-sectional areas must be determined by considering the probability of a specific element to be absent from the final solution. For example, if Amin = Amax and the critical cross-sectional area h is almost zero, the probability of any member being present in the final structure is approximately 50%. In discrete optimisation, the user defines the number of zero-force bars that are added to the available profile list to generate a reasonable probability of eliminating an element from the ground structure. 3. Firefly algorithm (FA) The Firefly algorithm (FA) is a very recent heuristic optimisation algorithm developed by Yang [42] and is inspired by the flashing behaviour of fireflies. According to Yang [42], FA optimisation has three idealised rules. (a) All fireflies are unisex, so that one firefly is attracted to other fireflies regardless of their sex. (b) Attractiveness is proportional to brightness, so for any two flashing fireflies, the less bright firefly will move towards the brighter firefly. Both attractiveness and brightness decrease as the distance between fireflies increases. If there is no firefly brighter than a particular firefly, that firefly will move randomly. (c) The brightness of a firefly is affected or determined by the landscape of the objective function. Based on these three rules, the basic steps of the FA can be summarised as the pseudo-code shown in Fig. 1 [42]. There are two essential components to FA: the variation of light intensity and the formulation of attractiveness. The latter is assumed to be determined by the brightness of the firefly, which in turn is related to the objective function of the problem under study.

Fig. 1. Pseudo-code of the Firefly algorithm (adapted from Yang [42]).

As light intensity and attractiveness decrease and the distance from the source increases, the variation of light intensity and attractiveness should be a monotonically decreasing function. For example, the light intensity can be: cr 2ij

Iðr ij Þ ¼ I0 e

ð3Þ

in which the light absorption coefficient c is a parameter of the FA and rij, is the distance between fireflies i and j at xi and xj, respectively, which can be defined as the Cartesian distance rij = kxi  xjk. Because a firefly’s attractiveness is proportional to the light intensity seen by other fireflies, it can be defined by:

bðr ij Þ ¼ b0 e

cr 2ij

ð4Þ

in which b0 is the attractiveness at r = 0. Finally, the probability of a firefly i being attracted to another, more attractive (brighter) firefly j is determined by cr 2ij

Dxi ¼ b0 e

ðxtj  xti Þ þ aei ;

xtþ1 ¼ xti þ Dxi ; i

ð5Þ

where t is the generation number, ei is a random vector (e.g., the standard Gaussian random vector in which the mean is 0 and the standard deviation is 1) and a is the randomisation parameter. The first term on the right-hand side of Eq. (5) represents the attraction between the fireflies and the second term is the random movement. In other words, Eq. (5) shows that a firefly will be attracted to brighter or more attractive fireflies and also move randomly. Eq. (5) indicates that the user must set parameters b0, c, a and the distribution of ei to apply the FA, and also shows that there are two limit cases when c is small or large. (a) If c approaches zero, the attractive and brightness are constants, and consequently, a firefly can be seen by all other fireflies. In this case, the FA reverts to the PSO. (b) f c approaches infinity, the attractiveness and brightness approach zero, and all fireflies are short-sighted or fly in a foggy environment, moving randomly. In this case, the FA reverts to the pure random search algorithm. Hence, the FA generally corresponds to the situation falling between these two limit cases. The next section describes the use of a numerical analysis to demonstrate the effectiveness of the FA at solving truss optimisation problems. 4. Numerical examples Standard test problems are useful for checking optimisation algorithms. The benchmark examples given in this section have been widely used for this purpose. Due to the stochastic nature of the FA, the final result can vary depending on the seed used for the random number generation. Yet there is no established benchmark criterion in the literature to evaluate the performance of metaheuristics in size, topology and shape optimisation of trusses. For instance, Hajela and Lee [26] and Tang et al. [29] have not mentioned the number of times the search was repeated and have presented only the best results obtained. Martini [41] has performed 10 optimisation runs of 4000 design cycles each. Moreover, some technical papers have not explained the adopted strategy in detail [25,28] and only the optimal results are furnished. With the goal of providing a statistical basis for further comparison, this paper presents the results of over 100 runs for each example. In this way, the average values and standard deviations are presented along with the optimal results. The problems are presented in increasing order of complexity, and the following set of parameters are used in all examples: h = 1  108, b0 = 1, c = 1, a = 0.5 and a value of ei that follows a uniform distribution between 0.5 and 0.5.

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Fig. 2. Eleven-bar truss benchmark example.

4.1. Eleven-bar truss example This truss is often used as a benchmark problem and has been applied in most studies in this field. The ground structure of this example is shown in Fig. 2 and the design parameters are given in Table 1. Two independent studies are performed: (i) size and topology optimisation and (ii) simultaneous size, shape and topology optimisation. 4.1.1. Size and topology optimisation Several researchers have studied this problem using GA, but have applied two different sets of discrete design variables. Rajan [25] and Tang et al. [29] have adopted cross-sectional areas from a set of 32 discrete values, whereas Hajela and Lee [26] and Deb and Gulati [28] have allowed the cross-sectional areas to vary within the range of 0.0–30.0 in.2 at increments of 1.0 in.2. Because the best known solution in the literature is 4912.15 lb, found by Deb and Gulati [28], the present paper attempts to reproduce the conditions of that study. Although Hajela and Lee [26] and Deb and Gulati [28] have studied the same problem, each study has employed a different approach to the solution. Hajela and Lee [26] have performed the optimisation in two steps. The goal of the first step is to search for kinematically stable topologies while disregarding the constraints. These topologies are then used as seeds to find the best cross-sectional member size. In contrast, Deb and Gulati [28] have conducted size and topology optimisation in only one step using a population size P = 220 and G = 225 generations, resulting in a total of 49,500 objective function evaluations (OFE). This paper intends to achieve the same accuracy as that of Deb and Gulati [28] while providing the designer with several optimal or near-optimal structural designs. In addition, the authors aspire to achieve these goals with fewer function evaluations than reference [28]. In other words, the authors employ n = 10 fireflies and S = 3000 searches, resulting in 30,000 OFE. The FA approach is capable of finding 46 feasible topologies (from an average of over 100 simulations), which demonstrates Table 1 Design parameters for the 11-bar truss benchmark example. Design parameter

Value

Modulus of elasticity Weight density Allowable stress in tension Allowable stress in compression Allowable y-displacement

104 ksi 0.1 lb/in.3 25 ksi 25 ksi 2 in.

the robustness of the multimodal procedure because the number of different topologies generated is one measure of its performance. In addition, the range of alternative solutions achieves very similar results when compared to the best solution of each run, another indication of the multimodal procedure’s effectiveness. Table 2 presents the normalised mean weight for the top five topologies, in which the results are again an average over 100 runs (the coefficient of variation is inside the parentheses). Fig. 3 shows three of the topologies typically generated during an optimisation run. Note that Topology A achieves the best weight. Fig. 3a shows that the best topology consists of only six members and five nodes. The cross-sectional areas and corresponding truss weights for the best designs over 100 runs obtained by this study and by Hajela and Lee [26] and Deb and Gulati [28] are shown in Table 3 and the convergence history is of the best topology is shown in Fig. 4. The mean value achieved in the present study is equal to 4919.77 lb and the coefficient of variation is 0.11%. The y-displacement approaches its limit for the nodes carrying loads, reaching 99.95% (intersection of members 6 and 9) and 99.85% (intersection of members 2, 4, and 6) of the allowable 2 in. displacement. Thus, the proposed optimisation scheme is capable of reproducing the weight obtained by Deb and Gulati [28], which has been the best result reported in the literature so far.

4.1.2. Size, shape and topology optimisation This problem has been studied by Rajan [25] and Balling et al. [38], who have employed the GA, and Martini [41], who has employed the Harmony Search (HS) optimisation algorithm. Shape is optimised by allowing the vertical coordinates of the three superior nodes to moves between 180 in. and 1000 in., considering the origin in the intersection of members 1–3. Because the nodal coordinates are continuous and the cross-sectional areas are taken from a set of 32 discrete variables [25], the problem is a mixed variable optimisation problem in that it addresses integer and continuous design variables simultaneously. Table 2 Normalised weight for the top five topologies of the size and topology 11-bar truss example. Normalised weight

FA

1st

2nd

3rd

4th

5th

1.000

1.038 (1.5%)

1.088 (1.3%)

1.101 (0.9%)

1.117 (1.2%)

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L.F.F. Miguel et al. / Advances in Engineering Software 56 (2013) 23–37

Topology A

10000

Weight (lb)

9000 8000 7000 6000 5000

Topology B

4000

0

1000

2000

3000

Generations Fig. 4. Convergence history for the size and topology 11-bar truss optimisation problem.

Table 4 Normalised weight for the top five topologies of the size, shape, and topology 11-bar truss example. Normalised weight

Topology C FA

Fig. 3. Samples of selected topologies determined for the size and topology 11-bar truss optimisation problem.

Table 3 Member areas of the optimised 11-bar truss benchmark example. Member number

Member areas (in.2) FA

Deb and Gulati [28]

Hajela and Lee [26]

2 3 4 5 6 9 Weight (lb)

24 20 6 30 16 21 4912.85

24 20 6 30 16 21 4912.85

24 21 6 28 16 22 4942.7

Rajan [25] has performed a unimodal study of this problem with a population size P = 40 and number of generations G = 96, while Balling et al. [38] has carried out a multimodal analysis with P = 1000 and G = 500, resulting in a total of 500,000 OFE. The latter has not eliminated the possibility of unstable topologies, allowing the algorithm to determine mechanisms. Martini [41] has employed 75 evaluations on the initialisation of harmony memory over 4000 design cycles, resulting in 4075 OFE in a multimodal analysis. The present paper uses n = 10 fireflies and S = 5000 searches. The multimodal procedure is capable of finding 62 feasible topologies (the average over 100 simulations). Meanwhile, Balling et al. [38] has found 26 feasible topologies, even though they do not eliminate the possibility of mechanisms, and Martini [41] has found less than 10 solutions. Table 4 presents the normalised weights of the top five topologies, in which the results are again an average over 100 runs (the coefficient of variation is inside

1st

2nd

3rd

4th

5th

1.000

1.028 (1.4%)

1.072 (2.8%)

1.102 (3.5%)

1.144 (3.4%)

the parentheses). Once again, the range of alternative solutions is very close to the best solution of each run. Despite the lower number of OFE, the best result over 100 runs in this study weighs 2705 lb, which is slightly better than the optimum weight determined by Balling et al. [38] (2736 lb) and Rajan [25] (3254 lb). Martini [41] has also presented a less desirable result, as expected because the number of OFE is much lower. The mean value achieved in this work is equal to 2893.45 lb and the coefficient of variation is 2.12%. Thus, the proposed scheme provides better results than the cited references. Fig. 5 shows three of the topologies typically generated during the optimisation run, of which Topology A corresponds to the best weight. The cross-sectional areas, tension members and corresponding weight of Topology A are listed in Table 5. Fig. 5a shows that the best topology is again composed of only six members and five nodes, and the convergence history for this case is shown in Fig. 6. The y-displacement approaches its limit for the nodes carrying loads, reaching 99.99% (intersection of members 6 and 9) and 99.85% (intersection of members 2, 4, and 6) of the allowable 2 in. displacement. From the results presented in this section, the authors can conclude that the proposed optimisation scheme is capable not only of reproducing the best results reported in the literature but also of improving the best design solution for this problem. 4.2. Thirty-nine-bar two-tiered truss example The second benchmark example is the single-span 39-bar, 12node, simply supported, two-tiered ground structure shown in Fig. 7. This structure has been studied before by Deb and Gulati [28], who have demonstrated one of the best GA procedures in the literature. Thus, it is interesting to test this problem again with the proposed scheme and compare the results to those of the previous study. Two independent studies are again performed: (i) size and topology optimisation and (ii) simultaneous size, shape and topology optimisation. However, the cross-sectional areas are treated as

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Topology A

5000

Weight (lb)

4500 4000 3500 3000 2500 2000

0

1000

2000

3000

4000

5000

Generations Topology B

Fig. 6. Convergence history for the size, shape and topology 11-bar truss optimisation problem.

properties and maximum allowable deflection are the same as those in the previous problem. The continuous cross-sectional areas vary between 0.225 and 2.25 in.2 and a critical area of e = 0.05 in.2 is considered.

Topology C

Fig. 5. Samples of selected topologies determined for the size, shape and topology 11-bar truss optimisation problem.

Table 5 Cross-sectional areas and tension members for the size, shape, and topology 11-bar truss optimisation problem. Member number

Member areas (in.2)

Stress (ksi)

2 3 4 5 6 9 Weight (lb)

11.50 2.88 5.74 11.50 7.22 13.50 2705.16

10.3 10.0 19.1 10.4 10.2 9.2

continuous variables in this case to properly compare the results to those in the literature. The overlapping members are shown laterally dislocated in the figure for visual clarity. Because the lateral symmetry around member 19 is assumed, the number of variables is reduced to 21. The allowable strength is 20 ksi and the material

4.2.1. Size and topology optimisation The procedure for this example consists of the optimisation of 21 continuous variables. Deb and Gulati [28] have found two optimised topologies. The first, corresponding to a simulation run with a population size of 630, consists of only 17 members and 10 nodes and produces a truss with a final weight of 198 lb. The second optimised truss weighing 196.546 lb is obtained by increasing the population size to 840. The resulting structure contains 19 members and 11 nodes, with a different topology than the first option, although both have a very similar total weight. It is impossible to know the total number of evaluations performed because the number of generations is not mentioned. While Deb and Gulati [28] have performed a unimodal analysis, the current study intends to achieve the same accuracy while attempting to obtain multimodal benefits—that is, to provide a set of high-quality options. Due to the great complexity of this optimisation problem and the large number of feasible topologies, the algorithm is truncated to propose only the top 100 topologies. This value, besides being sufficient for the purposes of the study, also lowers the required computational time. Table 6 presents the normalised weights of the top five topologies, the results of which are again an average over 100 runs. As in the previous examples, the range of alternative solutions has results that are very close to the best solution obtained from each run. The cross-sectional areas, tension members and corresponding truss weights obtained in this example are listed in Table 7. Fig. 8 shows three of the topologies typically generated during the optimisation run. Note that Topology A, corresponding to the best weight, contains 9 nodes and 15 members, which is less than both topologies presented by Deb and Gulati [28]. A typical convergence history of the best topology for this problem is shown in Fig. 9. The best result over 100 runs has a weight of 193.5472 lb, which is considerably lower than the solutions in the references mentioned above. The mean value obtained using the FA approach is equal to 221.68 lb and the coefficient of variation is 12.9%. Thus, the proposed scheme once again obtains the best result found in the literature to the best of the authors’ knowledge. 4.2.2. Size, shape and topology optimisation To carry out simultaneous size, shape and topology optimisation, selected nodal coordinates are taken as design variables in

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120in

120in

120in

120in

21 4

25 37

16 11

15

17 7

18

8

36

38 19

3

39

29 24

20 35

14

27

6 5

13 9

120in

32

28

10

34

12

1

2

20000lb

26 33

23

20000lb

30

31

120in

22

20000lb

Fig. 7. Two-tiered truss, 39-member benchmark example.

Table 6 Normalised weight for the top five topologies of the size, and topology 39-bar twotiered truss example.

Topology A

Normalised weight

FA

1st

2nd

3rd

4th

5th

1.000

1.024 (2.6%)

1.034 (2.4%)

1.049 (2.4%)

1.062 (2.6%)

Table 7 Cross-sectional areas and tension members for the size, and topology 39-bar twotiered truss optimisation problem. Member number

1 2 3 5 7 8 9 10 11 14 21 22 23 24 26 28 29 30 31 32 35 Weight (lb)

FA

Topology B

Deb and Gulati [28]

Member areas (in2)

Stress (ksi)

Member areas (in.2)

0.0500 0.7524 – 1.5001 – 0.2504 – 1.0647 1.0612 0.5604 1.0016 0.0500 0.7524 – 1.5001 – 0.2504 – 1.0647 1.0612 0.5604 193.5472

0 19.93514 – 19.99902 – 19.96994 – 19.92404 19.98894 19.95146 19.96714 0 19.93514 – 19.99902 – 19.96994 – 19.92404 19.98894 19.95146

– 0.751 0.051 1.502 0.052 0.251 0.051 1.061 1.063 0.559 1.005 – 0.751 0.051 1.502 0.052 0.251 0.051 1.061 1.063 0.559 196.546

addition to the 21 cross sections. Assuming symmetry and considering that constrained and load-carrying nodes must remain fixed and the highest node at the centre of the structure does not move laterally, it is possible to reduce the number of nodal coordinates to 7. Each of these seven nodes is allowed to move (120, 120) in. from its original position. The optimised configuration proposed by Deb and Gulati [28] has been obtained using a population size P = 1680 and 300 generations, resulting in 504,000 OFE. The resulting truss requires 15

Topology C

Fig. 8. Samples of selected topologies determined for the size and topology twotiered truss optimisation problem.

members and nine nodes and weighs 192.19 lb. As due to the complexity of this optimisation problem and the large number of feasible topologies, the algorithm is again truncated to propose only the top 100 topologies. Table 8 presents the normalised weights of the top five topologies, the results of which are again an average over 100 runs. As in the other cases, the range of alternative solutions is very close to the best solution from each run.

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Topology A

600

Weight (lb)

500

400

300

200

100

Topology B 0

1000

2000

3000

4000

5000

Generations Fig. 9. Convergence history for the size and topology two-tiered truss optimisation problem.

Table 8 Normalised weight for the top five topologies of the size, shape, and topology 39-bar two-tiered truss example. Normalised weight 2nd

3rd

4th

5th

1.000

1.025 (2.2%)

1.031 (2.3%)

1.044 (2.4%)

1.050 (2.4%)

The study uses n = 10 fireflies and S = 5000 searches, resulting in 50,000 OFE, which represents less than 10% of the computational effort of the work of Deb and Gulati [28]. Fig. 10 shows three of the topologies typically generated during the optimisation run. Note that Topology A, corresponding to the best weight, contains eight nodes and 13 members, which is less than in the topology produced by Deb and Gulati [28]. A typical convergence history of the best topology for this example is shown in Fig. 11, while the cross-sectional areas, tension members and corresponding truss weights obtained in this study are listed in Table 9. The best result over 100 runs achieved by the FA has a weight of 191.304 lb, which is once again better than that of the reference solution. The mean value achieved by the FA optimisation scheme is equal to 207.34 lb and the coefficient of variation is 5.3%. 4.3. Twenty-five-bar 3D truss example This 3D truss is also often used as a benchmark problem. The ground structure is shown in Fig. 12 and the details of the loading and member groupings are given in Tables 10 and 11. The allowable strength is 40 ksi in tension and compression, the material properties (modulus of elasticity and weight density) are the same as in the previous problem and the maximum allowable deflection is 0.35 in. in any direction for each node. The cross-sectional areas are chosen from a set D = (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, 2.5, 2.6, 2.8, 3.0, 3.2, 3.4) in. Two independent studies are performed: (i) size and shape optimisation and (ii) simultaneous size, shape and topology optimisation. 4.3.1. Size and shape optimisation Past researchers have studied this problem using the GA, including Tang et al. [29], Rajeev and Krishnamoorthy [17] and Wu and Chow [18]. The x-, y- and z-coordinates of nodes 3, 4, 5 and 6 and the x- and y-coordinates of nodes 7–10 are taken as design variables, while nodes 1 and 2 remains unchanged. Because double symmetry is required in both the x–z and y–z planes, the

Topology C

Fig. 10. Samples of selected topologies determined for the size, shape, and topology two-tiered truss optimisation problem.

800 700 600

Weight (lb)

FA

1st

500 400 300 200 100 0

0

1000

2000

3000

4000

5000

Generations Fig. 11. Convergence history for the size, shape, and topology two-tiered truss optimisation problem.

problem includes eight size and five configuration variables. The side constraints for the configuration variables are 20 in. 6 x4 6 60 in., 40 in. 6 x8 6 80 in., 40 in. 6 y4 6 80 in., 100 in. 6 y8 6 140 in. and 90 in. 6 z4 6 130 in. Because the nodal coordinates are continuous and the cross-sectional areas are taken from a set of 30 discrete variables, this problem is also a mixed variable optimisation problem in that it deals simultaneously with integer and continuous design variables. The three different GA codings proposed by Tang et al. [29] have been carried out using a population size P = 40 and 150 genera-

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Table 14 shows the results of the present study compared to those in the literature. Note that the best solution over 100 runs obtained by the present algorithm (116.58 lb) is better than that obtained by three of the four different GA approaches and slightly worse than the fourth (1.5%). The mean value achieved in the present study is equal to 139.16 lb and the coefficient of variation is 8.0%. Fig. 14 shows three of the topologies typically generated during the optimisation run, and the convergence history of the best topology is shown in Fig. 15.

Table 9 Cross-sectional areas and tension members for the size, shape, and topology 39-bar two-tiered truss optimisation problem. Member number

Present study Member areas (in.2)

Stress (ksi)

1 2 4 8 14 15 19 22 23 25 29 35 36

0.29465559 1.06482113 1.19137888 1.25548701 0.0505146 1.53242606 0.96093992 0.29465559 1.06482113 1.19137888 1.25548701 0.0505146 1.53242606

19.97512 19.71634 19.98304 19.96446 19.0254 19.94986 19.80172 19.97512 19.71634 19.98304 19.96446 19.0254 19.94986

Weight (lb)

191.304

4.4. Fifteen-bar planar truss example This 15-bar planar truss was also studied by [18,29] and it was adopted here to verify the capability of the FA multimodal optimisation approach under Euler’s buckling as a constraint. The ground structure is illustrated in Fig. 16, showing a vertical tip load of 10,000 lb applied on node 8. The allowable strength is 25 ksi and the material properties (modulus of elasticity and weight density) are the same as in the previous examples. The x- and y- coordinates of nodes 2, 3, 6 and 7 and the y-coordinate of nodes 4 and 8 are taken as design variables. However, nodes 6 and 7 are constrained to have the same x-coordinates of the nodes 2 and 3, respectively. Thus, the problem includes 15 size and eight configuration variables (x2 = x6, x3 = x7, y2, y3, y4, y6, y7, y8). The cross-sectional areas are chosen from a set D = (0.111, 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, 19.180) in. The side constraints for the configuration variables are 100 in. 6 x2 6 140 in., 220 in. 6 x3 6 260 in., 100 in. 6 y2 6 140 in., 100 in. 6 y3 6 140 in., 50 in. 6 y4 6 90 in., 20 in. 6 y6 6 20 in., 20 in. 6 y7 6 20 in., and 20 in. 6 y8 6 60 in. Four independent studies are performed: (i) size and shape optimisation without buckling constraints, (ii) simultaneous size, shape and topology optimisation without buckling constraints, (iii) size and shape optimisation with buckling constraints and, (iv) simultaneous size, shape and topology optimisation with buckling constraints. In the cases (i) and (ii) the allowable strength is 25 ksi for tension and compression.

tions, resulting in 6000 OFE. To properly compare the FA performance to this GA study, the present study uses n = 10 fireflies and S = 600 searches, resulting in the same 60,000 OFE. The best results over 100 runs achieved by the FA have a weight of 118.83 lb, which is better than that of the reference, as observed in Table 12. The mean value obtained with the FA optimisation scheme is equal to 132.3 lb and the coefficient of variation is 5.5%. The convergence history of the best topology is shown in Fig. 13. 4.3.2. Size, shape and topology optimisation In addition to the eight discrete size and five continuous configuration variables, all eight member groups are considered as topology variables. Thus, this is once again a mixed variable optimisation problem. The previously cited references have performed unimodal procedures, while the present paper intends to achieve multimodal benefits. The scheme is capable of finding 11 feasible topologies (an average over 100 simulations). Table 13 presents the normalised weights of the top five topologies, in which the results are again an average over 100 runs.

z

75 in

2 y

x

1 75 in

100 in

4 3

5 6

100 in

8

7 9 200 in 200 in

10

Fig. 12. Twenty-five 3D truss benchmark example.

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Table 10 Loading of 25 bar 3D truss.

600

Px (lb)

Py (lb)

Pz (lb)

1 2 3 6

1000 0 500 600

10,000 10,000 0 0

10,000 10,000 0 0

500

Weight (lb)

Node

400 300 200 100

Table 11 Nodes co-ordinates and member grouping of 25 bar 3D truss.

0 Node

x (in.)

1 2 3 4 5 6

37.5 37.5 37.5 37.5 37.5 37.5

y (in.) 0 0 37.5 37.5 37.5 37.5

z (in.)

Group

200 200 100 100 100 100

A1 A2 A3 A4 A5 A6

Member (end nodes)

0

100

200

300

400

500

600

Generations

7

100

100

0

A7

8

100

100

0

A8

9 10

100 100

100 100

0 0

1(1, 2) 2(1, 4), 3(2, 3), 4(1, 5), 5(2, 6) 6(2, 5), 7(2, 4), 8(1, 3), 9(1, 6) 10(3, 6), 11(4, 5) 12(3, 4), 13(5, 6) 14(3, 10), 15(6, 7), 16(4, 9), 17(5, 8) 18(3, 8), 19(4, 7), 20(6, 9), 21(5, 10) 22(3, 7), 23(4, 8), 24(5, 9), 25(6, 10)

Fig. 13. Convergence history for the size and shape twenty-five 3D truss optimisation problem.

Table 13 Normalised weight for the top five topologies of the size, shape, and topology twentyfive 3D truss example. Normalised weight

FA

1st

2nd

3rd

4th

5th

1.000

1.150 (15.3%)

1.488 (25.8%)

1.721 (33.6%)

2.150 (28.62%)

Table 12 Optimum size and shape solution for the 25 bar 3D truss problem. Design variables

Ref. [17]

Ref. [18]

Tang et al. [29]

FA

GA 1

GA 2

GA 3

A1 A2 A3 A4 A5 A6 A7 A8 X4 Y4 Z4 X8 Y8

0.1 1.8 2.3 0.2 0.1 0.8 1.8 3

0.1 0.2 1.1 0.2 0.3 0.1 0.2 0.9 41.07 53.47 124.6 50.8 131.48

0.1 0.2 0.9 0.2 0.2 0.1 0.2 1.1 24.87 62.39 117.88 42.36 129.46

0.2 0.1 1 0.1 0.1 0.1 0.1 1.1 39.327 61.296 115.906 65.477 135.905

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

0.1 0.1 0.9 0.1 0.1 0.1 0.1 1 37.32 55.74 126.62 50.14 136.40

Weight (lb)

546.01

136.2

136.09

130.2

124.94

118.83

Max stress (lb/in.2)

15589.7

15834.32

16043.5

18228.6

18830.23

Max displac. (in.)

0.347

0.3486

0.3395

0.35

0.35

4.4.1. Size and shape optimisation without buckling constraints Past researchers have studied this problem using the GA, including Tang et al. [29] and Wu and Chow [18]. Because the nodal coordinates are continuous and the cross-sectional areas are taken from a set of 30 discrete variables, this problem is also a mixed variable optimisation problem since it deals simultaneously with integer and continuous design variables. The three different GA proposed by Tang et al. [29] have been carried out using a population size P = 40 and 200 generations, resulting in 8000 OFE. To properly compare the FA performance to this GA study, the present study uses n = 10 fireflies and S = 800 searches, resulting in the same 8000 OFE. The best results over 100 runs achieved by the FA have a weight of 75.5 lb, which

Table 14 Optimum size, shape and topology solution for the twenty-five 3D truss problem. Design variables

Ref. [18]

Tang et al. [29]

FA

GA 1

GA 2

GA 3

A1 A2 A3 A4 A5 A6 A7 A8 X4 Y4 Z4 X8 Y8 Weight (lb) Max stress (lb/ in.2) Max displac. (in.)

0.1 0.2 1.1 0.2 0.3 0.1 0.2 0.9 41.07 53.47 124.6 50.8 131.48 136.2 15589.66

0 0.1 1 0 0 0.1 0.1 1.1 38.83 50.62 126.55 50.37 125.63 120.88 18840.45

0 0.1 0.9 0 0 0.1 0.1 1 39.91 61.99 118.23 53.13 138.49 114.74 17353.01

0 0.1 0.9 0 0 0.1 0.1 1.1 39.51 70.18 105.16 55.15 136.27 118.73 21240.45

0 0.1 1.1 0 0 0.1 0.1 0.9 38.50 64.35 112.87 49.13 134.94 116.58 19791.08

0.347

0.35

0.35

0.3494

0.35

is better than that of the reference, as observed in Table 15. The mean value obtained with the FA optimisation scheme is equal to 82.64 lb and the coefficient of variation is 2.96%. The convergence history of the best topology is shown in Fig. 17. 4.4.2. Size, shape and topology optimisation without buckling constraints In addition to the eight discrete size and five continuous configuration variables, all 15 member groups are considered as topology variables. Thus, this is once again a mixed variable optimisation problem. The previously cited references have performed unimodal procedures, while the present paper intends to achieve multimodal benefits. The scheme is capable of finding 75 feasible topologies

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1

6

6

32 8

32

7

8

54

7 54

9 9

24 24

1819 1819

17

23

17

23

15

16 16

15

25

14

25

14

2021

20 21

22 22

Topology A

Topology B

1 6 32 8

7 54

12 9 24 18 19

17

23

13 16

15

25

14 20 21 22

Topology C Fig. 14. Samples of selected topologies determined for the size, shape, and topology twenty-five 3D truss optimisation problem.

(an average over 100 simulations). Table 16 presents the normalised weights of the top five topologies, in which the results are again an average over 100 runs. Table 17 shows the results of the present study compared to those in the literature. Note that the best solution over 100 runs obtained by the present algorithm (74.33 lb) is once again better than the GA approaches. The mean value achieved in the present study is equal to 84.45 lb and the coefficient of variation is 6.49%. Fig. 18 shows three of the topologies typically generated during

the optimisation run, and the convergence history of the best topology is shown in Fig. 19. 4.4.3. Size and shape optimisation with buckling constraints Wu and Chow [18] have studied this problem using the GA and buckling constraints. In this study the member stresses are constrained to be below the Euler buckling stress:

rcr ¼

100EAi 8L2i

:

ð6Þ

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L.F.F. Miguel et al. / Advances in Engineering Software 56 (2013) 23–37

800

600

700

500

500

Weight (lb)

Weight (lb)

600

400 300 200 100 0

400 300 200 100

0

100

200

300

400

500

600

0

360in

2 11

10

13

12

15

14 8

5

800

Fig. 17. Convergence history for the size and shape 15 planar truss optimisation problem without buckling constraints.

Normalised weight 120in

9

FA

4

600

Table 16 Normalised weight for the top five topologies of the size, shape, and topology 15 planar truss without buckling example.

3

7

400

Generations

Fig. 15. Convergence history for the size, shape, and topology twenty-five 3D truss optimisation problem.

1

200

0

Generations

1st

2nd

3rd

4th

5th

1.000

1.121 (8.68%)

1.348 (20.35%)

1.472 (23.18%)

1.575 (24.86%)

6

P

Table 17 Optimum size, shape and topology solution for the 15 planar truss problem without buckling constraints.

Fig. 16. Fifteen-bar planar truss benchmark example.

Design variables Table 15 Optimum size and shape solution for the 15 planar truss problem without buckling constraints. Design variables

Ref. [18]

Tang et al. [29]

FA

GA 1

GA 2

GA 3

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)

1.174 0.954 0.44 1.333 0.954 0.174 0.44 0.44 1.081 1.333 0.174 0.174 0.347 0.347 0.44 123.189 231.595 107.189 119.175 60.462 16.728 15.565 36.645 120.528

1.081 0.954 0.111 1.174 0.539 0.539 0.954 0.22 0.539 0.287 0.539 0.111 0.287 0.539 0.27 134.94 252.439 125.894 100.371 80.724 10.388 0.503 28.517 106.007

0.954 0.954 0.111 1.174 2.697 0.539 0.111 0.111 0.111 0.539 0.111 0.111 0.539 0.539 0.111 139.667 220.678 115.733 106.469 53.01 16.362 12.259 43.689 100.327

1.081 0.539 0.287 0.954 0.954 0.22 0.111 0.111 0.287 0.22 0.44 0.44 0.111 0.22 0.347 133.612 234.752 100.449 104.738 73.762 10.067 1.339 50.402 79.82

0.954 0.539 0.220 0.954 0.539 0.220 0.111 0.111 0.287 0.440 0.440 0.220 0.220 0.270 0.220 114.967 247.040 125.919 111.067 58.298 17.564 5.821 31.465 75.55

Wu and Chow [18] carried out the optimisation using a population size P = 30. However, they did not mention the maximum number of generations. Aiming to maintain the parameters of the previous problem, the present study uses n = 10 fireflies and S = 800 searches, resulting in the same 8000 OFE. The best results over 100 runs achieved by the FA weights 138.06 lb, which is better

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)

Tang et al. [29]

FA

GA 1

GA 2

GA 3

0.954 0.954 0 1.333 0.44 0.44 0 0.111 0 0.539 0 0.111 0.539 0.539 0 139.15 255.03 110.74 104.07 – 5.58 4.5 24.59 80.59

1.081 0.954 0 1.081 0.539 0.539 0 0.111 0 0.347 0.27 0.111 0.539 0.539 0 110.7 225.52 124.76 114.67 – 8.26 0.56 36.77 78.64

1.081 0.539 0 1.081 0.954 0.44 0 0.141 0 0.27 0.27 0.539 0.141 0.44 0 111.85 242.45 104.02 109.22 – 10.82 11.13 48.84 77.84

0.954 0.539 0.141 0.954 0.539 0.287 0.141 0.000 3.813 0.440 0.440 0.220 0.220 0.347 0.141 112.027 247.076 137.514 116.776 50.162 10.905 3.179 48.825 74.33

than reached by the reference, as observed in Table 18. The mean value obtained with the FA optimisation scheme is equal to 154.21 lb and the coefficient of variation is 3.85%. The convergence history of the best topology is shown in Fig. 20.

4.4.4. Size, shape and topology optimisation without buckling constraints In addition to the eight discrete size and five continuous configuration variables, all 15 member groups are considered as topology variables. Eq. (6) is also adopted. Because this problem has not

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L.F.F. Miguel et al. / Advances in Engineering Software 56 (2013) 23–37

Topology A 1 10

Table 18 Optimum size and shape solution for the 15 planar truss problem with buckling constraints.

2

11

12

3

13

Design variables

15

9

6 4

5

Topology B 1 10

2

11

12

3

13

7

14 15

8

4

9

6

5

Topology C 1 10

2 12

11

Wu and Chow [18]

14

7

13

3 14 15

9

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)

FA (8000 OFE)

Size optimisation

Size and shape

2.8 2.8 4.805 2.697 2.697 1.488 1.488 0.111 2.697 2.697 1.333 2.8 1.333 1.488 3.565 – – – – – – – – 483.279

5.952 1.764 1.488 5.952 2.142 1.488 2.8 1.333 0.347 1.081 1.333 1.081 1.333 1.764 1.764 118.891 222.569 108.02 124.068 54.835 0.347 15.516 34.711 402.357

1.174 1.081 0.440 1.764 1.488 1.081 0.111 0.220 1.488 0.174 0.220 0.270 1.333 0.287 1.333 109.559 224.866 103.312 100.435 51.576 17.060 19.022 48.846 138.068

6

4

5

Fig. 18. Samples of selected topologies determined for the size, shape, and topology 15 planar truss optimisation problem without buckling constraints.

1000 800

Weight (lb)

800

Weight (lb)

600

600 400 200

400

0 200

0

200

400

600

800

Generations 0

0

200

400

600

800

Fig. 20. Convergence history for the size and shape 15 planar truss optimisation problem with buckling constraints.

Generations Fig. 19. Convergence history for the size, shape and topology 15 planar truss optimisation problem without buckling constraints.

Table 19 Normalised weight for the top five topologies of the size, shape, and topology 15 planar truss with buckling example. Normalised weight

been studied before, the results obtained could not be compared to the literature. Once again, the present paper intends to achieve multimodal benefits. The scheme is capable of finding 58 feasible topologies (an average over 100 simulations). Table 19 presents the normalised weights of the top five topologies, in which the results are again an average over 100 runs. The best results over 100 runs achieved by the FA have a weight of 125.229 lb, as observed in Table 20. The mean value achieved in the present study is equal to 152.61 lb and the coefficient of variation is 8.69%. Fig. 21 shows three of the topologies typically generated during the optimisation run, and the convergence history of the best topology is shown in Fig. 22.

FA

1st

2nd

3rd

4th

5th

1.000

1.281 (8.76%)

1.663 (18.03%)

1.936 (21.71%)

2.318 (24.16%)

4.5. Further comments In this section, we compare the results and computational cost of the optimisation of four different truss structures, totalizing ten optimisation cases. The summary of this comparison is: in seven cases the FA performed better (3 cases with 10% of computational cost and 5 cases with similar computational cost), in one case the FA performed similar (with 10% of the computational cost), in only one case the FA achieved the second best weight (with similar

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L.F.F. Miguel et al. / Advances in Engineering Software 56 (2013) 23–37

Table 20 Optimum size, shape and topology solution for the 15 planar truss problem with buckling constraints. FA (8000 OFE)

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)

0.954 0.954 0.111 2.142 1.081 1.333 0.111 0.220 1.174 0.440 0.000 0.000 1.764 0.539 0.000 131.996 233.985 115.401 116.888 53.065 13.937 11.605 51.895 125.229

Topology A 1

2 3

10

7

13

8

14

9

6 5

4

Topology B 1

2 3

10

7

13 12

8

14

700 600

Weight (lb)

Design variables

800

500 400 300 200 100

0

200

400

600

800

Generations Fig. 22. Convergence history for the size, shape and topology 15 planar truss optimisation problem with buckling constraints.

Table 21 Summary of the results. Example Deb and Balling Tang Wu and FA Gulati [28] et al. [38] et al. [29] Chow [18] Weight OFE Weight OFE Weight OFE Weight OFE Weight OFE Weight OFE Weight OFE Weight OFE Weight OFE Weight OFE

(lb) 4.1.1

4912.85 (49,500)

(lb) 4.1.2 (lb) 4.2.1 (lb) 4.2.2 (lb) 4.3.1 (lb) 4.3.2 (lb) 4.4.1 (lb) 4.4.2 (lb) 4.4.3 (lb) 4.4.4

2736 (500,000) 196.54 – 192.19 (504,000) 124.94 (6000) 114.74 (6000) 79.82 (8000) 77.84 (8000) 402.36 –

4912.85 (30,000) 2705.16 (50,000) 193.55 (50,000) 191.30 (50,000) 118.83 (6000) 116.58 (6000) 75.55 (8000) 74.33 (8000) 138.07 (8000) 125.23 (8000)

9

6 5

4

Topology C 1 10

2 11

3 7

13

8

14

9

6 4

5

Fig. 21. Samples of selected topologies determined for the size, shape, and topology 15 planar truss optimisation problem with buckling constraints.

computational cost) and in one case the FA was not compared because the example was not taken from the literature. To make this comparison even clearer for the reader, it is detailed in Table 21. These results demonstrate the effectiveness of the FA to deal with optimisation of truss structures.

structures in a single-stage procedure. The results show that the approach is especially suited to mixed variable optimisation problems, which is a typical scenario for such a problem. It is also shown that the presence of mechanisms can be disregarded as possible solutions by checking the positive definiteness of the stiffness matrix. Furthermore, the study performs a multimodal optimisation in that the procedure is capable of providing multiple optimal or near-optimal solutions in each run, resulting in a set of designs at the end of the optimisation stage. This capability is very attractive in practice because it is not always possible to account for constructional aspects or aesthetics in the fitness functions, but the multimodal optimisation allows the engineer to evaluate those options considering external requirements. The effectiveness of the FA at solving the simultaneous size, shape and topology optimisation of trusses is demonstrated through several benchmark problems, the results of which are similar or even better than those reported in the literature, with lower computational costs. These examples emphasise the capabilities of the proposed methodology in this field.

5. Conclusions

Acknowledgment

This paper employs the Firefly algorithm (FA) in the simultaneous optimisation of the size, shape, and topology of truss

The authors gratefully acknowledge the financial support of CNPq and CAPES.

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