Large-scale structural optimization using a fuzzy reinforced swarm intelligence algorithm

Advances in Engineering Software 142 (2020) 102790

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Research paper

Large-scale structural optimization using a fuzzy reinforced swarm intelligence algorithm

T

Ali Mortazavi Graduate School of Natural and Applied Science, Ege University, Izmir, Turkey

ARTICLE INFO

ABSTRACT

Keywords: Large-scale structure problems Metaheuristic optimization algorithms Fuzzy decision mechanism

In contrast with conventional structural optimization benchmark problems real size structures mostly contain a large number of members and their optimal design provide a serious challenging area for optimization methods. In this regard, current study deals with assessing the search performance of the recently developed fuzzy reinforced metaheuristic technique so called Interactive Fuzzy Search Algorithm on the optimization of large-scale structures. This method is a self-adaptive and parameter-free algorithm which applies a dual-module fuzzy decision mechanism to adjust its search behavior during the optimization process. This mechanism employs two nine rule fuzzy modules which permanently monitor the agents updating process and based on the governing conditions of the problem emphasize their exploration or exploitation search behavior. Attained results show that proposed method can adopt itself with the extensive search space of the studied problems. Form both accuracy and stability aspects Interactive Fuzzy Search Algorithm provides promising results on solving largescale structural optimization problems.

1. Introduction Structural optimization can be studied from different aspects, but it broadly concerns with the weight minimization of the structural systems. Along with developments in the computer technologies this class of optimization becomes the subject of many researches’ studies [1–8]. Several of these studies deal with measuring the search capability of different optimization techniques on solving some conventional benchmark problems with limited number of design variables [5,9–14]. However, in the practice, structures consist of much higher number of members. Thus, in the current study to consider more realistic condition, optimization of the large-scale structures is targeted. As the size of structure is raised, the optimization process may require enormous computational effort due to drastically increase in the number of decision variables. Since applied optimization technique plays important role on the solution process, development of more effective optimization methods is one of the significant ongoing research subjects in the field of the structural optimization [15–18]. Generally, optimization techniques are divided into two main groups as gradient and non-gradient based approaches. Gradient based approaches apply the gradient information of the objective function to determine the steps size and search direction specially in proximity of the starting point. These approaches have a fast convergence rate and high accuracy. Despite of their affirmative features, these methods have

two main shortcomings. Firstly, they are very sensitive to the starting point of the process which makes their performance highly dependent on initial running condition [19]. Secondly, they require a continuous (or semi-continuous) objective function and its different order gradients [9]. However, defining such a function for many engineering problems is very difficult or even impossible. To tackle these shortcomings, metaheuristic techniques can be applied as an alternative approach to solve complex optimization problems. Metaheuristic techniques are the non-gradient based approaches which are widely used for solving different types of engineering problems in the last decade [1,9,20–26]. These methods are generally based on the stochastic algorithms inspired from natural events or physical and social rules [27]. One can hierarchically sort some of them as Teaching and Learning Based Optimization (TLBO) models the knowledge transfer process in the class [28], Drosophila food-Search Optimization (DSO) inspired from food search mechanism of the insect with the same name [29], Virus Optimization Algorithm (VOA) imitates the viruses attacking mechanism to the living cells [30], Heat Transfer Search (HTS) models the heats transfer process between environments [31], Multi-Strategy Adaptive Particle Swarm Optimization (MAPSO) which improves the search capability of the original PSO approach [32], Water Wave Optimization (WWO) mimics the water waves phenomena [33], Interactive Search Algorithm (ISA) hybridizes the PSO based approach with a learning mechanism [34] and Butterfly

E-mail address: [email protected]. https://doi.org/10.1016/j.advengsoft.2020.102790 Received 21 December 2019; Received in revised form 26 January 2020; Accepted 21 February 2020 Available online 27 February 2020 0965-9978/ © 2020 Elsevier Ltd. All rights reserved.

Advances in Engineering Software 142 (2020) 102790

A. Mortazavi

Algorithm (IFSA) is employed as an optimizer tool. IFSA is a hybrid metaheuristic algorithm recently developed by the author of the current study [39]. This method utilizes a decision making mechanism included two fuzzy modules to adjust its search behavior according to the current conditions of the optimization problem. These modules permanently monitor the optimization process and adaptively quantify internal parameters for each agent based on its requirements. To comparatively evaluate each agent of the colony, IFSA uses a predefined normalized objective function. The performance of the proposed IFSA already has been verified on different types of typical mathematical and mechanical benchmark problems [39]. However, more complex optimization problems due to their extensive search spaces involve several local optima, so they demand more flexible and compatible search techniques. In this respect, in the current study adaptability and search performance of IFSA are verified on solving the large-scale structural optimization problems. The constrained search domains of these problems can challenge any optimization methods from different aspects. The rest of these manuscript is arranged as follows, in the next section the structural optimization mathematical formulation is given, in Section 2 IFSA method and its corresponding terms are explained, Section 4 is devoted to numeric examples and consequently related conclusion is reported in the final section.

Table 1 Fuzzy rules for tuning the tendency factor. Number of Rule 1 2 3 4 5 6 7 8 9

Input NOFi

τi

Output Δτi

S S S M M M L L L

S M L S M L S M L

LPV SPV ZV SPV ZV SNV ZV SNV LNV

Table 2 Fuzzy rules for tuning the inertia weight. Number of Rule

Input NOFi

ωi

Output Δωi

1 2 3 4 5 6 7 8 9

S S S M M M L L L

S M L S M L S M L

LPV SPV ZV LNV ZV ZV ZV SNV LNV

2. Structural optimization Generally, the structural optimization deals with searching the most proper configuration of a structural system to minimize the weight of the selected load-carrying mechanism. To reach feasible solutions some constraints (e.g. stress and displacement limitations) should be implemented to determine the boundaries of the search space. The objective function for such an optimization problem is defined as below:

Optimization (BO) mimics the butterfly mating mechanism of the butterflies [35]. Two important search behaviors of a metaheuristic technique are exploration and exploitation. Concisely, exploration is defined as the algorithm's ability to find a promising region of the search space, while exploitation designates its ability to perform a finer search inside the spotted region [36]. Establishing a proper balance between these two search behaviors plays an important role in the algorithm search capability. Generally, this aim is met via adjusting some predefined internal parameters [37] by performing a series of sensitivity analyses based on the governing condition of the anticipated problem. However, this process has two main drawbacks, firstly performing the sensitivity analyses depends on the governing conditions of the current problem and adjusting the internal parameters for a specific problem might cause the algorithm to lose its performance on solving other type of problems [38]. Secondly, since a certain problem should be repeatedly solved for different values of the internal parameters, sensitivity analysis is a tedious and time-consuming procedure. To mitigate these shortcomings, in the current study a self-adaptive and parameter-free algorithm so called Interactive Fuzzy Search

f(X) = gi (X)

m LA i=1 i i i

0, i = 1, ...,nc

(1)

where ρ, L and A are the material density, length and cross-sectional area of the ith structural member, respectively. Also, f(X) declares the weight of the structures and X=[x1,x2,...,xd] is the design vector in which d subscript shows the total number of decision variables; g(X) demonstrates the ith constraint of the problem while nc indicates the total number of constraints. 3. Interactive fuzzy search algorithm (IFSA) Interactive Fuzzy Search Algorithm (IFSA) is the parameter-free and self-adaptive metaheuristic optimization technique. This method is a swarm-based approach which has been recently developed by the current author [39]. IFSA utilizes two fundamental search patterns as tracking and interacting to navigate the agents of the colony. In the

Fig. 1. Membership functions for (a) NOFi (b) τiand (c) Δτi. 2

Advances in Engineering Software 142 (2020) 102790

A. Mortazavi

Fig. 2. Membership functions for (a) NOFi (b) ωiand (c) Δωi.

tracking pattern, each agent updates its location considering three main locations in the search space spotted with three certain agents as the best agent (XG), a location stored in the previous best memory of the colony (XP) and the position of the weighted agent (XW). In the interacting pattern, each agent updates its position based on the pairwise knowledge sharing with other agents in the colony. These patterns are mathematically defined in Eq. (2). IFSA applies a combination of these two patterns to guide the agents while the ratio of this combination is determined via its fuzzy decision mechanism. This fuzzy updating pattern is mathematically given in Eq. (3). Updating praradigms: Tracking paradigm t+ 1 Tr vij =

t i . vij +

t P 1ij . x qj

t x ij +

2ij .

tx G j

t xP qj

+ 3ij . txW j

t xP qj

Interacting paradigm t + 1v = 4ij . In ij t + 1v = 4ij . In ij

( txij

t x ) if f (X ) < f (X ) qj i j

( tx

t x ) if f (X ) ij i

qj

(2)

f (Xj)

Updating formulation: t + 1x

ij

t+1 =t x ij + µij .tIn+ 1 vij + µij .Tr vij

(3)

th W in this formulation, txW j is the j component of the weighted agent (X ). The weighted agent is the weighted average of all colony's agents and it is defined as follows: M w P i = 1 c¯i Xi

XW =

where, w ^w c¯iw = (c^i / iM = 1 ci ) w in which c^i =

1 1

max f (XkP ) k M

max f (XkP) k M

f (XiP)

min f (XkP) + 1 k M

i = 1, 2, …, M

(4)

w where M is the population size, c^i is the weighted coefficient of each particle, f (.) returns the problem objective function value, max (f (XkP)) and min (f (X kP)) , respectively are the worst and best

1 k

M

1 k

M

objective values of the colony, ε is a positive small number to prevent division by zero condition, if occurred (e.g. ε = 1E − 6). In Eqs. (2,3), superscripts t and t + 1 are indicate the current and updated condition of the corresponding components, respectively. The number of population and the problem dimension are shown by M and D, respectively. The subscripts i ∈ [1, 2, ..., M] q ∈ [1, 2, ..., M] and j ∈ [1, 2, ..., D]. The xijindicates the jth component of the ith agent. The coefficients ϕkij = Ck.rkijand rijare the random scalars with uniform distribution over interval [0,1] the i and j subscripts respectively show the agent's number and component's order and k = {1, 2, 3, 4}. The acceleration factors are defined as C1 = C3 = C4 = 1and C2 = 2. Fuzzy inertia weight factor (ωi) determines the amount of effect of previous motion of ith agent. Also, μij is the fuzzy decision factor which designated the proper ratio between interacting and tracking paradigms for the current agent during the optimization process. Adjusting the fuzzy

Fig. 3. The flowchart for IFSA.

3

Advances in Engineering Software 142 (2020) 102790

A. Mortazavi

Fig. 4. Schematic of the spatial 582-bar tower.

parameters is enplaned in the next section in details. The parameter of μij and its complement µ¯ ij are defined as below:

Table 3 Parameter setting for applied algorithms. Algorithm

Parameter

Value

PSO [42] TLBO [28] iPSO [43]

Coefficients of acceleration Teaching factor Tendency factor Internal coefficients Tendency factor Internal coefficients –

C1 = C2 = 2 TF = [1,2] α = 0.4 C1 = 2 and C2 = C3 = C4 = 1 τ = 0.4 C1 = 2 and C2 = C3 = 1 –

ISA [34] IFSA [39]

µij = rij. µ¯ ij = 1

(5)

i

µij

(6)

where rij ∈ [0, 1] is uniformly distributed random number and τi ∈ [0, 1]is the fuzzy tendency factor for ith agent. Considering Eq. (6), it is µ¯ ij 1. Applied fuzzy strategy to clear that if 0 ≤ μij ≤ τi then 1 i determine τiand ωiparameters is described in following section. 4

Advances in Engineering Software 142 (2020) 102790

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Table 4 Discrete cross-sections available for the sizing variables of the 582-bar tower problem. W-shape profile list taken from AISC code W27 W27 W27 W27 W27 W27 W27 W24 W24 W24 W24 W24 W24 W24 W24 W24 W24 W24 W21 W21

× × × × × × × × × × × × × × × × × × × ×

178 161 146 114 102 94 84 162 146 131 117 104 94 84 76 68 62 55 147 132

W21 W21 W21 W21 W21 W21 W21 W21 W21 W21 W21 W18 W18 W18 W18 W18 W18 W18 W18 W18

× × × × × × × × × × × × × × × × × × × ×

122 111 101 93 83 73 68 62 57 50 44 119 106 97 86 76 71 65 60 55

W18 W18 W18 W18 W16 W16 W16 W16 W16 W16 W16 W16 W16 W16 W16 W14 W14 W14 W14 W14

× × × × × × × × × × × × × × × × × × × ×

50 46 40 35 100 89 77 67 57 50 45 40 36 31 26 730 665 605 550 500

W14 W14 W14 W14 W14 W14 W14 W14 W14 W14 W14 W14 W14 W14 W14 W14 W14 W14 W14 W14

× × × × × × × × × × × × × × × × × × × ×

3.1. The fuzzy strategy in IFSA

f (Xi) f (XWorst)

f (XG) f (XG) +

(7)

where, NOFi shows the normalized objective function value for ith agent, f(Xi), f(XG)and f(XWorst)respectively indicate the objective function value for ith agent, the best agent and the worst agent in the colony. Also, ς is a very small positive scalar to prevent division by zero condition, if occurred. It is clear from Eq. (7) that for a minimization problem, lower value for NOF designates comparatively better optimal condition. To adaptively adjusting the tunable parameters (i.e. ωiandτi) two fuzzy decision variables as Δωiand Δτiare defined as follows: old i = i + and old + i = i

i

(8)

i

W14 W14 W14 W14 W14 W14 W14 W14 W14 W14 W14 W12 W12 W12 W12 W12 W12 W12 W12 W12

× × × × × × × × × × × × × × × × × × × ×

74 68 61 53 48 43 38 34 30 26 22 336 305 279 252 230 210 190 170 152

W12 W12 W12 W12 W12 W12 W12 W12 W12 W12 W12 W12 W12 W12 W12 W12 W12 W10 W10 W10

× × × × × × × × × × × × × × × × × × × ×

136 120 106 96 87 79 72 65 58 53 50 45 40 35 30 26 22 112 100 88

W10 × 77 W10 × 68 W10 × 60 W10 × 54 W10 × 49 W10 × 45 W10 × 39 W10 × 33 W10 × 30 W10 × 26 W10 × 22 W8 × 67 W8 × 58 W8 × 48 W8 × 40 W8 × 35 W8 × 31 W8 × 28 W8 × 24 W8 × 21

For fuzzy tendency factor (τi), as the first state, the condition that both NOFi and τi are small (S) respectively indicates that the current agent comparatively is in a good condition and its search process is on exploration mode. So, the algorithm should emphasize the exploitation search behavior to perform finer search in the spotted promising area of the search domain, which means Δτishould take large positive value (LPV) to amplify τi. As the second state, the condition that agent's NOFi and τi both are large, reveals that it is in a poor condition and its search process on the exploitation mode. To flee from this poor region the global exploration is required, so τishould be reduced, which means Δτishould take large negative value (LNV). For fuzzy inertia weight (ωi), as the first state, the condition that both NOFi and ωi are small (S) indicates that the agent is in a good condition and the current inertia weight has had an affirmative effect on its search process, so its effect can be even more increased by designating the positive large value (PLV) to Δωi. As the second state, the condition that both NOFi and ωi are large (L) reveals that the agent is in the poor condition and current inertia weight has had an negative effect on its search process so reducing in the effect of inertia weight (ωi) is entailed and subsequently Δωi is selected as the negative large value (NLV). Finally, to give a general scheme of the IFSA method its flowchart is provided in Fig. 3.

Based on the above discussion the fuzzy tendency factor (τi) and fuzzy inertia weight (ωi) play important role on the algorithm's search behavior. To determine the proper values for these parameters, the fuzzy mechanism of IFSA employs a dimensionless value so called Normalized Objective Function (NOFi) to comparatively evaluate each agent. Its formulation is mathematically given as below:

NOFi =

455 426 398 370 342 311 283 257 233 211 193 176 159 145 132 120 109 99 90 82

old

where, τi and τi are the updated and prior values of the fuzzy tendency factor, respectively. Also, ωi and oldωi respectively indicate the updated and prior values of the fuzzy inertia weight. The parameters of Δτi and Δωi designate the amount of changes in the corresponding parameters. The linguistic terms for specifying the values of these terms are defined and tabulated in Tables 1 and 2. In these tables for input membership function, Small, Medium and Large terms are abbreviated with S, M and L, respectively. For output fuzzy memberships, the abbreviation is defined as LNV: Large Negative Value, SNV: Small Negative Value, ZV: Zero Value, SPV: Small Positive Value, LPV: Large Positive Value. Membership functions for both ΔτiandΔωi are provided in Tables 1 and 2, respectively. The corresponding input and output membership functions for the fuzzification-defuzzification process are given in Fig. 1 and 2. It should be noted that the boundaries of these functions are specified based on the fundamental results of sensitivity analyses on different types of problems [39] also the type of membership functions (i.e. linear) is selected based on the results reported in [40]. To clarify the applied fuzzy strategy, two different probable circumstances are assumed and assessed for ith agent.

4. Numeric examples In this section three different design examples are considered to assess the performance of the proposed IFSA algorithm. As the first case the 582-bar tower structure with extended grouping is considered. The sizing variables of this example is selected from discrete set W-shape profiles and the stress limitations are implemented based on the AISC code [41]. The second example is devoted to optimizing the1262-bar tower structure which its sizing variables are selected among a continues search space. Finally, in the last example the 4666-bar truss tower is optimized while its sizing variables similar to the pervious case are taken from continuous search space. All examples are solved for 30 times to prevent any premature convergence. The machine with CORE i7™ CPU with 8 Mb of RAM installed is applied to adopt the analyses. The internal parameters for utilized algorithms in current study are set as below:

5

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A. Mortazavi

Table 5 Optimal results for the 582-bar tower problem with 58 groups. Design variables

Optimal cross-sectional areas (in2) PSO TLBO [9] [9]

iPSO [9]

ISA [9]

IFSA This study

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 Best Volume (m3) Mean Value (m3) Standard Deviation (m3) NSAs

W12 × 22 (6.48) W27 × 102 (30) W8 × 24 (7.08) W12 × 22 (6.48) W16 × 89 (26.2) W8 × 24 (7.08) W8 × 21 (6.16) W18 × 71 (20.8) W8 × 24 (7.08) W8 × 21 (6.16) W16 × 57 (16.8) W8 × 24 (7.08) W12 × 22 (6.48) W16 × 57 (16.8) W8 × 24 (7.08) W8 × 21 (6.16) W12 × 50 (14.7) W8 × 24 (7.08) W8 × 21 (6.16) W16 × 36 (10.6) W8 × 21 (6.16) W10 × 68 (20) W18 × 76 (22.3) W14 × 90 (26.5) W24 × 68 (20.1) W12 × 40 (11.8) W8 × 21 (6.16) W27 × 102 (30) W10 × 22 (6.49) W8 × 21 (6.16) W10 × 88 (25.9) W10 × 22 (6.49) W8 × 21 (6.16) W24 × 62 (18.2) W8 × 21 (6.16) W8 × 21 (6.16) W24 × 68 (20.1) W8 × 21 (6.16) W8 × 21 (6.16) W12 × 30 (8.79) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) 21.096 24.520 5.36 49,500

W8 × 21 (6.16) W24 × 84 (24.7) W8 × 24 (7.08) W8 × 21 (6.16) W14 × 74 (21.8) W8 × 24 (7.08) W8 × 21 (6.16) W18 × 71 (20.8) W8 × 24 (7.08) W8 × 21 (6.16) W12 × 50 (14.7) W8 × 24 (7.08) W8 × 21 (6.16) W12 × 50 (14.7) W8 × 24 (7.08) W8 × 21 (6.16) W12 × 45 (13.2) W8 × 24 (7.08) W8 × 21 (6.16) W16 × 36 (10.6) W8 × 21 (6.16) W10 × 68 (20) W14 × 61 (17.9) W18 × 76 (22.3) W21 × 83 (24.3) W12 × 40 (11.8) W8 × 21 (6.16) W18 × 76 (22.3) W8 × 24 (7.08) W8 × 21 (6.16) W24 × 62 (18.2) W10 × 22 (6.49) W8 × 21 (6.16) W10 × 54 (15.8) W8 × 21 (6.16) W8 × 21 (6.16) W12 × 40 (11.8) W8 × 21 (6.16) W8 × 21 (6.16) W10 × 39 (11.5) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) 19.991 21.115 0.901 18,500

W8 × 21 (6.16) W24 × 84 (24.7) W8 × 24 (7.08) W8 × 21 (6.16) W14 × 74 (21.8) W8 × 24 (7.08) W8 × 21 (6.16) W18 × 71 (20.8) W8 × 24 (7.08) W8 × 21 (6.16) W12 × 50 (14.7) W8 × 24 (7.08) W8 × 21 (6.16) W12 × 50 (14.7) W8 × 24 (7.08) W8 × 21 (6.16) W12 × 45 (13.2) W8 × 24 (7.08) W8 × 21 (6.16) W16 × 36 (10.6) W8 × 21 (6.16) W10 × 68 (20) W18 × 76 (22.3) W12 × 65 (19.1) W24 × 68 (20.1) W12 × 40 (11.8) W8 × 21 (6.16) W14 × 61 (17.9) W10 × 22 (6.49) W8 × 21 (6.16) W21 × 57 (16.7) W10 × 22 (6.49) W8 × 21 (6.16) W10 × 54 (15.8) W8 × 21 (6.16) W8 × 21 (6.16) W16 × 36 (10.6) W8 × 21 (6.16) W8 × 21 (6.16) W12 × 30 (8.79) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) 19.975 21.082 0.501 14,520

W8 × 21 (6.16) W24 × 84 (24.7) W8 × 24 (7.08) W8 × 21 (6.16) W14 × 74 (21.8) W8 × 24 (7.08) W8 × 21 (6.16) W18 × 71 (20.8) W8 × 24 (7.08) W8 × 21 (6.16) W12 × 50 (14.7) W8 × 24 (7.08) W8 × 21 (6.16) W12 × 50 (14.7) W8 × 24 (7.08) W8 × 21 (6.16) W12 × 45 (13.2) W8 × 24 (7.08) W8 × 21 (6.16) W16 × 36 (10.6) W8 × 21 (6.16) W10 × 68 (20) W14 × 61 (22.3) W18 × 76 (20) W21 × 83 (20.1) W12 × 40 (11.8) W8 × 21 (6.16) W18 × 76 (18.3) W8 × 24 (6.49) W8 × 21 (6.16) W24 × 62 (16.8) W10 × 22 (6.49) W8 × 21 (6.16) W10 × 54 (14.7) W8 × 21 (6.16) W8 × 21 (6.16) W12 × 40 (9.71) W8 × 21 (6.16) W8 × 21 (6.16) W10 × 39 (8.79) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) 19.973 20.889 0.489 8360

W12 × 22 (6.48) W10 × 88 (25.9) W8 × 24 (7.08) W12 × 22 (6.48) W16 × 89 (26.2) W8 × 24 (7.08) W8 × 21 (6.16) W18 × 71 (20.8) W8 × 24 (7.08) W8 × 21 (6.16) W12 × 50 (14.7) W8 × 24 (7.08) W12 × 22 (6.48) W12 × 50 (14.7) W8 × 24 (7.08) W8 × 21 (6.16) W12 × 45 (13.2) W8 × 24 (7.08) W8 × 21 (6.16) W16 × 36 (10.6) W8 × 21 (6.16) W10 × 68 (20) W18 × 76 (22.3) W10 × 68 (20) W24 × 68 (20.1) W12 × 40 (11.8) W8 × 21 (6.16) W18 × 76 (22.3) W10 × 22 (6.49) W8 × 21 (6.16) W16 × 57 (16.8) W10 × 22 (6.49) W8 × 21 (6.16) W16 × 57 (16.8) W8 × 21 (6.16) W8 × 21 (6.16) W24 × 68 (20.1) W8 × 21 (6.16) W8 × 21 (6.16) W12 × 30 (8.79) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) 20.501 21.298 0.882 20,000

4.1. A spatial 582-bar truss tower with extended grouping

I The vertical load on each node as −6.75 kips II The horizontal load on each node in x– direction as 1.12 kips III The horizontal load on each node in y– direction as 1.12 kips

In the current case as shown in Fig. 4a 582-bar spatial truss tower weight minimization is considered. The members of this structure are divided into 58 indecent groups [9]. This grouping is done maintaining the symmetry of the structure. This tower is subjected to three different loading conditions as follows:

The sizing variables are selected from discrete set of predefined wshape profiles defined in AISC–ASD list given in Table 3. The lower and upper boundaries for cross-sectional areas are set as 6.16 in2 6

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convergence histories diagrams for selected methods are plotted in Fig. 5. Convergence rates of the different algorithm also show that the fuzzy module by reducing the waste iterations (iterations without improvements) considerably can speed up of the convergence rate of the algorithm. Also, it should be noted that there is no any constraint violation occurred in the achieved optimal solution. Considering accuracy level of the solution, the convergence rate and stability of the process it can be concluded that the fuzzy module of the proposed IFSA works proper to adjust the search behavior of the algorithm. 4.2. A spatial 1262-bar truss tower In this section the 35 story space tower consists of 1262 members, as shown in three different sections in Fig. 6 is investigated as the second example. Maintaining symmetry of the structure the sizing variables are divided into 72 independent groups. The upper and lower bounds for the cross-sectional areas are selected as 100 in2 and 1 in2, respectively. The vertical and horizontal loads on the structure are as follows: Fig. 7 and 8 - VL: The vertical loads are applied as 3, 6 and 9 kips at each node in the first, second and third sections of the tower, respectively. - HL1: The horizontal loads are considered in the X direction as 1.5 and 1 kips at each node on the left and right sides of the tower, respectively. - HL2: The horizontal loads are given as 1 kips in the Y direction at each node on the back side,1 kips in the Y direction at each node on the front side

Fig. 5. Convergence diagrams for spatial 582-bar tower with extended grouping.

(39.74 cm2) and 215.0 in2 (1387.09 cm2), respectively. The nodal displacements in all directions is limited up to 3.15 in (8 cm). Based on the limitations given in AISC–ASD89 the stresses in structural members should be restricted as below: + i

= 0.6Fy i i < 0 i 1

i

=

The allowable displacement for superior nodes of the structure is limited up to 20 in (50.8 cm) in all directions which is about 1/250 of total height of the tower. The load combination for the structure is given as bellow:

0 (9) 2 i

2Cc2

Fy /

5 3

12 23

+ 2E 2 i

3 i

3 i 8Cc

for

for

8Cc3 i

Cc

i

(i) (ii) (iii) (iv)

< Cc (10)

where i and i+ indicate the compressive and tensile stresses, respectively. For compressive members, allowable value for σi− is a function of their slenderness ratio (λ). In this formulation, Cc is the critical slenderness ratio and it is defined as:

Cc =

2 2E Fy

=

k i li ri

(11)

300 for tension members 200 for compression members

loading loading loading loading

condition VL acting alone conditions VL and HL1 acing together conditions VL and HL2 acing together conditions VL, HL1 and HL2 acing all together

This problem is solved via TLBO, iPSO, ISA and IFSA methods and outcomes are reported together with results of non-gradient based simultaneous analysis and design (SAND) method [44] are considered. Obtained optimal values for sizing variables are competitively tabulated in Table 6. None of constraints are violated. It should be noted, due to problem's high number of variables, rather than reporting all methods’ results, the outcomes found by IFSA and SAND methods, as best results, are given in this table while statistical data for all methods are provided in Table 7. Also, the recorded convergence history for optimization process of the current example is provided in Fig. 9. Among the tested metaheuristic algorithms, IFSA can find the lightest structure as 3176,412 in3 through 13,790 number of structural analyses (NSAs). SAND method could find 3176,619 in3 for the current example. Based on the authors’ claim in the reference [44] the SAND technique as the gradient-based method can find “solutions close to the true solutions”. It indicates that in comparison with SAND, as a deterministic method, IFSA by providing an adaptive balance between exploitation and exploration search behaviors has performed a proper level of local search through the promising are of the search space and could spot better optimal solution.

According to AISC-ASD code, for tensile and compressive members, maximum allowable slenderness ratios should be lower or equal with 200 and 300, respectively. Calculation of slenderness and its limitation are provided below: i

The The The The

(12)

in which λi, ri and li designate the slenderness ratio, radius of gyration and length of member for the ith member, respectively. For compression members, in the case of violating slenderness ratio given criterion 12 2E the maximum allowable value of ( 23 2 ) for stress must not be exceeded i

(AISC–ASD, 1989). Table 4 The obtained optimal cross-sectional areas and statistical data attained by IFSA in comparison with the results obtained with other different methods are tabulated in Table 5. As can be seen IFSA provides the best solution among the other techniques. Also, it shows the most stable behavior among the other approaches. Required number of structural analyses (NSAs) indicate that IFSA needs the lowest computational cost. Also, IFSA finds the lightest system. The related

4.3. A spatial 4666-bar truss tower The 62-story truss tower which demonstrated in Fig. 9 is studied as the last example. This structure consists of 4666 members which form an inner and outer part of this structure. For this example, by considering symmetry of the structure the sizing variables are put into 238 7

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Fig. 6. Schematic for 1262-bar truss dome structure.

independent groups. The lower and upper limits for sizing variables are set as 1 in2 and 300 in2, respectively. Three loading conditions are defined for this structure below:

The displacement is limited to 37.5 in (92.25 cm) in all directions for the nodes of the top level of the tower. This limitation is nearly equal to 1/250 height of the tower. The structure is subjected to the following loading combinations:

- VL: The vertical loads are applied 6 kips at each node of the tower, respectively. - HL1: The horizontal loads are considered in the X direction as 1 kips at each node on both the left and right sides of the tower. - HL2: The horizontal loads are considered in the Y direction as 1 kips at each node on both the back and front sides of the tower.

(i) The loading condition VL acting alone (ii) The loading conditions VL and HL1 acing together (iii) The loading conditions VL, HL1 and HL2 acing all together This problem is solved by TLBO, iPSO, ISA and IFSA methods in the current study. Table 8 tabulates the optimal values accomplished by 8

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Fig. 7. Grouping for 1262-bar truss structure.

IFSA method. In the reported optimal solutions any constraint is not violated. It should be noted that due to excessive number of design variables, just the optimal variables obtained by IFSA (as the best results) are reported in this table. The statistical results for all verified methods is provided in Table 9. As deterministic approach SAND method finds the optimal structure volume as 21,387,963 in3 while IFSA can find the lighter structure with volume of 21,378,918 in3. Taking into account that the SAND is a gradient-based method and based on its authors’ claim it can find the exact (or very close to exact) global optimum, IFSA, as a non-gradient based method, can balance the search behavior of search process and perform proper local search inside the promising region the search domain. For more clarity, Fig. 10 provides the convergence history of optimization process for all tested methods. Taking into account the number of decision variables of this example, acquired standard deviation indicates that among all tested methods IFSA for the sake of its fuzzy decision mechanism shows the highest stability on the finding optimal solution.

Fig. 8. Optimization history for 1262-bar tower.

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Table 6 Optimal results for 1262-bar tower. Group no.

Sizing variables SAND [44]

IFSA This study

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

1 1 5.28641 2.9728 1 19.38372 4.51468 1 44.70364 6.00115 19.06477 15.29749 44.57285 9.3913 13.45314 36.94019 3.73817 7.78929 1 49.79923 4.30263 10.48146 1 63.70566

1 1 5.28605 2.9719 1 19.38446 4.51416 1 44.70402 5.99883 19.06435 15.2963 44.57104 9.39101 13.45284 36.94017 3.73715 7.78906 1 49.79844 4.30215 10.47485 1 63.70491

Group No.

Sizing variables SAND [44]

IFSA This study

25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

4.93936 22.66435 1 77.58933 5.70887 42.12632 61.93681 86.67255 3.162 32.81444 41.8811 28.22154 28.22154 30.63449 88.81648 5.20451 6.71387 5.19254 1 1 100 5.12347 3.96092 5.03295

4.93827 22.6635 1 77.58538 5.70531 42.12469 61.93686 86.67205 3.16194 32.81171 41.8812 28.22104 28.22104 30.63337 88.81606 5.20445 6.71374 5.18729 1 1 99.99993 5.12208 3.96098 5.03322

SAND [44]

TLBO This study

iPSO

ISA

IFSA

Best volume (in3) Mean volume (in3) OFEs Std. Dev. (in3)

3176,619 – – –

3999,320 4100,259 17,180 15,842

3984,985 4109,722 17,713 16,589

3856,666 3906,187 15,470 10,589

3176,412 3181,111 13,790 5254

Sizing variables SAND [44]

IFSA This study

49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72

1.57496 1 100 5.1591 3.99833 5.03982 1.51319 1 100 5.23421 2.7808 5.04361 1.20899 100 5.50973 10.70632 5.2288 30.16244 29.49249 100 1 100 15.14065 1

1.57465 1 100 5.15909 3.99856 5.0372 1.51305 1 99.99985 5.23395 2.78087 5.04309 1.20732 99.99992 5.50958 10.70514 5.22423 30.16348 29.49454 99.99958 1 99.99982 15.13963 1

outcomes are compared with a high accuracy deterministic method as simultaneous analysis and design (SAND). Achieved numerical results firstly, show that the accuracy of the solutions is improved even in comparison with the cited high precise deterministic method. Secondly, the convergence rate of the process is comparatively increased. Thirdly, the required number of structural analyses (NSAs) show that IFSA, thanks to its fuzzy adjusting module, by dropping the number of waste iterations (i.e. iterations without improvement) considerably reduces the required computational cost. Finally, considering the statistic results IFSA by providing comparatively lower standard deviation index shows proper stability on finding optimal solutions. Taking all of these features into account, IFSA via putting forward an adaptive search behavior yields promising results on solving the large-scale structural optimization problems.

Table 7 Statistical results for 1262-bar tower. Result

Group no.

5. Conclusion The current study deals with assessing the search capability of the recently developed fuzzy reinforced metaheuristic technique so called Interactive Fuzzy Search Algorithm (IFSA) on the optimization of the large-scale structures. The proposed IFSA is the fuzzy reinforced selfadaptive and parameter-free metaheuristic algorithm. It uses a compound search strategy consist of two navigation patterns as tracking and interacting. In the tracking pattern, agents try to enhance their locations employing some specific certain agents in the colony, while through the interacting pattern agents try to improve their positions via a pairwise knowledge sharing mechanism. Two nine-rule fuzzy modules of IFSA permanently monitor the search process and try to balance the exploration and exploitation search behaviors of the algorithm. The modules employ a predefined normalized objective function (NOFi) to evaluate each agent. The performance of the IFSA method is tested on the optimal design of three high-rise tower structures. The search spaces of these structures are considerably larger compared to the conventional benchmark problems also they contain both discrete and continuous decision variables. So, they not only challenge the proposed method on handling the extensive search domains but also cause the adaptability of IFSA on handling different types of variables (i.e. discrete and continuous) to be verified. To provide more comparative results selected cases are solved via three different metaheuristic approaches. Also, the attained

Author statement Authorship contributions: Please indicate the specific contributions made by each author (list the authors’ initials followed by their surnames, e.g., Y.L. Cheung). The name of each author must appear at least once in each of the three categories below. Category 1: Conception and design of study: A. Mortazavi; acquisition of data: A. Mortazavi; analysis and/or interpretation of data: A. Mortazavi. Category 2: Drafting the manuscript: A. Mortazavi; revising the manuscript critically for important intellectual content: A. Mortazavi; Category 3: Approval of the version of the manuscript to be published (the names of all authors must be listed): A. Mortazavi; Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Fig. 9. Schematic for 4666-bar truss dome structure.

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Table 8 Optimal design for 62-story tower. No.

Area

No.

Area

No.

Area

No.

Area

No.

Area

No.

Area

No.

Area

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

1 1 135.9613 300 1 300 1 1 1 212.2404 1 190.9517 1 1 300 300 4.20652 2.19102 3.53601 150.3467 32.65928 7.94425 1.53994 1 1 300 300 11.8165 9.8832 1 31.11265 11.38323 4.81419 1.72206

35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68

1 1 300 300 13.32064 13.65558 1 8.19403 1 1 1 1 1 300 277.6568 13.31618 14.04063 1 7.3662 1 1 1 1 1 300 242.5512 13.59717 13.22345 1 6.64486 1 1 1 1

69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102

1 300 205.4284 13.6293 12.49536 1 5.90102 1 1 1 1 1 300 167.8867 13.32577 12.00102 1 5.14393 1 1 1 1 1 300 131.3807 12.784 11.57698 1 4.38233 1 1 1 1 1

103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136

300 97.25517 12.066 11.13599 1 3.62127 1 1 1 1 1 300 66.84142 11.20017 10.66306 1 2.8621 1 1 1 1 1 300 41.53437 10.26226 10.13064 1 2.10454 1 1 1 1 1 288.2712

137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170

22.51553 9.42072 9.53328 1 1.34889 1 1 1 1 1 255.7447 10.06617 8.76217 8.94167 1 1 1 1 1 1 1 216.5244 3.36454 8.27004 8.31723 1 1 1 1 1 1 1 173.6193 1

171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204

7.76944 7.68716 1 1 1 1 1 1 1 131.0715 1 7.12455 7.00419 1 1.3822 1 1 1 1 1 92.38851 1 6.34854 6.23823 1 2.10296 1 1 1 1 1 58.68445 1 5.45939

205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238

5.3662 1 2.80499 1 1 1 1 1 30.72878 1 4.36836 4.26954 1 3.50105 1 1 1 3.19165 2.46887 11.20995 1 2.769 2.77823 1 4.06963 1 1.18099 1.07516 34.05819 4.09933 1 8.03458 2.00794 1

Table 9 Statistical results for 4666-bar tower. Sizing variables

SAND [44]

TLBO This study

iPSO

ISA

IFSA

Best volume (in3) Mean volume (in3) OFEs Std. Dev. (in3)

21,387,963 – – –

26,008,542 26,113,952 26,290 38,279

25,917,009 26,439,196 24,185 536,852

25,310,333 25,922,724 20,585 112,587

21,378,918 21,496,622 16,230 37,001

Acknowledgements All persons who have made substantial contributions to the work reported in the manuscript (e.g., technical help, writing and editing assistance, general support), but who do not meet the criteria for authorship, are named in the Acknowledgements and have given us their written permission to be named. If we have not included an Acknowledgements, then that indicates that we have not received substantial contributions from non-authors. References [1] Le D.T., Bui D-K, Ngo T.D., Nguyen Q.-.H., Nguyen-Xuan H. A novel hybrid method combining electromagnetism-like mechanism and firefly algorithms for constrained design optimization of discrete truss structures. Comput Struct. 2019;212:20–42. [2] Camp C.V., Farshchin M.Design of space trusses using modified teaching–learning based optimization. Eng Struct. 2014;62–63:87–97. [3] Hasançebi O., Teke T., Pekcan O. A bat-inspired algorithm for structural optimization. Comput Struct. 2013;128:77–90. [4] Degertekin S.O., Hayalioglu M.S. Sizing truss structures using teaching-learningbased optimization. Comput Struct. 2013;119:177–88. [5] Sonmez M.Artificial bee colony algorithm for optimization of truss structures. Appl Soft Comput. 2011;11:2406–18. [6] Gomes H.M.Truss optimization with dynamic constraints using a particle swarm algorithm. Expert Syst Appl. 2011;38:957–68.

Fig. 10. Optimization history for 4666-bar tower. 12

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