A modified symbiotic organisms search (mSOS) algorithm for optimization of pin-jointed structures

Accepted Manuscript Title: A modified symbiotic organisms search (mSOS) algorithm for optimization of pin-jointed structures Author: Dieu T.T. Do Jaehong Lee PII: DOI: Reference:

S1568-4946(17)30484-2 http://dx.doi.org/doi:10.1016/j.asoc.2017.08.002 ASOC 4392

To appear in:

Applied Soft Computing

Received date: Revised date: Accepted date:

14-4-2017 31-7-2017 1-8-2017

Please cite this article as: Dieu T.T. Do, Jaehong Lee, A modified symbiotic organisms search (mSOS) algorithm for optimization of pin-jointed structures, (2017), http://dx.doi.org/10.1016/j.asoc.2017.08.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Highlights •

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trade-off between exploration and exploitation. Seven benchmark pin-jointed optimization problems are tested to verify the effectiveness and

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robustness of the proposed algorithm. The mSOS significantly improves the convergence speed and the solution accuary.

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A modified symbiotic organisms search (mSOS) algorithm for optimization of pin-jointed structures, i.e, truss ones with discrete design variable and tensegrity ones with continuous design variables, is presented. Five modifications in all of the phases of the original SOS are performed to achieve a better

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Graphical abstract

mSOS

X best is replaced by X k

rand(-1,1) is replaced by rand(0.4,0.9)

Parasitism phase

an elitist selection technique

the exploitation ability

enhance the solution accuracy

improve the convergence rate

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the exploration ability

is eliminated

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BF1=BF2=1

Commensalism phase

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Mutualism phase

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pin-jointed structures

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A modified symbiotic organisms search (mSOS) algorithm for optimization of pin-jointed structures Dieu T. T. Do1 , Jaehong Lee∗

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Department of Architectural Engineering, Sejong University, 209 Neungdong-ro, Gwangjin-gu, Seoul 05006, Republic of Korea

Abstract

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The paper introduces a modified symbiotic organisms search (mSOS) algorithm to optimization of pin-jointed structures including truss and tensegrity ones. This approach is refined from the original SOS with five modifications in the following three phases: mutualism, commensalism and parasitism. In the mutualism one, benefit factors are suggested as 1 to equally represent the level of benefit to each organism, whilst the best organism is replaced by a randomly selected one to increase the global search capability. With the aim of improving the convergence speed, randomly created coefficients in the commensalism phase are restricted in the range [0.4, 0.9]. Additionally, an elitist technique is applied to this phase to filter the best organisms for the next generation as well. Finally, the parasitism phase is eliminated to simplify the implementation and reduce the time-consuming process. To verify the effectiveness and robustness of the proposed algorithm, five examples relating to truss weight minimization with discrete design variables are performed. Additionally, two examples regarding minimization a function of eigenvalues and force densities of tensergrity structures with continuous design variables are considered further. Optimal results acquired in all illustrated examples reveal that the proposed method requires fewer number of analyses than the original SOS and the DE, but still gaining high-quality solutions. Furthermore, the mSOS also outperforms numerous other algorithms in available literature in terms of optimal solutions, especially for problems with a large number of design variables. Keywords: Optimization; Modified symbiotic organisms search (mSOS); Pin-jointed structures; Truss structures; Tensegrity structures. 1. Introduction

Structural optimization aims to design structures under certain constraints to achieve better behavior and have a proper manufacturing cost. Due to its benefits, this field has received considerable attention of many researchers during the past decades. Various types of structures in many real engineering problems have been studied. Among them, pin-jointed structures consisting of truss [1] and tensegrity [2–7] ones have been frequently examined and considered as benchmark problems in numerous literature. A large number of optimization techniques have thus been proposed and applied successfully. In general, those techniques can be categorized into the following two major groups: (i) gradient∗ 1

Corresponding author. E-mail: [email protected] E-mail: [email protected]

Preprint submitted to Applied Soft Computing

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based methods, and (ii) non-gradient-based methods. Several typical approaches of the first group can be listed as optimality criterion (OC) [8, 9], sequential linear programming (SLP) [10–12], sequential quadratic programming (SQP) [13] and force method [14, 15], etc. Although the convergence rate of these approaches is fairly fast, a requirement of sensitivity analyses of the objective and constraints is always compelled. Its performances regarding mathematical analyses are quite expensive and complex, even impossible in many cases. Furthermore, obtained solutions are often trapped at local regions since the search capability focuses on only derivative information provided from sensitivity analyses. In order to overcome the above limitations, non-gradient-based methods in the second group which are also known as metaheuristic approaches have been developed. Several methods are utilized to optimize truss structures such as genetic algorithm (GA) [16, 17], particle swarm optimization (PSO) [18], mine blast algorithm (MBA) [19], harmony search (HS) [20], flower pollination algorithm [21], artificial bee colony algorithm (ABC) [22], differential evolution (DE) [23, 24], adaptive dimensional search (ADS) [25], firefly algorithm (FA) [26], symbiotic organisms search (SOS) [27], and several improved versions of metaheuristic algorithms [18, 24, 28–39], and so forth. Furthermore, several algorithms are employed to search form-finding of tensegrity structures that can be listed as GA [40–44], different evolution (DE) and modified differential evolution (mDE) [45], etc. Because stochastic searching techniques are utilized to randomly select candidate solutions in a given domain, sensitivity analyses are eliminated. As a result, a global optimal solution can be acquired without requiring much mathematical knowledge. Nevertheless, the convergence speed is relatively slow, and thus the time-consuming process is high. As can be seen that most of the aforementioned approaches deal with only continuous design variables. Therefore, they often encounter difficulties in meeting the essential requirements of practical engineering problems that take cross-section areas of truss members as discrete variables based on a certain design standard or pre-engineered products of manufacturers. Lee et al. [20] reported a technique to tackle such an issue, yet rounded-off solutions may be infeasible when the number of variables increases. Among the foregoing approaches, symbiotic organisms search (SOS) first developed by Cheng and Prayogo [27] for continuous design variables has been showed its good performance in engineering applications. This algorithm is based on the symbiotic interaction between organisms in the ecosystem and is free from turning parameters, and thus its stability is enhanced significantly. Additionally, it is also effective in finding a global optimal solution that has illustrated via numerous recently published studies. For instance, Panda et al. [46] successfully applied the SOS to multi-objective constrained optimization problems. The authors indicated that the SOS outperforms several other methods such as multi-gradient explorer (MGE) and multi-gradient pathfinder (MGP) in [47]. Abdullahi et al. [48] improved the SOS to tackle discrete design variables for optimal scheduling of tasks on cloud resources. Obtained results of that work revealed that the SOS is better than the PSO algorithm. Ye et al. [49] solved the capacitated vehicle routing problem by using the SOS algorithm with six newly developed versions. Several remarkable applications of the SOS to other problems could be found in [50–52]. However, analogously to other non-gradient-based algorithms, the SOS still demands a significant amount of computational cost in searching a global optimal solution, especially for truss optimization problems with a large number of discrete design variables. Accordingly, the simultaneous improvement on the convergence speed for the SOS with discrete design variables, but still ensuring the quality of solutions is a concern that has not been yet dealt with so far. Moreover, to the best knowledge of the authors, no paper 2 Page 4 of 46

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regarding its application to tensegrity structures has published. In this paper, a modified symbiotic organisms search (mSOS) algorithm is proposed for optimization of pin-jointed structures, namely for truss structures with discrete design variables, and tensegrity structures with continuous ones. The mSOS is modified from the original SOS algorithm with five modifications in all three phases including mutualism, commensalism and parasitism. In order to maintain a better balance between global and local search abilities, both mutualism and commensalism are refined. In particular, the mutualism phase modified aims to enhance the global search capability and restrict local optimal solutions. For this purpose, benefit factors are restricted in the range [0.4, 0.9] and the interaction between organisms is built based on randomly chosen organisms instead of the best organism. Moreover, by narrowing the search range of randomly generated coefficients, and applying an elitist selection technique [53] to the commensalism phase, the local search capability to increase the convergence speed is improved. Consequently, the parasitism phase is eliminated to simplify the optimization procedure and reduce the computational cost. The performance and robustness of the proposed mSOS algorithm are demonstrated through seven numerical examples covering truss and tensegrity structures. Obtained results in all considered examples indicate that the mSOS is superior to the SOS and several other methods in existing literature in terms of the quality of solution and convergence speed. Moreover, the proposed method not only inherits the advantages of the SOS but also is simpler than its original.

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2. Problem statement

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2.1. Truss optimization with discrete design variables The problem aims at minimizing the weight of a structural truss so that constraints on its behavior and design variables are satisfied. For this purpose, cross-section areas of structural truss members are considered as discrete variables that are often designed according to a given set of datum based on a certain standard. This problem can be mathematically stated as follows Minimize: weight (A) = A

e P

ρi li Ai , i = 1, 2, ..., e

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subject to: δmin ≤ δj ≤ δmax , j = 1, 2, ..., n σmin ≤ σi ≤ σmax , i = 1, 2, ..., e σkb ≤ σk ≤ 0, k = 1, 2, ..., nc Ai ∈ S = {A1 , A2 , ..., Ad }

(1)

where weight (A) is the objective function; A is the design variable vector including member cross-section areas Ai ; ρi and li are the material density and the length of the i th member, respectively; δj and σi are the jth nodal displacement and ith member stress, respectively; σkb is the allowable buckling stress in the ith compression member; S is an available discrete set of cross-section areas; e and n are the total number of members and nodes in the truss, respectively, and nc is the number of members subjected to compression. 2.2. Tensegrity optimization with continuous design variables The purpose of the problem is to find an equilibrium configuration (as known as formfinding or shape finding) of a tensegrity structure so that a function of eigenvalues and

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force densities is minimized. A mathematical statement of this problem is expressed in detail as follows Minimize: q

αβ,

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Subject to qiL ≤ qi ≤ qiU , i = 1, 2, ..., b,

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in which

i=1 b X j=1

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α=

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where αβ is the objective function. According to Koohestani [41], α denotes the sum of the first (d + 1) smallest eigenvalues of the force density matrix and its minimum value is zero; β stands for the variable regarding the force density qi ; q is the force density vector including continuous design variables, and qiL and qiU are the lower and upper bounds of the force density of the ith element, respectively. In order to transform a certain constrained optimization problem into a corresponding unconstrained problem, the non-death-penalty method which is one of the most frequently used constraint-handling approaches [54] is adopted in this work. Consequently, the above two problems can be rewritten in terms of the following expression, Xneq h Xnin 
i2 2 max 0, |hs | −  , max (0, gr ) +λ Minimize : fpenalty (A) = weight (A)+λ s=1 r=1 (4) th th where gr and hs are the r inequality and s equality constraints, respectively; nin and neq are the number of inequality and equality constraints, respectively; λ is the penalty parameter and set to be 105 ;  is the small positive tolerance for equality constraints and chosen as 10−4 in this study.

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3. The symbiotic organisms search (SOS) algorithm The symbiotic organisms search (SOS) algorithm is a population-based metaheuristic approach which was first proposed by Cheng and Prayoyo [27]. This algorithm simulates the symbiotic interaction schemes between organisms to survive and propagate in an ecosystem, where each organism is considered as a candidate solution in a search space. The effectiveness and robustness of this method are demonstrated for both benchmark and real-word problems [27, 46, 49, 55]. For a more detailed discussion, the interested readers are encouraged to consult the above literature. The SOS algorithm randomly initializes N organisms to generate the first ecosystem. New solutions are then updated in turn by using the mutualism, commensalism and parasitism phases. For subsequent iterations, only the above three phases are repeatedly done until satisfying stopping criteria to find the best organism. An overview of the algorithm is concisely introduced in the next subsequent sub-sections. 3.1. Mutualism phase A symbiotic relationship between two distinct species that produces individual benefits from the synergy is the so-called mutualism. Xi and Xj represent the i th and j th 4 Page 6 of 46

organisms of the ecosystem, respectively. In which, Xj is a randomly chosen organism in the ecosystem. Xi interacts with Xj to create new candidate solutions as follows Xinew = Xi + rand (0, 1) × (Xbest − Mutual Vector × BF 1) ,

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(6) Xjnew = Xj + rand (0, 1) × (Xbest − Mutual Vector × BF 2) , X i + Xj Mutual Vector = , (7) 2 where rand(0, 1) is a uniformly distributed random number in the range [0,1]; Xbest is the best organism in the ecosystem which represents the highest degree of adaption; BF 1 and BF 2 are the benefit factors randomly generated as either 1 or 2, these factors symbolize the level of benefit to each organism, and Mutual Vector expresses the relationship characteristic between two organisms Xi and Xj . Subsequently, Xinew and Xjnew are compared with Xi and Xj to choose the most fit organism in each pair, respectively. In this phase, new organisms are generated based on the best organism Xbest . This helps to enhance the local search ability or the exploitation capability.

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3.2. Commensalism phase Commensalism designates a symbiotic relationship between two distinct species in which one benefits and the other is unaffected. In this phase, a new organism is created by the interaction between two organisms Xi and Xj , where Xj is randomly chosen from the ecosystem. Since Xi receives benefits from the interaction while Xj is not, only an organism is newly created in this phase and is performed as follows:
Xinew = Xi + rand (−1, 1) × Xbest − Xj , (8)

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where rand (−1, 1) is a uniformly distributed random number in the range [−1, 1]. It can be seen in Eq. (8) the new candidate solution Xinew is also determined based on the best organism Xbest at the current iteration to increase its survival advantage. Therefore, this phase contributes to the exploitation ability of the algorithm as well. Finally, the objective function values of Xi and Xinew are compared to choose a better one. 3.3. Parasitism phase Parasitism denotes a symbiotic relationship between two distinct species in which a specie receives benefits from the interaction while the other is harmed. Parasite Vector representing a parasite organism is a duplication of Xi . Some components of the original Parasite Vector are then randomly modified to create a new candidate solution. A different organism Xj randomly selected from the ecosystem symbolizes a host of the parasite. If the objective function value of Parasite Vector is better than those of Xj , Xj is replaced by Parasite Vector. Otherwise, Xj is kept in the ecosystem. The parasitism phase plays an important role in improving the global search ability or the exploration capability of the algorithm. As previously mentioned in sub-sections 3.1 and 3.2, the mutualism and commensalism phases contribute to the exploitation capability of the algorithm. Accordingly, in order to maintain a balance between the exploration and exploitation abilities, the parasitism phase is required to guarantee the solution accuracy in the original SOS algorithm. Additionally, provided that this phase is removed from the optimization procedure, the SOS cannot yield a good swarm behavior that will be numerically proved in sub-section 5.1.1 5 Page 7 of 46

4. The modified symbiotic organisms search (mSOS) algorithm

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As indicated by Das et al. [56], the balance between exploration and exploitation abilities significantly affects the success of most of the population-based optimization algorithms. In which, the exploration capacity stands for the global search ability that strongly affects the accuracy of obtained optimal solution. The exploitation capability characterizes the local search ability that plays an important rule in impacting on the convergence of the optimization algorithm. Obviously, if the exploration capacity is superior to the other, a global optimal solution can be achieved, yet the convergence is slow. This is due to the fact that the algorithm must require a remarkable amount of computational cost for seeking an optimal solution in a given whole domain. Conversely, the algorithm converges quickly, but optimal solutions may occur. Therefore, provided that the above two abilities are adjusted to gain a better balance, the solution accuracy and the convergence speed can be obtained simultaneously. Although the original SOS algorithm is good at the global search capability, the limitation on the computational cost is existing. As observed, both mutualism and commensalism phases improve the exploitation ability of the algorithm, whilst the parasitism phase contributes to the exploration capability. Nevertheless, the last phase requires a considerable amount of computational cost to sever its search process. If the parasitism phase is eliminated to save computational cost, the SOS approach is easily trapped at local solutions. Therefore, the other phases must be refined to preserve the balance between exploitation and exploration capabilities. To overcome the above shortcomings of the original SOS approach, a modified symbiotic organisms search (mSOS) algorithm is proposed in this study. Five modifications in all of the three phases are described in detail in subsequent sub-sections.

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4.1. Modification of the mutualism phase The global search ability or the exploration capability is reinforced in this phase to find a global optimal solution. Provided that a new solution is updated relied on the best organism, the convergence rate often speeds up; however, local solutions may occur. Therefore, in order to improve the exploration ability, the best organism in Eqs. (5) and (6) is replaced by Xk which is randomly selected in the ecosystem so that Xi 6= Xj 6= Xk . This first modification helps to extend the search space and improve the diversity of the ecosystem. Accordingly, the proposed method can achieve the global optimal solution with high accuracy. Eqs. (5) and (6) can be now rewritten as follows Xinew = Xi + rand (0, 1) × (Xk − Mutual Vector × BF 1) ,

(9)

Xjnew = Xj + rand (0, 1) × (Xk − Mutual Vector × BF 2) ,

(10)

where Mutual Vector is defined as Eq. (7). As presented in the original SOS algorithm, the participating organisms will receive the partial or full benefit from their interaction, and thus two benefit factors BF 1 and BF 2 are randomly chosen as either 1 or 2. Obviously, if an organism fully receives benefits from the interaction, i.e. BF 1 = BF 2 = 2, the terms of Mutual Vector × BF 1 and Mutual Vector × BF 2 in Eqs. (9) and (10) will produce a larger diversity for the relationship characteristic between organism Xi and Xj compared with those of BF 1 = BF 2 = 1. This leads to a larger diversity for (Xk − Mutual Vector × BF 1) and (Xk − Mutual Vector × BF 2) as well. As a result, new solutions of Xinew and Xjnew may be out of feasible regions that increase the deviation of the objective function between the best organism and the whole ecosystem. Consequently, the stability of the 6 Page 8 of 46

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algorithm based on the evaluation of standard deviation decreases, and the algorithm also demands many more numbers of analyses to converge a global optimal solution. Therefore, in this study, each organism will receive the same part of benefits from the interaction. In other words, benefit factors BF 1 and BF 2 are fixed to be 1. Note that if BF 1 and BF 2 are randomly selected as either 1 or 2, the stability and computational cost of the approach will have intermediate results in comparison with the foregoing two cases. A detailed examination of these remarks is proven in sub-section 5.1.1. Note that apart from the above two changes, the rest of this phase is executed in the same manner as the original SOS.

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4.2. Modification of the commensalism phase As discussed above, the exploration ability was enhanced in the modified mutualism phase. Therefore, in order to achieve a better balance between exploration and exploitation abilities, two modifications are proposed in this phase to improve the exploitation ability. It is recognized that rand(−1, 1) coefficient in Eq. (8) plays an important role in controlling the convergence speed of optimization process. Its search range is given in the relatively wide interval from -1 to 1, this causes the search space of the current ecosystem to be enlarged further. As a result, the convergence speed is slow. To handle this issue, a narrower range in [0.4, 0.9] is suggested to save the computational cost, yet still ensuring the accuracy of optimal solutions. Its efficiency is demonstrated in sub-section 5.1.1. Accordingly, Eq. (8) in the original SOS is rewritten as
Xinew = Xi + rand (0.4, 0.9) × Xbest − Xj . (11)

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In the original commensalism phase, each organism Xinew is compared with the preinteraction organism Xi to choose a better organism for the next step. It is clear that the worse organism in each pair is not selected; however, it can be better than the other ones in the whole current ecosystem. Therefore, several candidate organisms can be neglected and the convergence speed is reduced. In order to overcome these drawbacks, an elitist selection technique first proposed by Padhye et al. [53] is applied to this phase to choose the best organisms for the next ecosystem. Furthermore, this technique has been sucessfully applied to numerous studies [24, 36, 38]. This scheme is succinctly summarized as follows. Firstly, the new organisms Xinew is combined with the old ones Xi to create a new ecosystem including 2N organisms. Next, only N best organisms are filtered to form a new one for the performance of the next ecosystem. Obviously, this organism inherits candidate solutions that can be ignored if the original selection way is done. Accordingly, the exploitation ability and convergence speed are improved due to the decreased search space. 4.3. Modification of the parasitism phase According to the original SOS, the parasitism phase is effective in addressing the exploration ability. Nonetheless, this phase causes higher computational cost due to the widen search space. This is because a new candidate solution Parasite Vector is generated by replacing some random elements of the organism Xi with other ones. Parasite Vector is then compared to another organism which is randomly selected in the ecosystem. However, with the afore-implemented modifications in the first two phases of the mSOS, the exploration and exploitation abilities are balanced. Therefore, the parasitism phase is removed to simplify the performance of the mSOS and save the computational time without

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the loss of the solution accuracy. Accordingly, the convergence speed of the mSOS is improved as well, yet still achieving a good swarm behavior as numerically investigated in sub-section 5.1.1. The whole procedure of the proposed algorithm can be summarized as follows: The main procedure of the proposed algorithm (mSOS)

(12)

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1 : Generate the initial ecosystem 2 : Evaluate the fitness for each organism in the ecosystem 3 : while delta > tolerance or M axEval is not reached do 4: for i = 1 to N do % Mutualism phase : 5 Select two organisms randomly, Xj , Xk , where j 6= k 6= i 6: Define Mutual Vector and BF 1 = BF 2 = 1 7: Xinew = Xi + rand (0, 1) × (Xk − Mutual Vector × BF 1) 8: Xjnew = Xj + rand (0, 1) × (Xk − Mutual Vector × BF 2) 9: Evaluate the fitness of the new solutions : 10 Accept the new solution if the fitness is better % Commensalism phase 11 : Define Xbest and select one organism randomly, Xj , where j 6= i
12 : Xinew = Xi + rand (0.4, 0.9) × Xbest − Xj 13 : Evaluate the fitness of the new solution 14 : end for % An elitist technique 15 : Eco consists of Xi , (i = 1, 2, ..., N ) Eco new consists of Xinew , (inew = 1, 2, ..., N ) Assign Total Eco = Eco ∪ Eco new 16 : Select N best organisms from Total Eco and assign to Eco 17 : Define f mean , fbest 18 : delta = fmean /fbest − 1 19 : end while

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where tolerance is the allowable error; MaxEval is the maximum number of finite element analyses, and N is the number of organisms in the ecosystem. 4.4. The mSOS with discrete design variables For truss optimization problems, a simple technique [57] is utilized to convert a continuous design variable into a corresponding discrete one by using the following rounding function   Xidiscrete = F ix Xicontinuous , (13) where Xi is the ith organism, and F ix(Xi ) is a function which rounds each member of Xi to the nearest discrete value. It is noticeable that new solutions updated by Eqs. (9), (10) and (11) are rounded to the nearest values by the above technique before estimating the fitness function. 5. Numerical examples In this section, five truss optimization problems with discrete design variables and two tensegrity ones with continuous design variables are investigated to illustrate the 8 Page 10 of 46

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effectiveness and robustness the proposed mSOS algorithm. For a succinct presentation, the influence of benefit factors of BF 1 and BF 2 in the mutualism phase, as well as the search range of randomly generated coefficients in the commensalism phase of the mSOS algorithm on the optimal solution is only examined for the first example, i.e. a 10-bar planar truss. Furthermore, the influence of the parasitism phase is investigated as well. From obtained results, appropriate values with respect to those parameters are suggested in all subsequent examples. For comparison purpose, gained results by the mSOS are compared with those of other algorithms available in the literature to verify the robustness of the proposed method in searching the global optimal solution. The original SOS and the DE are executed as a reference to demonstrate the effectiveness of the mSOS in dramatically reducing the computational cost in terms of the number of analyses. In all investigated examples, the population size N of 50 is chosen as that previously carried out in the original SOS [27]. The two-node linear bar element [58] is utilized for the truss analysis whereas the force density method [59] is employed for the tensegrity analysis. The iterative process will be stopped either when the relative error between the best objective function value and the mean objective function one of the whole ecosystem is less than or equal to tolerance or when the maximum number of structural analyses (MaxEval ) reaches. It can be found that if the value of tolerance is too large, attained optimal solutions have low accuracy. On the contrary, high-quality solutions are achieved, but the computational cost of the optimization process is high. In this study, a proper value of tolerance is chosen as 10−6 . MaxEval is set to be 300,000. Since the above three algorithms are meta-heuristic and stochastic, optimal results obtained by different runs are hence not exactly the same. To deal with this issue, each algorithm is run 30 independent times. Statistical results are reported including the best weight and the corresponding number of analyses, the worse weight, average weight, standard deviation and average numbers of analyses. The Matlab code of the original SOS algorithm is found in [27].

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5.1. Truss optimization with discrete design variables 5.1.1. A 10-bar planar truss structure A 10-bar planar truss structure displayed in Fig. 1 is optimized as the first example. The Young’s modulus E and material density ρ are taken as 104 ksi and 0.1 lb/in3 , respectively. Two vertically applied loads at nodes of 2 and 4 are 105 lbs. The stress constraint on all structural members is set to be ±25 ksi in both tension and compression. All nodal displacements are limited in the range [-2,2] in for both x and y directions. A given set of cross-section discrete values of S={1.62, 1.80, 1.99, 2.13, 2.38, 2.62, 2.88, 2.93, 3.09, 3.13, 3.38, 3.47, 3.55, 3.63, 3.84, 3.87, 3.88, 4.18, 4.22, 4.49, 4.59, 4.80, 4.97, 5.12, 5.74, 7.22, 7.97, 11.50, 13.50, 13.90, 14.20, 15.50, 16.00, 16.90, 18.80, 19.90, 22.00, 22.90, 26.50, 30.00, 33.50} in2 is utilized. Each element cross-section area is considered as a design variable. Various algorithms have been previously suggested to solve this example such as genetic algorithm (GA) [16], heuristic particle swarm optimization (HPSO) [18], mine blast algorithm (MBA) [19] and adaptive elitist differential evolution (aeDE) [24]. • Influence of both benefit factors (BF1 and BF2) and the search range of randomly generated coefficients on the optimal solution This part is devoted to investigating the influence of two benefit factors (BF 1 and BF 2) and the search range of randomly generated coefficients on the optimal solution. Firstly, BF 1 and BF 2 are investigated by considering the following three cases: either 9 Page 11 of 46

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1 or 2, 1 and 2. Obtained results are tabulated in Table 1. Although all those cases result in the same value of objective function, when both BF 1 and BF 2 are equal 1, the algorithm always yields better results in terms of the number of analyses and the standard deviation than the others. Moreover, also found from the table, the number of analyses and the standard deviation in the case of BF 1 = BF 2 = 2 are the worst. The results indicate that the mSOS converges faster and becomes more stable. This totally agrees with those explained in Section 4.1 Next, three different ranges of randomly generated coefficients in the commensalism phase are examined with BF 1=BF 2=1. Table 2 shows a comparison of gained results. It can be found that the obtained results with coefficients in the range [0.4, 0.9] are more stable than those of the others with the quite small standard deviation of 0.301. Moreover, the mSOS with the suggested parameter values has faster convergence speed against the SOS as well. From the obtained results in Tables 1 and 2, the unit values of BF 1, BF 2 in the mutualism phase and randomly generated coefficients within the range [0.4,0.9] in the commensalism phase are suitable parameters and recommended for the mSOS.

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• Influence of the parasitism phase

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Now, the influence of the parasitism phase on convergence histories of mean and best values obtained by the SOS and mSOS is investigated. As depicted in Fig. 2(a), when this phase is not considered in the optimization procedure, the convergence of the SOS failed although the optimal solution can be obtained. Nevertheless, the mean value yielded by the SOS without taking account of the parasitism phase is not almost gradually decreased to the best one. Whilst provided that this phase is included, the convergence of both values is achieved. Additionally, the number of analyses in this case is dramatically reduced as well, and thus the computational cost can be saved. From the above investigation, it can be concluded that this phase is necessary to the SOS in order to obtain a better swarm behavior. As shown in Fig. 2(b), the proposed mSOS still results in a good swarm behavior that is characterized by the convergence of both mean and best values to the same one. Furthermore, the mSOS significantly improves the computational cost in terms of the number of analyses in comparison with the SOS. This proves that this phase can be neglected for the mSOS as explained in sub-section 4.3. • Comparison with other methods in the literature Optimal results of the present methods are compared with those of the foregoing algorithms and summarized in Table 3. From this table, it can be seen that the optimal weight acquired by the mSOS agrees well with those given by the DE, SOS and aeDE, while the mSOS results in a better solution than other methods (5613.84 lb for the GA, 5531.98 lb for the HPSO and 5507.75 lb for the MBA). The mSOS only requires 5,850 analyses which are less than the HPSO (50,000 analyses), DE (17,500 analyses) and SOS (120,900 analyses). Although the convergence speed of the MBA and aeDE are faster than that of the mSOS (3,600 analyses for MBA and 2,380 analyses for aeDE), the mSOS is more stable than the MBA and aeDE with the smallest standard deviation (0 lb for mSOS, 11.38 lb for MBA and 20.7801 lb for aeDE). Since the objective function values of the DE obtained at the first 1,500 analyses are very large, the convergence of the DE, SOS and mSOS after those analyses are compared 10 Page 12 of 46

in Fig. 3. As can be seen from the figure, the mSOS converges to the optimal solution faster than the DE and SOS. From the figure, it can be seen that the SOS requires a large number of analyses with tolerance = 10−6 . Therefore, this is a limitation of the original SOS for 10-bar truss problem with discrete design variables. It can be concluded that the mSOS significantly improves the computational cost and the convergence of the original SOS, but still attaining the global optimal solution with high accuracy and reliability. Displacement and stress values obtained by the mSOS satisfy all allowable constraints as in the Fig. 4 .

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5.1.2. A 52-bar planar truss structure In this example, all members of the 52-bar planar truss structure as shown in Fig. 5 are divided into 12 groups corresponding to 12 design variables as follows: (1) A1 -A4 ; (2) A5 -A10 ; (3) A11 -A13 ,; (4) A14 -A17 ; (5) A18 -A23 ; (6) A24 -A26 ; (7) A27 -A30 ; (8) A31 -A36 ; (9) A37 -A39 ; (10) A40 -A43 ; (11) A44 -A49 , and (12) A50 -A52 . The Young’s modulus and material density are 2.07 × 105 MPa and 7860 kg/m3 , respectively. The allowable stress is ±180 MPa. This structure is subjected to the loads concluding Px =100 kN and Py =200 kN as identified in the above figure. The discrete design variables are selected from the American Institute of Steel Construction (ASIC) code as depicted in Table 4. Table 5 presents a comparison between results obtained by the present methods and the HS [20], HPSO [18], MBA [19] and aeDE [24]. As seen, the mSOS, SOS and DE yield the same optimal weight of 1899.654 kg, but the mSOS requires far fewer the number of analyses than the HS, HPSO, DE and SOS (7,950 analyses for the mSOS, 60,000 analyses for the HS, 150,000 analyses for the HPSO, 45,600 analyses for the DE and 300,000 analyses for the SOS) to converge the optimal solution. As seen, the convergence speed of the MBA and aeDE are faster than that of the mSOS (5,450 analyses for MBA and 3,720 analyses for aeDE), yet the mSOS is always more stable than the MBA and aeDE through the best value of standard deviation, namely 1.558 lb for mSOS, 4.09 lb for MBA and 6.679 lb for aeDE. Furthermore, the result obtained by the proposed algorithm is also better than those given by other methods, namely 1970.14kg for the HS, 1905.49 kg for the HPSO, 1902.61 kg for the MBA. Moreover, the optimization process of the SOS does not terminate with the stopping criterion of tolerance = 10−6 . It is clear that the mSOS dramatically saves the computational cost. Fig. 6 shows that all constraints of the 52-bar truss are satisfied by the mSOS. 5.1.3. A 200-bar planar truss structure A 200-bar truss structure displayed in Fig. 7 is examined herein. The Young’s modulus and material density are 30 psi and 0.283 lb/in3 , respectively. Stresses in all truss members cannot surpass ±10 ksi. This structure is subjected to three loading conditions as follows: (1) 1 kip in positive x−axis at nodes of 1, 6, 15, 20, 29, 43, 48 57, 62 and 71; (2) 10 kips in negative y−axis at nodes of 1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 15, 16, 17, 18, 19, 20, 22, 24, 26, 28, 29, 30, 31, 32, 33, 34, 36, 38, 40, 42, 43, 44, 45, 46, 47, 48, 50, 52, 54, 56, 58, 59, 60, 61, 62, 64, 66, 68, 70, 71, 72, 73, 74 and 75, and (3) both case 1 and case 2. This structure includes 29 design variables corresponding to 29 member groups whose values are chosen in the available set S={0.1, 0.347, 0.44, 0.539, 0.954, 1.081, 1.174, 1.333, 1.488, 1.764, 2.142, 2.697, 2.8, 3.131, 3.565, 3.813, 4.805, 5.952, 6.572, 7.192, 8.525, 9.3, 10.85, 3.131, 3.565, 4.805, 5.952, 6.572, 7.192, 8.525, 9.3, 10.85, 13.33, 14.29, 17.17, 19.18, 23.68, 28.08, 33.7} (in.2 ). Table 6 shows gained optimal solutions by the present methods and other approaches in the literature such as the IGA [60], HACOHS-T [61], ESASS [62] and aeDE [24]. From 11 Page 13 of 46

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the table, it can be seen that the acquired results by the present methods (27544.191 lb) are better than other methods (28544.01 lb for the IGA, 28030.2 lb for the HACOHS-T, 28075.488 lb for the ESASS and 27858.5 lb for the aeDE). Moreover, the mSOS requires fewer the number of analyses than the IGA, DE and SOS (21,675 analyses for the mSOS, 51,360 analyses for the IGA, 211,855 analyses for the DE and 300,000 analyses for the SOS). Note that the mSOS is still more stable than the aeDE, DE and SOS with the smallest standard deviation (481.590 lb for the aeDE, 122.382 lb for the DE, 297.119 lb for the SOS and 90.254 lb for the mSOS). Also observed from this Table, the IGA, HACOHS-T, ESASS and aeDE converge prematurely, these algorithms therefore tend to get local solutions. It is clear that the mSOS reduces computational cost. Moreover, Fig. 8 shows that no stress constraints given by the mSOS are violated.

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5.1.4. A 25-bar space truss structure A 25-bar space truss structure schematized in Fig. 9 is studied herein. The Young’s modulus and material density are 104 ksi and 0.1 lb/in3 , respectively. Allowable tensile and compression stresses are ±40 ksi. The limitation on all nodal displacements in both directions is ±0.35 in. The loads applied to the structure are tabulated in Table 7. All structural members are categorized into 8 design variables with respect to the following 8 groups: (1) A1 ; (2) A2 -A5 ; (3) A6 -A9 ; (4) A10 -A11 ; (5) A12 -A13 ; (6) A14 -A17 ; (7) A18 -A21 , and (8) A22 -A25 . A given set of datum of S={0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0, 2.1, 2.2, 2.3, 2.4, 2.6, 2.8, 3.0, 3.2, 3.4} (in.2 ) is utilized for design purpose. This example has been previously solved by several algorithms such as the SGA [28], HS [20], HPSO [18], MBA [19] and aeDE [24]. A comparison between outcomes achieved by the present methods and the aforementioned algorithms is summarized in Table 8. The optimal weight acquired by the mSOS agrees well with those of the others with 484.85 lb, and is better than the SGA with 486.29 lb. From the table, it can be seen that the mSOS only requires 4,200 analyses to obtain the optimal solution while the SGA, HS, HPSO, DE and SOS require 40,000, 13,523, 25,000, 9,500 and 78,300 analyses, respectively. In addition, as expected, the mSOS is always more stable than the MBA and aeDE with a better value of standard deviation. The weight convergence histories obtained by the DE, SOS and mSOS are shown in Fig. 10. As expected, the mSOS converges to the optimal solution faster than the DE and SOS. No constraints attained by the mSOS are violated as in Fig. 11. 5.1.5. A 160-bar space truss structure In the last example, a 160-bar space truss depicted in Fig. 12 is investigated to demonstrate the effectiveness and robustness of the mSOS in handling problems with a large number of design variables. The material density is equal to 0.00785 kg/cm3 and the Young’s modulus is 2.047 × 106 kgf/cm2 . 160 members are divided into 38 groups with regard to 38 design variables. Nodal coordinates and element data are listed in Table 9 and 10, respectively. The eight load cases given in Table 11 are considered. The buckling stress constraint on a member under compression is computed as follows (
2 1300 − kL/r /24, if kL/r ≤ 120,
2 σb = (14) 107 / kL/r , otherwise, in which L is the member length, r is the radius of gyration and k is the effective length factor assumed to be 1.0. The cross-section areas and the corresponding radii of gyration are respectively selected from S={1.84, 2.256, 2.66, 3.07, 3.47, 3.88, 4.79, 5.27, 5.75, 6.25, 12 Page 14 of 46

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6.84, 7.44, 8.06, 8.66, 9.40, 10.47, 11.38, 12.21, 13.79, 15.39, 17.03, 19.03, 21.12, 23.20, 25.12, 27.50, 29.88, 32.76, 33.90, 34.77, 39.16, 43.00, 45.65, 46.94, 51.00, 52.10, 61.82, 61.90, 68.30, 76.38, 90.60, 94.13} cm2 and r={0.47, 0.57, 0.67, 0.77, 0.87, 0.97, 0.97, 1.06, 1.16, 1.26, 1.15, 1.26, 1.36, 1.46, 1.35, 1.36, 1.45, 1.55, 1.75, 1.95, 1.74, 1.94, 2.16, 2.36, 2.57, 2.35, 2.56, 2.14, 2.33, 2.97, 2.54, 2.93, 2.94, 2.94, 2.92, 3.54, 3.6, 3.52, 3.51, 3.93, 3.92, 3.92} cm. Optimal results acquired by the current methods and the RGA [29], RBAS [30], SDR [63] and aeDE [24] are summarized in Table 12. As observed, the mSOS, SOS and aeDE provide the best result with 1336.634 kg while the others give larger weights, namely RGA (1337.442 kg), RBAS (1348.905 kg), SDR (1359.781 kg) and DE (1346.324 kg). Furthermore, the mSOS only requires 24,000 analyses to converge the optimal solution, whilst the RBAS, DE and SOS need 90,000, 300,000 and 300,000 analyses, respectively. From the obtained worst weight, mean weight and standard deviation values of the present methods, it can be seen that the mSOS is stable and effective. It is clear that the mSOS is effective and robust in treating problems with a large number of discrete design variables, yet still attaining high-quality solutions with significant reduction of the computational cost. Fig. 13 demonstrates that constraints of this structure obtained by the mSOS are not violated.

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5.2. Tensegrity optimization with continuous design variables 5.2.1. A two-dimensional hexagonal tensegrity structure Consider an initial hexagonal tensegrity structure as in the Fig. 14 that consists of six nodes and nine elements concluding three struts and six cables. The only initial information relating to the nodal connectivity and the member types are used to select relevant solutions for strut members in the range [-1,0]. In this structure, members are categorized into two groups (1) q1 −q6 and (2) q7 −q9 . The gained force density coefficients are demonstrated in Table 13, in which results obtained by the present methods are compared to those gained by Tibert et al. [64], Estrada et al. [65] and Tran et al. [66]. The result of the mSOS agrees well with those of the other algorithms. The mSOS converges faster than the DE and SOS (2,550 analyses for the mSOS, 10,150 analyses for the DE and 7,200 analyses for the SOS). 5.2.2. A three-dimensional truncated tetrahedral tensegrity structure A three-dimensional truncated tetrahedral tensegrity structure including twelve nodes and twenty-four members, which consists of eighteen cables and six struts, is examined as in Fig. 15. Members of this structure are divided into the following three groups: (1) twelve cables q1 −q12 ; (2) six cables q13 −q18 , and (3) six struts q19 −q24 . Table 14 presents a comparison of optimal results achieved by the present methods and the previous study of Koohestani et al. [67]. It is obvious that although all methods give the same force density coefficients and standard deviation, the mSOS only needs 2700 analyses to converge while the DE and SOS require 10250 and 7500 analyses, respectively. It can be concluded that the mSOS significantly reduces the computational cost but still attaining the global optimal solution. 6. Conclusions A new modified symbiotic organisms search (mSOS) algorithm with five modifications from the SOS in all of the three phases is successfully developed as well as applied to 13 Page 15 of 46

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pin-jointed optimization problems with both discrete and continuous design variables. In the mutualisum phase, both benefit factors are fixed as 1 to balance the level of benefit to each organism, whereas a randomly chosen organism is utilized to increase the exploration ability in lieu of the best one. Accordingly, the proposed algorithm can achieve the global optimal solution with high accuracy and reliability. Randomly generated coefficients in the commensalism phase are limited within the range [0.4, 0.9] instead of [−1, 1]. Furthermore, an elitist selection technique is also employed in this phase to extract the best organisms for the next generation. Both implementations aim at increasing the exploitation capability, and thus enhancing the convergence speed. From the above modifications, a better trade-off between the exploration and exploitation abilities is therefore achieved. Consequently, the parasitism phase is neglected to simplify the performance and save the computational cost for the whole optimization process. Through the presented formulation and investigated examples, several conclusions are summarized as follows

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(i) None of significant changes in the code structure of the mSOS is observed in comparison with the SOS as well as other population-based algorithms. The present method is thus relatively simple and easy to implement.

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(ii) The optimal solutions obtained by the mSOS are almost better than those of many algorithms available in the literature. This indicates that the mSOS is robust in searching the global optimal solution, even in problems with a large number of design variables.

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(iii) The mSOS always outperforms the SOS and the DE in terms of the computational time, but still achieving high-quality solutions.

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Acknowledgments

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The mSOS is a simple and robust algorithm, its extension to more complex engineering problems such as laminated composite or functionally graded structures, and so forth is hence very promising with high feasibility. In addition, with a fast convergence rate of the present method, topology optimization awaits further attention.

This research was supported by Grant (NRF -2017R1A4A1015660) from NRF (National Research Foundation of Korea) funded by MEST (Ministry of Education and Science Technology) of Korean government. References

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List of tables Table 1. Influence of benefit factors BF 1 and BF 2 on the optimal solution. 1 or 2

2

1 5490.737 7,200 5491.717 5490.787 0.219 7,350

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Best weight (lb) 5490.737 5490.737 Number of analyses 9,450 11,550 Worst weight (lb) 5507.758 5533.656 Mean weight (lb) 5491.589 5498.192 Standard deviation (lb) 3.806 12.015 Average numbers of analyses 9,600 12,600

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BF 1 and BF 2

[-1,1]

[0,1]

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Search range

us

Table 2. Influence of the search range of randomly generated coefficients on the optimal solution.

5490.737 7,350 5491.717 5490.836 0.301 7,500

Ac ce

pt

ed

M

Best weight (lb) 5490.737 5490.737 Number of analyses 9,150 7,050 Worst weight (lb) 5607.203 5536.965 Mean weight (lb) 5513.549 5502.438 Standard deviation (lb) 32.424 18.774 Average numbers of analyses 10,050 7,350

[0.4,0.9]

20 Page 22 of 46

cr us

M an

Table 3. Optimal results of the 10-bar planar truss structure obtained by different algorithms.

te

Weight (lb) Number of analyses Worst weight (lb) Mean weight (lb) Standard deviation (lb) Average numbers of analyses

Ac c

21

ep

A1 A2 A3 A4 A5 A6 A7 A8 A9 A10

d

Area (in.2)

Present

GA [16]

HPSO [18]

MBA [19]

aeDE [24]

33.5 1.62 22 15.5 1.62 1.62 14.2 19.9 19.9 2.62

30 1.62 22.9 13.5 1.62 1.62 7.79 26.5 22 1.8

30 1.62 22.9 16.9 1.62 1.62 7.97 22.9 22.9 1.62

5613.84 -

5531.98 50,000 3.8402 -

5507.75 3,600 5536.965 5527.296 11.38 -

DE

SOS

mSOS

33.5 1.62 22.9 14.2 1.62 1.62 7.97 22.9 22 1.62

33.5 1.62 22.9 14.2 1.62 1.62 7.97 22.9 22 1.62

33.5 1.62 22.9 14.2 1.62 1.62 7.97 22.9 22 1.62

33.5 1.62 22.9 14.2 1.62 1.62 7.97 22.9 22 1.62

5490.738 2,380 5549.204 5502.623 20.7801 2,550

5490.738 17,500 5491.717 5490.771 0.176 22,710

5490.738 120,900 5490.738 5490.738 0 217,365

5490.738 5,850 5490.738 5490.738 0 7,280

Page 23 of 46

in.2

mm2

No.

in.2

mm2

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

0.111 0.141 0.196 0.250 0.307 0.391 0.442 0.563 0.602 0.766 0.785 0.994 1.000 1.228 1.266 1.457 1.563 1.620 1.800 1.990 2.130 2.380 2.620 2.630 2.880 2.930 3.090 3.130 3.380 3.470 3.550 3.630

71.613 90.968 126.451 161.290 198.064 252.258 285.161 363.225 388.386 494.193 506.451 641.289 645.160 792.256 816.773 939.998 1008.385 1045.159 1161.288 1283.868 1374.191 1535.481 1690.319 1696.771 1858.061 1890.319 1993.544 2019.351 2180.641 2238.705 2290.318 2341.931

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 59 60 61 62 63 64

3.840 3.870 3.880 4.180 4.220 4.490 4.590 4.800 4.970 5.120 5.740 7.220 7.970 8.530 9.300 10.850 11.500 13.500 13.900 14.200 15.500 16.000 16.900 18.800 19.900 22.000 22.900 24.500 26.500 28.000 30.000 33.500

2477.414 2496.769 2503.221 2696.769 2722.575 2896.768 2961.284 3096.768 3206.445 3303.219 3703.218 4658.055 5141.925 5503.215 5999.988 6999.986 7419.340 8709.660 8967.724 9161.272 9999.980 10322.560 10903.204 12128.008 12838.684 14193.520 14774.164 15806.420 17096.740 18064.480 19354.800 21612.860

us

an

M

ed

pt

Ac ce

cr

No.

ip t

Table 4. The available cross-section areas of the ASIC code.

22 Page 24 of 46

cr us

M an

Table 5. Optimal results of the 52-bar planar truss structure obtained by different algorithms.

te ep

Ac c

23

A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12

HS [20]

HPSO [18]

MBA [19]

aeDE [24]

45658.06 1161.29 645.16 3303.22 1045.16 494.193 2477.41 1045.16 285.161 1696.77 1045.16 641.289

4658.06 1161.29 363.225 3303.22 940 494.193 2283.71 1008.39 388.386 1283.87 1161.29 792.256

4658.06 1161.29 494.193 3303.22 940 494.193 2283.71 1008.39 494.193 1283.87 1161.29 494.193

1970.142 60,000 -

1905.49 150,000 -

1902.605 5,450 1912.646 1906.076 4.09 -

d

Area (mm2)

Weight (kg) Number of analyses Worst weight (kg) Mean weight (kg) Standard deviation (kg) Average numbers of analyses

Present DE

SOS

mSOS

4658.06 1161.29 494.193 3303.22 940 494.193 2283.71 1008.39 494.193 1283.87 1161.29 494.193

4658.06 1161.29 494.193 3303.22 940 506.451 2283.71 1008.39 388.386 1283.87 1161.29 506.451

4658.06 1161.29 494.193 3303.22 940 506.451 2283.71 1008.39 388.386 1283.87 1161.29 506.451

4658.06 1161.29 494.193 3303.22 940 506.451 2283.71 1008.39 388.386 1283.87 1161.29 506.451

1902.605 3,720 1925.714 1906.735 6.679 3,402

1899.654 45,600 1902.605 1901.228 1.472 54,202

1899.654 300,000 1902.605 1901.917 1.2483 300,000

1899.654 7,950 1903.64 1901.003 1.558 9,750

Page 25 of 46

Table 6. Optimal results of the 200-bar planar truss structure obtained by different algorithms. Present IGA [60]

HACOHS-T [61]

ESASS [62]

aeDE [24]

A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16 A17 A18 A19 A20 A21 A22 A23 A24 A25 A26 A27 A28 A29

0.347 1.081 0.1 0.1 2.142 0.347 0.1 3.565 0.347 4.805 0.44 0.44 5.952 0.347 6.572 0.954 0.347 8.525 0.1 9.3 0.954 1.764 13.3 0.347 13.33 2.142 4.805 9.3 17.17

0.1 1.081 0.347 0.1 2.142 0.347 0.1 3.131 0.1 4.805 0.44 0.1 5.952 0.1 6.572 0.539 1.174 8.525 0.1 9.3 1.333 0.539 13.33 1.174 13.33 2.697 3.565 8.525 17.17

0.1 0.954 0.1 0.1 2.142 0.347 0.1 3.131 0.1 4.805 0.347 0.1 5.952 0.1 6.572 0.44 0.539 7.192 0.44 8.525 0.954 1.174 10.85 0.44 10.85 1.764 8.525 13.33 13.33

0.1 0.954 0.347 0.1 2.142 0.347 0.1 3.131 0.347 4.805 0.539 0.347 5.952 0.1 6.572 0.954 0.44 8.525 0.1 9.3 0.954 1.081 13.33 0.539 14.29 2.142 3.813 8.525 17.17

Weight (lb) Number of analyses Worst weight (lb) Mean weight (lb) Standard deviation (lb) Average numbers of analyses

28544.01 51,360 -

28030.2 20,000 -

mSOS

0.1 0.954 0.44 0.1 2.142 0.347 0.1 3.131 0.1 4.805 0.44 0.44 5.952 0.1 6.572 0.954 0.347 8.525 0.1 9.3 0.954 1.174 13.33 0.44 13.33 2.142 3.813 8.525 17.17

0.1 0.954 0.1 0.1 2.142 0.347 0.1 3.131 0.1 4.805 0.44 0.1 5.952 0.1 6.572 0.954 0.347 8.525 0.1 9.3 0.954 1.174 13.33 0.44 13.33 2.142 3.813 8.525 17.17

0.1 0.954 0.44 0.1 2.142 0.347 0.1 3.131 0.1 4.805 0.44 0.44 5.952 0.1 6.572 0.954 0.347 8.525 0.1 9.3 0.954 1.174 13.33 0.44 13.33 2.142 3.813 8.525 17.17

27544.191 233,300 27918.949 27653.828 122.382 211,855

27544.191 300,000 28283.476 27768.287 297.119 300,000

27544.191 20,700 27825.545 27629.818 90.254 21,675

cr

us

an M

ed

SOS

27858.5 12,325 29415.000 28425.871 481.590 11,644

Ac ce

pt

28075.488 11,156 -

DE

ip t

Area (in.2)

24 Page 26 of 46

ip t cr us an

Loads

Py (kips)

Pz (kips)

1 0 0.5 0.6

-10 -10 0 0

-10 -10 0 0

Ac ce

pt

1 2 3 6

Px (kips)

ed

Nodes

M

Table 7. Loads applied to the 25-bar space truss structure.

25 Page 27 of 46

cr us

M an

Table 8. Optimal results of the 25-bar space truss structure obtained by different algorithms.

Area (in.2)

te ep

Weight (lb) Number of analyses Worst weight (lb) Mean weight (lb) Standard deviation (lb) Average numbers of analyses

Ac c

26

SGA [28]

HS [20]

HPSO [18]

MBA [19]

aeDE [24]

0.1 0.5 3.4 0.1 1.5 0.9 0.6 3.4

0.1 0.3 3.4 0.1 2.1 1 0.5 3.4

0.1 0.3 3.4 0.1 2.1 1 0.5 3.4

0.1 0.3 3.4 0.1 2.1 1 0.5 3.4

486.29 40,000 -

484.85 13,523 -

484.85 25,000 0.02664 -

484.85 2,150 485.048 484.885 0.072 -

d

A1 A2 A3 A4 A5 A6 A7 A8

Present DE

SOS

mSOS

0.1 0.3 3.4 0.1 2.1 1 0.5 3.4

0.1 0.3 3.4 0.1 2.1 1 0.5 3.4

0.1 0.3 3.4 0.1 2.1 1 0.5 3.4

0.1 0.3 3.4 0.1 2.1 1 0.5 3.4

484.85 1,440 486.100 485.014 0.273 1,678

484.85 9,500 484.85 484.85 0 11,633

484.85 484.85 78,300 4,200 485.05 485.05 484.87 484.86 0.058 0.035 174,739 5,965

Page 28 of 46

cr us

Y-Coord

Z-Coord

d

−105 −105 105 105 −93.929 −93.929 93.929 93.929 −82.859 −82.859 82.859 82.859 −71.156 −71.156 71.156 71.156 −60.085 −60.085 60.085 60.085 −49.805 −49.805 49.805 49.805 0 −40

te

−105 105 105 −105 −93.929 93.929 93.929 −93.929 −82.859 82.859 82.859 −82.859 −71.156 71.156 71.156 −71.156 −60.085 60.085 60.085 −60.085 −49.805 49.805 49.805 −49.805 −214 −40

Ac c

27

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

X-Coord

ep

Node no

M an

Table 9. Nodal coordinate data of the 160-bar space truss structure.

0 2000 0 0 175 175 175 175 350 350 350 350 535 535 535 535 710 710 710 710 872.5 872.5 872.5 872.5 1027.5 1027.5

Node no 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

X-Coord 40 214 40 −40 −40 40 40 −40 −40 40 −207 40 −40 −40 40 40 −40 −26.592 26.592 26.592 −26.592 −12.737 13.737 13.737 −12.737 0

Y-Coord 40 0 40 40 −40 −40 40 40 −40 −40 0 40 40 −40 −40 40 40 −26.592 −26.592 26.592 26.592 −12.737 −12.737 12.737 12.737 0

Page 29 of 46

Z-Coord 1027.5 1027.5 1027.5 1027.5 1105.5 1105.5 1105.5 1105.5 1256.5 1256.5 1256.5 1256.5 1256.5 1346.5 1346.5 1346.5 1346.5 1436.5 1436.5 1436.5 1436.5 1526.5 1526.5 1526.5 1526.5 1615

Table 10. Element data of the 160-bar space truss structure.

18 17 19 18 20 19 17 20 21 22 23 24 22 21 24 23 23 22 21 24 26 27 29 30 27 26 30 29 29 27 26 30 27 29 30 26 26 28 30 28

8 8 8 8 8 8 8 8 9 9 9 9 10 10 10 10 11 11 11 11 12 12 12 12 13 13 13 13 14 14 14 14 15 15 15 15 16 16 16 16

81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120

25 28 28 25 26 27 29 30 26 27 29 30 27 29 30 26 26 27 31 32 33 34 33 32 31 34 32 33 34 31 37 37 37 37 35 36 38 39 35 36

31 32 33 34 31 32 33 34 32 31 34 33 33 32 31 34 29 30 35 36 38 39 39 35 36 38 38 36 35 39 35 39 40 43 40 41 42 43 38 39

17 17 17 17 18 18 18 18 19 19 19 19 20 20 20 20 21 21 22 22 22 22 23 23 23 23 24 24 24 24 25 25 26 26 27 27 27 27 28 28

121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160

1

2

36 38 39 35 40 41 42 43 35 36 38 39 40 41 42 43 40 41 42 43 44 45 46 44 44 45 46 47 45 46 47 44 48 49 50 48 48 49 50 51

40 41 42 43 41 42 43 40 36 38 39 35 44 45 46 47 45 46 47 44 45 46 47 47 48 49 50 51 48 49 50 51 49 50 51 51 52 52 52 52

ip t

13 14 14 15 15 16 16 13 17 18 19 20 17 18 19 20 18 19 20 17 21 22 23 24 21 22 23 24 22 23 24 21 26 27 29 30 25 27 25 29

2

E. No.

cr

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 69 70 71 72 73 74 75 76 77 78 79 80

1

Node Ai

us

1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 6 6 6 6 6 6 6 6 7 7 7 7

E. No.

an

5 6 7 8 6 5 7 6 8 7 5 8 9 10 11 12 10 9 11 10 12 11 9 12 13 14 15 16 14 13 15 14 16 15 13 16 17 18 19 20

2

Node Ai

ed

1 2 3 4 1 2 2 3 3 4 4 1 5 6 7 8 5 6 6 7 7 8 8 5 9 10 11 12 9 10 10 11 11 12 12 9 13 14 15 16

1

pt

2

E. No.

Ac ce

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

1

Node Ai

M

Node E. No.

Ai 29 29 29 29 30 30 30 30 31 31 31 31 32 32 32 32 33 33 33 33 34 34 34 34 35 35 35 35 36 36 36 36 37 37 37 37 38 38 38 38

28 Page 30 of 46

ip t cr

3

4

Py

Pz

Load case

Node

52 37 25 28 52 37 25 28 52 37 25 28 52 37 25 28

an

-868 0 -491 -996 0 -546 -1091 0 -546 -1091 0 -546 -493 1245 -363 -996 0 -546 -1091 0 -546 -1091 0 -546 -917 0 -491 -951 0 -546 -1015 0 -546 -1015 0 -546 -917 0 -546 -572 1259 -428 -1015 0 -546 -1015 0 -546

M

5

6

ed

2

52 37 25 28 52 37 25 28 52 37 25 28 52 37 25 28

Px

pt

1

Node

7

8

Ac ce

Load case

us

Table 11. Load cases applied to the 160-bar space truss structure. Px

Py

Pz

-917 0 -491 -951 0 -546 -1015 0 -546 -636 1259 -428 -917 0 -491 -572 1303 -428 -1015 0 -546 -1015 0 -546 -917 0 -491 -951 0 -546 -1015 0 -546 -636 1303 -428 -498 1460 -363 -951 0 -546 -1015 0 -546 -1015 0 -546

29 Page 31 of 46

Table 12. Optimal results of the 160-bar space truss structure obtained by different algorithms. Present RBAS [30]

SDR [63]

aeDE [24]

A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16 A17 A18 A19 A20 A21 A22 A23 A24 A25 A26 A27 A28 A29 A30 A31 A32 A33 A34 A35 A36 A37 A38

19.03 5.27 19.03 5.27 19.03 5.75 15.39 5.75 13.79 5.75 5.75 13.79 6.25 5.75 2.66 7.44 1.84 8.66 2.66 3.07 2.66 8.06 5.27 6.25 5.75 1.84 4.79 2.66 3.47 1.84 2.26 3.88 1.84 1.84 3.88 1.84 1.84 3.88

19.03 5.27 19.03 5.27 19.03 5.75 15.39 5.75 13.79 5.75 5.75 12.21 6.25 5.75 3.47 7.44 1.84 9.4 2.66 3.47 3.07 8.06 5.27 6.25 5.75 2.26 4.79 3.07 3.47 1.84 3.88 3.88 1.84 2.26 3.88 2.26 3.47 3.88

19.03 5.27 19.03 5.27 19.03 5.27 17.03 6.25 13.79 6.25 5.75 12.21 6.84 5.75 2.66 7.44 1.84 8.66 2.66 3.07 2.66 8.06 5.27 7.44 6.25 1.84 4.79 2.66 3.47 1.84 2.26 3.88 1.84 1.84 3.88 1.84 1.84 3.88

19.03 5.27 19.03 5.27 19.03 5.75 15.39 5.75 13.79 5.75 5.75 12.21 6.25 5.75 3.88 7.44 1.84 8.66 2.66 3.07 2.66 8.06 5.27 6.25 5.75 2.26 4.79 2.66 3.47 1.84 2.26 3.88 1.84 1.84 3.88 1.84 1.84 3.88

1337.442 -

1348.905 90,000 1401.6323 1367.5275 90,000

us

an

M

ed

pt Ac ce

Weight (kg) Number of analyses Worst weight (kg) Mean weight (kg) Standard deviation (kg) Average numbers of analyses

DE

SOS

mSOS

19.03 5.27 19.03 5.27 19.03 5.75 15.39 5.75 13.79 5.75 5.75 13.79 6.25 5.75 3.07 7.44 1.84 8.66 2.66 3.07 3.07 8.06 5.75 6.25 5.75 1.84 5.27 2.66 3.47 2.26 2.26 3.88 2.26 1.84 3.88 1.84 2.66 3.88

19.03 5.27 19.03 5.27 19.03 5.75 15.39 5.75 13.79 5.75 5.75 12.21 6.25 5.75 3.88 7.44 1.84 8.66 2.66 3.07 2.66 8.06 5.27 6.25 5.75 2.26 4.79 2.66 3.47 1.84 2.26 3.88 1.84 1.84 3.88 1.84 1.84 3.88

19.03 5.27 19.03 5.27 19.03 5.75 15.39 5.75 13.79 5.75 5.75 12.21 6.25 5.75 3.88 7.44 1.84 8.66 2.66 3.07 2.66 8.06 5.27 6.25 5.75 2.26 4.79 2.66 3.47 1.84 2.26 3.88 1.84 1.84 3.88 1.84 1.84 3.88

1346.324 300,000 1396.231 1376.422 16.619 300,000

1336.634 300,000 1344.445 1339.784 2.547 300,000

1336.634 24,000 1343.821 1338.245 2.080 25,530

ip t

RGA [29]

cr

Area (cm2)

1359.781 -

1336.634 23,925 1410.611 1355.875 18.805 21,265

30 Page 32 of 46

cr us

M an

Table 13. Optimal results of the two-dimensional hexagonal tensegrity structure obtained by different algorithms.

q1 -q6 q7 -q9

Tibert [64]

Estrada [65]

Tran [66]

1.0 -0.5

1.0 -0.5

-

-

DE

SOS

mSOS

1.0 -0.5

1.0 -0.5

1.0 -0.5

1.0 -0.5

-

1.148e-15 3.435e-16 1.148e-15 10,150 7,200 2,550 1.148e-15 1.148e-15 1.148e-15 1.148e-15 1.148e-15 1.148e-15 0.0 0.0 0.0 9,772 8,070 2,550

Ac c

31

ep

te

d

Best αβ Number of analyses Worst αβ Mean αβ Standard deviation Average number of analyses

Present

Page 33 of 46

Table 14. Optimal results of the truncated tetrahedral tensegrity structure obtained by different algorithms. Present Koohestani [67]

DE

SOS

mSOS

1.0 1.0 -0.574609

1.0 1.0 -0.574609

1.0 1.0 1.0 1.0 -0.574609 -0.574609

Best αβ Number of analyses Worst αβ Mean αβ Standard deviation Average number of analyses

-

1.898e-15 10,250 1.898e-15 1.898e-15 0.0 9,789

1.898e-15 7,500 1.898e-15 1.898e-15 0.0 6,495

ip t

q1 -q12 q13 -q18 q19 -q24

Ac ce

pt

ed

M

an

us

cr

1.898e-155 2,700 1.898e-15 1.898e-15 0.0 2,580

32 Page 34 of 46

List of figures 360 in.

360 in.

1

3

5 (7)

(9) (6)

(5) (10)

(8)

360 in.

(2)

(1)

ip t

(4)

(3)

6

2

4

P

cr

P

Ac ce

pt

ed

M

an

us

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

33 Page 35 of 46

ip t cr us an

M

(a) The SOS algorithm

6000

pt

Ac ce

Weight (lb)

5900

5800

Best value Mean value

ed

6100

5700

5600

5500

5400 1000

2000

3000

4000 5000 Number of analyses

6000

7000

8000

(b) The mSOS algorithm

Fig. 2. The influence of the parasitism phase on mean and best values obtained by the SOS and mSOS for the 10-bar planar truss structure.

34 Page 36 of 46

ip t cr us

6200

DE SOS mSOS

6100

an

5900

M

5800

5600 5500

2

4 6 8 Number of analyses

10

12

14 4 x 10

Ac ce

5400 0

ed

5700

pt

Weight (lb)

6000

Fig. 3. Weight convergence history of the 10-bar planar truss structure obtained by the DE, SOS and mSOS.

35 Page 37 of 46

2 Allowable displacement Optimal displacement

1.5

ip t

0.5

cr

0 -0.5

us

Displacment (in.)

1

-1

-2 0

2

an

-1.5

4

6 8 Node number

10

12

M

(a) Displacement constraint results

ed

30

20

pt

0

Ac ce

Stress (ksi)

10

Allowable stress Optimal stress

-10

-20

-30 0

1

2

3

4

5 6 7 Element number

8

9

10

(b) Stress constraint results

Fig. 4. Constraint results of the 10-bar truss structure at the optimal design computed by the mSOS.

36 Page 38 of 46

Y

(50)

(51)

(52)

(44)

(46)

(48)

(28) (32)

(29)

(30) (36)

(34) (25)

(24)

(15)

(14)

(26) (22)

(20)

(18)

(16)

(17) (23)

(21)

(19)

6

5

7 (12) (7)

(11) (5) (2)

(1)

X 2

(4)

(10)

3

2m

4 2m

ed

2m

8

(13) (9)

(3) (8)

(6)

12

11

10

9

ip t

(35)

(33)

(27)

16 (39)

an

3m

15 (38)

(31)

3m

(49)

14 (37)

20

Py Px

(43)

(47)

(45)

13

3m

(42)

(41)

(40)

1

19

Py Px

cr

18

Py Px

M

3m

17

Px

us

Py

pt

Fig. 5. A 52-bar planar truss structure.

200

Allowable stress Optimal stress

Ac ce

150

Stress (MPa)

100

50 0

-50

-100 -150 -200 0

10

20

30

40 50 60 Element number

0

10

20

30

Fig. 6. Stress constraint results of the 52-bar truss structure at the optimal design computed by the mSOS.

37 Page 39 of 46

4 x l1

(26)

(6) (7)

7

(18)

(8)

(19)

(27) (28) (29)

15

3

(2) (9) (10) (11)

8 (20) 16

(21)

(30) (31) (32)

(12) (13) (14)

10(22) 11

(23)

(33) (34) (35)

17

(40)

5

(4) (15) (16)

12(24) 13 18

(41)

(17)

(25)

(36) (37)

14 (38)

19

(42)

(179) (178)

(171)

64(172) 65

(173)

66(174) 67

(175)

68(176) 69

(177)

(182) (185) (188) (181) (184) (187) 72 73 74 (183) (186) (189) (180) (191) (192) (193) (194)

Ac ce

71

63

(170)

(195)

(196)

(197)

(198)

(199)

70 (190)

75

(200)

l3

62

pt

ed

M

10 x l2

an

us

(39)

9

4

(3)

ip t

(5)

6

2

(1)

cr

1

y

O

x

76

77

l1=240 in, l2=144 in, l3=360 in.

Fig. 7. A 200-bar planar truss structure.

38 Page 40 of 46

4

x 10 1

Allowable stress Load case 1 Load case 2 Load case 3

ip t

0

cr

Stress (psi)

0.5

us

-0.5

-1 20

40

60

80 100 120 Element number

140

160

180

200

an

0

75

pt

1

(8)

2 (3)

(2)

(5)

3

(4) (10)

6

75

(6) (12)

in .

(7) (13)

4

(23) (11)

(14)

(22)

10

(1)

(9)

in.

in.

(15)

100 in.

Ac ce

75

Z

100 in.

ed

M

Fig. 8. Stress constraint results of the 200-bar truss structure at the optimal design computed by the mSOS.

5 (18)

(24) Y

(20)

(19)

7

(21)

(17)

(25) (16)

8

20 0i n.

9

. 200 in

X

Fig. 9. A 25-bar space truss structure.

39 Page 41 of 46

ip t cr us

510

DE SOS mSOS

an

505

M

Weight (lb)

500

495

ed

490

2

4

6 8 Number of analyses

10

12

14 4 x 10

Ac ce

480 0

pt

485

Fig. 10. Weight convergence history of the 25-bar space truss structure obtained by the DE, SOS and mSOS.

40 Page 42 of 46

0.3

Allowable displacement

ip t

0.1

cr

0

-0.1

us

Displacment (in.)

0.2

-0.2

0

5

10

an

-0.3

15 Node number

20

25

30

M

(a) Displacement constraint results 50

10 0

Ac ce

Stress (ksi)

20

Allowable stress

pt

30

ed

40

-10 -20 -30 -40

-50 0

5

10 15 Element number

20

25

(b) Stress constraint results

Fig. 11. Constraint results of the 25-bar truss structure at the optimal design computed by the mSOS.

41 Page 43 of 46

ip t cr us

52

an

37

25

Ac ce

pt

ed

M

28

3

4

1

2

Fig. 12. A 160-bar space truss structure.

42 Page 44 of 46

1 Allowable limit Load case 1 Load case 2 Load case 3 Load case 4

ip t

cr

0.6

0.4

us

Stress ratio

0.8

0 0

20

40

60 80 100 Element number

120

140

160

M

(a)

ed

1

Allowable limit Load case 5 Load case 6 Load case 7 Load case 8

pt

0.8

0.6

Ac ce

Stress ratio

an

0.2

0.4

0.2

0

0

20

40

60 80 100 Element number

120

140

160

(b)

Fig. 13. Stress constraint results of the 160-bar truss structure at the optimal design computed by the mSOS.

43 Page 45 of 46

6

5

(5)

(6) (7)

4

(8)

(1)

(9)

(2)

3

us

2

(3)

cr

1

ip t

(4)

pt

ed

M

an

Fig. 14. A two-dimensional hexagonal tensegrity structure.

1

3 (11)

(10)

Ac ce

2

(12)

(17) (18)

(16)

9

(1)

7

(24)

(20) (22)

11

(21)

4

(19) (6)

(2) (13)

(9)

(23)

(3)

(15) (5)

(8)

(7)

10

8

(14)

12

5 6

(4)

Fig. 15. A three-dimensional truncated tetrahedral tensegrity structure.

44 Page 46 of 46