A Backtracking Search Algorithm for the Simultaneous Size, Shape and Topology Optimization of Trusses

Rafael R. Souza Leandro F. Fadel Miguel Rafael H lopez André J. Torii Letícia F.F. Miguel About the authors

Abstract

This paper presents a Backtracking Search Optimization algorithm (BSA) to simultaneously optimize the size, shape and topology of truss structures. It focuses on the optimization of these three aspects since it is well known that the most effective scheme of truss optimization is achieved when they are simultaneously considered. The minimization of structural weight is the objective function, imposing displacement, stress, local buckling and/or kinematic stability constraints. The effectiveness of the BSA at solving this type of optimization problem is demonstrated by solving a series of benchmark problems comparing not only the best designs found, but also the statistics of 100 independent runs of the algorithm. The numerical analysis showed that the BSA provided promising results for the analyzed problems. Moreover, in several cases, it was also able to improve the statistics of the independent runs such as the mean and coefficient of variation values.

Keywords:
Backtracking Search algorithm; truss optimization; topology optimization; size optimization; shape optimization; mixed variable problem

1 INTRODUCTION

Optimization of trusses has been widely studied over the past decades, mostly because of the growing requirement from the industry for more economical design. In fact, several real life structures, or at least part of them, may be modelled using truss elements, such as roofs, towers, bridges, and so on. The difficulty of solving the optimization of truss structures may range from very simple problems to complex ones.

For example, a truss size optimization problem with continuous design variables and stress constraints is a linear and convex problem (considering a unique load case), which is rather simple to be solved. If shape variables are added to the problem, it becomes nonlinear and it may be nonconvex (Achtziger 2007Achtziger, W. (2007). On simultaneous optimization of truss geometry and topology. Structural and Multidisciplinary Optimization 33: 285-305. DOI 10.1007/s00158-006-0092-0
https://doi.org/10.1007/s00158-006-0092-...
, Torii et al. 2011Torii, A.J., Lopez, R. H., Luersen, M. (2011). A local-restart coupled strategy for simultaneous sizing and geometry truss optimization, Lat. Am. J. Solids Stru., 8, 335-349., Torii et al. 2012Torii, A. J.; Lopez, R. H.; Biondini, F. (2012). An approach to reliability-based shape and topology optimization of truss structures. Eng. Optimiz. (Print) 44: 37-53. DOI: 10.1080/0305215X.2011.558578.
https://doi.org/10.1080/0305215X.2011.55...
). The difficulty of the problem may be further increased by including, for example, topology and discrete design variables. The latter usually comes from limitations of the manufacturing process, e.g. the available cross sectional areas in a manufacturer catalogue.

The so-called metaheuristic algorithms, are a class of optimization methods able to handle these difficulties, regarding the truss optimization problems. These algorithms can also be found in different applications, such as in Rojas et. al. (2012), Koide et. al. (2013Koide, R. M., de França, G.V. Z., Luersen, M. A. (2013) an ant colony algorithm applied to lay-up optimization of laminated composite plates. Latin American Journal of Solids and Structures. 10(3).), Lobato and Steffen Jr (2014), Behrooz et. al. (2016Behrooz, A., Mohammad, R. K. R., Shahram Y., Mahdi T. (2016) A heuristic algorithm based on line-up competition and gener-alized pattern search for solving integer and mixed integer non-linear optimization problems. Latin American Journal of Solids and Structures. 13(2).). Examples of these algorithms are the Genectic Algorithms (GA), Particle Swarm Optimization (PSO), Ant colony (AC) and others.

It is important to point out that lighter structures may be designed when the engineer considers the combined effect of size, shape and topology variables. However, most of the papers found in the literature regarding truss optimization using metaheuristics focused on one or two of these aspects. Indeed, metaheuristic algorithms have been extensively applied to the size optimization of trusses such as Goldberg and Samtani (1986Goldberg, D.E., Samtani, M.P. (1986). Engineering optimization via genetic algorithms. Proceedings of the Ninth Conference on Electronic Computations, ASCE, Birmingham, Alabama, pp. 471-482, ), Adeli and Kamal (1991Adeli, H., Kamal, O. (1991). Efficient optimization of plane trusses. Adv. Eng Softw. 13(3): 116-122.), and Rajeev and Krishnamoorthy (1992Rajeev, S., Krishnamoorthy,C.S. (1992). Discrete optimization of structures using genetic algorithms, J. Struct. Eng. 118 (5): 1233-1250.), Bland (2001Bland, J.A. (2001). Optimal structural design by ant colony optimization. Eng Optimiz. 33(4): 425-443. DOI:10.1080/03052150108940927
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), Lee et al. (2005Lee, K.S., Geem,Z.W., Lee,S.-H.O. ,Bae,K.-W. (2005). The harmony search metaheuristic algorithm for discrete structural optimization. Eng. Optimiz. 37(7): 663-684.), Kaveh and Talatahari (2009aKaveh, A. and Talatahari, S. (2009a). Size optimization of space trusses using Big Bang-Big Crunch algorithm. Comput. Struct. (87, 17-18): 1129-1140. DOI: 10.1016/j.compstruc.2009.04.011.
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), Kaveh and Talatahari (2009b)Kaveh, A. and Talatahari, S. (2009b) . Particle swarm optimizer, ant colony strategy and harmony search scheme hybridized for optimization of truss structures. Comput. Struct. (87, 5-6), 267-283. DOI: 10.1016/j.compstruc.2009.01.003.
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, Sonmez (2011Sonmez, M. (2011). Discrete optimum design of truss structures using artificial bee colony algorithm. Struct. Multidiscip. O. 43:85-87. DOI 10.1007/s00158-010-0551-5.
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), Kazemzadeh Azad et al. (2014Kazemzadeh Azad, S., Hasançebi, O., Saka, M.P. (2014). Guided stochastic search technique for discrete sizing optimization of steel trusses: A design-driven heuristic approach. Computers and Structures. 134, 62-74.), Sadollaha et al. (2015Sadollaha, A., Eskandarb, H., Bahreininejadc, A., Kim, J.H. (2015). Water cycle, mine blast and improved mine blast algorithms for discrete sizing optimization of truss structures. Comput. Struct. (149):1-16. DOI: 10.1016/j.compstruc.2014.12.003
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), Kaveh and Mahdavi (2014)Kaveh, A. and Mahdavi., V.R. (2014). Colliding Bodies Optimization method for optimum design of truss structures with continuous variables. Adv, Eng. Softw. (70): 1-12. DOi: 10.1016/j.advengsoft.2014.01.002.
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, Kaveh et al. (2014)Kaveh, A., Sheikholeslamib, R., Talataharic, S., Keshvari-Ilkhichib, M. (2014). Chaotic swarming of particles: A new method for size optimization of truss structures. Adv, Eng. Softw. (67): 136-147. DOI: 10.1016/j.advengsoft.2013.09.006.
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, Degertekin (2012Degertekin, S.O. (2012). Improved harmony search algorithms for sizing optimization of truss structures. Comput. Struct. (92-93): 229-241. DOI: 10.1016/j.compstruc.2011.10.022.
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), Degertekin and Hayalioglu (2013)Degertekin, S.O. and Hayalioglu, M.S. (2013). Sizing truss structures using teaching-learning-based optimization. Comput. Struct. (119): 177-188. DOI: 10.1016/j.compstruc.2012.12.011.
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, Zhanga et al. (2014Zhanga, Y., Houa, Y. and Liua, S. (2014). A new method of discrete optimization for cross-section selection of truss structures. Eng. Optimiz. 46(8): 1052-1073. DOI: 10.1080/0305215X.2013.827671.
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), Hasançebi and Kazemzadeh Azad (2014Hasançebi, O. and Kazemzadeh Azad, S. (2014). Discrete size optimization of steel trusses using a refined big bang-big crunch algorithm. Eng. Optimiz. 46 (1): 61-83. DOI: 10.1080/0305215X.2012.748047.
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), Kazemzadeh Azad and Hasançebi (2015a)Kazemzadeh Azad, S., Hasançebi, O. (2015a). Discrete sizing optimization of steel trusses under multiple displacement constraints and load cases using guided stochastic search technique. Structural and Multidisciplinary Optimization. 52, 383-404., Hasançebi and Kazemzadeh Azad (2015)Hasançebi, O., Kazemzadeh Azad, S. (2015). Adaptive Dimensional Search: A New Metaheuristic Algorithm for Discrete Truss Sizing Optimization. Computers and Structures. 154, 1-16., Kazemzadeh Azad and Hasançebi (2014)Kazemzadeh Azad, S., Hasançebi, O. (2014). An Elitist Self-Adaptive Step-Size Search for Structural Design Optimization. Applied Soft Computing, 19, 226-235. to name just a few.

Papers regarding size and shape optimization can be found at Wu and Chow (1995Wu, S.J., Chow,P.T., (1995). Integrated discrete and configuration optimization of trusses using genetic algorithms. Comput. Struct. 56(4):695-702.), Salajegheh and Vanderplaats (1993Salajegheh, E., Vanderplaats, G. (1993). Optimum design of trusses with discrete sizing and shape variables. Struct. Optimiz. 6:79-85.), Galante (1996Galante, M. 1996. Genetic Algorithms as an Approach to Optimize real world structures. Int J Numer. Meth. Eng. (39): 361-82. DOI: 10.1002/(SICI)1097-0207(19960215)39:3<361::AID-NME854>3.0.CO;2-1.
https://doi.org/10.1002/(SICI)1097-0207(...
), Son and Yang (1996), Jármai et al. (2004Jármai, K., Snyman, J.A., Farkas, J. (2004). Application of novel constrained optimization algorithms to the minimum volume design of planar CHS trusses with parallel chords. Eng. Optimiz. 36 (4): 457-471. DOI: 10.1080/03052150410001686495.
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), Agarwal and Raich (2006)Agarwal, P and Raich, A.M. Design and optimization of steel trusses using genetic algorithms, parallel computing, and human-computer interaction Structural Engineering and Mechanics.23(4): 325-337., Kelesoglu (2007Kelesoglu., O. (2007). Fuzzy Multiobjective Optimization of Truss-structures using genetic algorithms. Adv. Eng. Softw. (38):717-21. DOI: 10.1016/j.advengsoft.2007.03.003.
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), Miguel and Fadel Miguel (2012Miguel, L.F.F; Fadel Miguel, L. F. (2012) Shape and size optimization of truss structures considering dynamic constraints through modern metaheuristic algorithms. Expert Systems with Applications. 39, 9458-9467.), Kaveh and Zolghadr (2014Kaveh, A. and Zolghadr, A. (2014) A new PSRO algorithm for frequency constraint truss shape and size optimization. Structural Engineering and Mechanics. 52(3):445-468. DOI: 10.12989/sem.2014.52.3.445.
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), Li (2014Li, J-P. (2014). Truss topology optimization using an improved species-conserving genetic algorithm. Eng. Optimiz. 47(1):107-128. DOI: 10.1080/0305215X.2013.875165.
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), among others.

However, when it comes to the simultaneous size, shape and topology optimization (SSTO) a limited number of papers is found, for example: Pereira et. al. (2004Pereira, J. T., Fancello, E. A., & Barcellos, C. S. (2004). Topology optimization of continuum structures with material failure constraints. Structural and Multidisciplinary Optimization, 26(1-2), 50-66.), Tand et al. (2005), Miguel et al. (2012Miguel, L.F.F; Fadel Miguel, L. F. (2012) Shape and size optimization of truss structures considering dynamic constraints through modern metaheuristic algorithms. Expert Systems with Applications. 39, 9458-9467.), Miguel et al. (2013)Miguel, L.F.F., Lopez, R.H., Miguel, L.F.F. (2013). Multimodal size, shape and topology optimization of truss structures using the firefly algorithm. Adv Eng. Softw. (56) 23-37. DOI: 10.1016/j.advengsoft.2012.11.006.
https://doi.org/10.1016/j.advengsoft.201...
, Grierson and Pak (1993Grierson, D.E., Pak, W.H. (1993). Optimal sizing, geometrical and topological design using a genetic algorithms. Struct. Optimization (6): 151-9.), Rajan (1995Rajan, S. D. (1995). Sizing, shape, and topology design optimization of trusses using genetic algorithms. J. Struct. Eng. 121(10): 1480-1487.), Hajela and Lee (1995Hajela,P., Lee, E. (1995). Genetic algorithms in truss topological optimization. Int. J. Solids Struct 32 (22): 3341-3357.), Shrestha and Ghaboussi (1998)Sherestha, S.M., Ghaboussi, J, (1998). Evolution of optimum structural shapes using genetic algorithms. J. Struct. Eng. 123(11): 1331-8., Deb and Gulati (2001Deb, K., Gulati, S. (2001). Design of truss-structures for minimum weight using genetic algorithms. Finite Elem. Anal. Des. 37: 447-465.), Ahari et al. (2014)Ahrari, A., Atai, A.A., Deb, K. (2014). Simultaneous topology, shape and size optimization of truss structures by fully stressed design based on evolution strategy. Eng. Optimiz. DOI:10.1080/0305215X.2014.947972
https://doi.org/10.1080/0305215X.2014.94...
, Kaveh and Ahmadi (2014Kaveh, A and Ahmadi, B. (2014). Sizing, geometry and topology optimization of trusses using force method and supervised charged system search. Structural Engineering and Mechanics. 50(3):362-282. DOI: 10.12989/sem.2014.50.3.365
https://doi.org/10.12989/sem.2014.50.3.3...
), Kutyłowski and Rasiak (2014)Kutyłowski R.a and Rasiak, B.b (2014). The use of topology optimization in the design of truss and frame bridge girders. Structural Engineering and Mechanics. 51(1): 67-88. DOI: 10.12989/sem.2014.51.1.067
https://doi.org/10.12989/sem.2014.51.1.0...
, Gonçalves et al. (2015)Goncalves, M. S., Lopez, R. H. , Miguel, L. F. F. (2015).Search group algorithm: a new metaheuristic method for the optimization of truss structures. Comput. Struct., 153:165-184. DOI: 10.1016/j.compstruc.2015.03.003.
https://doi.org/10.1016/j.compstruc.2015...
. This fact motivates the development of optimization methods and their application to the SSTO problem.

In this context, a Backtracking Search algorithm (BSA) (Civicioglu 2013Civicioglu, P. (2013). Backtracking search optimization Algorithm for numerical optimization problems. Appl. Math. Comp. 219:8121-8144. DOI: 10.1016/j.amc.2013.02.017.
https://doi.org/10.1016/j.amc.2013.02.01...
. Miguel et al. 2015Miguel, L. F. F.; Fadel Miguel, L. F.; Lopez, R. H. (2015). Simultaneous Optimization of Force and Placement of Friction Dampers Under Seismic Loading. Eng. Optimiz., accepted for publication.) is presented in this paper for the simultaneous SSTO of truss structures in the presence of discrete and continuous design variables. The selection of the BSA was mainly based on two aspects:

(a) it was shown to outperform several algorithms in unconstrained optimization (Civicioglu, 2013Civicioglu, P. (2013). Backtracking search optimization Algorithm for numerical optimization problems. Appl. Math. Comp. 219:8121-8144. DOI: 10.1016/j.amc.2013.02.017.
https://doi.org/10.1016/j.amc.2013.02.01...
) and presented promising results for the optimization of transmission line towers structures, studied by Souza et al. (2016Souza, R.R., Miguel, L.F.F., Lopez, R.H., Miguel, L.F.F, Torii, A.J. (2016) A procedure for the size, shape and topology optimization of transmission line tower structures. Engineering Structures, 111, 162-184.);

(b) it has only one parameter to be set, different from other metaheuristic algorithms. Indeed, one of the main drawbacks of metaheuristic algorithms is that their efficiency depends on the tuning of many parameters usually by trial and error (Lopez et al. 2009aLopez, R. H., Luersen, M.A., Cursi, E.S. (2009a). Optimization of laminated composites considering different failure criteria. Compos. Part B-Eng 40:731-740. DOI: 10.1016/j.compositesb.2009.05.007.
https://doi.org/10.1016/j.compositesb.20...
, Lopez et al. 2009bLopez, R.H., Luersen, M.A., Cursi, E.S. (2009b). Optimization of Hybrid Laminated Composites using a Genetic Algorithm. J. Braz. Soc. Mech. Sci. 31(3): 269-278.).

This paper is organized as follows: in Section 2, the truss optimization problem is stated. In Section 3, the framework of the BSA is presented. Then, a set of benchmark problems regarding the SSTO of trusses is analyzed in Section 4, and finally Section 5 presents the main conclusions drawn from this work.

2 PROBLEM FORMULATION

The problem is formulated similarly to Miguel et al. (2013Miguel, L.F.F., Lopez, R.H., Miguel, L.F.F. (2013). Multimodal size, shape and topology optimization of truss structures using the firefly algorithm. Adv Eng. Softw. (56) 23-37. DOI: 10.1016/j.advengsoft.2012.11.006.
https://doi.org/10.1016/j.advengsoft.201...
), where the topology is optimized through a ground structure concept, in which members and nodes are allowed to be eliminated in order to vary the truss topology.

Simultaneously, the algorithm performs the size optimization of the truss by changing the cross-sectional area (A ∊ Rm ) of the remaining structural members and the shape optimization by modifying the nodal coordinates of the nodes q modeled as design variables (ξRq' ). This optimization procedure searches for the minimum structural weight of the truss subjected to stress, displacement and kinematic stability constraints. For convenience of notation, the design variables A and ξ are grouped into the vector x =[A 1,...,Am , ξ 1 ,..., ξq' ]. Thus, the optimization problem can be posed as:

(1)

Where nv = m+q' which is the number of design variables of the optimization problem, W stands for the structural weight, m is the number of members in design, ρ is the specific weight of the bar material, lj is the length of the jth bar (which is represented by the nodal distances). σj is the stress of the jth bar, while σt j and σc j are respectively the maximum allowable stress in tension and compression of the jth bar; δk and δ max k are the displacement and maximum allowable displacement of the kth node, respectively. Ω is the discrete set comprised by np available cross sectional areas; A min j and A max j are the lower and upper bounds of the cross-sectional area of the jth ar, respectively; and ξmin i and ξmax i are the lower and upper bounds of the allowable movement of the ith nodal coordinate treated as design variable, respectively.

The displacement constraints (G2) are formulated by considering that the deflection in a specified coordinate direction of a node must be lower than an allowable displacement chosen by the designer. Likewise, the stress in the element must be lower than the allowable stress in tension and compression of the material (G3). However, if buckling constraints are included in the formulation, the allowable stress in compression is the lowest value between the maximum stress in compression of the material and the critical stress given by the Euler's equation (see Eq.(5) in section 5.3). Finally, kinematic stability (G1) is achieved by checking the positive definiteness of the stiffness matrix of each solution, by evaluating the determinant of the stiffness matrix, i.e. computationally, it should be higher than a very small number, e.g.10-7. If it is not, a penalization of 1020 is applied to this solution. Thus, an unstable design is not analyzed and the constraints according to stress and/or displacement are not evaluated. It is important to highlight that every individual generated by the algorithm is counted as an objective function evaluation, even when the truss analyses is not performed (unstable topologies).

If the stability constraint is satisfied, the member forces and node displacements are calculated. Then, if one or more of the displacement and/or stress constraints are violated (constraints G2 and G3 in Eq. (1)), a penalty Pt is added to the objective function value of the current design. In this case, the penalty magnitude is proportional to the violation, and takes the form:

(2)

in which h is a positive parameter defined according to de characteristics and magnitude of each example, i is equal to t if the member is in tension and i is equal to c if the member is in compression. Finally, constraints on the allowable cross sectional areas and node positions are addressed by a coding approach. These bounds are imposed by not sampling infeasible designs in the computer code. Thus, the resulting optimization problem to be solved by the BSA is: stands for the absolute value,

(3)

3 BACKTRACKING SEARCH ALGORITM (BSA)

The BSA is multi-agent based evolutionary algorithm developed by Civicioglu (2013Civicioglu, P. (2013). Backtracking search optimization Algorithm for numerical optimization problems. Appl. Math. Comp. 219:8121-8144. DOI: 10.1016/j.amc.2013.02.017.
https://doi.org/10.1016/j.amc.2013.02.01...
) able to solve unconstrained non-convex optimization problems. It is thus employed in this paper to address the optimization problem given by Eq. 3.

An overview of the algorithm is presented in Figure 1. In the initialization step, the initial population of BSA is randomly generated. Then, the population is perturbed, using the "direction/length" of the perturbation; the parameter (mr ); the scale factor α and the matrix M. Because of the perturbation process, some variables may extrapolate the boundaries of the design domain; in this case, these variables are randomly regenerated. In the final step, the fitness values are evaluated in order to select the new population. The process is repeated until the stop criteria is met.

Figure 1:
Pseudocode of BSA.

For a more detailed description of the BSA, the reader is referred to Civicioglu (2013Civicioglu, P. (2013). Backtracking search optimization Algorithm for numerical optimization problems. Appl. Math. Comp. 219:8121-8144. DOI: 10.1016/j.amc.2013.02.017.
https://doi.org/10.1016/j.amc.2013.02.01...
) and Souza et al. (2016Souza, R.R., Miguel, L.F.F., Lopez, R.H., Miguel, L.F.F, Torii, A.J. (2016) A procedure for the size, shape and topology optimization of transmission line tower structures. Engineering Structures, 111, 162-184.)

In the original description of the BSA, the author claimed that this algorithm had only one parameter to be adjusted, the mix rate (mr ). It was only possible to have this one parameter due to the following aspects: (i) fixing the probability of occurrence of cases 1 and 2 in the construction of both Pold and M, and (ii) fixing the length of the perturbation, i.e. the value of α. Of course, these parameters may also be adjusted accordingly to the designer needs. In fact, on this paper the mix rate (mr ) was kept unchanged from the original version of BSA, while the scale factor α, was the parameter adjusted, based on its performance on each studied example.

4 BSA FOR SIMULTANEOUS TOPOLOGY, SHAPE AND SIZE OPTIMIZATION

The BSA was originally proposed for unconstrained continuous optimization problems. Thus, in order to apply it to a constrained and discrete problem, some modifications are necessary.

The constraints can be divided into two categories: the first one (G1, G2, and G3) regards the structural behavior (Section 2) and the second one (G4 and G5) regards the range of the design variables (Section 2). The former is addressed by the algorithm through a penalization strategy described by Eq. 2 and Eq. 3 of the Section 2. The scheme allows the optimization process to avoid undesirable parts of the domain, penalizing each design proportionally to its violation. The latter is addressed by setting upper and lower bounds to the range of the variables, which will be used by the BSA. Thus, the range is applied on the "Initialization" phase and also at the end of the "Construction of the trial or perturbed population Ppert . Hence, the constrained optimization problem can be transformed in an unconstrained optimization problem.

In order to deal with discrete variables, an operator is employed at the very beginning of the structural analysis code. In this step, the continuous values generated by the BSA are rounded into discrete ones. For instance, in the size optimization, the continuous variables can be rounded into each 1.0 in 2 or, still, into an integer value which corresponds to a value on a discrete set of cross-sectional areas. Then, the BSA only deals with continuous variables. It is important to highlight that this strategy does not demand any modifications on the BSA itself since the operator is applied within the structural analysis code.

Furthermore, this strategy it is easily applicable to any other optimization algorithms originally developed for unconstrained continuous problems. The approach employed on this paper has been successfully applied by other researches, such as Miguel and Fadel Miguel (2012Miguel, L.F.F; Fadel Miguel, L. F. (2012) Shape and size optimization of truss structures considering dynamic constraints through modern metaheuristic algorithms. Expert Systems with Applications. 39, 9458-9467.), Miguel et al. (2013)Miguel, L.F.F., Lopez, R.H., Miguel, L.F.F. (2013). Multimodal size, shape and topology optimization of truss structures using the firefly algorithm. Adv Eng. Softw. (56) 23-37. DOI: 10.1016/j.advengsoft.2012.11.006.
https://doi.org/10.1016/j.advengsoft.201...
and Gonçalves et al. (2015)Goncalves, M. S., Lopez, R. H. , Miguel, L. F. F. (2015).Search group algorithm: a new metaheuristic method for the optimization of truss structures. Comput. Struct., 153:165-184. DOI: 10.1016/j.compstruc.2015.03.003.
https://doi.org/10.1016/j.compstruc.2015...
.

In order to address the simultaneous optimization of size, shape and topology of trusses, a single-stage procedure is performed, in which all variables including topology, shape and size are determined simultaneously. Here, we employed different approaches for eliminating members when the cross-section areas are continuous or discrete. These approaches are detailed in the following.

In the case of continuous cross-sectional areas, during the optimization process, a specific member is eliminated from the ground structure following the criteria proposed by Deb and Gulati (2001Deb, K., Gulati, S. (2001). Design of truss-structures for minimum weight using genetic algorithms. Finite Elem. Anal. Des. 37: 447-465.). The cross-sectional area of a member is compared to a specific critical cross-sectional area. If the member area is smaller than this specific value, this element is eliminated from the ground structure. This method defines how different topologies can be obtained in a continuous optimization procedure. Note that the critical cross-sectional area and the lower (A min) and upper (A max) bounds of the cross-sectional areas must be determined by considering the probability of a specific element to be absent from the final solution. For example, if A min = -A max and the critical cross-sectional area is zero, the probability of any member being present in the final structure is approximately 50 %.

In the case of discrete cross-sectional areas, the user defines the number of zeros that are added to the available profile set Ω, where a member addressed to a zero cross-sectional area, is eliminated from the ground structure. As well as the values of A min, A max and the critical area for the continuous case, the number of zeros added to the set Ω is defined in order to generate a reasonable probability of eliminating an element from the ground structure. It is important to highlight that the supports and nodes carrying loads cannot be disregarded in the final solution.

Finally, the results presented on the following Tables and Figures are feasible solutions (i.e. a solution that respects all the imposed constraints) achieved by the optimization algorithm. The results are compared regarding the weight of the best design, mean value and coefficient of variation of the 100 independent runs. The additional information about the best designs (found by the present study and the ones reported in the literature) are presented in the Appendix A.

4.1 Eleven-Bar Truss Example

This truss has been extensively used as a benchmark problem and appears in several papers on truss optimization. The ground structure for this example is shown in Figure 2 and the design parameters are given in Table 1. Here, we perform the simultaneous SSTO of this structure.

Table 1:
Design parameters for the eleven-bar truss problem.

Figure 2:
Eleven-bar truss benchmark example.

This problem was studied by Rajan (1995Rajan, S. D. (1995). Sizing, shape, and topology design optimization of trusses using genetic algorithms. J. Struct. Eng. 121(10): 1480-1487.) and Balling et al. (2006Balling, R.J., Briggs, R.R., Gillman, K. (2006). Multiple optimum size/shape/topology designs for skeletal structures using a genetic algorithm., J Struct. Eng. 132(7): 1158-1165. DOI:10.1061/ ASCE0733-9445 2006132:7 1158) who employed the GA, Martini (2011Martini, K. (2011). Harmony Search Method for Multimodal Size, Shape, and Topology Optimization of Structural Frameworks. J. Struct. Eng. 137:11. DOI: 10.1061/(ASCE)ST.1943-541X.0000378.
https://doi.org/10.1061/(ASCE)ST.1943-54...
) who employed the Harmony Search (HS), and Miguel et al. (2013Miguel, L.F.F., Lopez, R.H., Miguel, L.F.F. (2013). Multimodal size, shape and topology optimization of truss structures using the firefly algorithm. Adv Eng. Softw. (56) 23-37. DOI: 10.1016/j.advengsoft.2012.11.006.
https://doi.org/10.1016/j.advengsoft.201...
) who employed the FA. In this study, the truss shape is optimized allowing the vertical coordinates of the three superior nodes (1, 2 and 3, Figure 2) to move between 457.2 cm and 2540 cm. The cross-sectional areas, for each of the 11 bars, come from the discrete set Ω = (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10.45159, 11.61288, 15.35481, 16.90319, 18.58061, 19.93544, 20.19351, 1.80641, 23.41931, 24.77414, 24.96769, 26.96769, 28.96768, 30.96768, 32.06445, 33.03219, 37.03218, 46.58055, 51.41925, 74.19340, 87.09660, 89.67724, 91.61272, 99.99980, 103.22560, 121.29008, 128.38684, 141.93520, 147.74164, 170.96740, 193.54800, 216.12860)cm 2. Thus, the design vector can be written as x = [A 1 , A 2 ,...A 11 , ξ 1, ξ 2, ξ 3]. Note that the first 10 cross-section areas from the set Ω are null values. Where a member addressed to a null area is eliminated from the structure.

Because the nodal coordinates are continuous variables and the cross-sectional areas are taken from a set of 42 discrete variables, the problem is a mixed variable optimization problem in that it simultaneously addresses discrete and continuous design variables.

In this example, the stopping criterion for each search was set as the number of objective function evaluations, i.e. OFE = 50000, to compare the results obtained here to the ones of Miguel et al. (2013Miguel, L.F.F., Lopez, R.H., Miguel, L.F.F. (2013). Multimodal size, shape and topology optimization of truss structures using the firefly algorithm. Adv Eng. Softw. (56) 23-37. DOI: 10.1016/j.advengsoft.2012.11.006.
https://doi.org/10.1016/j.advengsoft.201...
), which found the best design in the literature and provided statistics for a series of runs of the FA.

The parameters in this example were set as npop = 10, it max = 5000, the penalization term h was set as 108 and the scale factor α was set as 3 randn, where randn is random number from a standard normal distribution between 0 and 1. The results obtained by the BSA are presented in Table 2 and compared to the results of other algorithms taken from the literature. Figures 3 illustrates the topology of the optimized structure. Figure 4 showsthe converge curve of the best result found with BSA and FA (Miguel et al. 2013Miguel, L.F.F., Lopez, R.H., Miguel, L.F.F. (2013). Multimodal size, shape and topology optimization of truss structures using the firefly algorithm. Adv Eng. Softw. (56) 23-37. DOI: 10.1016/j.advengsoft.2012.11.006.
https://doi.org/10.1016/j.advengsoft.201...
), one can notice that despite the similar results, the BSA converged significantly faster.

Table 2:
Results for the size, shape and topology optimization of the eleven-bar truss.

Figure 3:
Optimized configuration for the size, shape and topology optimization problem of the eleven-bar.

Figure 4:
Convergence curves for the size, shape and topology optimization of the eleven-bar truss.

The best result over 100 runs in this study weights 1227.070 kg, which is slightly worse than the value found by Miguel et al. (2013Miguel, L.F.F., Lopez, R.H., Miguel, L.F.F. (2013). Multimodal size, shape and topology optimization of truss structures using the firefly algorithm. Adv Eng. Softw. (56) 23-37. DOI: 10.1016/j.advengsoft.2012.11.006.
https://doi.org/10.1016/j.advengsoft.201...
) (1227.040 kg), but better than the optimum weight determined by Balling et al. (2006Balling, R.J., Briggs, R.R., Gillman, K. (2006). Multiple optimum size/shape/topology designs for skeletal structures using a genetic algorithm., J Struct. Eng. 132(7): 1158-1165. DOI:10.1061/ ASCE0733-9445 2006132:7 1158) (1241.029 kg), Rajan (1995Rajan, S. D. (1995). Sizing, shape, and topology design optimization of trusses using genetic algorithms. J. Struct. Eng. 121(10): 1480-1487.) (1475.999 kg) and Martini (2011Martini, K. (2011). Harmony Search Method for Multimodal Size, Shape, and Topology Optimization of Structural Frameworks. J. Struct. Eng. 137:11. DOI: 10.1061/(ASCE)ST.1943-541X.0000378.
https://doi.org/10.1061/(ASCE)ST.1943-54...
) (1315.418 kg). The mean value and COV provided by the BSA are equal to 1265.654 kg and 2.02%, respectively, which are better than the ones found by the FA of Miguel et al. (2013)Miguel, L.F.F., Lopez, R.H., Miguel, L.F.F. (2013). Multimodal size, shape and topology optimization of truss structures using the firefly algorithm. Adv Eng. Softw. (56) 23-37. DOI: 10.1016/j.advengsoft.2012.11.006.
https://doi.org/10.1016/j.advengsoft.201...
(1268.285 kg and 2.12%, respectively). Thus, the proposed scheme improved the mean and COV results than the cited references.

4.2 Thirty-Nine-Bar Two-Tiered Truss Example

The second benchmark example is the single-span 39-bar, 12-node, simply supported, two-tiered ground structure shown in Figure 5. This structure was studied before by Deb and Gulati (2001Deb, K., Gulati, S. (2001). Design of truss-structures for minimum weight using genetic algorithms. Finite Elem. Anal. Des. 37: 447-465.), Luh and Lin (2008Luh, G.C., Lin, C.Y. (2008). Optimal design of truss structures using ant algorithm. Structural and Multidisciplinary Optimization, 36(4), 365-379.), Wu and Tseng (2010Wu, C.Y., Tseng, K.Y. (2010). Truss structure optimization using adaptive multi-population differential evolution. Structural and Multidisciplinary Optimization, 42(4), 575-590.), Lu and Lin (2011)Luh, G.C., Lin, C.Y. (2011). Optimal design of truss-structures using particle swarm optimization. Computers & Structures, 89(23), 2221-2232., Miguel et al. (2013Miguel, L.F.F., Lopez, R.H., Miguel, L.F.F. (2013). Multimodal size, shape and topology optimization of truss structures using the firefly algorithm. Adv Eng. Softw. (56) 23-37. DOI: 10.1016/j.advengsoft.2012.11.006.
https://doi.org/10.1016/j.advengsoft.201...
) and Ahrari et al. (2014Ahrari, A., Atai, A.A., Deb, K. (2014). Simultaneous topology, shape and size optimization of truss structures by fully stressed design based on evolution strategy. Eng. Optimiz. DOI:10.1080/0305215X.2014.947972
https://doi.org/10.1080/0305215X.2014.94...
).

Figure 5:
Two-tiered truss, thirty-nine-member benchmark example (Miguel et al 2013Miguel, L.F.F., Lopez, R.H., Miguel, L.F.F. (2013). Multimodal size, shape and topology optimization of truss structures using the firefly algorithm. Adv Eng. Softw. (56) 23-37. DOI: 10.1016/j.advengsoft.2012.11.006.
https://doi.org/10.1016/j.advengsoft.201...
).

Two independent studies are performed: (i) size and topology optimization and (ii) simultaneous SSTO. However, the cross-sectional areas are treated as continuous variables in this case to properly compare the results to those of the literature. The overlapping members are shown laterally dislocated in the figure for visual clarity. Because the vertical symmetry around member 19 is assumed, the number of variables is reduced to 21. The allowable strength is 137.895 MPa; the material properties and maximum allowable deflection are the same as those in the previous problem. The stopping criterion is set to OFE = 50000. The BSA parameters employed in both cases were npop = 20 and it max = 2500, the penalization term h was set as 108, and the scale factor α was set as 1/gamrnd(1,0.5), where gamrnd(1,0.5) is a random number from a gamma distribution with shape parameter 1 and scale parameter 0.5.

4.2.1 Size and Topology Optimization

The procedure for this example consists of the optimization of 21 continuous variables. Each size variable is allowed to vary between -1.4516∙10-4 and 1.45161∙10-3, (in square meters). Where, for values below 3.2258∙10-5, the respective member is eliminated from the ground structure. The design vector can be written x = [A 1 , A 2 ,...A 21]. The results found by this study, and others provided by the literature, for this example are listed in Table 3. Figure 6 shows the topology of the best solution given by the BSA, which contains 10 nodes and 18 members. The convergence curves of the BSA and the FA (from Miguel et al. 2013Miguel, L.F.F., Lopez, R.H., Miguel, L.F.F. (2013). Multimodal size, shape and topology optimization of truss structures using the firefly algorithm. Adv Eng. Softw. (56) 23-37. DOI: 10.1016/j.advengsoft.2012.11.006.
https://doi.org/10.1016/j.advengsoft.201...
) for this problem are shown in Figure 7.

Table 3:
Results for the size and topology optimization of the eleven-bar truss.

Figure 6:
Optimized configuration for the size and topology optimization of the thirty-nine-member truss.

Figure 7:
Convergence curves for the size and topology optimization of the thirty-nine-member truss.

The best result over 100 runs weights 87.637 kg, which is slightly heavier than the 87.549 kg found by Lu and Lin (2011)Luh, G.C., Lin, C.Y. (2011). Optimal design of truss-structures using particle swarm optimization. Computers & Structures, 89(23), 2221-2232.. The mean value obtained using the BSA is equal to 93.284 kg and the coefficient of variation is 5.28%. The coefficient of variation and mean value are only compared with Miguel et al. 2013Miguel, L.F.F., Lopez, R.H., Miguel, L.F.F. (2013). Multimodal size, shape and topology optimization of truss structures using the firefly algorithm. Adv Eng. Softw. (56) 23-37. DOI: 10.1016/j.advengsoft.2012.11.006.
https://doi.org/10.1016/j.advengsoft.201...
, since the other studies do not provide this information.

4.2.2 Size, Shape and Topology Optimization

To carry out simultaneous SSTO, selected nodal coordinates are taken as design variables in addition to the 21 cross sectional areas. Assuming symmetry and considering that constrained and load-carrying nodes must remain fixed, and the highest node at the center of the structure does not move laterally, it is possible to reduce the number of nodal coordinates to 7. Each of these 7 nodes is allowed to move (-304.8, 304.8) cm from its original position (Figure 5). The design vector can then be written as: x = [A 1 , A 2 ,...A 21 , ξ 1, ξ 2, ξ7 ].

Figure 8 illustrates the topology of the best design found by the BSA over the 100 runs. Note that this topology contains 11 nodes and 20 members. Figure 9 presents the results reached by the present study and others found in the literature are listed in Table 4. The best result over 100 runs achieved by the BSA weights 86.753 kg, which is slightly lighter than the one found by Miguel et al. (2013Miguel, L.F.F., Lopez, R.H., Miguel, L.F.F. (2013). Multimodal size, shape and topology optimization of truss structures using the firefly algorithm. Adv Eng. Softw. (56) 23-37. DOI: 10.1016/j.advengsoft.2012.11.006.
https://doi.org/10.1016/j.advengsoft.201...
), but heavier in comparison to the result presented by Ahrari et al. (2014Ahrari, A., Atai, A.A., Deb, K. (2014). Simultaneous topology, shape and size optimization of truss structures by fully stressed design based on evolution strategy. Eng. Optimiz. DOI:10.1080/0305215X.2014.947972
https://doi.org/10.1080/0305215X.2014.94...
), which is 82.091 kg. The mean value and COV achieved by the BSA optimization scheme are equal to 100.590 kg and 4.60%, respectively, while the ones of the FA are 94.478 kg and 5.3%, respectively. Hence, in this example, the BSA was able to improve the COV.

Figure 8:
Optimized configuration for the SSTO of the thirty-nine-member truss.

Figure 9:
Convergence curves for the size, shape and topology optimization of the thirty-nine-member truss.

Table 4:
Results for the size, shape and topology optimization of the thirty-nine-member truss.

Figure 9 shows the convergence curves for the best results found with the BSA and FA (from Miguel et al. 2013Miguel, L.F.F., Lopez, R.H., Miguel, L.F.F. (2013). Multimodal size, shape and topology optimization of truss structures using the firefly algorithm. Adv Eng. Softw. (56) 23-37. DOI: 10.1016/j.advengsoft.2012.11.006.
https://doi.org/10.1016/j.advengsoft.201...
), similarly to previous examples the BSA converged faster, in comparison to the FA. Furthermore, in Figure 10 is shown the convergence curves for the cases 4.2.1 and 4.2.2. Where the better convergence in the earlier iterations for the size and topology case can be associated to the increase of possibilities provided by the geometrical variations allowed in case 4.2.2.

Figure 10:
Convergence curves for the size and topology and size, shape and topology optimization of the thirty-nine-member truss.

4.3 Fifteen-Bar Planar Truss Example

This fifteen-bar planar truss was studied by Wu and Chow (1995Wu, S.J., Chow,P.T., (1995). Integrated discrete and configuration optimization of trusses using genetic algorithms. Comput. Struct. 56(4):695-702.), Tang et al. (2005Tang, W., Tong,L., Gu,Y., (2005). Improved genetic algorithm for design optimization of truss structures with sizing, shape and topology variables. International Journal for Numerical Methods in Engineering 62:1737-1762.), Rahami et al. (2008), Miguel et al. (2013Miguel, L.F.F., Lopez, R.H., Miguel, L.F.F. (2013). Multimodal size, shape and topology optimization of truss structures using the firefly algorithm. Adv Eng. Softw. (56) 23-37. DOI: 10.1016/j.advengsoft.2012.11.006.
https://doi.org/10.1016/j.advengsoft.201...
) and Gholaizadeh (2013). The ground structure is illustrated in Figure 11, showing a vertical tip load P = 44482.2161 N applied on node 8. The allowable strength of the material is 172.369 MPa for both tension and compression, and the material properties (modulus of elasticity and weight density) are the same as in the previous examples. The x and y coordinates of nodes 2, 3, 6 and 7 and the y-coordinate of the nodes on bar9 are taken as design variables. However, nodes 6 and 7 are constrained to have the same xcoordinates of the nodes 2 and 3, respectively. Thus, the problem includes fifteen size and eight shape variables (x2=x6, x3=x7, y2, y3, y4, y6, y7, y8).

Figure 11:
Fifteen-bar planar truss benchmark example.

The cross-sectional areas are chosen from the set Ω = (0.716, 0.910, 1.123, 1.419, 1.742, 1.852, 2.239, 839, 3.477, 6.155, 6.974, 7.5748.600, 9.600, 11.381, 13.819, 17.400,18.064, 20.200, 23.000, 24.600, 1.000, 38.400, 42.400, 46.400, 55.000, 60.000, 70.000, 86.000, 92.193, 110.774, 123.742) cm². The side constraints for the shape variables are 254 cm ≤ x2 ≤ 355.6 cm, 558.8 cm ≤ x3 ≤ 660.4 cm, 254 cm ≤ y2 ≤ 355.6 cm, 254 cm ≤ y3 ≤ 355.6 cm, 127 cm ≤ y4 ≤ 228.6 cm, -50.8 cm ≤ y6 ≤ 50.8 cm, -50.8 cm ≤ y7 ≤ 50.8 cm, and 50.8 cm ≤ y8 ≤ 50.8 cm.

Two independent studies are performed: (i) size and shape optimization, (ii) simultaneous SSTO. On both cases buckling constraints are considered.

For all examples the design vector can be written as x = [A 1 , A 2 ,...A 15 , ξ 1, ξ 2, ξ8 ]. In order to perform the topology optimization10 null cross-sectional areas are added to the available set Ω. If a size variable Aj assumes a null value, the respective member is eliminated from the structure.

The BSA parameters were set in these two studies as npop = 8 and it max = 1000, resulting in 8000 OFE. The penalization term was set as 108 and the scale factor α was set as 1/gamrnd(1,0.5).

The member stresses are constrained to be below the Euler buckling stress, as shown in Eq. 4:

(4)

4.3.1 Size and Shape Optimization with Buckling Constraints

Wu and Chow (1995Wu, S.J., Chow,P.T., (1995). Integrated discrete and configuration optimization of trusses using genetic algorithms. Comput. Struct. 56(4):695-702.) and Miguel et al. (2013Miguel, L.F.F., Lopez, R.H., Miguel, L.F.F. (2013). Multimodal size, shape and topology optimization of truss structures using the firefly algorithm. Adv Eng. Softw. (56) 23-37. DOI: 10.1016/j.advengsoft.2012.11.006.
https://doi.org/10.1016/j.advengsoft.201...
) have studied this problem using the GA and FA, respectively.

Wu and Chow (1995Wu, S.J., Chow,P.T., (1995). Integrated discrete and configuration optimization of trusses using genetic algorithms. Comput. Struct. 56(4):695-702.) carried out the optimization using a population of 30 individuals. However, they did not mention the maximum number of generations. Miguel et al. (2013Miguel, L.F.F., Lopez, R.H., Miguel, L.F.F. (2013). Multimodal size, shape and topology optimization of truss structures using the firefly algorithm. Adv Eng. Softw. (56) 23-37. DOI: 10.1016/j.advengsoft.2012.11.006.
https://doi.org/10.1016/j.advengsoft.201...
) employed 10 fireflies and 800 iterations, resulting in 8000 OFE.

The BSA results together with a comparison with the GA and FA are presented in Table 5. The best design found by the BSA is illustrated in Figure 12. Figure 13 shows the convergence curves of the best results found with the BSA and FA (from Miguel et al. (2013Miguel, L.F.F., Lopez, R.H., Miguel, L.F.F. (2013). Multimodal size, shape and topology optimization of truss structures using the firefly algorithm. Adv Eng. Softw. (56) 23-37. DOI: 10.1016/j.advengsoft.2012.11.006.
https://doi.org/10.1016/j.advengsoft.201...
)).

Table 5:
Results for the size and shape optimization with buckling constraint of the 15-bar truss.

Figure 12:
Optimized configuration for the size and shape optimization with buckling constraint of the 15-bar truss.

Figure 13:
Convergence curves for the best result for the size and shape optimization with buckling constraint of the 15-barr truss.

The best results over 100 runs achieved by the BSA weights 59.641 kg, which is lighter than the ones of the GA and FA. The mean value and COV reached by the BSA are equal to 64.049 kg and 3.96%, respectively. Thus, on the one hand, the BSA reduced the mean value of the independent runs in 9.21%, and on the other hand, the COV was around 2.7% higher than the one of the FA.

4.3.2 Size, Shape and Topology Optimization with Buckling Constraints

As proposed by Miguel et al. (2013Miguel, L.F.F., Lopez, R.H., Miguel, L.F.F. (2013). Multimodal size, shape and topology optimization of truss structures using the firefly algorithm. Adv Eng. Softw. (56) 23-37. DOI: 10.1016/j.advengsoft.2012.11.006.
https://doi.org/10.1016/j.advengsoft.201...
), in addition to the eight discrete size and five continuous configuration variables, all fifteen member groups are considered as topology variables in this example. In order to employ the topology optimization, simultaneously to the size and shape optimization, 10 null areas are added to the set of possible cross-sectional areas. Thus, when a null area is addressed, the respective member is eliminated from the structure.

The results of the BSA and FA are presented in Table 6, while the topology of the best design is illustrated in Figure 14. Figure 15 shows the convergence curves of the best results found with the BSA and FA (from Miguel et al. (2013Miguel, L.F.F., Lopez, R.H., Miguel, L.F.F. (2013). Multimodal size, shape and topology optimization of truss structures using the firefly algorithm. Adv Eng. Softw. (56) 23-37. DOI: 10.1016/j.advengsoft.2012.11.006.
https://doi.org/10.1016/j.advengsoft.201...
)), one can notice that the BSA presented a faster convergence in comparison to the FA. Furthermore, Figure 16 presents the converge curves for cases 4.3.1 and 4.3.2, where it is possible to obverse that the increase of possible solutions (provided by the topology variations allowed) delayed the convergence of the algorithm, however achieve a better result.

Table 6:
Results for the size, shape and topology optimization with buckling constraint of the 15-bar truss.

Figure 14:
Optimized configuration for the size, shape and topology optimization with buckling constraint of the 15-bar truss.

Figure 15:
Convergence curves for the best result for the size, shape and topology optimization with buckling constraint of the 15-bar truss.

Figure 16:
Convergence curve for the best result for the size and shape and size, shape and topology optimization with buckling constraint of the 15-bar truss.

The best design over 100 runs achieved by the BSA weights 54.418 kg, which is 3.48% lighter than the best design provided by the FA. The mean value and COV reached in the present study are equal to 60.187 kg and 7.88%, respectively. Thus, the BSA reduced the mean value by 13.05% and the COV by 0.81%, when compared to the FA. The results for this case show that the BSA was capable of not only provide the lightest design ever found for this problem, but also lower mean and COV values.

4.4 Twenty-Five-Bar 3D Truss Example

This example was studied before by Rajeev and Krishnamoorthy (1992Rajeev, S., Krishnamoorthy,C.S. (1992). Discrete optimization of structures using genetic algorithms, J. Struct. Eng. 118 (5): 1233-1250.), Wu and Chow (1995Wu, S.J., Chow,P.T., (1995). Integrated discrete and configuration optimization of trusses using genetic algorithms. Comput. Struct. 56(4):695-702.), Tang et al. (2005Tang. W., Tong. L., Gu, Y., (2005). Improved genetic algorithm for design optimization of truss structures with sizing, shape and topology variables. Int. J. Numer. Meth. Eng (62) 1737-1762.),Rahami et al. (2008), Miguel et al. (2013Miguel, L.F.F., Lopez, R.H., Miguel, L.F.F. (2013). Multimodal size, shape and topology optimization of truss structures using the firefly algorithm. Adv Eng. Softw. (56) 23-37. DOI: 10.1016/j.advengsoft.2012.11.006.
https://doi.org/10.1016/j.advengsoft.201...
), Gholaized (2013) and Ahrari Atai and Deb (2014Ahrari, A., Atai, A.A., Deb, K. (2014). Simultaneous topology, shape and size optimization of truss structures by fully stressed design based on evolution strategy. Eng. Optimiz. DOI:10.1080/0305215X.2014.947972
https://doi.org/10.1080/0305215X.2014.94...
).The ground structure is shown in Figure 17. The mechanical properties and the loading conditions are presented in Table 7, while the nodes coordinates and member grouping are presented in Table 8. The structure is subject to a stress constraint of 275.8 MPa for each member, and a displacement constraint of 0.89 cm for each node.

Figure 17:
Twenty-five-bar 3D truss example.

Table 7:
Mechanical properties and loading conditions for the twenty-five-bar 3D truss example.

Table 8:
Nodes coordinates and member grouping for the twenty-five-bar 3D truss example.

During the optimization the cross-sectional areas are chosen from the set Ω = (0.645, 1.290, 1.936, 2.581, 3.226, 3.871, 4.512, 5.161, 5.807, 6.452, 7.097, 7.742, 8.387, 9.032, 9.677, 10.323, 10.968, 11.613, 12.258, 12.903, 13.548, 14.193, 14.839, 15.484, 16.129, 16.774, 18.065, 19.355, 21.935) cm2.

Two independent studies were performed: (i) size and shape optimization and (ii) simultaneous size, shape and topology optimization.

The BSA parameters were set in these two studies as npop = 10 and it max = 600, resulting in 6000 OFE. The penalization term was set as 108 and the scale factor α was set as 4·randg, where randg is a scalar random value chosen from a gamma distribution with unit scale and shape. The statistical results presented, for both cases, are extracted from 100 independent runs of the algorithm.

4.4.1 Size and Shape Optimization

For the size variation, the member groups are allowed to assume the cross-sectional areas from the set Ω. The x, y, and z coordinates of nodes 3, 4, 5 and 6 and the x and y coordinates of nodes 7-10 are taken as design variables, while nodes 1 and 2 remains unchanged. Due to symmetry of the structure there are five shape variables, with the following bounds: 50.8 cm ≤ x4≤ 152.4 cm, 101.6 cm ≤ x8≤ 203.2 cm, 101.6 cm ≤ y4≤ 203.2 cm, 254 cm ≤ y8≤ 355.6 cm and 228.6 cm ≤ z4≤ 330.2 cm. Thus, the design vector can be written as x = [A 1 , A 2 ,...A 8 , ξ 1, ξ 2,... ξ 5].

The results for this case are presented in Table 9, the best result found with the BSA was 54.02 kg, which slightly worse compared to Miguel et al. (2013Miguel, L.F.F., Lopez, R.H., Miguel, L.F.F. (2013). Multimodal size, shape and topology optimization of truss structures using the firefly algorithm. Adv Eng. Softw. (56) 23-37. DOI: 10.1016/j.advengsoft.2012.11.006.
https://doi.org/10.1016/j.advengsoft.201...
) and Gholaized (2013). Figure 18 presents the convergence curves for the BSA and FA and Figure 19 illustrates the best design found with BSA.

Table 9:
Results for the size and shape optimization of the Twenty-five-bar 3D truss example.

Figure 18:
Convergence curves for the size and shape optimization of the twenty-five-bar 3D truss example.

Figure 19:
Optimized configuration for the size and shape optimization of the twenty-five-bar 3D truss example.

4.4.1 Size, Shape and Topology Optimization

In order to perform the topology variation each one of the eight member groups are allowed to be eliminated during the optimization process. To do so, 20 null areas are added to the set of possible cross-sectional areas Ω.

The results for this case are shown in Table 10, the BSA presented a lower C.O.V compared with the FA, however the best result (53.94 kg) and mean weight (66.84 kg) were worst compared to previous works. The configuration of the best result found with the BSA and the convergence curves of the BSA and FA are presented in Figures 20 and 21, respectively. Additionally, Figure 22 presents the convergences curves for the size and shape and size, shape and topology optimization cases. Similarly to the previous examples, the BSA presented a faster convergence in the early iterations.

Table 10:
Results for the size, shape and topology optimization of the Twenty-five-bar 3D truss example.

Figure 20:
Optimized configuration for the size, shape and topology optimization of the twenty-five-bar 3D truss example.

Figure 21:
Convergence curves for the size and shape optimization of the twenty-five-bar 3D truss example.

Figure 22:
Convergence curve for the best result for the size and shape and size, shape and topology optimization of the twenty-five-bar 3D truss example

5 CONCLUDING REMARKS

This paper presented a Backtracking Search Algorithm (BSA) for the simultaneous SSTO of truss structures. It focused on the optimization of these three aspects since it is well known that the most effective scheme of truss optimization is when one considers them simultaneously. In addition, in most of the examples, we deal concurrently with continuous and discrete design variables, due to the difficulties that they impose.

The effectiveness and robustness of the algorithm were demonstrated through a series of benchmark problems taken from the literature. Not only the best design obtained by each algorithm was compared, but also the statistics of the best designs found over 100 independent runs of the BSA. This statistical analysis is of paramount importance due to the stochastic nature of metaheuristic algorithms. Hence, the results presented here may serve as a more formal basis of comparison for future researchers analyzing the same problems with other optimization methods.

From the numerical analysis, it was shown that the BSA provided the best design ever found for several of the analyzed problems. Moreover, in some cases, it was also able to improve the statistics of the independent runs such as the mean and COV values. These results emphasize the capabilities of the BSA in this field and motivate its further development as well its application to real engineering problems.

Acknowledgements

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

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APPENDIX A

On this section, the information regarding the best designs, for the studied examples, are presented on the following Tables.

Table A1:
Cross-sectional area values in cm² and weight in kg for the size, shape and topology optimization of the eleven-bar truss, on Section5.1.

Table A2:
Cross-sectional area values in cm² and weight in kg for the size and topology optimization of the thirty-nine-member truss, on Section 5.2.1.

Table A3:
Cross-sectional area values in cm² and weight in kg for the size, shape and topology optimization of the thirty-nine-member truss, on Section 5.2.2.

Table A4:
Cross sectional area (cm²) and weight (kg) for the size and shape optimization with buckling constraint of the 15-bar truss, on Section 5.3.1.

Table A5:
Cross sectional area (cm²) and weight (kg) for the size, shape and topology optimization with buckling constraint of the 15-bar truss.

Table A6:
Results for the size and shape optimization of the Twenty-five-bar 3D truss example.

Table A7:
Results for the size, shape and topology optimization of the Twenty-five-bar 3D truss example.

Publication Dates

  • Publication in this collection
    Dec 2016

History

  • Received
    18 May 2016
  • Reviewed
    14 Sept 2016
  • Accepted
    20 Sept 2016
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