The natural frequencies of a truss structure can match with its resonant frequencies. For example, if a time-varying force with a frequency equivalent to one of the natural frequencies is employed to the system, this will result in immense amplitude vibrations that risk putting the truss structure in jeopardy. This is why when designing a truss structure, it’s important to calculate and limit the natural frequencies to a given range. Generally, a truss optimization problem with frequency constraints has the cross-sectional areas of members and/or nodal coordinates of joints as the sizing and layout variables, respectively. In such a problem, the first several natural frequencies will be limited to given upper and/or lower bounds.

Truss optimization with frequency constraints is very difficult due to its high nonlinear property and high dimensions. But in view of its importance in the real structural design practice, many efforts have been made to cope with it. In this paper, we will present an effective heuristic algorithm for the truss optimization with frequency constraints. The new algorithm called the medalist learning algorithm (MLA) is inspired by the learning behavior of individuals in a group. The process of learning is divided into learning stages. In each stage, individuals will first identify the excellent performers called medalists in this study according to their performances and then conduct individual learning. Medalists will carry out the self-improving trials to improve their performances. But a common learner will learn from his/her past experience and imitate the medalists to improve his/her later performance. In each learning stage, a learning efficiency will be generated using a learning curve which comes from the Sigmoid function. The learning efficiency will used to balance the exploitation and exploration capacities of the algorithm. We will employ six well-known truss optimization problems with frequency constraints to verify the efficiency of the new algorithm by comparing the results obtained by the MLA with those reported previously in literature.

In the following, we will introduce the related studies done in the past several decades. We will categorize them into two basic classes: truss optimization without frequency constraints and truss optimization with frequency constraints. We further group the studies in the former class into three subclasses: earlier efforts with classical methods, truss optimization using genetic algorithm, and optimization using a variety of heuristics. All the studies will be presented in the order of the publishing time.

The first class is the truss optimization without frequency constraints. For this problem, the earlier efforts often applied various classical optimization methods. Toakley(1968)[1] proposed to consider the finite range of discrete member sizes in the structural design methods. Aldo and Robert (1971)[2] proposed a branch-and-bound algorithm for member-sizing optimization problems with discrete variables. Schmit and Fleury (1980)[3] extended the approximation concepts and dual methods to solve structural synthesis problems involving a mix of discrete and continuous sizing variables. Templeman and Yates (1983)[4] proposed a linear programming approach to the discrete optimum design of trusses. Duan (1986)[5] presented a 0–1 simplex method for the optimum design of trusses with discrete member sizes based on the Templeman’s algorithm. Ringertz (1988)[6] proposed a branch and bound algorithm for the discrete structural optimization based on Lagrangean relaxation and rounding off the continuous solution. Templeman (1988)[7] compared several methods for the optimum design of trusses and pointed out that difference of the nature of the discrete design problem from that of the continuous design problem. Hansen and Vanderplaats (1990)[8] presented an efficient method for truss configuration optimization utilizing an approximate structural analysis based on first-order Taylor series expansions of the member forces. Salajegheh and Vanderplaats (1993)[9] presented a branch and bound method with the approximation of responses, such as forces and displacements, for optimizing truss structures with discrete design variables.

To improve the efficiency and avoid trapping at local optima, researchers started to implement various genetic algorithms to solving truss optimization problems without frequency constraints. Rajeev and Krishnamoorthy (1992)[10] proposed a genetic algorithm for discrete optimization of truss structures and designed a penalty-based transformation for handling the constraints. Wu and Chow (1995)[11] proposed a modified genetic algorithm based on the Rajeev and Krishnamoorthy (1992)[10] for the sizing optimization of trusses with discrete variables. Rajeev and Krishnamoorthy (1997)[12] proposed two genetic algorithms-based methods for the size and configuration optimization and the topology optimization, respectively. Hasançebi and Erbatur (2001)[13] introduced two methodologies, “annealing perturbation” and “adaptive reduction of the design space”, in conjunction with genetic algorithms for the simultaneous treatment of size, shape and topology variables in the optimum design of space trusses. Kaveh and Kalatjari (2004)[14] solved the size and geometry optimization of trusses subsequently with an improved genetic algorithm. Tang et al. (2005)[15] presented an improved genetic algorithm to minimize weight of truss with sizing, shape and topology discrete and continuous variables. Rahami et al. (2008)[16] implemented the combination of energy and force method with the genetic algorithm as an optimization tool to the sizing, geometry and topology optimization of trusses.

The success of application of various genetic algorithms encouraged many researchers to try other more efficient heuristic algorithms in this field. Lee et al. (2005)[17] applied the harmony search heuristic algorithm to discrete truss structural size optimization. Kaveh and Talatahari (2009)[18] proposed the discrete heuristic particle swarm ant colony optimization (DHPSACO) to optimize truss structures with discrete variables. Li et al. (2009)[19] presented a heuristic particle swarm optimizer (HPSO) algorithm for truss structures with discrete variables based on the standard particle swarm optimizer (PSO) and the harmony search (HS) scheme. Gholizdeh et al. (2011)[20] proposes a modified harmony search (MHS) algorithm based on elitism for size and shape optimization of structures. Sadollah et al. (2012)[21] presented a novel optimization method called mine blast algorithm (MBA inspired from the explosion of mine bombs in real world for the optimization of truss structures with discrete variables. Gholizadeh (2013)[22] proposed an efficient hybrid optimization algorithm for layout optimization of truss structures by integrating the computational merits of the cellular automata (CA) and the particle swarm optimization (PSO). Miguel et al. (2013)[23] presented an efficient single-stage Firefly-based algorithm (FA) to simultaneously optimize the size, shape and topology of truss structures. Kaveh and Mahdavi (2014)[24] implemented the meta-heuristic algorithm Colliding Bodies Optimization (CBO) which is inspired by the laws of one-dimensional collision for the optimization of truss structures with discrete sizing variables. Dede and Ayvaz (2015)[25] proposed a new meta-heuristic algorithm called teaching–learning-based optimization (TLBO) for the size and shape optimization of structures. Ho-Huu et al. (2015)[26] proposed a novel discrete variables handling technique and integrated it into original improved constrained differential evolution (ICDE) to give a so-called Discrete-ICDE (D-ICDE) for solving layout truss optimization problems. Kaveh and Mahdavi (2015)[27] proposed a two-dimensional colliding bodies algorithm for optimal design of truss structures. Ho-Huu et al. (2016)[28] proposed an adaptive elitist differential evolution (aeDE) for optimization of truss structures with discrete design variables. Mirjalili and Lewis (2016)[29] implemented the whale optimization algorithm to the truss sizing optimization. Kalatjari and Talebpour (2017)[30] presented an improved ant colony algorithm with the proposed sampling search space method for the optimization of skeletal structures. Khatibinia and Yazdani (2018)[31] introduced an accelerated multi-gravitational search algorithm (AMGSA) that exhibits an improved convergence rate for size optimization of truss structures. Degertekin et al. (2019)[32] presented a novel Jaya algorithm formulation for discrete sizing, layout and topology optimization of truss structures under stress and displacement constraints. Jaya algorithm was also used to handle the effect of the post tensiong force on the behavior of cable-stayed bridges by Atmaca et al.(2022)[33], and to optimize the 3D reinforced concrete shear walls and frames by Aslay and Dede (2022)[34]. Han et al. (2019)[35] presented an extension of the basic truss layout optimization, in which the use of various materials is considered. And developed a novel improved version of the particle swarm optimization algorithm for solving this problem. Kaveh and Seddighian (2020)[36] proposed a new enriched version of the firefly algorithm for the multi-material layout and connectivity optimization problems of truss structures. Awad (2021)[37] presented a political optimization algorithm for the sizing optimization of truss structures. Bouzouiki et al. (2021)[38] presented a new non-uniform Cellular Automata algorithm, which is based on non-identical cells and a modified Fully Stressed Design (FSD) and Fully Utilized Design methods (FUD) approach for the minimum weight optimization problem of truss structures including both stress and displacement constraints. Jafari et al. (2021)[39] presented an efficient hybrid algorithm based on the particle swarm optimizer (PSO) and the cultural algorithm (CA) for the optimal design of truss structures. Jawad et al. (2021)[40] proposed a heuristic dragonfly algorithm for optimal design of truss structures with discrete variables. Jawad et al. (2021)[41] proposed an artificial bee colony algorithm for the sizing and layout optimization of truss structures. Pierezan et al. (2021)[42] applied the chaotic coyote algorithm to the truss sizing optimization problems. Renkavieski and Parpinelli (2021)[43] applied a self-adaptive algorithm based on Differential Evolution to truss size optimization with continuous variables. Ha et al. (2022)[44] proposed a parallel differential evolution with cooperative multi-search strategy for sizing truss optimization. Liu and Xia (2022)[45] introduced a hybrid intelligent genetic algorithm based on deep neutral network for truss sizing optimization. Altay et al. (2022)[46] applied the modified salp swarm algorithm to the size optimization of planar truss systems. Vu-Huu et al. (2023)[47] presented an improved bat algorithms for the size optimization of truss structures.

The truss optimization with frequency constraints is the second class. Different from the problems in the first class, except the calculation of the structural stiffness matrix, the truss optimization with frequency constraints needs to calculate the mass matrix so as to obtain the natural frequencies. The problems in the second class is more complicated than those in the first class in terms of structural analysis. Felix (1981)[48] solved as a multi-level numerical optimization problem the three-dimensional truss design problem subject to constraints on: member stresses, Euler buckling, joint displacements and system natural frequencies. Kaveh and Zolghadr (2012)[49] proposed a hybridization of the Charged System Search and the Big Bang-Big Crunch algorithms with trap recognition capability for the frequency constraint structural optimization. Khatibinia and Naseralavi (2014)[50] presented an orthogonal multi-gravitational search algorithm (OMGSA) to solve truss optimization on shape and sizing with frequency constraints. Kaveh and Ghazaan (2015)[51] presented two optimization algorithms including the Particle Swarm Optimization with an Aging Leader and Challengers (ALC-PSO) and the HALC-PSO that transplants harmony search-based mechanism to ALC-PSO as a variable constraint handling approach for finding the optimal mass of truss structures with natural frequency constraints. Vu (2015)[52] employed differential evolution method combined with finite element code to find the optimal cross-section sizes and node coordinates of trusses with natural frequency constraints. Farshchin et al. (2016)[53] introduced a multi-class teaching–learning-based optimization (MC-TLBO) technique for truss structural optimization with frequency constraints. Ho-Huu et al. (2016)[54] presented an improved differential evolution algorithm based on an adaptive mutation scheme for the optimal design of truss structures with frequency constraints. Pham (2016)[55] proposed an enhanced differential evolution based on adaptive directional mutation and nearest neighbor comparison for the truss optimization with frequency constraints. Tejani et al. (2016)[56] presented the modified symbiotic organisms search(SOS) algorithm for the structure optimization problems with the shape and size variables along with the Frequency constraints. Kanarachos et al. (2017)[57] proposed a smell and visual contrast-based fruit fly optimization algorithm for truss optimizations with frequency constraints. Ho-Huu et al. (2018)[58] proposed an improved differential evolution based on roulette wheel selection for shape and size optimization of truss structures with frequency constraints. Lieu et al. (2018)[59] proposed an adaptive hybrid evolutionary firefly algorithm for shape and size optimization of truss structures with frequency constraints. Degertekin et al. (2021)[60] presented a parameter free Jaya algorithm for truss sizing-layout optimization under natural frequency constraints. Lemonge et al. (2021)[61] adopted a modified version of the Differential Evolution called the Third Evolution Step Differential Evolution (GDE3) for the multi-objective truss structural optimization considering natural frequencies of vibration and global stability. Nguyen-Van et al. (2021)[62] introduced a hybrid differential evolution and symbiotic organisms search algorithm for size and shape optimization of truss structures under multiple frequency constraints. Dede et al. (2022)[63] applied Rao algorithms to size optimization of truss domes with different scales and found that the Rao-2 algorithm outperformed the Rao-1 algorithm. Y1lmaz and Dede (2023)[64] integrated the non-dominant sorting into the Rao algorithms and used the resulting new algorithm to address the time–cost trade-off problems of construction project planning. In addition to natural frequencies, defect damage and hybrid uncertainties should be considered in the structure design process. Li et al. (2023)[65] proposed a concurrent reliability-based topology optimization design method to address this problem. Wang et al. (2022)[66] utilized evidence theory to process uncertainty parameters and evaluate the reliability of the structural strength performance.

Though so many methods have been proposed in this field, the comparison among them commonly shows that no algorithm can handle all the problems better than all the others. According to the “NO FREE LUNCH THEORM”[67], we can only expect that an algorithm provides best solutions to some problems, but performs mediocrely on other problems. This is the reason why we still need to search more algorithms for the truss optimization today.

The purpose of this paper is to present an effective swarm intelligent heuristic algorithm for the truss optimization with frequency constraints. The remainder of this paper is organized as follows. Section 2 presents the truss optimization model with the frequency constraints. Section 3 introduces the medalist learning algorithm in detail. Section 4 applies the new algorithm to six benchmark truss optimization problems and compares the results given by many algorithms. Section 5 sums up the main contents of this paper and points out several potential directions for the future research.