Advances in Metaheuristic Algorithms for Optimal Design of Structures

In today’s extremely competitive world, human beings attempt to exploit the maximum output or profit from a limited amount of available resources. In engineering design, for example, choosing design variables that fulfill all design requirements and have the lowest possible cost is concerned, i.e. the main objective is to comply with basic standards but also to achieve good economic results. Optimization offers a technique for solving this type of problems.

Particle Swarm Optimization (PSO) algorithms are nature-inspired population-based metaheuristic algorithms originally accredited to Eberhart, Kennedy and Shi [1, 2]. The algorithms mimic the social behavior of birds flocking and fishes schooling. Starting form a randomly distributed set of particles (potential solutions), the algorithms try to improve the solutions according to a quality measure (fitness function). The improvisation is preformed through moving the particles around the search space by means of a set of simple mathematical expressions which model some inter-particle communications. These mathematical expressions, in their simplest and most basic form, suggest the movement of each particle towards its own best experienced position and the swarm’s best position so far, along with some random perturbations. There is an abundance of different variants using different updating rules.

This chapter consists of two parts. In the first part an optimization algorithm based on some principles from physics and mechanics, which is known as the Charged System Search (CSS) [1]. In this algorithm the governing Coulomb law from electrostatics and the Newtonian laws of mechanics. CSS is a multi-agent approach in which each agent is a Charged Particle (CP). CPs can affect each other based on their fitness values and their separation distances. The quantity of the resultant force is determined by using the electrostatics laws and the quality of the movement is determined using Newtonian mechanics laws. CSS can be utilized in all optimization fields; especially it is suitable for non-smooth or non-convex domains. CSS needs neither the gradient information nor the continuity of the search space.

This chapter consists of two parts. In first part, the standard Magnetic Charged System Search (MCSS) is presented and applied to different numerical examples to examine the efficiency of this algorithm. The results are compared to those of the original charged system search method [1].

Nature has provided inspiration for most of the man-made technologies. Scientists believe that dolphins are the second to humans in smartness and intelligence. Echolocation is the biological sonar used by dolphins and several kinds of other animals for navigation and hunting in various environments. This ability of dolphins is mimicked in this chapter to develop a new optimization method. There are different metaheuristic optimization methods, but in most of these algorithms parameter tuning takes a considerable time of the user, persuading the scientists to develop ideas to improve these methods. Studies have shown that metaheuristic algorithms have certain governing rules and knowing these rules helps to get better results. Dolphin Echolocation takes advantages of these rules and outperforms many existing optimization methods, while it has few parameters to be set. The new approach leads to excellent results with low computational efforts [1].

This chapter presents a novel efficient metaheuristic optimization algorithm called Colliding Bodies Optimization (CBO), for optimization. This algorithm is based on one-dimensional collisions between bodies, with each agent solution being considered as the massed object or body. After a collision of two moving bodies having specified masses and velocities, these bodies are separated with new velocities. This collision causes the agents to move toward better positions in the search space. CBO utilizes simple formulation to find minimum or maximum of functions; also it is independent of parameters [1].

In this chapter a newly developed metaheuristic method, so-called Ray Optimization, is presented. Similar to other multi-agent methods, Ray Optimization has a number of particles consisting of the variables of the problem. These agents are considered as rays of light. Based on the Snell’s light refraction law when light travels from a lighter medium to a darker medium, it refracts and its direction changes. This behavior helps the agents to explore the search space in early stages of the optimization process and to make them converge in the final stages. This law is the main tool of the Ray Optimization algorithm. This chapter consists of three parts.

The Big Bang–Big Crunch (BB–BC) method developed by Erol and Eksin [1] consists of two phases: a Big Bang phase, and a Big Crunch phase. In the Big Bang phase, candidate solutions are randomly distributed over the search space. Similar to other evolutionary algorithms, initial solutions are spread all over the search space in a uniform manner in the first Big Bang. Erol and Eksin [1] associated the random nature of the Big Bang to energy dissipation or the transformation from an ordered state (a convergent solution) to a disorder or chaos state (new set of solution candidates).

In this chapter, a metaheuristic method so-called Cuckoo Search (CS) algorithm is utilized to determine optimum design of structures for both discrete and continuous variables. This algorithm is recently developed by Yang [1], Yang and Deb [2, 3], and it is based on the obligate brood parasitic behavior of some cuckoo species together with the Lévy flight behavior of some birds and fruit flies. The CS is a population based optimization algorithm and similar to many others metaheuristic algorithms starts with a random initial population which is taken as host nests or eggs. The CS algorithm essentially works with three components: selection of the best by keeping the best nests or solutions; replacement of the host eggs with respect to the quality of the new solutions or Cuckoo eggs produced based randomization via Lévy flights globally (exploration); and discovering of some cuckoo eggs by the host birds and replacing according to the quality of the local random walks (exploitation) [2].

In this chapter an optimization method is presented based on a socio-politically motivated strategy, called Imperialist Competitive Algorithm (ICA). ICA is a multi-agent algorithm with each agent being a country, which is either a colony or an imperialist. These countries form some empires in the search space. Movement of the colonies toward their related imperialist, and imperialistic competition among the empires, form the basis of the ICA. During these movements, the powerful Imperialists are reinforced and the weak ones are weakened and gradually collapsed, directing the algorithm towards optimum points. Here, ICA is utilized to optimize the skeletal structures which is based on [1, 2].

In nature complex biological phenomena such as the collective behavior of birds, foraging activity of bees or cooperative behavior of ants may result from relatively simple rules which however present nonlinear behavior being sensitive to initial conditions. Such systems are generally known as “deterministic nonlinear systems” and the corresponding theory as “chaos theory”. Thus real world systems that may seem to be stochastic or random, may present a nonlinear deterministic and chaotic behavior. Although chaos and random signals share the property of long term unpredictable irregular behavior and many of random generators in programming softwares as well as the chaotic maps are deterministic; however chaos can help order to arise from disorder. Similarly, many metaheuristics optimization algorithms are inspired from biological systems where order arises from disorder. In these cases disorder often indicates both non-organized patterns and irregular behavior, whereas order is the result of self-organization and evolution and often arises from a disorder condition or from the presence of dissymmetries. Self-organization and evolution are two key factors of many metaheuristic optimization techniques. Due to these common properties between chaos and optimization algorithms, simultaneous use of these concepts can improve the performance of the optimization algorithms [1]. Seemingly the benefits of such combination is a generic for other stochastic optimization and experimental studies confirmed this; although, this has not mathematically been proven yet [2].

In this chapter a multi-objective optimization algorithm is presented and applied to optimal design of large-scale skeletal structures [1]. Optimization is a process in which one seeks to minimize or maximize a function by systematically choosing the values of variables from/within a permissible set. In recent decades, a vast amount of research has been conducted in this field in order to design effective and efficient optimization algorithms. Besides, the application of the existing algorithms to engineering design problems has also been the focus of many studies (Gou et al. [2]; Lee and Geem [3]; Gero et al. [4]). In a vast majority of structural design applications, including previous studies (Kaveh and Talatahari [5]; Kaveh and Talatahari [6]; Kaveh and Talatahari [7]; Kaveh and Rahami [8]), the fitness function was based on a single evaluation criterion. For example, the total weight or total construction cost of a steel structural system has been frequently employed as the evaluation criterion in structural engineering applications. But in the practical optimization problems, usually more than one objective are required to be optimized, such as, minimum mass or cost, maximum stiffness, minimum displacement at specific structural points, maximum natural frequency of free vibration, maximum structural strain energy. This makes it necessary to formulate a multi-objective optimization problem, and look for the set of compromise solutions in the objective space. This set of solutions provides valuable information about all possible designs for the considered engineering problem and guides the designer to make the best decision. The application of multi-objective optimization algorithms to structural problems has attracted the interest of many researchers. For example, in (Mathakari et al. [9]) Genetic algorithm is employed for optimal design of truss structures, or in (Liu et al. [10]) Genetic algorithm is utilized for multi-objective optimization for performance-based seismic design of steel moment frame structures, and in (Paya et al. [11]) the problem of design of RC building frames is formulated as a multi-objective optimization problem and solved by simulated annealing. In all these studies, some well-known multi-objective algorithms have been applied to structural design problems. However, the approach which has attracted the attention of many researchers in recent years is to utilize a high-performance multi-objective optimization algorithm for structural design problems. For example, in (Su et al. [12]) an adaptive multi-island search strategy is incorporated with NSGA-II for solving the truss layout optimization problem, or in (Ohsaki et al. [13]) a hybrid algorithm of simulated annealing and tabu search is used for seismic design of steel frames with standard sections, and in (Omkar et al. [14]) the specific version of particle swarm optimization is utilized to solve design optimization problem of composite structures which is a highly multi-modal optimization problem. These are only three examples of such studies.