# An Investigation of Particle Swarm Optimization Topologies in Structural Damage Detection
## 1 Introduction  
In the last two decades, vibration-based structural damage detection (SDD) has received considerable attention since the dynamic responses of engineering structures are easy to obtain. The main idea is that the structural dynamic responses can be regarded as a function of its physical parameters (mass, stiffness, and damping). Hence, the presence of damage leads to the change of modal parameters, such as natural frequencies, mode shapes, and structural flexibility. The objective of SDD is to infer the physical parameters from the measured response data to locate and quantify structural damage. Therefore, vibration-based SDD belongs to a typical inverse problem [1, 2]. Nowadays, various popular meta-heuristic algorithms have been developed to solve this inverse problem in SDD. When compared with the traditional optimization algorithms [3], such as [PERSON]'s method and the gradient descent method, they are effective and robust in coping with uncertainties and incomplete information. The meta-heuristic algorithms [4] mainly simulate the physical laws or biological phenomena in nature, and they use simple coding methods to represent various practical problems. In different meta-heuristic algorithms, the PSO algorithm has received extensive attention because of its few parameters, fast convergence, and easy implementation. The PSO algorithm is a population-based meta-heuristic that was proposed by [PERSON] and [PERSON] [5]. It is based on simulating the foraging behavior of bird flocking, and it has been widely used in many fields, such as benchmark function optimization [6], image processing [7], scheduling decision [8], and engineering [9]. However, it is well-known that the classicalPSO algorithm yields premature convergence. Consequently, various PSO variants have been proposed to alleviate this phenomenon in different research fields. When dealing with the SDD problem, the improvement of the algorithm mainly includes hybrid PSO schemes [10; 11; 12] and multi-stage PSO schemes [13; 14; 15]. In all of these recent works in SDD, only the Global topology is considered, which is the most commonly used one. In the Global topology, all of the particles are connected. Each particle adjusts its position according to its self-experience and global social experience. In addition to the Global topology, other topologies [16; 17; 18] have also been developed to enhance the performance of the PSO algorithm. A proper PSO topology can alleviate premature convergence and improve search efficiency [19]. Therefore, it is crucial to study the effects of PSO topologies on the performance of the studied problems. [PERSON] [19] examined the influences of four topologies, namely Global, Local, Four Clusters, and Von Neumann, on the population diversity in benchmark functions. The experimental results demonstrated that the Ring topology performs better than others. [PERSON] [20] analyzed the performances of Global, Local, and Von Neumann topologies in Neural Networks training, but all of these three topologies exhibit overfitting and non-convergent behaviors. [PERSON] et al. studied the effects of five topologies (Global, Local, Von Neumann, Wheel, and Four Clusters) on PSO performance in extreme learning machines [21]. The results showed that the Global topology is more promising than other topologies. From the literature, it can be concluded that there is no one best topology for all issues, and the performances of PSO topologies depend significantly on the studied problems. However, to the best of our knowledge, there is no research on the effects of different PSO topologies in SDD. Therefore, this paper investigates the performances of eight PSO topologies in the SDD of a cantilever beam. This paper is organized, as follows: in Section 2, the basic principles of SDD are briefly introduced. Section 3 presents the definition of the basic PSO and the characteristics of eight topologies. In Section 4, the performances of different topologies are compared under single-damage and multi-damage scenarios of a cantilever beam.