PSI - Issue 78

Shahin Sayyad et al. / Procedia Structural Integrity 78 (2026) 277–284

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Particle Swarm Optimization (PSO), introduced by (Kennedy & Eberhart, 1995), is a population-based stochastic optimization method inspired by the social behavior and movement patterns of animals such as birds, fish, and insects. In PSO, each member of the swarm, referred to as a particle, represents a potential solution and explores the search space in pursuit of the optimal solution, analogous to animals searching for the most fertile feeding ground. Each particle stores information about its own best-found position as well as either the best-known position within the entire swarm (global best or gbest ) or within its local neighbourhood (local best or lbest ). The position of each particle is updated iteratively based on its own experience and that of its neighbours. PSO is particularly well-suited for continuous optimization problems, requires only a few control parameters, and has been effectively applied in various fields, including finite element model calibration. Based on the information the particle gathers, it decides on the speed of movement to the new position. Position, , is the solution reached by the i-th particle out of a total of m swarm particles in the n-th iteration. The position of each particle is defined by coordinates within an n-dimensional space, where s denotes the number of variables. These variables, = { 1 , 2 ,…, } , constitute the solution for particle i at iteration n , with i =1,2,…, p . The particle’s velocity, meanwhile, is described by the rate at which its position changes over time. The primary steps in the PSO are outlined as follows (Ereiz et al., 2022): • Setting the number of particles and initializing algorithm constants, including particle positions and velocities. • Defining the objective function, which calculates the difference between each particle's current position and the target position. • Tracking and updating the best position for each particle as well as the overall best position achieved by any particle in the swarm. • Updating the velocity of the particle swarm according to the equation: + 1 = + +1 , where +1 is the velocity, + 1 is the position in the iteration n +1, and is the position in iteration n. The velocity is calculated with the next equation: 1 - - 1 1 2 2 n n n n v wv C rand Pbest x C rand Gbes x ij ij ij ij tj ij  + = +          +   (4) • In the previous equation, C 1 and C 2 represent learning factors. These factors are positive weighting coefficients used to balance the influence of individual and social experiences. rand 1 and rand 2 are random numbers between zero and one, while Pbest ij and Gbest j are the best positions achieved by the i-th agent closest to the target since the beginning of the process. w is the inertia coefficient avoids trapping in local minima. • Update the position of each particle based on social behavior to match the environment by constantly returning to the most promising identified region. • Repeat steps 1 – 5 until the termination criteria are met. In this study, the Pyswarm library (Miranda, 2018) was adopted to implement particle swarm optimization for model calibration. The algorithm searched the solution space by using a swarm of 50 particles and operating for up to 250 iterations. With 15 parameters and specified lower and upper bounds, each particle represented a potential solution. 3. Results and Discussion In the present study, a two-phase approach was followed to validate and assess the applicability of GA and PSO in calibrating the finite element model of bell tower. In the first phase, the output results from the initial finite element analysis, including the natural frequencies of the first 10 and 5 modes along with their corresponding mode shapes, were considered as reference (experimental-like) data. This step aimed to evaluate the effectiveness of both algorithms in estimating the model parameters. Provided that the algorithms could estimate parameters closely matching those of the initial model, they would be regarded as effective. In the second phase, after confirming the acceptable performance of the algorithms, actual experimental data were used as the target for the model calibration process. In the first case, only the objective function of Eq. 1, based solely on the natural frequencies of the structure, was employed in the optimization process by both algorithms. Since mode shapes were excluded from this objective function, the influence of higher modes was addressed by incorporating the first ten modes to enhance the accuracy of the results. In the second scenario, the first five vibration modes of the structure were subjected to the objective