PSI - Issue 78

Shahin Sayyad et al. / Procedia Structural Integrity 78 (2026) 277–284 281 (Eskew & Jang, 2017). The experimental and numerical modal vectors for the i th mode are denoted by and . The Modal Assurance Criterion (MAC) is a calculation used in modal analysis to assess the degree of similarity between two mode shapes, whose value ranges from one (or 100%), indicating identical mode shapes, where all points move the same, to almost zero, suggesting very different mode shapes. For mode shapes that differ, the MAC value will be less than one.

            2 exp * * * exp exp exp exp t num i i t t i i i i         

    exp , num i  

 

  

(3)

MAC i 

=

   

   



2.4.1. Genetic Algorithm: Overview and Application to the Case Study The Genetic Algorithm (GA) is a metaheuristic optimization technique inspired by natural evolution and Darwin’s principle of natural selection that was initially developed by (Holland, 1992) and (Goldberg, 1989) and then introduced by (Levin & Lieven, 1998) for calibrating finite element models. A population of possible solutions (individuals) evolves to improve its average fitness through specialized operators, mimicking natural processes. To encode the parameters (genes) of the problem that will change during optimization, the first step is to define and represent a chromosome for the problem being studied. Every chromosome represents an individual, and the population is made up of all individuals. Genes, or parameters, can be represented as continuous variables , discrete variables , or binary variables. To allow for in-depth investigation, the initial population, which is either generated at random or through quasi-random methods, must be evenly distributed throughout the parameter space to enable thorough exploration. The population is processed by evaluating each individual’s objective function values, followed by ranking and selection. During the ranking phase, the results of the evaluation are inspected, and the population is ranked according to fitness. sss Selection creates a “mating pool” of promising individuals by cloning high -fitness individuals and discarding low-fitness ones. Selection can be made by ranking the fitness of each solution and selecting the best solution or by ranking a randomly selected sample of the population (computational efficiency) using methods such as uniform order selection, stochastic tournament selection, and roulette wheel selection. Afterwards, crossover (or recombination) produces new offspring by combining two parent solutions, typically with a crossover probability p c . An elitist strategy, in which the top N individuals are always included in the next population (without undergoing crossover operators), can be employed to improve convergence. After generating the new population, a mutation is applied to select individuals by randomly altering certain genes based on a probability p m . This process helps maintain diversity within the population, though it can disrupt convergence if applied excessively. Thus, careful calibration of mutation type and rate is necessary, as excessive mutation may disrupt convergence. The process proceeds by evaluating the newly created population. As the most favorable genetic material propagates from one generation to the next, the standard deviation of population fitness decreases, and if a single global optimum is reached, the parameter standard deviation similarly approaches zero. Termination criteria are necessary to conclude the process. Typically, the process can be terminated when (i) a specified maximum number of generations have been created, (ii) the current population reaches a minimum fitness standard deviation, or (iii) a maximum number of consecutive generations show no improvement in the solution. The finite element model of the San Giuseppe bell tower was calibrated using a GA, implemented through the PyGAD library (Gad, 2024). The algorithm uses a population of 50 individuals that has evolved over 200 generations to optimize 15 parameters, each of which is bounded within specified lower and upper limits presented in Table 3. Two parents were selected using a roulette wheel selection strategy based on fitness, and offspring were created by randomly combining the genes of both parents using uniform crossover. In order to add variability and prevent premature convergence, random mutation was employed with a probability of 0.1. By keeping one top individual in each generation, elitism was used to maintain high-performing solutions.

2.4.2. Particle Swarm Optimization: Overview and Application to the Case Study