# Optimization of Composite Structures with Thin Rigid Fibers Using Bioinspired Algorithms
## 3 The Particle Swarm Optimizer  
Like immune algorithms, particle swarm optimizers are designed based on principles seen in the natural world. PSOs mimic the social behaviors of animals traveling and residing in groups. The animals move around in three-dimensional space to modify where they eat and stay, to find a favorable location for breeding, or to avoid predators. Numerous types of insects living in swarms, fish swimming in schools, birds flying in flocks, and mammals in herds may all be identified. [PERSON] published a simulation of bird flocking [43]. According to his assumption, only three fundamental conditions--collision avoidance, neighbor velocity matching, and flock centering--can lead to this type of coordinated motion. These three principles' computer implementation displayed incredibly lifelike flocking behavior, splitting ahead of obstacles and reuniting after avoiding them in three dimensions. Fish shoals were the subject of similar observations. Further observations and simulations of bird and fish behavior produced, in effect, more accurate and more precisely formulated conclusions [44]. The results of this biological examination were used by [PERSON] and [PERSON] [45], who proposed a particle swarm optimizer--PSO. This program finds a solution to an n-variable optimization problem by realizing the directed motion of the particles in j-dimensional space. PSO employs an iterative methodology. The location of one individual (particle) is determined based on its earlier experience and the experience of the whole group (swarm). Furthermore, the capacity for memorization and subsequent return to previously identified regions with favorable characteristics allows the particles to adjust to their new surroundings. Finding ever better sites in the search space--that is, areas in the natural environment that serve as feeding or hatching grounds, for  
Figure 3: The idea of the crowding mechanism. example--is the foundation of the PSO optimization process. In this research, an algorithm known as constricted continuous PSO--which uses a constant constriction coefficient and continuous representation of the design variables--has been applied. With this method, every particle oscillates in the search space between its neighbor's best position and its own prior best position in an attempt to discover new best locations throughout its route. When the swarm is tiny (consists of a few or tens of particles), it can be presumed that every particle stays in the vicinity of the particle that is currently being considered. In this instance, we may presume that the global neighborhood version and the swarm's most optimal location to date--the swarm leader's present location--are taken into consideration (Figure 4). The position of the \(i\)-th particle is changed by stochastic velocity \(\mathrm{v}_{i}\), which is dependent on the particle distance from its earlier best position and the position of the swarm leader. This approach is given by the following equations:  
\[v_{ij}(k+1)=wv_{ij}(k)+\phi_{1j}(k)\big{[}q_{ij}(k)-d_{ij}(k)\big{]}+\phi_{2j}(k )\big{[}q_{ij}(k)-d_{ij}(k)\big{]} \tag{8}\]  
\[d_{ij}(k+1)=d_{ij}(k)+v_{ij}(k+1),i=1,2,\ldots,m\;;\;j=1,2,\ldots,n \tag{9}\]  
where  
\(\phi_{1j}(k)=c_{1}r_{1j}(k)\); \(\phi_{2j}(k)=c_{2}r_{2j}(k)\),  
\(m\)--the number of the particles;  
\(n\)--the number of design variables (problem dimension);  
\(w\)--the inertia weight;  
\(c_{1}\), \(c_{2}\)--the acceleration coefficients;  
\(r_{1}\), \(r_{2}\)--random numbers with a uniform distribution [0, 1];  
\(d_{i}(k)\)--the position of the i-th particle in the k-th iteration step;  
\(v_{i}(k)\)--the velocity of the i-th particle in the k-th iteration step;  
\(q_{i}(k)\)--the best found position of the i-th particle found thus far;  
\(\hat{q}_{i}(k)\)--the best position found thus far by the swarm, the position of the swarm leader;  
\(k\)--the iteration step. Equations (8) and (9) contain certain parameters that affect the speed and efficiency of the optimization process.