# An Investigation of Particle Swarm Optimization Topologies in Structural Damage Detection
## 4 Numerical Simulations for SDD of the Cantilever Beam  
Figure 11 is a 2D finite element model of the cantilever beam structure. The length of the beam is 1 m, and thirty identical finite elements are considered. Table 1 provides the physical parameters of the beam structure. Figure 10: Rotation skill example. Figure 9: Multi-Ring topology. Single damage and multiple damages are all considered to evaluate the performance of different PSO topologies. Nine damage cases with different sites and degrees are assumed in the paper, as displayed in Table 2. According to the number of damaged elements, those scenarios are classified into three categories: type I is composed of cases with one damage element; type II has cases with two damage elements; and, type III is the scenarios with three damage elements. The parameter settings of the PSO algorithm are outlined, as follows:  
1. Initialization: the positions of the initial population are randomly created in the search space, and the initial velocities of the population are set to zero to prevent the swarm explosion at the beginning of the algorithm [26]. 2. Boundary conditions: according to the guidance in [27,28], the following restricted boundary condition is used, and \(Vmax=Xmax\): \[if\quad x_{ij} >X_{max} if\quad v_{ij}>V_{max}\] \[x_{ij} =X_{max} \quad v_{ij}=V_{max}\] \[else\ if\quad x_{ij} <X_{min} else\ if\quad v_{ij}<-V_{max}\] \[x_{ij} =X_{min} \quad v_{ij}=-V_{max}\]
3. \(\omega\), \(c_{1}\), \(c_{2}\), \(N\), and \(t_{max}\): the population size \(N\) is set to 100 and the maximum number of iterations is \(t_{max}=200\); the inertia weight \(\omega\) is linearly decreasing with the number of \begin{table}
\begin{tabular}{c c c c c} \hline
**Young Modulus** & **Poisson Ratio** & **Density** & **Width** & **Thickness** \\
**E (Pa)** & \(\
u\) & \(\rho\) **(Kg\(\cdot\)m\({}^{-3}\))** & **w (m)** & **d (m)** \\ \hline \(2.0\times 10^{11}\)\(\mathrm{N/m^{2}}\) & 0.33 & 7850 & \(2.49\times 10^{-2}\) m & \(5.3\times 10^{-3}\) m \\ \hline \end{tabular}
\end{table} 
Table 1: The physical parameters of the beam structure. \begin{table}
\begin{tabular}{c c c c} \hline
**Damage Scenarios** & **Types** & **Elements** & **Severity** \\ \hline
1 & I & 8 (Front) & 20\% \\
2 & & 15 (Middle) & 30\% \\
3 & & 25 (End) & 50\% \\
4 & II & 5, 6 (Neighbor) & 10\%, 10\% \\
5 & & 13, 18 (Symmetrical) & 20\%, 50\% \\
6 & & 15, 25 & 20\%, 30\% \\
7 & III & 5, 6, 25 & 30\%, 30\%, 20\% \\
8 & & 10, 16, 22 & 30\%, 30\%, 30\% \\
9 & & 15, 19, 20 & 10\%, 10\%, 10\% \\ \hline \end{tabular}
\end{table} 
Table 2: Damage scenarios for the cantilever beam. Figure 11: The Euler–Bernoulli cantilever beam model. iterations from 0.9 to 0.4; the time-varying acceleration coefficient strategy for \(c_{1}\) and \(c_{2}\) is formulated as [12]:  
\[c_{1}^{t} =c_{1}^{ini}+(c_{1}^{fin}-c_{1}^{ini})\frac{t}{t_{max}}\] \[c_{2}^{t} =c_{2}^{ini}+(c_{2}^{fin}-c_{2}^{ini})\frac{t}{t_{max}}\]  
where \(c_{1}^{ini},c_{1}^{fin},c_{2}^{ini},c_{2}^{fin}\) are constants and their values are \(2.5,0.5,0.5,2.5\), respectively, and \(t\) is the current iteration number. 4. Calculation accuracy: in view of the limitation of measurement accuracy in the experiment, the calculation accuracy of the algorithm adopts \(1\times 10^{-4}\). In this section, performances of the PSO with eight different topologies are evaluated on multiple damage scenarios of the cantilever beam. The experimental simulations are conducted using MATLAB, and the simulation environment is: Inter(R) Core(TM)i5-7600 CPU @ 244 3.50 GHz RAM: 16.0 GB. The success rate and mean iterations are selected as the performance measures for evaluating the performances of eight PSO topologies. One hundred trials are performed for each damage scenario shown in Table 2. A trial is defined to be successful when the minimum fitness value is obtained.