## 3 Case Study
### FEM Creation and Updating
#### 3.3.2 Update Model Parameters through Particle Swarm Optimisation (PSO) Algorithm  
The particle swarm optimization (PSO) algorithm was established and developed on the principles of swarm intelligence, in order to find solutions for optimisation problems in a particular search space [56; 57]. To understand the PSO algorithm better, observe a simple example of a flock of birds foraging. The foraging space is now the entire three-dimensional space. At the beginning of the search, the whole flock flies in a certain direction, which can be very random. However, after a period of searching, some individuals in the herd began to find a place containing food. Depending on the amount of food discovered, the individual sends a signal to the other individuals searching in the vicinity. This signal propagates throughout the population. Based on the information received, each individual will adjust both their flight direction and speed in the direction of where there is the most food. Such communication is often viewed as a phenotype of herd intelligence. This mechanism helps the whole flock of birds to find out where there is the most food in extremely large search spaces. In swarm optimization, each particle searches a space by itself, remembering the best value, and informing other individuals. Other instances will receive the information and decide to continue the search or report its location so that other instances continue to act. Consequently, values in the search space are obtained quickly and accurately. There are two parameters that are particularly important: the location of an instance, and the search velocity. These two parameters are expressed through the formulas for updating the position and updating the velocity of the instance:  
\[x^{i}(t+1)=x^{i}(t)+v^{i}(t+1) \tag{27}\]  
\[v^{i}(t+1)=\mathrm{w}v^{i}+C_{1}r_{1}(p^{i}(t)-x^{i}(t))+C_{2}r_{2}(G_{\mathrm{ best}}-x^{i}(t)) \tag{28}\]  
where \(x^{i}\) is the position of instance \(i\) at different times (\(t\) and \(t+1\)); \(v^{i}\) is the speed of individual \(i\); \(\mathrm{w}\) is the parameter of inertial weight; \(C_{1}\) and \(C_{2}\) represent the population's cognitive coefficient; \(r_{1}\) and \(r_{2}\) are random numbers in the range [0, 1]; \(p^{i}(t)\) is the best position of each individual; \(G_{\text{best}}\) is the best location of the entire population. Each individual is characterised using its velocity vector and its position in space. To evaluate the similarity between the FE model and the practical structure, an objective function was built based on the natural frequency and the mode shape of the structure:  
\[Fitness=\sum_{k=1}^{n}\left[1-MAC(\varphi_{k},\widetilde{\varphi}_{k})\right]+ \sum_{k=1}^{n}\frac{\omega_{k}^{2}}{\widetilde{\omega}_{k}^{2}}=\sum_{k=1}^{n }\left[1-\frac{\left(\widetilde{\varphi}_{k}^{T}\varphi_{k}\right)^{2}}{ \left(\varphi_{k}^{T}\varphi_{k}\right)\left(\varphi_{k}^{T}\widetilde{\varphi }_{k}\right)}\right]+\sum_{k=1}^{n}\frac{\left(\omega_{k}-\widetilde{\omega}_{ k}\right)^{2}}{\widetilde{\omega}_{k}^{2}} \tag{29}\]  
In the case of Chuong Duong bridge, determining material parameters required many experiments. At the same time, the masses of nonstructural parts were also difficult to determine accurately. Some uncertain parameters have been selected to update the numerical model. Based on experience, Table 6 shows the range of variation for the uncertain parameters. The updated uncertainty parameters and FE model after using the PSO are presented in Table 7.  
The results after updating show that updating the model lowered the discrepancy between the calculated and measured natural frequencies, and the MAC value reached a good level (Table 8). The MAC values of the FE model before and after the update are shown in Figure 13. After being updated, the accuracy of the model increased a lot. These MAC values show good agreement between the FE and the actual models.