## 2 Optimization Methods and Design Procedure  
### SOM Optimization Method  
This paper proposes a structural calculator and optimizer method (SOM) based on professional bridge software and programming software to optimize cable force by seeking external data storage and transfer media. The selection of bridge-specific software for the structural calculator ensures its broad applicability and professionalism. The selection of programming software as the optimizer allows organic integration with the swarm intelligence algorithm. However, the optimizer itself does not have the function to realize the synergy between mathematics and mechanics, so it is necessary to consider introducing the concept of mechanics into the optimizer. The influence matrix method can well accomplish this connection between mathematics and mechanics. Based on the concept of generalized influence matrix, [PERSON] et al. [5] derived the influence matrix method for cable force optimization of cable-stayed bridges. Under the condition that the structure meets the principle of linear superposition, the change of internal force (such as cable force) of the structure has the following relationship with the structural response:  
\[[I]\{X\}=\{C\} \tag{1}\]  
where \([I]\) is the influence matrix, and \(\{X\}\) is the adjustment vector, which refers to the column vector composed of n independent elements specified in the structure that can be adjusted to change the adjustment vector. \(\{C\}\) is the regulated vector, which refers to the column vector composed of \(m\) (\(m>n\)) independent elements on the cross section of the structure, generally referring to the structural response of internal force, stress, or displacement of the cross section.  
It is assumed that we have obtained a set of relatively reasonable cable force values \(\{X_{1}\}\) and the corresponding vertical displacement of main beam \(\{C_{0}\}\) for cable-stayed bridges according to traditional methods (the response of other structures follow the same principle). The influence matrix of the cable force corresponding to the vertical displacement of the main beam is noted as \([I_{y}]\). When the change of cable force value \(\{\Delta X\}\) becomes \(\{X_{1}\}\), the change of the vertical displacement of the main beam is denoted as \(\{\Delta C\}\) and becomes \(\{C_{1}\}\). When the span of a cable-stayed bridge is small and the structure is under elastic stress, the condition of the linear superposition principle is satisfied. At this point there is the following quantitative relationship:  
\[[I_{y}]\{\Delta X\}=\{\Delta C\} \tag{2}\]  
also known as  
\[[I_{C}]\{X_{1}-X_{0}\}=\{C_{1}-C_{0}\} \tag{3}\]  
By incorporating the above Equation (3) into the optimizer, the corresponding vertical displacement of the main beam can be obtained from the initial values of the cable force and the vertical displacement of the main beam when the position of the particle is updated (either by a regular update or by variation operation). Then, the objective function related to the vertical displacement of the main beam can be solved to evaluate the particles. It is worth noting that the premise of using the influence matrix method is that the response of the structure before and after the cable force adjustment satisfies the linear superposition principle, so the influence matrix is built into the optimizer in the field of cable force adjustment of small- and medium-span cable-stayed bridges to perform better. The primary roles of the cable force optimizer, the structure calculator, and the various parts of the data transfer medium are shown in Figure 1. The intelligent optimization algorithm uses particle swarm optimization, which has been proven to have excellent performance in cable force optimization. However, it needs to be further improved to optimize its performance.  
Figure 1: Collaboration between Midas/Civil and MATLAB.