## 3 Correlated Mode Auto-Pairing and Adaptive Evolution Screening Method
### Population Evolution Screening Mechanism Provided by an Evolutionary Algorithm  
The objective function value of each individual can be determined by the correlated mode auto-pairing strategy. In addition, a global optimization algorithm is needed to search the minimum value of the objective function. As mentioned in Section 1, the evolutionary algorithm has two central characters: population search strategy and information exchange among individuals in the population. Thus, the population evolution screening mechanism is particularly suitable for the CMPs adaptively screening before each iteration and the global optimal solution searching. In evolutionary iteration, the updating parameters \(\theta^{(r)}\) can be randomly generated  
\[\theta^{(r)}=\left[ \begin{array}{cccc}\theta^{(r,1)}&\theta^{(r,2)}&\cdots& \theta^{(r,s)}\end{array} \right] \tag{14}\]  
where \"\(r\)\" and \"\(s\)\" as superscripts represent generation number and individual number, respectively. Then, the analytical modal analyses are performed based on the FE model, and natural frequencies \(\mathbf{f}_{A}\) and mode shapes \(\boldsymbol{\varphi}_{A}\) for the \(r^{\text{th}}\) generation can be obtained:  
\[\left\{ \begin{array}{cccc}\mathbf{f}_{A}^{(r)}=\left[\begin{array}{cccc} \mathbf{f}_{A}^{(r,1)}&\mathbf{f}_{A}^{(r,2)}&\cdots&\mathbf{f}_{A}^{(r,s)} \end{array}\right]\\ \boldsymbol{\varphi}_{A}^{(r)}=\left[\begin{array}{cccc}\boldsymbol{\varphi }_{A}^{(r,1)}&\boldsymbol{\varphi}_{A}^{(r,2)}&\cdots&\boldsymbol{\varphi}_{A}^ {(r,s)}\end{array}\right]\end{array} \right. \tag{15}\]  
Based on experimental natural frequencies \(\mathbf{f}_{E}\) and mode shapes \(\boldsymbol{\varphi}_{E}\), the FRE and MAC matrix for the \(r^{\text{th}}\) generation can be calculated to provide the base for the following auto-pairing process. \[\left\{ \begin{array}{cccc}\mathbf{FRE}^{(r)}=\left[\begin{array}{cccc} \mathbf{FRE}^{(r,1)}&\mathbf{FRE}^{(r,2)}&\cdots&\mathbf{FRE}^{(r,s)}\end{array} \right]\\ \mathbf{MAC}^{(r)}=\left[\begin{array}{cccc}\mathbf{MAC}^{(r,1)}&\mathbf{MAC }^{(r,2)}&\cdots&\mathbf{MAC}^{(r,s)}\end{array}\right]\end{array} \right. \tag{16}\]  
According to the correlated mode auto-pairing strategy, the one-to-one CMPs for the \(r^{\text{th}}\) generation can be determined by the given MAC threshold in Figure 1, taken as 0.5 in this paper. \[\mathbf{CMP}^{(r)}=\left\{ \begin{array}{cccc}\mathbf{CMP}^{(r,1)}&\mathbf{CMP }^{(r,2)}&\cdots&\mathbf{CMP}^{(r,s)}\end{array} \right\} \tag{17}\]  
According to Equation (11) to Equation (13), the objective function value for the \(i^{\text{th}}\) individual in the \(r^{\text{th}}\) generation is given by  
\[J^{(r,t)}=\sum_{k=1}^{N_{E}}J^{(r,j)}_{rk} \tag{18}\]  
The objective function values of all the individuals in the \(r^{\text{th}}\) generation can be expressed as  
\[\mathbf{J}^{(r)}=\left[ \begin{array}{cccc}J^{(r,1)}&J^{(r,2)}&\cdots&J^{(r,s )}\end{array} \right] \tag{19}\]  
Based on the objective function values \(\mathbf{J}^{(r)}\), rank and evaluation of all the individuals in the \(r^{\text{th}}\) generation can be performed. Then \(\theta^{(r+1)}\), \(\mathbf{CMP}^{(r+1)}\), and \(\mathbf{J}^{(r+1)}\) of the next generation can be generated by exchanging individual genes with evolutionary operators such as crossover, mutation, and selection. \[\left\{ \begin{array}{cccc}\Theta^{(r+1)}=\left[\begin{array}{cccc}\theta^ {(r+1,1)}&\theta^{(r+1,2)}&\cdots&\theta^{(r+1,s)}\end{array}\right]\\ \mathbf{CMP}^{(r+1)}=\left\{\begin{array}{cccc}\mathbf{CMP}^{(r+1,1)}& \mathbf{CMP}^{(r+1,2)}&\cdots&\mathbf{CMP}^{(r+1,s)}\end{array}\right\}\\ \mathbf{J}^{(r+1)}=\left[\begin{array}{cccc}J^{(r+1,1)}&J^{(r+1,2)}&\cdots&J^{ (r+1,s)}\end{array}\right]\end{array} \right\} \tag{20}\]  
Combined with the correlated mode auto-pairing strategy, the updating parameters \(\theta\) and CMPs will simultaneously evolve toward the ideal values due to the schema theoremand building block hypothesis [35].