## 3 Correlated Mode Auto-Pairing and Adaptive Evolution Screening Method
### Correlated Mode Auto-Pairing Strategy  
According to Equations (3) and (4), the matrix \(\mathbf{FRE}\) and \(\mathbf{MAC}\) can be expressed as  
\[\mathbf{FRE}=\left[\begin{array}{cccc}FRE_{11}&FRE_{12}&\cdots&FRE_{1N_{E}} \\ FRE_{21}&FRE_{22}&\cdots&FRE_{2N_{E}}\\ \vdots&\vdots&\ddots&\vdots\\ FRE_{N_{A}1}&FRE_{N_{A}2}&\cdots&FRE_{N_{A}N_{E}}\end{array}\right] \tag{9}\]  
\[\mathbf{MAC}=\left[\begin{array}{cccc}MAC_{11}&MAC_{12}&\cdots&MAC_{1N_{E}} \\ MAC_{21}&MAC_{22}&\cdots&MAC_{2N_{E}}\\ \vdots&\vdots&\ddots&\vdots\\ MAC_{N_{A}1}&MAC_{N_{A}2}&\cdots&MAC_{N_{A}N_{E}}\end{array}\right] \tag{10}\]where \(N_{A}\) and \(N_{E}\) are the number of analytical and experiential modes, respectively.  
As mentioned above, when natural frequencies or mode shapes change dramatically over iterations for complex structures, the constant \(k^{\prime}\) in Equation (6) determined by pre-analysis cannot be adjusted adaptively even though a better correlation occurs. In order to auto-pair and adaptively screen out the CMPs based on the one-to-one symbiotic relationship, correlated and penalized subitems are introduced as follows:  
For case 1\(\leftrightarrow\)1 and case 1\(\leftrightarrow\)n, the \(k^{\prime}\)th analytical mode is auto-paired with the \(k^{\text{th}}\) experimental mode, where \(MAC_{k^{\prime}k}\) is maximum and greater than a given MAC threshold in the \(k^{\text{th}}\) column in **MAC**. Then, the correlated subitem of the objective function \(J_{k^{\prime}k}\) in Equation (6) can be calculated by  
\[J_{k^{\prime}k}=w_{f,k}\cdot\left(\kappa_{f,k}\cdot FRE_{k^{\prime}k}\right)^ {2}+w_{\varphi,k}\Big{[}\kappa_{\varphi,k}\cdot\left(1-MAC_{k^{\prime}k}\right) \Big{]}^{2} \tag{11}\]  
where \(w_{f,k}\) and \(w_{\varphi,k}\) are taken as unity to give equal weight for each mode. The penalized function \(\kappa\) can be selected or specified based on concrete issues. As given in Equation (12), a sigmoid function flipped horizontally was taken in this paper to accelerate the convergence. As shown in Figure 1, when \(MAC_{k^{\prime}k}\) is close to 0, the objective function value is multiplied with a bigger \(\kappa\) (next to 3 here) to play a penalized role. The decreasing rate is much more significant in the middle region to urge \(MAC_{k^{\prime}k}\) forward to a larger value. Until \(MAC_{k^{\prime}k}\) is close to 1, \(\kappa\) approaches 1 to remain unpenalized.  
\[\kappa_{f,k}=\kappa_{\varphi,k}=1+\frac{2}{1+\exp[10-20(1-MAC_{k^{\prime}k})]} \tag{12}\]  
For case 0\(\leftrightarrow\)n, none of the \(N_{A}\) analytical modes can be correlated with the \(k^{\text{th}}\) experimental mode by the given MAC threshold. Constant penalized values \(\overline{\text{FRE}}\) and \(\overline{\text{MAC}}\), 100% and 0, respectively, in this paper, are set to guarantee equal numbers of the subitems in the objective function. Then, the penalized subitem can be calculated by  
\[\overline{J_{k^{\prime}k}}=w_{f,k}\cdot\left(\kappa_{f,k}\cdot\overline{ \text{FRE}}\right)^{2}+w_{\varphi,k}\Big{[}\kappa_{\varphi,k}\cdot\left(1- \overline{\text{MAC}}\right)\Big{]}^{2} \tag{13}\]  
According to Equations (11) and (13), all the subitems of the objective function can be determined by correlated mode auto-pairing strategy. In particular, when the updating parameters approach the ideal values during iteration, some analytical modes may be correlated with the experimental modes corresponding to penalized subitems. That is to say, case 0\(\leftrightarrow\)n may transform into case 1\(\leftrightarrow\)1 or case 1\(\leftrightarrow\)n, then the subitems calculated by Equation (13) can switch to correlated ones calculated by Equation (11). All the potential of analytical or experimental modes can be fully exploited in the updating procedure.  
Figure 1: Penalized function value and its derivation used in this paper.