## 3 Identification of Deformation Modes
### Cross-Section Deformation Modes
#### 3.2.1 Basic Algorithms  
Here, some basic algorithms were employed to handle the eigen matrices. For each eigen matrix \(\mathbf{x}^{(k)}\), one should check the participation \(\rho_{i}\) of a PDM \(i\) in advance, since not all of them are effective. The participation \(\xi_{i}\) is defined as  
\[\xi_{i}=\frac{\left\|\mathbf{x}_{i}{}^{(k)}\right\|_{\infty}}{\max\left( \left\|\mathbf{x}_{1}{}^{(k)}\right\|_{\infty},\left\|\mathbf{x}_{2}{}^{(k)} \right\|_{\infty},\cdots,\left\|\mathbf{x}_{i}{}^{(k)}\right\|_{\infty}, \cdots,\left\|\mathbf{x}_{N}{}^{(k)}\right\|_{\infty}\right)} \tag{14}\]where \(\|\|_{\infty}\) is the infinite norm of a vector. In practice, the PDM with a participation lower than the threshold \(\xi_{0}=0.001\) are ignored since it has little impact on the structural performance. The PDM with participtions larger than 0.001 are referred to as effective amplitude function vectors (EAFV). To address the first issue DCAF, the Pearson correlation coefficient was employed to distinguish the relationship between the two amplitude function vectors. The correlation coefficient \(r_{i,j}\) between \(\mathbf{x}_{i}{}^{(k)}\) and \(\mathbf{x}_{j}{}^{(k)}\) is defined as  
\[r_{i,j}=\frac{\text{cov}\big{(}\mathbf{x}_{i}{}^{(k)},\mathbf{x}_{j}{}^{(k)} \big{)}}{\sqrt{\text{var}\big{[}\mathbf{x}_{i}{}^{(k)}\big{]}\text{var}\big{[} \mathbf{x}_{j}{}^{(k)}\big{]}}} \tag{15}\]  
where the symbols cov() and var[] are the covariance and the variance, respectively. A coefficient \(r_{i,j}=1\) or \(-1\) is the sign of collinear amplitude function vectors. Generally, PDM with a mutual correlation coefficient of \(1\) or \(-1\) are deemed to be describe the same structural behaviour and can thus be assembled together to form a final deformation mode after AWDM. For the second issue AWDM, the weight \(\lambda_{i,j}\) of a PDM \(j\) is assigned as  
\[\lambda_{i,j}=\frac{\|\mathbf{x}_{j}{}^{(k)}\|_{\infty}}{\|\mathbf{x}_{i}{}^{ (k)}\|_{\infty}} \tag{16}\]  
where \(\mathbf{x}_{i}{}^{(k)}\) is the amplitude function selected as the standard reference (the unit weight). Notice that the amplitude function vectors in Equation (16) should be collinear EAFV, and that the standard reference is usually confined to the one having the maximum participation. The third issue DAFC confronts the situation that PDM \(j\) may participate in several structural behaviours, being characterized by correlation coefficients equal to neither \(1\) nor \(-1\). The situation can be further divided into two cases according to the type of deformation families involved. In the first case, the amplitude function vectors of the Dominant family are collinear, but an amplitude function vector \(\mathbf{x}_{j}{}^{(k)}\) of other families should be decomposed into two components \(\beta_{1}\) and \(\beta_{2}\). Then, the two components can be obtained as  
\[\beta_{1}=\frac{\text{dot}\Big{(}\mathbf{x}_{j}{}^{(k)},\mathbf{x}_{i}{}^{(k) }\Big{)}}{\text{dot}\Big{(}\mathbf{x}_{i}{}^{(k)},\mathbf{x}_{i}{}^{(k)}\Big{)} }\mathbf{x}_{i}{}^{(k)},\ \beta_{2}=\mathbf{x}_{j}{}^{(k)}-\beta_{1} \tag{17}\]  
where dot() is the inner product of \(\mathbf{x}_{i}{}^{(k)}\) and \(\mathbf{x}_{j}{}^{(k)}\), and \(\mathbf{x}_{i}{}^{(k)}\), as the standard reference, should only participate in this structural behaviour. Thus, the derived component \(\beta_{1}\) is bound to be collinear with \(\mathbf{x}_{i}{}^{(k)}\), while the residual component \(\beta_{2}\) is linearly independent. The procedure is referred to as DAFC 1 (the first case of DAFC). In the second case, the amplitude function vectors of the Dominant family are not collinear, with mutual correlation coefficients close to \(1\) or \(-1\). It indicates two structural behaviours, I and II (usually two Classic modes or a Classic mode and a Primary mode) involved, and another procedure is needed to decompose both the two amplitude function vectors \(\mathbf{x}_{i}{}^{(k)}\) and \(\mathbf{x}_{j}{}^{(k)}\) into two components.