## 3 Identification of Deformation Modes
### Cross-Section Deformation Modes
#### 3.2.1 Basic Algorithms
Naturally, the two vectors can be expressed as  
\[\mathbf{x}_{i}{}^{(k)}=\beta_{i,\text{I}}+\beta_{i,\text{II}},\ \mathbf{x}_{j}{}^{(k)}=\beta_{j,\text{I}}+\beta_{j,\text{II}} \tag{18}\]  
where the symbol \(\beta\) indicates amplitude function components and the subscripts I and II denote different structural behaviours. Furthermore, the two components belonging to a same structural behaviour should be collinear, reading  
\[\beta_{i,\text{I}}=\mu_{1}\beta_{j,\text{I}},\ \beta_{i,\text{II}}=\mu_{2}\beta_{j, \text{II}} \tag{19}\]where \(\mu_{1}\) and \(\mu_{2}\) are the undetermined coefficients. In fact, the proportion of amplitude functions for a Classic mode is predictable. The point is to identify the type of the classic mode involved. This problem was discussed in a previous paper [37], where the Classic modes were recognized through the signs of benchmark points. Thus, Equation (18) and Equation (19) yield  
\[(\mu_{2}-\mu_{1})\beta_{i,\text{II}}=\mathbf{x}_{j}{}^{(k)}-\mu_{1}\mathbf{x}_ {i}{}^{(k)} \tag{20}\]  
which means the component \(\beta_{i,\text{II}}\) is collinear with the linear combination \(\mathbf{\chi}_{j}{}^{(k)}-\mu_{1}\mathbf{x}_{i}{}^{(k)}\). Therefore, an orthogonalization procedure can be used to further decompose all the amplitude function components. Based on this, the components can be obtained as  
\[\beta_{i,\text{I}}=\mathbf{\chi}_{i}{}^{(k)}-\frac{\text{dot}\left(\mathbf{x}_ {j}{}^{(k)}-\mu_{1}\mathbf{x}_{i}{}^{(k)},\mathbf{x}_{j}{}^{(k)}\right)}{ \text{dot}\left(\mathbf{x}_{j}{}^{(k)}-\mu_{1}\mathbf{x}_{i}{}^{(k)},\mathbf{ x}_{j}{}^{(k)}-\mu_{1}\mathbf{x}_{i}{}^{(k)}\right)}\Big{(}\mathbf{x}_{j}{}^{(k)} -\mu_{1}\mathbf{x}_{i}{}^{(k)}\Big{)},\ \beta_{i,\text{II}}=\mathbf{x}_{i}{}^{(k)}-\beta_{i,\text{I}}, \tag{21}\]  
\[\beta_{j,\text{I}}=\mathbf{x}_{j}{}^{(k)}-\frac{\text{dot}\left(\mathbf{x}_{j }{}^{(k)}-\mu_{1}\mathbf{x}_{i}{}^{(k)},\mathbf{x}_{j}{}^{(k)}\right)}{\text{ dot}\left(\mathbf{x}_{j}{}^{(k)}-\mu_{1}\mathbf{x}_{i}{}^{(k)},\mathbf{x}_{j}{}^{(k)} -\mu_{1}\mathbf{x}_{i}{}^{(k)}\right)}\Big{(}\mathbf{x}_{j}{}^{(k)}-\mu_{1} \mathbf{x}_{i}{}^{(k)}\Big{)},\ \beta_{j,\text{II}}=\mathbf{x}_{j}{}^{(k)}-\beta_{j,\text{I}}. \tag{22}\]  
The process is referred to as DAFC 2 (the second case of DAFC). In the process of DAFC, a possible problem is how to decide whether a residual component should be further studied or just ignored. Hence, the residual participation \(\xi_{\text{r}}\) is defined as the indicator. When the residual component \(\beta_{2}\) obtained from DAFC satisfies the condition  
\[\xi_{\text{r}}=\frac{\|\beta_{2}\|_{\infty}}{\text{max}\Big{(}\|\mathbf{x}_{1 }{}^{(k)}\|_{\infty},\|\mathbf{x}_{2}{}^{(k)}\|_{\infty},\cdots,\|\mathbf{x}_ {i}{}^{(k)}\|_{\infty},\cdots,\|\mathbf{x}_{N}{}^{(k)}\|_{\infty}\Big{)}} \leq\xi_{0} \tag{23}\]  
then the component \(\beta_{2}\) can be ignored. Otherwise, it should be assembled with other PDM. Finally, the shape function \(\eta(s)\) of an identified deformation mode can be obtained as  
\[r_{i,j}=\frac{\text{cov}\big{(}\mathbf{x}_{i}{}^{(k)},\mathbf{x}_{j}{}^{(k)} \big{)}}{\sqrt{\text{var}\big{[}\mathbf{x}_{i}{}^{(k)}\big{]}\text{var}\big{[} \mathbf{x}_{j}{}^{(k)}\big{]}}} \tag{15}\]0  
where \(\theta_{j}(s)\) is the shape function of a PDM \(j\), \(\mathbf{x}_{i}{}^{(k)}\) is the amplitude function vector selected as the standard reference and all the amplitude function vectors related are collinear EAFV. One should bear in mind that the algorithms above can handle two vectors each time. In practice, a certain number of amplitude functions are processed in pairs according to a well-planned sequence. Details are discussed in the following section.