## 5 Deformation Analysis and Reconstruction
### Inverse Finite Element Method  
The core of the iFEM algorithm is to construct an inverse element and construct the error functional between the analytical strain and measured strain based on the inverse element. The purpose of reconstructing the structural deformation is realized by finding the minimum value of the functional. [PERSON] et al. (2016) developed a four-node inverse quadrilateral shell (iQS4) element which has six degrees of freedom (DOFs) per node as shown in Figure 10. Then, the element nodal displacement vector \(\mathbf{u}^{i}=[\mathbf{u}_{1}^{i}\ \mathbf{u}_{2}^{i}\ \mathbf{u}_{3}^{i}\ \mathbf{u}_{4}]^{\text{T}}\) can be expressed as \(\mathbf{u}_{q}^{i}=[u_{q}\ v_{q}\ v_{q}\ \theta_{\alpha q}\ \theta_{yy}]^{\text{T}}\), (\(q=1,\,2,\,3,\,4\)). According to the linear strain-displacement relations and the plane-stress assumption \(\sigma_{\text{z}}=0\) within the implication that the transverse-normal strain \(\varepsilon_{\text{zz}}\) does not contribute to the strain energy, the strain components can be combined to describe the membrane strain \(e(\mathbf{u}^{i})\), bending curvatures \(\mathbf{k}(\mathbf{u}^{i})\), and transverse shear strain \(\mathbf{g}(\mathbf{u}^{i})\). Then, the strain-displacement relations are described as  
\[ \begin{bmatrix}\varepsilon_{xx}&\varepsilon_{yy}&\gamma_{xy} \end{bmatrix} ^{\text{T}}=e\mathbf{\left(\mathbf{u}^{i}\right)}+\mathbf{z}\mathbf{k}\left(\mathbf{u }^{i}\right)^{i},\ \begin{bmatrix}\gamma_{xz}&\gamma_{yz} \end{bmatrix} ^{\text{T}}=g\left(\mathbf{u}^{i}\right) \tag{5}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Central Angle \(\gamma\)**} & \multicolumn{2}{c}{**Elevation Angle \(\beta\)**} & \multicolumn{2}{c}{**Differences**} \\ \cline{2-5}  & **Tests (a)** & **Simulation (b)** & **Theory (c)** & **a and c** & **b and c** \\ \hline
0.0\({}^{\circ}\) & 90\({}^{\circ}\) & 90.00\({}^{\circ}\) & 90.00\({}^{\circ}\) & 0 & 0 \\
12.5\({}^{\circ}\) & 80\({}^{\circ}\) & 80.13\({}^{\circ}\) & 80.43\({}^{\circ}\) & 0.53\% & 0.37\% \\
20.0\({}^{\circ}\) & 75\({}^{\circ}  
The deformation reconstruction based on the iFEM algorithm requires the surface measurement strain data. Considering the discrete strain measurement method shown in Figure 11, the strain sensors are arranged on the top and bottom surfaces of the structure. The measured membrane strains \(\mathbf{e}^{\varepsilon}\) and bending curvatures \(\mathbf{k}^{\varepsilon}\) that correspond to their analytic counterparts \(\mathbf{e}(\mathbf{u}^{i})\) and \(\mathbf{k}(\mathbf{u}^{i})\) given by Equation (5) can be determined from the measured surface strain as  
\[\mathbf{e}^{\varepsilon}=\frac{1}{2}\left\{ \begin{array}{c}\varepsilon_{xx}^{+}+ \varepsilon_{xx}^{-}\\ \varepsilon_{yy}^{+}+\varepsilon_{yy}^{-}\\ \gamma_{xy}^{+}+\gamma_{xy}^{-}\end{array} \right\},\,\mathbf{k}^{\varepsilon}= \frac{1}{2h}\left\{ \begin{array}{c}\varepsilon_{xx}^{+}-\varepsilon_{xx}^{- }\\ \varepsilon_{yy}^{+}-\varepsilon_{yy}^{-}\end{array} \right\} \tag{6}\]  
The key formulas of the iFEM algorithm are derived from the extreme value of the least-squares functional between the analytical strain in Equation (5) and measured strain in Equation (6). In terms of the iQS4 element, the functional takes the form as  
\[\mathbf{\Phi}\Big{(}\mathbf{u}^{i}\Big{)}=w_{m}\Big{\|}\mathbf{e}\Big{(}\mathbf{u}^{i}\Big{)}- \mathbf{e}^{\varepsilon}\Big{\|}^{2}+w_{b}\Big{\|}\mathbf{k}\Big{(}\mathbf{u}^{i}\Big{)}- \mathbf{k}^{\varepsilon}\Big{\|}^{2}+w_{s}\Big{\|}\mathbf{g}\Big{(}\mathbf{u}^{i}\Big{)}- \mathbf{g}^{\varepsilon}\Big{\|}^{2} \tag{7}\]  
Figure 11: Discrete surface strain measurement. Figure 10: The four-node quadrilateral shell element (iQS4). (a) Element local coordinates, (b) Isoparametric coordinates. The unknown variable of the functional is the element nodal displacement vector \(u^{i}\).