## 5 Deformation Analysis and Reconstruction
### Deformation Reconstruction by iFEM  
Due to the over-constraint of the back frame in a specific motion state, there may be discontinuous angular displacement at the connection between the panels and the shell. Exploiting the advantage of the iFEM algorithm that it does not need to consider the physical parameters such as external load during deformation reconstruction, the full-field deformation reconstruction of a statically indeterminate structure is realized by combining the sub-region division and iFEM algorithm. The reconstruction steps are as follows:  
1. The measured discrete in situ strains in the iFEM algorithm are derived from the surface strains of FEM.
2. The back frame structure is divided into three sub-regions according to the boundary constraints, namely two side panels and a curved shell.
3. The iFEM algorithm is utilized to reconstruct the deformation of two side panels with fixed constraints.
4. The nodal displacements of two side panels at the joints are derived as the boundary constraints of the curved shell and input data in the iFEM code, then the deformation of the curved shell is reconstructed.
5. Integrating the reconstructed deformation of all sub-regions, the full-field deformation of the statically indeterminate structure is reconstructed.  
The same mesh discretization as FEM and the above reconstruction steps are performed for the back frame structure. The measuring points are arranged at the center of each element and the discrete in situ strains in Figure 11 are derived from the strain results of FEM. The total displacement in four rotation states inverted by the iFEM algorithm are depicted in Figure 14. In addition, the inverse accuracy in four rotation states is demonstrated by the reference solution given by FEM, and the differences between iFEM results and FEM solutions are displayed in Table 5.  
Further, another coarse mesh discretization method with 546 iQS4 elements and 3564 DOFs is adopted for the case at \(\beta=90^{\circ}\). The in situ strains of this inverse finite element (FE) model are also derived from the surface strains of FEM. Figure 15 displays the total displacement fields of the coarse inverse FE model. And the differences between iFEM results and FEM solutions are also listed in Table 5.  
From Figures 13 and 14, it is clear that there is little difference between FEM solutions and iFEM reconstruction results. By comparing the deformation contours at \(\beta\) = 90\({}^{\circ}\) in Figures 13a, 14a, and 15, it is apparent that the full-field deformations reconstructed by two inverse FE models are highly consistent with the deformation calculated by FEM. With reference to the average reconstruction errors, the percentage errors of total rotation reconstruction are larger than that of total displacement reconstruction. With the increase in the rotation trend of the back frame, the reconstruction accuracy of the iFEM algorithm generally decreases to a certain degree. There are some greater errors of the inverse FE model in Figure 15 than in Figure 14 due to the coarser mesh of iQS4 elements. And it is indicated in Table 5 that the maximum relative percentage error of this coarse model is 6.41%, which appears in total rotation. Nevertheless, the average differences are available within the error range. These results suggest that the iFEM algorithm has an excellent shape-sensing performance for the back frame of the FCRA.