# A Refined Shape Sensing Method for Skin Antenna Structure Based on Inverse Finite Element Method
## 3 Numerical and Experimental Examples  
The deformation of the antenna reflector is the main factor that affects pointing accuracy In order to examine the accuracy of iFEM for reconstructing the deformation of the antenna plate, an analysis between numerical and experiment on the skin antenna plate was presented. A rectangular plate has a length of 0.560 m, a height of 0.202 m, and a uniform thickness of 3.2 mm. The plate has an elastic modulus of 1.0 GPa and a Poisson's ratio of 0.3 and is meshed 675 quadrilateral elements in analysis software; however, the structure was divided only two inverse quadrilateral elements (shown Figure 2). In Figure 2, the numbers in red indicate the element nodes, and the square letters in black denote the sampling points, respectively. First, the deformation of sampling points can be obtained by iFEM. Second, based on these results, the whole deformation field can be reconstructed by the interpolation method proposed above. The deformations with two different boundary constraints are analyzed--namely, the left side of the structure--and both ends are fixed respectively. With the condition of complete constraints at line connected node 1 and 2, the concentrated force of 10N is applied positive at nodes 5 and negative 10N at nodes 8 in the z direction concurrently. The nodes' deformations calculated by iFEM together with FEM are shown in Table 1.  
Figure 2: Inverse finite elements model of structure. Based on the values of nodes 3-8 by iFEM, one can obtain the sample points A-H by the general interpolation method. Moreover, acting the nodes A-H together with 5 and 8 as sampling points, one can regenerate the full-field deformation with the proposed interpolation framework. The deformation of the whole structure from FEM and iFEM are presented in Figure 3.  
Figure 3 shows the deformed shapes of the plate obtained by different approaches shown from three different colors lines. The x-axis indicates the points of structure-expressed finite numbers, and y-axis is the deformation. According to the various interpolation means, iFEM(a) reflects the deformation trend of the whole structure according to the Equation (13), and iFEM(b) does that by Equation (15), respectively. In order to quantitatively analyze the validity of the proposed method, the evaluation indexes mean error of nodes (\(ME\)) and root mean square (\(RMS\)) are introduced. \[ME=\frac{1}{m}\sum_{i=1}^{m}\left|\left(X_{i}^{FEM}-Y_{i}^{iFEM}\right)\right| \tag{17}\]  
\[RMS=\sqrt{\frac{\sum_{i=1}^{n}\left(X_{i}^{FEM}-Y_{i}^{iFEM}\right)^{2}}{n}} \tag{18}\]  
where, \(X_{i}^{FEM}\) indicates the value provided by the finite element method, \(Y_{i}^{iFEM}\) is the value from the inverse finite element method, \(m\) and \(n\) are the number of nodes and sample points, respectively. Considering the values from FEM as references, Table 1 lists the results from iFEM. The maximum deformation of the node along the z direction is 50.2 mm, and the maximum percentage difference is 8. The reconstruction \(ME\) is 1.87 mm, and the RMS of the reconstruction accuracy is 2.57 mm from iFEM(a). However, based on the proposed method, the RMS value of the deformation reconstruction accuracy is 1.0 mm. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Node**} & \multicolumn{2}{c}{**Z Directional Deformation (mm)**} \\ \cline{2-4}  & **FEM** & **iFEM** & **Error** \\ \hline
3 & 11.3 & 11.1 & 0.2 \\
4 & 11.3 & 11.0 & 0.3 \\
5 & 5.4 & 5.60 & 0.2 \\
6 & 50.1 & 48.1 & 2.1 \\
7 & 50.1 & 50.0 & 0.1 \\
8 & 29.3 & 26.6 & 2.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Results of nodes deformation. Figure 3: Deformation of the plate structure. With case two, lines connected nodes 1 and 2, and at the same time, 6 and 7 are fixed completely, so the concentrated force of 10N at node 5 and 10N at nodes 8 are applied in the positive z direction.