# Morphological Reconstruction for Variable Wing Leading Edge Based on the Node Curvature Vectors
## 2 NCV-CPM Methodology
### Reconstruction of Wing Leading-Edge Curvature and Strain Field Based on Node Curvature Vectors  
Based on the curvature calculation method using discrete actual sensing points as described in Section 2.1, curvature information and strain information at positions where strain sensors are not attached cannot be obtained. Therefore, this section proposes a method for reconstructing the curvature field and strain field of the wing leading edge based on solving the curvature node vector of arc segments. This method can obtain curvature and strain distributions, corresponding to virtual sensing points on the wing's leading edge--positions where actual sensors are not deployed--thereby achieving an equivalent expansion of curvature/strain sensing data. Based on this, it is possible to increase the density of segment partition without increasing the number of actual sensor measurement points, thereby improving the accuracy of morphological reconstruction, as shown in Figure 2.  
Figure 2: Schematic diagram of arc division density optimization. The core idea of the method is as follows: Firstly, according to the discrete layout scheme of actual measurement points, the wing leading-edge section is divided into a finite number of arc segments, as illustrated in Figure 2. To enhance the order of the curvature fitting function within each segment, an internal node \(r\) is added within each arc segment, except at the two ends [26], as depicted in Figure 3.  
Next, the construction of the curvature node vector for each arc segment is undertaken. Taking the \(m\)th arc segment as an example, the curvature information from the two endpoints and one internal node of this segment is utilized to construct a curvature node vector, denoted as \(q_{Nade}\), as shown in Equation (5). \[q_{Node}= \begin{bmatrix}q_{1}&q_{2}&q_{r}\end{bmatrix} ^{T} \tag{5}\]  
The theoretical curvature information at any point within the segment can be obtained by computing the curvature vector and curvature functions (\(N_{1}\), \(N_{2}\), \(N_{r}\)), as illustrated in Equation (6). By substituting the arc length information s of the discrete measurement points into this equation, the theoretical curvature of the discrete measurement points can be calculated. \[q(s)=\sum_{i=1,2,r}N_{i}q_{Node},\ \left\{ \begin{array}{l}N_{1}=2\xi^{2}-3 \xi+1\\ N_{2}=2\xi^{2}-\xi\\ N_{r}=4\xi-4\xi^{2}\end{array} \right.\ \ \ \ \ \ (\xi=s/L_{e}\in[0,\ 1]) \tag{6}\]  
where \(s\) represents the arc length parameter, and \(L_{e}\) denotes the arc length of the segment. Furthermore, Equation (6) is transformed into a matrix form that incorporates the curvature node vector \(q_{Node}\) of the arc segment:  
\[q(q_{Node})=Cq_{Node},\ \ \ \ C=[N_{1}\ \ \ N_{2}\ \ \ N_{r}] \tag{7}\]  
Based on the curvature \(q_{m}\) calculated from Section 2.1 for the \(m\)th discrete actual measurement point, combined with the theoretical curvature of this point derived from the above analysis, a least squares error function model between the two is established, yielding the following [27]:  
\[\Phi_{e}(q_{Node})=\left\|q(q_{Node})-q_{m}\right\|^{2} \tag{8}\]  
By taking the derivative of the error function model concerning the curvature vector and setting it to zero, the curvature stiffness equation is obtained, as shown in Equation (9). \[\frac{\partial\Phi_{e}(q_{Node})}{q_{Node}}=k_{e}q_{Node}-f_{e}=0 \tag{9}\]  
Figure 3: Arc segment node definition. where \(k_{e}\) represents the curvature stiffness matrix, and \(f_{e}\) denotes the curvature load matrix, as illustrated in Equation (10). \[\left\{ \begin{array}{l}k_{e}=\sum\limits_{i=1}^{n}\int\frac{\frac{iL_{e}}{ \int_{(-1)L_{e}}^{iL_{e}}}}{\frac{iL_{e}}{n}}\left(C^{T}C\right)\mathrm{d}X\\ f_{e}=\sum\limits_{i=1}^{n}\int\frac{\frac{iL_{e}}{\int_{(-1)L_{e}}^{iL_{e}}}}{ \frac{iL_{e}}{n}}\left(C^{T}q_{m}\right)\mathrm{d}X\end{array} \right. \tag{10}\]  
Subsequently, based on the segmentation scheme of the arc segments, the curvature stiffness equations for each segment are assembled into the overall curvature stiffness equation.