# Morphological Reconstruction for Variable Wing Leading Edge Based on the Node Curvature Vectors
## 2 NCV-CPM Methodology
### Optimization of Virtual Sensing Point Quantity Based on Particle Swarm Algorithm  
According to the particle swarm optimization algorithm, the sum of the virtual perception points and the actual measurement points in Section 2.2 is defined as the number of particles. By iteratively adjusting the number of particles, the optimal number of virtual perception points is sought, aiming to improve the accuracy of wing leading-edge propagation reconstruction without increasing the number of actual sensors. Considering the symmetry of the variable curvature wing leading-edge structure and the randomness of deformation, both actual and virtual perception points are arranged in an equidistant layout. The optimization process is as follows. Firstly, based on the sensor layout initialization scheme, several actual strain sensors are uniformly distributed on the wing leading-edge surface. Combined with the method described in Section 2.2, approximate global measurements of the leading-edge surface curvature and strain distribution characteristics are obtained. Secondly, utilizing the particle swarm optimization algorithm, along with Equation (14), the number of particles \(n\) is updated [30; 31], yielding the following:  
\[\left\{ \begin{array}{l}n(t+1)=n(t)+v(t+1)\\ v(t+1)=w(t)v(t)+C_{1}r_{1}\big{(}p_{g}(t)-n(t)\big{)}\end{array} \right. \tag{14}\]  
where \(t\) represents the iteration count of particles, \(n_{i}(t)\) denotes the number of measurement points at the \(t\)th iteration of particles, \(n_{i}(t\)+1) represents the number of measurement points at the \(t\)+1 th iteration of particles, \(v_{i}(t\)+1) signifies the updating velocity at the \(t\)+1 th iteration  
Figure 5: Coordinate increment calculation of arc segment. of particles, and \(p_{g}(t)\) stands for the optimal number of measurement points obtained in the previous t iterations of particles. \(C_{1}\) is the weighting coefficient for particles tracking their own historical best value, typically set to 2, and \(r_{1}\) is a uniformly distributed random number within the interval \([0,1]\). \(w(t)\) represents the updating velocity coefficient at the \(t\)th iteration of particles, as shown in Equation (15). \[w(t)=w_{\max}-iter\times\frac{w_{\max}-w_{\min}}{iter_{\max}} \tag{15}\]  
where \(iter_{\max}\) represents the maximum iteration count for the algorithm, and \(iter\) denotes the current iteration count. Typically, \(w_{\max}\) is set to 0.9, and \(w_{\min}\) is set to 0.4. Furthermore, following the method described in Section 2.2, curvature and strain information corresponding to different particle positions is obtained after each iteration. Building upon this information and combining it with the wing leading-edge propagation reconstruction method outlined in Section 2.3, the curvature accuracy is computed and compared for each iteration of particle positions. This process facilitates the determination of the optimal number of virtual perception points aimed at wing leading-edge propagation reconstruction, as depicted in Figure 6.  
To evaluate the effectiveness of the proposed method for wing leading-edge propagation reconstruction, the Euclidean distance \(d(i)\) between each validation point on the wing leading edge and the coordinate origin is defined, as shown in Equation (16). \[d(i)=\sqrt{x(i)^{2}+y(i)^{2}} \tag{16}\]  
where \(d(i)\) denotes the Euclidean distance between the \(i\)th validation point and the coordinate origin, \(x(i)\) represents the deformed \(X\)-coordinate of the \(i\)th validation point, and \(y(i)\) denotes the deformed \(Y\)-coordinate of the \(i\)th validation point. Based on this, the absolute error, relative error, root mean square error, and mean relative error of the reconstruction for each validation point on the wing leading edge can be represented as follows:  
\[\left\{