## 3 Dynamic Force-Thermal Follow-Up Loading Method for Multi-Stage Telescopic Wings
### Dynamic Multi-Point Distribution Equivalence Method for Aerodynamic Loads
The Lagrangian function is constructed using the method of Lagrange multipliers, as follows:  
\[F(\lambda,\lambda_{x},\lambda_{y})=\sum\limits_{k=1}^{n}(\frac{P_{k}^{2}l_{k} ^{3}}{6E}-\lambda p_{k}-\lambda_{x}P_{k}\overline{x_{k}}-\lambda_{y}P_{k} \overline{y_{k}}) \tag{5}\]  
where \(\lambda\), \(\lambda_{x}\), and \(\lambda_{y}\) are the Lagrange operators for the static equivalent condition construction function, \(\overline{x_{k}}\) represents the difference in the x-coordinate between loading point k and the aerodynamic grid, and \(\overline{y_{k}}\) represents the difference in the y-coordinate between loading point k and the aerodynamic grid. Taking the derivative and setting it equal to zero yields the following:  
\[P_{k}l_{k}^{3}=\lambda+\lambda_{x}\cdot\overline{x_{k}}+\lambda_{y}\cdot \overline{y_{k}} \tag{6}\]  
By combining Formula (3), we can obtain the following:  
\[\left[ \begin{array}{l}\sum\limits_{k=1}^{n}l_{k}^{-3}&\sum\limits_{k=1}^{n} \overline{x_{k}}l_{k}^{-3}&\sum\limits_{k=1}^{n}\overline{y_{k}}l_{k}^{-3}\\ \sum\limits_{k=1}^{n}\overline{x_{k}}l_{k}^{-3}&\sum\limits_{k=1}^{n}\chi_{k} ^{2}l_{k}^{-3}&\sum\limits_{k=1}^{n}\overline{x_{k}}l_{k}^{-3}\\ \sum\limits_{k=1}^{n}\overline{y_{k}}k_{k}^{-3}&\sum\limits_{k=1}^{n}\overline {x_{k}}l_{k}^{-3}&\sum\limits_{k=1}^{n}\chi_{k}^{2}l_{k}^{-3}\\ \sum\limits_{k=1}^{n}\overline{y_{k}}k_{k}^{-3}&\sum\limits_{k=1}^{n}\overline {x_{k}}\overline{y_{k}}l_{k}^{-3}&\sum\limits_{k=1}^{n}y_{k}^{2}l_{k}^{-3}\\ \end{array} \right]\times\left\{ \begin{array}{l}\lambda\\ \lambda_{x}\\ \lambda_{y}\\ \end{array} \right\}=\left\{ \begin{array}{l}P_{o}\\ 0\\ 0\\ 0\\ \end{array} \right\} \tag{7}\]  
By solving \(\lambda\) and substituting it into Equation (4), the aerodynamic load of each aerodynamic grid point o\({}_{\text{uq}}\) (u = 1, 2, 3,..., i; q = 1, 2, 3,..., j) is distributed to the corresponding structural grid points. Using the above method, the aerodynamic loads on the wing surface are equivalently allocated to the predefined structural grid points. The above multi-point distribution method completes the load equivalence of aerodynamic grid loads on the wing surface to the predefined loading points at a specific moment. Building on this, a dynamic multi-point distribution method is proposed by incorporating the mesh reconstruction technique, enabling time-varying loads with changing areas to be equivalently distributed to fixed predefined loading points. The dynamic multi-point distribution method operates as follows: The entire motion process is discretized into a finite number of time steps at regular intervals, converting the dynamic process into a series of quasi-static processes. At each time step, the aerodynamic grid on the wing surface is re-meshed to achieve dynamic grid updates. The multi-point distribution method is then applied at each time step to allocate the aerodynamic grid loads to the predefined loading points. The loads throughout the discretized process are  
Figure 16: Schematic of the multi-point distribution method. interpolated and fitted, resulting in the load distribution across all loading points during the motion. This distribution ensures that the equivalent conditions of force and moment are satisfied at all times while minimizing the additional strain energy introduced by load equivalence. As shown in Figure 17, during the wing deformation process, the entire process is discretized into t\({}_{1}\) to t\({}_{n}\), representing n finite moments. At each moment, the wing profile undergoes dynamic grid updates. The selection of the time step \(\Delta t\) depends on the adopted time discretization strategy, which is typically determined based on the maximum wave speed of the material c\({}_{\text{max}}\) and the minimum mesh size L\({}_{\text{min}}\).