## 4 Numerical Method
### FEM Module Based on N-R Iterative Method  
The deformation size of the flexible skin is 1\(\sim\)2 orders of magnitude higher than its own thickness, and there is a strong geometric nonlinear effect. Under this condition, the finite element method generally has higher prediction accuracy than the analytical method.  
Firstly, the skin is discretized into finite elements, and then the appropriate displacement function is selected to describe the displacement distribution law in the element. Proceeding to the next step, the coordinate of any point is derived from the node displacement, and the force on the node is obtained. Based on the principle of virtual work, the relationship between node force and node displacement can be constructed:  
\[[F]^{e}=[K]^{e}[e]^{e} \tag{5}\]  
where \([F]^{e}\) is the node force matrix, \([e]^{e}\) is the node displacement matrix, and \([K]^{e}\) is the element stiffness matrix.  
Through the overall force balance conditions and boundary conditions, the characteristics of each element are superimposed into a whole, and the overall finite element equation is established:  
\[[F]=[K][e] \tag{6}\]  
where \([F]\) is the load component array, \([e]\) is the displacement of all nodes, and \([K]\) is the overall stiffness matrix.  
For nonlinear problems, the stiffness matrix \([K]\) is constantly changing with the change of displacement, and the solution of the stiffness matrix requires multiple iterations of updates. The iterative methods mainly include direct iterative method, Newton iterative method and incremental method. The solution in this chapter adopts Newton-Raphson iteration method, which realizes iteration through linearization of nonlinear equations. Every iteration will regenerate the tangent stiffness matrix until the result converges.  
On the basis of the above finite element equations, the boundary constraints are introduced to solve the overall finite element equations, and the node displacement components are obtained.  
In this section, the M3D3 element is used for the relevant calculation of the membrane, taking the large deformation effect into account, and the stress vector follows the normal direction of the element during the iteration. The M3D3 element is a three-node triangular membrane, which has high accuracy for predicting membrane deformation. The membrane structure is supported by wing ribs, leading edge foam, trailing edge strips, and other structures. These solid structures adopt the C3D8R element, which is an eight-node linear brick with reduced integration and hourglass control. The ribs on both sides of the wing are fixed, as portrayed in Figure 20.  
As portrayed in Figure 21, the finite element method based on the N-R iterative method is used to simulate the deformation of membrane structure when v = 6.4 m/s and a = 4\({}^{\circ}\). It can be observed from the comparison with Figures 10a and 11a that the numerical calculation method can predict the deformation trend of the membrane, although it cannot predict the amplitude of the upper surface deformation accurately. The predicted value of the maximum convexity of the upper surface is 0.87 mm, which is smaller than the 1.41 mm observed in the experiment. The predicted value of the maximum concave amplitude of the lower surface is 3.06 mm, which is near to the 3.12 mm obtained by the experiment.