## 3 Numerical Validation and Experiments
### Initial Deformation of Panel Components  
During the positioning and clamping process, the panel components, skin and stringers, are just subjected to the initial kinematic boundary conditions listed in Equations (9) and (36) without the joining force. Substituting boundary conditions in Tables 3 and 4 into the Fourier series expansion of shell displacement functions and beam deformation expressions, unknown coefficients \({}^{m}j_{i}\) and \(C_{i}\) can be derived. Deformations of shell, beam I and beam II are provided by the proposed method, as shown in Figures 4 and 5a. FE models of shell and beams are established by using commercial software ABAQUS(r) 6.14 Standard with S8R element and B33 elements, respectively. Approximate element size is 25 mm and the grid number of FE model is over 1440. Figure 4: Displacement and rotation of the deformed shell middle surface. (**a**) Magnitude of displacement; (**b**) displacement in direction of \(\gamma_{i}\) (**c**) rotation around axis \(\alpha\); (**d**) rotation around axis \(\beta\). Besides, positions of points to be joined on the stringers I and II are predicted by solving a system of equations \(u^{st}+x_{p}=x_{p}^{theoretical}\) and \(v^{st}+y_{p}=y_{p}^{theoretical}\) by the Newton-Raphson approach. Substituting the solutions \((x_{p},y_{p})\) and \(z_{p}=13.062\) into Equation (35), deflections \(w_{st}\) of the points to be joined are obtained, as shown in Figure 5b. Based on the theoretical arc lengths of skin, \(\beta^{theoretical}=\theta_{b}R+l_{j}\), the actual positions of points to be joined on shell are obtained by solving a system of equations \(u-\varphi_{b}h/2+\alpha_{p}=\alpha_{p}^{theoretical}\) and \(v+\varphi_{u}h/2+\beta_{p}=\beta_{p}^{theoretical}\). Solutions \((\alpha_{p},\beta_{p})\) are substituted into Equation (A2), deflections \(w\) of the points to be joined on skin are calculated. In the practical aircraft assembly process, actual clearance between the panel components \(\Delta_{w}=w-w^{st}\) is measured to describe the relative deformation variation directly. A comparison of the predicted clearance and measured data is made to analyze the relative deformation components in the directions of surface normal, as shown in Figure 6.  
Figure 5: Deformation of stringers and presented deflections of points to be joined. (**a**) Deformation of beam I and II along the centroid locus of cross-section; (**b**) positions and deflection of drilling points on the stringers I and II. Figure 6: Clearances between the points to be joined on stringers and adjacent skin surface. Figures 4 and 5 indicate that shell and beam deformation results provided by the proposed method are consistent with FEM result. As seen in Figure 4a, the shell displacement has increased nearly fourfold from initial variation 0.814 mm to 2.89 mm. Initial variations are obtained by \(\sqrt{\left(\Delta\tau_{1}-\alpha\right)^{2}+\Delta\tau_{2}{}^{2}+\Delta\tau_{3} {}^{2}}\), the variables of which are listed in Table 3. The functional boundary conditions cause the dramatical changes in the displacement. Maximum deviation of shell displacement between FEM and the proposed results are 0.196 mm which can verify the accuracy of the semi-analytical solutions. Figure 4b shows that deflection \(w\) in the direction of surface normal increases considerably before a decline when \(\beta<0.2b\) and then decreases finally. Displacements \(w\) and rotation \(\varphi_{u}\) distribute almost symmetrically with respect to section \(\beta=b/2\). It is noted that \(\varphi_{u}\) and \(\varphi_{v}\) provided by the FEM have distortion near the edge node while the proposed semi-analytical solution has a more stable value, as seen in Figure 4c,d. As seen in Figure 5a, displacements \(w^{b}\) in \(z\)-axis direction have relatively smaller values compared with the large deflection in \(y\)-axis direction. When the drilling operation is conducted by the automated robot without compensation, the positions of drilling points are skewed, as shown in Figure 5b.